text
stringlengths 6
128k
|
|---|
# Conditional Mutual Information Constrained Deep Learning for Classification
En-Hui Yang, Shayan Mohajer Hamidi∗, Linfeng Ye∗, Renhao Tan, and Beverly Yang
∗ Authors contributed equally.
En-Hui Yang, Shayan Mohajer Hamidi, Linfeng Ye and Renhao Tan are with the
Department of Electrical and Computer Engineering, University of Waterloo,
Waterloo, ON N2L 3G1, Canada (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]).
Beverly Yang is with the NBK Institute of Mining Engineering, University of
British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]).
###### Abstract
The concepts of conditional mutual information (CMI) and normalized
conditional mutual information (NCMI) are introduced to measure the
concentration and separation performance of a classification deep neural
network (DNN) in the output probability distribution space of the DNN, where
CMI and the ratio between CMI and NCMI represent the intra-class concentration
and inter-class separation of the DNN, respectively. By using NCMI to evaluate
popular DNNs pretrained over ImageNet in the literature, it is shown that
their validation accuracies over ImageNet validation data set are more or less
inversely proportional to their NCMI values. Based on this observation, the
standard deep learning (DL) framework is further modified to minimize the
standard cross entropy function subject to an NCMI constraint, yielding CMI
constrained deep learning (CMIC-DL). A novel alternating learning algorithm is
proposed to solve such a constrained optimization problem. Extensive
experiment results show that DNNs trained within CMIC-DL outperform the state-
of-the-art models trained within the standard DL and other loss functions in
the literature in terms of both accuracy and robustness against adversarial
attacks. In addition, visualizing the evolution of learning process through
the lens of CMI and NCMI is also advocated.
###### Index Terms:
Alternating minimization, concentration and separation, conditional mutual
information, cross entropy, deep learning.
## I Introduction
In recent years, deep neural networks (DNNs) have been applied in a wide range
of applications, revolutionizing fields like computer vision, natural language
processing, and speech recognition [1, 2]. Typically, a DNN consists of
cascaded non-linear layers that progressively produce multi-layers of
representations with increasing levels of abstraction, starting from raw input
data and ending with a predicted output label. The success of DNNs is largely
attributable to their ability to learn these multi-layers of representations
as features from the raw data through a deep learning (DL) process.
Putting its neural architecture aside, a classification DNN is,
mathematically, a mapping from raw data $x\in\mathbb{R}^{d}$ to a probability
distribution $P_{x}$ over the set of class labels, predicting an output label
$\hat{y}$ with probability $P_{x}(\hat{y})$. Given a pair of random variables
$(X,Y)$, the distribution of which governs either a training set or testing
set, where $X\in\mathbb{R}^{d}$ represents the raw data and $Y$ is the ground
truth label of $X$, the prediction performance of the DNN is often measured by
its error rate
$\epsilon=\Pr\\{\hat{Y}\not=Y\\},$
where $\hat{Y}$ is the label predicted by the DNN with probability
$P_{X}(\hat{Y})$ in response to the input $X$. The accuracy of the DNN is
equal to $1-\epsilon$. The error rate is further upper bounded by the average
of the cross entropy between the conditional distribution of $Y$ given $X$ and
$P_{X}$ (see Section II). To have better prediction performance, a DL process
is then applied to minimize the error rate $\epsilon$ or its cross entropy
upper bound [1, 2].
Although the error rate of a DNN is its most important performance as far as
its prediction is concerned, focusing entirely on the error rate is not
enough, and can actually lead to several problems. First, the error rate of a
DNN depends not only on the DNN itself, but also on the governing joint
distribution of $(X,Y)$. When a DNN has a small error rate for one governing
joint distribution of $(X,Y)$, it does not necessarily imply that it would
have a small error rate for another governing joint distribution of $(X,Y)$,
especially when two distributions are quite different. This is essentially
related to the well-known overfitting and robustness problems [2, 3, 4, 5].
Second, even when a DNN works well across different governing distributions of
$(X,Y)$, it remains a block box to us, especially when its architecture is
huge. We don’t know why it works and how it works. Its error rate does not
reveal any useful information about the intrinsic mapping structure such as
the intra-class concentration and inter-class separation of the DNN in its
output probability distribution space.
To gain deep insights into the intrinsic mapping structure of a DNN as a
mapping from $x\in\mathbb{R}^{d}$ to $P_{x}$, in this paper we introduce
information quantities from information theory [6] to measure intra-class
concentration and inter-class separation of the DNN. Specifically, we propose
to use the conditional mutual information (CMI) $I(X;\hat{Y}|Y)$ between $X$
and $\hat{Y}$ given $Y$ as the measure for the intra-class concentration of
the DNN as a mapping $x\in\mathbb{R}^{d}\to P_{x}$. For each class label $y$,
the conditional mutual information $I(X;\hat{Y}|Y=y)$ between $X$ and
$\hat{Y}$ given $Y=y$ tells how all output probability distributions $P_{X}$
given $Y=y$ are concentrated around its “centroid”, the conditional
probability distribution $P_{\hat{Y}|Y=y}$. The smaller $I(X;\hat{Y}|Y=y)$ is,
the more concentrated all output probability distributions $P_{X}$ given $Y=y$
are around its centroid. We further introduce another information quantity
(see Section II) to measure the inter-class separation of the DNN as a mapping
$x\in\mathbb{R}^{d}\to P_{x}$. Define the ratio between $I(X;\hat{Y}|Y)$ and
the inter-class separation as the normalized conditional mutual information
(NCMI) between $X$ and $\hat{Y}$ given $Y$. One may interpret CMI and NCMI as
certain mapping structure traits of the DNN. Then in addition to its error
rate, the DNN can also be evaluated in terms of its CMI and NCMI.
Equipped with our new concepts of CMI and NCMI, we further evaluate popular
DNNs pretrained in the literature over ImageNet in terms of their respective
CMI and NCMI. It turns out that their validation accuracies over the ImageNet
validation data set are more or less inversely proportional to their NCMI
values. In other words, even though these DNNs have different architectures
and different sizes, their error rates and NCMI values have more or less a
positive linear relationship. Indeed, the correlation between the error rate
and NCMI is above $0.99$. This implies that given a DNN architecture, one may
be able to further improve the effectiveness of DL by simultaneously
minimizing the error rate (or cross entropy upper bound) and NCMI of the DNN
during the learning process, where the error rate and NCMI represent the
prediction performance and the concentration/separation mapping structure
performance of the DNN, respectively. This in turn motivates us to modify the
standard DL framework to minimize the standard cross entropy function subject
to an NCMI constraint, yielding CMI constrained deep learning (CMIC-DL). To
this end, a novel alternating learning algorithm is further proposed to solve
such a constrained optimization problem. Extensive experiment results show
that DNNs trained within CMIC-DL outperform the state-of-the-art models
trained within the standard DL and other loss functions in the literature in
terms of both accuracy and robustness against adversarial attacks.
The remainder of this paper is organized as follows. In Section II, we
formally introduce the concepts of CMI and NCMI to measure intra-class
concentration and inter-class separation structure performance of a DNN when
it is viewed as a mapping from $x\in\mathbb{R}^{d}$ to $P_{x}$. In Section
III, we use NCMI to evaluate and compare popular DNNs pretrained in the
literature over ImageNet. These DNNs have different architectures and
different sizes. Section IV is devoted to the full development of CMIC-DL. In
Section V, extensive experiment results are presented and compared with the
prior art in the literature; visualizing the evolution of learning process
through the lens of CMI and NCMI is also advocated. Finally, conclusions are
drawn along with some open problems in Section VI.
## II Performance of DNNs: Concentration and Separation
A DNN can be described either by its neural architecture along with its
connection weights, the number of which can be in billions, or by its
mathematical mapping from $x\in\mathbb{R}^{d}$ to $P_{x}$. Both perspectives
are useful. In this and next sections, we will take the second perspective and
regard a DNN simply as a mapping $x\in\mathbb{R}^{d}\to P_{x}$. Before
formally introducing CMI and NCMI, we set up notation to be used throughout
the paper.
### II-A Notation
For a positive integer $K$, let $[K]\triangleq\\{1,\dots,K\\}$. Assume that
there are $C$ class labels with $[C]$ as the set of class labels. Let ${\cal
P}([C])$ denote the set of all probability distributions over $[C]$. For any
two probability distributions $P_{1},P_{2}\in{\cal P}([C])$, the cross entropy
of $P_{1}$ and $P_{2}$ is defined as
$H(P_{1},P_{2})=\sum_{i=1}^{C}-P_{1}(i)\ln P_{2}(i),$ (1)
where $\ln$ denotes the logarithm with base $e$; the Kullback–Leibler (KL)
divergence (or relative entropy) between $P_{1}$ and $P_{2}$ is defined as
$D(P_{1}||P_{2})=\sum_{i=1}^{C}P_{1}(i)\ln{P_{1}(i)\over P_{2}(i)}.$ (2)
For any $y\in[C]$ and $P\in{\cal P}([C])$, write the cross entropy of the one-
hot probability distribution corresponding to $y$ and $P$ as
$H(y,P)=-\ln P(y).$ (3)
Given a DNN: $x\in\mathbb{R}^{d}\to P_{x}$, let $\mathbf{\theta}$ denote its
weight vector consisting of all its connection weights; whenever there is no
ambiguity, we also write $P_{x}$ as $P_{x,\mathbf{\theta}}$, and $P_{x}(y)$ as
$P(y|x,\mathbf{\theta})$ for any $y\in[C]$.
### II-B Error Rate
Fix a DNN: $x\in\mathbb{R}^{d}\to P_{x}$. As before, let $(X,Y)$ be a pair of
random variables representing the raw input data and the corresponding ground
truth label; let $\hat{Y}$ be the label predicted by the DNN with probability
$P_{X}(\hat{Y})$ in response to the input $X$, that is, for any input
$x\in\mathbb{R}^{d}$ and any $\hat{y}\in[C]$
$P(\hat{Y}=\hat{y}|X=x)=P_{x}(\hat{y})=P(\hat{y}|x,\mathbf{\theta}).$ (4)
Note that $Y\to X\to\hat{Y}$ forms a Markov chain in the indicated order.
Therefore, given $X=x$, $Y$ and $\hat{Y}$ are conditionally independent.
The error rate of the DNN for $(X,Y)$ is equal to
$\epsilon=\Pr\\{\hat{Y}\not=Y\\}$
which can be upper bounded by the average of the cross entropy of the
conditional probability distribution of $Y$ given $X$,
$P_{Y|X}=P_{Y|X}(\cdot|X)$, and $P_{X}$, as shown in the following theorem.
###### Theorem 1.
For any DNN: $x\in\mathbb{R}^{d}\to P_{x}$ and any $(X,Y)$,
$\epsilon\leq\mbox{$\bf E$}_{X}\left[H(P_{Y|X},P_{X})\right]$ (5)
where $\mbox{$\bf E$}_{X}$ denotes the expectation with respect to $X$.
###### Proof:
Let $I_{\\{\hat{Y}\not=Y\\}}$ denote the indicator function of the event
$\\{\hat{Y}\not=Y\\}$. Then
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle\Pr\\{\hat{Y}\not=Y\\}$
$\displaystyle=$ $\displaystyle\mbox{$\bf E$}[I_{\\{\hat{Y}\not=Y\\}}]$
$\displaystyle=$ $\displaystyle\mbox{$\bf E$}_{X}\left[\mbox{$\bf
E$}[I_{\\{\hat{Y}\not=Y\\}}|X]\right]$ $\displaystyle=$
$\displaystyle\mbox{$\bf
E$}_{X}\left[1-\sum_{i=1}^{C}P_{Y|X}(i|X)P_{X}(i)\right]$ $\displaystyle=$
$\displaystyle\mbox{$\bf
E$}_{X}\left[\sum_{i=1}^{C}P_{Y|X}(i|X)(1-P_{X}(i))\right]$
$\displaystyle\leq$ $\displaystyle\mbox{$\bf
E$}_{X}\left[\sum_{i=1}^{C}-P_{Y|X}(i|X)\ln P_{X}(i)\right]$ (7)
$\displaystyle=$ $\displaystyle\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X})\right]$ (8)
where (II-B) follows from the fact that $Y$ and $\hat{Y}$ are conditionally
independent given $X$, and (7) is due to the inequality $\ln z\leq z-1$ for
any $z>0$. This completes the proof of Theorem 1. ∎
Given $X=x$, what happens if the DNN outputs instead the top one label
$\hat{Y}^{*}$
$\hat{Y}^{*}=\operatornamewithlimits{arg\,max}_{i\in[C]}P_{x}(i)?$
In this case, the error rate of the DNN for $(X,Y)$ is equal to
$\epsilon^{*}=\Pr\\{\hat{Y}^{*}\not=Y\\}$
which can also be upper bounded in terms of $\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X})\right]$.
###### Corollary 1.
For any DNN: $x\in\mathbb{R}^{d}\to P_{x}$ and any $(X,Y)$,
$\epsilon^{*}\leq C\epsilon\leq C\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X})\right].$ (9)
###### Proof:
$\displaystyle\epsilon^{*}$ $\displaystyle=$
$\displaystyle\Pr\\{\hat{Y}^{*}\not=Y\\}$ $\displaystyle=$
$\displaystyle\mbox{$\bf E$}_{X}\left[1-P_{Y|X}(\hat{Y}^{*}|X)\right]$
$\displaystyle\leq$ $\displaystyle C\mbox{$\bf
E$}_{X}\left[P_{X}(\hat{Y}^{*})(1-P_{Y|X}(\hat{Y}^{*}|X))\right]$
$\displaystyle\leq$ $\displaystyle C\mbox{$\bf
E$}_{X}\left[\sum_{i=1}^{C}P_{X}(i)(1-P_{Y|X}(i|X))\right]$ $\displaystyle=$
$\displaystyle C\mbox{$\bf
E$}_{X}\left[1-\sum_{i=1}^{C}P_{Y|X}(i|X)P_{X}(i)\right]$ $\displaystyle=$
$\displaystyle C\epsilon\leq C\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X})\right],$ (11)
where (II-B) follows from the fact that $P_{X}(\hat{Y}^{*})\geq 1/C$, and (11)
is due to (II-B) and (8). ∎
In view of Theorem 1 and Corollary 1, no matter which form of error rate
$\epsilon$ or $\epsilon^{*}$ is used, minimizing the average of the cross
entropy $\mbox{$\bf E$}_{X}[H(P_{Y|X},P_{X})]$ would have an effect to reduce
$\epsilon$ and $\epsilon^{*}$. This provides mathematical justifications for
the use of the average of the cross entropy $\mbox{$\bf
E$}_{X}[H(P_{Y|X},P_{X})]$ as an objective function or a major component
thereof in DL and knowledge distillation, where $P_{Y|X}$ is approximated by
the one-hot probability vector corresponding to $Y$ in DL[1, 2], and by the
output probability distribution of the teacher in knowledge distillation [7,
8, 9].
Figure 1: The mappings from the label space to the input space, and from the
input space to the output space of a DNN. Here caricatures are used to depict
label and input spaces, where each of the three instances in the label space
are mapped to two instances in input space according to
$P_{X|Y}(\cdot|Y=y_{i})$, for $i\in\\{1,2,3\\}$. On the other hand, the figure
for the output space is obtained from a real example, where for the ResNet56
model trained on CIFAR-100 dataset, the output probability vectors
corresponding to all validation sample instances from three randomly-picked
classes are projected over the two-dimensional probability simplex.
### II-C Concentration
The error rates $\epsilon$ and $\epsilon^{*}$ of the DNN:
$x\in\mathbb{R}^{d}\to P_{x}$ for $(X,Y)$ do not provide any useful
information on the intrinsic mapping structure of the DNN in the probability
distribution space ${\cal P}([C])$. Two important mapping structure properties
the DNN: $x\in\mathbb{R}^{d}\to P_{x}$ possesses, are its intra-class
concentration and inter-class separation in the space ${\cal P}([C])$. In this
and next subsections, we formally introduce information quantities to quantify
these two mapping structure properties, respectively.
Visualize the DNN: $x\in\mathbb{R}^{d}\to P_{x}$ according to Fig. 1. Given
$Y=y$, $y\in[C]$, the input data $X$ is conditionally distributed according to
the conditional distribution $P_{X|Y}(\cdot|y)$ and then mapped into $P_{X}$,
a random point in the space ${\cal P}([C])$. The instances (or realizations)
of this random point $P_{X}$ form a cluster in the space ${\cal P}([C])$. The
centroid of this cluster is the average of $P_{X}$ with respect to the
conditional distribution $P_{X|Y}(\cdot|y)$, which is exactly the conditional
distribution of $\hat{Y}$ given $Y=y$
$P_{\hat{Y}|y}=\mbox{$\bf E$}[P_{X}|Y=y].$ (12)
Measure the “distance” between each $P_{X}$ and the centroid $P_{\hat{Y}|y}$
by their KL divergence $D(P_{X}||P_{\hat{Y}|y})$. Then the average of KL
divergence $D(P_{X}||P_{\hat{Y}|y})$ with respect to the conditional
distribution $P_{X|Y}(\cdot|y)$ is equal to
$\displaystyle\mbox{$\bf E$}\left[D(P_{X}||P_{\hat{Y}|y})|Y=y\right]$ (13)
$\displaystyle=$ $\displaystyle\mbox{$\bf
E$}\left[\left(\sum_{i=1}^{C}P_{X}(i)\ln{P_{X}(i)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right)\left|Y=y\right.\right]$ $\displaystyle=$
$\displaystyle\sum_{x}P_{X|Y}(x|y)\left[\sum_{i=1}^{C}P(\hat{Y}=i|x)\times\right.$
$\displaystyle\left.\ln{P(\hat{Y}=i|x)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right]$ $\displaystyle=$ $\displaystyle
I(X;\hat{Y}|y),$ (14)
where $I(X;\hat{Y}|y)$ is the conditional mutual information between $X$ and
$\hat{Y}$ given $Y=y$. (Please refer to [6] for the notions of mutual
information and conditional mutual information.) In (13), $X$ is assumed to be
discrete; if $X$ is continuous, then the average $\sum_{x}P_{X|Y}(x|y)$ should
be replaced by the integral
$\int_{x}dP_{X|Y}(x|y).$
Note that (14) is due to the fact that $Y\to X\to\hat{Y}$ forms a Markov
chain.
The information quantity $I(X;\hat{Y}|y)$ quantifies the concentration of the
cluster formed by the instances of the random point $P_{X}$ given $Y=y$ around
its centroid $P_{\hat{Y}|y}$. Averaging $I(X;\hat{Y}|y)$ with respect to the
distribution $P_{Y}(y)$ of $Y$, we get the conditional mutual information
$I(X;\hat{Y}|Y)$ between $X$ and $\hat{Y}$ given $Y$:
$\displaystyle I(X;\hat{Y}|Y)$ $\displaystyle=$
$\displaystyle\sum_{y\in[C]}P_{Y}(y)I(X;\hat{Y}|y)$ (15) $\displaystyle=$
$\displaystyle\mbox{$\bf E$}\left[D(P_{X}||P_{\hat{Y}|Y})\right]$
$\displaystyle=$
$\displaystyle\sum_{y}\sum_{x}P(x,y)\left[\sum_{i=1}^{C}P(\hat{Y}=i|x)\times\right.$
$\displaystyle\left.\ln{P(\hat{Y}=i|x)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right].$
The CMI $I(X;\hat{Y}|Y)$ can then be regarded as a measure for the intra-class
concentration of the DNN: $x\in\mathbb{R}^{d}\to P_{x}$ for $(X,Y)$.
In practice, the joint distribution $P(x,y)$ of $(X,Y)$ may be unknown. To
compute the CMI $I(X;\hat{Y}|Y)$ in this case, one may approximate $P(x,y)$ by
the empirical distribution of a data sample
$\\{(x_{1},y_{1}),(x_{2},y_{2}),\cdots,(x_{n},y_{n})\\}$. For any $y\in[C]$,
let
$n_{y}=|\\{(x_{j},y_{j}):y_{j}=y,1\leq j\leq n\\}|,$ (16)
where $|S|$ denotes the cardinality of a set $S$, and
$P_{y}={1\over n_{y}}\sum_{(x_{j},y_{j}):y_{j}=y}P_{x_{j}}.$ (17)
Then $I(X;\hat{Y}|Y)$ can be computed as follows
$\displaystyle I(X;\hat{Y}|Y)$
$\displaystyle=\sum_{y\in[C]}\sum_{(x_{j},y_{j}):y_{j}=y}{1\over
n}D(P_{x_{j}}||P_{y})$ $\displaystyle={1\over
n}\sum_{j=1}^{n}D(P_{x_{j}}||P_{y_{j}}).$ (18)
### II-D Separation and NCMI
Let $(U,V)$ be a pair of random variables independent of $(X,Y)$, and having
the same joint distribution as that of $(X,Y)$. With reference to Fig. 1, we
define the following information quantity111Other information quantities can
also be defined and used as a measure for the inter-class separation of the
DNN: $x\in\mathbb{R}^{d}\to P_{x}$, which will be explored in Appendix B.
Although they are more or less equivalent, the information quantity $\Gamma$
defined here is more convenient for the selection of hyper parameters in our
proposed CMIC deep learning.
$\Gamma=\mbox{$\bf E$}\left[I_{\\{Y\not=V\\}}H(P_{X},P_{U})\right],$ (19)
and use $\Gamma$ as a measure for the inter-class separation of the DNN:
$x\in\mathbb{R}^{d}\to P_{x}$. It is clear that the larger $\Gamma$ is, the
further apart different clusters are from each other on average.
Ideally, we want $I(X;\hat{Y}|Y)$ to be small while keeping $\Gamma$ large.
This leads us to consider the ratio between $I(X;\hat{Y}|Y)$ and $\Gamma$:
$\hat{I}(X;\hat{Y}|Y)\mbox{$\
\stackrel{{\scriptstyle\Delta}}{{=}}$}{I(X;\hat{Y}|Y)\over\Gamma}.$ (20)
We call $\hat{I}(X;\hat{Y}|Y)$ the normalized conditional mutual information
between $X$ and $\hat{Y}$ given $Y$.
In case where the joint distribution $p(x,y)$ of $(X,Y)$ is unknown, it can be
approximated by the empirical distribution of a data sample
$\\{(x_{1},y_{1}),(x_{2},y_{2}),\cdots,(x_{n},y_{n})\\}$. In parallel with
(18), $\Gamma$ can be computed in this case as follows:
$\Gamma={1\over
n^{2}}\sum_{j=1}^{n}\sum_{k=1}^{n}I_{\\{y_{j}\not=y_{k}\\}}H(P_{x_{j}},P_{x_{k}}),$
(21)
from which and (18), $\hat{I}(X;\hat{Y}|Y)$ can be computed accordingly.
### II-E Related Works
In the literature, intra-class concentration and inter-class separation of a
DNN have been mainly investigated in the feature space corresponding to the
penultimate layer of the DNN, and largely treated in an ad-hoc manner in a
deep learning process or algorithm. Specifically, it was observed numerically
in [10, 11, 12] that DNNs concentrate features of each class around their
separated mean. This observation was further analyzed in [13] under the
Gaussian mixture model assumption about features. In [14, 15, 16, 17, 18] and
references therein, different loss functions including the so-called center
loss, contrastive center loss, orthogonal project loss, constrained center
loss, and their variants, all of which are defined in the feature space, were
proposed and used in the respective learning processes to improve the intra-
class concentration and inter-class separation of such trained DNNs.
In contrast, in this paper we investigate the intra-class concentration and
inter-class separation of a DNN in its output probability distribution space
${\cal P}([C])$, where the DNN is viewed as a mapping from
$x\in\mathbb{R}^{d}$ to $P_{x}$. This perspective allows us to introduce
information quantities, CMI, $\Gamma$, and NCMI, to quantify the intra-class
concentration and inter-class separation of each DNN. In addition, our
introduced CMI and NCMI can also be regarded as additional performance metrics
for any DNN, which are in parallel with the error rate performance metric, are
independent of any learning process, and represent mapping structure
properties of a DNN. As additional performance metrics, they can be used to
evaluate and compare different DNNs regardless of the architectures and sizes
of DNNs.
Another related work in the sense of introducing information theoretic ideas
into DL is the so-called coded deep learning (CDL) [19], where information
theoretic coding ideas are embedded into the inner workings of DL. The
purposes of CDL are to eliminate essentially floating operations of a coded
DNN during its inference time and efficiently compress the coded DNN while
maintaining or even improving the error rate of the coded DNN.
In the next section, CMI and NCMI $\hat{I}(X;\hat{Y}|Y)$ will be used to
evaluate and compare popular DNNs pre-trained over ImageNet in the literature.
## III NCMI Vs. Accuracy
The popular DNNs we selected for evaluation according to their respective CMI
and NCMI are ResNet-$\\{18,34,50,101,152\\}$ [20], VGG-$\\{11,13,16,19\\}$
[21], EfficientNet-$\\{\text{B0},\text{B1},\text{B2},\text{B3}\\}$ [22], Wide-
ResNet-$\\{50,101\\}$ [23], MobileNet-V3-$\\{\text{small},\text{large}\\}$
[24], and AlexNet [25]. They are all pre-trained on ImageNet dataset and
obtained from the Pytorch official
website222https://pytorch.org/vision/stable/models.html..
Table I lists the values of CMI, $\Gamma$, and NCMI of the selected DNNs,
which are calculated, according to (18), (21), and (20), over the ImageNet
validation set, along with their respective error rate $\epsilon^{*}$. From
Table I, it is clear that within the same family, as the model size increases,
the CMI value decreases. This shows that larger models have more compact
clusters in the output probability space ${\cal P}([C])$. For the $\Gamma$
value, although the general trend is that within the same family, the $\Gamma$
value increases as the model size gets larger, there does exist an exception.
Note that for the EfficientNet family, the smallest model EfficientNet-B0 has
the largest $\Gamma$ value.
Now turn our attention to the NCMI value. From Table I, it follows that as the
model size within the same family increases, the NCMI value decreases as well.
Even more interesting is the relationship between the NCMI and error rate
$\epsilon^{*}$. Across all models evaluated, as the NCMI value decreases, so
does the error rate $\epsilon^{*}$. To make the relationship between the NCMI
and error rate $\epsilon^{*}$ more transparent, Figure 2 illustrates the
relationship graphically. From Figure 2, it seems that the NCMI and error rate
$\epsilon^{*}$ have a positive linear relationship; indeed, the Pearson
correlation coefficient $\rho$ [26] between them is $\rho=0.9929$, strongly
supporting the former statement. As such, the NCMI value of a DNN can be used
to gauge the prediction performance of the DNN.
To conclude this section, let us draw some analogies. If a DNN is analogized
with a student, then the error rate and NCMI of the DNN can be analogized with
the testing score of the student in an exam and certain trait of the student,
respectively. In a way similar to using the trait of the student to predict
the student’s testing performance, one can also use the NCMI value of the DNN
to predict the DNN’s testing performance.
TABLE I: CMI, $\Gamma$, and NCMI values over the validation set of some pre-
trained models on ImageNet dataset along with their error rate $\epsilon^{*}$,
where the DNNs from the same family are highlighted by the same color.
Models | CMI | $\Gamma$ | NCMI | Error rate $\epsilon^{*}$ | Models | CMI | $\Gamma$ | NCMI | Error rate $\epsilon^{*}$
---|---|---|---|---|---|---|---|---|---
ResNet18 | 0.999 | 9.891 | 0.101 | 0.302 | AlexNet | 1.331 | 9.830 | 0.135 | 0.434
ResNet34 | 0.902 | 9.919 | 0.090 | 0.266 | EfficientNet-B0 | 0.692 | 9.433 | 0.073 | 0.220
ResNet50 | 0.815 | 9.929 | 0.082 | 0.238 | EfficientNet-B1 | 0.661 | 9.114 | 0.072 | 0.213
ResNet101 | 0.779 | 9.948 | 0.078 | 0.226 | EfficientNet-B2 | 0.639 | 9.224 | 0.069 | 0.193
ResNet152 | 0.749 | 9.953 | 0.075 | 0.216 | EfficientNet-B3 | 0.627 | 9.365 | 0.067 | 0.180
VGG11 | 0.959 | 9.899 | 0.096 | 0.296 | Wide-ResNet50 | 0.749 | 9.935 | 0.075 | 0.215
VGG13 | 0.930 | 9.909 | 0.094 | 0.284 | Wide-ResNet101 | 0.734 | 9.937 | 0.073 | 0.211
VGG16 | 0.878 | 9.925 | 0.088 | 0.266 | MobileNet-V3-Small | 1.088 | 9.898 | 0.110 | 0.323
VGG19 | 0.860 | 9.930 | 0.086 | 0.257 | MobileNet-V3-Large | 0.922 | 9.956 | 0.092 | 0.259
$\displaystyle
J_{\mathcal{B}}\left(\lambda,\beta,\theta,\\{Q_{c}\\}_{c\in[C]}\right)$
$\displaystyle=\frac{1}{|\mathcal{B}|}\sum_{(x,y)\in\mathcal{B}}H(y,P_{x,\mathbf{\theta}})+\lambda{1\over|\mathcal{B}|}\sum_{(x,y)\in\mathcal{B}}D(P_{x,\mathbf{\theta}}||Q_{y})-\beta\frac{1}{|\mathcal{B}|^{2}}\sum_{(x,y),(u,v)\in\mathcal{B}}I_{\\{y\not=v\\}}H(P_{x,\mathbf{\theta}},P_{u,\mathbf{\theta}}).$
(22) Figure 2: The error rate vs NCMI value over the validation set of popular
pre-trained models on ImageNet dataset. The sizes of the circles represent the
sizes of respective models in terms of the number of model parameters; the
larger the circle, the larger the model.
## IV CMIC Deep Learning
The discussions in the above section suggest a new way of learning. In the
learning process, instead of minimizing the average of cross entropy
$\mbox{$\bf E$}_{X}\left[H(P_{Y|X},P_{X})\right]$ alone, one also needs to
look after the NCMI $\hat{I}(X;\hat{Y}|Y)$. This leads to a new form of
learning framework dubbed CMI constrained deep learning (CMIC-DL), which is
described next.
### IV-A Optimization Problem Formulation
In CMIC-DL, the optimization problem to be solved is as follows:
$\displaystyle\min_{\mathbf{\theta}}~{}$ $\displaystyle\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]$ s.t.
$\displaystyle\hat{I}(X;\hat{Y}|Y)=r,$ (23)
where $r$ is a positive constant. By interpreting $\hat{I}(X;\hat{Y}|Y)$ as a
rate, and $\mbox{$\bf E$}_{X}\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]$ as
a distortion, the above optimization problem resembles the rate distortion
problem in information theory [6, 27, 28]. By rewriting the constraint in
(IV-A), and using the Lagrange multiplier method, the constrained optimization
problem in (IV-A) could be formulated as the following unconstrained one
$\displaystyle\min_{\mathbf{\theta}}~{}$ $\displaystyle\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]$ $\displaystyle+\lambda
I(X;\hat{Y}|Y)-\beta\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}H(P_{X,\mathbf{\theta}},P_{U,\mathbf{\theta}})\right],$
(24)
where $\lambda>0$ is a scalar, and $\beta=\lambda r$.
Note that in view of (15), the CMI $I(X;\hat{Y}|Y)$ in (IV-A) depends on
$P_{\hat{Y}|Y}$, which, for $Y=y$, is the average of $P_{X,\mathbf{\theta}}$
with respect to the conditional distribution $P_{X|Y}(\cdot|y)$ (see (12)). As
such, the unconstrained optimization problem in its form (IV-A) is not
amenable to numerical solutions. To overcome this, we first convert it into a
double unconstrained minimization problem by introducing a dummy distribution
$Q_{y}\in{\cal P}([C])$ for each $y\in[C]$, as shown in the following theorem,
which will be proved in Appendix A.
###### Theorem 2.
For any $\lambda>0$ and $\beta>0$,
$\displaystyle\min_{\mathbf{\theta}}~{}$ $\displaystyle\left\\{\mbox{$\bf
E$}_{X}\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]\right.$
$\displaystyle\left.+\lambda I(X;\hat{Y}|Y)-\beta\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}H(P_{X,\mathbf{\theta}},P_{U,\mathbf{\theta}})\right]\right\\}$
$\displaystyle=$
$\displaystyle\min_{\mathbf{\theta}}~{}\min_{\\{Q_{c}\\}_{c\in[C]}}~{}\left\\{\mbox{$\bf
E$}[H(P_{Y|X},P_{X,\mathbf{\theta}})+\lambda
D(P_{X,\mathbf{\theta}}||Q_{Y})]\right.$ $\displaystyle\left.-\beta\mbox{$\bf
E$}[I_{\\{Y\not=V\\}}H(P_{X,\mathbf{\theta}},P_{U,\mathbf{\theta}})]\right\\}.$
(25)
In practice, the joint distribution $P(x,y)$ of $(X,Y)$ may be unknown. In
this case, to solve (2) numerically, one may approximate $P(x,y)$ by the
empirical distribution of a data sample (such as a mini-batch in the DL
process)
$\mathcal{B}=\\{(x_{i_{1}},y_{i_{1}}),(x_{i_{2}},y_{i_{2}}),\cdots,(x_{i_{m}},y_{i_{m}})\\}$,
and $P_{Y|X}$ by the one-hot probability distribution corresponding to $Y$.
Accordingly, the objective function in the double minimization (2) can be
approximated by
$J_{\mathcal{B}}\left(\lambda,\beta,\theta,\\{Q_{c}\\}_{c\in[C]}\right)$ shown
in (22) (on the top of the page).
### IV-B Algorithm for Solving the Optimization in (2)
Having addressed how to approximate the objection function in the double
minimization (2), we are now ready to present an algorithm for solving (2). In
fact, by reformulating the single minimization problem as a double
minimization problem, Theorem 2 lends us an alternating algorithm that
optimizes $\mathbf{\theta}$ and $\\{Q_{c}\\}_{c\in[C]}$ alternatively to
minimize the objective function in (2), given that the other is fixed.
Given $\\{Q_{c}\\}_{c\in[C]}$, $\mathbf{\theta}$ can be updated using the same
strategy as in the conventional DL through stochastic gradient descent
iterations over mini-batches, where the training set is divided into $B$ mini-
batches $\\{\mathcal{B}_{b}\\}_{b\in[B]}$ with each batch of size
$|\mathcal{B}|$. Given $\mathbf{\theta}$, how is $\\{Q_{c}\\}_{c\in[C]}$
updated? This is where differences arise. In view of (12) and (32), the
optimal $\\{Q_{c}\\}_{c\in[C]}$ given $\mathbf{\theta}$ is equal to
$Q_{c}=P_{\hat{Y}|y=c}=\sum_{x}P(x|y=c)P_{x,\mathbf{\theta}},$ (26)
for any $c\in[C]$. Therefore, to update $\\{Q_{c}\\}_{c\in[C]}$ given
$\mathbf{\theta}$, we construct, at each iteration, $C$ mini-batches
$\\{\mathfrak{B}_{c}\\}_{c\in[C]}$ in the following manner: to make
$\mathfrak{B}_{c}$, $\forall c\in[C]$, we randomly sample $|\mathfrak{B}_{c}|$
instances from the training samples whose ground truth labels are $c$. It then
follows from (26) that for any $c\in[C]$, $Q_{c}$ is updated as333To update
$\\{Q_{c}\\}_{c\in[C]}$, we may use momentum to make the updation more stable
and less noisy.
$Q_{c}=\frac{\sum_{x\in\mathfrak{B}_{c}}P_{x,\mathbf{\theta}}}{|\mathfrak{B}_{c}|}.$
(27)
The procedure for solving the optimization problem (2) is now summarized in
Algorithm 1, where we use $(\cdot)^{t}_{c,b}$ to indicate class $c$ at the
$b$-th batch updation during the $t$-th epoch. We also use $(\cdot)^{t}_{c,B}$
as $(\cdot)^{t}_{c}$ whenever necessary, and set
$(\cdot)^{t}_{c,0}=(\cdot)^{t-1}_{c}$.
Algorithm 1 The proposed alternating algorithm for solving the optimization
problem in (2)
0: The training set $\mathcal{T}$, all mini-batches
$\\{\mathcal{B}_{b}\\}_{b\in[B]}$, number of epochs $\mathit{T}$, $\lambda$,
and $\beta$.
1: Initialization:Initialize $\theta^{0}$ and $\\{Q_{c}^{0}\\}_{c\in[C]}$.
2: for $t=1$ to $T_{max}$ do
3: for $b=1$ to $B$ do
4: _$[$ Updating $\theta$$]$_: Fix $\\{Q_{c,b-1}^{t}\\}_{c\in[C]}$. Update
$\theta^{t}_{b-1}$ to $\theta^{t}_{b}$ by using (stochastic) batch gradient
descent over the loss function
$J_{\mathcal{B}_{b}}\left(\lambda,\beta,\theta^{t}_{b-1},\\{Q_{c,b-1}^{t}\\}_{c\in[C]}\right)$.
5: _$[$ Updating $\\{Q_{c}\\}_{c\in[C]}$$]$_: Fix $\theta^{t}_{b}$. Construct
mini-batches $\\{\mathfrak{B}_{c}\\}_{c\in[C]}$ from $\mathcal{T}$. Update
$Q_{c,b-1}^{t}$ to $Q_{c,b}^{t}$, $\forall c\in[C]$, according to (27), i.e.,
$\displaystyle
Q_{c,b}^{t}=\frac{\sum_{x\in\mathfrak{B}_{c}}P_{x,\mathbf{\theta}^{t}_{b}}}{|\mathfrak{B}_{c}|}.$
(28)
6: end for
7: end for
8: return model parameters $\theta^{T}$.
## V Experiment Results
To demonstrate the effectiveness of CMIC-DL and compare it with some state-of-
the-art alternatives, we have conducted a series of experiments. Specifically,
we have performed experiments on two popular image classification datasets,
namely CIFAR-100 [29] and ImageNet [25]. In Subsections V-A and V-B, we
present their respective accuracy results. In Subsection V-C, we explore how
to visualize the concentration and separation of a DNN, which is made possible
by viewing the DNN as a mapping from $x\in\mathbb{R}^{d}$ to $P_{x}$; using
such a visualization method, the concentration and separation of ResNet-56
trained within our CMIC-DL framework are then compared with those of ResNet-56
trained within the standard DL framework.
In the literature, a deep learning process is typically analyzed
experimentally through the evolution curve of its error rate. With our newly
introduced performance metrics, CMI, $\Gamma$ (separation), and NCMI, the
learning process can also be analyzed through the evolution curves of CMI,
$\Gamma$, and NCMI, which show interestingly how the mapping structure in
terms of CMI, $\Gamma$, and NCMI evolves over the course of learning process.
In Subsection V-D, we use ResNet-56 as an example, and illustrate and compare
the evolution curves of CMI, $\Gamma$, NCMI, and error rate within our CMIC-DL
framework vs within the standard DL framework. Lastly, in Subsection V-E, we
evaluate the robustness of models trained within our CMIC-DL framework against
two different adversarial attacks, and show that in comparison with the
standard DL, CMIC-DL improves the robustness of DNNs as well.
### V-A Experiments on CIFAR-100
CIFAR-100 dataset contains 50K training and 10K test colour images of size
$32\times 32$, which are labeled for 100 classes.
$\bullet$ Models: To show the effectiveness of CMIC-DL, we have conducted
experiments on three different model architectural families. Specifically, we
have selected (i) three models from ResNet family [20], namely
ResNet-$\\{32,56,110\\}$; (ii) VGG-13 from VGG family [21]; and (iii) Wide-
ResNet-28-10 from Wide-ResNet family [23].
$\bullet$ Benchmarks: We evaluate the performance of the DNNs trained via
CMIC-DL against those trained by conventional cross entropy loss (CE), center
loss (CL) [16] which promotes clustering the features, focal loss (FL) [30]
which uses regularization, large-margin Gaussian Mixture (L-GM) loss [31]
which imposes margin constraints, and orthogonal projection loss (OPL) [18]
which imposes orthogonality in the feature space.
$\bullet$ Training settings: We have deployed an SGD optimizer with a momentum
of 0.9, a weight decay of 0.0005, and a batch size of 64. We have trained the
models for 200 epochs, and adopted an initial learning rate of 0.1, which is
further divided by 10 at the 60-th, 120-th and 160-th epochs. To have a fair
comparison, we have reproduced the results of all the benchmark methods using
their respective best hyper-parameters reported in their original papers. In
addition, in Algorithm 1, we set $\\{Q_{c}^{0}(i)\\}_{c\in[C]}=\frac{1}{C}$,
for $i\in[C]$, use $|\mathfrak{B}_{c}|=8$, $\forall c\in[C]$, and also update
$Q_{c,b}^{t}$ using the momentum of 0.9999.
The results are summarized in Table II. As seen, the models trained within our
CMIC-DL framework outperform those trained by the benchmark methods.
Importantly, the improvement is consistent across the models from different
architectural families, showing that CMIC-DL can effectively train DNNs from
different families. As a rule of thumb, compared to the CE method, CMIC-DL
yields DNNs with almost 1.3% higher validation accuracy for the ResNet models.
Furthermore, in Table III we report the NCMI values $\hat{I}(X;\hat{Y}|Y)$,
over the validation set, for the models we trained in Table II, where we use
the notation $\hat{I}_{\text{Loss}}$ to denote the NCMI value when the
underlying DNN is trained using “Loss” method. As observed,
$\hat{I}_{\text{CMIC}}$ has the smallest value compared to the other
counterparts.
In addition, in Table IV, we report the $\lambda^{*}$ and $\beta^{*}$ values
for which we obtained the best validation accuracies. As observed, the
$\lambda^{*}$ and $\beta^{*}$ values are almost the same for all the models.
TABLE II: The validation accuracies (%) of different models trained by CMIC-DL and different benchmark methods over CIFAR-100 dataset, which are averaged over three different random seeds, and where Bold and underlined values denote the best and second best results, respectively. Loss | Res32 | Res56 | Res110 | VGG13 | WRN-28-10
---|---|---|---|---|---
CL | 70.23 | 72.70 | 74.20 | 74.50 | 80.97
FL | 71.62 | 73.20 | 74.35 | 74.53 | 81.24
LGM | 71.50 | 73.06 | 74.39 | 74.57 | 81.29
OPL | 71.03 | 72.60 | 73.98 | 74.11 | 81.12
CE | 70.90 | 72.40 | 73.79 | 73.77 | 80.93
CMIC | 72.24 | 73.66 | 75.08 | 74.62 | 81.63
TABLE III: The respective NCMI values, measured over the validation set, of the models trained in Table II via different benchmark methods. The values are averaged over thee different runs. Loss | Res32 | Res56 | Res110 | VGG13 | WRN-28-10
---|---|---|---|---|---
$\hat{I}_{\text{CL}}$ | 0.057 | 0.045 | 0.0395 | 0.0395 | 0.0309
$\hat{I}_{\text{FL}}$ | 0.053 | 0.046 | 0.0393 | 0.0399 | 0.0312
$\hat{I}_{\text{LGM}}$ | 0.054 | 0.047 | 0.0390 | 0.0398 | 0.0310
$\hat{I}_{\text{OPL}}$ | 0.056 | 0.050 | 0.0397 | 0.0402 | 0.0314
$\hat{I}_{\text{CE}}$ | 0.057 | 0.053 | 0.0402 | 0.0408 | 0.0317
$\hat{I}_{\text{CMIC}}$ | 0.051 | 0.042 | 0.0382 | 0.0392 | 0.0303
TABLE IV: Hyper-parameters, $\lambda^{*}$ and $\beta^{*}$, that were used in
CMIC-DL in Table II.
Params. | Res32 | Res56 | Res110 | VGG13 | WRN-28-10
---|---|---|---|---|---
($\lambda^{*}$,$\beta^{*}$) | (0.7,0.4) | (0.7,0.4) | (0.7,0.2) | (0.8,0.3) | (0.7,0.4)
### V-B Experiments on ImageNet
ImageNet is a large-scale dataset used in visual recognition tasks, containing
around 1.2 million training samples and 50,000 validation images.
$\bullet$ Models: We have conducted experiments on two models from ResNet
family, namely ResNet-18 and ResNet-50.
$\bullet$ Benchmarks: We evaluate the performance of CMIC-DL against CE and
OPL.
$\bullet$ Training settings: We have deployed an SGD optimizer with a momentum
of 0.9, a weight decay of 0.0001, and a batch size of 256. We have trained the
models for 90 epochs, and adopted an initial learning rate of 0.1, which is
further divided by 10 at the 30-th and 60-th epochs. In Algorithm 1, we set
$\\{Q_{c}^{0}(i)\\}_{c\in[C]}=\frac{1}{C}$, for $i\in[C]$, use
$|\mathfrak{B}_{c}|=8$, $\forall c\in[C]$, and also update $Q_{c,b}^{t}$ using
the momentum of 0.9999.
The top-$\\{1,5\\}$ accuracies are reported in Table V. As seen, in comparison
with the CE method, CMIC-DL increases the top-1 validation accuracy for
ResNet-18 and ResNet-50 by 0.56% and 0.37%, respectively. The improvement is
also consistent for the top-5 validation accuracy.
The hyper parameters $(\lambda^{*},\beta^{*})$ used in CMIC-DL for ResNet-18
and ResNet-50 are $(0.6,0.1)$ and $(0.6,0.2)$, respectively. The corresponding
NCMI values are $\hat{I}_{\text{CE}}=0.110$ and $\hat{I}_{\text{CMIC}}=0.102$
for ResNet-18, and $\hat{I}_{\text{CE}}=0.091$ and
$\hat{I}_{\text{CMIC}}=0.088$ for ResNet-50.
TABLE V: The validation accuracies (%) of different models trained by CMIC-DL and different benchmark methods on ImageNet dataset. | ResNet-18 | ResNet-50
---|---|---
Method | top-1 | top-5 | top-1 | top-5
CE (Baseline) | 69.91 | 89.08 | 76.15 | 92.87
OPL | 70.27 | 89.60 | 76.32 | 93.09
CMIC | 70.47 | 89.96 | 76.52 | 93.44
### V-C Concentration and Separation Visualization
In this subsection, we explore how to visualize concentration and separation
of a DNN. Consider the data set CIFAR-100. To visualize concentration and
separation of a DNN in a dimension reduced probability space, we randomly
select three class labels. Restrict ourselves to a subset consisting of all
validation sample instances with labels from the three selected labels. Given
a DNN, feed each validation sample instance from the subset into the DNN, keep
only three logits corresponding to the three selected labels, and then convert
these three logits into a 3 dimension probability vector through the softmax
operation. Following these steps in the indicated order, the DNN then maps
each validation sample instance from the subset into a 3 dimension probability
vector. Further project the 3 dimension probability vector into the 2
dimension simplex. Then the concentration and separation properties of the DNN
for the three selected classes can be more or less visualized through the
projected 2 dimension simplex.
Using the above visualization method, Fig. 3 compares the concentration and
separation properties of ResNet-56 trained within our CMIC-DL framework with
those of ResNet-56 trained within the standard CE framework. From Fig. 3, it
is clear that the three clusters in the case CMIC-DL are more concentrated
than their counterparts in the case of CE, and also further apart from each
other than their counterparts in the case of CE. Again, this is consistent
with the NCMI values reported in Table III.
Figure 3: Visualization and comparison of concentration and separation:
ResNet56 trained via CE vs ResNet56 trained via CMIC, where different shapes
indicate different classes.
### V-D Evolution of CMI, $\Gamma$, NCMI, and error rate
In this subsection, we analyze and visualize a learning process within either
our CMIC-DL framework or the conventional CE-based DL framework through the
lens of CMI, $\Gamma$, NCMI, and error rate. Fig. 4 shows the evolution curves
of CMI, $\Gamma$, NCMI, and error rate over the validation set during the
course of training ResNet-56 on CIFAR-100 dataset in each case, where the
training setup is the same as that used in Subsection V-A, and we use
$\lambda=0.7$ and $\beta=0.4$ in the case of CMIC-DL.
As seen in Fig. 4a, the CMI value in both CE and CMIC-DL cases is small at the
beginning of the training (epoch zero). This is because at the beginning, all
clusters in the output probability distribution space $\cal{P}([C])$ stick
around together, as shown from the separation distance curve (see Fig. 4b),
and probability distributions within each cluster are not separated at all.
After the training starts and for the first a few epochs, the clusters move
away from each other; during the course of movement, probability distributions
within each cluster move in different speed, and become separated. As such,
both the values of CMI and $\Gamma$ increase. Indeed, this is shown in Fig. 4a
and Fig. 4b. Hereafter, the clusters continue to move away from each other,
while at the same time, probability distributions within each cluster tend to
move together. Thus the $\Gamma$ value continues to increase, while the CMI
value decreases, as shown again in Fig. 4a and Fig. 4b.
The above summarizes the general behaviour of the CMI and $\Gamma$ evolution
curves in both CE and CMIC-DL cases. Let us now examine the differences
between them. From Fig. 4a, it is clear that the CMI evolution curve in the
case of CMIC-DL always remains below its counterpart in the CE case. On the
other hand, as shown in Fig. 4b, although initially the $\Gamma$ value
increases faster in the CE case than in the CMIC-DL case, after the first a
few epochs, the rate of increase in $\Gamma$ value is consistently higher in
the CMIC-DL case than in the CE case to the extent that the $\Gamma$ value in
the CMIC-DL case surpasses its counterpart in the CE case in the late stage of
the learning process.
From Fig. 4c and Fig. 4d, we can see that once the learning process is more or
less stabilized, both the NCMI value and error rate in the CMIC-DL case are
consistently smaller than their counterparts in the CE case. Once again, this
is consistent with our observation in Fig. 2: the smaller the NCMI value, the
lower the error rate. In conjunction with the visualization method discussed
in Subsection V-C, we have created a video available at
https://youtu.be/G0fDwv6o9Ek to illustrate the learning process during the
course of training ResNet-56 on CIFAR-100 dataset in each of the CE and NMIC-
DL cases through the lens of CMI and $\Gamma$, where concentration and
separation are shown for three randomly selected classes, and the evolution
curves of CMI and $\Gamma$ are shown for all classes.
(a) CMI ($I$).
(b) Separation ($\Gamma$).
(c) NCMI ($\hat{I}$).
(d) Error rate ($\epsilon^{*}$).
Figure 4: The evolution curves of (a) CMI, (b) $\Gamma$, (c) NCMI, and (d)
error rate over the course of training ResNet-56 over CIFAR-100 dataset using
CE and CMIC frameworks.
### V-E Robustness against adversarial attacks
As a by-product, we would expect that DNNs trained within the CMIC-DL
framework are more robust against adversarial attacks, in comparison with
their counterparts trained within the standard CE-based DL framework. This is
because when a DNN is trained within our CMIC-DL framework, its clusters in
its output probability distribution space are more compact, and also further
separated from each other, in comparison with its counterpart trained within
the standard CE-based DL framework. As such, it is harder for an adversary to
craft a perturbation which, when added to a clean sample, would result in an
attacked sample falling into a cluster with a different label. Our purpose in
this subsection is to confirm this by-product. To this end, we have performed
the following experiments.
$\bullet$ Dataset: We have used MNIST dataset [32] comprising of 10-class
handwritten digits.
$\bullet$ Model: We have selected a simple DNN with three convolutional and
one fully connected layers.
$\bullet$ Attacks: Two white-box attacks have been selected, where the
adversary has an access to the gradients of the underlying model.
Specifically, FGSM [3] and PGD attack [5] with 5 iterations were employed with
attack perturbation budgets
$\|\epsilon\|_{\infty}=\\{0.05,0.10,0.15,0.20,0.25,0.30,0.35\\}$.
$\bullet$ Training settings: We have deployed an SGD optimizer with a batch
size of 64. We have trained the models for 15 epochs and adopted an step
learning rate annealing with decay factor of 0.7. The hyper parameters were
selected to be $\lambda^{*}=2$ and $\beta^{*}=9$ in our CMIC-DL framework due
to the fact that the classification task over MNIST dataset is far simpler
than that over CIFAR-100 and ImageNet dataset.
Fig. 5 illustrates the resulting trade-offs between robust accuracy and
perturbation budget. From Fig. 5, it is clear that the DNN trained within the
CMIC-DL framework is more robust against both FGSM and PGD attacks, in
comparison with its counterpart trained within the standard CE-based DL
framework, thus confirming the by-product. In addition, the clean accuracy for
the models trained within the CE-based DL and CMIC-DL frameworks are 99.14%
and 99.21%, respectively, showcasing that the accuracy over the benign samples
is not sacrificed for a higher robust accuracy.
(a)
(b)
Figure 5: The robustness of a simple DNN over MNIST dataset trained within the
conventional CE-based DL and CMIC-DL frameworks against (a) FGSM attack and
(b) PGD attack with 5 iteraions, respectively.
We conclude this subsection by pointing out that although CMIC-DL can improve
the robustness of DNNs trained therein against adversarial attacks, CMIC-DL
itself is not a framework for adversarial training. In our future work, we
will fully address CMIC adversarial training by extending the performance
metrics of CMI, $\Gamma$ (separation), and NCMI to the new concepts of robust
CMI, robust separation, and robust NCMI.
## VI Conclusion
Viewing a DNN as a mapping from $x\in\mathbb{R}^{d}$ to $P_{x}$, in this paper
we have introduced conditional mutual information (CMI) and normalized
conditional mutual information (NCMI) as new performance metrics of the DNN to
measure the intra-class concentration and inter-class separation of the DNN.
As new performance metrics, CMI and NCMI are in parallel with error rate. We
then have used CMI and NCMI to evaluate and compare DNNs of different
architectures and sizes. It turns out that NCMI and error rate have
essentially a positive linear relationship with their correlation $\geq 0.99$.
As such, the NCMI value of a DNN can be used to gauge the prediction
performance of the DNN.
Based on NCMI, we have then developed a learning framework called CMI
constrained deep learning (CMI-DL) within which the conventional cross entropy
function is minimized subject to a NCMI constraint. An novel alternating
learning algorithm has been further proposed to solve such a constrained
optimization problem. Extensive experiment results consistently show that DNNs
trained within the CMIC-DL framework outperform those trained using the other
DL benchamrk methods discussed in the paper. In addition, with CMI and NCMI as
performance metrics for measuring the concentration and separation of a DNN,
the learning process of the DNN can also be analyzed and visualized through
the evolution of CMI and NCMI.
Open problems include (1) how to extend CMI and NCMI to define concepts of
robust CMI, robust separation, and robust NCMI; (2) how to extend CMIC-DL to
robust CMIC-DL to fully address adversarial training; (3) how to use CMI to
help estimate the conditional probability distribution of $Y$ given $X$; and
(4) the investigation of minimizing NCMI alone without using the standard
cross entropy objective function by modifying a predictor. These problems will
be addressed in the future.
## Appendix A Proof of Theorem 2
Since $\lambda>0$ and $\beta>0$, it suffices to show that
$I(X;\hat{Y}|Y)=\min_{\\{Q_{c}\\}_{c\in[C]}}\mbox{$\bf
E$}[D(P_{X,\mathbf{\theta}}||Q_{Y})].$ (29)
To this end, we apply (15) to get the following:
$\displaystyle
I(X;\hat{Y}|Y)=\sum_{y}\sum_{x}P(x,y)\left[\sum_{i=1}^{C}P(\hat{Y}=i|x,\mathbf{\theta})\times\right.$
$\displaystyle\left.\ln{P(\hat{Y}=i|x,\mathbf{\theta})\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right]$
$\displaystyle=\sum_{y}\sum_{x}P(x,y)\left[\sum_{i=1}^{C}P(\hat{Y}=i|x,\mathbf{\theta})\times\left[\ln{P(\hat{Y}=i|x,\mathbf{\theta})\over
Q_{y}(i)}\right.\right.$ $\displaystyle\left.\left.+\ln{Q_{y}(i)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right]\right]$
$\displaystyle=\sum_{y}\sum_{x}P(x,y)D(P_{x,\mathbf{\theta}}||Q_{y})+\sum_{y}\sum_{x}P(x,y)\times$
$\displaystyle\left[\sum_{i=1}^{C}P(\hat{Y}=i|x,\mathbf{\theta})\ln{Q_{y}(i)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right]$ $\displaystyle=\mbox{$\bf
E$}[D(P_{X,\mathbf{\theta}}||Q_{Y})]$
$\displaystyle+\sum_{y}P(y)\left[\sum_{i=1}^{C}P_{\hat{Y}|y}(\hat{Y}=i|y)\ln{Q_{y}(i)\over
P_{\hat{Y}|y}(\hat{Y}=i|Y=y)}\right]$ $\displaystyle=\mbox{$\bf
E$}[D(P_{X,\mathbf{\theta}}||Q_{Y})]-\mbox{$\bf E$}[D(P_{\hat{Y}|Y}||Q_{Y})]$
$\displaystyle\leq\mbox{$\bf E$}[D(P_{X,\mathbf{\theta}}||Q_{Y})],$ (30)
for any $Q_{y}\in{\cal P}([C]),y\in[C]$, where the inequality above is due to
the nonnegativity of KL divergence. Thus
$I(X;\hat{Y}|Y)\leq\min_{\\{Q_{c}\\}_{c\in[C]}}\mbox{$\bf
E$}[D(P_{X,\mathbf{\theta}}||Q_{Y})].$ (31)
On the other hand, (30) becomes an equality whenever
$Q_{c}=P_{\hat{Y}|y=c},\forall c\in[C].$ (32)
This, together with (30), implies (29), and hence completes the proof of
Theorem 2.
## Appendix B Other Information Quantities for Separation
In this Appendix, we explore other information quantities which can also be
defined and used as a measure for the inter-class separation of the DNN:
$x\in\mathbb{R}^{d}\to P_{x}$. Specifically, two more information quantities
$\Gamma^{\prime}$ and $\Gamma^{\prime\prime}$ are introduced and compared with
$\Gamma$ defined in (19). Although they are more or less equivalent, $\Gamma$
is more convenient for selecting hyper parameters in our CMIC-DL framework.
### B-A Information Quantity $\Gamma^{\prime}$
A possible information quantity for measuring inter-class separation can be
defined as follows
$\displaystyle\Gamma^{\prime}=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{X}||P_{U})\right],$ (33)
where the cross entropy function $H(P_{X},P_{U})$ in (19) is replaced by the
KL divergence $D(P_{X}||P_{U})$. To connect $\Gamma^{\prime}$ with CMI and
$\Gamma$, we simplify $\Gamma^{\prime}$ as follows:
$\displaystyle\Gamma^{\prime}$ $\displaystyle=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P(\hat{Y}=i|X)\ln{\frac{P(\hat{Y}=i|X)}{P(\hat{Y}=i|U)}}\right]$
$\displaystyle=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P(\hat{Y}=i|X)\left(\ln{\frac{P(\hat{Y}=i|X)}{P_{\hat{Y}|Y}(\hat{Y}=i|Y)}}\right.\right.$
$\displaystyle\left.\left.+\ln{\frac{P_{\hat{Y}|Y}(\hat{Y}=i|Y)}{P(\hat{Y}=i|U)}}\right)\right]$
$\displaystyle=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{X}||P_{\hat{Y}|Y})\right]$
$\displaystyle+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P(\hat{Y}=i|X)\ln{\frac{P_{\hat{Y}|Y}(\hat{Y}=i|Y)}{P(\hat{Y}=i|U)}}\right]$
(34) $\displaystyle=\mbox{$\bf
E$}\left[(1-P(Y))D(P_{X}||P_{\hat{Y}|Y})\right]$ (35)
$\displaystyle+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P_{\hat{Y}|Y}(\hat{Y}=i|Y)\ln{\frac{P_{\hat{Y}|Y}(\hat{Y}=i|Y)}{P(\hat{Y}=i|U)}}\right]$
(36) $\displaystyle=\mbox{$\bf
E$}\left[(1-P(Y))D(P_{X}||P_{\hat{Y}|Y})\right]$ $\displaystyle+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{\hat{Y}|Y}||P_{U})\right],$ (37)
where (35) is due to the fact that $V$ is independent of $(X,Y)$, and (36)
follows from the independence of $(X,Y)$ and $(U,V)$ and the Markov chain
$Y\to X\to\hat{Y}$.
Note that the first expectation in (37) is related to the CMI
$I(X;\hat{Y}|Y)$. Indeed, when $P(Y)$ is equal to a constant, i.e., $1/C$,
which is true in most empirical cases, it follows from (15) that
$\mbox{$\bf E$}\left[(1-P(Y))D(P_{X}||P_{\hat{Y}|Y})\right]=(1-{1\over
C})I(X,\hat{Y}|Y),$
which, together with (37), implies that
$\Gamma^{\prime}=(1-{1\over C})I(X,\hat{Y}|Y)+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{\hat{Y}|Y}||P_{U})\right].$ (38)
Plugging (38) into the optimization problem in (IV-A), we get the following
optimization problem
$\displaystyle\min_{\mathbf{\theta}}~{}\mbox{$\bf E$}_{X}$
$\displaystyle\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]+\left(\lambda-\left(\beta-\frac{\beta}{C}\right)\right)I(X;\hat{Y}|Y)$
$\displaystyle-\beta\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{\hat{Y}|Y}||P_{U,\mathbf{\theta}})\right].$
(39)
Thus, if $\Gamma^{\prime}$ was used as a measure for inter-class separation,
then it would cancel out part of the CMI, making the selection of hyper
parameters $\lambda$ and $\beta$ become harder.
### B-B Information Quantity $\Gamma^{\prime\prime}$
Equations (38) and (B-A) suggest that one might use the following information
quantity as a measure for inter-class separation instead
$\displaystyle\Gamma^{\prime\prime}=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}D(P_{\hat{Y}|Y}||P_{U})\right].$ (40)
In fact, $\Gamma^{\prime\prime}$ has a descent physical meaning in the sense
that it measures the average of distances between the output distributions of
the DNN in response to input sample instances and the centroids of the
clusters with different ground truth labels.
To connect $\Gamma^{\prime\prime}$ with CMI and $\Gamma$, we further simplify
$\Gamma^{\prime\prime}$ as follows
$\displaystyle\Gamma^{\prime\prime}$ $\displaystyle=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P(\hat{Y}=i|X)\ln{\frac{P_{\hat{Y}|Y}(\hat{Y}=i|Y)}{P(\hat{Y}=i|U)}}\right]$
(41) $\displaystyle=\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}H(P_{X},P_{U})\right]$ $\displaystyle+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P(\hat{Y}=i|X)\ln
P_{\hat{Y}|Y}(\hat{Y}=i|Y)\right]$ $\displaystyle=\Gamma+\mbox{$\bf
E$}\left[I_{\\{Y\not=V\\}}\sum_{i=1}^{C}P_{\hat{Y}|Y}(\hat{Y}=i|Y)\ln
P_{\hat{Y}|Y}(\hat{Y}=i|Y)\right]$ (42) $\displaystyle=\Gamma-\mbox{$\bf
E$}\left[(1-P(Y))H(P_{\hat{Y}|Y},P_{\hat{Y}|Y})\right].$ (43)
In the above, (41) follows from (34) and (37), (42) is due to the fact that
$X$ is independent of $V$, and $Y\to X\to\hat{Y}$ forms a Markov chain, and
(43) is attributable to the independence of $V$ and $Y$.
Note again that the second term in (43) is related to the CMI
$I(X;\hat{Y}|Y)$. Indeed, when $P(Y)$ is equal to a constant, i.e., $1/C$,
which is true in most empirical cases, it follows that
$\displaystyle\mbox{$\bf
E$}\left[(1-P(Y))H(P_{\hat{Y}|Y},P_{\hat{Y}|Y})\right]$ (44) $\displaystyle=$
$\displaystyle(1-{1\over C})H(\hat{Y}|Y)$ $\displaystyle=$
$\displaystyle(1-{1\over C})\left[I(X;\hat{Y}|Y)+H(\hat{Y}|X,Y)\right]$
$\displaystyle=$ $\displaystyle(1-{1\over
C})\left[I(X;\hat{Y}|Y)+H(\hat{Y}|X)\right],$
where $H(W|Z)$ denotes the Shannon conditional entropy of the random variable
$W$ given the random variable $Z$, and (44) is due to the Markov chain $Y\to
X\to\hat{Y}$. Combining (44) with (43) yields
$\Gamma^{\prime\prime}=\Gamma-(1-{1\over
C})\left[I(X;\hat{Y}|Y)+H(\hat{Y}|X)\right].$ (45)
Plugging (45) into the optimization problem in (IV-A), we get the following
optimization problem
$\displaystyle\min_{\mathbf{\theta}}~{}\mbox{$\bf E$}_{X}$
$\displaystyle\left[H(P_{Y|X},P_{X,\mathbf{\theta}})\right]+\left(\lambda+\left(\beta-\frac{\beta}{C}\right)\right)I(X;\hat{Y}|Y)$
$\displaystyle+\beta(1-{1\over C})H(\hat{Y}|X)-\beta\Gamma.$ (46)
Thus, if $\Gamma^{\prime\prime}$ was used as a measure for inter-class
separation, then it would further enhance the effect of the CMI, making the
selection of hyper parameters $\lambda$ and $\beta$ become harder as well.
## Acknowledgments
This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada under Grant RGPIN203035-22, and in part by the
Canada Research Chairs Program.
## References
* [1] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” _nature_ , vol. 521, no. 7553, pp. 436–444, 2015.
* [2] I. Goodfellow, Y. Bengio, and A. Courville, _Deep learning_. MIT press, 2016.
* [3] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,” _arXiv preprint arXiv:1412.6572_ , 2014.
* [4] N. Carlini and D. Wagner, “Towards evaluating the robustness of neural networks,” in _2017 ieee symposium on security and privacy (sp)_. Ieee, 2017, pp. 39–57.
* [5] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models resistant to adversarial attacks,” in _International Conference on Learning Representations_ , 2018.
* [6] T. M. Cover, _Elements of information theory_. John Wiley & Sons, 1999.
* [7] G. Hinton, O. Vinyals, J. Dean _et al._ , “Distilling the knowledge in a neural network.”
* [8] K. Zheng and E.-H. Yang, “Knowledge distillation based on transformed teacher matching,” in _To be submitted to International Conference on Learning Representations_. International Conference on Learning Representations, ICLR, 2024.
* [9] A. K. Menon, A. S. Rawat, S. Reddi, S. Kim, and S. Kumar, “A statistical perspective on distillation,” in _International Conference on Machine Learning_. PMLR, 2021, pp. 7632–7642.
* [10] E. Oyallon, “Building a regular decision boundary with deep networks,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 5106–5114.
* [11] V. Papyan, “Traces of class/cross-class structure pervade deep learning spectra,” _The Journal of Machine Learning Research_ , vol. 21, no. 1, pp. 10 197–10 260, 2020.
* [12] V. Papyan, X. Han, and D. L. Donoho, “Prevalence of neural collapse during the terminal phase of deep learning training,” _Proceedings of the National Academy of Sciences_ , vol. 117, no. 40, pp. 24 652–24 663, 2020.
* [13] J. Zarka, F. Guth, and S. Mallat, “Separation and concentration in deep networks,” in _International Conference on Learning Representations_ , 2020\.
* [14] Y. Wen, K. Zhang, Z. Li, and Y. Qiao, “A comprehensive study on center loss for deep face recognition,” _International Journal of Computer Vision_ , vol. 127, pp. 668–683, 2019.
* [15] Z. Shi, H. Wang, and C.-S. Leung, “Constrained center loss for convolutional neural networks,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2021.
* [16] Y. Wen, K. Zhang, Z. Li, and Y. Qiao, “A discriminative feature learning approach for deep face recognition,” in _Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11–14, 2016, Proceedings, Part VII 14_. Springer, 2016, pp. 499–515.
* [17] C. Qi and F. Su, “Contrastive-center loss for deep neural networks,” in _2017 IEEE international conference on image processing (ICIP)_. IEEE, 2017, pp. 2851–2855.
* [18] K. Ranasinghe, M. Naseer, M. Hayat, S. Khan, and F. S. Khan, “Orthogonal projection loss,” in _Proceedings of the IEEE/CVF international conference on computer vision_ , 2021, pp. 12 333–12 343.
* [19] S. M. Hamidi and E.-H. Yang, “Coded deep learning: framework and algorithms,” _Submitted to IEEE transactions on pattern analysis and machine intelligence_.
* [20] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
* [21] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv:1409.1556_ , 2014.
* [22] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for convolutional neural networks,” in _International conference on machine learning_. PMLR, 2019, pp. 6105–6114.
* [23] S. Zagoruyko and N. Komodakis, “Wide residual networks,” _arXiv preprint arXiv:1605.07146_ , 2016.
* [24] L. Zhao and L. Wang, “A new lightweight network based on mobilenetv3.” _KSII Transactions on Internet & Information Systems_, vol. 16, no. 1, 2022\.
* [25] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” _Advances in neural information processing systems_ , vol. 25, 2012.
* [26] I. Cohen, Y. Huang, J. Chen, J. Benesty, J. Benesty, J. Chen, Y. Huang, and I. Cohen, “Pearson correlation coefficient,” _Noise reduction in speech processing_ , pp. 1–4, 2009.
* [27] T. Berger, “Rate distortion theory. englewood clis,” 1971.
* [28] E.-H. Yang, Z. Zhang, and T. Berger, “Fixed-slope universal lossy data compression,” _IEEE Trans. Inf. Theory_ , vol. 43, no. 5, pp. 1465–1476, 1997.
* [29] A. Krizhevsky, G. Hinton _et al._ , “Learning multiple layers of features from tiny images,” 2009.
* [30] T.-Y. Lin, P. Goyal, R. Girshick, K. He, and P. Dollár, “Focal loss for dense object detection,” in _Proceedings of the IEEE international conference on computer vision_ , 2017, pp. 2980–2988.
* [31] W. Wan, Y. Zhong, T. Li, and J. Chen, “Rethinking feature distribution for loss functions in image classification,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 9117–9126.
* [32] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” _Proceedings of the IEEE_ , vol. 86, no. 11, pp. 2278–2324, 1998.
| En-Hui Yang (M’97-SM’00-F’08) received the B.S. degree in applied
mathematics from Huaqiao University, China, Ph.D. degree in mathematics from
Nankai University, China, and Ph.D. degree in electrical engineering from the
University of Southern California, USA, in 1986, 1991, and 1996, respectively.
Since June 1997, he has been with the Department of Electrical and Computer
Engineering, University of Waterloo, ON, Canada, where he is currently a
Professor and Canada Research Chair, and the founding Director of the Leitch-
University of Waterloo multimedia communications lab. A co-founder of
SlipStream Data Inc. (now a subsidiary of BlackBerry) and the founder of
BicDroid Inc., he currently also serves as an Executive Council Member of
China Overseas Friendship Association, an Expert Advisor for the Overseas
Chinese Affairs Office of the State Council of China, a Board Governor of the
University of Waterloo, a Board Trustee of Huaqiao University, a member of
IEEE Founders Medal Committee, and advisors for other national and provincial
bodies. His current research interests are: multimedia compression, digital
communications, information theory, source and channel coding, image and video
coding, deep learning, big data analytics, and information security. Dr. Yang
is a recipient of several awards and honors, a partial list of which includes
the 2021 IEEE Eric E. Sumner Award, the prestigious Inaugural Premier’s
Catalyst Award in 2007 for the Innovator of the Year; the 2007 Ernest C.
Manning Award of Distinction, one of the Canada’s most prestigious innovation
prizes; the 2013 CPAC Professional Achievement Award; the 2014 IEEE
Information Theory Society Padovani Lecture; and the 2014 FCCP Education
Foundation Award of Merit. With over 230 papers and more than 230
patents/patent applications worldwide, his research work has benefited people
over 170 countries through commercialized products, video coding open sources,
and video coding standards. He is a Fellow of the Canadian Academy of
Engineering and a Fellow of the Royal Society of Canada: the Academies of
Arts, Humanities and Sciences of Canada. He served, inter alia, as a review
panel member for the International Council for Science; a general co-chair of
the 2008 IEEE International Symposium on Information Theory; an Associate
Editor for IEEE Transactions on Information Theory; a Technical Program Vice-
Chair of the 2006 IEEE International Conference on Multimedia and Expo (ICME);
the Chair of the award committee for the 2004 Canadian Award in
Telecommunications; a Co-Editor of the 2004 Special Issue of the IEEE
Transactions on Information Theory; and a Co-Chair of the 2003 Canadian
Workshop on Information Theory.
---|---
| Shayan Mohajer Hamidi received the B.Sc. degree from Sharif University,
Tehran, Iran, in 2016, and the MASc. degree from the University of Waterloo,
Waterloo, ON, Canada, in 2018, both in electrical engineering. He was a
research assistant with the CST lab at the University of Waterloo from 2018 to
2020. He is currently working toward his Ph.D. degree in Electrical
Engineering at the University of Waterloo. His current research interests
include machine learning, optimization, information theory.
---|---
| Linfeng Ye received the B.Sc. degree from Xi’an University of Science and
Technology, Xi’an, China in 2020, in microelectronics, and the MEng. degree
from the University of Waterloo, Waterloo, ON, Canada, in 2021, in electrical
engineering. He is currently working towards his MASc. degree in electrical
engineering at University of Waterloo. His current research interests include
machine learning, and information theory.
---|---
| Renhao Tan received the Bachelor of Advanced Computing (Honours) degree
from the Australian National University, Canberra, ACT, Australia, in 2021. He
is currently pursuing a MASc. degree in Electrical & Computer Engineering at
University of Waterloo, Waterloo, ON, Canada. His research interests include
machine learning, computer vision and data mining.
---|---
| Beverly Yang received the BASc degree in Geological Engineering from the
University of Waterloo in 2020. She is currently pursuing a PhD in mining
engineering at the University of British Columbia. Her research interests
include rock mechanics and machine learning.
---|---
|
# Unified theory of the nonlinear Schrödinger equation
David B. Reinhardt<EMAIL_ADDRESS><EMAIL_ADDRESS>German
Aerospace Center (DLR), Institute of Quantum Technologies, Wilhelm-Runge-
Straße 10, 89081 Ulm, Germany Dean Lee Facility for Rare Isotope Beams and
Department of Physics and Astronomy, Michigan State University, MI 48824, USA
Wolfgang P. Schleich Institut für Quantenphysik and Center for Integrated
Quantum Science and Technology (IQST), Universität Ulm, D-89069 Ulm, Germany
Hagler Institute for Advanced Study at Texas A$\&$M University, Texas A$\&$M
AgriLife Research, Institute for Quantum Science and Engineering (IQSE), and
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843-4242, USA Matthias Meister<EMAIL_ADDRESS><EMAIL_ADDRESS>German Aerospace Center (DLR), Institute of Quantum
Technologies, Wilhelm-Runge-Straße 10, 89081 Ulm, Germany
(July 14, 2023)
###### Abstract
The nonlinear Schrödinger equation (NLSE) is a rich and versatile model, which
in one spatial dimension has stationary solutions similar to those of the
linear Schrödinger equation as well as more exotic solutions such as solitary
waves and quantum droplets. We present a unified theory of the NLSE, showing
that all stationary solutions of the cubic-quintic NLSE can be classified
according to a single number called the cross-ratio. Any two solutions with
the same cross-ratio can be converted into one another using a conformal
transformation, and the same also holds true for traveling wave solutions. In
this way we demonstrate a conformal duality between solutions of cubic-quintic
NLSEs and lower-order NLSEs. The same analysis can be applied to the Newtonian
dynamics of classical particles with polynomial potentials. Our framework
provides a deeper understanding of the connections between the physics of the
NLSE and the mathematics of algebraic curves and conformal symmetry.
Introduction – The nonlinear Schrödinger equation (NLSE) is ubiquitous in
physics, where it plays a key role in plasma physics [1, 2, 3], hydrodynamics
[4, 5, 6], degenerate quantum gases [7, 8] and light propagation in nonlinear
fiber optics [9, 10, 11, 12]. Understanding the possible solutions of the NLSE
is therefore of great importance for a large variety of purposes whether they
are application-oriented or fundamental. In this Letter we point out a
conformal duality between different classes of solutions and even different
orders of the NLSE. This conformal mapping provides a unified picture of the
cubic- and the cubic-quintic NLSE and even establishes a direct link to the
linear Schrödinger equation. Moreover, our method allows us to systematically
classify the complete solution spaces of these equations.
The linear Schrödinger equation typically features oscillating and constant-
amplitude solutions which have their counterparts in the NLSE. However, there
also exist solutions which are uniquely nonlinear such as solitary waves [13,
14, 15, 16, 17, 18, 19] which are of versatile interest in physics [20, 21,
22, 23, 24]. Considering (multiple) higher-order self-modulating terms like in
the cubic-quintic NLSE drastically expands the solution space allowing for
instance for bright and dark soliton pairs [25] and solitons with power law
tail decay [26]. Although the different polynomial NLSEs have been studied in
great detail [27, 28, 29, 30, 31, 32, 33, 34, 35, 36] there exists so far no
unified theory linking their solution spaces.
In this work, we identify a large family of conformal dualities for the one-
dimensional time-independent cubic-quintic NLSE. These dualities allow us to
establish conformal maps between solutions of the cubic-quintic NLSE and even
to conformally reduce the cubic-quintic to the cubic NLSE and the linear
Schrödinger equation, highlighting that the lower-order equations essentially
are conformal limiting cases of the cubic-quintic NLSE. Conformal dualities
are of particular interest in physics [37, 38, 39] with famous instances being
the Kramers–Wannier duality in statistical mechanics [40] relating high-and
low temperature limits of the free energy in certain Ising models and the
Montonen–Olive duality [41] a generalization of the electro-magnetic symmetry
of Maxwell’s equations in quantum field theory.
The new and useful insights into the conformal duality of NLSEs presented in
this Letter can directly be transferred to Newtonian mechanics relating the
motion of classical particles in various harmonic and anharmonic potentials.
In fact, there is a remarkable similarity between the solutions of the NLSE
and Newtonian dynamics. Finally, our theory provides a fundamental
understanding of two-and three-body contact interactions being inherently
related in one-dimensional Bose-Einstein condensates [42, 43, 44, 45]
described by higher-order Gross-Pitaevskii equations [46, 47, 48, 14, 49, 17].
Nonlinear Schrödinger equation – We consider the dimensionless time-
independent cubic-quintic NLSE
$\displaystyle\left(-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+a_{3}\absolutevalue{\psi}^{2}+\frac{a_{4}}{2}\absolutevalue{\psi}^{4}\right)\psi=a_{2}\psi$
(1)
in one spatial dimension of coordinate $x$, where $\psi=\psi\left(x\right)$ is
the complex-valued wave function and $a_{2}$, $a_{3}$, and $a_{4}$ are
constants 111The indices $j$ of the coefficients $a_{j}$ are chosen such that
they correspond to the exponents of the different powers of $\sigma$ of the
polynomial P defined in Eq. (3). By omitting position-dependent potentials, we
focus on the homogeneous case with either box or periodic boundary conditions.
The amplitude-phase representation
$\psi\equiv\sqrt{\sigma}\exp\left(i\phi\right)$ casts Eq. (1) into the
differential equations [28, 30, 25]
$\displaystyle\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x}\right)^{2}=P\left(\sigma\right)$
(2)
for the density $\sigma=\sigma(x)$ with the quartic polynomial
$\displaystyle
P\left(\sigma\right)\equiv\frac{4}{3}a_{4}\,\sigma^{4}+4a_{3}\,\sigma^{3}-8a_{2}\,\sigma^{2}-16a_{1}\,\sigma-4a_{0}$
(3)
and
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}$
$\displaystyle=\pm\frac{\sqrt{a_{0}}}{\sigma}$ (4)
for the phase $\phi=\phi(x)$. Here $a_{0},a_{1}$ are constants of integration
and the different signs in Eq. (4) refer to the two possible directions of the
flow induced by the phase gradient.
Obviously, the order of the polynomial $P=P(\sigma)$ directly depends on the
leading nonlinearity in Eq. (1) yielding a cubic or quartic polynomial in the
case of the cubic ($a_{4}=0$) or cubic-quintic NLSE, while the polynomial is
quadratic for the linear Schrödinger equation ($a_{3}=a_{4}=0$).
Classification of solutions – The stationary solutions of Eq. (1) are in
general determined [30] by the polynomial $P$, defined by Eq. (3), and its
discriminant $\Delta=a_{4}^{6}\prod_{j\neq k}(\sigma_{j}-\sigma_{k})$ given by
the roots $\sigma_{j}$ of $P$. Depending on the sign of $\Delta$, three
classes of solutions can be identified: (i) simple complex conjugated roots
($\Delta<0$), (ii) multiple roots ($\Delta=0$), or (iii) only simple roots
($\Delta>0$) of $P$.
In order to discuss the roots $\sigma_{j}$ of $P$ it is convenient to
introduce the tuple notation ($r_{4}$, $r_{3}$, $r_{2}$, $r_{1}$), where every
entry $r_{m}$ denotes the number of roots at order $m$. For instance
$(0,0,0,4)$ labels a polynomial with four simple real roots as displayed in
Fig. 1a, while a polynomial with two simple real roots and two simple complex-
conjugated roots as shown in Fig. 1d is labeled by
$(0,0,0,2+2_{{\mathbb{C}}})$.
Figure 1: Conformal mapping between two realizations of the cubic-quintic NLSE
with discriminant $\Delta>0$ (a–c) and $\Delta<0$ (d–f). (a,d) Polynomials
$P(\sigma)$ and $\tilde{P}(\tilde{\sigma})$, defined by Eq. (2), with four
simple roots $\sigma_{j}$ (a) or two simple and two complex roots
$\tilde{\sigma}_{j}$ (d), respectively. (b,e) Oscillating phase-space
trajectories corresponding to real (blue) and complex (red) solutions
determined by $P$ in (a,d). The two cases (a–c) and (d–f) are related by a
conformal transformation, Eq. (5), which maps the positions of the roots, the
polynomials, and the corresponding phase-space orbits into each other. (c,f)
Complex density plane of $P$ and $\tilde{P}$ shown in (a,d) illustrating the
positions of the roots $\sigma_{j}$ and $\tilde{\sigma}_{j}$ (black dots) as
well as the argument of the phase of the density $\sigma$ (color map). The
lines of constant real and imaginary part of the density $\sigma$ form a
square grid (c) which is mapped into a grid of circles (f) by the conformal
transformation due to changing the sign of $\Delta$. The roots $\sigma_{j}$
are thus mapped from a straight line (c) to a circle (f) with counter-
clockwise orientation starting from the first real root. Likewise, the
cloverleaf-shaped boundary $Q$ shown in (c) is the inverse image of the square
boundary shown in (f) while the center of the angle-shaped region in (f)
corresponds to the point at infinity in (c).
The explicit solutions of the stationary NLSE are obtained by direct
integration of Eqs. (2) and (4) in the region between two neighboring real
roots. Consequently, oscillatory solutions of Eq. (2) occur between two
neighboring simple real roots which define the minimum and maximum density of
the oscillation as displayed by the three closed phase-space orbits in Fig. 1b
which originate from the polynomial in Fig. 1a.
Complex conjugate roots with finite imaginary parts can therefore not be the
turning points of such solutions, but instead deform the resulting orbits
spanned between other real roots as illustrated in Fig. 1e. Due to the finite
order of $P$, there is thus only one oscillatory orbit possible for the
polynomial shown in Fig. 1d.
For polynomials with a multiple root (shown in Fig. 2b for the case (0,0,1,2),
solitonic and other more exotic solutions emerge 222See Supplemental Material
at the end of the document for details on the conformal transformation of the
NLSE and explicit expressions of the solutions discussed in context of the
conformal reduction shown in Fig. 2. The Supplemental Material cites Refs.
[52, 59, 58].. In fact, the multiple root acts as a bifurcation point in phase
space constituting a separatrix for the phase-space trajectories separating
the two other solution classes [25]. Moreover, there always exists a constant
amplitude solution at the density value of the multiple root.
Finally, the outer density regions which are restricted by only one real root
typically lead to unbounded solutions with poles. For instance the light
orange shaded region of the polynomial displayed in Fig. 2c yields such an
unbounded solution [51].
Consequently, the sign of the discriminant $\Delta$ and thus the nature of the
roots of $P$ not only determine the character and shape of the resulting
solutions, but also the total number of different solutions for a given set of
parameters. Indeed, according to Eq. (2) physically meaningful real solutions
require $P(\sigma)>0$ between the roots considered, in addition to any
restrictions set by the boundary conditions of the system under study, while
for $P(\sigma)<0$ complex density solutions emerge. Hence, this approach
enables a straightforward and systematic classification of all possible
stationary solutions of higher-order NLSEs.
Conformal duality – In the phase space $(\sigma,\sigma^{\prime})$ with
$\sigma^{\prime}\equiv\mathrm{d}\sigma/\mathrm{d}x$, the differential equation
Eq. (2) constitutes an elliptic curve. A key characteristic of elliptic curves
is the possibility to transform their underlying algebraic equation by
rational transformations [52]. Strikingly, in the case of the NLSE the Möbius
transformation can be adapted to the differential equation Eq. (2) leading to
the conformal map
$\displaystyle\sigma\left(x\right)=\frac{A\,\tilde{\sigma}\left(\tilde{x}\right)+B}{C\,\tilde{\sigma}\left(\tilde{x}\right)+D}$
(5)
of the densities $\sigma$ and $\tilde{\sigma}$ with the generally complex-
valued coefficients $A,B,C,D$. In contrast to the mapping of elliptic curves,
here the spatial coordinate $x$ also needs to be transformed according to the
affine transformation $x=x_{0}+\left(AD-BC\right)\tilde{x}$, where $x$ can
become complex-valued and $x_{0}$ is a constant.
The duality, Eq. (5), relates any two physical systems with the same real-
valued cross-ratio $k^{2}$ which is an invariant of the transformation
determined by the roots $\sigma_{j}$ of $P$ [51]. Note that the conformal
character of the Möbius transformation will preserve the angles in the complex
density plane by mapping every straight line of constant density into another
line or circle of constant density, and vice versa.
The gradient of the phase, determined by Eq. (4), enjoys a similar
transformation [51]
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}=\pm\sqrt{a_{0}}\,\dfrac{D\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,C}{B\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,A}$
(6)
with the very same coefficients $A,B,C,D$ as in Eq. (5). As a result, the
combination of Eqs. (5) and (6) provides the complete conformal mapping of the
differential equations under study. Hence, different realizations of the
cubic-quintic NLSE are conformally related establishing a fundamental
connection between their solution spaces. In particular, these transformations
also apply to the solutions of density $\sigma$ and phase $\phi$ themselves
such that the complete complex wave function $\psi$ can be conformally mapped.
We emphasize that the conformal duality remains intact for traveling-wave
solutions of the NLSE such as solitary waves subjected to a velocity boost.
This effect is a direct consequence of the Galilean covariance of the NLSE
allowing to obtain arbitrary many additional solutions through the application
of Galilei-transformations [51].
Conformal mapping and reduction of the NLSE – Depending on the choice of the
transformation coefficients different scenarios can be realized. Indeed, the
conformal map, Eq. (5), directly relates different quartic polynomials with
each other and therefore their solution spaces. In this case the ratio $A/C$
must not match the value of any of the roots of the involved polynomials to
preserve their quartic order. Real-valued coefficients $A,B,C,D$ connect
polynomials within a given solution class, while complex coefficients enable
to change the solution class corresponding to a change of sign of $\Delta$.
Figure 1 shows an intriguing example of the latter case, where a
$(0,0,0,4)$-polynomial is mapped to one classified by
$(0,0,0,2+2_{{\mathbb{C}}})$. Despite the fact that the graphs of the
polynomials $P$ and $\tilde{P}$ (a,d) and their phase-space orbits (b,e)
appear quite distinct in Fig. 1 they are intimately connected as visualized by
the density maps (c,f). Here, the conformal character of the transformation
manifests itself by transforming the straight line connecting all four simple
roots in Fig. 1c to a circle which passes again through all (now partly
complex) roots in Fig. 1f. In the same way, the rectangular boundary of the
density plot in Fig. 1f is mapped into the cloverleaf-shaped boundary
displayed in Fig. 1c.
Figure 2: Conformal reduction from the cubic-quintic (b) to the cubic NLSE (c)
and the linear Schrödinger equation (a). Exemplary polynomials $P$ (green) and
$-P$ (orange), Eq. (2), with (a) two simple roots (black dots), (b) two simple
roots $\sigma_{3}$, $\sigma_{4}$ and one double root $\sigma_{1,2}$ (encircled
black dot), or (c) one simple and one double root. By moving the roots
$\sigma_{4}$ (or $\sigma_{1,2}$) to plus (minus) infinity the cubic-quintic
NLSE can be reduced to the cubic NLSE or the linear Schrödinger equation,
respectively. The green (soliton solutions) and orange (oscillating solutions)
fillings show which of the density regions (and corresponding solutions) are
mapped into each other featuring similar characteristics. The light shaded
green and orange areas in (a) and (c) illustrate unbounded solutions.
Moreover, as depicted in Fig. 2, it is possible to conformally reduce the
cubic-quintic NLSE to either the cubic NLSE or the linear Schrödinger equation
by mapping either an outer simple root or an outer double root to plus or
minus infinity, respectively. In these cases the ratio $A/C$ must match the
value of the roots to be moved. As a consequence, the overall degree of the
polynomial is reduced by one (simple root moved) or two (double root moved).
Analogously, the linear Schrödinger equation with an energy eigenvalue of zero
is obtained by removing a triple root.
By reducing the degree of the polynomial the solution space changes based on
Eq. (5) and as illustrated in Fig. 2: (i) one unbounded solution vanishes
because the root constituting its minimum or maximum density has been removed,
(ii) a bound solution becomes unbound since it is now only restricted by one
root, and (iii) the remaining solutions get transformed, but keep their main
characteristics as their roots retain their order.
The case shown in Fig. 2 highlights two prominent solitonic solutions, namely
the flat-top soliton [14] (green shaded area in b) and the elementary bright
soliton [53] (green shaded area in c) which are both governed by a hyperbolic
cosine in the denominator of their density profile. By the transformation from
the cubic-quintic to the cubic NLSE only the prefactor in front of the
hyperbolic cosine gets changed such that both solutions are quite similar
[51].
Likewise, the oscillatory solution in the cubic-quintic case (orange area in
Fig. 2b) governed by a cosine in the denominator as well changes its prefactor
when transformed. However, in this case the corresponding solution of the
cubic case (light orange area in Fig. 2c) becomes unbound due to the now
different prefactor and has thus completely changed its character by the
transformation.
Fascinatingly, the solutions of the green and orange regions in Fig. 2 are
also interconnected by a tranformation that maps a real position coordinate
$x$ to a purely imaginary position $\tilde{x}$ changing the functional
dependency from a hyperbolic sine (green) to a trigonometric sine (orange).
Effectively, this transformation thus flips the overall sign of the polynomial
from $P$ to $-P$. In this way all the solutions of the cubic and cubic-quintic
NLSE as well as the linear Schrödinger equation are fundamentally connected.
Connection to Newtonian mechanics – Besides the importance of the unified
theory of the NLSE, the conformal duality discussed in this Letter has also
strong implications for the dynamics of classical particles subjected to
anharmonic conservative potentials in Newtonian mechanics. Indeed, it is well-
known that the NLSE formally constitutes a classical Hamiltonian system for
the density $\sigma$ with the Hamiltonian function [54]
$\displaystyle\mathcal{H}(\sigma^{\prime},\sigma)\equiv\frac{1}{2}{\sigma^{\prime\,}}^{2}+U\left(\sigma\right)$
(7)
with the potential $U=U\left(\sigma\right)$. Here, the density $\sigma$ will
be analogous to the classical position, while the spatial coordinate $x$
corresponds to time in classical mechanics.
By constraining the energy of $\mathcal{H}$ to the value of $-4a_{0}$ and
considering the potential $U\left(\sigma\right)\equiv
2a_{0}-P\left(\sigma\right)/2$ one can recover [25] Eq. (2). Hence, in this
analogy the nonlinearites in Eq. (1) directly correspond to anharmonic
contributions in $U$ with the cubic or cubic-quintic NLSE giving rise to a
cubic or quartic potential, respectively, while the linear Schrödinger
equation yields a harmonic potential as usual.
The conformal map, Eq. (5), now allows us to transform the Hamiltonian, Eq.
(7), of a classical particle and consequently its underlying equations of
motions. Thus, we can map a double-well problem to another double-well
problem, or carry out the conformal reduction from a quartic to a cubic,
quadratic, linear or constant potential by mapping a simple, double, triple or
quadruple root of the potential to plus or minus infinity.
As a result, soliton solutions, as shown in Fig. 2, in our classical mechanics
analogy correspond to an oscillation with infinitely long period where the
particle in phase space approaches a bifurcation point similar to a
mathematical pendulum, where the angular coordinate approaches the unstable
fixed point at $\pi$ radians.
Analogously, unbounded solutions are the counterpart of scattering states of
the corresponding classical potentials. Hence, the ideas and concepts for
treating physics problems of classical particles in anharmonic potentials have
a direct correspondence to those required for the NLSE. Remarkably, both
systems enable the conformal mapping of their solutions within and between
different solution classes.
Conclusion – In summary we have provided a unified picture of the NLSE by
establishing a conformal duality between the solution spaces of the cubic-
quintic and cubic NLSE as well as the linear Schrödinger equation. This
connection gives rise to novel and elementary understanding of the physics of
nonlinear systems, in particular, when comparing the effect of nonlinearities
of different degrees.
Our results apply to stationary and travelling-wave solutions of the NLSE and
remain valid even under Galilean transformations. We therefore expect our
findings to have a wide variety of applications that include the dynamics of
solitons and their dual counterparts, mode structures in nonlinear fiber
optics, hydrodynamic wave-dynamics, and the interplay of two- and three-body
interactions in quasi-1D Bose-Einstein condensates as utilized for atomtronics
devices [55, 56].
In addition, the conformal duality can be employed for Newtonian mechanics
allowing us to classify and relate numerous different dynamical systems caused
by anharmonic conservative potentials. Finally, our algebraic-geometric
classification scheme can straightforwardly be extended to even higher order
NLSEs such as the cubic-quintic-septic NLSE [57] to search for new physics in
the form of exotic solutions that require strong nonlinearities of this kind.
Acknowledgments – We thank M.A. Efremov for fruitful discussions and helpful
suggestions. D.L. acknowledges financial support from the U.S. Department of
Energy (DE-SC0021152, DE-SC0013365, DE-SC0023658, SciDAC-5 NUCLEI
Collaboration). W.P.S. is grateful to Texas A$\&$M University for a Faculty
Fellowship at the Hagler Institute for Advanced Study at the Texas A$\&$M
University as well as to the Texas A$\&$M AgriLife Research.
## References
* Bohm and Gross [1949a] D. Bohm and E. P. Gross, Phys. Rev. 75, 1851 (1949a).
* Bohm and Gross [1949b] D. Bohm and E. P. Gross, Phys. Rev. 75, 1864 (1949b).
* Hasegawa [1975] A. Hasegawa, _Plasma Instabilities and Nonlinear Effects_ (Springer, Berlin, Heidelberg, 1975).
* Zakharov [1968] V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).
* Peregrine [1983] D. H. Peregrine, The ANZIAM Journal 25, 16 (1983).
* Kuznetsov _et al._ [1986] E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
* Dalfovo _et al._ [1999] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
* Giorgini _et al._ [2008] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008).
* Haus and Wong [1996] H. A. Haus and W. S. Wong, Rev. Mod. Phys. 68, 423 (1996).
* Kivshar and Luther-Davies [1998] Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
* Lederer _et al._ [2008] F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
* Copie _et al._ [2020] F. Copie, S. Randoux, and P. Suret, Rev. Phys. 5, 100037 (2020).
* Zakharov and Shabat [1971] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971), [Sov. Phys. JETP 34, 62 (1972)].
* Bulgac [2002] A. Bulgac, Phys. Rev. Lett. 89, 050402 (2002).
* Muryshev _et al._ [2002] A. Muryshev, G. V. Shlyapnikov, W. Ertmer, K. Sengstock, and M. Lewenstein, Phys. Rev. Lett. 89, 110401 (2002).
* Konotop and Pitaevskii [2004] V. V. Konotop and L. Pitaevskii, Phys. Rev. Lett. 93, 240403 (2004).
* Petrov and Astrakharchik [2016] D. S. Petrov and G. E. Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016).
* Zhou _et al._ [2021] Y. Zhou, H. Meng, J. Zhang, X. Li, X. Ren, X. Wan, Z. Zhou, J. Wang, X. Fan, and Y. Shi, Sci. Rep. 11, 11382 (2021).
* Seidel _et al._ [2022] T. G. Seidel, S. V. Gurevich, and J. Javaloyes, Phys. Rev. Lett. 128, 083901 (2022).
* Burger _et al._ [1999] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 83, 5198 (1999).
* Denschlag _et al._ [2000] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, Science 287, 97 (2000).
* Strecker _et al._ [2002] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature 417, 150 (2002).
* Becker _et al._ [2008] C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K. Sengstock, Nat. Phys. 4, 496 (2008).
* Kibler _et al._ [2015] B. Kibler, A. Chabchoub, A. Gelash, N. Akhmediev, and V. E. Zakharov, Phys. Rev. X 5, 041026 (2015).
* Crosta _et al._ [2011] M. Crosta, A. Fratalocchi, and S. Trillo, Phys. Rev. A 84, 063809 (2011).
* Hayata and Koshiba [1995] K. Hayata and M. Koshiba, Phys. Rev. E 51, 1499 (1995).
* Akhmediev _et al._ [1987] N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Theor. Math. Phys. 72, 809 (1987).
* Gagnon [1989] L. Gagnon, J. Opt. Soc. Am. A 6, 1477 (1989).
* Pushkarov and Tanev [1996] D. Pushkarov and S. Tanev, Opt. Commun. 124, 354 (1996).
* Schürmann [1996] H. W. Schürmann, Phys. Rev. E 54, 4312 (1996).
* Serkin and Hasegawa [2000] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
* Carr _et al._ [2000a] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev. A 62, 063610 (2000a).
* Carr _et al._ [2000b] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev. A 62, 063611 (2000b).
* Wamba _et al._ [2016] E. Wamba, A. Pelster, and J. R. Anglin, Phys. Rev. A 94, 043628 (2016).
* Khawaja and Sakkaf [2019] U. A. Khawaja and L. A. Sakkaf, _Handbook of Exact Solutions to the Nonlinear Schrödinger Equations_ (IOP Publishing, Bristol, 2019).
* Liu _et al._ [2021] Y.-Y. Liu, W.-D. Li, and W.-S. Dai, J. Phys. Commun. 5, 015011 (2021).
* Burkhardt and Choi [1992] T. Burkhardt and J.-Y. Choi, Nucl. Phys. B 376, 447 (1992).
* Nussinov _et al._ [2015] Z. Nussinov, G. Ortiz, and M.-S. Vaezi, Nucl. Phys. B 892, 132 (2015).
* Ares _et al._ [2016] F. Ares, J. G. Esteve, F. Falceto, and A. R. de Queiroz, J. Stat. Mech. 2016, 043106 (2016).
* Kramers and Wannier [1941] H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941).
* Montonen and Olive [1977] C. Montonen and D. Olive, Phys. Lett., B 72, 117 (1977).
* Schreck _et al._ [2001] F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. 87, 080403 (2001).
* Görlitz _et al._ [2001] A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, Phys. Rev. Lett. 87, 130402 (2001).
* Greiner _et al._ [2001] M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 87, 160405 (2001).
* Meyrath _et al._ [2005] T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, Phys. Rev. A 71, 041604 (2005).
* Kolomeisky _et al._ [2000] E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, Phys. Rev. Lett. 85, 1146 (2000).
* Abdullaev _et al._ [2001] F. K. Abdullaev, A. Gammal, L. Tomio, and T. Frederico, Phys. Rev. A 63, 043604 (2001).
* Salasnich _et al._ [2002] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 (2002).
* Cardoso _et al._ [2011] W. B. Cardoso, A. T. Avelar, and D. Bazeia, Phys. Rev. E 83, 036604 (2011).
* Note [1] The indices $j$ of the coefficients $a_{j}$ are chosen such that they correspond to the exponents of the different powers of $\sigma$ of the polynomial P defined in Eq. (3).
* Note [2] See Supplemental Material at the end of the document for details on the conformal transformation of the NLSE and explicit expressions of the solutions discussed in context of the conformal reduction shown in Fig. 2. The Supplemental Material cites Refs. [52, 59, 58].
* Bateman [1953] H. Bateman, _Higher Transcendental Functions [Volumes I-III]_ , Vol. 2 (McGraw-Hill Book Company, New York, 1953).
* Pethick and Smith [2002] C. Pethick and H. Smith, _Bose-Einstein Condensation in Dilute Gases_ (Cambridge University Press, Cambridge, 2002).
* Dauxois and Peyrard [2006] T. Dauxois and M. Peyrard, _Physics of Solitons_ (Cambridge University Press, Cambridge, 2006).
* Amico _et al._ [2021] L. Amico _et al._ , AVS Quantum Science 3, 039201 (2021).
* Amico _et al._ [2022] L. Amico, D. Anderson, M. Boshier, J.-P. Brantut, L.-C. Kwek, A. Minguzzi, and W. von Klitzing, Rev. Mod. Phys. 94, 041001 (2022).
* Reyna and de Araújo [2017] A. S. Reyna and C. B. de Araújo, Adv. Opt. Photon. 9, 720 (2017).
* Byrd and Friedman [1971] P. F. Byrd and M. D. Friedman, _Handbook of Elliptic Integrals for Engineers and Scientists_ (Springer, Berlin, Heidelberg, 1971).
* Milne [1911] J. J. Milne, _An Elementary Treatise on Cross-Ratio Geometry, with Historical Notes_ (Cambridge University Press, Cambridge, 1911).
Supplemental material for
Unified theory of the nonlinear Schrödinger equation
David B. Reinhardt1, Dean Lee2, Wolfgang P. Schleich3,4 and Matthias Meister1
1German Aerospace Center (DLR), Institute of Quantum Technologies,
Wilhelm-Runge-Straße 10, 89081 Ulm, Germany
2Facility for Rare Isotope Beams and Department of Physics and Astronomy,
Michigan State University, MI 48824, USA
3Institut für Quantenphysik and Center for Integrated Quantum Science and
Technology (IQST),
Universität Ulm, D-89069 Ulm, Germany
4Hagler Institute for Advanced Study at Texas A$\&$M University, Texas A$\&$M
AgriLife Research, Institute for Quantum Science and Engineering (IQSE), and
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843-4242, USA
(Dated: July 14, 2023)
## I Conformal duality of the nonlinear Schrödinger equation
In this supplemental material we derive the transformation of the gradient of
the phase from the Möbius transformation of the density and further provide
more insights into the transformation of the differential equations with
quartic, cubic, and quadratic polynomial dependency in the density. In
addition, the cross-ratio of the transformation is discussed in more detail.
We conclude this supplement by providing explicit expressions for the
conformal reduction of typical oscillating solutions of the NLSE and of the
solitonic solutions shown in Fig. 2. In this context we also present the
involved analytical solutions of the NLSE.
### Basics of elliptic curves
A common way to define elliptic- and hyper-elliptic curves is to introduce the
relation
$\displaystyle Y^{2}=P(X)$ (8)
between the algebraic variables $X$ and $Y$ with the polynomial
$\displaystyle P(X)\equiv\sum_{n=0}^{N}\alpha_{n}X^{n}$ (9)
of order $N$ with coefficients $\alpha_{n}$. Note that for elliptic curves
$N=3,4$, while for hyper-elliptic curves $N>4$ [58].
A key feature of elliptic or hyper-elliptic curves is the possibility to
transform them into other elliptic or hyper-elliptic curves by rational
transformations [52]. For instance the bi-rational transformations
$\displaystyle X=\frac{A\tilde{X}+B}{C\tilde{X}+D}$ (10)
and
$\displaystyle Y=\frac{\tilde{Y}}{\left(CX+D\right)^{\frac{N}{2}}}$ (11)
connect both algebraic variables $X$ and $Y$ with the corresponding new
variables $\tilde{X}$ and $\tilde{Y}$ through the coefficients $A,B,C$ and
$D$.
### I.1 Conformal transformation of the density and the phase gradient
In the case of the nonlinear Schrödinger equation (NLSE) discussed here we
identify the variables $X$ and $Y$ with $\sigma$ and
$\sigma^{\prime}\equiv\mathrm{d}\sigma/\mathrm{d}x$ such that the
transformation of the density is given by the relation
$\displaystyle\sigma\left(x\right)=\frac{A\,\tilde{\sigma}\left(\tilde{x}\right)+B}{C\,\tilde{\sigma}\left(\tilde{x}\right)+D}$
(12)
being of linear fractional type.
In contrast, the transformation of the derivative of the density
$\sigma^{\prime}$ is of nonlinear fractional type as is the mapping between
the algebraic variables $Y$ and $\tilde{Y}$, Eq. (11). Nevertheless, in case
of the NLSE the transformation of $\sigma^{\prime}$ is self-consistently
obtained by differentiation of Eq. (12)
$\displaystyle\frac{\mathrm{d}\sigma\left(x\right)}{\mathrm{d}x}$
$\displaystyle=\frac{\mathrm{d}\tilde{x}}{\mathrm{d}x}\frac{AD-
BC}{\left(C\,\tilde{\sigma}\left(\tilde{x}\right)+D\right)^{2}}\frac{\mathrm{d}\tilde{\sigma}\left(\tilde{x}\right)}{\mathrm{d}\tilde{x}}$
(13)
$\displaystyle=\frac{1}{\left(C\,\tilde{\sigma}\left(\tilde{x}\right)+D\right)^{2}}\frac{\mathrm{d}\tilde{\sigma}\left(\tilde{x}\right)}{\mathrm{d}\tilde{x}}\,,$
(14)
where we have used the linear coordinate transformation $x=x_{0}+\left(AD-
BC\right)\tilde{x}$ between $x$ and $\tilde{x}$ in the second step.
The origin of the Möbius transformation of the phase gradient
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}$
$\displaystyle=\pm\frac{\sqrt{a_{0}}}{\sigma}$ (15)
is rooted in its reciprocal dependence on the density $\sigma$. Indeed, by
inserting the conformal map of the density, Eq. (12), into Eq. (15) and
dividing both the numerator and denominator by the density $\tilde{\sigma}$ we
immediately obtain the transformation
$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}=\pm\sqrt{a_{0}}\,\dfrac{D\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,C}{B\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,A}\,,$
(16)
of the phase gradient, where $\sqrt{\tilde{a}_{0}}$ is the flux in the
transformed system. Thus, the phase gradient also transforms according to a
Möbius transform with the same coefficients $A,B,C,D$ used for the density.
Consequently, the transformations Eq. (12) and Eq. (16) of the density and
gradient of the phase together enable a complete conformal mapping of the
complex field $\psi=\sqrt{\sigma}\,\mathrm{e}^{\mathrm{i}\phi}$ of the NLSE.
### Conformal transformation of the polynomial equation
The stationary solutions of the NLSE are governed by the differential equation
(2) which features a polynom $P(\sigma)$ of fourth order on the right-hand
side. In order to analyze this equation it is convenient to consider the
factored form
$\displaystyle\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x}\right)^{2}$
$\displaystyle=a_{4}\,\left(\sigma-\sigma_{1}\right)\left(\sigma-\sigma_{2}\right)\left(\sigma-\sigma_{3}\
\right)\left(\sigma-\sigma_{4}\ \right)$ (17)
where $\sigma_{j}$ are the roots of $P$.
By applying the transformations Eq.(12) and (14) the transformed differential
equation is given by
$\displaystyle\left(\frac{\mathrm{d}\tilde{\sigma}}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=a_{4}\Big{(}\left(A-\sigma_{1}C\right)\sigma+B-\sigma_{1}D\Big{)}\cdot\Big{(}\left(A-\sigma_{2}C\right)\sigma+B-\sigma_{2}D\Big{)}\cdot\Big{(}\left(A-\sigma_{3}C\right)\sigma+B-\sigma_{3}D\Big{)}\cdot\Big{(}\left(A-\sigma_{4}C\right)\sigma+B-\sigma_{4}D\Big{)}$
$\displaystyle=\tilde{a}_{4}\,\Big{(}\tilde{\sigma}-\tilde{\sigma}_{1}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{2}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{3}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{4}\Big{)}$
(18)
with
$\displaystyle\tilde{a_{4}}\equiv
a_{4}\,C^{4}\left(\frac{A}{C}-\sigma_{1}\right)\left(\frac{A}{C}-\sigma_{2}\right)\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\,,$
(19)
the new roots
$\displaystyle\tilde{\sigma}_{j}\equiv\frac{D\sigma_{j}-B}{-C\sigma_{j}+A}$
(20)
and $A/C\neq\sigma_{j}$. In general the conformal character of the
transformation maps straight lines to circles and vice versa such that the new
and original roots will always lie on a straight line or a circle. By choosing
a specific set of coefficients $A,B,C$ and $D$ Eq. (I) can be used to relate
two different solutions of the cubic-quintic NLSE as illustrated in Fig. 1.
Moreover, the quartic polynomial can conformally be reduced to a cubic
polynomial, describing the cubic NLSE, by mapping one of the outer roots to
plus or minus infinity as shown in Fig. 2. In this case $A/C$ needs to be
equal to the value of the root to be mapped. If for instance the first root
$\sigma_{1}$ is mapped to minus infinity ($A/C=\sigma_{1}$) the resulting
differential equation can be written as
$\displaystyle\left(\frac{\mathrm{d}\rho}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=\tilde{a}_{3}\left(\rho-\rho_{1}\right)\left(\rho-\rho_{2}\right)\left(\rho-\rho_{3}\right)$
(21)
$\displaystyle=\tilde{a}_{3}\rho^{3}+\tilde{a}_{2}\rho^{2}+\tilde{a}_{1}\rho+\tilde{a}_{0}$
(22)
with
$\displaystyle\tilde{a}_{3}\equiv
a_{4}C^{2}\left(\frac{A}{C}-\sigma_{2}\right)\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\left(BC-
AD\right)\,.$ (23)
Here we have introduced the density $\rho$ the of the cubic NLSE as to
distinguish between the various cases.
In the case that the quartic polynomial, Eq. (2), exhibits a vanishing
discriminant $\Delta$ multiple roots might emerge. If one of the outer roots
is a double root, the degree of the polynomial can be reduced by two, giving
rise to a simple parabola corresponding to the linear Schrödinger equation. By
choosing e.g. $A/C=\sigma_{1,2}$ we obtain a conic curve
$\displaystyle\left(\frac{\mathrm{d}\lambda}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=e_{2}\,\left(\lambda-\lambda_{1}\right)\left(\lambda-\lambda_{2}\right)$
(24) $\displaystyle=e_{2}\lambda^{2}+e_{1}\lambda+e_{0}$ (25)
describing the stationary states of the linear Schrödinger equation, where
$\displaystyle e_{2}\equiv
a_{4}\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\left(BC-
AD\right)^{2}$ (26)
and the density is labeled by $\lambda$.
### I.2 Details of the cross-ratio
In the mathematics literature [59], the cross-ratio is sometimes also called
the anharmonic ratio of four co-linear or concyclic complex numbers
$z_{1},z_{2},z_{3}$ and $z_{4}$ and is typically defined as
$\displaystyle\left(z_{1},z_{2};z_{3},z_{4}\right)=\frac{z_{3}-z_{1}}{z_{3}-z_{2}}\frac{z_{4}-z_{2}}{z_{4}-z_{1}}\equiv\Lambda\,.$
(27)
Actually, this definition is not unique as the value will be unaltered for
pairwise interchange of all points . Furthermore, all other possibilities of
partitioning the four points can eventually be related to the cross-ratio
$\Lambda$ defined in Eq. (27).
For instance, the solutions of the “W”-shaped graph of the polynomial $P$
shown in Fig. 1a employ a cross-ratio or elliptic modulus given by the
relation
$\displaystyle k^{2}$
$\displaystyle\equiv\frac{\sigma_{4}-\sigma_{3}}{\sigma_{4}-\sigma_{2}}\frac{\sigma_{2}-\sigma_{1}}{\sigma_{3}-\sigma_{1}}=\frac{\Lambda-1}{\Lambda}\,.$
(28)
Similarly, the solutions of a “M”-shaped graph use the relation
$\displaystyle
k^{{\prime}^{2}}=\frac{\sigma_{3}-\sigma_{2}}{\sigma_{4}-\sigma_{2}}\frac{\sigma_{4}-\sigma_{1}}{\sigma_{3}-\sigma_{1}}=1-k^{2}=\frac{1}{\Lambda}$
(29)
often referred to as the complementary elliptic modulus [58]. In case of a
multiple root of the polynomial $P$, the cross-ratio employed in solutions
given in terms of Jacobi elliptic functions, will either correspond to the
trigonometric limit where $k^{2}=0$ or the hyperbolic limit with $k^{2}=1$.
### Galilean invariance of the conformal duality
#### I.2.1 Velocity boost for the time-dependent nonlinear Schrödinger
equation
Here, we discuss the Galilean covariance of the cubic-quintic NLSE by showing
that any Galilean transformation leaves the underlying equation of motion
invariant. We consider a solution $\psi\left(x,t\right)$ of the time-dependent
cubic-quintic NLSE
$\displaystyle i\frac{\partial}{\partial
t}\psi\left(x,t\right)=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+a_{3}\absolutevalue{\psi\left(x,t\right)}^{2}+a_{4}\absolutevalue{\psi\left(x,t\right)}^{4}\right)\psi\left(x,t\right)$
(30)
in the rest frame $\mathcal{F}$ described by the coordinates $x$ and $t$.
Applying a boost with the dimensionless velocity $u$ introduces the moving
frame $\mathcal{F}^{\prime}$ with coordinates $x^{\prime}$ and $t^{\prime}$
which are related to the rest frame via
$\displaystyle x$ $\displaystyle=x^{\prime}+ut^{\prime}$ (31) $\displaystyle
t$ $\displaystyle=t^{\prime}\,.$ (32)
To obtain the wave function $\psi^{\prime}\left(x^{\prime},t^{\prime}\right)$
in the boosted frame $\mathcal{F}^{\prime}$ we consider the transformation
$\displaystyle\psi\left(x,t\right)=\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$
(33)
of the wavefunction $\psi\left(x,t\right)$ in the rest frame with a phase
containing the boost velocity and the additional kinetic energy due to the
boost in the moving frame $\mathcal{F}^{\prime}$.
Next, we will show that the Galilean transformation of the wavefunction and
coordinates leaves the NLSE in Eq. (30) formally unchanged. For this purpose,
we consider the transformations of the derivatives in Eq. (30) which are given
by
$\displaystyle\frac{\partial}{\partial t}=\frac{\partial}{\partial
t^{\prime}}\cdot\frac{\partial t^{\prime}}{\partial
t}+\frac{\partial}{\partial x^{\prime}}\cdot\frac{\partial
x^{\prime}}{\partial t}=\frac{\partial}{\partial
t^{\prime}}-u\frac{\partial}{\partial x^{\prime}}$ (34)
and
$\displaystyle\frac{\partial}{\partial x}=\frac{\partial}{\partial
x^{\prime}}\frac{\partial x^{\prime}}{\partial x}+\frac{\partial}{\partial
t^{\prime}}\frac{\partial t^{\prime}}{\partial x}=\frac{\partial}{\partial
x^{\prime}}.$ (35)
By substituting these derivatives into Eq. (30) and also inserting the
transformation law for the wave function, Eq. (33), we obtain the differential
equation
$\displaystyle i$ $\displaystyle\left(\frac{\partial}{\partial
t^{\prime}}-u\frac{\partial}{\partial
x^{\prime}}\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$
(36)
$\displaystyle=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{4}\
\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$
describing the evolution of the boosted wave function
$\psi^{\prime}\left(x^{\prime},t^{\prime}\right)$ in the moving frame
$\mathcal{F}^{\prime}$.
By performing the partial derivatives on both sides of the equation and
factoring out the global phase factor
$\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$ the differential
equation in the moving frame reduces to
$\displaystyle i\frac{\partial}{\partial
t^{\prime}}\psi^{\prime}\left(x^{\prime},t^{\prime}\right)=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{4}\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\,.$
(37)
The fact that Eqs. (30) and (37) have the same form proves the Galilean
invariance of the cubic-quintic NLSE.
#### I.2.2 Impact on stationary solutions
In order to derive the impact of the velocity boost on stationary solutions,
we start with the usual ansatz
$\displaystyle\psi\left(x,t\right)=\psi\left(x,0\right)\exp\left(-ia_{2}t\right)$
(38)
for the time-independent wave function $\psi(x,0)$ in the rest frame
$\mathcal{F}$ with eigenvalue $a_{2}$ determined by Eq. (1).
Applying the transformation, Eq. (33), to both the time-dependent and time-
independent wave functions in Eq. (38) yields the relation
$\displaystyle\psi^{\prime}\left(x^{\prime},t^{\prime}\right)=\psi^{\prime}\left(x^{\prime},0\right)\exp\left(-i\left(a_{2}+\frac{u^{2}}{2}\right)t^{\prime}\right)\,,$
(39)
where we have also used the relation $t=t^{\prime}$, Eq. (32).
Inserting this result into the time-dependent NLSE in the boosted frame, Eq.
(37) allows us to obtain the time-independent NLSE in the boosted frame
$\displaystyle\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},0\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},0\right)}^{4}\right)\psi^{\prime}\left(x^{\prime},0\right)$
$\displaystyle=a_{2}^{\prime}\psi^{\prime}\left(x^{\prime},0\right)\,,$ (40)
where
$\displaystyle a_{2}^{\prime}\equiv a_{2}+\frac{u^{2}}{2}.$ (41)
Consequently, the boost to the moving frame adds a velocity-dependent term to
the eigenvalue $a_{2}^{\prime}$ of the stationary solution. This is just the
kinetic energy associated with the Galilean boost.
In the main text, we have shown that any two stationary solutions with the
same cross-ratio can be converted into each other using conformal
transformations. This fundamental result can now be extended to traveling wave
solutions. We simply take the dual stationary solutions and perform Galilean
boosts to produce the desired traveling waves.
### Explicit solutions for the conformal duality and reduction of the
nonlinear Schrödinger equation
#### I.2.3 Oscillating solutions
As a first application of the conformal duality of the NLSE we show how
typical oscillating solutions of the cubic and the cubic-quintic NLSE are
connected via the transformation, Eq. (12). For this purpose we consider the
real-valued solution
$\displaystyle\rho\left(x_{2}\right)$
$\displaystyle=\left(\rho_{2}-\rho_{1}\right)\operatorname{sn}^{2}\left(\kappa_{2}x_{2},k\right)+\rho_{1},$
(42)
which oscillates between the density values $\rho_{1}$ and $\rho_{2}$
corresponding to the roots of a cubic polynomial $P(\rho)$ with an “N”-shaped
graph. Here $\operatorname{sn}$ refers to the Jacobi elliptic sine function
with $\kappa_{2}=a_{3}\sqrt{\rho_{3}-\rho_{1}}/2$ and
$k^{2}=(\rho_{2}-\rho_{1})/(\rho_{3}-\rho_{1})$.
We now demonstrate the conformal duality between this solution and another
solution for a quartic polynomial $P(\sigma)$. By applying the solution of the
cubic NLSE, Eq. (42), to the transformation Eq. (12), and choosing
$A/C=\sigma_{1}<0$, to pull a root from minus infinity, we obtain the solution
$\displaystyle\sigma\left(x_{3}\right)$
$\displaystyle=\frac{\sigma_{2}\eta-\sigma_{1}\operatorname{sn}^{2}\left(\kappa_{3}\,x_{3},k\right)}{\eta-\operatorname{sn}^{2}\left(\kappa_{3}\,x_{3},k\right)},$
(43)
of the cubic-quintic NLSE. Here,
$\eta=\frac{\sigma_{3}-\sigma_{1}}{\sigma_{3}-\sigma_{2}}>1$ and
$\kappa_{3}=\kappa_{2}/(AD-BC)$ such that the solution given by Eq. (43)
oscillates between the densities $\sigma_{2}$ and $\sigma_{3}$ corresponding
to the roots of a quartic polynomial $P(\sigma)$ with a “W”-shaped graph.
Direct integration of the quartic polynomial of the cubic-quintic NLSE yields
exactly the same solution as given by Eq. (43) proving the validity of the
conformal transformation.
The mapping coefficients of this conformal reduction are given by
$\displaystyle A$
$\displaystyle\equiv\frac{\sigma_{1}}{\rho_{2}-\rho_{1}}\qquad$ $\displaystyle
B\equiv-\sigma_{2}\eta-\sigma_{1}\frac{\rho_{1}}{\rho_{2}-\rho_{1}}$ (44)
$\displaystyle C$ $\displaystyle\equiv\frac{1}{\rho_{2}-\rho_{1}}\qquad$
$\displaystyle D\equiv\eta-\frac{\rho_{1}}{\rho_{2}-\rho_{1}}.$ (45)
The coordinates are related via the linear map $x_{3}=\left(AD-
BC\right)x_{2}+\text{const.}$, where in this example the constant is set to
zero for simplicity.
#### I.2.4 Solitonic solutions
Here, we provide the explicit transformations of the conformal reduction shown
in Fig. 2. For the bright soliton solutions we naturally need a focusing or
attractive nonlinearity with $\tilde{a}_{3}<0$ while the flat-top soliton
likewise requires a focusing cubic nonlinearity $a_{3}<0$ and a defocusing
quintic nonlinearity $a_{4}>0$. The analytical solutions for all orange and
green shaded areas in Fig. 2 given by Eqs. (46) and (47), sorted according to
the appearance in the figure from left to right, are given by
$\displaystyle\lambda\left(x_{1}\right)$
$\displaystyle=\left(\lambda_{2}-\lambda_{1}\right)\sin^{2}\left(\kappa_{1}x_{1}\right)+\lambda_{1}\qquad$
$\displaystyle\sigma\left(x_{3}\right)=\sigma_{0}\eta\frac{1}{1+\sqrt{1-\eta}\cos\left(2\kappa_{3}x_{3}\right)}\qquad$
$\displaystyle\rho\left(x_{2}\right)=\rho_{0}\frac{1}{1+\cos\left(2\kappa_{2}x_{2}\right)}$
(46) $\displaystyle\tilde{\lambda}\left(\tilde{x}_{1}\right)$
$\displaystyle=\left(\lambda_{2}-\lambda_{1}\right)\sinh^{2}\left(\kappa_{1}\tilde{x}_{1}\right)+\lambda_{1}\qquad$
$\displaystyle\tilde{\sigma}\left(\tilde{x}_{3}\right)=\sigma_{0}\eta\frac{1}{1+\sqrt{1-\eta}\operatorname{cosh}\left(2\kappa_{3}\tilde{x}_{3}\right)}\qquad$
$\displaystyle\tilde{\rho}\left(\tilde{x}_{2}\right)=\rho_{0}\frac{1}{1+\cosh\left(2\kappa_{2}\tilde{x}_{2}\right)}\,,$
(47)
where $\kappa_{1}\equiv\sqrt{\absolutevalue{e_{2}}}$, $\rho_{0}\equiv
4\tilde{a}_{2}/\tilde{a}_{3}$,
$\kappa_{2}\equiv\sqrt{\absolutevalue{\tilde{a}_{2}}}$,
$\sigma_{0}\equiv-3a_{3}/2a_{4}$,
$\kappa_{3}\equiv\sqrt{\absolutevalue{a_{2}}}$, and
$\eta\equiv-8a_{2}a_{4}/3a_{3}^{2}<1$. Again $\lambda$ refers to the density
of the linear Schrödinger equation, $\rho$ corresponds to the density of the
cubic NLSE, and $\sigma$ gives the density of the cubic-quintic NLSE.
Now the transformation coefficients $A,B,C,D$, Eq. (12), of the conformal
reduction from the cubic-quintic to the cubic NLSE, connecting the solutions
$\sigma\left(x_{3}\right)$ and $\rho\left(x_{2}\right)$ or
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ and
$\tilde{\rho}\left(\tilde{x}_{2}\right)$, can be determined. For the
particular set of solutions given by Eqs. (46) and (47) the phase is
independent of position as they require a root at the origin such that the
coefficient $a_{0}=0$ in Eq. (2). As a consequence, the transformation
coefficient $B$ needs to be zero by construction in order to conformally
relate theses solutions. In addition, this transformation requires
$AD=\frac{\kappa_{2}}{\kappa_{3}}>0$ and
$A/C=\sigma_{0}\left(1+\sqrt{1-\eta}\right)=\sigma_{4}$. Ultimately, the
transformation coefficients are thus given by
$\displaystyle A$
$\displaystyle\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\rho_{0}\sigma_{0}\eta}{\sqrt{1-\eta}}}\qquad$
$\displaystyle B\equiv 0$ (48) $\displaystyle C$
$\displaystyle\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\rho_{0}}{\sigma_{0}}}\frac{1}{1+\sqrt{1-\eta}}\sqrt{\frac{\eta}{\sqrt{1-\eta}}}\qquad$
$\displaystyle
D\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\sqrt{1-\eta}}{\rho_{0}\sigma_{0}\eta}}.$
(49)
As the roots of the polynomial $P$ and $-P$ are identical the transformation
coefficients for the conformal reduction from $\sigma\left(x_{3}\right)$ to
$\rho\left(x_{2}\right)$ (orange shaded areas in Fig. 2) are formally
identical to those required for the reduction from
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ to
$\tilde{\rho}\left(\tilde{x}_{2}\right)$ (green shaded areas in Fig. 2).
Hence, the coefficients given by Eqs. (48) and (49) apply to both cases.
Finally, in order to connect the solutions of a polynomial $P$ to those of the
inverse polynomial $-P$ one can for instance employ the coefficients
$A=D=(1+i)/\sqrt{2},\,B=C=0$ which formally do not change the densities
according to Eq. (12), but instead yield the coordinate transformation
$x=\mathrm{i}\tilde{x}$. By transforming back to a real coordinate $\tilde{x}$
the imaginary unit is absorbed in the functional dependency of the density
inducing the change from a trigonometric function to a hyperbolic function. In
this way the conformal partnering of the trigonometric oscillating solution
${\sigma}\left({x}_{3}\right)$ and the the resting droplet solution
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ given by Eqs. (46) and (47) can be
realized.
|
$e^{k}(j)$ is a canonical vector (see Sec. II.1). Since $s_{k}=W^{i}_{k}u_{i}$
is the outgoing strength and
$s_{k}e^{k}(j)x_{j}(t)=s_{k}e^{k}(j)\delta^{i}_{j}x_{i}(t)$, the diffusion
equation can be written in more compact form as:
$\displaystyle\frac{dx_{j}(t)}{dt}=-\mathcal{D}L^{i}_{j}x_{i}(t)\,,$ (66)
where $L^{i}_{j}=W^{l}_{k}u_{l}e^{k}(j)\delta^{i}_{j}-W^{i}_{j}$ is the
combinatorial Laplacian tensor [chung1997]. It is worth remarking here that
this tensor differs from the normalized random walk Laplacian introduced in
the previous section, although in some cases – as for a classical random walk
– they are related by the simple relationship
$\tilde{L}_{j}^{i}=(D^{-1})^{i}_{k}L^{k}_{j}$. The solution of Eq. (66) is
given by $x_{j}(t)=x_{i}(0)e^{-\mathcal{D}L^{i}_{j}t}$, similarly to what we
have seen for continuous-time random walks. However, at variance with random
walks, it can be shown that in the stationary regime, the solution takes the
form $x_{j}(\infty)\propto u_{j}$, i.e., one will find exactly the same
fraction of the quantity in each node, uniformly distributed.
We might wonder how fast diffusion happens. Since the Laplacian matrix is
semi-positive definite, it is possible to show that it can be decomposed as
$\displaystyle L^{i}_{j}=Q^{i}_{h}\Lambda^{h}_{k}(Q^{-1})^{k}_{j},$ (67)
where $Q^{i}_{h}$ is a matrix whose columns are the eigenvectors of the
Laplacian and $\Lambda^{h}_{k}$ is a diagonal matrix whose entries are the
Laplacian’s eigenvalues. This algebraic feature allows us to prove that
$e^{-\mathcal{D}L^{i}_{j}t}=Q^{i}_{h}(e^{-\mathcal{D}\Lambda
t})^{h}_{k}(Q^{-1})^{k}_{j}$, highlighting that the exponential decay of
$x_{j}(t)$ is dominated by the smallest positive eigenvalue, which usually is
indicated by $\Lambda_{2}$. Therefore, the diffusion temporal scale is given
by $\tau\approx 1/\Lambda_{2}$.
Figure 35: Continuous-time diffusion on a duplex network, i.e., a multiplex
consisting of two layers, obtained from two Erdős–Rényi networks with
independently wiring probabilities $p_{1},p_{2}\in[0,1]$, used to connect two
pairs of nodes within the same layer. Diffusion speed is quantified by the
second smallest eigenvalue of the Laplacian tensor, $\Lambda_{2}$: depending
on the wiring probabilities, diffusion in the duplex can be faster (left-hand
side panel) or slower (right-hand side panel) than in each layer separately.
The condition for enhanced diffusion to happen is given by
$\Lambda_{2}^{\text{multiplex}}\geq\max\\{\Lambda_{2}^{\text{layer
1}},\Lambda_{2}^{\text{layer 2}}\\}$: a sharp change in the behavior of the
characteristic temporal scale can be observed for varying weight of the inter-
layer connections (left and right panels), with a clear transition between two
distinct regimes above a certain critical value of inter-layer coupling. The
middle panel shows when the enhanced diffusion condition holds (encoded by
colors), while varying the wiring probabilities. Figure from [de2016physics].
In the case of interconnected multilayer networks, the diffusion equation has
been generalized by means of the supra-adjacency matrix [gomez2013diffusion]
and the tensorial formulation [de2013mathematical]. In this new setup, a
quantity can diffuse through inter-layer connections as well. If we indicate
by $X_{i\alpha}(t)$ the rank-2 state tensor at time $t$, then the multilayer
diffusion equation can be written as
$\displaystyle\frac{dX_{j\beta}(t)}{dt}=M^{i\alpha}_{j\beta}X_{i\alpha}(t)-M^{i\alpha}_{k\gamma}U_{i\alpha}E^{k\gamma}(i\beta)X_{i\beta}(t)\,,$
(68)
where $U_{i\alpha}=u_{i}u_{\alpha}$ and
$E^{k\gamma}(i\beta)=e^{k}(i)e^{\gamma}(\beta)$. If we define the multilayer
combinatorial Laplacian tensor as
$L^{i\alpha}_{j\beta}=M^{l\epsilon}_{k\gamma}U_{l\epsilon}E^{k\gamma}(j\beta)\delta^{i\alpha}_{j\beta}-M^{i\alpha}_{j\beta},$
(69)
the diffusion equation can be written more compactly as
$\displaystyle\frac{dX_{j\beta}(t)}{dt}=-L^{i\alpha}_{j\beta}X_{i\alpha}(t)\,,$
(70)
whose solution is given by
$X_{j\beta}(t)=X_{i\alpha}(0)e^{-L^{i\alpha}_{j\beta}t}$, a clear
generalization of the result obtained in the case of single-layer networks.
Also in this case, the second smallest eigenvalue $\Lambda_{2}$ – calculated
from the supra-adjacency matrix representation – governs the speed of
diffusion [de2013mathematical, sole2013spectral, gomez2013diffusion], leading
to interesting phenomena (see Fig. 35 for details).
##### Topological transition with diffusive processes
Figure 36: Algebraic phase transition. In (a) and (e), the intra-layer
connectivity of a duplex (i.e., a multiplex consisting of 2 layers) is shown,
where the color of each node refers to the value of the corresponding
component in the Fiedler vector, i.e., the eigenvector associated with the
second smallest eigenvalue $\lambda_{2}$. In (a) $p=p^{*}=0.602$ and in (e)
$p=0.603$, In the middle column is displayed the eigenvalue $\lambda_{2}$ (b),
the inner product $\langle v_{A}|v_{B}\rangle$ (c), and, in (d), inner
products of $|v_{A}\rangle$ and $|v_{B}\rangle$ with the unit vector, i.e.,
the sum of their elements (see the text for details). Figure reproduced from
[radicchi2013abrupt].
An important question concerns the emergence of such effects in spreading
processes, but not only. One can adapt the dynamical rules of models defined
in monolayers to encompass the fact that the topology is more complex and
this, as we will see, can lead to substantially different behaviors than those
observed in isolated networks. Another point of view, though, is to ask why
and when a certain model could effectively behave as in an isolated network
even though it runs in an a layered structure, or in the other way around, why
and when the multilayer dimension dominates, thus disregarding any role of the
individual networks of the layers. In some special cases, such as
interconnected multiplex networks with identical coupling, it is possible to
show the existence of two distinct regimes as a function of the inter-layer
coupling strength [radicchi2013abrupt], highlighting how the multilayer
structure can influence the outcome of several physical processes. By
considering a duplex network – i.e., a multiplex with 2 layers, $A$ and $B$ –
and by using spectral properties of multilayer systems, it was found an abrupt
transition between the two aforementioned regimes as a function of $p$, the
coupling strength. The two regimes are inferred by analyzing the behavior of
eigenvectors and eigenvalues of the supra-Laplacian matrix and are clearly
distinguishable, separated by a critical point $p^{*}$. For $p\leq p^{*}$, the
second smallest eigenvalue111111Note that here we are using $\lambda_{2}$
instead of $\Lambda_{2}$, as in the previous section, to keep the same
notation of the original paper by [radicchi2013abrupt]. $\lambda_{2}$, which
is associated with a plethora of network properties, is independent of the
structure of the layers and hence the dynamical processes can be studied
separately, while in the regime $p>p^{*}$, $\lambda_{2}$ tends to a value
independent of $p$, i.e., depending only on the details of the intra-layer
networks.
The eigenvector $|v\rangle$ associated with $\lambda_{2}(p)$ can be split into
$|v_{A}\rangle$ and $|v_{B}\rangle$, which correspond to the elements of
$|v\rangle$ associated with nodes of networks $A$ and $B$, respectively. It
can be proved [radicchi2013abrupt] that for, $p\leq p^{*}$,
$|v_{A}\rangle=-|v_{B}\rangle=0$ holds, where
$|v_{A}\rangle=\pm\frac{1}{\sqrt{2N}}|1\rangle$, while for $p>p^{*}$ we have
$\langle v_{A}|1\rangle=\langle v_{B}|1\rangle=0$. Physically, this means that
in the subcritical regime the layers are structurally independent whereas in
the supercritical regime the interlayer connection dominates, imposing the
same sign in the eigenvector for nodes across networks and alternating the
sign for nodes in the same layer. Therefore, the algebraic phase transition
can be visualized in several ways. Panels in Fig. 36 show the behavior of the
eigenvectors in the subcritical and supercritical regions. Here,
$\lambda_{2}(p)$ displays a singular point $p^{*}$ at which the first
derivative is not continuous, a sign of an abrupt transition. For $p\leq
p^{*}$, $\lambda_{2}$ grows as $2p$, while in the other regime it tends to the
value that would take for a weighted superposition of the two layers $A$ and
$B$, whose Laplacian is $(1/2)(\mathcal{L}_{A}+\mathcal{L}_{B})$. This
abruptness can be observed more directly in the middle and lower panels, which
show the behavior of $\langle v_{A}|v_{B}\rangle$, $\langle v_{A}|1\rangle$
and $\langle v_{B}|1\rangle$ as a function of $p$.
### IV.2 Synchronization processes
Synchronization is an emergent phenomenon of a population of dynamically
interacting units that, usually with a second-order phase transition
[nelson1977recent, stanley1999scaling, dorogovtsev2008critical], start
operating in a collective, coherent way. Synchronization phenomena may be
found in biology, sociology, and ecology and include birds flocking, fireflies
flashing, people singing, and neurons spiking, just to mention a few examples.
Once a fully synchronized state is reached, the (linear) stability of such a
state is tested by studying the effects of a small perturbation of the system
state. This approach was introduced in [Pecora1998a] for simple network
configurations and has been widely used and extended to complex network
topologies. Here we briefly introduce the framework of Master Stability
Function for a network of oscillators and then extend the formalism to
multilayer networks. For a more exhaustive description see [Boccaletti2018,
Arenas2008].
It is worth remarking that, in the following, we will use the more traditional
vector notation where $\mathbf{x}(t)$ indicates the state of the system at
time $t$. In some cases, we will also use the Kronecker product operator
$\otimes$, as is usual in equations governing synchronization dynamics. Our
choice is to help the reader link these concepts to the original studies which
introduced them. Nevertheless, the whole section could consider the tensorial
formalism, where the system state is indicated by the rank-1 tensor
$x_{\ell}(t)$ and where products such as $\mathbf{A}\otimes\mathbf{B}$ are
indicated as $A^{\alpha}_{\beta}B^{\gamma}_{\delta}$.
Let us consider a network of $N$ identical oscillators in an $m$-dimensional
space, where, in the absence of any interaction, the dynamics of each node $i$
is described by:
$\dot{\textbf{x}_{i}}=\textbf{F}(\textbf{x}_{i}),\qquad
i=1,2,...,N;\textbf{x}_{i}\in\mathbb{R}^{m}.$ (71)
We introduce an interaction between oscillators due to the fact that they are
coupled in an unweighted network specified by the adjacency matrix
$\textbf{A}=\\{A_{ij}\\}$ and we define the output function
$\textbf{H}(\textbf{x})$ as the function that governs the interaction between
nodes. We also assume that the coupling between the oscillators is diffusive,
that is the effect that node $j$ has on node $i$ is proportional to the
difference between $\textbf{H}(\textbf{x}_{j})$ and
$\textbf{H}(\textbf{x}_{i})$. Then, the evolution of the state of node $i$ is
given by:
$\dot{\textbf{x}_{i}}=\textbf{F}(\textbf{x}_{i})+\sigma\sum_{i=1}^{N}A_{ij}[\textbf{H}(\textbf{x}_{j})-\textbf{H}(\textbf{x}_{i})]=\textbf{F}(\textbf{x}_{i})-\sigma\sum_{j=1}^{N}L_{ij}\textbf{H}(\textbf{x}_{j}),$
(72)
where L is the Laplacian matrix and $\sigma$ is the coupling strength .
##### Stability of synchronized states
The MSF approach to test the stability of a fully synchronized state,
described in Appendix A, was extended to multilayer complex systems
[DelGenio2016]. In this work, it is considered a network with $M$ different
layers, each layer representing a different kind of interaction between nodes.
Eq. (72) is thus extended to describe the dynamics of the whole system:
$\dot{\textbf{x}_{i}}=\textbf{F}(\textbf{x}_{i})-\sum_{\alpha=1}^{M}\sigma_{\alpha}\sum_{j=1}^{N}L^{(\alpha)}_{ij}\textbf{H}^{(\alpha)}(\textbf{x}_{j}),$
(73)
where $\alpha$ is the index accounting for layers. We obtain the $M$-parameter
equation describing the time evolution of perturbation error:
$\dot{\boldsymbol{\xi}_{i}}=\left[J\textbf{F}(\textbf{s})-\sum_{\alpha=1}^{M}\sigma_{\alpha}\lambda^{(\alpha)}_{i}J\textbf{H}_{\alpha}(\textbf{s})\right]\boldsymbol{\xi}_{i},$
(74)
where $\boldsymbol{\xi}_{i}$ is the eigenmode associated with the eigenvalue
$\lambda_{i}$ of L. As in the case of a dynamics evolving on top of a single
layer, in a multilayer system the stability of the synchronized state is
completely specified by the sign of the maximum conditional Lyapunov exponent
$\Lambda_{max}$. In particular, it is also found that stability of the
complete synchronization state may be reached even if each layer, taken
individually, is unstable: a very interesting feature for practical
application.
It is worth noting that, together with complete synchronization, a network may
exhibit other forms of synchronization where clusters of nodes have a
synchronized dynamics but different clusters evolve on distinct time
evolutions. This type of synchronization is called clustered synchronization
(CS) and it has been well studied in terms of cluster formation, stability and
role of network symmetries [Nicosia2013, Pecora2014, Sorrentino2016]. CS has
been also studied in multiplex [Jalan2016] and, more recently, in multilayer
[DellaRossa2020] networks, and cluster stability has been tested as a function
of intra- and inter-layer symmetries. To describe the evolution of the
perturbation error for complex synchronization patterns, such as in cluster
synchronization, it is useful to rewrite Eq. (105) with a more compact
formalism. To this end, we write the state vector as a vector of vectors,
$\textbf{X}=(\textbf{x}_{1};\textbf{x}_{2};...;\textbf{x}_{N})$, and the
variational equation assumes the form:
$\delta\dot{\textbf{X}}=\left[\textbf{I}_{N}\otimes
J\textbf{F}(\textbf{s})-\sigma\textbf{L}\otimes
J\textbf{H}(\textbf{s})\right]\delta\textbf{X},$ (75)
where $\textbf{I}_{N}$ is the identity matrix and $\otimes$ is the Kronecker
product. Eq. (75) can be decoupled into N independent equations by
diagonalizing L. However, to deal with cluster synchronization and multilayer
interactions we have to further generalize Eq. (75). The variational equations
for complex synchronization patterns on generalized networks has the form
[zhang2020]:
$\delta\dot{\textbf{X}}=\left[\sum_{l=1}^{L}\textbf{D}^{(l)}\otimes
J\textbf{F}(\textbf{s}^{l})-\sum_{l=1}^{L}\sum_{\alpha=1}^{M}\sigma_{\alpha}\textbf{L}^{(\alpha)}\textbf{D}^{(l)}\otimes
J\textbf{H}^{(\alpha)}(\textbf{s}^{l})\right]\delta\textbf{X}$ (76)
where the identity matrix has been replaced by the diagonal matrix
$\textbf{D}^{(l)}$, whose generic element $D_{ii}^{(l)}=1$ if node $i$ belongs
to $l$th dynamical cluster and $D_{ii}^{(l)}=0$ otherwise.
In a recent paper [zhang2020], it was established that to optimally decouple
Eq. (76), the matrices encoding the synchronization pattern and the
interaction pattern, that is $\\{\textbf{D}^{(l)}\\}$ and
$\\{\textbf{L}^{(\alpha)}\\}$, should be simultaneously diagonalized. In this
work, an algorithm was also developed to find the finest simultaneous block
diagonalization.
##### Synchronization in a network of phase oscillators
One of the first approaches to describe phase synchronization in an ensemble
of oscillators on multiplex networks was done in 2015 [Gambuzza2015], to
investigate the synchronization of indirectly coupled units using a system
composed by two layers, where the top layer was made of disconnected
oscillators and the bottom one, modeling the medium, consisted of oscillators
coupled according to a given topology and with a characteristic natural
frequency. Each node of the multiplex was modeled as a Stuart-Landau (SL)
oscillator, that is an oscillator with amplitude as well as phase dynamics.
Therefore, the Kuramoto model (KM, see Appendix B) can be retrieved as a
limiting case when the amplitude dynamics vanishes. The Kuramoto order
parameter can be generalized to the multiplex framework as:
$r^{\alpha\beta}_{ij}=\big{|}\big{\langle}e^{i[\theta^{\alpha}_{i}(t)-\theta^{\beta}_{j}(t)]}\big{\rangle}_{t}\big{|},$
(77)
while intra- and inter-layer coherence, respectively, can be also defined by
$r^{\alpha}=\frac{1}{N(N-1)}\sum^{N}_{i,j=1}r^{\alpha\alpha}_{ij},$ (78)
and
$r^{\alpha\beta}=\frac{1}{N}\sum^{N}_{j=1}r^{\alpha\beta}_{jj}.$ (79)
By studying a population of $N$ disconnected oscillators, indirectly coupled
through an inhomogeneous medium, authors have shown the onset of intra-layer
synchronization without inter-layer coherence, i.e. a state in which the nodes
of a layer are synchronized between them without being synchronized with those
of the other layer (see Fig. 37, left panel). Synchronization of units that
are not connected requires the presence of an amplitude dynamics as the regime
of intra-layer synchronization is not observed in purely phase oscillators,
such as those in the KM.
Figure 37: Left: Two-layer multiplex network with one-to-one coupling between
the layers. In the top layer ($\alpha$) the nodes only interact with those in
the bottom one whereas in the bottom layer ($\beta$) the nodes also interact
with other members of the same layer. Middle: Kuramoto order parameters
($r^{\alpha}$) and ($r^{\alpha\beta}$) vs. coupling coefficients
$\lambda=\lambda_{\alpha\beta}=\lambda_{\beta}/5$. Continuous lines refer to a
multilayer network of Stuart-Landau oscillators with $a=1$, whereas the dashed
ones to purely phase oscillators ($a\xrightarrow{}\infty$). Right:
Synchronization transitions in two-layer networks with a fraction of the nodes
adaptively controlled by a local order parameter $f=1$. Squares and circles
(triangles and stars) refer to the values of $R1$ ($R2$), and the insets show
the corresponding dependence of the width of hysteretic loop $d$ on $f$. Left
and middle panels readapted from [Gambuzza2015], right panel from [Zhang2015]
As previously mentioned, not only were continuous second-order transitions
observed in an ensemble of networked phase oscillators, but examples of an
abrupt first-order transition, named explosive synchronization (ES), were also
observed [Gomez-Gardenes2011, Leyva2013]. It was first observed in a network
of oscillators presenting a positive correlations between natural frequencies
and the degree of the nodes. Moreover, ES was also studied in systems where a
local order parameter for the $i$-th oscillator is defined [Zhang2015]:
$r_{i}(t)e^{i\Phi(t)}=\frac{1}{k_{i}}\sum_{j=1}^{k_{i}}\sin(\theta_{j}),$ (80)
and where the phase dynamics is expressed as:
$\dot{\theta_{i}}=\omega_{i}+\sigma\alpha_{i}\sum_{j=1}^{N}A_{ij}\sin(\theta_{j}-\theta_{i}).$
(81)
The overall amount of phase coherence in the network is measured by means of
the global order parameter $R$:
$Re^{i\Psi}=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_{j}},$ (82)
where $0\leq R\leq 1$ and $\Psi$ denotes the average phase.
Explosive synchronization onset has been reported in a system of two
interdependent networks (see Fig. 37, right panel), with the same size and
where nodes on the two layers are coupled in a one-to-one correspondence, so
that a group of oscillators in the first layer is controlled by the local
order parameters of the corresponding nodes on the second layer, and vice-
versa [Zhang2015]. Nevertheless, it was shown that ES is a property of a
generic multilayer network as long as some microscopic suppressive rule can
prevent the formation of the giant synchronization cluster which characterizes
second-order transitions.
### IV.3 Game dynamics: cooperation processes
Human cooperation may be intended as a collective behavior that emerges as the
result of the interactions among individuals. In the past few years
cooperation has been studied in social sciences with methods of statistical
physics [Perc2017], in particular Monte Carlo methods and the theory of
collective behavior of interacting particles near phase-transition points.
That approach has proven very valuable for understanding cooperation and its
spatio-temporal dynamics. The mathematical framework used to study human
cooperation is usually evolutionary game theory, which quantitatively
describes social interactions using example games and formalizes the concept
of the social dilemma, intended as the conflictual choice that an individual
has between doing what is best for society or doing what is best for
themselves.
Figure 38: a) Mean final cooperation as function of the temptation to defect
$T$, for several multiplex networks with different numbers of layers $M$. b)
Multiplex with $M=2$ layers, where cooperation is studied as a function of
synergy factor $r$ and edge overlap $\omega$. Dashed lines separate regions
with either full defection $c=0$ (IA) or full cooperation $c=1$ (IB), from the
region of continuously evolving coexistence of cooperators and defectors
$0<c<1$ (II). Panel a) readapted from [gomez2012evolution] and panel b)
readapted from [battiston2017determinants]. c) Mean final cooperation (color
coded) for the Prisoner’s Dilemma as a function $M$ and the strength of degree
correlations $\nu$. d) Mean final cooperation as a function of the initial
density of cooperators $c_{0}$ and the strength of correlations $\nu$. Panels
c) and d) readapted from [Kleineberg2018].
Evolution of cooperation strongly depends on the population interaction
network: individuals who have the same strategy are more likely to interact
[Nowak1992, Nowak2010], effectively creating resilient cooperative clusters in
a structured population, a phenomenon named network reciprocity. When the
edges that determine the interaction among individuals were fixed on a time
scale, it was demonstrated [Santos2006] that in heterogeneous populations –
modeled by networks with degree distributions exhibiting a power-law behavior
– the sustainability of cooperation is simpler to achieve than in
homogeneously structured populations. Here we address the problem of how human
cooperation emerges in multiplex networks, with different interaction layers
that can account for different kinds of social ties an individual may be
involved in.
In particular, we consider a Prisoner’s Dilemma game implemented in a set of
$M$ interdependent networks, characterized by an average degree $\langle
k\rangle$, each of them containing the same number $N$ of nodes. Each
individual is represented by one node in each of the layers and we define a
set of adjacency matrices $\\{A^{\alpha}\\}$ so that $A_{ij}^{\alpha}=1$ when
two nodes are connected in layer $\alpha$ and $A_{ij}^{\alpha}=0$ otherwise.
At each time step, for each of the $k_{i}^{\alpha}$ games played, each
individual $i$ facing a cooperator collects a payoff $\pi_{ij}=R$ or
$\pi_{ij}=T$ when playing as a cooperator or as a defector, respectively.
Conversely, if $i$ faces a defector, $i$ collects a payoff $\pi_{ij}=S$ or
$\pi_{ij}=P$ playing as a cooperator or as a defector, respectively. For game
parameters it holds that $S<P<R<T$. The aggregated payoff of node $i$ is given
by the sum of the payoffs $\pi_{i}$ over all layers. Furthermore, after each
round of the game, individuals update their strategies with a rule that can
use the global knowledge about the benefits of the neighbors or be random
[Gomez-Gardenes2011b].
Finally, it worth noting that cooperation is also studied in public good
games, where the Prisoner’s Dilemma is played in overlapping groups of
individuals: cooperators contribute with a “cost” $d$ to the public good,
while defectors do not contribute and the total amount in the common pool of
each group is multiplied by a factor $r$ and distributed equally among all
members of the group [Santos2008a].
Previous research [gomez2012evolution] demonstrated that the resilience of
cooperative behavior can be enhanced by the multiplex structure through the
simultaneous formation of correlated clusters of cooperators across different
layers, i.e. through multiplex network reciprocity. However, it was also
proven [battiston2017determinants] that, to gain benefits from multiplex
structure, a high topological overlap is needed or, in other words,
individuals have to be similarly linked across different layers (see Fig. 38,
panels a) and b)). Other studies [Kleineberg2018] investigated the role of
degree correlation $\nu$ between nodes in different layers for the emergence
of cooperation. They found that in the absence of degree correlations,
increasing the number of layers only leads to mild changes. However, if degree
correlations are present we observe a mean final cooperation of $c=0.5$, and
this value is nearly independent of the game payoff parameters. This mechanism
is called _topological enslavement_ and can be understood by considering that,
if degree correlations are strong, hubs dominate the game dynamics, since they
have the potential to earn higher payoffs (because they play more games) and
they are more likely to be selected by other nodes as imitation candidates.
Furthermore, topological enslavement implies that the outcome of the evolution
of the system is determined by the initial conditions (see Fig. 38, panels c)
and d)).
Finally, as anticipated in Sec. III.5, layer-layer correlations might have a
deep impact on the dynamics on the top of multilayer systems. This is the case
for game dynamics, where cooperation might be hampered, rather than enhanced,
by specific correlation patterns combining assortative and disassortative
degree mixing across layers [wang_pre14a, duh_njp19] (see Fig. 39 for
details).
Figure 39: Fraction of cooperators as a function of the payoff of a cooperator
when playing with a defector ($S$) and the payoff of a defector when facing a
cooperator ($T$). The interaction layer is subject to disassortative mixing
(assortative coefficient $A_{I}<0$) while the updating network is subject to
assortative mixing (disassortative coefficient $A_{U}>0$). Results show the
disruptive effect of symmetry breaking for the evolution of cooperation.
Parameter values are $A_{I}=-0.1$ and $A_{U}=0.1$ (Left) and $A_{I}=-0.2$ and
$A_{U}=0.2$ (Right). Figure from [wang_pre14a].
### IV.4 Interdependent processes
The second class of dynamical processes on multilayer networks is the one of
“interdependent” or “coupled” dynamics, consisting of systems characterized by
different processes on each layer. The inter-layer interactions couple such
processes and are responsible for emerging phenomena which could not be
detected in a single-layer network framework (see Tab. 3).
As in the case of the structure, it is possible to introduce a unifying
framework in terms of a dynamical SNXI decomposition, outlined in Eq. (8), to
describe dynamics on multilayer networks. Let $x^{[l]}_{i\alpha}$ (where
$l\in{1,2,...,C}$) denote the $l$-th component of a $C-dimensional$ vector
$\mathbf{x}_{i\alpha}$ that represents the state of node $i$ in layer
$\alpha$. The most general (and possibly nonlinear) dynamics governing the
evolution of each state is given by the systems of equations
$\displaystyle\dot{\mathbf{x}}_{i\alpha}(t)$
$\displaystyle=F_{i\alpha}(\mathbf{X}(t))=\sum_{\beta=1}^{L}\sum_{j=1}^{N}f_{i\alpha}^{j\beta}(\mathbf{X}(t))$
(83)
$\displaystyle=\underbrace{\sum_{\beta=1}^{L}\sum_{j=1}^{N}f_{i\alpha}^{j\beta}(\mathbf{X}(t))\delta_{\alpha}^{\beta}\delta_{i}^{j}+\sum_{\beta=1}^{L}\sum_{j=1}^{N}f_{i\alpha}^{j\beta}(\mathbf{X}(t))\delta_{\alpha}^{\beta}\left(1-\delta_{i}^{j}\right)}_{\text{intra-
layer dynamics }}$
$\displaystyle+\underbrace{\sum_{\beta=1}^{L}\sum_{j=1}^{N}f_{i\alpha}^{j\beta}(\mathbf{X}(t))\left(1-\delta_{\alpha}^{\beta}\right)\delta_{i}^{j}+\sum_{\beta=1}^{L}\sum_{j=1}^{N}f_{i\alpha}^{j\beta}(\mathbf{X}(t))\left(1-\delta_{\alpha}^{\beta}\right)\left(1-\delta_{i}^{j}\right)}_{\text{inter-
layer dynamics }}$
$\displaystyle=\underbrace{f_{i\alpha}^{i\alpha}(\mathbf{X}(t))}_{\text{self-
interaction }}+\underbrace{\sum_{j\neq
i}f_{i\alpha}^{j\alpha}(\mathbf{X}(t))}_{\text{endogenous interaction
}}+\underbrace{\sum_{\beta\neq\alpha}\sum_{j\neq
i}f_{i\alpha}^{j\beta}(\mathbf{X}(t))}_{\text{exogenous interaction
}}+\underbrace{\sum_{\beta\neq\alpha}f_{i\alpha}^{i\beta}(\mathbf{X}(t))}_{\text{intertwining
}}$
$\displaystyle=\mathbb{S}_{i\alpha}(\mathbf{X}(t))+\mathbb{N}_{i\alpha}(\mathbf{X}(t))+\mathbb{X}_{i\alpha}(\mathbf{X}(t))+\mathbb{I}_{i\alpha}(\mathbf{X}(t))$
where
$\mathbf{X}(t)\equiv\left(\mathbf{x}_{11},\mathbf{x}_{21},\ldots,\mathbf{x}_{N1},\mathbf{x}_{12},\mathbf{x}_{22},\ldots,\mathbf{x}_{N2},\ldots,\mathbf{x}_{1L},\mathbf{x}_{2L},\ldots,\mathbf{x}_{NL}\right)$
and we did not use the tensorial formalism to make explicit the contributions
of each term in the governing equation. In fact, this equation could also be
compactly written as
$\displaystyle\dot{x}_{j\beta}(t)=F[x_{j\beta}(t)]=\mathbb{S}[x_{j\beta}(t)]+\mathbb{N}[x_{j\beta}(t)]+\mathbb{X}[x_{j\beta}(t)]+\mathbb{I}[x_{j\beta}(t)].$
(84)
We call this equation “dynamical $\mathbb{SNXI}$ decomposition”. Similarly to
the structural decomposition in Eq. (8), we have decoupled the different
contributions of intra-layer and inter-layer dynamics, allowing us to classify
different dynamical processes in terms of the corresponding dynamical
$\mathbb{SNXI}$ components. The peculiar behavior of the interdependent
processes is ascribed to the exogenous and intertwining components. Although
the most-studied examples come from mixed spreading processes, which are
crucial for understanding phenomena such as the spreading dynamics of two
concurrent diseases in two-layer multiplex networks [Sanz2014,
salehi2015spreading, Dickison2012epidemics, Cozzo2013social, DeArruda2017],
and the spread of diseases coupled with the spread of information or behavior
[wang2015coupled, funk2015, Granell2013, lima2015disease, Funk2009,
granell2014competing], other types of dynamical interdependence are attracting
a growing interest.
##### Coupling diffusion with synchronization
An illustrative and pedagogical example has been proposed by
[nicosia2017collective]. In this work, the authors examine the interdependent
dynamics of the two processes presented in the two previous sections, namely,
diffusion and synchronization. They propose a model that mimics the interplay
between the neural activity and energy transport in brain regions, from which
a rich collection of behaviors emerges. These processes evolve in the two
layers of a multilayer network and are related to each other by the
correspondence between layers, which in this case is realized through the
functional relation between the parameters governing the two processes and the
state variables. In particular, the dynamics of the entire system is governed
by the following equations:
$\left\\{\begin{aligned}
\dot{x}_{i}&=F_{\omega_{i}}\left(\mathbf{x},A^{[1]}\right)\\\
\dot{y}_{i}&=G_{\chi_{i}}\left(\mathbf{y},A^{[2]}\right)\end{aligned}\quad
i=1,2,\ldots,N\right.$ (85)
where $\mathbf{x}=\left\\{x_{1},x_{2},\ldots,x_{N}\right\\}\in\mathbb{R}^{N}$
and $\mathbf{y}=\left\\{y_{1},y_{2},\ldots,y_{N}\right\\}\in$ $\mathbb{R}^{N}$
denote the states of the two dynamical processes, while the topologies of the
two layers are encoded in the adjacency matrices
$A^{[1]}=\left\\{a_{ij}^{[1]}\right\\}$ and
$A^{[2]}=\left\\{a_{ij}^{[2]}\right\\},$ respectively, such that
$a_{ij}^{[1]}=1\left(a_{ij}^{[2]}=1\right)$ if a link exists between nodes $i$
and $j$ in the first (second) layer, and $a_{ij}^{[1]}=0$
$\left(a_{ij}^{[2]}=0\right)$ otherwise. The evolution of the system depends
on a set of parameters $\omega$ and $\chi$, which in turn depend on the state
of the nodes in the adjacent layer:
$\begin{array}[]{l}\dot{\omega}_{i}=f\left(\omega_{i},y_{i}\right)\\\
\dot{\chi}_{i}=g\left(\chi_{i},x_{i}\right)\end{array}\quad i=1,2,\ldots N$
(86)
The authors assign to functions $F_{\omega_{i}}$ and $G_{\chi_{i}}$ a Kuramoto
dynamic and a continuous-time random walk, respectively. Subsequently, they
assign the functions $f$ and $g$ in Eqs. (86), respectively, relating the
frequency $\omega_{i}$ of the oscillator $i$ at layer 1 to the state $y_{i}$
at layer 2 , and the bias property $\chi_{i}$ of the random walkers at layer 2
to the oscillator phase $x_{i}$ at layer $1$. The natural frequency
$\omega_{i}$ of the oscillator $i$ evolves relaxing to values proportional to
the fraction of random walkers at the replica node $i$ in the other layer.
Analogously, the random walks are biased toward (away from) strongly
synchronized nodes.
Note that the coupling between the two layers is tunable through two
parameters $\lambda$ and $\alpha$ that represent, respectively, the intensity
of the influence of the random walk on the oscillators and vice versa. This
setup completely defines an interdependent process on a multilayer network:
depending on the coupling strengths $\lambda$ and $\alpha$ the collective
behavior exhibits special dynamics (see Fig. 40).
Figure 40: Coupling diffusion with synchronization dynamics. a) Distribution
$P(y_{i})$ of steady-state random walker fractions $y_{i}$ at layer $2$ for
$\alpha=1.0$, when the oscillators at layer $1$ are incoherent ($\lambda=0.1$,
top, blue) and synchronized ($\lambda=0.4$ bottom, red). b) Synchronization
phase diagram showing the level of synchronization as a function of coupling
$\lambda$ and bias exponent $\alpha$. The bistable region is colored in white.
Figure adapted from [nicosia2017collective].
The random walkers are homogeneously distributed in the incoherent state,
while in the synchronized state the distribution is heterogeneous. Conversely,
at certain values of the tuning parameters the system encounters an explosive
synchronization, or a bistability region characterized by a hysteretic
behavior. Similar phase transitions can also be observed in single-layer
networks under certain conditions, as reported in [acebron2005kuramoto,
martens2009exact, buendia2021broad]. Importantly, the multilayer network model
offers a parsimonious explanation of the emergence of these collective
phenomena, considering explicitly the intertwined nature of the dynamics.
##### Coupling epidemics spreading with awareness diffusion
Many different phenomena in nature can be described, in their essence, by the
results of constructive or destructive relations between two or numerous
parts. For instance, the mutualistic or competitive relation between dynamical
systems gives rise to a wealth of fascinating behaviors.
The dynamics occurring in one layer can have positive or negative feedbacks on
another: for example, human behavior (e.g., information awareness) can inhibit
the spread of disease; social mixing between classes and mobility may produce
abrupt changes in the critical properties of the epidemic onset; cooperation
emerges where the classical expectation was defection.
A generalization of these results can be found in [danziger2019dynamic], in
which the authors describe some universal features of interdependent systems
with coupled dynamics. There is an interesting relation between the collective
behavior emerging from percolation processes and from the ones arising in
interdependent systems. In such processes, abrupt transitions and critical
dynamics may arise in certain circumstances and, in dynamically coupled
systems, other interesting phenomena may be also observed, such as hysteresis
and multistability, with functionality of nodes in one layer influencing the
functionality of their replicas in the other layers. In [danziger2019dynamic]
the functionality is quantified by an order parameter, which plays a role in
the coupling strength between the nodes in different layers, with the order of
a node dynamics affecting the order of its neighbors. This fact is the
dynamical counterpart of interdependent percolation, where the functionality
of a node is related to its belonging to the mutual giant connected component
(see Sec. IV.5 for details). It turns out that the dynamic interdependence
increases the vulnerability of the system [danziger2016vulnerability], as in
the case of percolation processes.
Figure 41: Two (left) reciprocally enhanced and (right) reciprocally
inhibited disease-spreading processes of susceptible–infected–susceptible
type. The colors in the figure represent the prevalence levels of the diseases
at a steady state of Monte Carlo simulations. Note the emergence of a curve of
critical points (at a “metacritical point”) in which the spreading in one
layer depends on the spreading in the other. Figure from [de2016physics].
Epidemic models are another important example of systems in which
interdependencies play a crucial role (see [wang2015coupled, de2016physics]
for a review) and, also in this case, hysteretic behavior with abrupt
transitions may occur. This means that, for instance, explosive pandemics with
no early-warning can suddenly appear and, in a similar implosive way, can
disappear. Unfortunately, the value of the infection rate which can eradicate
the disease must be much lower than the value that triggered it
[danziger2019dynamic]. Other interesting behaviors arise in the case of
competitive diseases, where mutual exclusion or endemic coexistence may
spontaneously occur.
Figure 41 shows two phase diagrams of disease incidence of reciprocally
enhanced (right) and inhibited (left) disease-spreading processes. In the
figure is highlighted the existence of a curve of critical points that
separate endemic and non-endemic phases of the disease. Moreover, the curve
that separates the endemic and non-endemic phases, is in turn divided into two
parts: one in which the critical properties of one spreading process are
independent of the other (straight dashed line), and one in which the critical
properties of one spreading process do depend on those of the other layer
(solid curve). The point at which this crossover occurs is called a
metacritical point.
Figure 42: SIS-SIS interacting diseases model. Left: multiplex representation
of the diseases spreading. Each individual is present in both the layers and
can be infected by one (or both) of the diseases, as indicated in the central
panel. Right: possible transitions between the different states of the SIS
model. The variables represent the densities of individuals of each type
having degree equal to $k$ in the first layer and degree $l$ in the second.
Figure from [Sanz2014].
In fact, traditional epidemic models can describe in details the spreading of
a single disease in different realistic situations, whereas the multilayer
representation can extend the possibilities to spreading processes with
interacting diseases. In [Sanz2014] the authors propose a framework to
describe the spreading dynamics of two concurrent diseases (see Fig.42). Using
SIS-SIS and SIR-SIR models with appropriate interlayer couplings describing
the influence of one disease on the other, they derive the epidemic threshold
for the two interacting diseases, showing that the onset of a disease’s
outbreak is conditioned to the prevalence levels of the other disease.
Epidemic thresholds can be influenced not only by the presence of a second
disease, but also by the individual’s change of awareness about an ongoing
epidemic. In [Granell2013], a multilayer network couples the dynamics of
disease spreading in a social network and the diffusion of awareness among
actors. The two layers are coupled in a multiplex as illustrated in Fig. 43.
The process of spreading of awareness (UAU) is akin to an SIS process, where
in place of susceptible (S) there are unaware (U) and in place of infected (I)
there are aware (A) actors. The probability of being infected is influenced by
the state of awareness of the individuals. The authors found the existence of
a metacritical point in which the state of awareness of the individuals can
control the onset of an epidemic.
Figure 43: Multiplex representation of awareness-disease spreading. The
spreading of awareness occurs in the upper layer, whereas the spreading of the
diseases takes place in the lower layer. Figure from [Granell2013].
##### Coupling game dynamics with opinion dynamics
Figure 44: Polarization of opinions and strategies in a multiplex with 5,000
nodes, in the presence of angular correlations between the layers. Top:
comparison between the two layers; the color is the time average of the state
of each node. Bottom: evolution of the density of cooperators in the angular
bins used to compute the interlayer correlations. Numeric labels indicate the
clusters of nodes that adopt the same strategy. Figure from
[amato2017interplay].
The multilayer representation of the dynamics of complex systems can also be
used to shed light on real world social dilemmas. The influence of factors
acting on different layers might explain the emergence of particular patterns
of cooperation between social agents. In [amato2017interplay] the authors
present a model coupling evolutionary game dynamics and opinion dynamics,
regarded as two processes evolving on distinct layers of a multiplex network.
To model the game dynamics on the first layer the authors adopt a replicator,
in which the individuals (nodes) copy the strategy of one of their neighbors
with a probability that is so much higher the higher the neighbors’ payoff
compared to themselves. The possible states are ”cooperate” and ”defeat”. The
opinion dynamics on the second layer is modeled using the voter model, where
individuals (nodes) adopt the opinion of a randomly selected neighbor with a
certain probability. The opinions of the nodes can be ”cooperate” or ”defeat”,
as in the game layer. The authors assume that imitating a cooperative opinion
is more likely than imitating a defection. This can be interpreted as the
influence of media campaigns or broadcasting agents. Depending on the specific
parametrization of both the game and the opinion dynamics, this model gives
rise to fascinating dynamical behaviors in which equilibria of different types
exist for the game dynamics. The impact of social influence on the decision of
individuals is conveyed by the interlayer coupling, which is encoded in the
parameter $\gamma$, representing the tendency of individuals to act in
agreement with their proclamations: the nodes in one layer copy their own
state from the other layer with probability $\gamma$.
The main result is that this model is sufficient to explain the emergence of
cooperation in scenarios where the pure game dynamics predicts defection. This
is due both to the intertwined dynamics and to the multilayer structure
itself. In fact, the authors proved that the geometric correlation between
layers has a significant impact on the stability of the system. Importantly,
under certain conditions of correlation between layers, the system can reach a
polarized metastable state (see Fig. 44). This result can explain the observed
polarization in real world social systems.
Ultimately, the emergence of cooperation in unexpected conditions is due to
the interplay between the coupled dynamics of strategies and opinions, the
complex topologies of the networks upon which the dynamics exert, and the
structural relations between the layers. Missing one of these elements could
hinder the right interpretation of the complex behavior of such systems.
##### Coupling epidemics spreading with social integration
A multilayer perspective on epidemic spreading can be a valuable support to
design more educated strategies to reduce disease risk. A recent example comes
from [bosetti2020heterogeneity], which integrates social dynamics, human
mobility and epidemic spreading to assess the risk of measles outbreak in
Turkey. During the past decade, Turkey has received more than 3.5 million
refugees coming from Syria. The levels of immunization of the two populations
are considerably different. The outbreak risk is analysed through a multilayer
transmission model, which takes into account the different levels of
immunization in the two populations, along with the mobility pattern and the
level of social integration. The main result of the study is that, in the case
of heterogeneous immunization, high levels of social interaction can
drastically reduce the spatial spread and incidence of a disease. This
apparently counter-intuitive result is due to the fact that the high
immunization coverage of one population (Turkish citizens) can shield the
other (Syrian refugees) from getting exposed to the infection, as an effect of
herd immunity. The network structure is defined by dividing Turkey into
patches corresponding to administrative areas that represent the nodes. The
layers of the network are encoded by Turkish $(T)$ and Syrian refugee $(R)$
populations. On this network, social dynamics, mobility and epidemic spreading
happen simultaneously.
Figure 45: (A) Model scheme. Prefectures of Turkey are the nodes of a network
of geographic patches. Turkish and Syrian populations are encoded by two
colors and move between patches following the inferred mobility pathways. The
two populations encode two layers, where social dynamics and epidemic
spreading happen simultaneously. (B) Mobility of Syrian refugees (upper) and
Turkish citizens (lower) inferred from mobile phone data. Figure from
[bosetti2020heterogeneity].
The authors denotes with $c_{ki}^{(p)}(p\in T,R)$ the elements of a matrix
$\mathbf{C}^{(p)}$ encoding the number of people belonging to population
$p\in\\{T,R\\}$ traveling from patch $k$ to patch $i$, and with $\alpha$ the
fraction of Syrian contacts with Turkish citizens. The force of infection for
each population in the $i$-th patch depends on the contribution of all patches
in the country:
$\displaystyle\begin{array}[]{l}\lambda_{i}^{(T)}\left(\alpha,\mathbf{C}^{(T)},\mathbf{C}^{(R)}\right)=\beta_{T}\sum\limits_{k=1}^{L}\left[\underbrace{c_{ki}^{(T)}\frac{I_{k}^{(T)}}{N_{k}^{(T)}}}_{\text{Endogenous
}}+\underbrace{\alpha
c_{ki}^{(R)}\frac{I_{k}^{(R)}}{N_{k}^{(R)}}}_{\text{Exogenous }}\right]\\\
\lambda_{i}^{(R)}\left(\alpha,\mathbf{C}^{(T)},\mathbf{C}^{(R)}\right)=\beta_{R}\sum\limits_{k=1}^{L}\left[\underbrace{\alpha
c_{ki}^{(T)}\frac{I_{k}^{(T)}}{N_{k}^{(T)}}}_{\text{Exogenous
}}+\underbrace{c_{ki}^{(R)}\frac{I_{k}^{(R)}}{N_{k}^{(R)}}}_{\text{Endogenous
}}\right]\end{array}$
where $\beta_{p}=\beta/P_{i}^{(p)}(\alpha,c)$ is the transmission rate for
population $p$ and $P_{i}^{(p)}(\alpha,c)$ is an appropriate normalization
factor. From this equation we can easily recognise the structure of Eq. (83),
where the contributions to the force of infection come from an endogenous
term, accounting for the infectivity due to individuals from the same
population, and an exogenous term, accounting for the infectivity due to the
other population. The parameter $\alpha$ is the level of social mixing and can
be changed according to different social integration scenarios. This is the
parameter which plays the role of the coupling between the layers. An
illustrative representation of the model is shown in Fig. 45.
The epidemic transmission dynamics is eventually regulated by the following
SIR dynamical model
$\displaystyle\left\\{\begin{aligned}
\dot{S}_{i}^{1}&=-\lambda_{i}^{1}(\alpha,c)S_{i}^{1}\\\
\dot{S}_{i}^{2}&=-\lambda_{i}^{2}(\alpha,c)S_{i}^{2}\\\
\dot{I}_{i}^{1}&=\lambda_{i}^{1}(\alpha,c)S_{i}^{1}-\gamma I_{i}^{1}\\\
\dot{I}_{i}^{2}&=\lambda_{i}^{2}(\alpha,c)S_{i}^{2}-\gamma I_{i}^{2}\\\
\dot{R}_{i}^{1}&=\gamma I_{i}^{1}\\\ \dot{R}_{i}^{2}&=\gamma
I_{i}^{2}\end{aligned}\right.,$
whose analysis suggests that the incidence of the measles can be reduced up to
90% in the case of very high levels of integration [bosetti2020heterogeneity].
Despite the different contexts, the multilayer dynamics can have positive or
negative feedbacks, leading to interdependence between the corresponding
critical points of the dynamics. As a consequence, two different regimes
exist: i) one in which the critical properties of one process depend on those
of the other, and (ii) in which the critical properties are independent of the
other. The two regimes are separated by a metacritical point, where a
crossover occurs (see a recent review from [de2016physics] and Fig. 41). This
regime-shift is analogous to the one occurring in percolation processes, which
is presented in the next section.
### IV.5 Percolation
In this section we switch our attention to percolation
[stauffer2018introduction], where the goal is to analyse how different
properties of the network change as we remove some of its nodes or links. The
properties we are interested in are typically topological, such as the size of
the largest connected component or the distribution of small connected
components, and they are used as a first proxy to assess the functionality of
a system exposed to failures or attacks. Failures are modelled as random
removals, whereas attacks assume some a priori knowledge of the network, where
the elements are ranked given some criterion —frequently topological (degree,
betweenness, etc.) [albert2000error, cohen2001breakdown], albeit not strictly
necessary [artime2020effectiveness, artime2021percolation]— and removed
accordingly. We aim at presenting the basic mathematical framework to address
the problem of multilayer percolation and showcase some applications. Good
reviews to expand on what we present here can be found in
[bianconi2018multilayer, lee2015towards].
The first step towards a description of multilayer percolation processes is to
consider the generalization of the generating function methodology, frequent
in single-layer networks [newman2001random]. In [leicht2009percolation],
random percolation is studied in general multilayer networks described by the
set of degree distributions $\\{p_{k_{1}k_{2}\ldots k_{L}}^{\alpha}\\}$, where
$\alpha$ is the layer label and $k_{\beta}$ denotes the number of links toward
nodes in layer $\beta$. Expressions for the size of the giant component of
each layer $S^{\alpha}$ can be written as a function of the generating
functions. This framework allows us to consider percolation on correlated
multilayer networks, leading to non-trivial results. For example, in
[lee2012correlated] correlated duplexes of Erdős–Rényi networks are studied.
When degrees across layers are maximally anti-correlated – i.e., hubs in one
layer are low degree nodes in the other layer – the giant component appears at
a link density considerably higher than the value for which it appears in
uncorrelated duplexes. The giant component exists, though, for any nonzero
link density if interlayer degrees are maximally positively correlated. In
other words, the more correlated the degrees are, the fewer edges are needed
to make a macroscopic structure emerge. When it comes to attacks on the most
connected nodes, the latter statement is true up to a certain intra-layer mean
degree (assuming it is the same for both layers), after which the behavior is
reversed and maximally negative correlated networks become more robust
[min2014network].
This framework disregards any functional characteristics of the layers and,
from a phenomenological point of view, continuous phase transitions are always
found. This is no longer true if the nature of the layers is taken into
account, for example via the interdependence of the nodes or antagonistic
interactions. These more realistic scenarios define new conditions for a node
to remain in the network, and need alternative, more function-oriented metrics
to describe the state of the system. A widely accepted choice is the mutually
connected component [buldyrev2010catastrophic, son2012percolation] already
defined in Sec. III.2, which is the set of nodes that are connected within
each and every layer simultaneously. The most striking result is that when the
giant mutual component (GMC) is computed in interdependent networks, the
percolation phase transition changes its nature, becoming a discontinuous one
[son2012percolation]. This has serious implications for the robustness of the
system, since the disintegration occurs abruptly, i.e., it is hard to
anticipate. Mathematically, the idea is as follows.
Let us assume an edge-colored multigraph, let $p_{k_{1}\ldots k_{L}}$ be the
probability that a node has degree $k_{\alpha}$ to other nodes within the
layer $\alpha$, and let $q_{k_{1}\ldots k_{L}}$ be the corresponding excess
degree distribution. We indicate by $w_{\alpha}$ the probability that a node
does not belong to the GMC via a link in layer $\alpha$. Hence,
$w_{\alpha}^{k_{\alpha}}$ gives the probability that the node does not belong
to the GMC via any of its neighbors in layer $\alpha$. The condition to belong
to the GMC is that the node has to be connected to it in all the layers, i.e.,
the size of the GMC is proportional to
$\prod_{\alpha=1}^{L}(1-w_{\alpha}^{k_{\alpha}})$. We just need to rescale by
the occupation probability $\phi$ and average over the degree distribution,
yielding
$\displaystyle M$
$\displaystyle=\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}p_{k_{1}\ldots
k_{L}}\prod_{\alpha=1}^{L}(1-w_{\alpha}^{k_{\alpha}}).$ (88)
To compute $w_{\alpha}$, we first note that $1-w_{\alpha}$ is the probability
that a node at the end of a link in layer $\alpha$ belongs to the GMC. For
this to happen any of its remaining $k-1$ neighbors in layer $\alpha$ are in
the GMC as well. Moreover, due to the condition of mutual connectivity, in
every other layer, the node needs to belong to the GMC via any of its
neighbors. Rescaling by $\phi$ because the node needs to be present in the
network, and averaging, we obtain
$1-w_{\alpha}=\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}q_{k_{1}\ldots
k_{L}}\left(1-w_{\alpha}^{k_{\alpha}-1}\right)\prod_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{L}(1-w_{\beta}^{k_{\beta}}).$ (89)
By inserting the solutions $\\{w_{\alpha}\\}$ of this system of equations into
Eq. (88) we readily obtain the size of the giant mutual component. See Fig. 46
to appreciate the emergence of the abrupt transition of the GMC for multiplex
systems with Erdős–Rényi networks in the layers. The discontinuity is also
present when considering multiplexes of scale-free networks but, at odds with
single-layer scale-free percolation, the occupation probability $\phi_{c}$ is
finite, making them more vulnerable to random failures than the single-layer
network [baxter2012avalanche]. Targeted interventions in the network can be
easily modeled as well by including a degree-dependent occupation probability
$\phi_{k_{1}\ldots k_{L}}$ inside the sums [min2014network,
zhao2016robustness].
The dependence of the mutual component $M$ shown in Fig. 46 for Erdős–Rényi
multiplexes is shared for other intra-layer topologies as well. That is, when
the number of layers increases, the value of the mean degree at which the
discontinuity appears becomes larger, thus broadening the parameter region
where the network is not functional. Moreover, the height of the discontinuity
jump becomes larger too, thus making the transition harder to anticipate when
going from the supercritical to the subcritical region. In light of these
results, the more layers, the more fragile the system is, which might seem
paradoxical from an evolutionary point of view: why would a system organize
itself in a layered structure if that reduces its robustness? In
[radicchi2017redundant], Radicchi and Bianconi provided a potential answer to
this conundrum by proposing a model of multilayer percolation that relaxes the
condition for functionality of the nodes. They argue that a node can be
functional, i.e., it is not removed, as far as it is functioning in at least a
pair of layers. This new condition for functionality allows them to conclude
that the addition of extra layers boosts the robustness of the system.
Figure 46: Percolation phase transition in a multiplex formed by intralayer
Erdős–Rényi networks, with the same mean degree $\langle k\rangle$. On the
left, we display $M$, which is the solution of $M=\phi(1-e^{-\langle k\rangle
M})^{L}$, as a function of the mean degree, for different number of layers,
with the occupation probability fixed to $\phi=1$. On the right, we study the
emergence of the giant mutual component for partial multiplexes, where $q$ is
the fraction of interdependent nodes. In this case $M$ is the solution of
$M=\phi(1-e^{-\langle k\rangle M})(1-qe^{-\langle k\rangle M})$. In the inset
we show the height of the jump of the order parameter at the transition point,
along with the theoretical value of the tricritical point (vertical dashed
line).
The abrupt nature of the transition induced by Eq. (89) holds as far as $L>1$,
see Fig. 46. When $L=1$ the mutual component coincides with the standard giant
component, so the transition is continuous. We cannot interpolate continuously
from $L=1$ to $L=2$ to understand how the nature of the phase transition
changes. However, there are other variables that we can tune to go from an
effective single-layer system to a multiplex.
The first of these variables is the multiplexity parameter. It might occur
that in real multiplex networks only a fraction $q$ of all nodes shares the
functional dependency across layers, hence a natural question is what is the
nature of the transition as a function of $q$. Does it suffice to have a non-
zero fraction of dependent nodes to observe the discontinuity, or, on the
contrary, is there a finite threshold, only above which we observe an abrupt
transition? To answer this question, let us focus on duplexes. If a node in
one of the layers has a dependency link, which occurs with probability $q$,
then the condition to belong to the GMC is the same as discussed earlier.
Instead, if the node does not have a dependency link, which occurs with
probability $1-q$, then it belongs to the GMC as far as any of its neighbors
of its very same layer belong to the component. Therefore, we can write
$\displaystyle M_{\alpha}$
$\displaystyle=q\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}p_{k_{1}\ldots
k_{L}}\prod_{\beta=1}^{L}(1-w_{\beta}^{k_{\beta}})$
$\displaystyle+(1-q)\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}p_{k_{1}\ldots
k_{L}}w_{\alpha}^{k_{\alpha}}.$ (90)
The set of probabilities $\\{w_{\alpha}\\}$ obeys the equations
$\displaystyle 1-w_{\alpha}$
$\displaystyle=q\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}q_{k_{1}\ldots
k_{L}}\left(1-w_{\alpha}^{k_{\alpha}-1}\right)\prod_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{L}(1-w_{\beta}^{k_{\beta}})$
$\displaystyle+(1-q)\phi\sum_{k_{1}=0}^{\infty}\ldots\sum_{k_{L}=0}^{\infty}q_{k_{1}\ldots
k_{L}}(1-w_{\alpha}^{k_{\alpha}-1}).$ (91)
Focusing again on a duplex of Erdős–Rényi networks, we see in Fig. 46 that
there is a finite tricritical point $q_{c}$ at which the transition changes
its order, confirming that, in general, a multiplex needs a certain level of
interdependency between layers to experience the discontinuous transition.
Since many infrastructural networks are embedded in a two-dimensional space,
in a similar fashion one might ask what type of transition we encounter when
coupling low-dimensional networks, that alone show continuous transitions, in
a multiplex. It turns out that in this case the transition does not change its
order [son2011percolation, berezin2013comment].
The second variable that allows us to interpolate between multiplexes and
monoplexes is the link overlap across layers. For complete overlapping all
layers are equal and the problem is reduced to percolation of a single-layer,
showing a continuous transition. How much overlap do we need to observe the
abrupt transition? Different approaches and techniques have been used to
answer this question, and accordingly slightly different phenomenology has
been discovered. Although the details of the phase diagram depend on the
number of layers and the degree distributions, Hu and coathors have found that
the transition stays abrupt as far as we do not have complete overlapping
[hu2013percolation]. In [cellai2013percolation] it has been reported that a
critical value of the edge overlap exists that changes the nature of the phase
transition from a hybrid first-order to a continuous one. See
[cellai2016message] for the generalization of the theory to an arbitrary
number of layers. If the edge overlap is combined with other topological
correlations, the phase diagram displays multiple and recursive hybrid phase
transitions [baxter2016correlated]. In [min2015link], the problem of link
overlap has been formulated in a way that the discontinuous transitions
display hysteresis. The role of the edge overlap, together with other
topological correlations, has been also explored in the problem of multilayer
optimal percolation [osat2017optimal], that is, in the identification of the
smallest set of nodes that, when removed, cause the largest damage in the
network [santoro2020optimal]. This problem is known to be NP-hard in single-
layer networks [kempe2003maximizing], and although there is no equivalent
proof in multilayer architectures, the intuition indicates that it is so as
well. On this basis, heuristic approaches to identify the critical subset of
nodes predominate. In [santoro2020optimal], the efficiency of $20$ of these
strategies is evaluated. The authors find that when no structural correlations
exist, a family of Pareto-efficient strategies based on both structural
descriptors and multiobjective optimization is the best at dismantling the
network. However, when evaluated in real multilayer networks that present non-
trivial correlations, the variability in performance changes from one
dismantling strategy to another, suggesting that a fair assessment of
multilayer robustness requires a comparison between strategies.
Beyond node interdependency as a condition for functioning, other interesting
mechanisms have been considered in the literature. One of them is antagonistic
relations, where a node in one layer is functional — that is, it has not been
removed — only if its replicas are not [zhao2013percolation]. Examples of this
kind of competitive or non-cooperative relations might be relevant, for
example, in biological networks. Interestingly, it has been found that when
percolation is considered in this scenario, the abrupt transition shown in
what follows persists, but displays as well, at variance with the transition
of the mutual component introduced earlier, the typical hysteresis and bi-
stability behaviors of equilibrium thermal first-order phase transitions
[goldenfeld2018lectures]. Other mechanisms have followed to include these
behaviors too, e.g., in [min2014multiple, min2015link].
Note that all of these results derived with the machinery of generating
functions come with some underlying assumptions that are very rarely met when
studying real networks: (i) the network is tree-like, (ii) it has infinite
size and (iii) the quantities of interest are computed as an average over the
ensemble of networks with given degree distribution, although, in reality, we
only have access to one supra-adjacency matrix, among all the possible ones
that a graph model could generate. As a consequence, the analytical
predictions might deviate from the actual process of percolation, obtained for
example via simulations. Analytical approaches have been proposed to overcome
points (ii) and (iii) in [radicchi2015percolation, bianconi2016percolation],
where a percolation theory of multiplex and interdependent networks is
introduced that takes as input the adjacency matrix instead of the degree
distribution. Furthermore, in a real network we may need to assess the
robustness under perturbation scenarios related to node metadata. For example,
in [baggio2016multiplex] it is addressed, among other things, how a multiplex
network capturing the flow of subsistence-related goods and services among
households in several Alaskan Natives communities responds to perturbations
involving targeted removals of specific resources by category, e.g.,
terrestrial, marine or riverine, as a representation of natural disasters. An
analytical framework to take into account non-topological features in the
robustness of a network has been recently developed for single layers
[artime2021percolation], but, at present, there is not a generalization for
multilayer networks.
### IV.6 Cascade failures
The percolation model has the limitation that failures and attacks are treated
statically. A more complete description of multilayer robustness and
resilience would include their time evolution, in order to better understand
under which conditions small perturbations can trigger global network-wide
effects, the so-called _cascades_. In this section we review some of the most
emblematic models for multilayer cascade failures.
The seminal work of Buldyrev and colleagues was one of the first to deal with
the propagation of failures across interdependent multiplexes
[buldyrev2010catastrophic]. The motivation for such a study was the 2003 Italy
blackout, for which it was hypothesized that the power grid and the Internet
network, the latter acting as a supervisory control and data acquisition
system, were interdependent, and that failures in the power stations hampered
the Internet communication and further propagated the malfunction across the
system, see top panels of Fig. 47).
However, the interest in the amplification of small perturbations throughout
the network is much more general, finding eventual applications in areas such
as biochemistry, where metabolic pathways interact in complex ways with key
tissues, or finance, where different banks might share the same asset in their
balance sheet [huang2013cascading], among many other examples. In
[buldyrev2010catastrophic] was introduced the concept of a mutually connected
component from a dynamical perspective, at odds with its static definition
presented in the previous section, see bottom panels of Fig. 47. It was shown
that these cascades evolving in multiplex networks yield a discontinuous phase
transition in the size of the giant mutual component
[buldyrev2010catastrophic], meaning that the failure of one single node from
an apparently healthy, functional infrastructure can generate the emergence of
a global cascade that collapses the entire system. It was later shown that
these cascading failures can be mapped to static percolation in multiplexes
[son2012percolation]. In fact, many of the results presented in Sec. IV.5 are
recovered at the final state of the cascade propagation of this type of model.
Figure 47: Top: evolution of a cascade of failures in a real interdependent
system composed by a power network (on the map) and an Internet network
(shifted above) and involved in the Italian blackout of September 2003. In a,
failure of a power station makes the Internet nodes to stop functioning,
indicated in red. In green is shown the nodes that disconnect from the largest
connected component in the next cascade step. These nodes will fail
immediately after ($b$), introducing a feedback of failures back in the power
network. Those nodes that have been isolated (green nodes in $b$) are the ones
that will fail in the next event $(c)$, inducing further interdependent
failures in the Internet network. Bottom: we sketch this process for further
clarification. In d, an initial attack to one node in network $\mathsf{A}$
occurs. In e, the attacked node is removed, along with its links, and the
corresponding interdependent node in network $\mathsf{B}$, with its links, is
also removed. In f, the actual cascade starts. All the $\mathsf{B}$-links
between $\mathsf{B}$-nodes connected interdependently to different
$\mathsf{A}$-clusters are removed. In d, the same rule applies but for links
in network $\mathsf{A}$: all $\mathsf{A}$-links between $\mathsf{A}$-nodes are
deleted if the interdependent nodes do not belong to the same
$\mathsf{B}$-cluster. Steps f and g are repeated, propagating the failures
back and forth, until the cascade cannot further evolve. The remaining
connected components at the end of the propagation of failures coincide with
the mutually connected components introduced in Sec IV.5. Figures from
[buldyrev2010catastrophic].
Despite the fact that the dynamical propagation of cascades needs a more
convoluted mathematical treatment than percolation, these techniques have been
very flexible at adapting to variations of the original work of Buldyrev, so
that the propagation of cascades is modeled in more realistic scenarios. We
briefly review some of them in the following, and refer the interested reader
to the more complete reviews [kenett2014network, shekhtman2016recent,
valdez2020cascading].
After [buldyrev2010catastrophic], the problem of dependency-based cascades has
been generalized to $L$ layers in [gao2012networks], allowing analytical,
closed solutions in certain interdependent setups. The problem of failure
propagation in multiplexes with partial interdependency, where not all nodes
have dependency connections, has been addressed in [Parshani2010reducing]. If
the fraction of interdependent nodes is decreased enough, the transition is no
longer abrupt but becomes second-order. In order to narrow the range of
parameters for which the abrupt transition occurs, different strategies to
choose the autonomous nodes, those without interdependencies, have been
proposed [schneider2013towards]. It turns out that selecting those with the
highest degree or highest betweenness, significantly reduces the likelihood of
an abrupt collapse [schneider2013towards]. Intra-layer correlations can be
included in the analysis as well. In [huang2013robustness], interdependent
multiplexes with a tunable average number of single links and an average
number of triangles per node are analyzed, finding that, for fixed average
degree, the higher clustered networks are less robust than those with lower
clustering. Intra-layer correlations might be coupled as well to inter-layer
ones: in [reis2014avoiding] a flexible model including both types of
correlations is studied, and the maximization of robustness is addressed as a
function of the tunable strength of the correlations and the failure
propagation dynamics, showing that the theoretical results match well with the
experimental results of coupled functional brain modules.
Beyond cascades driven by topological failures, there are other mechanisms via
which small perturbations can drive a multilayer network to collapse. In many
problems related to social sciences, such as the adoption of fads, the
diffusion of norms and innovations, and the changes in the collective
attention in a population, models of behavioral contagion are used to model
the decisions of the agents. Agents need to decide between two alternative
options/actions and the influence of their neighbors is crucial in the final
choice. A simple way to encompass this influence is by setting an activation
threshold. Initially all nodes are inactive but a controlled small fraction.
An inactive agent changes her state and becomes active only when the fraction
of her active neighbors is larger than a threshold. This process might require
several activation steps before reaching a frozen state. In [watts2002simple]
was given the range of parameters for which such global cascades, measured as
the number of active users when the process stops, emerge in monoplexes,
turning out that they are only possible when the network is neither too sparse
nor too dense. The generalization of this threshold model in multiplexes was
addressed in [brummitt2012multiplexity], where a propagation rule is proposed
such that an agent activates as far as the fraction of active neighbors in at
least one layer exceeds the threshold. Following this rule, multiplexity
widens the ranges of parameters in which global cascades can occur, therefore
increasing the network’s vulnerability. Interestingly, isolated layers in
which, owing to their topological properties, global cascades would not be
observed, when multiplex-coupled, cooperate in a way that facilitates agent
activation and therefore lead to cascades.
Figure 48: The power grid is a paradigmatic example of a network susceptible
to overload cascades of failures due to the energy transported along the
links. We see on the top an aggregated representation of the multilayer power
grid of the USA, where each layer (link color) corresponds to different
voltage ranges. In red is depicted the planned interconnections to transport
wind power. On the bottom, the Bak-Tang-Wiesenfeld sandpile model is simulated
in a duplex of random regular graphs. For a cascade starting in one of the
networks (network $a$), is shown the probability of observing a final cascade
size in $a$, $T_{aa}$, and in network $b$, $T_{ab}$, as a function of the
probability of interconnections $p$ between nodes in $a$ and in $b$. Both
$T_{aa}$ and $T_{a}=1/2(T_{aa}+T_{ab})$ display a minimum, indicating that an
isolated network can reduce the damage caused by a cascade by interconnecting
to another network, but only up to a certain level of interconnectedness
$p^{*}$. Figures readapted from [brummitt2012suppressing]. Figure 49:
Robustness of multilayer networks exposed to overload failures due to nonlocal
load redistribution. We show the behavior of $\langle S\rangle$, the size of
the largest connected component of the network at the end of the cascade, as a
function of the tolerance parameter $\alpha$ (see text). The faster $\langle
S\rangle$ grows, the more robust is the system because there the range of
tolerance values for which it disintegrates is smaller. Three mechanisms to
increase the robustness are presented. In $(a)$, the comparison between the
multilayer network and its aggregated counterpart. In $(b)$, dependence on the
number of layers. In $(c)$, $\langle S\rangle$ for different values of the
multiplexity parameter (fraction of nodes participating simultaneously in a
duplex), with the value indicated in the legend. Figure readapted from
[artime2020abrupt].
Another mechanism that can induce cascades is load redistribution due to
overloads. Arguably, the most famous stylized model accounting for this type
of dynamics is the Bak-Tang-Wiesenfeld sandpile model [bak1988self], a
paradigmatic example of self-organized criticality used in a variety of
contexts [jensen1998self]. Each node has an internal, discrete variable, the
load. At each unit of time the load of a node selected uniformly at random
increases in one unit. When the load reaches a threshold, assumed to be
degree, the node redistributes its load to its neighbors. The neighbors might,
in turn, exceed their capacity and reallocate their load to their own
neighbors, hence propagating the cascade. Once all nodes have their own load
below the threshold, the random addition of load again is restarted. To avoid
inundation of the system, a frequently used strategy is to dissipate load at a
certain rate when it is reallocated. In [brummitt2012suppressing], the BTW
model is used in the context of multilayer power grids, shedding light on the
benefits and dangers of the level of interconnectivity between the layers.
Surprisingly, it is found that there is an optimal value of the
interconnectivity for which the chance to observe large cascades is minimum,
see Fig. 48. This is because the addition of interlayer links between highly
isolated power grids helps mitigate the cascades by creating a reservoir that
absorbs excess load, a sort of alternative dissipative mechanism. This
behavior, though, is valid up to a certain point, from which the further
addition of interlayer links is no longer beneficial since it creates more
loaded systems due to larger thresholds and, therefore, chances of larger
cascades, as well as it creates a positive feedback due to the reentrant new
paths of load into a layer. Regarding the critical properties of the BTW
model, it has been found that multiplexity does not alter the mean-field
scaling behavior observed in monoplexes [lee2012sandpiles].
To close this section, we discuss the relevant, but still quite unexplored,
case of nonlocal cascade propagation. One might argue that nonlocality can be
realized in spatially extended systems by allowing interlayer dependencies
between different locations of the space [li2012cascading]. Yet, under this
description the failures propagate via first neighbors, in the interdependent
sense. In fact, all types of cascades discussed earlier evolve locally via
first-neighbors, something that need not to be true in real systems, as
happened in the 1996 disturbance of the Western Systems Coordinating Council
(WSCC) system [NERC2002system], in the 2003 blackout in the northeastern USA
[nerc2004technical] or in the air-traffic disruption due to the eruption of
the Icelandic volcano Eyjafjallajökull [eurocontrol2010ashcloud]. A plausible
description of this phenomenon is based on the load-capacity model of Motter
and Lai [motter2002cascade], where a load, defined as the number of shortest
paths crossing a node, and a constant capacity, a factor $1+\alpha$ larger
than the initial load, are assigned to every node. When the load of a node
exceeds its capacity, the node fails. An initial perturbation in the form of
node removal is applied to the network, that globally modifies the loads and
allows subsequent failures to occur not necessarily close to the prior
failure. If during this process nodes that overload, they also fail,
propagating the cascade. This model has been investigated recently in
[artime2020abrupt], with the finding that the size of the largest connected
component at the end of the cascade suffers an abrupt jump when the tolerance
parameter $\alpha$ is increased. Moreover, since the load redistribution
depends significantly on the topology of the network, the average path length
is identified as a metric that correlates well with robustness. Based on this,
the article proposes different strategies to increase the robustness of the
network, such as adding new layers or reducing the level of multiplexity (see
Fig. 49). Unlike the other cascades phenomena introduced in this section,
which can be analytically treated with generating functions or multi-type
branching processes, nonlocal cascades do not have a solid mathematical
machinery to be described, and this certainly represents a challenge for the
future.
## V Frontiers
### V.1 Kinematic geometry
The geometric approach [boguna2020network] to network analysis has garnered
growing interest in the past two decades. Here we focus on the geometry of
network-driven processes on multilayer networks, a family of kinematic
geometries generalizing the diffusion geometry introduced in
[de2017diffusion].
The structure of a wide variety of real-world complex systems is modular and
hierarchical [guimera2005functional] and the effect of these large scale
properties on the dynamics of such systems has been studied, during the past
decade. It has been shown that complex systems with such a mesoscale
organization [fortunato2010community] are characterized by topological scales
[arenas2006synchronization], exhibiting the emergence of functional clusters
which might be different from topological ones.
In [de2017diffusion] the authors investigate the multiscale functional
geometry of monoplexes to characterize functional clusters. This approach
defines the diffusion distance between any pair of units in a networked
system, based on random walk dynamics, shown to correspond, among others, to
the phase deviation of coupled oscillators close to metastable synchronization
state and consensus dynamics. The diffusion distance for single-layer networks
is a key concept to define a kinematic geometry based on network-driven
processes and it can be calculated as the $L^{2}-$norm of the difference
between rows of the propagator $e^{-t\tilde{\mathbf{L}}}$:
$D_{t}(i,j)=\left\lVert\mathbf{p}(t|i)-\mathbf{p}(t|j)\right\rVert_{2}.$ (92)
Exploiting the fact that diffusion geometry is based on random walk dynamics,
it is possible to extend it to the realm of multilayer networks. At variance
with walks on edge-colored networks presented in Sec. IV.1, where one can
obtain the transition rules for the multigraph as a weighted average of the
transition probabilities in each distinct layer, see Eq. (61), on a multilayer
network the walk type determines the probability of a random walker jumping
across and within layers (see Eq. 62 and Tab. 1). Consequently, in the edge-
colored case diffusion distances between nodes can be obtained directly from
Eq. (92), while for multilayers we have to introduce a diffusion distance
between state nodes.
Regardless of the random walk type, let us indicate the probability of finding
a random walker at a given node and layer at time $t$ by $p_{j\beta}(t)$.
Similarly to Eq. (92) we define the diffusion distance between state nodes
$(i,\alpha)$ and $(j,\beta)$ as
$D_{t}^{2}((i,\alpha),(j,\beta))=\sum\limits_{k,\gamma}(p_{k\gamma}(t|(i,\alpha))-p_{k\gamma}(t|(j,\beta)))^{2},$
(93)
where probabilities are conditional to using the corresponding state nodes as
the origins of random walkers at time $t=0$. One can summarize this supra-
distance matrix $\mathbf{D}_{t}=\left(D_{t}((i,\alpha),(j,\beta))\right)$ into
an $N\times N$ matrix by encoding the diffusion distance among the physical
nodes, across all the layers. Intuitively, resembling the parallel sum of
resistances in electrical circuits leading to the equivalent resistance, the
_equivalent diffusion distance_ can be written as
$D_{t}^{\text{eq}}(i,j)=\left(\sum_{\alpha=1}^{L}\frac{1}{D_{t}((i,\alpha),(j,\alpha))}\right)^{-1}.$
(94)
Usually, the functional distance (supra-)matrices are rescaled in $[0,1]$,
normalizing each by its maximum value to allow for comparisons. If,
additionally, one is interested in the most persistent patterns, the diffusion
distance is averaged over time and called the average diffusion distance. The
corresponding supra-distance matrix is indicated by $\bar{\mathbf{D}}_{t}$.
The diffusion distance between nodes is highly dependent on the type of random
walk dynamics, its propagation time, the topology of layers and the layer-
layer correlations. To better understand the effects of dynamics and
topological variations, let us consider three classes of two-layer networks,
namely:
* •
Barabasi-Albert layers with preferential attachment of 4 links;
* •
Watts-Strogatz layers, with rewiring probability 0.2;
* •
Girvan-Newman model layers, where the inter-community connectivity probability
is 1, whereas cross-group connections exist with probability 0.05.
Additionally, we consider the five random walks of Tab. 1. For the three
cases, the link overlap across layers – i.e., the fraction of links present in
both layers among the same pairs of nodes [de2015structural]— is fixed to 10%.
As shown in Fig. 50, the diffusive walk on a scale-free topology leads to a
high level of mixed pathways across layers, while in small-world systems,
nodes tend to shape more distinct functional clusters.
Figure 50: Average diffusion distance supra-matrices $\bar{\mathbf{D}}_{t}$
for different combinations of multilayer topologies and random walk dynamics
(see the text for details). Figure reprinted with permission from
[bertagnolli2020diffusion]. Copyright (2021) by the American Physical Society.
The functional geometry framework has also been used to analyze empirical
systems in [bertagnolli2020diffusion], where the authors highlighted
dissimilarities in the diffusion spaces of the public transportation of London
and of the social network of Noordin terrorists, by looking at the projections
of the spaces and at the results of a Mantel’s test [mantel1967detection]
applied to supra-distance matrices.
Finally, it is worth mentioning that one can use other walks instead of random
ones: this is the case of walks that result from powers of the adjacency
matrix and provide the basis for _communicability_ , a measure recently used
to introduce a geometric framework with applications to single and multiplex
networks [Estrada2019].
### V.2 Statistical physics of multilayer systems
#### V.2.1 Classical ensembles
To study the properties of an observed multilayer network, comparison with
null models is often essential. The maximum entropy approach is one of the
standard ways to obtain the required null models, in terms of ensembles of
networks exhibiting one (or more) specific property of the observed network,
while being maximally random with respect to the other properties. For
instance, in the case of single layer networks, one of the most famous null
models is the configuration model (CM) [molloy1995critical] which is an
ensemble of networks with the same degree sequence as the observed one. The
unbiased probability distribution of the members of this ensemble is indicated
as $P(G)$ and must maximize the Shannon information entropy
$S=-\sum\limits_{G\in\mathcal{G}}P(G)\log{P(G)}.$ (95)
It is worth mentioning that such a probability distribution does not
correspond to a physical process related to the second law of thermodynamics.
However, the discussed entropy maximization approach is mimicking the
mathematical machinery of statistical physics and, in the case of a fixed
degree sequence, it leads to a microcanonical ensemble where all members of
the CM have equal probability.
Depending on the type of study, the mentioned constraint of a fixed degree
sequence can be relaxed to obtain other ensembles [cimini2019]. For instance,
by fixing the average degree instead of the full degree sequence, we achieve
the grand-canonical ensemble of random graphs. Furthermore, if our knowledge
is limited to the degree distribution from which the degree sequence is
sampled, one obtains the hypersoft configuration model
[Krioukov2020MaxEntropy] which is a hypercanonical ensemble of random networks
all drawn from the fixed distribution (see Fig. 51).
Figure 51: Schematic of sampling the network, represented by $G$, from
hypercanonical, grandcanonical, and microcanonical ensembles discussed in the
text. Figure from [Krioukov2020MaxEntropy].
Similarly, maximum entropy approaches have been used to study ensembles of
non-interconnected multiplex networks. Assume an edge-colored multigraph $M$
with $L$ layers, each forming a network $G^{(\ell)}$ ($\ell=1,2,...N$).
Remarkably, when there is no correlation between the layers, the probability
of observing $M$ can be obtained as the product of the probabilities of the
layers:
$\displaystyle P(M)=\prod\limits_{\ell=1}^{L}P(G^{(\ell)})$ (96)
and the corresponding Shannon entropy becomes the summation of Shannon
entropies of layers [bianconi2013statistical]:
$\displaystyle S=\sum\limits_{\ell=1}^{L}S^{(\ell)}.$ (97)
By imposing the soft constraints – e.g., fixing the average degree instead of
degree sequence – and using the Lagrangian multipliers method, one can obtain
the canonical multiplex ensemble with probability $P_{C}(M)$ maximizing the
Shannon entropy:
$\displaystyle
P_{C}(M)=\frac{e^{-\sum\limits_{\mu}\lambda_{\mu}F_{\mu}(M)}}{Z_{C}}=\prod\limits_{\ell=1}^{L}P_{C}(G^{(\ell)}),$
(98)
where $\lambda_{\mu}$ corresponds to the Lagrangian multipliers and
$F_{\mu}(M)$ determine the constraints on the network (e.g., average degree).
Note that, in presence of layer-layer correlations, we have
$P(M)\neq\prod\limits_{\ell=1}^{L}P(G^{(\ell)})$ leading to other probability
distributions extensively studied in [bianconi2013statistical].
#### V.2.2 Quantum-like ensembles
Complex systems include a wide variety of physical attributes and dynamics.
The interactions between their units vary, from the electrochemical signals
traveling among neurons in the human brain to the transport of goods between
different areas of a urban ecosystem and the spreading of an infectious
pathogen between individuals of a society. Regardless of the nature of these
interactions, they can be described as information exchange. In fact, complex
systems resemble one another in the way they handle information. Therefore, to
understand how these systems operate, from a physical point of view,
investigating their information dynamics is crucial.
It is important to note that the information flow between the units is
regulated by the underlying structure, depending on the local neighborhood in
certain classes of networks and on long-range communication between distant
components in other topologies. At the same time, it is essential to consider
the coupling between the structure and the dynamical processes governing the
flow of information, such as diffusion, random walks, synchronization and
consensus and, since it is often difficult to understand a system in terms of
microscopic features, a framework providing a statistical description of the
system might be relevant for applications.
Recently, a statistical field theory of complex information dynamics has been
introduced to unify a range of dynamical processes governing the evolution of
information on top of static or time-varying structures [de2020SFT]. This
framework describes the interactions among the units in terms of _information
streams_ , a set of operators that determines the direction of information
flow and provides a statistical ensemble to construct the statistical physics
of complex information dynamics. In fact, the formalism allows for defining
density matrices— i.e., statistical average of streams— from which a variety
of descriptors can be derived. Of course, density matrix formalism comes from
quantum mechanics, where vectors were found insufficient to represent the
mixed states and encode the pairwise coherence between quantum states.
Similarly, properties of networked systems cannot be fully described by
vectors or distribution functions, without information loss. For instance, a
system’s structure is often encoded in two dimensional data structures, being
the adjacency matrix, except for trivial symmetries like chains or paths.
Therefore, the counterpart of the quantum density matrix has been introduced
for complex networks, where off-diagonal elements provide a proxy for
interactions between the nodes, instead of the coherence between the states.
So far, this framework has only been used to analyze classical networks. In
the following, we provide a direct application of the theoretical framework to
multilayer systems.
Figure 52: Schematic illustration of the spectral entropy for different
classes of networks has been presented. Remarkably, the entropic distances
provided by the framework can be used to compare network similarity and
characterize the network topology with high accuracy. Figure from
[de2016spectral].
##### Redundancy and reducibility of multilayer systems
As discussed previously, the constituents of complex systems must exchange
information efficiently in order to function properly. However, a deep
understanding of the structure’s role in enhancing or hindering the transport
properties – such as navigability [boguna2009navigability] – continues to
elude us, especially in the case of multilayer systems [gao2012networks,
de2013mathematical, gomez2013diffusion, radicchi2013abrupt,
kivela2014multilayer, boccaletti2014structure, de2016physics] where an
efficient flow might be hindered by the lack of synchronization between
different layers [gallotti2014anatomy] or the redundancy of pathways across
the layers. There are a number of methods to reduce multiplex networks by
structurally merging their most similar layers, based on information-theoretic
frameworks [de2015structural]. Despite their success, these methods are mostly
based on heuristics and have been proved inaccurate under specific
circumstances [de2020EnhanceTransport].
Enhancing the flow distribution in multilayer systems is challenging, since
adding links to the layers (e.g., highways, tubes, flights, synapses, etc.)
comes with a cost. Interestingly, for multilayer networks with interlinks,
changing the weights of interlinks can enhance the diffusion on top of the
networks [gomez2013diffusion]. When acting on the structure is not an option,
it has been shown that one can still enhance the transport properties (see
Fig. 53) using the statistical physics of complex information dynamics by
coupling layers dynamically, in a way that a dynamical process can not
distinguish the functionally coupled layers – e.g., airlines proposing shared
flights to their customers – while evolving. The functional reduction includes
coupling the layers with high similarity that are responsible for the
redundant diffusion pathways in the system, in order to identify the (sub)set
with maximally diverse layers [de2020EnhanceTransport].
Figure 53: Schematic illustration comparing structural against functional
reduction of a multilayer network consisting of $L=4$ layers. The procedure is
similar for both approaches, but the relevant difference is that while
structural reducibility alters the topology of the system, function
reducibility allows us to functionally couple layers without altering their
structure. Figure from [de2020EnhanceTransport].
Diffusive processes, such as random walks, have been used extensively to model
the information transport within complex structures [masuda2017random]. Here,
we consider random walk dynamics governed by the normalized Laplacian
$\hat{\tilde{\mathbf{L}}}=\langle\hat{\tilde{\mathbf{L}}}^{(\ell)}\rangle$
given by
$\hat{\tilde{\mathbf{L}}}=\hat{\mathbf{I}}-\langle\hat{\mathbf{T}}^{(\ell)}\rangle$
playing the role of the quasi-Hamiltonian (see Eq. (61)). The density matrix
can be obtained for random walks on multiplex networks as
$\hat{\boldsymbol{\rho}}(t)=\frac{e^{-t\hat{\tilde{\mathbf{L}}}}}{Z(t)}$.
Interestingly, it has been shown that the partition function, which is
encoding the amount of the trapped field, is proportional to the average
return probability of random walk dynamics: $Z(t)=N\mathcal{R}(t)$, where
$\mathcal{R}(t)=N^{-1}\sum\limits_{i=1}^{N}e^{-t\lambda_{i}}$ is the average
return probability and $\lambda_{i}$ is the $i$–th eigenvalue of
$\hat{\tilde{\mathbf{L}}}$. Intuitively, the average return probability is
high when the structural symmetries and abundance of redundant diffusion
pathways slows down the information propagation between the units. Thus, it is
expected that breaking the structural regularities by adding long-range
interactions [watts1998collective] or increasing the diversity of diffusion
pathways across layers can lead to faster information flow and and lower
$Z(t)$.
Multiplexity of interactions among units generates non-trivial dynamical
correlations between layers that have no counterpart when layers are
considered in isolation. This important difference can be characterized in
terms of average entropy distance between the multiplex and its layers. This
measure, named intertwining, is defined by
$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\langle\mathcal{D}_{KL}(\hat{\boldsymbol{\rho}}||\hat{\boldsymbol{\rho}}^{(\ell)})\rangle=\frac{1}{L}\sum\limits_{\ell=1}^{L}\mathcal{D}_{KL}(\hat{\boldsymbol{\rho}}||\hat{\boldsymbol{\rho}}^{(\ell)})$
(99)
where
$\mathcal{D}_{KL}(\hat{\boldsymbol{\rho}}||\hat{\boldsymbol{\rho}}^{(\ell)})=Tr[\hat{\boldsymbol{\rho}}(\log_{2}\hat{\boldsymbol{\rho}}-\log_{2}\hat{\boldsymbol{\rho}}^{(\ell)})]$
is the quantum-like Kullback-Leibler (KL) divergence between layer $\ell$ and
a multiplex system as a whole.
Directly from intertwining, a fundamental inequality between the partition
function of a multiplex system as whole and the partition functions of its
layers can be derived. This inequality is important, as it relates the
transport phenomena of multiplex system and layers, through average dynamical
trapping (i.e., the partition function):
$\displaystyle Z(t)\leq\prod\limits_{\ell=1}^{L}Z^{(\ell)}(t)^{1/L},$ (100)
where equality holds if and only if all the layers are the same. Using
dynamical trapping as a measure of transport, Eq. (100) shows that a multiplex
network has better transport properties than the geometric average of layers,
adding an advantage to multilayer structures.
Furthermore, being reminiscent of statistical physics of particles, the
equality defines the non-interacting scenario where layer-layer correlations
do not alter the underlying dynamics: the entropy $S^{(\ell)}(t)$ of each
layer is calculated separately and the overall entropy is given by their
average $\mathcal{S}_{nint}(t)=\langle S^{(\ell)}(t)\rangle$. Conversely, any
topological alteration of the non-interacting scenario introduces a dynamical
correlation between layers, requiring the exploration of layers to gather more
information about the system: in this case the network consists of interacting
layers, where the diffusion dynamics on the whole multiplex network is
considered to measure the entropy $\mathcal{S}_{int}(t)$. To obtain another
form of Eq. (99), a mean-field approximation of the Von Neumann entropy is
given by
$\displaystyle\mathcal{S}^{MF}(t)=\frac{1}{\log
2}\left(t\frac{Z(t)-1}{Z(t)}+\log Z(t)\right),$ (101)
and can be used to prove that layer-layer interactions lower the system’s
entropy ($\mathcal{S}_{int}(t)\leq\mathcal{S}_{nint}(t)$) and partition
function $Z(t)$. Normalizing the intertwining by its upper bound, for values
of time $t$ sufficiently large and in absence of isolated state nodes, Eq.
(99) reduces to the relative intertwining:
$\displaystyle\mathcal{I}^{*}(t)=1-\frac{\mathcal{S}_{int}(t)}{\mathcal{S}_{nint}(t)},$
(102)
which is bounded between 0 (i.e., the layers are redundant) and 1 (i.e., the
layers are diverse and the system is irreducible).
We can show how intertwining is proportional to the functional diversity of
layers. The Laplacian matrix of the multiplex network is given by
$\hat{\tilde{\mathbf{L}}}=\langle\hat{\tilde{\mathbf{L}}}^{(\ell)}\rangle$.
Therefore, the Laplacian matrix of each layer
$\hat{\tilde{\mathbf{L}}}^{(\ell)}$ can be written as a perturbation of
multiplex Laplacian
$\hat{\tilde{\mathbf{L}}}^{(\ell)}=\hat{\tilde{\mathbf{L}}}+\Delta\hat{\tilde{\mathbf{L}}}^{(\ell)}$,
reflected in its eigenvalues as
$\lambda_{i}^{(\ell)}=\lambda_{i}+\Delta\lambda_{i}^{(\ell)}$ $(i=0,1,...N)$.
It is straightforward to show that
$\frac{1}{N}\sum\limits_{i=1}^{N}\Delta\lambda_{i}^{(\ell)}=\overline{\Delta\lambda^{(\ell)}}=0$
and that $\overline{\Delta\lambda^{(\ell)^{2}}}\geq 0$, the latter quantifying
the influence of the perturbation to each layer. The average of the variance
across all layers $\overline{\langle(\Delta\lambda^{(\ell)})^{2}\rangle}$
provides a measure of the overall spectral diversity of layers which,
interestingly, is demonstrated to be proportional to the relative
intertwining:
$\displaystyle\mathcal{I}^{\star}(t)$ $\displaystyle\approx$
$\displaystyle\frac{t^{2}}{2}\overline{\left\langle(\Delta\lambda^{(\ell)})^{2}\right\rangle}.$
(103)
This proves the sensitivity of intertwining to the spectral diversity of
layers. Furthermore, the partition functions of layers can be written in terms
of perturbations such as $Z^{(\ell)}(t)=Z(t)+\Delta Z^{(\ell)}(t)$, leading to
$\displaystyle\mathcal{I}^{\star}(t)$ $\displaystyle\approx$
$\displaystyle\frac{\overline{\Delta Z^{(\ell)}}(t)}{Z(t)-1};\
\overline{\Delta Z^{(\ell)}}(t)=\frac{1}{L}\sum\limits_{\ell=1}^{L}\Delta
Z^{(\ell)}(t).$ (104)
Equations (103) and (104) provide a fundamental result: they show that by
minimizing the partition function of the system one maximizes the relative
intertwining while favoring the maximum functional diversity of layers.
Additionally, it has been shown that an inverse proportionality holds between
the diffusion time ($1/\lambda_{2}$) and intertwining, as further evidence for
the role of intertwining in characterizing the transport properties.
These theoretical findings have been applied to a broad range of synthetic and
empirical systems, providing a transparent framework for coupling the most
similar layers of multiplex networks, in order to improve their transport
properties including dynamical trapping, diffusion time, and navigability
[de2020EnhanceTransport].
## VI Conclusions
Network Science is one of the greatest achievements of the 21st century,
paving the way toward a mathematical approach for the analysis of disparate
complex systems, and allowing us to find regularities in apparently disordered
connectivity patterns. The past decade has seen the flourishing of analytical
techniques and models exploiting, or characterizing, the inherent
multidimensionality of empirical systems, from multiplexity – i.e., the
existence of distinct types of relationships among the same set of actors or
units – to interdependency – i.e., the existence of structural or functional
connections among sets of actors or units of a different nature. Such multiple
dimensions are nowadays easily encoded into layers of information.
Figure 54: Multilayer representation of a cell. Layers: i) regulatory
interactions involving RNA and protein expression, ii) protein-protein
interactions involved in signaling and responsible cell function, iii)
metabolic interactions with reactions and pathways crucial for cell function.
Figure from [vermeulen2020exposome].
Most of the advances in this direction are described in some detail or
referred to in this work. Starting from the mathematical representation of
multilayer networks (Chap. 2) we have introduced structural descriptors for
units, layers and the whole system (Chap. 3), providing an overview of the
micro- and meso-scale organization of such systems. We have discussed the rich
spectrum of phenomena, with no classical counterparts, related to dynamical
processes on the top of the networks and their intertwining (Chap. 4), guiding
the reader towards two promising research areas for the future, namely network
geometry and information dynamics (Chap. 5), although many other exciting sub-
field are emerging, e.g., higher-order modeling and analysis
[lambiotte2019networks, battiston2020networks].
At this point, the reader should be sufficiently familiar with multilayer
network science and, to conclude, we would like to make a quick journey
through the most recent applications of its paradigm, moving across different
spatial scales and ranging from cells to societies.
The first stop of this journey is exactly a cell, which can vary in diameter
between $10^{-6}$ m and $10^{-4}$ m (note that a DNA double helix is about
$10^{-8}$ m wide). The cell can be seen as a multilayer system consisting of
three interdependent layers (see Fig. 54). Here, applications are mostly
related to the emerging field of systems biology and network medicine,
promising to use biomolecular interactions to develop a deeper knowledge of
biology across scales with the ultimate goal to better understand diseases, to
prevent them, and to treat them with medical drugs that reduce side effects.
The second stop requires a big jump of about three orders of magnitude, to
explore one of the most famous multicellular organisms with a small-scale
neural system: the Caenorhabditis elegans, a nematode of about $10^{-3}$ m.
The nervous system of this small worm consists of synaptic and neuropeptide
interactions which can be seen as interconnected layers. Here, the multilayer
perspective provides the opportunity to better understand how the integration
between hard-wired synaptic or junctional circuits and extrasynaptic signals
can modulate the large-scale behavior of the worm [bentley2016multilayer].
At a larger scale, around $10^{-1}$ m, another emblematic neural system,
namely the human brain, is currently being characterized in terms of how its
structural and functional connectivity evolves over time, e.g., while
performing a specific task, or across groups, unraveling the existence of
modular and hierarchical structures which would remain hidden under the lens
of less sophisticated models and analytical techniques [bassett2017network].
In parallel, the functional connectivity of the human brain can be stratified
by frequency bands (see Fig. 55) where specific correlations or causal
relationships between regions of interests appear [de2017multilayer].
Multilayer analysis can be used to better characterize the relative importance
of all brain regions [williamson2021multilayer] and to enhance the accuracy in
discriminating between healthy and unhealthy patients starting from imaging
information [de2016mapping].
Figure 55: Multilayer representation of a human brain. Both the 3D and layered
visualization encode the functional brain of a schizophrenic subject (11 non-
overlapping frequency-band layers between 0.01 and 0.23 Hz). Figure from
[de2017multilayer], readapted from [de2016mapping].
At larger scales, on the order of hundreds of meters ($10^{2}$m), cooperative
systems like that of dolphins, can be characterized with respect to distinct
types of interactions allowing us to get unprecedented insights about the
underlying social organization and group dynamics. At similar spatial scales,
another emblematic example, accounting for the socio-spatial interdependence
typical of many other networks, concerns the organization of ecological
systems (Fig. 56): for instance, it has been recently shown that the
importance of dispersers for an ecosystem is better captured by multilayer
measures of importance rather than by standard metrics
[timoteo2018multilayer].
Figure 56: Aggregate (top) and multilayer (bottom) representations of a real
seed-dispersal network. Reproduced from [timoteo2018multilayer] under Creative
Commons Attribution 4.0 International License.
At the human scale, relatively small-scale social systems
($10^{4}\leavevmode\nobreak\ $m) have been studied to better understand the
impact of external shocks on social structure and dynamics. For instance, it
has been shown that multilayer modeling can disentangle the different social
ties that conform to the proximity interaction networks of extant hunter-
gatherer societies, identifying social relationships with a key role in the
spread and accumulation of culture [migliano2017characterization]. In another
study, the analysis of mixed economies in three villages of Alaska unraveled
that factors related to climate change, such as global warming, have a non-
negligible effect on household structure, but the most important factors for
vulnerability are indeed due to social shift rather than resource depletion
[baggio2016multiplex].
Let us jump by three more orders of magnitude and discuss systems at the
planetary scale: the ones based on information exchange thanks to large-scale
communication infrastructures, such as the Internet. At this scale the
activity of a complex system might be very frenetic; think about an online
social media platform, where millions or billions of users worldwide
continuously produce content to be shared, which quickly travels and bounces
from one country to another. For instance, it is interesting to study how
rumors and memes spread on these systems, as recently proposed in
[d2019spreading] to capture the behavior of users who post information from
one social media platform to another and to provide a plausible explanation
for the heavy-tailed distributions of meme popularity that is usually observed
in empirical data.
Recent applications also include the analysis of trade networks and their
nested and modular structure [torreggiani2018identifying, a2018unfolding,
alves2019nested]. Additionally, as anticipated, multilayer models of financial
networks have been proposed very recently, to better explain how the financial
distress of one country can ignite a global financial crisis, like the one in
2008. By considering distinct asset types as layers where countries (nodes)
exchange financial assets (links), [del2020multiplex] have highlighted the
importance of both intra-layer and inter-layer connectivity contributions to
the propagation of contagions (Fig. 57).
Figure 57: Multilayer representation of a global financial network. Layers are
asset types, nodes are countries and links are cross-country financial
relationships. Figure from [del2020multiplex].
The long journey summarized in the last few pages of this work allowed us to
stress the importance of multilayer modeling across a broad spectrum of
disciplines, including cell biology, neuroscience, ecology and social
sciences, spanning 10 orders of magnitude from a spatial scale of about
$10^{-6}$ m to one of about $10^{4}$ m. The theoretical and computational
framework presented in this work provides researchers and practitioners with a
versatile and unified tool kit to shed light and gain new insights on the
complexity of natural and artificial systems.
## Appendix A Master Stability Function (MSF) formalism
To test the stability of the synchronized state s we study how the
perturbation error $\delta\textbf{x}_{i}(t)=\textbf{x}_{i}(t)-\textbf{s}(t)$
evolves. By assuming small perturbations, we can write the variational
equation for $\delta\textbf{x}_{i}$:
$\delta\textbf{x}_{i}=J\textbf{F}(\textbf{s})\delta\textbf{x}_{i}-\sigma
J\textbf{H}(\textbf{s})\sum_{j=1}^{N}L_{ij}\delta\textbf{x}_{j}$ (105)
where $J\textbf{F}$ and $J\textbf{H}$ are the Jacobian of F and H,
respectively.
To find the solution of Eq. (105) we have to project $\delta\textbf{x}$ into
the eigenspace formed by the eigenvectors of the Laplacian matrix L, obtaining
a decomposition of the time evolution of perturbation error into $N$ decoupled
eigenmodes:
$\dot{\boldsymbol{\xi}_{i}}=[J\textbf{F}(\textbf{s})-\sigma\lambda_{i}J\textbf{H}(\textbf{s})]\boldsymbol{\xi}_{i},\quad
i=1,...,N$ (106)
where $\boldsymbol{\xi}_{i}$ is the eigenmode associated with the eigenvalue
$\lambda_{i}$ of L. By indicating as $\Lambda_{max}$ the maximum Lyapunov
exponent associated with the system of equations (106), then we can write the
time evolution of $\boldsymbol{\xi}$ as $|\boldsymbol{\xi}|\sim
e^{\Lambda_{max}t}$ and, finally, we found that a necessary condition for the
stability of a synchronized state is that $\Lambda_{max}<0$. The expression of
$\Lambda_{max}$ as a function of a generic parameter
$\alpha_{i}=\sigma\lambda_{i}$ is named the Master Stability Function (MSF),
and it usually assumes negative values in an interval
$\alpha\in(\alpha_{1},\alpha_{2})$. It means that, for a fixed coupling
strength $\sigma$, a network can reach and maintain complete synchronization
only if its structure, defined by its Laplacian matrix, is such that:
$\alpha_{1}<\sigma\lambda_{2}\leq\sigma\lambda_{\leq}...\leq\sigma\lambda_{N}<\alpha_{2}.$
(107)
or, equivalently,
$R\equiv\frac{\lambda_{N}}{\lambda_{2}}<\frac{\alpha_{2}}{\alpha_{1}},$ (108)
used, for instance, in [sole2013spectral] to unravel the existence of an
optimal value for the synchronizability of a multilayer system.
## Appendix B Kuramoto model on networks
The first approach to model collective synchronization considers a population
of coupled limit-cycle oscillators whose natural frequencies are drawn from
some prescribed distribution and exert a phase-dependent influence on each
others. In formulae, we can write the Kuramoto model (KM) [Strogatz2000,
Arenas2008] as a system of $N$ oscillators, whose instantaneous phases
$\theta_{i}$ are described by the equation:
$\dot{\theta_{i}}=\omega_{i}+\frac{\sigma}{N}\sum_{j=1}^{N}\sin(\theta_{j}-\theta_{i}),$
(109)
where $\sigma$ is the coupling constant and $\omega_{i}$ is the natural
frequency of oscillator $i$, chosen from an unimodal distribution $g(\omega)$.
We can use a order parameter that describes the transition: it is defined as a
macroscopic measure that quantifies the collective rhythm produced by the
whole population:
$r(t)e^{i\Phi(t)}=\frac{1}{N}\sum_{j=1}^{N}\sin(\theta_{j}),$ (110)
where $\Phi(t)$ is the average phase, and $0\leq r(t)\leq 1$ measures the
phase coherence, where the two extreme values correspond to phase locked
($r=1)$ or incoherent oscillators. Equation (109) is then rewritten in terms
of the order parameter, and the equation for the instantaneous phase reduces
to
$\dot{\theta_{i}}=\omega_{i}+\sigma r\sin(\theta_{i}),$ (111)
which has two types of long-term behaviours. Oscillators for which
$|\omega_{i}|\leq\sigma r$ are phase-locked and form a mutually synchronized
cluster. Oscillators with frequencies in the tails of $g(\omega)$
distribution, where $|\omega_{i}|>\sigma r$ holds, are drifting with respect
to the synchronized cluster.
The Kuramoto model is then generalized on networks by including in Eq. (109)
information about network connectivity:
$\dot{\theta_{i}}=\omega_{i}+\sum_{j=1}^{N}\sigma_{ij}A_{ij}\sin(\theta_{j}-\theta_{i}).$
(112)
## Appendix C Transitions in multilayer systems
Type of phase transition | Mode | Reference
---|---|---
Enhanced diffusion | C | [gomez2013diffusion]
Emergence of multiplexity | D | [radicchi2013abrupt]
Table 2: Phase transitions (C: continuous, D: discontinuous, H: hybrid) in
multilayer networks due to the algebraic, topological and simple dynamics
described in Sec. IV.1.
Type of dynamics | Mode | Reference
---|---|---
Synchronization | C | [Gambuzza2015]
Explosive synchronization with local order parameter | D | [Zhang2015]
Synchronization of oscillators with random walks | C & D | [nicosia2017collective]
Interacting diseases | C | [Sanz2014]
Epidemic onset control by raising of awareness | C | [Granell2013]
Explosive pandemics with no early-warning | D | [danziger2019dynamic]
Cooperative behaviour in coupled competitive games – social influence | C | [amato2017interplay]
Cooperative behaviour in Prisoner’s Dilemma game | C | [gomez2012evolution]
Cooperative behaviour in Public Good game | C | [battiston2017determinants]
Table 3: Phase transitions in multilayer networks due to the simple and
couple dynamics described in Sec. IV.2, Sec. IV.3 and Sec. IV.4.
Type of network | Mode | Reference
---|---|---
Multilayer network | C | [leicht2009percolation]
Multiplex | D | [son2012percolation]
Partial Multiplex | C & D | [son2012percolation]
Multiplex with overlap | C, D & H | [hu2013percolation, baxter2016correlated, cellai2016message]
Multiplex with spatially embedded networks | C & D | [son2011percolation, bashan2013extreme, grassberger2015percolation]
Bond Perc. on Multiplex | C | [hackett2016bond]
Multiplex & Local cascades | D | [buldyrev2010catastrophic]
Multiplex & Non-local cascades | D | [artime2020abrupt]
Table 4: Type of transitions for different scenarios in percolation and
cascades propagation.
|
# Guided Sampling-based Evolutionary Deep Neural Network for Intelligent Fault
Diagnosis
Arun K. Sharma Student Member, IEEE and Nishchal K. Verma Senior Member,
IEEE Arun K. Sharma and Nishchal K. Verma are with the Dept. of Electrical
Engineering, Indian Institute of Technology, Kanpur, India. e-mail:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract
The diagnostic performance of most of the deep learning models is greatly
affected by the selection of model architecture and their hyperparameters. The
main challenges in model selection methodologies are the design of
architecture optimizer and model evaluation strategy. In this paper, we have
proposed a novel framework of evolutionary deep neural network which uses
policy gradient to guide the evolution of DNN architecture towards maximum
diagnostic accuracy. We have formulated a policy gradient-based controller
which generates an action to sample the new model architecture at every
generation. The best fitness obtained is used as a reward to update the policy
parameters. Also, the best model obtained is transferred to the next
generation for quick model evaluation in the NSGA-II evolutionary framework.
Thus, the algorithm gets the benefits of fast non-dominated sorting as well as
quick model evaluation. The effectiveness of the proposed framework has been
validated on three datasets: the Air Compressor dataset, Case Western Reserve
University dataset, and Paderborn university dataset.
Nowadays even if there are a lot of advancements in computational
intelligence, intelligent fault diagnosis requires good expertise to design
the diagnostic model. This is because the performance of most of the
diagnostic models depends on model selection and hyperparameters. The proposed
method of guided sampling-based evolutionary deep neural network non-dominated
sorting-based evolutionary algorithm to optimize the model architecture. The
model architecture is sampled by using a reward-based policy to force the
evolutionary algorithm to get a model architecture that can give maximum
accuracy. Therefore, the proposed method is the best solution for the model
architecture selection without any human expertise while ensuring the best
possible diagnostic performance.
Neural architecture search, Intelligent fault diagnosis, Deep neural network,
Non-dominated sorting algorithm, Policy gradient.
## 1 Introduction
With the advancement in modern computational technology, machine learning
based intelligent fault diagnosis has become an integral part of almost all
industrial sectors. Intelligent fault diagnosis refers to the preventive
maintenance of industrial machines using machine learning-based data analysis
and class detection [1, 2, 3, 4, 5, 6]. Data-driven fault diagnosis of
industrial machines faces many challenges: (i) labeled data availability as
running the machine in a faulty state with real-time load is not possible,
(ii) training the deep learning algorithms with limited availability of
labeled dataset, (iii) model selection best suited for the dataset under
consideration. A deep neural network (DNN) has been commonly used for
intelligent fault diagnosis due to its high capability of non-linear feature
transformation [7, 8]. However, training the DNN from scratch for every change
in the operating condition of the machine becomes an uneconomical process. To
solve such problems, many researchers have suggested a variety of domain
adaptation methods[9, 10, 11, 12, 13, 14, 15, 16]. [11], [14] and [15] uses
labeled source dataset from the laboratory machine and the unlabeled target
dataset from the test machine to train a model capable to diagnose the fault
on the test dataset from the target machine. A quick learning mechanism to
train the model in the target domain by transforming an already trained model
on source data. All these methods work well only if a suitable architecture is
selected based on trial and error and some prior experience with the dataset.
In practice, it becomes a big challenge for selecting and training the best
suitable architecture with every change in the dataset due to variations in
machine operating conditions and fault types.
Therefore, our main objective is to investigate and develop an algorithm that
can find a best suitable architecture for the given dataset with limited
computational resources and computational time. As the performance of most of
the deep learning algorithms is very much affected by the hyper-parameters,
research on hyper-parameter optimization or neural architecture search (NAS)
has gained very much attention nowadays [17, 18]. Based on architecture
optimization strategy, NAS methods are mainly categorized as: (i) random
search and grid search [19], (ii) surrogate model-based optimization [20],
(iii) reinforcement learning [21], (iv) genetic algorithm [22, 23, 24, 25,
26], (v) gradient descent [27], and (vi) hybrid algorithms [28]. In most of
the aforementioned methods, the biggest challenge for architecture search is
the model evaluation because of the complex training mechanism for most of the
deep learning models [18]. In genetic algorithm-based NAS methods, the
evolution of architecture takes many days of GPU. For example, regularized
evolution of image classifier [29] with 450 K40 GPU takes 3150 GPU days.
Architecture optimization using non-dominated sorting genetic algorithm II
(NSGA-II [30]) has also gained much attention due to the fast sorting
framework [31, 26]. EvoN2N [26] uses the concept of knowledge transfer for
fitness evaluation in the NSGA-II based framework. The quick fitness
evaluation with fast NSGA-II makes the algorithm faster compared to the state-
of-the-art evolutionary methods. In this paper, we extend the work of EvoN2N
[26] to introduce a guided sampling-based evolution that makes the convergence
faster while exploiting the search space with the help of a reward-based
controller. The key contributions of this work are highlighted below
1. i)
We have formulated the guided sampling-based evolution of DNN architecture
(GS-EvoDNN) using policy gradient.
2. ii)
We have introduced a mean and variance term to exploit the search space for
the model optimization.
3. iii)
We have formulated an update law for mean and variance term using policy
gradient.
4. iv)
We have adopted the knowledge transfer mechanism to initialize the weight
matrices of the model at a generation by using the best model from the
previous generation.
The rest of the article is organized as follows. Table 1 summarizes the
symbols commonly used throughout the literature except index variables on
summation and loop variables in algorithmic steps. Section 2 defines the
objective problem. Section 3 briefly discusses the related works and the
theoretical backgrounds. Section 4 explains the implementation details of the
proposed framework of GS-EvoDNN. Section 5 discussed the effectiveness of the
proposed framework for fault diagnosis under various load and operating
conditions of the machine. And finally, Section 6 concludes the whole paper.
Table 1: List of Symbols
Symbol | Description
---|---
$\mathcal{D}^{tr}$, $\mathcal{D}^{val}$, & $\mathcal{D}^{te}$ | Dataset, training, validation & test dataset
$\textbf{X}\in\Re^{(n_{s}\times n_{f})}$, $\textbf{y}\in\Re^{n_{s}}$ | Input data, Output labels
$n_{s}$ & $n_{f}$ | Number of samples & features
$C$ | number class of the dataset
$\Psi_{t}$ | DNN model at generation $t$
$\Psi_{t}^{\dagger}$ | Best DNN model at generation $t$
$\Lambda_{t},\,\mathcal{R}_{t}$ | Fitness matrix, Rank at generation $t$
$P_{t},\,Q_{t}$ | Population, Offspring at generation $t$
$n_{p}$ | number hidden layers in $p^{th}$ model
$h_{k}$ | Number nodes in $k^{th}$ hidden layer
## 2 Problem Statement
Let the training dataset, validation dataset, and test dataset are
$\mathcal{D}^{tr}=\left(\textbf{X}^{tr},\textbf{y}^{tr}\right)$,
$\mathcal{D}^{val}=\left(\textbf{X}^{val},\textbf{y}^{val}\right)$, and
$\mathcal{D}^{te}=\left(\textbf{X}^{te},\textbf{y}^{te}\right)$, respectively
where $\textbf{X}\in\Re^{(n_{s}\times n_{f})}$ be the input data with $n_{s}$
samples & $n_{f}$ features and $\textbf{y}\in\Re^{n_{s}}$ be the corresponding
output label. The objective of optimal DNN architecture search for fault
classification is mathematically be formulated as
$\displaystyle\Psi^{\dagger}=\mathcal{H}\left(P,\;\mathcal{D}^{tr},\;\mathcal{D}^{val}\right)$
(1)
$\displaystyle\hat{y}^{te}\;=\;\mathcal{F}\left(\Psi^{\dagger},\;X^{te}\right)$
(2)
where $\mathcal{H}(.)$ denotes the optimization function to get the best model
$\Psi^{\dagger}$ with optimal parameters for the training dataset and
$\mathcal{F}(.)$ is the feed-forward DNN function which predicts the fault
class $\hat{\textbf{y}}^{te}$ for the test data $\textbf{X}^{te}$.
## 3 Related Works and Theoretical Background
### 3.1 Deep Neural Network (DNN)
The deep neural network (DNN): a multi-layered neural network is the most
popular technique for pattern recognition via non-linear feature
transformation in multiple stages [32]. From the training point of view, DNN
can be considered as two parts: Stack of a given number of auto-encoders (
also called stacked auto-encoder: SAE) [33] and a classifier usually softmax
classifier as output layer. First, a greedy layer unsupervised training is
used to train each of the auto-encoder (AE) in the SAE. Then, the SAE stacked
with the classifier at the end layer is fine-tuned using a labeled training
dataset. The SAE with softmax classifier (DNN model $\Psi$) is depicted in
Fig. 1.
Figure 1: SAE with softmax classifier (DNN model: $\Psi$)
### 3.2 Intelligent Fault Diagnosis
Recently, with the advent of advanced machine learning techniques and the
availability of fast computational resources, the data-driven intelligent
fault diagnosis method has gained much popularity, [1, 2, 4, 3]. In these
methods, various machine learning techniques are utilized to learn the
specific signature of the recorded signals like current, vibration,
temperature, etc., and thereafter identify the existence of machinery fault
using the test samples. Neural network (NN) [34], Support vector machine (SVM)
[35, 36], and random forest (RF) classifier [5] have been very effectively
used for intelligent fault diagnosis and have been proved to be the baseline
method for pattern recognition. But the diagnostic performances by these
methods are reduced due to high sparsity and low-quality features in the
dataset [37]. The intelligent fault diagnosis using deep learning methods has
gained much attention due to its capability of multi-scale hierarchical
feature transformation and large dimensional data handling, [7, 8, 14].
However, using a deep neural network for fault diagnosis faces a major
challenge of training from scratch for every new operating condition of the
machines. The recent trend of using deep transfer learning methods for domain
adaptation has been very effective for fault diagnosis under changeable
operating conditions [9, 10, 12, 13, 38, 15, 16]. However, the diagnosis
performance by these methods is very much affected by the selection of the
architecture of the deep neural network and the other hyper-parameters.
### 3.3 Neural Architecture Search (NAS)
The main objective of NAS methods is to obtain the optimal architecture in a
given search space with the best model performance [18]. There are three
important aspects of the NAS methods: (i) formulation of the search space,
(ii) the architecture optimizer, and (iii) the model evaluation. Formulation
of search space defines the design format for the model architecture. It can
be categorized into four groups: (i) cell-based (ii) entire-structured, (iii)
morphism-based, and (iv) hierarchical search space. The most important aspect
of NAS methods is the model evaluation as it is computationally very expensive
to train each model during the search process and evaluate on the unseen
dataset. To accelerate the evolution, various mechanism have been suggested
for the model evaluation [18, 29, 39, 40, 20, 41, 42, 43]. K. Kandasamy,
et.al, [41] suggested learning the curve extrapolation for the model
performance evaluation instead to train and evaluate the actual architecture.
H. Pham et. al, [43] proposed the method of parameter sharing for the faster
training and evaluation of the model architecture.
Another important aspect of NAS methods is the architecture optimizer (AO).
The objective automatic AO is to automatically guide the model architecture
search in a direction to get the best suitable model for a given dataset. The
AO methods adopted by various researchers can be categorized as (i) random
search (RS) (ii) grid search (GS), (iii) surrogate model-based optimization
(SMBO), (iv) gradient descent (GD), (v) reinforcement learning (RL), (vi)
genetic algorithms (GA), and (vii) hybrid methods. In the RS method, the
search optimizer tries different architecture randomly from the defined search
space, [19], whereas, in GS, the search method uses a grid to sample and
evaluate the model architecture [40]. SMBO methods use Basian optimization
[20, 41, 42] or neural networks [44] as a surrogate model of the objective
function to obtain the most promising solution (the model architecture).
Gradient descent-based method uses softmax function to find the optimal
architecture over a continuous and differentiable search space [27]. In RL-
based NAS [21, 45], a controller (usually, a recurrent neural network)
generate an action to sample a new architecture. The observation (state) & the
reward from the environment is used to update the controller policy to
generate new architecture samples. Here, the training & validation process of
the sampled neural network is treated as the environment that returns back the
validation accuracy. GA-based NAS [22, 23, 24, 25, 46, 31], use heuristic
search to find the best performing architecture over a given search space. In
these methods, heuristically sampled neural architectures are trained and
evaluated using the convention neural training methods and the performance
metrics are used as fitness for evolution to obtain the optimal architecture.
The main challenge of these methods is the fitness evaluation of the
individual model All of these methods of AO have their own merits and
demerits. The hybridization of two of the above methods may give a significant
improvement in the search efficiency, called the hybrid method of AO [47, 28,
48].
### 3.4 Policy Gradient
Policy gradient (PG) is a tool to optimize the controller policy for
reinforcement learning algorithm [49, 50]. The controller policy is the
parameterized function that defines the learning agent’s way to act on the
environment to get maximum reward Fig. 2. The reward defines the good or bad
effect of the action taken by the policy towards the fulfillment of the
optimal objective. Let the parameter vector be $\theta_{t}$, then the
parameterized function policy is represented as
$\pi_{\theta_{t}}(a_{t}|s_{t})$, where $a_{t}$ and $s_{t}$ represent action
and state at given time $t$. Let the action $a_{t}$ produces reward $r_{t+1}$
from the environment, then the trajectory of state, action and reward can be
represented as
$\left((s_{0},a_{0},r_{1}),(s_{1},a_{1},r_{2}),....(s_{t},a_{t},r_{t+1})\right)$.
The policy parameter $\theta_{t}$ can be updated using policy gradient as
$\theta_{t+1}=\theta_{t}+\eta_{t}\nabla_{\theta_{t}}\mathcal{J}(\theta_{t})$
(3)
where $\eta_{t}$ denotes the learning rate at time $t$, usually a constant
real number. $\nabla_{\theta_{t}}\mathcal{J}(\theta_{t})$ denotes the policy
gradient and can be calculated using expected of cumulative reward ${U}_{t}$
over the time $t$ as follows.
$\displaystyle\nabla_{\theta_{t}}\mathcal{J}(\theta_{t})=\nabla_{\theta_{t}}E\left[{U}_{t}\right]=\nabla_{\theta_{t}}\int_{t}\pi(\tau)r(\tau)d(\tau)$
(4)
$\displaystyle=E\left[r(\tau).\nabla_{\theta_{t}}\log\pi_{\theta_{t}}(\tau)\right]$
(5)
$\nabla_{\theta_{t}}\mathcal{J}(\theta_{t})=\frac{1}{N}\sum_{k=1}^{N}r(k)\left(\sum_{t=1}^{T}\nabla_{\theta_{t}}\log\pi_{\theta_{t}}\left(a_{t}|a_{t-1:1};\theta_{t}\right)\right)$
(6)
## 4 Proposed Framework
In this section, the proposed framework of guided sampling-based evolutionary
DNN (GS-EvoDNN) is described in detail. The Fig 2 shows the schematic of the
workflow of GS-EvoDNN. In the figure, DNN architecture optimization in the
NSGA-II framework constitute the environment. The fitness of the best model is
termed as the reward. The sorted fitness of all the individuals in the
population is treated as the state of the controller. The controller policy
$\pi_{\theta_{t}}$ generates an action $a_{t}=[m_{t},\sigma_{t}]$, where
$m_{t}$ and $\sigma_{t}$ be the mean and variance for the sampling of DNN
architecture at generation $t$. Given the training dataset $\mathcal{D}^{tr}$
and the validation dataset $\mathcal{D}^{val}$, the algorithmic steps for the
GS-EvoDNN is presented in Algorithm 1. Our contributions are highlighted in
Algorithm 1 and are further discussed in the following sections.
Figure 2: Flow Diagram of guided sampling based NSGA-II. Figure 3: Variable-
length gene encoding strategy. Algorithm 1 GS-EvoDNN: The Main Framework
Input: $\mathcal{D}^{tr}\,\&\,\mathcal{D}^{val}$ = training & validation
datasets, $n_{R}\,\&\,h_{R}=$ Maximum depth & width of the DNN respectively
Output: $\Psi^{\dagger}$ = best model after the termination or last generation
of the evolution.
1:$t\longleftarrow 0$ //Set generation count ($t$) = 0;
2:$[m,\;\sigma]\longleftarrow$ Compute mean and variance from allowable range
for depth ($n_{R}$) and width ($h_{R}$) of the DNN.
3:$\textbf{$P_{0}$}\longleftarrow\textbf{GuidedPop}(m,\sigma,N_{p})$
//Generate $N_{p}$ number of populations using Algorithm 2.
4:$\Psi_{0}\longleftarrow$ Initialize weight matrices of the first model
($P_{0}\\{1\\}$) by small random numbers.
5:$\Lambda,\,\Psi_{1}^{\dagger}\longleftarrow\textbf{FitnessEval}(P_{0},\mathcal{D}^{tr},\mathcal{D}^{val},\Psi_{0})$
//Evaluate fitness of all individuals in $P_{0}$ using the Algorithm 3.
6:$\mathcal{R}\longleftarrow NonDominatedSorting(\Lambda_{1})$ //Assign rank
using non-dominated sorting [30].
7:$P_{1}\longleftarrow SelectParents(P_{0},\;\mathcal{R})\,$ //Select parents
bybinary tournament selection, [30].
8:${\Lambda}_{s}\\{1\\}\longleftarrow\Lambda$ //Store the fitness History.
9:$[m,\;\sigma]\longleftarrow\,\textbf{UpdateMeanVar}(P_{1},\;{\Lambda}_{s})$
//Update mean and variance term using the Algorithm 4.
10:$Q_{1}\longleftarrow\textbf{CrossoverMutation}(P_{1},m,\sigma))$ //Apply
crossover and mutation on $P$ using Algorithm 5.
11:$t\longleftarrow\;t+1$ //Update the generation count
12:while termination condition is false do
13:$S_{t}\longleftarrow(P_{t}\cup Q_{t})$ //Combine the parent population
($P_{t}$) & the child population ($Q_{t}$).
14:$\Lambda,\,\Psi_{t+1}^{\dagger}\longleftarrow\textbf{FitnessEval}(S_{t},\mathcal{D}^{tr},\mathcal{D}^{val},\Psi_{t}^{\dagger})$
//Evaluate fitness of all individuals in $S_{t}$ using the Algorithm 3.
15:$\mathcal{R}\longleftarrow NonDominatedSorting(\Lambda)$ //Assign rank by
non-dominated sorting of fitness $\Lambda$, [30].
16:$\mathcal{K}\longleftarrow CrowdingDistances(S_{t},\mathcal{R},\Lambda)$
//Find crowding distances of individuals in population set $S_{t}$, [30].
17:$P_{t+1}\longleftarrow SelectParentsByRankDist(S_{t},\mathcal{K},\Lambda)$
//Select parents by crowding distance and rank, [30].
18:$\Lambda_{s}\\{t+1\\}\longleftarrow\Lambda$ //Store the fitness History.
19:$[m,\;\sigma]\longleftarrow\,\textbf{UpdateMeanVar}(S_{t},\;\Lambda_{s})$
//Update mean and variance term using the Algorithm 4.
20:$Q_{t+1}\longleftarrow\textbf{CrossoverMutation}(P_{t+1},m,\sigma)$ //Apply
crossover and mutation on $P$ using Algorithm 5.
21: if Termination condition is true then
22: Exit
23: else
24: $t\longleftarrow t+1$ //Update the generation counter
25: end if
26:end while
27:Return: $\textrm{{Best
Model}}:\;\Psi^{\dagger}\longleftarrow\Psi_{t+1}^{\dagger}$ //Best model of
the last generation.
### 4.1 Population Sampling using Mean and Variance
AO includes depth and width variation in a defined search space. The real-
coded gene encoding strategy is adopted to encode the depth as the number of
genes (length of a chromosome) and the number of nodes in a hidden layer as
the value of a gene as shown in Fig. 3. Let $n_{R}$ & $h_{R}$ be the maximum
depth and width of the DNN, then the search space is defined as $[1\;n_{R}]$ &
$[1\;h_{R}]$ for depth and the width variations. The mean and variance
$m=[m_{1},m_{2}]$ & $\sigma=[\sigma_{1},\sigma_{2}]$) are initialized as
$m_{1}=(1+n_{R})/2,\;m_{2}=(1+h_{R})/2$ and
$\sigma_{1}=(n_{R}-1)/2,\;\sigma_{2}=(h_{R}-1)/2$.
Algorithm 2 GuidedPop: Population Sampling
Input: $N$ = Population size, $m=[m_{1},m_{2}]=$ mean &
$\sigma=[\sigma_{1},\sigma_{2}]=$ variance.
Output: $P$ = Population with $N$ chromosomes.
1:$H\longleftarrow$ generate $N$ Gaussian numbers with $m_{1}$ and
$\sigma_{1}$.
2:for p = 1 : N do
3:$h\longleftarrow H(p)$ : depth of $p^{th}$ chromosome
4:$tmp\longleftarrow$ generate $h$ Gaussian numbers with $m_{2}$ and
$\sigma_{2}$.
5:$P\\{p\\}\longleftarrow$ convert all numbers in $tmp$ to nearest integers.
6:end for
7:Return $P$
### 4.2 Fitness Evaluation
Fast model evaluation is the most important requirement for NAS, especially
when the evolutionary algorithm is used as an AO strategy. If the best model
at a generation is transferred for initialization of the DNN weight matrices
in the next generation, it makes the training and evaluation of the models
faster. The quick learning mechanism suggested in [16] is adopted for the
fitness evaluation as shown in Fig. 4. For the first generation, DNN models
are initialized randomly and trained using Limited-Broyden-Fletcher-Goldfarb-
Shanno (LBFGS) [51] algorithm. For next-generation and onward, the best model
obtained is transformed (Fig. 4) to initialize the models followed by fine-
tuning with the LBFGS algorithm for a few iterations only. If a model
$\Psi^{t}$ at generation $t^{th}$ has weight matrix $W^{t}$, the
classification loss $\mathcal{J}$ for a $C$ problem can be defined in term of
$[w,\,b]\in W^{t}$ as
$\displaystyle\mathcal{J}(W^{t})=\frac{1}{n^{s}}\left[\sum_{k=1}^{n^{s}}\sum_{i=1}^{C}I[y_{k}=c_{i}]\log{\frac{e^{(w_{i}^{T}f(x_{p})+b_{i})}}{\sum_{i=1}^{C}e^{(w_{i}^{T}f(x_{p}^{)}+b_{i})}}}\right]$
(7)
where, $f(x)=\Phi(wx+b)$ is the h-level features representation of DNN,
$y_{k}$ be the output label of the $k^{th}$ data sample and $c_{i}$ denotes
the $i^{th}$ class. The classification accuracy ($CA$) of fine-tuned model for
the validation data is returned as the fitness of that model.
Figure 4: Fitness evaluation strategy. Algorithm 3 FitnessEval: Fitness
Evaluation
Input: $P$ = Population with population size $N_{p}$,
$(\mathcal{D}^{tr},\mathcal{D}^{val})$ = Training & validation data.
Output: $\Lambda$ = Fitness matrix and $\Psi^{\dagger}$ = Best model.
1:$t\longleftarrow$ current generation
2:for $p\,=\,1\,:\,N_{p}$ do
3: $\Psi_{t}\longleftarrow\textrm{N2N}(\Psi_{t}^{\dagger})$ // Transform
$\Psi_{t}^{\dagger}$ as $p^{th}$ model using N2N transformation as depicted in
step-1 of Fig. 4.
4: Fine-tune the model ($\Psi_{t}$) on $\mathcal{D}^{tr}$ to minimize Eq. (7).
5: $\Lambda(p)\longleftarrow$ Find $CA$ of $\Psi_{t}$ on dataset
$\mathcal{D}^{val}$.
6:end for
7:$\Psi_{t}^{\dagger}\longleftarrow\textrm{Best model}$ // Find the model with
maximum $CA$ and minimum number of model parameters.
8:Return $\Lambda,\;\Psi_{t}^{\dagger}$
### 4.3 Update Mean and Variance using PG
Mean ($m$) and variance ($\sigma$) terms are used for sampling of a new
population as illustrated in Section-4.1. Here, we design a PG-based update
laws for $m$ & $\sigma$ such that the best fitness ($\max(\Lambda)$) is
maximized. At any generation $t$, $\max(\Lambda)$ is termed as reward,
$a_{t}=[m,\sigma]$ is termed as action, and weighted average of fitness
($\Lambda$) is termed as state of the policy. Thus, the policy generates
$[m^{T},\sigma^{T}]$ to guide the evolution for faster and better convergence.
For the design simplification, let us assume that the action is generated by a
deterministic policy as
$a_{t}=f(\theta_{t})=\frac{1}{1+e^{-\theta_{t}}}$ (8)
where, policy parameter $\theta_{t}$ is selected such that it controls
$m\in\Re^{2}$ (mean of depth and width of DNN) and $\sigma\in\Re^{2}$
(variance for depth and width of DNN). The parameter $\theta$ is updated by
the policy gradient in (3) calculated using gradient of expected total reward
$U_{t}$ derived in (6). The total cumulative reward is calculated using
fitness matrix $\Lambda_{t}$ at generation $t$ as in (9).
$U_{t}\;=\;\sum_{i=1}^{t}\frac{\max(\Lambda_{i})-\max(\Lambda_{i-1}}{\max(\Lambda_{i-1})}$
(9)
The algorithmic steps for the implementation of policy gradient based update
of $a_{t}=[m,\sigma]$ at generation $t$ is summarised in Algorithm 4.
Algorithm 4 UpdateMeanVar: Update $m$ & $\sigma$ using PG
Input: $P_{t}$ = Current population, $\Lambda$ = Fitness matrix,
$a_{t-1}=[m,\sigma]$ = Initial mean & variance term.
Output: $a_{t}=[m,\sigma]$ = Updated mean & variance.
1:$N=$ Number of models in $P_{t}$, $\alpha=$ Learning rate
2:$\bar{n}\longleftarrow$ Compute average depth of models in $P_{t}$
3:$\Omega=[\omega_{p}]_{p=1}^{N}$ //Generate a set of weights $\omega_{p}$
such that $\sum_{p=1}^{N}\omega_{p}=0$ and
$\omega_{1}>\omega_{2}>\,....\,>\omega_{N}$
4:$\Lambda_{t}^{sorted},idx$ = sort($\Lambda_{t}$, ‘descending’)
5:$P_{t}\longleftarrow$ Sort $P_{t}$ according to $idx$.
6:$n_{p}=$ no. of hidden layers (depth) in $p^{th}$ model.
7:$\delta_{p}=\max(H_{p})-\min(H_{p})$ //$H_{p}=$ set of nodes in hidden
layers of $p^{th}$ model
8:if $t\leq 1$ then
9:
$m=\sum_{j=1}^{N}\left[n_{p}\omega_{p}\;\;\;\frac{1}{n_{p}}\sum_{k=1}^{n_{p}}h_{kp}\omega_{p}\right]$
//$h_{kp}\in H_{p}$.
10:
$\sigma=\sum_{j=1}^{N}\left[(\bar{n}-n_{p})\omega_{p}\;\;\;\delta_{p}\omega_{p}/2\right]$
11:else
12: $\theta_{t-1}=\ln{\left[a_{t-1}/(1-a_{t-1})\right]}$
13: $U_{t}\longleftarrow$ Compute cumulative reward using (9).
14:
$s_{m}=\sum_{j=1}^{N}\left[n_{p}\omega_{p}\;\;\;\frac{1}{n_{p}}\sum_{k=1}^{n_{p}}h_{kp}\omega_{p}\right]$
15:
$s_{\sigma}=\sum_{j=1}^{N}\left[(\bar{n}-n_{p})\omega_{p}\;\;\;\delta_{p}\omega_{p}/2\right]$
16: $s_{t}=[s_{m}^{T}\;s_{\sigma}^{T}]$
17:
$\theta_{t}\longleftarrow\theta_{t-1}+\alpha*\frac{1}{N}\sum_{k=1}^{N}U_{t}E[\nabla\log{\pi_{\theta}(s_{t}|a_{t-1};\theta)}]$
18: $a_{t}={1}/{\left(1+e^{-\theta_{t}}\right)}$
19:end if
20:Return $a_{t}$
### 4.4 Crossover and Mutation
For the optimal search of the network architecture, a combination of
exploration and exploitation strategies is adopted. The guided sampling-based
generation of new population exploits the search space to force the evolution
towards maximum accuracy. To avoid local convergence, $N$ number of
individuals are sampled using $m$ and $\sigma$ based on Gaussian distribution,
and also $N$ number of parent populations are selected using crowding distance
and rank from the current generation. After that, the two populations are
merged to create a double-sized mating pool. Now, the two-step crossover
operator introduced in [26] is applied. The two steps are (i) single point
depth crossover (SPDC) for depth variation and (ii) common depth simulated
binary crossover (CDSBC) for gene value (width) alteration. The two-step
crossover method is depicted in Fig. 5. The whole process of offspring
generation is provided in Algorithm 5:
Figure 5: Two-steps crossover for chromosomes with different length. Algorithm
5 Offspring Generation: Crossover and Mutation
Input: $P$ = Parent population, $p_{c}$ = Crossover probability,
$(m,\sigma)$ = mean & variance term for sampling.
Output: $Q$ = Offspring population.
1:$Q\longleftarrow$ GuidedPop($m,\sigma,N,n_{R},h_{R}$) //Generate $N$
offspring populations using Algorithm 2.
2:$I_{c}=$ generate random indices of $p_{c}*100$% members from $P$.
3:for $i\in I_{c}$ do
4: Select $P_{1}=P\\{i\\}\;\&\;P_{2}=Q\\{i\\}$.
5: Find lengths ($n_{1},\;n_{2}$) of $P_{1},\;P_{2}$
6: Set a point $h<\min(n_{1},n_{2})$ on $P_{1},\;P_{2}$.
7: $C_{1},\,C_{2}\longleftarrow$ SPDC of $P_{1},\;P_{2}$ at point $h$ (as
depicted in step-1 of Fig. 5)
8: $\bar{C}_{1},\bar{C}_{2}\longleftarrow$ CDSBC for genes of the common depth
portion of $C_{1},\,C_{2}$ as depicted in step-2 of Fig 5.
9: Replace $Q\\{i\\}$ by $\bar{C}_{1}$ or $\bar{C}_{2}$.
10:end for
11:Return $Q$
## 5 Experimental Results and Discussion
The efficacy of the proposed framework of GS-EvoDNN is demonstrated on fault
diagnosis dataset under different operating conditions taken from (i) Air
compressor fault data [52], (ii) Paderborn University (PBU) bearing fault data
[53], and (iii) CWRU bearing fault data [54].
### 5.1 Experimental Setup
#### 5.1.1 Air Compressor Fault Data [52]
The air compressor data contains acoustic signal recorded on single stage
reciprocating type air compressor driven by an $5hp$ induction motor installed
at the workshop, EE Department, IIT Kanpur. Data were recorded in eight
different cases: healthy and seven different faulty states of the air
compressor valve. Therefore, the dataset has 8 classes: (i) Healthy (H), (ii)
Leakage Inlet Valve (LIV), (iii) Leakage Outlet Valve (LOV), (iv) Non-Return
Valve (NRV), (v) Piston Ring (PR), (vi) Flywheel (F), (vii) Rider Belt (RB),
and (viii) Bearing (B). For each class, 225 measurements were taken with 50k
samples in each measurement.
#### 5.1.2 PBU Bearing Fault Data [53]
PBU bearing fault data is the collection of time-series signals recorded on
electrical machine operating under wide variation of shaft load and rotational
speed. The four Load and speed settings (LS) are LS1: N09_M07_F10 (speed = 900
rpm, torque = 0.7 Nm & radial force = 1000 N), LS2: N15_M01_F10 (speed = 1500
rpm, torque = 0.1 Nm & radial force = 1000 N), LS3: N15_M07_F04 (speed = 1500
rpm, torque = 0.7 Nm & radial force = 400 N), and LS4: N15_M07_F10 (speed =
1500 rpm, torque = 0.7 Nm & radial force = 1000 N). Total of 32
experimentation with 6 healthy, 12 artificially damaged, and 14 damaged by
long run accelerated tests were conducted to record current, vibration signal,
radial forces, torque, and bearing temperature. The recorded signals contains
two types of faults: inner race (IR) fault and outer race (OR) fault.
#### 5.1.3 CWRU Bearing Fault Data [54]
The CWRU bearing fault data provided by Case Western Reserve University (CWRU)
has been a widely used benchmark dataset for bearing fault diagnosis. It
contains vibration signal recorded at drive-end (DE) and fan-end (FE) of
bearing artificially seeded with inner race fault (IR), outer race (OR), and
rolling element ball (B) faults of variable fault diameters (F.D.) (from 0.007
to 0.028 inches). The bearing test rig setup details can be found in [54].
### 5.2 Segmentation and Data Processing
The recorded time-series signals contain a huge number of samples which is not
suitable for training the DNN. To make the dimension of the time-series
signals compatible with the DNN, we adopt a segmentation rule with the segment
length of approximately $1/4^{th}$ of data points recorded per revolution.
Here, we have selected segmentation lengths of 100, 200, & 400 for the CWRU
dataset, Air compressor dataset, and PBU dataset respectively. Also, the time-
series signals are usually unstructured and not to the scale. Therefore, we
have applied the min-max normalization technique to scale down the dataset to
$[0,\,1]$. The min-max normalization also removes the effect of outlier
points. If for some cases, the outlier points carry some important
information, then the z-score minimization technique may be used to make the
dataset well-structured [16].
### 5.3 Dataset Preparation
For the study of fault diagnosis with the proposed GS-EvoDNN, we prepare the
training, the validation, & the testing dataset under various operating
conditions described below.
Case-1 (T1): From Air Compressor Dataset, $7$ different cases of binary
classes and one case of multi-class diagnosis are investigated as listed in
table 2. For each class, 4 measurement files (having 50k samples/file) are
merged to create a sample of size $1000\times 200$ per class taking $200$
points as segment length.
Case-2 (T2): From CWRU FE Dataset, multi-class diagnosis with class name
healthy (H), inner race (IR), outer race (OR), and ball element (B) are
considered under different load ($1$, $2$, & $3$ hp) conditions and different
fault diameters (FD) ($7$, $14$, & $21$ mil). For each FD (for example, $7$
mil), dataset from all three load conditions are prepared. Thus, the fault
diagnosis on total of $9$ cases are presented as listed in table 3.
Case-3 (T3 & T4): From PBU Dataset, two different cases are considered (i) T3:
artificially damaged bearing fault and (ii) T4: bearing fault due to long
accelerated test. In both cases, multi-class diagnosis with three classes
namely H-OR-IR is studied under four load settings (LS) as listed in table 4.
Now for the training, the validation, & the testing, each of the above dataset
is split into three portions: $64\%$ train ($\mathcal{D}^{tr}$), $20\%$ test
($\mathcal{D}^{te}$), and $16\%$ validate ($\mathcal{D}^{val}$) datasets.
Table 2: Air Compressor Dataset (T1): Diagnostic Performance in term of
Classification Accuracy
Class | SVM [35] | DNN [7] | DTL [13] | DAFD [12] | N2N [16] | EvoDCNN [25] | EvoN2N[26] | GS-EvoDNN
---|---|---|---|---|---|---|---|---
W. D. A. | D. A.
H-LIV | 99.75 | 96.25 | 99.00 | 99.75 | 99.50 | 99.75 | 100.00 | 100.00 | 100.00
H-LOV | 98.25 | 95.75 | 99.25 | 99.66 | 99.25 | 99.25 | 99.75 | 99.75 | 100.00
H-PR | 98.25 | 93.25 | 93.30 | 98.75 | 97.75 | 98.75 | 98.25 | 99.75 | 99.75
H-B | 98.25 | 98.50 | 98.75 | 98.75 | 96.75 | 98.75 | 98.75 | 99.75 | 100.00
H-F | 99.25 | 99.25 | 99.00 | 98.75 | 99.25 | 99.25 | 99.25 | 100.00 | 100.00
H-NRV | 98.75 | 99.00 | 99.00 | 99.75 | 99.00 | 99.25 | 99.25 | 100.00 | 100.00
H-RB | 98.25 | 98.25 | 98.25 | 99.00 | 99.75 | 99.75 | 99.25 | 100.00 | 100.00
H-ALL | 97.75 | 99.25 | 99.00 | 99.00 | 99.25 | 99.25 | 99.75 | 99.75 | 100.00
S. D. | 0.65 | 2.16 | 2.00 | 0.46 | 1.02 | 0.38 | 0.57 | 0.13 | 0.09
Table 3: CWRU FE Dataset (T2): Diagnostic Performance in term of
Classification Accuracy
Class | F.D. | Load | SVM [35] | DNN [7] | DTL [13] | DAFD [12] | N2N [16] | EvoDCNN [25] | EvoN2N[26] | GS-EvoDNN
---|---|---|---|---|---|---|---|---|---|---
W. D. A. | D. A.
H-IR-OR-B | DE 7 mil | 1hp | 88.12 | 96.69 | 96.56 | 97.94 | 98.94 | 98.94 | 99.60 | 100.00 | 100.00
2hp | 98.12 | 95.94 | 93.44 | 96.12 | 97.12 | 98.12 | 99.60 | 100.00 | 100.00
3hp | 99.10 | 98.75 | 98.75 | 98.44 | 99.44 | 99.44 | 99.70 | 100.00 | 100.00
DE 14 mil | 1hp | 99.10 | 94.75 | 96.88 | 97.19 | 99.19 | 99.67 | 100.00 | 100.00 | 100.00
2hp | 98.10 | 95.31 | 92.19 | 95.69 | 97.69 | 98.69 | 98.12 | 99.12 | 99.60
3hp | 99.25 | 96.88 | 94.69 | 97.62 | 99.33 | 98.62 | 98.44 | 98.84 | 100.00
DE 21 mil | 1hp | 96.88 | 86.56 | 84.69 | 89.62 | 95.62 | 96.62 | 93.75 | 98.75 | 100.00
2hp | 88.44 | 85.31 | 82.19 | 86.69 | 90.69 | 90.69 | 90.10 | 95.37 | 98.85
3hp | 92.19 | 86.56 | 79.38 | 88.06 | 91.06 | 92.06 | 92.81 | 95.81 | 98.81
S. D. | 4.62 | 5.25 | 7.06 | 4.66 | 3.46 | 3.32 | 3.69 | 1.81 | 0.51
Table 4: CA for Target-2 & Target-3 Dataset and for very limited samples of
Target-2 & Target-3 Dataset
Class | Data- L.S. | SVM [35] | DNN [7] | DTL [13] | DAFD [12] | N2N [16] | EvoDCNN [25] | EvoN2N[26] | GS-EvoDNN
---|---|---|---|---|---|---|---|---|---
W. D. A. | D. A.
H-OR-IR | T3-L1 | 94.25 | 96.92 | 96.92 | 96.92 | 98.64 | 98.94 | 99.25 | 99.75 | 100.00
T3-L2 | 90.00 | 93.58 | 95.00 | 94.58 | 95.12 | 96.12 | 99.58 | 99.83 | 99.75
T3-L3 | 87.17 | 91.92 | 93.33 | 92.08 | 94.44 | 94.44 | 97.50 | 97.70 | 100.00
T3-L4 | 87.17 | 93.15 | 93.75 | 94.17 | 97.19 | 95.28 | 100.00 | 100.00 | 100.00
T4-L1 | 95.00 | 97.50 | 97.50 | 98.33 | 98.33 | 99.17 | 99.17 | 100.00 | 100.00
T4-L2 | 92.83 | 95.92 | 96.50 | 96.33 | 96.33 | 96.33 | 98.60 | 99.15 | 100.00
T4-L3 | 94.83 | 94.67 | 93.33 | 94.17 | 95.72 | 96.33 | 98.60 | 99.15 | 100.00
T4-L4 | 94.83 | 95.33 | 95.00 | 95.83 | 95.69 | 95.69 | 93.33 | 98.75 | 99.75
S. D. | 3.41 | 1.92 | 1.65 | 1.95 | 1.50 | 1.68 | 2.13 | 0.79 | 0.10
### 5.4 Implementation Details
For the implementation of the proposed framework of GS-EvoDNN, the initial
parameters are selected as: population size ($N$) = 100, crossover probability
($P_{c}$) = 0.5, and the maximum number of generations is set to very high
usually at 50. Also, the termination criteria are set as either the validation
accuracy reaches 100% or it does not change continuously for 3 generations.
The allowable ranges for the variation of depth and width are selected as
$n_{R}\in[1,\;10]$ and $h_{R}\in[10,\;400]$ respectively. The learning rate
$\alpha=0.1$. The GS-EvoDNN framework is applied to the training dataset and
the best model obtained is tested for the test dataset under all cases (T1,
T2, T3, & T4) described in section 5.3. The classification accuracies ($CA$)
are tabulated in tables 2, 3, & 4.
For comparison and analysis, the state-of-the-art methods for intelligent
fault diagnosis best reported in various literature are selected as support
vector machines (SVM) [35], deep neural network (DNN) [7], deep transfer
learning (DTL) based on sparse autoencoder [13], Deep neural network for
domain Adaptation in Fault Diagnosis (DAFD) [12], Net2Net without domain
adaptation (N2N_WDA) [16], Net2Net with domain adaptation (N2N_DA) [16],
evolutionary deep CNN (EvoDCNN) [25], and evolutionary Net2Net (EvoN2N)[26].
The DNN, DTL, and DAFD are trained with hidden sizes as $(70-50-20)$. The
initial and hyper parameters for EvoDCNN and EvoN2N are kept same as mentioned
above. The same dataset (T1, T2, T3, & T4) are used to train and test all
these methods using the procedure suggested in the corresponding cited
references. The diagnostic performance in term of $CA$ are tabulated in tables
2, 3, & 4. The standard deviation (S.D.) of $CA$ calculated over the variation
in the operating conditions is also tabulated to compare the result deviation
with the change in the operating conditions.
Figure 6: TI in term of $\overline{CA}$ for (i) T1: Air Compressor dataset,
(ii) T2: CWRU dataset, (iii) T3: PBU dataset with single point fault, and (iv)
T4: PBU dataset with distributed fault. Figure 7: Confusion matrix for dataset
T4-L1 (Table 4): class label {‘1’, ‘2’, ‘3’} represents the class name {‘H’,
‘OR’, ‘IR’}.
### 5.5 Discussion
The diagnostic performances by the proposed method and the selected state-of-
the-art methods conclude the following important points.
1. 1.
The $CA$ comparison in tables 2, 3, & 4 reveal that the diagnostic
performances are very much affected by the model selection. The DNN model with
the best suitable architecture for the given dataset gives $CA$ up to almost
100% whereas other methods with pre-selected architecture fail to perform
well.
2. 2.
Considering SVM [35] as the baseline diagnostic method, We evaluate the
transfer improvement ($TI$) in term of average $CA$ for the dataset T1, T2,
T3, & T4 separately. If the average $CA$ is denoted as $\overline{CA}$, the
$TI$ is defined as $TI=\overline{CA}-\overline{CA}_{b}$, where
$\overline{CA}_{b}$ is the average $CA$ by SVM. The $TI$ graph shown in Fig. 6
shows the performance improvement of the proposed framework in comparison with
the state-of-the-art methods and the baseline method ‘SVM’.
3. 3.
The performance comparison between EvoN2N [26] and the proposed method shows
that architecture optimization is greatly affected by the guided sampling with
policy gradient framework.
4. 4.
Fig. 7 demonstrate the classification performance by confusion chart matrices
for one of the dataset (T4-L1: table 4). Classifications with 100% accuracies
are highlighted by blackened diagonal elements where for other cases, diagonal
elements are highlighted by gray shade.
5. 5.
The comparison between EvoDCNN [25], EvoN2N [26], & the proposed GS-EvoN2N
reveal that the DNN model with the best architecture performs better than the
CNN model for fault diagnosis applications with the segmented dataset
(described in section 5.2.
### 5.6 Complexity Analysis
The worst complexities in one iteration of the entire algorithm 1 are
contributed by (i) fitness evaluation of the DNN model and (ii) the non-
dominated sorting. The complexity of the fitness evaluation of DNN is mainly
contributed by parameter optimization by L-BFGS which is $O(N_{I}*n^{2})$,
where $n\&N_{I}$ be the total number of parameters and number of iterations
required to fine-tune the DNN model. The non-dominated sorting algorithm has a
complexity of $O(MN_{p}^{2})$, where $M\&N_{p}^{2}$ be the number of
objectives and the population size respectively. Since $M$ is very small
compared to $N_{I}$, therefore, the time complexity for GS-EvoDNN with
population size $N_{p}$ is given by $O(N_{I}N_{p}n^{2})$, where $n=$ total
number of parameters in one model and $N_{I}$ = number of iterations taken for
model training.
## 6 Conclusions
In this article, we have formulated a guided sampling-based evolutionary
algorithm for DNN architecture search. The proposed framework uses the concept
of policy gradient to sample the new population to force the evolution towards
the maximization of classification performance. The best model obtained at any
generation is transferred to the next generation to initialize the model
evaluation which makes the entire algorithm faster. Therefore, this method is
very good in terms of faster evolution and faster convergence while ensuring
global convergence by update policy of mean and variance. The validation using
dataset under various cases from Air Compressor data, CWRU data, and PBU data
prove that the proposed framework is capable to obtain the best model to get
diagnostic performance almost up to 100% accuracy. This method can also be
employed for the architecture optimization of the CNN model with image
classification applications.
## References
* [1] S. Nandi, H. A. Toliyat, and X. Li, “Condition monitoring and fault diagnosis of electrical motors—a review,” _IEEE Transactions on Energy Conversion_ , vol. 20, no. 4, pp. 719–729, Dec 2005.
* [2] A. Siddique, G. S. Yadava, and B. Singh, “A review of stator fault monitoring techniques of induction motors,” _IEEE Transactions on Energy Conversion_ , vol. 20, no. 1, pp. 106–114, March 2005.
* [3] S. Yin, S. X. Ding, X. Xie, and H. Luo, “A review on basic data-driven approaches for industrial process monitoring,” _IEEE Transactions on Industrial Electronics_ , vol. 61, no. 11, pp. 6418–6428, 2014.
* [4] X. Chen, S. Wang, B. Qiao, and Q. Chen, “Basic research on machinery fault diagnostics: Past, present, and future trends,” _Front. Mech. Eng._ , vol. 13, p. 264–291, 2018.
* [5] Z. Chen, F. Han, L. Wu, J. Yu, S. Cheng, P. Lin, and H. Chen, “Random forest based intelligent fault diagnosis for pv arrays using array voltage and string currents,” _Energy conversion and management_ , vol. 178, pp. 250–264, 2018.
* [6] A. K. Sharma, V. Singh, N. K. Verma, and J. Liu, “Condition based monitoring of machine using mamdani fuzzy network,” in _2018 Prognostics and System Health Management Conference (PHM-Chongqing)_ , Oct 2018, pp. 1159–1163.
* [7] Y. Qi, C. Shen, D. Wang, J. Shi, X. Jiang, and Z. Zhu, “Stacked sparse autoencoder-based deep network for fault diagnosis of rotating machinery,” _IEEE Access_ , vol. 5, pp. 15 066–15 079, 2017.
* [8] R. Zhao, R. Yan, Z. Chen, K. Mao, P. Wang, and R. X. Gao, “Deep learning and its applications to machine health monitoring,” _Mechanical Systems and Signal Processing_ , vol. 115, pp. 213 – 237, 2019.
* [9] S. J. Pan, I. W. Tsang, J. T. Kwok, and Q. Yang, “Domain adaptation via transfer component analysis,” _IEEE Transactions on Neural Networks_ , vol. 22, no. 2, pp. 199–210, Feb 2011.
* [10] M. Long, J. Wang, G. Ding, S. J. Pan, and P. S. Yu, “Adaptation regularization: A general framework for transfer learning,” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 26, no. 5, pp. 1076–1089, 2014.
* [11] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. March, and V. Lempitsky, “Domain-adversarial training of neural networks,” _Journal of Machine Learning Research_ , vol. 17, no. 59, pp. 1–35, 2016.
* [12] W. Lu, B. Liang, Y. Cheng, D. Meng, J. Yang, and T. Zhang, “Deep model based domain adaptation for fault diagnosis,” _IEEE Transactions on Industrial Electronics_ , vol. 64, no. 3, pp. 2296–2305, March 2017.
* [13] L. Wen, L. Gao, and X. Li, “A new deep transfer learning based on sparse auto-encoder for fault diagnosis,” _IEEE Transactions on Systems, Man, and Cybernetics: Systems_ , vol. 49, no. 1, pp. 136–144, Jan 2019.
* [14] L. Guo, Y. Lei, S. Xing, T. Yan, and N. Li, “Deep convolutional transfer learning network: A new method for intelligent fault diagnosis of machines with unlabeled data,” _IEEE Transactions on Industrial Electronics_ , vol. 66, no. 9, pp. 7316–7325, Sep. 2019.
* [15] D. Wei, T. Han, F. Chu, and M. J. Zuo, “Weighted domain adaptation networks for machinery fault diagnosis,” _Mechanical Systems and Signal Processing_ , vol. 158, p. 107744, 2021.
* [16] A. K. Sharma and N. K. Verma, “Quick learning mechanism with cross-domain adaptation for intelligent fault diagnosis,” 2021.
* [17] G. Wang, J. Qiao, J. Bi, W. Li, and M. Zhou, “Tl-gdbn: Growing deep belief network with transfer learning,” _IEEE Transactions on Automation Science and Engineering_ , vol. 16, no. 2, pp. 874–885, April 2019\.
* [18] X. He, K. Zhao, and X. Chu, “Automl: A survey of the state-of-the-art,” _Knowledge-Based Systems_ , vol. 212, p. 106622, 2021.
* [19] J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” _J. Mach. Learn. Res._ , vol. 13, pp. 281–305, Feb. 2012.
* [20] F. Hutter, H. H. Hoos, and K. Leyton-Brown, “Sequential model-based optimization for general algorithm configuration,” in _Learning and Intelligent Optimization_ , C. A. C. Coello, Ed. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011, pp. 507–523.
* [21] B. Baker, O. Gupta, N. Naik, and R. Raskar, “Designing neural network architectures using reinforcement learning,” _CoRR_ , vol. abs/1611.02167, 2016.
* [22] S. M. R. Loghmanian, H. Jamaluddin, R. Ahmad, R. Yusof, and M. Khalid, “Structure optimization of neural network for dynamic system modeling using multi-objective genetic algorithm,” _Neural Computing and Applications_ , vol. 21, no. 6, pp. 1281–1295, 2012.
* [23] C. Wang, C. Xu, X. Yao, and D. Tao, “Evolutionary generative adversarial networks,” _IEEE Transactions on Evolutionary Computation_ , vol. 23, no. 6, pp. 921–934, 2019.
* [24] Y. Sun, B. Xue, M. Zhang, and G. G. Yen, “A particle swarm optimization-based flexible convolutional autoencoder for image classification,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 30, no. 8, pp. 2295–2309, Aug 2019.
* [25] Y. Sun, B. Xue, M. Zhang, and G. G. Yen, “Evolving deep convolutional neural networks for image classification,” _IEEE Transactions on Evolutionary Computation_ , vol. 24, no. 2, pp. 394–407, 2020.
* [26] A. K. Sharma and N. K. Verma, “Transfer learning based evolutionary deep neural network for intelligent fault diagnosis,” 2021.
* [27] H. Liu, K. Simonyan, and Y. Yang, “Darts: Differentiable architecture search,” in _International Conference on Learning Representations_ , 2019\.
* [28] Z. Yang, Y. Wang, X. Chen, B. Shi, C. Xu, C. Xu, Q. Tian, and C. Xu, “Cars: Continuous evolution for efficient neural architecture search,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 1829–1838.
* [29] E. Real, A. Aggarwal, Y. Huang, and Q. V. Le, “Regularized evolution for image classifier architecture search,” _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, no. 01, pp. 4780–4789, Jul. 2019. [Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/4405
* [30] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” _IEEE Transactions on Evolutionary Computation_ , vol. 6, no. 2, pp. 182–197, 2002.
* [31] Z. Lu, I. Whalen, Y. Dhebar, K. Deb, E. D. Goodman, W. Banzhaf, and V. N. Boddeti, “Multiobjective evolutionary design of deep convolutional neural networks for image classification,” _IEEE Transactions on Evolutionary Computation_ , vol. 25, no. 2, pp. 277–291, 2021.
* [32] G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” _Science_ , vol. 313, no. 5786, pp. 504–507, 2006\.
* [33] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle, “Greedy layer-wise training of deep networks,” in _Advances in Neural Information Processing Systems 19_ , B. Schölkopf, J. C. Platt, and T. Hoffman, Eds. MIT Press, 2007, pp. 153–160.
* [34] H. Su and K. T. Chong, “Induction machine condition monitoring using neural network modeling,” _IEEE Transactions on Industrial Electronics_ , vol. 54, no. 1, pp. 241–249, 2007.
* [35] A. Widodo and B.-S. Yang, “Support vector machine in machine condition monitoring and fault diagnosis,” _Mechanical Systems and Signal Processing_ , vol. 21, no. 6, pp. 2560–2574, 2007.
* [36] X. Yan and M. Jia, “A novel optimized svm classification algorithm with multi-domain feature and its application to fault diagnosis of rolling bearing,” _Neurocomputing_ , vol. 313, pp. 47–64, 2018.
* [37] S. D. Juan Jose, et al., “Multifault diagnosis method applied to an electric machine based on high-dimensional feature reduction,” _IEEE Transactions on Industry Applications_ , vol. 53, no. 3, pp. 3086–3097, 2016.
* [38] X. Li, W. Zhang, and Q. Ding, “Cross-domain fault diagnosis of rolling element bearings using deep generative neural networks,” _IEEE Transactions on Industrial Electronics_ , vol. 66, no. 7, pp. 5525–5534, July 2019\.
* [39] B. Zoph, V. Vasudevan, J. Shlens, and Q. V. Le, “Learning transferable architectures for scalable image recognition,” in _Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit._ , 2018, pp. 8697–8710.
* [40] A. Hundt, V. Jain, and G. D. Hager, “sharpdarts: Faster and more accurate differentiable architecture search,” _CoRR_ , vol. abs/1903.09900, 2019. [Online]. Available: http://arxiv.org/abs/1903.09900
* [41] K. Kandasamy, W. Neiswanger, J. Schneider, B. Póczos, and E. P. Xing, “Neural architecture search with bayesian optimisation and optimal transport,” in _Proceedings of the 32nd International Conference on Neural Information Processing Systems_ , ser. NIPS’18, 2018, p. 2020–2029.
* [42] J. S. Bergstra, R. Bardenet, Y. Bengio, and B. Kégl, “Algorithms for hyper-parameter optimization,” in _Advances in Neural Information Processing Systems 24_ , J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger, Eds. Curran Associates, Inc., 2011, pp. 2546–2554.
* [43] H. Pham, M. Guan, B. Zoph, Q. Le, and J. Dean, “Efficient neural architecture search via parameters sharing,” in _International Conference on Machine Learning_. PMLR, 2018, pp. 4095–4104.
* [44] R. Luo, F. Tian, T. Qin, E. Chen, and T.-Y. Liu, “Neural architecture optimization,” in _Advances in Neural Information Processing Systems_ , S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, Eds., vol. 31. Curran Associates, Inc., 2018.
* [45] B. Zoph and Q. V. Le, “Neural architecture search with reinforcement learning,” _arXiv preprint arXiv:1611.01578_ , 2016.
* [46] J. Sun, X. Liu, T. Bäck, and Z. Xu, “Learning adaptive differential evolution algorithm from optimization experiences by policy gradient,” _IEEE Transactions on Evolutionary Computation_ , vol. 25, no. 4, pp. 666–680, 2021\.
* [47] Y. Chen, G. Meng, Q. Zhang, S. Xiang, C. Huang, L. Mu, and X. Wang, “Renas: Reinforced evolutionary neural architecture search,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 4787–4796.
* [48] Y. Sun, H. Wang, B. Xue, Y. Jin, G. G. Yen, and M. Zhang, “Surrogate-assisted evolutionary deep learning using an end-to-end random forest-based performance predictor,” _IEEE Transactions on Evolutionary Computation_ , vol. 24, no. 2, pp. 350–364, 2019.
* [49] R. J. Williams, “Simple statistical gradient-following algorithms for connectionist reinforcement learning,” _Machine Learning_ , vol. 8, no. 3-4, p. 229–256, May 1992.
* [50] R. Sutton and A. Barto, _Reinforcement Learning: An Introduction, second edition_ , ser. Adaptive Computation and Machine Learning series. MIT Press, 2018.
* [51] J. Nocedal and S. J. Wright, _Large-Scale Unconstrained Optimization_. New York, NY: Springer New York, 2006, pp. 164–192.
* [52] N. K. Verma, R. K. Sevakula, S. Dixit, and A. Salour, “Intelligent condition based monitoring using acoustic signals for air compressors,” _IEEE Transactions on Reliability_ , vol. 65, no. 1, pp. 291–309, March 2016\.
* [53] C. Lessmeier, J. Kimotho, D. Zimmer, and W. Sextro, “Condition monitoring of bearing damage in electromechanical drive systems by using motor current signals of electric motors: A benchmark data set for data-driven classification,” _European Conf., PHM Society, Bilbao (Spain)_ , vol. 3, no. 1, 2016.
* [54] W. A. Smith and R. B. Randall, “Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study,” _Mechanical Systems and Signal Processing_ , vol. 64, pp. 100–131, 2015.
|
construct some interesting hermitian line bundles on it, and explain the relations between them.
\subsection{A special Shimura variety}
\label{ss:special shimura}
To attach a Shimura variety to our fixed $V$, choose relevant (Definition \ref{def:relevant}) hermitian spaces $(W_0,h_0)$ and $(W,h)$ of signatures $(1,0)$ and $(n-1,1)$, respectively, in such a way that
\begin{equation}\label{hermitian hom}
V \iso \Hom_\kk(W_0 , W)
\end{equation}
as $$-hermitian spaces, where the hermitian form $⟨- , - ⟩$ on the right satisfies
\begin{equation}\label{basic hom hermitian}
\langle x,y\rangle \cdot h_0(w_0,w'_0 ) = h( x(w_0) , y(w'_0) )
\end{equation}
for all $w_0 , w_0'∈W_0$ and $x,y∈V$.
\begin{remark}
Such $W_0$ and $W$ always exist: take $W_0=\kk$ with its norm form, and $W=V$.
They are not uniquely determined by $V$, but their strict similarity classes are.
\end{remark}
As in \S 2.2 of [32], if $S$ is a connected $_$-scheme and
\[
(A_0,A) \in \mathcal{M}_{W_0}(S) \times \mathcal{M}_{W} (S),
\]
then $__(A_0,A)$ carries a positive definite hermitian form
\begin{equation}\label{KR hermitian}
\langle x,y\rangle = \psi_0^{-1} \circ y^\vee \circ \psi\circ x \in \End_{\co_\kk}(A_0) \iso \co_\kk,
\end{equation}
where $ψ_0 :A_0 →A_0^∨$ and $ψ:A→A^∨$ are the principal polarizations.
%If $S=\Spec(F)$ with $F$ an algebraically closed field and $\ell \neq\mathrm{char}(F)$ is prime, we can repeat this construction with $A_0$ and $A$ replaced by their $\ell$-adic Tate modules to endow $\Hom_{\co_\kk} ( T_\ell(A_0) , T_\ell(A) )$ with an $\co_{\kk,\ell}$-valued hermitian form.
%If $S=\Spec(\C)$ may similarly endow $\Hom_{\co_\kk} ( H_1(A_0,\Z) , H_1(A,\Z) )$ with an $\co_\kk$-valued hermitian form, and then
%\Hom_{\co_\kk} ( T_\ell(A_0) , T_\ell(A) ) \iso \Hom_{\co_\kk} ( H_1(A_0,\Z) , H_1(A,\Z) ) \otimes_\Z \Z_\ell.
As in \S 2.3 of [13], there is an open and closed substack
\begin{equation}\label{moduli inclusion}
\mathcal{S}_V \subset \mathcal{M}_{W_0} \times_{\co_\kk} \mathcal{M}_{W}
\end{equation}
characterized by its points valued in algebraically closed fields, which are those pairs
\[
(A_0,A)\in \mathcal{M}_{W_0} (F) \times \mathcal{M}_{W} (F)
\]
for which there exists an isometry
\[
V\otimes \Q_\ell \iso \Hom_{\co_\kk} \big( T_\ell(A_0) , T_\ell(A) \big) \otimes \Q_\ell
\]
of $_ℓ$-hermitian spaces for every $ℓ≠char(F) $.
Here the hermitian form on the right is defined as in \eqref{KR hermitian}.
When $F=$ this is equivalent to the existence of an isometry of $$-hermitian spaces
\[
V \iso \Hom_{\kk} \big( H_1(A_0 , \Q) , H_1(A , \Q) \big) .
\]
\begin{remark}
When $n$ is even the inclusion \eqref{moduli inclusion} is an isomorphism.
\end{remark}
\begin{remark}\label{rem:projection fiber}
The projection
\mathcal{S}_V \to \mathcal{M}_W
is a finite \'etale surjection, and the fiber over a geometric point $x \in \mathcal{M}_W(\F)$ satisfies
\[
\sum_{ y \in \mathcal{S}_{V,x}(\F) } \frac{1}{ |\Aut(y) | } = \frac{ |\mathrm{CL}(\kk) |} { | \co_\kk^\times|}
\cdot \begin{cases}
1 & \mbox{if $n$ is even} \\
2^{1-o(D)} & \mbox{if $n$ is odd}.
\end{cases}
\]
\end{remark}
\begin{remark}\label{rem:honest shimura}
The stack $\mathcal{S}_V$ is denoted $\mathcal{S}_\Kra$ in [13].
Later we will want to vary $V$, and so we have included it in the notation to avoid confusion.
As explained in [loc.~cit.], the generic fiber of $\mathcal{S}_V$ is a Shimura variety
for the subgroup $G \subset \mathrm{GU}(W_0) \times \mathrm{GU}(W)$ of pairs $(g_0,g)$ for which the similitude factors of the two components are equal.
\end{remark}
\begin{definition}\label{def:special exceptional}
If $n\ge 2$, define the \emph{exceptional divisor}
\[
\mathrm{Exc}_V \subset \mathcal{S}_V
\]
as the pullback of the exceptional divisor \eqref{basic exceptional} via $ \mathcal{S}_V \to \mathcal{M}_{(n-1,1)}$. Equivalently, it is defined by the cartesian diagram
\begin{equation}\label{special exceptional}
\xymatrix{
{\mathrm{Exc}_V } \ar[r] \ar[d] & { \mathcal{S}_V } \ar[d] \\
{\mathrm{Sing}_{(n-1,1)} } \ar[r] & { \mathcal{M}^\Pap_{(n-1,1)} }.
\end{equation}
\end{definition}
\begin{definition}[{[32]}]
\label{def:KR}
For any positive $m\in \Z$, the \emph{Kudla-Rapoport divisor} $\mathcal{Z}_V(m)$ is the $\co_\kk$-stack classifying triples $(A_0,A,x)$ consisting of a pair
\[
(A_0,A) \in \mathcal{S}_V(S)
\]
and an
x\in \Hom_{\co_\kk}(A_0,A)
satisfying $\langle x,x\rangle=m$.
\end{definition}
The natural forgetful morphism
\[
\mathcal{Z}_V(m) \to \mathcal{S}_V
\]
is finite and unramified, with image of codimension $1$. We denote again by $𝒵_V(m)$ the image of this morphism, viewed as a divisor on $𝒮_V$ in the usual way (that is to say, each irreducible component of $𝒵_V(m)$ contributes an irreducible component of the image, counted with multiplicity equal to the length of the local ring of its generic point).
Denote by
\[
\bar{\mathcal{Z}}_V(m)\to \bar{\mathcal{S}}_V
\]
the normalization of $𝒵_V(m) →𝒮̅_V$, and denote in the same way its image, viewed as a divisor on $𝒮̅_V$. In other words, take the Zariski closure of $𝒵_V(m)$.
Loosely speaking, each Kudla-Rapoport divisor is a union of unitary Shimura varieties associated to $$-hermitian spaces of signature $(n-2,1)$. In \S \ref{s:KR divisors} we will make this more precise, at least when $m$ is a prime split in $$.
\begin{definition}
Let $N$ be a positive integer, and let $S$ be an $\co_\kk$-scheme with $N\in \co_S^\times$.
A \emph{level $N$-structure} on a pair $(A_0,A) \in \mathcal{S}_V(S)$ consists of level $N$-structures $(\eta_0,\xi_0)$ and $(\eta,\xi)$ on $A_0$ and $A$, in the sense of Definition \ref{def:kramer-pappas level}, such that $ \xi_0 = \xi$.
\end{definition}
Adding level structure to pairs $(A_0,A)$ defines a finite \'etale cover
\[
\mathcal{S}_V(N) \to \mathcal{S}_{ V /\co_\kk[1/N] },
\]
and \eqref{moduli inclusion} lifts to a canonical open and closed immersion
\[
\mathcal{S}_V (N) \subset \mathcal{M}_{W_0} (N) \times_{\co_\kk[1/N]} \mathcal{M}_W (N).
\]
Define a toroidal compactification
\begin{equation}\label{compact inclusion}
\bar{\mathcal{S}}_V (N) \subset \mathcal{M}_{W_0} (N) \times_{\co_\kk[1/N]} \bar{\mathcal{M}}_W (N)
\end{equation}
as the Zariski closure of $𝒮_V(N)$, or, equivalently, as the normalization of
\[
\mathcal{S}_V (N) \to \bar{\mathcal{M}}_{W/ \co_\kk[1/N]}.
\]
When $N=1$ we abbreviate this to $𝒮̅_V$.
\begin{remark}\label{rem:special compact level}
As \eqref{compact inclusion} is an open and closed immersion, and $\mathcal{M}_{W_0} (N)$ is finite \'etale over $\co_\kk[1/N]$, the compactification $\bar{\mathcal{S}}_V (N)$ inherits all the nice properties of $\bar{\mathcal{M}}_W (N)$. In particular, Proposition \ref{prop:full compactification} holds word-for-word with $\mathcal{M}_W$ replaced by $\mathcal{S}_V$, and the same is true of the entire discussion of arithmetic intersection theory in \S \ref{ss:chow} and \S \ref{ss:volumes}.
\end{remark}
\subsection{Construction of hermitian line bundles}
The construction \eqref{hodge metric} associates to the universal CM elliptic curve $A_0 →ℳ_W_0 = ℳ_(1,0)$ its metrized Hodge bundle
\begin{equation}\label{elliptic hodge}
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{M}_{W_0} } \in \widehat{\Pic}(\mathcal{M}_{W_0} ) .
\end{equation}
Similarly, the universal $A →ℳ_W$ determines a metrized Hodge bundle
\begin{equation}\label{basic hodge}
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W } \in \widehat{\Pic}(\mathcal{M}_W ) .
\end{equation}
Pulling back the universal objects via projection to the two factors in \eqref{moduli inclusion} yields a universal pair $(A_0,A)$ of polarized abelian schemes (of relative dimensions $1$ and $n$) over $𝒮_V$.
As in \S 2.4 of [13] there is a \emph{metrized line bundle of modular forms}
\begin{equation}\label{metrized taut}
\widehat{\taut}_V \in \widehat{\Pic}(\mathcal{S}_V).
\end{equation}
The line bundle underlying \eqref{metrized taut} has inverse
\begin{equation}\label{dual taut}
\taut_V^{-1} = \Lie(A_0) \otimes \Lie(A) / \mathcal{F} ,
\end{equation}
where $ℱ ⊂(A)$ is the universal hyperplane satisfying Kr\"amer's signature condition (\S \ref{ss:basic moduli}).
The hermitian metric is defined as in \S 7.2 of [13]:
if we use Proposition 2.4.2 of [13] to identify
\begin{equation}\label{taut realization}
\taut_{V,z} \subset \Hom_{\kk} ( H_1(A_{0,z} ,\Q) , H_1(A_z,\Q) ) \otimes_\Q\C \iso V\otimes_\Q\C
\end{equation}
at a complex point $z∈𝒮_V()$, the line $_V,z$ is isotropic with respect to the $$-bilinear extension of the $$-bilinear form $ [ x,y] = Tr_/ ⟨x,y⟩$
on $V$, and
\begin{equation}\label{taut metric}
\| s \|^2 = - \frac{ [s,\overline{s}] }{ 4\pi e^\gamma }
\end{equation}
for any $s∈_V,z$.
Note that our $_V$ is denoted $ω$ in [13, 14].
\begin{proposition}\label{prop:taut-hodge1}
The hermitian line bundles \eqref{basic hodge} and \eqref{metrized taut} lie in the subgroups
\begin{align*}
\widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK ) \subset \widehat{\Pic}(\mathcal{M}_W )
\quad \mbox{and} \quad
\widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK ) \subset \widehat{\Pic}(\mathcal{S}_V ),
\end{align*}
respectively, of \eqref{pre-log injection}.
\end{proposition}
\begin{proof}
The extension to $\bar{\mathcal{M}}_W$ of the line bundle
\[
\omega^\mathrm{Hdg}_{A / \mathcal{M}_W } \iso \det( \Lie(A) )^{-1}
\]
underlying \eqref{basic hodge} is part of \eqref{hyperplane extension}.
The fact that the hermitian metric has a pre-log singularity along the boundary is a special case of
Theorem 6.16 of [12].
Note that this also uses Proposition 3.2 of [16], which shows that any rank one log-singular hermitian vector bundle in the sense of [12] is also a pre-log-singular hermitian line bundle in the sense of Definition 1.20 of [6].
This proves the claim for \eqref{basic hodge}, and the proof for \eqref{metrized taut} is the same.
\end{proof}
Pulling back the hermitian line bundles \eqref{elliptic hodge} and \eqref{basic hodge} via projection to the two factors in \eqref{compact inclusion}, we obtain three hermitian line bundles
\begin{equation}\label{three bundles}
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{S}_V },\,
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{S}_V },\,
\widehat{\mathcal{L}}_V
\in \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK ).
\end{equation}
In the remaining subsections we will make explicit the relations between them.
The reader may wish to skip directly to Theorem \ref{thm:taut-hodge compare} for the main results.
\subsection{An application of the Chowla-Selberg formula}
We will prove that, up to numerical equivalence (Definition \ref{def:numerical}), the first line bundle in \eqref{three bundles} is just the trivial line bundle with a constant metric.
The constant defining the metric is an interesting quantity in its own right.
Recall the Chowla-Selberg formula: the Faltings height, normalized as in \S 5.2 of [14],
of any elliptic curve with CM by $_$ is
\begin{equation}\label{faltings}
h^\mathrm{Falt}_\kk =
- \frac{1}{2} \frac{L'(0,\eps)}{L(0,\eps)} - \frac{1}{4} \log(4\pi^2 D ) .
\end{equation}
\begin{proposition}\label{prop:easy numerical}
The metrized line bundles
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{S}_V } \in \widehat{\Pic}(\mathcal{S}_V )
\quad \mbox{and}\quad ( 0 , C_1) \in \widehat{\Pic}(\mathcal{S}_V )
\]
are numerically equivalent, where $C_1 = \log(2\pi) + 2 h^\mathrm{Falt}_\kk$, and we are using the notation of \eqref{constant metrics}.
\end{proposition}
\begin{proof}
Fix an $N\ge 3$. As $\mathcal{M}_{W_0}(N)$ is finite \'etale over $\co_\kk[1/N]$, there is an isomorphism
\[
\mathcal{M}_{W_0}(N) \iso \bigsqcup_i \mathcal{X}_i
\]
with each $\mathcal{X}_i \iso \Spec(\co_{\kk_i} [ 1/N])$ for a finite field extension $\kk_i/\kk$ unramified outside $N$.
For each $\mathcal{X}_i$, consider the metrized Hodge bundle
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{X}_i} \in \widehat{\Pic}(\mathcal{X}_i )
\]
of the universal $A_0 \to \mathcal{X}_i$. The underlying line bundle can be identified with an element in the ideal class group of $\co_{\kk_i}[1/N]$, and hence some power of it admits a trivializing section
\[
s_i \in H^0\big( \mathcal{X}_i , ( \omega^\mathrm{Hdg}_{A_0/\mathcal{X}_i})^{\otimes d_i} \big)
\iso
H^0( A_0 , \Omega^{\otimes d_i} _{ A_0 / \mathcal{X}_i } ).
\]
Comparing \eqref{hodge metric} with the definition of the Faltings height, normalized as in \S 5.2 of [14], shows that
\begin{align*}
\frac{ -1}{ \# \mathcal{X}_i(\C) } \sum_{ x\in \mathcal{X}_i(\C) } \log\| s_{i,x}\|^2
% & = \frac{ -1}{ \# \mathcal{X}_i(\C) } \sum_{ x\in \mathcal{X}_i(\C) } \log \left| \frac{1}{2\pi} \int_{ A_{0,x} (\C)} s_{i,x} \wedge \overline{s}_{i,x} \right| \\
& = d_i \cdot C_1,
\end{align*}
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$.
Letting $d$ be the least common multiple of all $d_i$'s, we see that
\[
\big( \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)} \big)^{\otimes d} \in \widehat{\Pic}( \bar{\mathcal{S}}_V(N) )
\]
admits a trivializing section $s$ with $-\log\| s\|^2$ constant on every connected component of $\bar{\mathcal{S}}_V(N)(\C)$, and such that the average value of $-\log\| s\|^2$ over any $\Aut(\C/\kk)$-orbit of components is $d\cdot C_1$,
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$.
If we represent
\[
d \cdot \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)} \in \widehat{\mathrm{CH}}^1( \bar{\mathcal{S}}_V(N) )
\]
by the arithmetic divisor
\widehat{\mathrm{div}}(s) = ( 0 , - \log\| s\|^2),
then for any
\[
(\mathcal{Z}_N , g_N) \in \widehat{\mathrm{CH}}^{n-1}( \bar{\mathcal{S}}_V(N) )
\]
the vanishing of the Chern form of $-\log\| s\|^2$ implies the $*$-product formula
\[
[ - \log\| s\|^2] * g_N = - \log\| s\|^2 \wedge \delta_{\mathcal{Z}_N},
\]
which implies the intersection formula
\[
d \cdot \widehat{\deg}_N\big( (\mathcal{Z}_N , g_N) \cdot
\widehat{\omega}^\mathrm{Hdg} _{A_0/\mathcal{S}_V(N)} \big)
- \sum_{ x \in \mathcal{Z}_N(\C) } \log\| s_x\|^2
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$.
Let $L\subset \C$ be a finite Galois extension of $\kk$ large enough that all complex points of $\mathcal{Z}_N$ are defined over $L$, and rewrite the equality above as
\[
d \cdot \widehat{\deg}_N \big( (\mathcal{Z}_N , g_N) \cdot
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)} \big)
\frac{-1}{ [ L:\kk] } \sum_{ \substack{ x \in \mathcal{Z}_N(L) \\ \sigma \in \Gal(L/\kk) } } \log\| s_{x^\sigma} \|^2.
\]
The right hand side is $d C_1 \cdot \# \mathcal{Z}_N(\C)$, and hence
\[
\widehat{\deg}_N \big( (\mathcal{Z}_N , g_N) \cdot \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)} \big)
= C_1 \cdot \# \mathcal{Z}_N(\C)
= \widehat{\deg}_N \big( (\mathcal{Z}_N , g_N) \cdot (0,C_1) \big)
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$.
Varying $N$ shows that
\[
\widehat{\deg}\big( \widehat{\mathcal{Z}} \cdot \widehat{\omega}^\mathrm{Hdg} _{A_0/\mathcal{S}_V} \big)
\widehat{\deg}\big( \widehat{\mathcal{Z}} \cdot (0,C_1) \big)
\]
for every $\widehat{\mathcal{Z}} \in \widehat{\mathrm{CH}}^{n-1}( \bar{\mathcal{S}}_V )$,
and the claim follows.
\end{proof}
\subsection{Another hermitian line bundle}
In order to relate the hermitian line bundles of \eqref{three bundles}, we recall from \S 5.1 of [14] a fourth hermitian line bundle.
Denote by
\[
H^1_\mathrm{dR}(A) = \mathbb{R}^1\pi_* \Omega^\bullet_{A/\mathcal{S}_V}
\]
the first relative algebraic deRham cohomology of $π: A→𝒮_V$, a rank $2n$ vector bundle on $𝒮_V$.
The action of $_$ on $A$ induces an action on $H_1^dR(A) = H^1_dR(A)^∨$, which is locally free of rank $n$ over $_⊗__S$.
\[
H_1^\mathrm{dR}(A) \to \mathcal{V}
\]
denotes the largest quotient on which the action of $_$ is through the structure morphism $_→_𝒮_V$, then $𝒱$ is a rank $n$ vector bundle on $𝒮_V$, equipped with
a morphism
\[
\det(\mathcal{V})^{-1} \to \bigwedge\nolimits^n H^1_{\mathrm{dR}}(A)
\to H^n_{\mathrm{dR}}(A) .
\]
Given a complex point $z∈𝒮_V()$ and a vector
s_z ∈(𝒱_z)^-1,
we view $s_z$ as an element of $H^n(A_z,)$, and define
\[
\| s_z \|^2 = \left| \int_{A_z (\C)} s_z \wedge \overline{s}_z \right| .
\]
This defines a hermitian metric on $(𝒱)^-1$, and hence we obtain a hermitian line bundle
\begin{equation}\label{det bundle}
\det(\mathcal{V}) \in \widehat{\Pic}(\mathcal{S}_V) .
\end{equation}
\begin{proposition} \label{prop:full bundle compare}
Assume $n\ge 2$.
Recalling the exceptional divisor of Definition \ref{def:special exceptional} and the notation \eqref{constant metrics}, we have the equality
\[
2 \widehat{\taut}_V
= \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+ 2 \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}
+ \det( \mathcal{V}) + ( \mathrm{Exc}_V , C_2)
\]
in $\widehat{\Pic}(\mathcal{S}_V)$, where
\[
C_2 = 2 \log\left( \frac{2 e^\gamma}{ D} \right) + (2-n) \log(2\pi).
\]
In particular, $\det(\mathcal{V}) \in \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK )$ by \eqref{three bundles}.
\end{proposition}
\begin{proof}
This is a restatement of Proposition 5.1.2 of [14], keeping in mind that the metric on
\[
\det(\Lie(A))^{-1} = \omega^\mathrm{Hdg}_{A / \mathcal{S}_V}
\]
used in [loc.~cit.] differs from \eqref{hodge metric} by a power of $2\pi$.
\end{proof}
It is an observation of Gross [18] that the hermitian line bundle $(𝒱)$
behaves in the generic fiber, for all arithmetic purposes, like the trivial bundle with a constant metric.
This observation was extended to integral models in
\S 5.3 of [14], whose results are the basis of the following proposition.
\begin{proposition}\label{prop:gross numerical}
The Chern form of $\det(\mathcal{V})$ is identically $0$.
If $n>2$ then, up to numerical equivalence,
\[
\det(\mathcal{V}) = ( 0 , C_3 ),
\]
\[
C_3 = (4-2n) h_\kk^\mathrm{Falt} + \log( 4\pi^2 D).
\]
\end{proposition}
\begin{proof}
Fix an integer $N\ge 3$.
As in Theorem 1 of [18], the $\C$-vector space $H^0 ( \mathcal{S}_V(N)_{/\C} , \det(\mathcal{V}) )$ has dimension $1$, and the norm of any global section is a locally constant function on $\mathcal{S}_V(N)(\C)$. The vanishing of the Chern form follows.
Moreover, one can choose a nonzero global section $t$ defined over a finite Galois extension $\kk'/\kk$,
and then for every $\kk$-algebra embedding
$\sigma:\kk' \to \C$ the norm $\| t^\sigma \|$ must again be locally constant.
If $n>2$ then\footnote{The assumption $n>2$ is mistakenly omitted in Theorem 5.3.1 of [14], whose proof requires that $\mathcal{M}^\Pap_{(n-1,1)}$ has geometrically normal fibers.
The normality is a theorem of Pappas when $n>2$, but is false when $n=2$.}
Theorem 5.3.1 of [14] allows us to choose $t$ so that it extends to
a nowhere vanishing section
\[
t \in H^0 \big( \mathcal{S}_V(N)_{/\co_{\kk'}[1/N] } , \det(\mathcal{V}) \big) .
\]
Setting $d=[\kk':\kk]$ and taking the tensor product of all $\Gal(\kk'/\kk)$-conjugates of $t$, we obtain a section
\[
s \in H^0( \mathcal{S}_V(N) , \det(\mathcal{V})^{\otimes d})
\]
such that $\mathrm{div}(s) =0$, and such that $-\log\| s\|$ is locally constant.
\[
c_X = -\log\| s\|^2
\]
denote its value on the connected component $X \subset \mathcal{S}_V(N)_{/\C}$.
Fix a finite Galois extension $L/\kk$ contained $\C$ large enough that every component
$X \subset \mathcal{S}_V(N)_{/\C}$ is defined over $L$.
We may further enlarge $L$ to assume that each $X$ admits an $L$-point
\[
x \in X(L) \subset \mathcal{S}_V(N)(L)
\]
that extends to
\[
\underline{x} : \Spec( \co_L[1/N] ) \to \mathcal{S}_V(N).
\]
For example, start by fixing a complex point $x\in X(\C)$ corresponding to a pair $(A_0,A)$ for which $A$ has complex multiplication. Then enlarge $L$ so that both $A_0$ and $A$ (along with their level $N$-structures) are defined over $L$ and have everywhere good reduction.
Applying Corollary 5.3.2 and Proposition 5.3.3 of [14] to $\underline{x}$, we find
\[
\frac{-1}{ [ L:\kk] } \sum_{ \sigma \in \Gal(L/\kk) } \log\| s_{x^\sigma} \|^2
= d C_3
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$, and hence
\[
\frac{1}{ [ L:\kk] } \sum_{ \sigma \in \Gal(L/\kk) } c_{X^\sigma} = d C_3
\]
up to the same ambiguity.
We have now shown that the average value of $-\log\|s\|^2$ over any $\Aut(\C/\kk)$ orbit of connected components in $\mathcal{S}_V(N)(\C)$ is $d C_3$,
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$. The proposition follows from this, by the same argument used in the proof of Proposition \ref{prop:easy numerical}.
\end{proof}
\subsection{Comparison of hermitian line bundles}
%\label{ss:bundle comparisons}
We now come to the main results of \S \ref{s:special shimura bundles}.
The exceptional divisor of Definition \ref{def:special exceptional} is a recurring nuisance,
in part because it has nontrivial arithmetic intersection with $_V$.
This will be explored more fully in \S \ref{ss:exceptional volume} below.
The following proposition allows us to avoid this nuisance by slightly modifiying $_V$, and also clarifies the relation between the second and third line bundles in \eqref{three bundles}.
\begin{theorem}\label{thm:taut-hodge compare}
Assume $n\ge 2$.
Recalling the notation \eqref{constant metrics}, the hermitian line bundle
\[
\widehat{\tautmod}_V \define 2 \widehat{\taut}_V - (\mathrm{Exc}_V,0) \in \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK )
\]
enjoys the following properties.
\begin{enumerate}
\item
Every irreducible component $E\subset\mathrm{Exc}_V$ satisfies
\[
(E,0) \cdot \widehat{\tautmod}_V =0
\]
in $ \widehat{\mathrm{CH}}^2( \bar{\mathcal{S}}_V , \mathscr{D}_\BKK )_\Q$, as well as the height relation
\[
\mathrm{ht}_{\widehat{\tautmod}_V } (E) =0 .
\]
The same equalities hold if we replace $\widehat{\tautmod}_V$ with $\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}$ or $\widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}$.
\item
We have the equality of Chern forms
\[
\chern( \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V} ) =2 \chern( \widehat{\taut}_V ) =
\chern( \widehat{\tautmod}_V ) .
\]
\item
If $n>2$ then, up to numerical equivalence,
\[
\widehat{\tautmod}_V
\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V} + ( 0 , C_0(n) )
\in \widehat{\mathrm{Pic}}(\mathcal{S}_V) ,
\]
where $C_0(n)$ is the constant of Theorem \ref{thm:intro main}. In particular, up to numerical equivalence,
\[
2 \widehat{\taut}_V
\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V} + ( \mathrm{Exc}_V , C_0(n) ) .
\]
\end{enumerate}
\end{theorem}
\begin{proof}
For (1), they key observation is that the abelian scheme $A \to \mathcal{S}_V$ is a pullback via the vertical arrow on the right in \eqref{special exceptional}.
In particular, $\omega^\mathrm{Hdg}_{A/\mathcal{S}_V}$ is isomorphic to the pullback of
\[
\omega^\mathrm{Hdg}_{A/\mathcal{M}^\Pap_{(n-1,1)}} \in \Pic ( \mathcal{M}^\Pap_{(n-1,1)} ) .
\]
If $\mathcal{M}^\Pap_{(n-1,1)}$ were a scheme we could trivialize this latter line bundle over a Zariski open neighborhood of the ($0$-dimensional) singular locus.
This would pull-back to a trivialization of $\omega^\mathrm{Hdg}_{A/ \mathcal{S}_V}$ over an open neighborhood of $\mathrm{Exc}_V$, and in particular over an open neighborhood of
any irreducible component $E\subset \mathrm{Exc}_V$.
To account for the stackiness, simply fix an integer $N\ge 3$
and apply the same reasoning with level $N$-structure to see that
$\omega^\mathrm{Hdg}_{A/\mathcal{S}_V(N)}$ is trivial in some Zariski open neighborhood of
\[
E(N) = E \times_{ \mathcal{S}_V} \mathcal{S}_V(N).
\]
This implies the arithmetic intersection formula
( E(N) , 0 ) \cdot \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V(N)} =0
and varying $N$ proves
\[
( E , 0 ) \cdot \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} =0 .
\]
The line bundle \eqref{det bundle} is also a pullback via the vertical arrow on the right in \eqref{special exceptional}, hence the same argument shows
\[
(E,0) \cdot \det(\mathcal{V}) =0.
\]
The proof of Proposition \ref{prop:easy numerical} shows that, for any $N\ge 3$,
there is a positive multiple of
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)}
\in \widehat{\mathrm{CH}}^1( \bar{ \mathcal{S}} _V (N) , \mathscr{D}_\BKK )
\]
that can be represented by a purely archimedean arithmetic divisor $(0,g)$.
Any such arithmetic divisor satisfies
(E(N) ,0 ) \cdot (0,g) =0,
and varying $N$ shows that
\[
(E ,0 ) \cdot \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V} =0.
\]
Rewriting the relation of Proposition \ref{prop:full bundle compare} as
\begin{equation}\label{Kalt}
\widehat{\tautmod}_V
= \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+ 2 \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}
+ \det( \mathcal{V}) + ( 0 , C_2) ,
\end{equation}
we have shown that the right hand side has trivial arithmetic intersection with $(E,0)$, and hence so does the left hand side. This proves the first equality of (1). The second is a formal consequence of this and \eqref{height degree}.
The first equality of (2) follows from \eqref{Kalt},
as the Chern forms of the final three terms on the right vanish by Lemma \ref{lem:numerical basics}, Proposition \ref{prop:easy numerical}, and Proposition \ref{prop:gross numerical}.
The second equality of (2) is clear from the definitions.
Claim (3) also follows from \eqref{Kalt}, using Proposition \ref{prop:easy numerical}, Proposition \ref{prop:gross numerical}, and the equality
$ C_0 (n) = 2C_1 + C_2+ C_3$.
\end{proof}
\begin{remark}\label{rem:taut-hodge volume}
If $n>2$ then part (3) of Theorem \ref{thm:taut-hodge compare} implies
\begin{align*}
2^n \cdot \widehat{\mathrm{vol}} ( \widehat{\tautmod}_V )
& = \widehat{\mathrm{vol}} \big( \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+ ( 0 , C_0(n) ) \big) \\
& =
\widehat{\mathrm{vol}} \big( \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V} \big) + n C_0(n) \int_{\mathcal{S}_V(\C) } \chern( \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V} )^{n-1}
\end{align*}
where we have used Lemma \ref{lem:numerical basics} for the first equality, and Lemma \ref{lem:trivial volume shift} for the second.
In Proposition \ref{prop:taut-hodge strong volume} we will show that this equality also holds when $n=2$.
\end{remark}
\subsection{Volume of the exceptional divisor}
\label{ss:exceptional volume}
In this subsection we assume $n ≥2$, and fix an irreducible ($=$ connected) component $E$ of the exceptional divisor of Definition \ref{def:special exceptional}.
Recalling the notation \eqref{constant metrics}, our goal is to compute the arithmetic volume of
\[
(E,0) \in \widehat{\Pic}( \bar{\mathcal{S}}_V ).
\]
By definition of the exceptional divisor, there is a commutative diagram with cartesian squares
\begin{equation*}%\label{singular point}
\xymatrix{
{E} \ar[r] \ar[d] & { e } \ar[d] \\
{\mathrm{Exc}}_V \ar[r] \ar[d] & { \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathrm{Sing}_{(n-1,1)} } \ar[d] \\
{\mathcal{S}_V} \ar[r] & { \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{(n-1,1)} }
\end{equation*}
in which $e$ is a connected component of the $0$-dimensional reduced and irreducible $_$-stack
$ℳ_(1,0) ×__ Sing_(n-1,1)$. In particular, $e$ is supported in a single characteristic $p|D$, and admits a presentation as a stack quotient
\[
e \iso \Delta \backslash \Spec(\F_\mathfrak{p}'),
\]
in which $𝔭 ⊂_$ is the prime above $p$, $_𝔭'$ is a finite extension of $_𝔭=_/𝔭$, and $Δ$ is a finite group acting on $_𝔭'$.
Define a rational number
\[
m_E \define \sum_{ z\in e(\F_\mathfrak{p}^\alg) } \frac{ 1 }{ |\Aut(z)|} = \frac{[ \F_\mathfrak{p}' : \F_\mathfrak{p} ] }{ |\Delta| } .
\]
\begin{proposition}\label{prop:projective intersection}
The iterated intersection
\[
\widehat{\taut}_V^{-1} \cdots \widehat{\taut}_V^{-1} \cdot (E,0) \in \widehat{\mathrm{CH}}^n(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)
\]
has arithmetic degree $\widehat{m}_E = m_E \log(p)$
\end{proposition}
\begin{proof}
After Remark \ref{rem:singular description}, one might expect that $E$ is isomorphic to the projective space of hyperplanes in an $n$-dimensional vector space.
In particular, there should be a universal hyperplane $\mathcal{F} \subset \co_E^n$, with the property that the restriction of $\taut_V^{-1}$ to $E$ is isomorphic to $\co_E^n /\mathcal{F}$.
The obstruction to this being true is due to stacky issues, which can be removed by fixing an integer $N\ge 3$ prime to $p$ and adding level $N$-structure to the universal pair $(A_0,A)$ over $e$.
Let $e(N) \to e$ be the scheme classifying such level structures, and consider the cartesian diagram
(this is the definition of the upper left corner)
\[
\xymatrix{
{ E(N) } \ar[r]\ar[d] & { e(N) } \ar[d] \\
{ E } \ar[r] & { e .}
\]
The scheme $e(N)$, being a reduced scheme finite over $\F_\mathfrak{p}$, is a disjoint union of finitely many spectra of finite extensions of $\F_\mathfrak{p}$.
Fix a connected component $E' \subset E(N)$, and let
\[
\Spec(\F_\mathfrak{p}') = e' \subset e(N)
\]
be the connected component below it.
As in Remark \ref{rem:singular description}, $E'$ is precisely the projective space over $e'$ classifying hyperplanes $ \mathcal{F} \subset \Lie(A|_{e'})$.
Moreover, after fixing a trivialization $\Lie(A_0|_{e'}) \iso \F_\mathfrak{p}'$, the isomorphism \eqref{dual taut} identifies
\[
\taut_V^{-1} |_{E'} = \Lie(A|_{E'}) / \mathcal{F} \in \Pic(E') \iso \mathrm{CH}^1(E')
\]
where $\mathcal{F} \subset \Lie(A|_{E'})$ is the universal hyperplane.
A routine exercise then shows that the iterated intersection
\[
( \taut_V^{-1} |_{E'}) \cdots (\taut_V^{-1} |_{E'}) \in \mathrm{CH}^{n-1}(E') \iso \Z
\]
is represented by the cycle class of any $\F_\mathfrak{p}'$-valued point of $E'$.
In other words, the cycle class of any section to $E' \to e'$.
Now allow the connected component $E' \subset E(N)$ to vary. If we fix
any section to $E(N) \to e(N)$, and use it to view $e(N)$ as a $0$-cycle on $E(N)$, then
\[
e(N) = ( \taut_V^{-1} |_{E(N)}) \cdots (\taut_V^{-1} |_{E(N)}) \in \mathrm{CH}^{n-1}(E(N)) .
\]
This implies the arithmetic intersection formula
\[
( e(N) , 0 ) = \widehat{\taut}_V^{-1} \cdots \widehat{\taut}_V^{-1} \cdot ( E(N) , 0 )
\in \widehat{\mathrm{CH}}^n( \bar{\mathcal{S}}_V(N) , \mathscr{D}_\BKK )_\Q.
\]
Finally, recalling \eqref{degree normalization}, we deduce
\[
\widehat{\deg} \big( \widehat{\taut}_V^{-1} \cdots \widehat{\taut}_V^{-1} \cdot ( E , 0 ) \big)
\frac{ \# e(N) (\F_\mathfrak{p}^\alg) }{d_N} \log(p)
\sum_{ z\in e(\F_\mathfrak{p}^\alg) } \frac{\log(p)}{ |\Aut(z)|}
\]
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$. Varying $N$ completes the proof.
\end{proof}
\begin{corollary}\label{cor:exceptional volume}
The hermitian line bundle $(E,0) \in \widehat{\Pic}(\bar{\mathcal{S}}_V )$ has arithmetic volume
\[
\widehat{\vol} ( E,0) = (-2)^{n-1} \cdot \widehat{m}_E .
\]
\end{corollary}
\begin{proof}
Part (1) of Theorem \ref{thm:taut-hodge compare} implies the second equality in
\[
(E,0)\cdot (E,0) = (\mathrm{Exc}_V,0) \cdot (E,0) = 2 \widehat{\taut}_V \cdot (E,0)
\in \widehat{\mathrm{CH}}^2(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)_\Q.
\]
This implies the iterated intersection formula
\[
(E,0)\cdots (E,0)
= 2^{n-1} \widehat{\taut}_V \cdots \widehat{\taut}_V \cdot (E,0)
\in \widehat{\mathrm{CH}}^n(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)_\Q ,
\]
and claim follows using Proposition \ref{prop:projective intersection}.
\end{proof}
\section{Kudla-Rapoport divisors at split primes}
\label{s:KR divisors}
Assume $n > 2$, and let $𝒮_V$ be the Shimura variety \eqref{moduli inclusion} associated to a $$-hermitian space $V$ of signature $(n-1,1)$ containing a self-dual lattice.
Fix a prime $p$ split in $$, and factor
\[
p\co_\kk = \mathfrak{p} \overline{\mathfrak{p}}.
\]
We explain how the Kudla-Rapoport divisor $𝒵_V(p) →𝒮_V$ of Definition \ref{def:KR}
is related to a Shimura variety $𝒮_V'$ with $V'$ of signature $(n-2,1)$.
\subsection{Statement of the results}
\label{ss:split divisors main}
Let $V'$ denote the $$-hermitian space of signature $(n-2,1)$ whose local invariants satisfy
\begin{equation}\label{V'}
\mathrm{inv}_\ell(V') = ( p, -D)_\ell \cdot \mathrm{inv}_\ell(V)
\end{equation}
for all places $ℓ≤∞$. Equivalently, $V'$ is the orthogonal complement to the $$-span of any
$x∈V$ of hermitian norm $⟨x,x⟩=p$.
\begin{lemma}\label{lem:lower hermitian}
The hermitian space $V'$ admits a self-dual $\co_\kk$-lattice.
\end{lemma}
\begin{proof}
Suppose $\ell$ is a rational prime unramified in $\kk$. If $\ell \neq p$ then $p\in \Z_\ell^\times$, and hence is a norm from the unramified extension $\kk_\ell$. If $\ell =p$ then $p$ is again a norm from $\kk_\ell \iso \Q_p \times \Q_p$. In either case, $(p,-D)_\ell=1$, and so the local invariants of $V$ and $V'$ agree at all unramified primes.
In general, a $\kk$-hermitian space over $\kk$ admits a self-dual $\co_\kk$-lattice if and only if it has local invariant $1$ at every prime unramified in $\kk$.
As $V$ satisfies this condition by hypothesis, so does $V'$.
\end{proof}
Lemma \ref{lem:lower hermitian} allows us to form the $_$-stacks
\[
\mathcal{S}_{V'} \subset \bar{\mathcal{S}}_{V'} \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}_{(n-2,1)}
\]
analogous to
\[
\mathcal{S}_V \subset \bar{\mathcal{S}}_V\subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}_{(n-1,1)},
\]
but in one dimension lower. The stack $𝒮_V'$ is endowed with its own hermitian line bundle $_V'$ as in \eqref{metrized taut}, its own exceptional divisor $Exc_V'$ as in Definition \ref{def:special exceptional}, and its own universal pair $(A_0',A')$ of polarized abelian schemes with $_$-actions.
The following theorems, whose proofs will occupy the remainder of \S \ref{s:KR divisors}, lie at the core of the inductive arguments of \S \ref{s:volumes}.
\begin{theorem}\label{thm:height descent 1}
If we set
\[
\widehat{\tautmod}_V = 2 \widehat{\taut}_V - ( \mathrm{Exc}_V , 0) \in \widehat{\Pic} ( \bar{\mathcal{S}}_V , \mathscr{D}_\BKK )
\]
as in Theorem \ref{thm:taut-hodge compare}, and similarly with $V$ replaced by $V'$, then
\[
\int_{ \mathcal{Z}_V(p) (\C) } \chern( \widehat{\tautmod}_V)^{n-2}
= ( p^{n-1}+1) \vol_\C( \widehat{\tautmod}_{V'} ).
\]
Moreover, there is a rational number $a(p)$ such that
\[
\frac{ \mathrm{ht}_{ \widehat{\tautmod}_V } (\bar{\mathcal{Z}}_V(p)) } { p^{n-1}+1 }
= \widehat{\vol} (\widehat{\tautmod}_{V'} ) + a(p) \log(p) .
\]
\end{theorem}
\begin{theorem}\label{thm:height descent 2}
There is a rational number $b(p)$ such that
\begin{align*}
\frac{ \mathrm{ht}_{\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}} (\bar{\mathcal{Z}}_V(p)) }{ p^{n-1}+1 }
& = \widehat{\vol} ( \widehat{\omega}^\mathrm{Hdg}_{A'/\mathcal{S}_{V'}} ) + b(p) \log(p) \\
& \quad + (1-n) \left( \frac{L'(0,\eps)}{L(0,\eps)} +\frac{\log(D)}{2} \right)
\vol_\C( \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} ) .
\end{align*}
\end{theorem}
The proofs are rather long, so we summarize now the key steps.
The central idea is to make precise the impressionistic relation
\[
\mathcal{Z}_V(p) ``=" (p^{n-1}+1) \cdot \mathcal{S}_{V'} ,
\]
by decomposing
\begin{equation}\label{flavor decomp}
\mathcal{Z}_V(p)_{/\co_\kk[1/p]} =
\mathcal{U}_0 \sqcup \mathcal{U}_{\mathfrak{p}} \sqcup \mathcal{U}_{\overline{\mathfrak{p}}}
\end{equation}
as a disjoint union of open and closed substacks (Proposition \ref{prop:split divisor flavors} below).
The stacks on the right hand side are related to $𝒮_V'$ by closed immersions
\[
i_\mathfrak{p} : \mathcal{S}_{V' / \co_\kk[1/p]} \to \mathcal{U}_{\mathfrak{p}} ,\qquad
i_{\overline{\mathfrak{p}}} : \mathcal{S}_{V' / \co_\kk[1/p]} \to \mathcal{U}_{\overline{\mathfrak{p}}},
\]
and by a diagram
\[
\xymatrix{
{ \mathcal{S}_{V' / \co_\kk[1/p]} } & { \mathcal{T}_{V'} } \ar[l] \ar[r]^{i_0} & { \mathcal{U}_0 }
\]
in which the leftward arrow is a finite \'etale surjection of degree $p^n-1-1$, and the rightward arrow is a closed immersion.
See \eqref{tau cover} for the definition of the stack $ 𝒯_V' $.
Recalling the exceptional locus of Definition \ref{def:special exceptional},
denote by
\[
\mathcal{S}^{\nonexc}_V = \mathcal{S}_V \smallsetminus \mathrm{Exc}_V
\]
the (open) nonexceptional locus of $𝒮_V$, set
\[
\mathcal{Z}^{\nonexc}_{V}(p) = \mathcal{Z}_{V}(p) \times_{\mathcal{S}_{V}} \mathcal{S}^\nonexc_V ,
\]
and make the same definitions with $V$ replaced by $V'$.
\[
\mathcal{U}^\nonexc_\square = \mathcal{U}_\square \cap \mathcal{Z}^\nonexc_V(p)
\]
for $□∈{ 0, 𝔭 ,𝔭 }$, and
\[
\mathcal{T}^\nonexc_{V'} = \mathcal{T}_{V'} \times_{\mathcal{S}_{V'}} \mathcal{S}_{V'}^\nonexc .
\]
We will show that the closed immersions $i_0$, $i_𝔭$, and $i_𝔭$ are close to being isomorphisms. More precisely, $i_0$ restricts to an isomorphism
\[
i_0 : \mathcal{T}^\nonexc_{V'} \iso \mathcal{U}^\nonexc_0,
\]
$i_𝔭$ and $i_𝔭$ restrict to isomorphisms
\[
i_\mathfrak{p} : \mathcal{S}^\nonexc_{V' / \co_\kk[1/p]} \iso \mathcal{U}^\nonexc_{\mathfrak{p}}
\quad \mbox{and}\quad
i_{\overline{\mathfrak{p}}} : \mathcal{S}^\nonexc_{V' / \co_\kk[1/p]} \iso \mathcal{U}^\nonexc_{\overline{\mathfrak{p}}}.
\]
After taking compactifications into account, both theorems above will follow easily from these isomorphisms.
Note that it suffices to work only over the nonexceptional locus, as part (1) of Theorem \ref{thm:taut-hodge compare} guarantees that the hermitian line bundles in questions have trivial arithmetic intersection with all components of the exceptional divisor (which is why we work with $𝒦_V$ in Theorem \ref{thm:height descent 1} instead of $ℒ_V$).
%If $F$ is an algebraically closed field, it follows from the local model calculations of Pappas [45] that an $F$-point
% (A_0,A) \in \mathcal{S}_{V} (F)
%lies in nonexceptional locus if and only if the action of $\sqrt{-D} \in \End(A)$ on $\Lie(A)$ is nonzero.
%See \S 2.3, and especially Theorem 2.3.2, of [13] for a more thorough discussion.
\subsection{Decomposing the Kudla-Rapoport divisor}
The decomposition \eqref{flavor decomp} is a geometric reflection of a result of linear algebra.
As in \eqref{hermitian hom}, choose an isomorphism
\[
V \iso \Hom_\kk(W_0,W)
\]
in which $(W_0,h_0)$ and $(W,h)$ are relevant hermitian spaces of signatures $(1,0)$ and $(n-1,1)$.
Fix self-dual $_$-lattices $𝔞_0 ⊂W_0$ and $𝔞 ⊂W$.
Any vector
\[
x\in \Hom_{\co_\kk}(\mathfrak{a}_0, \mathfrak{a}) \subset V
\]
with $⟨x,x⟩=p$ determines an orthogonal decomposition
\[
W = \tilde{W}_0 \oplus W'
\]
with $W̃_0 = x(W_0)$, and a corresponding decomposition
\[
V = \kk x \oplus V'
\]
with $V'= _(W_0 ,W')$.
\begin{lemma}\label{lem:lattice trifurcation}
The $\co_\kk$-lattice
\tilde{\mathfrak{a}}_0=\mathfrak{a} \cap \tilde{W}_0
satisfies exactly one of
\[
\tilde{\mathfrak{a}}_0 = x(\mathfrak{a}_0) ,\qquad \tilde{\mathfrak{a}}_0 =\mathfrak{p}^{-1} x(\mathfrak{a}_0), \qquad
\tilde{\mathfrak{a}}_0=\overline{\mathfrak{p}}^{-1} x(\mathfrak{a}_0).
\]
\end{lemma}
\begin{proof}
Use $x$ to identify $W_0 = \tilde{W}_0$, and hence $\mathfrak{a}_0 \subset \tilde{\mathfrak{a}}_0 \subset \mathfrak{a}$.
Recalling the symplectic forms
\[
e_0: W_0 \times W_0 \to \Q ,\qquad e: W \times W \to \Q
\]
of \eqref{symplectic}, the relation \eqref{basic hom hermitian} implies that $e|_{W_0} = p \cdot e_0$.
The inclusion
\[
e_0 ( p \tilde{\mathfrak{a}}_0 , \mathfrak{a}_0)
= e ( \tilde{\mathfrak{a}}_0 , \mathfrak{a}_0) \subset e( \mathfrak{a} , \mathfrak{a}) =\Z,
\]
together with the self-duality of $\mathfrak{a}_0$ under $e_0$,
shows that $p \tilde{\mathfrak{a}}_0 \subset \mathfrak{a}_0$. If equality held we would have
\[
e_0( \mathfrak{a}_0 , \mathfrak{a}_0 ) = p^{-1} e( \mathfrak{a}_0 , \mathfrak{a}_0)
\subset p e( \tilde{\mathfrak{a}}_0,\tilde{\mathfrak{a}}_0)\subset p e(\mathfrak{a},\mathfrak{a}) \subset p \Z,
\]
contradicting $\mathfrak{a}_0$ being self-dual under $e_0$.
As $\tilde{\mathfrak{a}}_0$ is $\co_\kk$-stable with
$\mathfrak{a}_0 \subset \tilde{\mathfrak{a}}_0 \subsetneq p^{-1} \mathfrak{a}_0$, it is
$\mathfrak{a}_0$, $\mathfrak{p}^{-1} \mathfrak{a}_0$, or $\overline{\mathfrak{p}}^{-1} \mathfrak{a}_0$.
\end{proof}
\begin{proposition}\label{prop:split divisor flavors}
Let $(A_0,A,x)$ be the universal triple over $\mathcal{Z}_V(p)_{/\co_\kk[1/p]}$, so that
x\in \Hom_{\co_\kk}(A_0,A)
satisfies $\langle x,x\rangle =p$.
There is a decomposition \eqref{flavor decomp} into open and closed substacks in which
\begin{itemize}
\item
$\mathcal{U}_0$ is the locus of points where $\ker(x)=0$,
\item
$\mathcal{U}_\mathfrak{p}$ is the locus of points where $\ker(x) = A_0[\mathfrak{p}]$,
\item
$ \mathcal{U}_{\overline{\mathfrak{p}}}$ is the locus of points where $\ker(x) = A_0[ \overline{\mathfrak{p}}]$.
\end{itemize}
\end{proposition}
\begin{proof}
Recalling \eqref{KR hermitian}, the relation $\langle x,x\rangle =p$ implies that multiplication-by-$p$ factors as
\[
A_0 \map{x} A \iso A^\vee \map{x^\vee} A_0^\vee \iso A_0,
\]
and so $\ker(x) \subset A_0[p]$. Both of these group schemes are finite \'etale over $\mathcal{Z}_V(p)_{/\co_\kk[1/p]}$, which implies that each of $\mathcal{U}_0$, $\mathcal{U}_\mathfrak{p}$, and $\mathcal{U}_{\overline{\mathfrak{p}}}$ is open and closed.
It only remains to prove that every geometric point $s \to \mathcal{Z}_V(p)_{/\co_\kk[1/p]}$ is contained in one of them.
Abbreviate $T_0= T_p(A_{0 s})$ and $T=T_p(A_s)$.
Applying the snake lemma to the diagram
\[
\xymatrix{
0 \ar[r] & {T_0 } \ar[r] \ar[d]_{x_s} & { T_0\otimes \Q_p } \ar[r] \ar[d]_{x_s} & { A_{0 s}[p^\infty]} \ar[r] \ar[d]_{x_s} & 0 \\
0 \ar[r] & {T } \ar[r] & { T \otimes \Q_p } \ar[r] & { A_{s}[p^\infty]} \ar[r] & 0 ,
\]
and using the vertical arrow on the left to identify $T_0 \subset T$, we find that
\[
\ker(x_s) \iso \tilde{T}_0 / T_0,
\]
where $\tilde{T}_0 = T \cap T_{0 \Q_p}$.
After fixing an isomorphism $\Z_p \iso \Z_p(1)$ of \'etale sheaves on $s$, there are unique $\co_{\kk,p}$-valued hermitian forms $h_0$ and $h$ on $T_0$ and $T$, respectively, related to the Weil pairings $e_0$ and $e$ by \eqref{symplectic}. Thus we may apply Lemma \ref{lem:lattice trifurcation} with $\mathfrak{a}_0$ and $\mathfrak{a}$ replaced by $T_0$ and $T$, to see that $\tilde{T}_0$ must be one of $T_0$, $\mathfrak{p}^{-1} T_0$, or $\overline{\mathfrak{p}}^{-1} T_0$. These three cases correspond to $\ker(x_s)$ being trivial, $A_{0s}[\mathfrak{p}]$, or $A_{0s}[\overline{\mathfrak{p}}]$.
\end{proof}
\subsection{Analysis of $\mathcal{U}_\mathfrak{p}$}
In this subsection we study the structure of the substack $𝒰_𝔭$ of \eqref{flavor decomp},
and make explicit its relation to $𝒮_V'$.
The analogous analysis of $𝒰_𝔭$ is obtained by replacing $𝔭$ by $𝔭$ everywhere.
Return to the situation of Lemma \ref{lem:lattice trifurcation}, so that
$x∈__(𝔞_0,𝔞)$ with $⟨x,x⟩=p$ determines an orthogonal decomposition
\[
W = \tilde{W}_0 \oplus W' .
\]
Set $𝔞̃_0 = 𝔞∩W̃_0$.
\begin{lemma}\label{lem:pi linear algebra}
If $\tilde{\mathfrak{a}}_0 =\mathfrak{p}^{-1} x(\mathfrak{a}_0)$, there is an orthogonal decomposition
\[
\mathfrak{a} = \tilde{\mathfrak{a}}_0 \oplus \mathfrak{a}' \subset W
\]
in which $\mathfrak{a}' = \mathfrak{a} \cap W'$.
\end{lemma}
\begin{proof}
If we use $x$ to identify $W_0 = \tilde{W}_0$, the assumption $\tilde{\mathfrak{a}}_0=\mathfrak{p}^{-1}\mathfrak{a}_0$ implies that $\tilde{\mathfrak{a}}_0$ is self-dual with respect to the hermitian form $h|_{ \tilde{W}_0 }= p h_0$. The desired decomposition then follows by elementary linear algebra.
\end{proof}
The lemma suggests that if $(A_0,A,x) ∈𝒰_𝔭(S)$ for an $_[1/p]$-scheme $S$, then $x:A_0 →A$ should determine an $_$-linear splitting
\begin{equation}\label{pi product}
A = \tilde{A}_0 \times A'
\end{equation}
of principally polarized abelian schemes. Indeed, this is the case.
If we set
\[
\tilde{A}_0 = A_0 / A_0[\mathfrak{p}]
\]
and recall that $(x) = A_0[𝔭]$, the morphism $x :A_0 →A$ factors as
\[
A_0 \to \tilde{A}_0 \map{y} A
\]
for some $y∈__(Ã_0, A) $ satisfying $⟨y,y⟩=1$.
In other words, the composition
\[
\tilde{A}_0 \map{y} A \iso A^\vee \map{y^\vee} \tilde{A}_0^\vee \iso \tilde{A}_0
\]
is the identity. This implies that the composition
\[
A \iso A^\vee \map{y^\vee} \tilde{A}_0^\vee \iso \tilde{A}_0 \map{y} A
\]
is a Rosati-fixed idempotent in $__(A)$, and $A$ admits a unique splitting \eqref{pi product}
of principally polarized abelian schemes over $S$ such that this idempotent is the projection to the first factor.
Apply the above construction to the universal triple $(A_0,A,x)$ over $𝒰_𝔭$, and
recall that $A$ comes equipped with an $_$-stable hyperplane $ℱ ⊂(A)$ satisfying Kr\"amer's signature condition.
Using the decomposition
\[
\Lie(A) = \Lie(\tilde{A}_0) \oplus \Lie(A')
\]
of vector bundles on $S$, denote by
the largest closed substack over which $(Ã_0 ) ⊂ℱ$.
\begin{proposition}\label{prop:i_p construction}
There is a canonical isomorphism
\[
i_\mathfrak{p} : \mathcal{S}_{V'/\co_\kk[1/p]} \iso \mathcal{U}_\mathfrak{p}^\dagger.
\]
\end{proposition}
\begin{proof}
As above, let $S$ be an $\co_\kk[1/p]$-scheme.
Given a point $(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\dagger(S)$, the splitting \eqref{pi product} determines
\[
A' \in \mathcal{M}_{( n-2,1)}(S),
\]
where we have endowed $A'$ with the $\co_\kk$-stable hyperplane
\[
\mathcal{F}' = \mathcal{F} / \Lie(\tilde{A}_0 ) \subset \Lie(A)/ \Lie(\tilde{A}_0 ) \iso \Lie(A').
\]
satisfying Kr\"amer's condition.
The pair $(A_0,A')$ defines an $S$-point of
\[
\mathcal{S}_{V'} \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}_{( n-2,1)},
\]
and we have now constructed a morphism
\begin{equation}\label{i_p inverse}
\mathcal{U}^\dagger_\mathfrak{p} \map{ (A_0,A,x) \to (A_0,A')} \mathcal{S}_{V'/\co_\kk[1/p]}.
\end{equation}
Conversely, start with an $S$-point
\[
(A'_0 , A') \in \mathcal{S}_{V'}(S) .
\]
First define elliptic curves $A_0 = A_0'$ and $\tilde{A}_0 = A_0 /A_0[\mathfrak{p}]$.
Then define an abelian scheme $A$ by \eqref{pi product}, and endow $A$ with its product principal polarization and product $\co_\kk$-action.
Recalling that $A'$ comes equipped with a hyperplane $\mathcal{F}' \subset \Lie(A')$ satisfying Kr\"amer's condition, we endow $A$ with the hyperplane
\[
\mathcal{F} = \Lie( \tilde{A}_0 ) \oplus \mathcal{F}' \subset \Lie(A).
\]
It is easy to check that $A$, with its extra data, defines an $S$-point of $\mathcal{M}_{(n-1,1)}$, and that
$(A_0,A)$ defines an $S$-point of the open and closed substack
\[
\mathcal{S}_V \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}_{(n-2,1)}.
\]
If we define $x\in \Hom_{\co_\kk}(A_0,A)$ as the composition
\[
A_0 \to \tilde{A}_0 \hookrightarrow \tilde{A}_0 \times A' =A,
\]
where the first arrow is the quotient map, then $\langle x ,x \rangle =p$ and $\ker(x) =A_0[\mathfrak{p}]$.
The triple $(A_0,A,x)$ defines an $S$-point of $\mathcal{U}^\dagger_\mathfrak{p}$, and the morphism
\[
\mathcal{S}_{V'/\co_\kk[1/p]} \map{ (A'_0 , A') \to (A_0,A,x) } \mathcal{U}^\dagger_\mathfrak{p}
\]
is inverse to \eqref{i_p inverse}.
\end{proof}
Proposition \ref{prop:i_p construction} gives us a commutative diagram
\begin{equation}\label{i_p pullback}
\xymatrix{
{ \mathcal{S}_{V'/\co_\kk[1/p] } } \ar[r]\ar[d] & { \mathcal{S}_{V/\co_\kk[1/p] } } \ar[d] \\
{ \big( \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-2,1) } \big)_{/\co_\kk[1/p]} } \ar[r] & { \big( \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-1,1) } \big)_{/\co_\kk[1/p]} }
\end{equation}
in which the top horizontal arrow is the composition
\[
\mathcal{S}_{V'/\co_\kk[1/p]} \map{i_\mathfrak{p}} \mathcal{U}^\dagger_\mathfrak{p} \hookrightarrow \mathcal{Z}_V(p)_{/\co_\kk[1/p]} \to \mathcal{S}_{V/\co_\kk[1/p]} ,
\]
and the bottom horizontal arrow sends $(A_0' , A') ↦(A_0,A)$, where
\begin{equation}\label{i_p pullback explicit}
A_0 = A_0' , \quad \tilde{A}_0 = A_0 / A_0[\mathfrak{p}] , \quad A = \tilde{A}_0 \times A'.
\end{equation}
Both horizontal arrows are finite and unramified.
\begin{proposition}\label{prop:i_p pullbacks}
The homomorphism
\[
\widehat{\Pic} ( \mathcal{S}_{V/\co_\kk[1/p]} )_\Q \to \widehat{\Pic} ( \mathcal{S}_{V'/\co_\kk[1/p]} )_\Q
\]
induced by \eqref{i_p pullback} sends
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_V } \mapsto \widehat{\omega}^\mathrm{Hdg}_{A'_0/ \mathcal{S}_{V'} } + \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'} } ,
\]
where $(A_0' , A')$ is the universal pair over $\mathcal{S}_{V'}$. The same map also sends
\[
\widehat{\taut}_V \mapsto \widehat{\taut}_{V'}
\quad \mbox{and}\quad
(\mathrm{Exc}_V,0) \mapsto (\mathrm{Exc}_{V'} ,0 ).
\]
\end{proposition}
\begin{proof}
It follows from \eqref{i_p pullback explicit} that the top horizontal arrow in \eqref{i_p pullback} sends
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_V } \mapsto
\widehat{\omega}^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} }
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'} }.
\]
As we have inverted $p$ on the base, the degree $p$ quotient map $A_0' \to \tilde{A}_0$ induces an isomorphism on Lie algebras, and hence an isomorphism
\[
\omega^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} } \iso \omega^\mathrm{Hdg}_{ A'_0/ \mathcal{S}_{V'} } .
\]
This isomorphism does not respect the metrics \eqref{hodge metric}, but one can easily check that
\[
\widehat{\omega}^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} }
\widehat{\omega}^\mathrm{Hdg}_{ A'_0/ \mathcal{S}_{V'} } + (0 , - \log(p) )
\]
as elements of $\widehat{\Pic}( \mathcal{S}_{V'/\co_\kk[1/p]})$. The correction term $(0 , - \log(p) )$ is torsion in the arithmetic Picard group, as
\[
2 ( 0 , -\log(p) ) = (\mathrm{div}(p) , - \log(p^2) ) = 0,
\]
and the first claim follows.
For the second claim, let $S$ be an $\co_\kk[1/p]$-scheme, fix an $S$-point
$(A_0' , A') \in \mathcal{S}_{V'}(S)$, and let $(A_0,A) \in \mathcal{S}_V(S)$ be its image under the top horizontal arrow in \eqref{i_p pullback}.
Tracing through the construction of $i_\mathfrak{p}$ in Proposition \ref{prop:i_p construction}, we see that
\[
\Lie(A) / \mathcal{F} \iso \Lie(A') / \mathcal{F}'.
\]
It follows immediately from this and \eqref{dual taut} that $\taut_V|_S \iso \taut_{V'}|_S$.
At a complex point $s \in S(\C)$ we have a decomposition
\[
H_1(A_s , \Q) = H_1( \tilde{A}_{0s} ,\Q) \oplus H_1( A_s' , \Q),
\]
which induces an isometric embedding
\[
\Hom_\kk( H_1(A_{0s} ,\Q ) , H_1( A_s' , \Q) ) \subset \Hom_\kk( H_1(A_{0s} ,\Q ) , H_1( A_s , \Q) )
\]
of $\kk$-hermitian spaces. Recalling \eqref{taut realization}, this isometric embedding restricts to the isomorphism
$\taut_{V',s} \iso \taut_{V,s}$ just constructed, which therefore preserves the metrics defined by \eqref{taut metric}.
Finally, we show that under the top horizontal arrow in \eqref{i_p pullback}, the exceptional divisor $\mathrm{Exc}_V \subset \mathcal{S}_V$ pulls back to the exceptional divisor $\mathrm{Exc}_{V'} \subset \mathcal{S}_{V'}$.
These divisors are, by Definition \ref{def:special exceptional}, the pullbacks of the singular loci
\begin{align}
\mathcal{M}_{(1,0)} \times_{\co_\kk} \mathrm{Sing}_{ (n-1,1) }
& \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-1,1) } \label{first singular} \\
\mathcal{M}_{(1,0)} \times_{\co_\kk} \mathrm{Sing}_{ (n-2,1) }
& \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-2,1) } \label{second singular}
\end{align}
of \eqref{singular} under the vertical arrows in \eqref{i_p pullback}, and so it suffices to prove that the first singular locus \eqref{first singular} pulls back to the second singular locus \eqref{second singular} under the bottom horizontal arrow in \eqref{i_p pullback}.
Moreover, as each of \eqref{first singular} and \eqref{second singular} is a reduced $\co_\kk$-stack of dimension $0$, and as the bottom horizontal arrow in \eqref{i_p pullback} is finite and unramified, it suffices to check this on the level of geometric points.
This is an easy exercise in linear algebra, using the characterization of the singular locus found in
Remark \ref{rem:singular description}.
%So, let $F$ be an algebraically closed field $F$ of characteristic dividing $D$, fix an $F$-point
%(A_0',A') \in \mathcal{M}_{(1,0)}(F) \times \mathcal{M}^\Pap_{ (n-2,1) }(F) ,
%and let
%(A_0,A) \in \mathcal{M}_{(1,0)}(F) \times \mathcal{M}^\Pap_{ (n-1,1) }(F)
%be its image under the bottom horizontal arrow in \eqref{i_p pullback}.
%By \eqref{i_p pullback explicit}, there in an $\co_\kk$-linear isomorphism of $F$ vector spaces
%\Lie(A) = \Lie( \tilde{A}_0 ) \oplus \Lie(A').
%Consider the endomorphism $\sqrt{-D} \in \End( \tilde{A}_0 )$. As $F$ has characteristic dividing $D$, the square of this endomorphism annihilates the $1$-dimensional space $\Lie(\tilde{A}_0)$, which implies that $\sqrt{-D}$ also annihilates it.
%Now we use Pappas's characterization of the singular locus [45] (see also Theorem 2.3.2 of [13]): the pair $(A_0',A')$ lies in \eqref{second singular} if and only if $\sqrt{-D} \in \End(A')$ annihilates the Lie algebra of $A'$.
%This hold if and only if $\sqrt{-D} \in \End(A)$ annihilates the Lie algebra of $A$, which holds if and only if $(A_0,A)$ lies in \eqref{first singular}.
% Thus \eqref{first singular} pulls back to \eqref{second singular} under \eqref{i_p pullback}, completing the proof.
\end{proof}
\begin{proposition}\label{prop:i_p nonexceptional}
The nonexceptional locus $ \mathcal{U}_\mathfrak{p}^\nonexc \subset \mathcal{U}_\mathfrak{p}$ satisfies
\begin{equation*}
%\label{dagger open}
\mathcal{U}_\mathfrak{p}^\nonexc \subset \mathcal{U}_\mathfrak{p}^\dagger,
\end{equation*}
and the isomorphism of Proposition \ref{prop:i_p construction} restricts to an isomorphism
\[
\mathcal{S}^\nonexc_{V' / \co_\kk[1/p] } \iso
\mathcal{U}^\nonexc_{\mathfrak{p}} .
\]
\end{proposition}
\begin{proof}
Suppose $S$ is an $\co_\kk[1/p]$-scheme,
\[
(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\nonexc(S),
\]
and let $A'$ be the abelian scheme of \eqref{pi product}.
In particular, \[\Lie(A) = \Lie( \tilde{A}_0 ) \oplus \Lie(A').\]
The natural map
\[
\mathcal{U}_\mathfrak{p}^\nonexc \to \mathcal{M}_{(n-1,1)} \smallsetminus \mathrm{Exc}_{(n-1,1)}
\]
endows $A$ with a hyperplane $\mathcal{F} \subset \Lie(A)$, and results of Kr\"amer, as summarized in Theorem 2.3.3 of [13], shows that this hyperplane is actually determined by the $\co_\kk$-action on $A$ using the following recipe:
If we fix a $\pi \in \co_\kk$ such that $\co_\kk= \Z[\pi]$, and let
\[
\overline{\epsilon}_S = \overline{\pi} \otimes 1 - 1\otimes i_S(\overline{\pi}) \in \co_\kk \otimes_\Z \co_S,
\]
where $i_S : \co_\kk \to \co_S$ is the structure map, then
\[
\mathcal{F} = \mathrm{ker}( \overline{\epsilon}_S : \Lie(A) \to \Lie(A) ).
\]
On the other hand, the action of $\co_\kk$ on the Lie algebra of $A_0$ is through the structure morphism $i_S$, and so the same is true of $\tilde{A}_0 = A_0 / A_0[\mathfrak{p}]$.
This implies that $\overline{\epsilon}_S$ annihilates $ \Lie( \tilde{A}_0 )$, and hence
\Lie( \tilde{A}_0 ) \subset \mathcal{F}.
Having shown
\[
(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\dagger(S),
\]
the first claim of the proposition is proved.
The second claim follows from the first and Proposition \ref{prop:i_p pullbacks}.
\end{proof}
\begin{proposition}\label{prop:i_p toroidal}
The closed immersion $i_\mathfrak{p} : \mathcal{S}_{V'/\co_\kk[1/p]} \to \mathcal{U}_\mathfrak{p}$ of Proposition \ref{prop:i_p construction} extends uniquely to a proper morphism
\[
\bar{\mathcal{S}}_{V'/\co_\kk[1/p] } \to \bar{\mathcal{U}}_\mathfrak{p},
\]
where the codomain is defined as the Zariski closure of
\[
\mathcal{U}_\mathfrak{p} \subset \bar{\mathcal{Z}}_V(p)_{/\co_\kk[1/p]}.
\]
or, equivalently, as the normalization of
$ \mathcal{U}_\mathfrak{p} \to \bar{\mathcal{S}}_{V/\co_\kk[1/p]}.$
\end{proposition}
\begin{proof}
For ease of notation, we omit all subscripts $\co_\kk[1/p]$ in the proof.
As the exceptional locus of $\bar{\mathcal{S}}_{V'}$ does not meet the boundary, it suffices to construct the extension over its complement
\[
\bar{\mathcal{S}}_{V'}^\nonexc = \bar{\mathcal{S}}_{V'} \smallsetminus \mathrm{Exc}_{V'} .
\]
Consider the universal pair $(A_0,A')$ over $\mathcal{S}_{V'}^\nonexc$, and let
\[
x: A_0 \to A = \tilde{A}_0 \times A'
\]
be as in the proof of Proposition \ref{prop:i_p construction}. In other words, the triple $(A_0,A,x)$ over $\mathcal{S}_{V'}^\nonexc$ determines the isomorphism
i_\mathfrak{p} : \mathcal{S}_{V'}^\nonexc \iso \mathcal{U}_\mathfrak{p}^\nonexc.
of Proposition \ref{prop:i_p nonexceptional}.
The discussion of \S \ref{ss:toroidal} shows that $A'$ extends to a semi-abelian scheme over $\bar{\mathcal{S}}_{V'}^\nonexc$.
The elliptic curve $A_0$ extends to an elliptic curve $\bar{\mathcal{S}}_{V'}^\nonexc$, and so the same is true of its quotient $\tilde{A}_0$.
It follows that $A$ extends to a semi-abelian scheme over $\bar{\mathcal{S}}_{V'}^\nonexc$, and the extension property used in the proof of Lemma \ref{lem:normalized compactification} provides us with a morphism
\[
i_\mathfrak{p} : \bar{\mathcal{S}}_{V'}^\nonexc \to \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}^\Pap_{(n-1,1)}
\]
taking values in the open subscheme
\[
\mathcal{M}_{(1,0)} \times_{\co_\kk} \big( \bar{\mathcal{M}}^\Pap_{(n-1,1)} \smallsetminus \mathrm{Sing}_{(n-1,1)} \big)
\iso
\mathcal{M}_{(1,0)} \times_{\co_\kk} \big( \bar{\mathcal{M}}_{(n-1,1)} \smallsetminus \mathrm{Exc}_{(n-1,1)} \big).
\]
This provides us with a morphism
\[
i_\mathfrak{p} : \bar{\mathcal{S}}_{V'}^\nonexc \to \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}_{(n-1,1)} ,
\]
taking values in the open and closed substack $\bar{\mathcal{S}}_V$.
We now have a commutative diagram of solid arrows
\[
\xymatrix{
{ \mathcal{S}_{V'}^\nonexc } \ar[r]^{i_\mathfrak{p}} \ar[d] & { \bar{\mathcal{U}}_\mathfrak{p} } \ar[d] \\
{ \bar{\mathcal{S}}_{V'}^\nonexc } \ar[r]<EMAIL_ADDRESS> { \bar{\mathcal{S}}_V }
\]
in which the vertical arrow on the right is integral, and hence affine, by its construction as a normalization; see Lemma 29.53.4 of [51].
To complete the proof of the proposition, it suffices to show that there is a unique dotted arrow making the diagram commute.
This immediately reduces to the corresponding claim for affine schemes, in which
we are given homomorphisms of rings
\[
\xymatrix{
{ R' } & { B } \ar[l]<EMAIL_ADDRESS>\\
{ \bar{R}' } \ar[u] & { A } \ar[l] \ar[u]
\]
with $B$ integral over $A$, and $\bar{R}' \subset R'$ is an inclusion of normal domains with the same field of fractions.
The image of any $b\in B$ under $B \to R'$ is integral over $\bar{R}'$, hence contained in $\bar{R}'$.
Thus there is a unique dotted arrow making the diagram commute.
\end{proof}
\subsection{Two lemmas on abelian schemes}
For use in the next subsection, we prove two lemmas on abelian schemes.
The first is a criterion for determining when a polarization descends to a quotient.
The second is a criterion for the existence of an extension as a semi-abelian scheme.
\begin{lemma}\label{lem:polarization descent}
Suppose $f : X\to Y$ is an isogeny of abelian schemes over a Noetherian scheme $S$, and $\psi_X : X\to X^\vee$ is a polarization whose degree $d$ is invertible in $\co_S$. The following are equivalent.
\begin{enumerate}
\item
The kernel of $f$ is contained in $\mathrm{ker}(\psi_X)$, and is totally isotropic with respect to the
Weil pairing
\[
e : \mathrm{ker}(\psi_X) \times \mathrm{ker}(\psi_X) \to \mu_d.
\]
\item
There exists a (necessarily unique) polarization $\psi_Y : Y \to Y^\vee$ making the diagram
\[
\xymatrix{
{ X } \ar[r]^{\psi_X} \ar[d]_f & { X^\vee } \\
{ Y } \ar[r]_{\psi_Y} & { Y^\vee } \ar[u]_{f^\vee}
\]
\end{enumerate}
\end{lemma}
\begin{proof}
This is routine.
See, for example, Proposition 10.4 of [46].
%We may assume that $S$ is connected, and fix a geometric point $s \to S$. For each $p\mid d$ set
%V_p(X_s) = T_p(X_s) \otimes_{\Z_p} \Q_p.
%The polarization $\psi_X$ induces a $p$-adic Weil pairing
%e_p : V_p(X_s) \times V_p(X_s) \to \Q_p(1),
%and identifies $T_p(X_s^\vee)$ with the dual lattice of $T_p(X_s)$ with respect to \eqref{p-weil}.
%Moreover, we may identify
%\mathrm{ker}(\psi_X)_s \iso T_p(X_s^\vee)/T_p(X_s)
%in such a way that the Weil pairing on the left hand side agrees with the pairing
% T_p(X_s^\vee)/T_p(X_s) \times T_p(X_s^\vee)/T_p(X_s) \to (\Q_p/\Z_p)(1).
%induced by \eqref{p-weil}.
%If we now use $f:X\to Y$ to identify $V_p(X_s) \iso V_p(Y_s)$, the claim is that both properties of the proposition are equivalent to
%\item [(3)]
%for every prime $p\mid d$, the pairing \eqref{p-weil} is $\Z_p(1)$-valued on $T_p(Y_s)$.
\end{proof}
\begin{lemma}\label{lem:semiextension}
Suppose $S$ is a normal Noetherian scheme, $U\subset S$ is a dense open subscheme, and
$f : X \to Y$ is an isogeny of abelian schemes over $U$ whose degree $d$ is invertible in $\co_S$.
If $X$ extends to a semi-abelian scheme over $S$ then so does $Y$, and both extensions are unique.
\end{lemma}
\begin{proof}
The uniqueness claim follows from Proposition I.2.7 of [15], so we only need to prove the existence of a semi-abelian extension of $Y$.
Consider the morphisms
\mathrm{ker}(f) \to X[d] \to U.
The second arrow and the composition are both \'etale, as $d\in \co_U^\times$, and so the first arrow is also \'etale by \cite[\href{https://stacks.math.columbia.edu/tag/02GW}{Tag 02GW}]{stacks-project}.
As the first arrow is also a closed immersion, we deduce that $\mathrm{ker}(f) \subset X[d]$ is a union of connected components.
If $X^* \to S$ denotes the semi-abelian extension of $X$, the multiplication map $[d] : X^* \to X^*$ is quasi-finite and flat. Indeed, by Lemma 37.16.4 of [51] flatness can be checked fiberwise on $S$, where it follows from Proposition 3.11 of Expos\'e VI.B of [50], and the surjectivity of $[d]$ on any extension of an abelian variety by a torus.
The group scheme $X^*[d] \to S$ is therefore quasi-finite and flat.
Using our assumption that $d\in \co_S^\times$, and Lemma 29.36.8 of [51], we deduce that it is \'etale.
As we assume that $S$ is normal, so is $X^*[d]$. As the connected components of a normal scheme are the same as its irreducible components, it follows that the Zariski closure of $\mathrm{ker}(f)$ in $X^*[d]$ is an open and closed subscheme $\mathrm{ker}(f)^* \subset X^*[d]$. In particular $\mathrm{ker}(f)^*\to S$ is quasi-finite \'etale.
By Lemma IV.7.1.2 of [44], the fppf quotient $Y^* = X^*/\mathrm{ker}(f)^*$ provides a semi-abelian extension of $Y$ to $S$.
\end{proof}
\subsection{Analysis of $\mathcal{U}_0$}
In this subsection we study the substack $𝒰_0$ of \eqref{flavor decomp}, and make explicit its relation with $𝒮_V'$.
Return to the situation of Lemma \ref{lem:lattice trifurcation}, so that
$x∈__(𝔞_0,𝔞)$ with $⟨x,x⟩=p$ determines an orthogonal decomposition
\begin{equation}\label{W split}
W = \tilde{W}_0 \oplus W' .
\end{equation}
Set $𝔞̃_0 = 𝔞∩W̃_0$.
\begin{lemma}\label{lem:0 linear algebra}
Assume that $\tilde{\mathfrak{a}}_0 = x(\mathfrak{a}_0)$.
If we set $\mathfrak{b} = \mathfrak{a} \cap W'$ and let $\mathfrak{b}^\vee \subset W' $ be its dual lattice with respect to $h|_{ W' }$, the $\co_\kk$-lattice
\[
\mathfrak{a}'= \mathfrak{p} \cdot \mathfrak{b}^\vee + \mathfrak{b} ,
\]
is self-dual with respect to $h|_{W'}$. Moreover, there are inclusions of $\co_\kk$-lattices
\begin{equation}\label{lattice correspondence}
\tilde{\mathfrak{a}}_0 \oplus \mathfrak{a}'
\stackrel{p}{\supset} \tilde{\mathfrak{a}}_0 \oplus \mathfrak{b}
\stackrel{p^2}{\subset} \mathfrak{a}
\end{equation}
in $W$ of the indicated indices, and a canonical $\co_\kk$-linear injection
\begin{equation}\label{lattice t}
\mathfrak{p}^{-1} \tilde{\mathfrak{a}}_0/ \tilde{\mathfrak{a}}_0 \iso \mathfrak{p}^{-1} \mathfrak{b} / \mathfrak{b} \map{t} \mathfrak{p}^{-1} \mathfrak{a}'/ \mathfrak{a}' .
\end{equation}
\end{lemma}
\begin{proof}
Throughout the proof, we use $x$ to identify $W_0 = \tilde{W}_0$, so that $h|_{W_0} = p h_0$.
The projections to the two factors in \eqref{W split} induce isomorphisms
\[
\mathfrak{a} / ( \mathfrak{a}_0 \oplus \mathfrak{b} ) \to \mathfrak{a}_0^\vee / \mathfrak{a}_0,
\qquad
\mathfrak{a} /( \mathfrak{a}_0 \oplus \mathfrak{b} ) \to \mathfrak{b}^\vee / \mathfrak{b},
\]
where $\mathfrak{a}_0^\vee= p^{-1} \mathfrak{a}_0$ is the dual lattice of $\mathfrak{a}_0$ with respect to $h|_{W_0}$.
Composing these isomorphisms yields an $\co_\kk$-linear isomorphism
\begin{equation}\label{p trivialization}
p^{-1} \mathfrak{a}_0 / \mathfrak{a}_0 \iso \mathfrak{b}^\vee / \mathfrak{b}
\end{equation}
respecting the $p^{-1}\co_\kk/\co_\kk$-valued hermitian forms induced by $h|_{W_0}= ph_0$ and $h|_{W'}$.
It follows that $\mathfrak{p} \cdot ( \mathfrak{b}^\vee / \mathfrak{b} ) \subset \mathfrak{b}^\vee / \mathfrak{b}$
is maximal isotropic with respect to $h|_{W'}$, and so $\mathfrak{a}'$
is self-dual under $h|_{W'}$.
Consider the inclusions $\mathfrak{b} \subset \mathfrak{a}' \subset \mathfrak{b}^\vee$ of $\co_\kk$-modules.
The first is an isomorphism everywhere locally except at $\overline{\mathfrak{p}}$, while the second is an isomorphism everywhere locally except at $\mathfrak{p}$.
In particular, the first induces an isomorphism
\[
\mathfrak{p}^{-1} \mathfrak{b} / \mathfrak{b} \iso \mathfrak{p}^{-1} \mathfrak{a}'/\mathfrak{a}' .
\]
Restricting \eqref{p trivialization} to an isomorphism of $\mathfrak{p}$-torsion submodules defines \eqref{lattice t}.
The inclusions of \eqref{lattice correspondence}, with the indicated indices, are clear from what we have said.
\end{proof}
Just as Lemma \ref{lem:0 linear algebra} is more complicated than Lemma \ref{lem:pi linear algebra}, the analysis of $𝒰_0$ is more complicated than that of $𝒰_𝔭$.
Let $S$ be an $_[1/p]$-scheme.
Given Lemma \ref{lem:0 linear algebra}, one expects that any $S$-point $(A_0,A,x) ∈𝒰_0(S)$ should determine a diagram of abelian schemes with $_$-actions
\begin{equation}\label{abelian correspondence}
\xymatrix{
{ \tilde{A}_0 \times A' } & &{ \tilde{A}_0 \times B} \ar[ll]_{\deg = p} \ar[rr]^{\deg = p^2} & & { A}
\end{equation}
in which the arrows are $_$-linear isogenies of the indicated degrees.
There should also be a canonical $_$-linear closed immersion
\begin{equation}\label{t construct}
\tilde{A}_0[\mathfrak{p}] \iso B[\mathfrak{p}] \map{t} A'[ \mathfrak{p} ],
\end{equation}
and the morphism $x:A_0 →A$ should factor as
\[
A_0 \iso \tilde{A}_0 \hookrightarrow \tilde{A}_0 \times B \to A.
\]
Moreover, these abelian schemes should come with polarizations with the following properties.
The elliptic curve $Ã_0$ is equipped with its unique polarization of degree $p^2$,
$B$ is equipped with a polarization of degree $p^2$, and $A'$ is equipped with a principal polarization.
The pullback of the product polarization on $Ã_0 ×A'$ via the leftwards arrow in \eqref{abelian correspondence} is the product polarization on $Ã_0 ×B$, and similarly for the the pullback of the principal polarization on $A$ via the rightward arrow.
Here is how to construct the data above from $(A_0,A,x)$.
As in the proof of Proposition \ref{prop:split divisor flavors}, at any geometric point $s →S$ we may apply Lemma \ref{lem:0 linear algebra} with $𝔞_0$ and $𝔞$ replaced by the $p$-adic Tate modules of $A_0s$ and $A_s$ to obtain $_$-lattices
\begin{equation}\label{tate correspondence}
T_p( \tilde{A}_{0s}) \oplus T_p( A_s') \supset T_p( \tilde{A}_{0s}) \oplus T_p(B_s) \subset T_p(A_s).
\end{equation}
analogous to \eqref{lattice correspondence}, along with an $_$-linear injection
\begin{equation}\label{tate-level t}
\mathfrak{p}^{-1} T_p(\tilde{A}_{0s})/T_p(\tilde{A}_{0s}) \iso \mathfrak{p}^{-1} T_p(B_s)/T_p(B_s)
\map{t} \mathfrak{p}^{-1} T_p(A_s')/T_p(A_s').
\end{equation}
To be clear, we have not yet constructed the abelian schemes $Ã_0$, $A'$, and $B$, only lattices in $T_p(A_s)[1/p]$ that we choose to call $T_p(Ã_s)$, $T_p(A_s')$, and $T_p(B_s)$.
However, as $p∈_S^×$ and both inclusions in \eqref{tate correspondence} have finite index, there are abelian schemes $C$ and $D$ over $S$ with $_$-actions and $_$-linear isogenies
\[
\xymatrix{
{ D } & { C } \ar[r]^{p^2} \ar[l]_{p} & { A }
\]
of the indicated degrees that identify
\begin{equation}\label{tate-level decomp}
T_p(C_s) = T_p(\tilde{A}_{0s}) \oplus T_p(B_s) ,\qquad T_p(D_s) = T_p(\tilde{A}_{0s}) \oplus T_p(A_s').
\end{equation}
The condition that $⟨x,x⟩=p$ implies that
\[
A_0 \map{x} A \iso A^\vee \map{x^\vee} A_0^\vee \iso A_0
\]
is multiplication by $p$, which implies that the composition
\[
A \iso A^\vee \map{x^\vee} A_0^\vee\iso A_0 \map{x} A
\]
is a Rosati-fixed element $ α∈__(A)[1/p]$ such that $α∘α= pα$.
In particular, we obtain an idempotent
\[
p^{-1}\alpha \in \End_{\co_\kk}(A)[1/p]\iso \End_{\co_\kk}(C)[1/p]\iso \End_{\co_\kk}(D)[1/p],
\]
whose induced actions on $T_p(C_s)[1/p]$ and $T_p(D_s)[1/p]$ are just the projections to the first summands in \eqref{tate-level decomp}. It follows that we in fact have
\[
p^{-1}\alpha \in \End_{\co_\kk}(C) \iso \End_{\co_\kk}(D).
\]
These idempotents are the projections to the first factors for unique decompositions
\[
C= \tilde{A}_0 \times B ,\qquad D=\tilde{A}_0 \times A'
\]
of abelian schemes with $_$-actions, and the image of $x$ under
\[
\Hom_{\co_\kk}(A_0 , A) [1/p] \iso \Hom_{\co_\kk}(A_0 , C)[1/p] \to \Hom_{\co_\kk}(A_0 , \tilde{A}_0)[1/p]
\]
is an isomorphism $A_0 Ã_0$.
In particular, we now have $_$-linear isogenies \eqref{abelian correspondence}
such that taking $p$-adic Tate modules recovers \eqref{tate correspondence}, and \eqref{tate-level t} induces
the closed immersion \eqref{t construct}.
If we pullback the principal polarization via $Ã_0 ×B →A$, we obtain a polarization on $Ã_0 ×B$ of degree $p^4$.
As the idempotent $p^-1α$ constructed above is Rosati-fixed, this polarization of $Ã_0 ×B$ splits as the product of polarizations $Ã_0 →Ã_0^∨$ and $B→B^∨$.
By examining the induced Weil pairing on $p$-adic Tate modules, one can show that each polarization has kernel isomorphic, \'etale locally on $S$, to $_/(p)$, and that
\[
\ker(B\to A') = \ker( B\to B^\vee )[\overline{\mathfrak{p}}]
\]
is totally isotropic under the Weil pairing on $(B→B^∨)$.
Hence, by Lemma \ref{lem:polarization descent}, $B→B^∨$ descends to a principal polarization on $A'$.
This completes the construction of the abelian schemes \eqref{abelian correspondence} with all of their expected extra structure.
Now apply this construction to the universal triple $(A_0,A,x)$ over $𝒰_0$.
As $p∈_𝒰_0^×$, the $p$-power isogenies \eqref{abelian correspondence} induce an isomorphism
\begin{equation}\label{i_0 Lie}
\Lie(A) \iso \Lie( \tilde{A}_0) \times \Lie(A')
\end{equation}
of rank $n$ vector bundles on $𝒰_0$.
Recalling that $A$ comes equipped with an $_$-stable hyperplane $ℱ ⊂(A)$, denote by
\[
\mathcal{U}_0^\dagger \subset \mathcal{U}_0
\]
the largest closed substack over which $(Ã_0) ⊂ℱ$
Define a finite \'etale cover
\begin{equation}\label{tau cover}
\mathcal{T}_{V'} \to \mathcal{S}_{V'/\co_\kk[1/p]}
\end{equation}
of degree $p^n-1-1$ as follows:
for any $_[1/p]$-scheme $S$, let $𝒯_V'(S)$ be the groupoid of triples $(A'_0,A',t)$ in which
\[
(A'_0,A') \in \mathcal{S}_{V'}(S) \quad
\mbox{and} \quad
t : A'_0[ \mathfrak{p} ] \to A'[\mathfrak{p}]
\]
is an $_$-linear closed immersion of group schemes over $S$.
\begin{proposition}\label{prop:i_0 construction}
There is a canonical isomorphism $i_0 : \mathcal{T}_{V'} \iso \mathcal{U}_0^\dagger$.
\end{proposition}
\begin{proof}
The morphism $\mathcal{U}_0^\dagger \to \mathcal{T}_{V'}$ is essentially described above.
Suppose $S$ is an $\co_\kk[1/p]$-scheme and
\[
(A_0,A,x) \in \mathcal{U}_0^\dagger(S).
\]
We can use the isomorphism \eqref{i_0 Lie} to endow the abelian scheme
$A'$ of \eqref{abelian correspondence} with the rank one local direct summand
\[
\mathcal{F} ' = \mathcal{F} / \Lie( \tilde{A}_0) \subset \Lie(A) /\Lie( \tilde{A}_0) = \Lie(A')
\]
to obtain a point $A' \in \mathcal{M}_{(n-2,1)}(S)$. Setting $A_0'=\tilde{A}_0$,
this defines a morphism
\[
\mathcal{U}_0^\dagger \map{ (A_0,A,x) \mapsto (A'_0,A') } \mathcal{S}_{V' / \co_\kk[1/p] },
\]
and the closed immersion \eqref{t construct} defines the desired lift to
$\mathcal{U}_0^\dagger \to \mathcal{T}_{V'}$.
Now we construct the inverse.
Start with a point
(A'_0, A' , t) \in \mathcal{T}_{V'}(S).
Denote by $B$ the abelian scheme dual to
\[
B^\vee = A' / \mathrm{Im}(t).
\]
Using the principal polarization to identify $A'$ with its dual, the quotient map $A' \to B^\vee$ dualizes to a morphism $B \to A'$, and the composition
\[
B\to A' \to B^\vee
\]
is a degree $p^2$ polarization (it is the pullback via $B \to A'$ of the principal polarization on $A'$).
It has the property that each factor on the right hand side of
\[
\ker( B \to B^\vee ) = \ker(B \to B^\vee ) [\mathfrak{p}] \times \ker(B \to B^\vee ) [ \overline{\mathfrak{p}} ]
\]
has order $p$, and the second factor is the kernel of $B\to A'$. In particular, the induced map
$B[\mathfrak{p}] \to A'[\mathfrak{p}]$ is an isomorphism, and the closed immersion
\[
A'_0[\mathfrak{p}] \map{t} A'[\mathfrak{p}] \iso B[\mathfrak{p}]
\]
has image $\ker(B \to B^\vee) [\mathfrak{p}] $.
Setting $\tilde{A}_0 = A'_0$, the resulting isomorphism
\[
\tilde{A}_0[\mathfrak{p}] \iso \ker(B \to B^\vee) [\mathfrak{p}]
\]
admits a unique $\co_\kk$-linear extension to an isomorphism
\[
t : \tilde{A}_0[ p ] \iso \ker(B \to B^\vee)
\]
identifying the Weil pairings on source and target.
The antidiagonal subgroup
\[
\Delta \define \tilde{A}_0[p]\map{ a_0 \mapsto ( - a_0 , t (a_0) ) } \tilde{A}_0 \times B.
\]
is totally isotropic under the Weil pairing induced by the product polarization (where $\tilde{A}_0$ is given the polarization of degree $p^2$), which therefore descends, by Lemma \ref{lem:polarization descent}, to a principal polarization on the quotient
\[
A = (\tilde{A}_0\times B)/\Delta.
\]
As $p \in \co_S^\times$, there is an induced isomorphism of Lie algebras \eqref{i_0 Lie},
which allows us to define a corank one local direct summand $\mathcal{F} \subset \Lie(A)$ as the product
\[
\mathcal{F} = \Lie(\tilde{A}_0) \times \mathcal{F}'.
\]
This defines a point $A \in \mathcal{M}_{(n-1,1)}(S)$. Setting $A_0 = \tilde{A}_0$, the composition
\[
A_0=\tilde{A}_0 \hookrightarrow \tilde{A}_0 \times B \to A
\]
defines $x \in \Hom_{\co_\kk}(A_0,A)$ with $\langle x,x\rangle=p$
The above construction determines a morphism
\[
\mathcal{T}_{V'} \map{ (A'_0,A',t) \to (A_0,A,x) } \mathcal{Z}_V(p)
\]
taking values in the open substack $\mathcal{U}_0^\dagger$, which is inverse to the map $\mathcal{U}_0^\dagger \to \mathcal{T}_{V'}$ constructed above.
\end{proof}
Proposition \ref{prop:i_0 construction} gives us morphisms
\begin{equation}\label{i_0 pullback}
\xymatrix{
{ \mathcal{S}_{V'/\co_\kk[1/p] } }
& & { \mathcal{T}_{V'} } \ar[rr] \ar[ll]_{\eqref{tau cover}}
& & { \mathcal{S}_{V/\co_\kk[1/p] } }
\end{equation}
in which the arrow on the right is the (finite and unramified) composition
\[
{\mathcal{T}_{V'} } \map{i_0 } \mathcal{U}^\dagger_0 \hookrightarrow \mathcal{Z}_V(p)_{/\co_\kk[1/p]} \to \mathcal{S}_{V/\co_\kk[1/p]} ,
\]
sending $(A_0' , A' , t ) ↦(A_0,A)$, where $A_0=A_0'$, and $A$ is related to $A'$ by a diagram \eqref{abelian correspondence}.
\begin{proposition}\label{prop:i_0 pullbacks}
The hermitian line bundles
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}
\in \widehat{\Pic}( \mathcal{S}_{V' } )
\]
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_{V}} \in \widehat{\Pic}( \mathcal{S}_{V } )
have the same images under the homomorphisms
\[
\xymatrix{
{ \widehat{\Pic}( \mathcal{S}_{V'} ) } \ar[r] & { \widehat{\Pic} ( \mathcal{T}_{V'} ) _\Q }& { \widehat{\Pic}( \mathcal{S}_{V } ) } \ar[l]
\]
induced by \eqref{i_0 pullback}.
The same is true of
$(\mathrm{Exc}_{V'} ,0 )$ and $(\mathrm{Exc}_V,0)$, and of $\widehat{\taut}_{V'}$ and $\widehat{\taut}_V$.
\end{proposition}
\begin{proof}
If $(A'_0,A')$ denotes the pullback of the universal object via the left arrow in \eqref{i_0 pullback}, and $(A_0,A)$ denotes the pullback of the universal object via the right arrow in \eqref{i_0 pullback},
examination of the proof of Proposition \ref{prop:i_0 construction} shows that there is a quasi-isogeny
\[
f \in \Hom( A'_0 \times A' , A)[1/p]
\]
of degree $\deg(f)=p$.
As in the proof of Proposition \ref{prop:i_p pullbacks}, this implies that
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{T}_{V'}}
\widehat{\omega}^\mathrm{Hdg}_{ A'_0 / \mathcal{T}_{V'}}
\widehat{\omega}^\mathrm{Hdg}_{ A' / \mathcal{T}_{V'}} + (0 , -\log(p))
\]
as elements of $\widehat{\Pic}(\mathcal{T}_{V'} )$, and the term $(0,-\log(p))$ is torsion.
The first claim follows from this.
The remaining claims are essentially the same as in Proposition \ref{prop:i_p pullbacks}, and we leave the details to the reader.
\end{proof}
\begin{proposition}\label{prop:i_0 nonexceptional}
The nonexceptional locus $ \mathcal{U}_0^\nonexc \subset \mathcal{U}_0$ satisfies
\[
\mathcal{U}_0^\nonexc \subset \mathcal{U}_0^\dagger.
\]
Moreover, the morphism of Proposition \ref{prop:i_0 construction} restricts to an isomorphism
\[
\mathcal{S}^\nonexc_{V' / \co_\kk[1/p] } \iso
\mathcal{U}^\nonexc_0 .
\]
\end{proposition}
\begin{proof}
The proof is essentially the same as Proposition \ref{prop:i_p nonexceptional}, and the details are left to the reader.
\end{proof}
\begin{proposition}\label{prop:i_0 toroidal}
Denote by $\bar{\mathcal{T}}_{V'}$ the normalization of $ \mathcal{T}_{V'} \to \bar{\mathcal{S}}_{V'/\co_\kk[1/p] }$.
The closed immersion $i_0:\mathcal{T}_{V'} \to \mathcal{U}_0$ of Proposition \ref{prop:i_0 construction} extends to a proper morphism
\[
\bar{\mathcal{T}}_{V'} \to \bar{\mathcal{U}}_0,
\]
where the codomain is defined as the Zariski closure of
\[
\mathcal{U}_0 \subset \bar{\mathcal{Z}}_V(p)_{/\co_\kk[1/p]},
\]
or, equivalently, as the normalization of
$ \mathcal{U}_0 \to \bar{\mathcal{S}}_{V/\co_\kk[1/p]}$.
\end{proposition}
\begin{proof}
Consider the universal triple $(A_0,A',t)$ over $\mathcal{T}_{V'}$, and let $(A_0, A)$ be the pullback of the universal pair over $\mathcal{S}_V$ via the composition
\[
\mathcal{T}_{V'} \map{i_0} \mathcal{U}_0 \to \mathcal{S}_{V} .
\]
We know that $A'$ extends to a semi-abelian scheme over $\bar{\mathcal{T}}_{V'}$, obtained as a pullback via
\[
\bar{\mathcal{T}}_{V'} \to \bar{\mathcal{S}}_{V'} \to \bar{\mathcal{M}}_{(n-2,1)},
\]
and that $A_0$ extends to an elliptic curve over $\bar{\mathcal{T}}_{V'}$, obtained as a pullback via
\[
\bar{\mathcal{T}}_{V'} \to \bar{\mathcal{S}}_{V'} \to \mathcal{M}_{(1,0)},
\]
Using Lemma \ref{lem:semiextension} and the isogenies \eqref{abelian correspondence}, we deduce that $A$ also extends to a semi-abelian scheme over $\bar{\mathcal{T}}_{V'}$. With this extension in hand, the proof is essentially identical to that of Proposition \ref{prop:i_p toroidal}.
\end{proof}
\subsection{Proof of Theorems \ref{thm:height descent 1} and \ref{thm:height descent 2}}
Define an $_[1/p]$-stack
\[
\mathcal{Z}^\dagger_V(p) = \mathcal{T}_{V'} \sqcup \mathcal{S}_{V' / \co_\kk[1/p] } \sqcup \mathcal{S}_{V' /\co_\kk[1/p]} .
\]
Propositions \ref{prop:i_p construction} and \ref{prop:i_0 construction} provide us with a canonical closed immersion
\[
\mathcal{Z}^\dagger_V(p)
\map{i = i_0 \sqcup i_\mathfrak{p} \sqcup i_{\overline{\mathfrak{p}} }} \mathcal{U}_0 \sqcup \mathcal{U}_\mathfrak{p} \sqcup \mathcal{U}_{\overline{\mathfrak{p}}} = \mathcal{Z}_V(p)_{ / \co_\kk[1/p] }
\]
with image the closed substack $ 𝒰^†_0 ⊔𝒰^†_𝔭 ⊔𝒰^†_𝔭$. Hence we have a diagram
\begin{equation}\label{divisor correspondence}
\xymatrix{
{ \mathcal{Z}^\dagger_V(p) } \ar[rr]^{i} \ar[d]_\alpha & & { \mathcal{Z}_V(p)_{ / \co_\kk[1/p] } } \ar[d]^\beta \\
{ \mathcal{S}_{V'/\co_\kk[1/p]} } & & { \mathcal{S}_{V/\co_\kk[1/p] } }
\end{equation}
in which $α$ is a finite \'etale surjection of degree $p^n-1+1$, and $β$ is finite.
\begin{lemma}\label{lem:exceptional KR error}
As divisors on $ \mathcal{S}_{V/\co_\kk[1/p] }$, we have
\[
\mathcal{Z}_V(p)_{ / \co_\kk[1/p] } = \mathcal{Z}^\dagger_V(p) + E,
\]
where $\mathcal{Z}^\dagger_V(p)$ has no irreducible components contained in the exceptional divisor $\mathrm{Exc}_{V/\co_\kk[1/p]} \subset \mathcal{S}_{V/\co_\kk[1/p]}$ of Definition \ref{def:special exceptional}, and $E$ is supported entirely on the exceptional divisor.
\end{lemma}
\begin{proof}
The \emph{exceptional locus} of $ \mathcal{Z}^\dagger_V(p)$ is the preimage of the exceptional divisor under $\alpha$, and an irreducible component of it is \emph{exceptional} if it is contained in the exceptional locus.
Similarly, the \emph{exceptional locus} of $ \mathcal{Z}_V(p)_{ / \co_\kk[1/p] }$ is the preimage of the exceptional divisor under $\beta$, and an irreducible component of it is \emph{exceptional} if it is contained in the exceptional locus.
The final claims of Propositions \ref{prop:i_p nonexceptional} and \ref{prop:i_0 nonexceptional} show that $i$ restricts to an isomorphism between the non-exceptional loci of the source and target. In particular, it establishes a bijection between the generic points of non-exceptional irreducible components, and corresponding non-exceptional generic have local rings of the same (finite) length.
However, the morphism $\alpha$ is finite \'etale, so no irreducible component of the source can map into the exceptional divisor of the target.
Thus the closed immersion $i$ actually establishes a bijection between irreducible components of the source and non-exceptional irreducible components of the target, in a way that matches up their multiplicities. The claim follows immediately.
\end{proof}
Propositions \ref{prop:i_p toroidal} and \ref{prop:i_0 toroidal} imply that the diagram above extends to
\begin{equation}\label{compact divisor correspondence}
\xymatrix{
{ \bar{\mathcal{Z}}^\dagger_V(p) } \ar[rr]^i \ar[d]_\alpha & & { \bar{\mathcal{Z}}_V(p)_{ / \co_\kk[1/p] } } \ar[d]^\beta \\
{ \bar{\mathcal{S}}_{V'/\co_\kk[1/p]} } & & { \bar{\mathcal{S}}_{V/\co_\kk[1/p] } },
\end{equation}
where we have defined
\begin{equation}\label{dagger KR}
\bar{\mathcal{Z}}^\dagger_V(p) = \bar{\mathcal{T}}_{V'} \sqcup \bar{\mathcal{S}}_{V' / \co_\kk[1/p] } \sqcup \bar{\mathcal{S}}_{V' /\co_\kk[1/p]}.
\end{equation}
Equivalently, this is the normalization of
\[
\alpha: \mathcal{Z}^\dagger_V(p) \to \bar{\mathcal{S}}_{V'/\co_\kk[1/p] }.
\]
For any $N ≥1$ prime to $p$, we can add level structure to $𝒯̅_V'$ by defining
\[
\mathcal{T}_{V'}(N) = \mathcal{T}_{V'} \times_{\mathcal{S}_{V'}} \mathcal{S}_{V'}(N),
\]
and letting $𝒯̅_V'(N)$ be the normalization of
\[
\mathcal{T}_{V'}(N) \to \bar{\mathcal{T}}_{V' / \co_\kk[1/N]}.
\]
Equivalently, this is the normalization of
\[
\mathcal{T}_{V'}(N) \to \bar{\mathcal{S}}_{V'}(N)_{ / \co_\kk[ \frac{1}{Np} ]}.
\]
\begin{lemma}\label{lem:T compact}
The stack $\bar{\mathcal{T}}_{V'}(N)$ is regular, flat, and proper over $\co_\kk[\frac{1}{Np} ]$, and is a projective scheme if $N\ge 3$. It is smooth in a neighborhood of its boundary
\[
\partial \bar{\mathcal{T}}_{V'}(N) = \bar{\mathcal{T}}_{V'}(N) \smallsetminus \mathcal{T}_{V'}(N) ,
\]
which is a Cartier divisor smooth over $\co_\kk[\frac{1}{Np}]$.
\end{lemma}
\begin{proof}
The proof is similar to that of Proposition \ref{prop:full compactification}, and is again based on the results of [39] and [41].
Recall from Remark \ref{rem:honest shimura} that the generic fiber of $\mathcal{S}_{V'}$ is the Shimura variety associated to a reductive group $G'$ and a certain compact open subgroup of $G'(\A_f)$, and one can easily check that the generic fiber of the finite \'etale cover
\begin{equation}\label{alt level}
\mathcal{T}_{V'}(N) \to \mathcal{S}_{V'}(N)_{ /\co_\kk[ \frac{1}{Np} ]}
\end{equation}
is obtained by shrinking compact open subgroup.
Recall from \S \ref{ss:special shimura} that $\mathcal{S}_{V'}(N)$ is constructed as an open and closed substack
\[
\mathcal{S}_{V'}(N) \subset \mathcal{M}_{W_0}(N) \times_{\co_\kk[1/N]} \mathcal{M}_{W'}(N).
\]
In this construction, we may replace $ \mathcal{M}_{W'}(N)$ with the stack $\mathcal{M}^\Pap_{W'}(N)$ appearing in the proof of Proposition \ref{prop:full compactification} to obtain a blow-down
\[
\mathcal{S}^\Pap_{V'}(N) \subset \mathcal{M}_{W_0}(N) \times_{\co_\kk[1/N]} \mathcal{M}^\Pap_{W'}(N).
\]
Repeating the construction of \eqref{alt level} with $\mathcal{S}_{V'}(N)$ replaced by $\mathcal{S}^\Pap_{V'}(N)$ yields a finite \'etale cover
\[
\mathcal{T}^\Pap_{V'}(N) \to \mathcal{S}^\Pap_{V'}(N)_{ /\co_\kk[ \frac{1}{Np} ]},
\]
and $\mathcal{T}_{V'}(N)$ is the blow-up of the normal stack $\mathcal{T}^\Pap_{V'}(N)$ along its proper and $0$-dimensional locus of nonsmooth points.
As in the proof of Proposition \ref{prop:full compactification}, we now have morphisms
\[
\mathcal{T}^\Pap_{V'}(N) \to \mathcal{S}^\Pap_{V' }(N)_{/\co_\kk[ \frac{1}{Np} ]}
\to \mathcal{M}^\Pap_{W'}(N)_{/\co_\kk[ \frac{1}{Np} ]}
\to \bar{\mathcal{A}}_{/\co_\kk[ \frac{1}{Np} ]},
\]
and we may define $\bar{\mathcal{T}}^\Pap_{V'}(N)$ as the normalization of the composition.
We may now apply the results of [39] and [41] to deduce properties of $\bar{\mathcal{T}}^\Pap_{V'}(N)$ from properties of its interior. Arguing exactly as in the proof of Proposition \ref{prop:full compactification}, this compactification is normal and flat, and is a projective scheme if $N\ge 3$.
Its boundary is a smooth Cartier divisor. The nonsmooth locus has dimension $0$, and does not meet the boundary. As $\bar{\mathcal{T}}_{V'}(N)$ can be identified with the blow-up of $\bar{\mathcal{T}}^\Pap_{V'}(N)$ along its nonsmooth locus, it has all the same properties, and is also regular (being smooth near the boundary and regular in the interior).
\end{proof}
Lemma \ref{lem:T compact} and Remark \ref{rem:special compact level} provide us with an $_[ 1/Np ]$-stack
\[
\bar{\mathcal{Z}}^\dagger_V(p,N)
\define \bar{\mathcal{T}}_{V'}(N) \sqcup
\bar{\mathcal{S}}_{V'}(N)_{ / \co_\kk[\frac{1}{Np}] } \sqcup \bar{\mathcal{S}}_{V'}(N)_{/\co_\kk[ \frac{1}{Np} ] } ,
\]
with all the nice properties enumerated in Proposition \ref{prop:full compactification}. Thus \eqref{dagger KR} has its own theory of pre-log singular hermitian line bundles and Burgos-Kramer-K\"uhn arithmetic Chow groups
\[
\widehat{\CH}^d ( \bar{\mathcal{Z}}^\dagger_V(p) , \mathscr{D}_\BKK )
= \mil_{ \substack{ N\ge 3 \\ p \nmid N} } \widehat{\CH}^d ( \bar{\mathcal{Z}}^\dagger_V(p,N) , \mathscr{D}_\BKK ) ,
\]
exactly as in \S \ref{ss:chow}. These Chow groups include notions of arithmetic heights and volumes as in \S \ref{ss:volumes}, taking values in the abelian group $/ log(p)$.
One can repeat the construction of the diagram \eqref{compact divisor correspondence} with level structures, and so obtain pullbacks
\[
\xymatrix{
{ \widehat{\CH}^d( \bar{\mathcal{S}}_{V'} ,\mathscr{D}_\BKK ) } \ar[rr]^{\alpha^*} & &
{ \widehat{\CH}^d( \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK ) } & &
{ \widehat{\CH}^d( \bar{\mathcal{S}}_V ,\mathscr{D}_\BKK ) } \ar[ll]_{(\beta \circ i)^*}
\]
\[
\xymatrix{
{ \widehat{\Pic}( \bar{\mathcal{S}}_{V'} ,\mathscr{D}_\BKK )_\Q } \ar[rr]^{\alpha^*} & &
{ \widehat{\Pic}( \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK ) }_\Q & &
{ \widehat{\Pic}( \bar{\mathcal{S}}_V ,\mathscr{D}_\BKK )_\Q } \ar[ll]_{(\beta \circ i)^*}
\]
\begin{lemma}\label{lem:formal height induction}
If two pre-log singular hermitian line bundles
\[
\widehat{\mathcal{P}}' \in \widehat{\Pic}( \bar{\mathcal{S}}_{V'} , \mathscr{D}_\BKK)
\quad \mbox{and}\quad
\widehat{\mathcal{P}} \in \widehat{\Pic}( \bar{\mathcal{S}}_{V} , \mathscr{D}_\BKK)
\]
have the same pullback to $ \widehat{\Pic}( \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK ) _\Q$ then
\[
\int_{ \mathcal{Z}_V(p) (\C) } \chern( \widehat{\mathcal{P}})^{n-2}
= ( p^{n-1}+1) \int_{ \mathcal{S}_{V'} (\C) } \chern( \widehat{\mathcal{P}}' )^{n-2}.
\]
Moreover, there is an $a(p) \in \Q$ such that
\[
\mathrm{ht}_{\widehat{\mathcal{P}}} (\bar{\mathcal{Z}}_V(p))
(p^{n-1}+1 ) \cdot \widehat{\vol}( \widehat{\mathcal{P}}' ) + \mathrm{ht}_{\widehat{\mathcal{P}}} (E) + a(p) \log(p) ,
\]
where $E$ is the divisor of Lemma \ref{lem:exceptional KR error}.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{lem:exceptional KR error}, the closed immersion $i$ in \eqref{divisor correspondence}
restricts to an isomorphism of non-exceptional loci, and hence induces an isomorphism in the generic fiber.
\begin{align*}
\int_{ \mathcal{Z}_V(p) (\C) } \beta^* \chern( \widehat{\mathcal{P}})^{n-2}
& =
\int_{ \mathcal{Z}^\dagger_V(p) (\C) } (\beta \circ i)^* \chern( \widehat{\mathcal{P}})^{n-2} \\
& = \int_{ \mathcal{Z}^\dagger_V(p) (\C) } \alpha^* \chern( \widehat{\mathcal{P}}' )^{n-2} \\
& = ( p^{n-1}+1) \int_{ \mathcal{S}_{V'} (\C) } \chern( \widehat{\mathcal{P}}' )^{n-2} .
\end{align*}
The second claim follows from Lemma \ref{lem:exceptional KR error} and the equalities
\[
\mathrm{ht}_{ \widehat{\mathcal{P}}} (\bar{\mathcal{Z}}^\dagger_V(p))
\widehat{\vol}( (\beta \circ i)^*\widehat{\mathcal{P}} )
\widehat{\vol}( \alpha^*\widehat{\mathcal{P}}' )
(p^{n-1}+1 ) \cdot \widehat{\vol}( \widehat{\mathcal{P}}' )
\]
in $\R/\Q\log(p)$, where the first and last equalities are obtained directly by unpacking the definitions of pullbacks, heights, and arithmetic intersections in [11], and using the fact that $\alpha$ is (away from the boundary) a finite \'etale surjection of degree $p^{n-1}+1$.
%By Theorem 7.47 of [11], any arithmetic cycle class
%\widehat{\mathcal{D}} \in \widehat{\CH}^{n-1} ( \bar{\mathcal{S}}_{V'} , \mathscr{D}_\BKK )
%has a pullback
%f^* \widehat{\mathcal{D}} \in \widehat{\CH}^{n-1} ( \bar{\mathcal{Z}}^\dagger_V(p) , \mathscr{D}_\BKK ),
%which satisfies
%\begin{equation}\label{pullback degrees}
%\widehat{\deg} ( f^* \widehat{\mathcal{D}} )= (p^{n-1}+1 ) \cdot \widehat{\deg}( \widehat{\mathcal{D}} ).
%To justify \eqref{pullback degrees}, fix $N \ge 3$ prime to $p$, and represent the pullback of $\widehat{\mathcal{D}}$ to
%$\bar{\mathcal{S}}_{V'}(N)$ by an arithmetic $0$-cycle $(\mathcal{D} , g_\mathcal{D})$ with $\mathcal{D}$ supported in the interior $\mathcal{S}_{V'}(N)$. This can be done, for example, by applying Chow's moving lemma for projective varieties to the reductions of $ \bar{\mathcal{S}}_{V'}(N)$.
%The equality \eqref{pullback degrees} then follows immediately from the fact that
%f: \mathcal{Z}^\dagger_V(p,N) \to \mathcal{S}_{V'}(N)_{/\co_\kk[ \frac{1}{Np} ]}
% is a finite \'etale surjection of degree $p^{n+1}+1$.
% The final equality of \eqref{correspondence swap} follows by applying \eqref{pullback degrees} with $\widehat{\mathcal{D}}$ equal to the iterated self-intersection of $\widehat{\mathcal{P}}'$, and using the compatibility of arithmetic intersection with $f^*$.
\end{proof}
\begin{proof}[Proof of Theorems \ref{thm:height descent 1} and \ref{thm:height descent 2}]
Propositions \ref{prop:i_p pullbacks} and \ref{prop:i_0 pullbacks} imply that the hypotheses of
Lemma \ref{lem:formal height induction} are satisfied by
\[
\widehat{\mathcal{P}}' = \widehat{\tautmod}_{V'}
\quad \mbox{and} \quad
\widehat{\mathcal{P}} = \widehat{\tautmod}_{V}
\]
which therefore gives the equality
\[
\int_{ \mathcal{Z}_V(p) (\C) } \chern( \widehat{\tautmod}_V)^{n-2}
= ( p^{n-1}+1) \int_{ \mathcal{S}_{V'} (\C) } \chern( \widehat{\tautmod}_{V'} )^{n-2},
\]
and the equality
\[
\mathrm{ht}_{ \widehat{\tautmod}_V }(\bar{\mathcal{Z}}_V(p))
( p^{n-1}+1 ) \widehat{\vol} (\widehat{\tautmod}_{V'} )
+ \mathrm{ht}_{ \widehat{\tautmod}_V }(E)
\]
up to a rational multiple of $\log(p)$. The second term on the right vanishes by claim (1) of Theorem \ref{thm:taut-hodge compare}, completing the proof of Theorem \ref{thm:height descent 1}.
Similarly, Propositions \ref{prop:i_p pullbacks} and \ref{prop:i_0 pullbacks} show that the hypotheses of Lemma \ref{lem:formal height induction} are satisfied by
\[
\widehat{\mathcal{P}}' = \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}
\quad \mbox{and} \quad
\widehat{\mathcal{P}} = \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} ,
\]
and so
\[
\mathrm{ht}_{ \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} } (\bar{\mathcal{Z}}_V(p))
(p^{n-1}+1 ) \cdot \widehat{\vol}( \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} ) + \mathrm{ht}_{ \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} } (E)
\]
up to a rational multiple of $\log(p)$. One again, the second term on the right vanishes by claim (1) of Theorem \ref{thm:taut-hodge compare}.
For the first term on the right, Proposition \ref{prop:easy numerical} and Lemmas \ref{lem:numerical basics} and \ref{lem:trivial volume shift} imply
\begin{align*}
\widehat{\vol}( \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} )
& =
\widehat{\vol}( (0 , C_1) +
\widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} ) \\
&= \widehat{\vol}( \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} )
+ (n-1) C_1 \int_{\mathcal{S}_{V'}(\C) } \chern( \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}} )^{n-2} ,
\end{align*}
\[
= \log(2\pi) + 2h^\mathrm{Falt}_\kk
= - \frac{L'(0,\eps)}{L(0,\eps)} - \frac{\log(D)}{2} .
\]
This proves Theorem \ref{thm:height descent 2}.
\end{proof}
\section{Borcherds products}
\label{s:borcherds}
We continue to work with the Shimura variety $𝒮_V$ of \eqref{moduli inclusion}
associated to a $$-hermitian space $V$ of signature $(n-1,1)$, and now assume $n≥2$.
After explaining the connection between the complex orbifold $𝒮_V()$ and the Shimura variety \eqref{eq:X} associated to the unitary group $(V)$, we will use the results of \S \ref{ss:green functions} to construct Green functions for certain linear combinations of the Kudla-Rapoport divisors $𝒵_V(m) →𝒮_V$.
We then recall the arithmetic theory of Borcherds products on $𝒮_V$ from [13],
and show that one can produce Borcherds products whose divisors are linear combinations of only those $𝒵_V(p)$ with $p$ a prime congruent to $1$ modulo $D$, up to a linear combination of vertical divisors which can be computed explicitly.
\subsection{Green functions and Borcherds products}
\label{ss:general borcherds}
%We now combine Theorem~\ref{thm:int} with the formulas of \S~\ref{s:eisenstein} to derive an explicit formula for the integrals of automorphic Green functions as in \eqref{eq:greenint}.
%For simplicity we restrict ourselves to the case when the principal part of the input harmonic Maass form is supported on the trivial coset (up to the constant term). Such input forms can be obtained as lifts of scalar valued harmonic Maass forms for $\Gamma_0(D)$.
\[
V\iso \Hom_\kk(W_0,W)
\]
as in \S \ref{ss:special shimura}, fix self-dual $_$-lattices
$𝔞_0 ⊂W_0$ and $𝔞 ⊂W$ as in \S \ref{ss:basic moduli}, and define a self-dual $_$-lattice
\begin{equation}\label{lattice choice}
L = \Hom_{\co_\kk}(\mathfrak{a}_0 , \mathfrak{a}) \subset V.
\end{equation}
The subgroup $G⊂GU(W_0) ×GU(W)$ of Remark \ref{rem:honest shimura} acts on both $W_0$ and $W$ via unitary similitudes, and we denote by $K ⊂G(_f)$ the largest compact open subgroup fixing the lattices $𝔞_0$ and $𝔞$.
Recalling the hermitian symmetric domain $𝒟$ of \S \ref{ss:u(v) shimura}, we identify
\[
\mathcal{S}_V(\C) \iso G(\Q) \backslash \mathcal{D} \times G(\A_f) / K
\]
as in \S 2 of [13].
The group $G$ also acts on $V$, defining a surjective homomorphism
\[
G \to H=\Uni(V)
\]
with kernel the diagonally embedded $Res_/ 𝔾_m$.
Denoting again by $K$ the image of the above compact open subgroup under $G(_f) →H(_f)$,
and recalling the complex Shimura variety \eqref{eq:X}, we obtain a finite cover
\begin{equation}\label{cover}
\mathcal{S}_V(\C) \to \mathrm{Sh}_K(H,\mathcal{D}).
\end{equation}
Let $H^∞_2-n(D,^n)$ denote the space of $$-valued harmonic
Maass forms $f$ of weight $2-n$, level $Γ_0(D)$, and character $^n$ such that
\begin{itemize}
\item
$f$ is bounded at all cusps of $\Gamma_0(D)$ different from the cusp $\infty$,
\item
$f$ has polynomial growth at $\infty$, in sense that there is a
\[
P_f = \sum_{m\leq 0} c^+(m)q^m \in \C[q^{-1}]
\]
such that $f-P_f=o(1)$ as $q$ goes to $0$.
\end{itemize}
Such a harmonic Maass form has a Fourier expansion analogous to \eqref{eq:fourierf} with Fourier coefficients $c^±(m)∈$.
Fix an $f∈H_2-n^∞(D,^n)$ with Fourier coefficients $c^±(m)$.
This form can be lifted to a vector valued harmonic Maass form, in the sense of \S \ref{ss:green functions}, by setting
\begin{align}
\label{eq:lift}
\tilde f=\sum_{\gamma\in \Gamma_0(D)\backslash \SL_2(\Z)} (f|_{2-n} \gamma) (\omega_L(\gamma)^{-1}\varphi_0)
\in H_{2-n}(\omega_L) ,
\end{align}
where $φ_0 ∈S_L = [L'/L]$ is the characteristic function of $0 ∈L'/L$.
We denote the Fourier coefficients of $f̃$ by $c̃^±(m,μ)$ for $μ∈L'/L$ and $m∈$.
The coefficients of $f̃$ can be computed in terms of the coefficients of $f$, and for $m<0$ we have
\begin{align}
\label{eq:coeffrel}
\tilde c^{+}(m,\mu) =\begin{cases}
c^+(m) & \text{if $\mu=0$}\\
0&\text{if $\mu\neq 0$}
\end{cases}
\end{align}
as in Proposition 6.1.2 of [13] or \S 5 of [47].
Under the covering map \eqref{cover}, the divisors $Z(m)$ and the hermitian line bundle $ℒ$ of \S \ref{ss:u(v) shimura} pull pack to the divisors $𝒵_V(m)$ and $ℒ_V$ of \S \ref{ss:special shimura}. This allows us to apply the construction \eqref{eq:AutoGreen} to the vector valued form \eqref{eq:lift} to obtain a Green function
$Φ(z,h,f̃)$ for the analytic divisor
\[
Z(f) = \sum_{m>0} c^+(-m) Z(m) \in \mathrm{Div}_\C( \mathrm{Sh}_K(H,\mathcal{D}) )
\]
of \eqref{eq:zf}, which we pull back to a Green function $Φ_V(f)$ for the divisor
\[
\mathcal{Z}_V(f)= \sum_{m>0} c^+(-m) \mathcal{Z}_V(m) \in \mathrm{Div}_\C ( \mathcal{S}_V ) .
\]
If $n>2$, or if $n=2$ and $V$ is anisotropic, then
\begin{equation}\label{S_V green integral}
\vol_\C(\widehat{\taut}_V )^{-1} \int_{ \mathcal{S}_V(\C) } \Phi_V (f) \chern( \widehat{\taut}_V )^{n-1}
= \sum_{ m>0 } c^+(-m)B'(m, 0 ,s_0)
\end{equation}
by Theorem \ref{thm:int} and \eqref{eq:coeffrel}. Here, as always, $s_0=(n-1)/2$.
Now consider the subspace of weakly holomorphic forms
\[
M_{2-n}^{!,\infty}(D,\eps^n) \subset H^\infty_{2-n}(D,\eps^n) .
\]
These are meromorphic modular forms of the indicated weight, level, and character that are holomorphic outside the cusp $∞$ of $Γ_0(D)$. If we fix an
\begin{equation}\label{input form}
f(\tau) = \sum_{m \gg -\infty} c(m) q^m \in M_{2-n}^{ ! , \infty}( D , \eps^n )
\end{equation}
in such a way that $c(m) ∈$ for all $m$ then,
after possibly replacing $f$ by a nonzero integer multiple, Theorem 5.3.1 of [13] provides us with a Borcherds product $ψ(f)$.
This is a rational section of the hermitian line bundle $_V^⊗k(f)$ on $𝒮̅_V$
\begin{equation}\label{borcherds norm}
\Phi_V(f) = - \log\| \psi(f) \|^2,
\end{equation}
whose divisor (at least if $n>2$) is
\begin{equation}\label{naive Bdiv}
\bar{\mathcal{Z}}_V(f) = \sum_{m>0} c(-m) \bar{\mathcal{Z}}_V(m)
\end{equation}
plus an explicit (but complicated) linear combination of boundary components and vertical divisors in characteristics $p|D$.
\begin{remark}\label{rem:lifted weight}
The integer $k(f)$, called the \emph{weight} of the Borcherds product, is given as follows:
Applying the construction \eqref{eq:lift} yields a weakly holomorphic vector valued form
\[
\tilde{f} = \sum_{ \substack{ m\in \Q \\ m \gg -\infty } } \tilde{c}(m) q^m \in M_{2-n}^!(\omega_L)
\]
as in \eqref{eq:fourierf}, with coefficients $\tilde{c}(m) \in S_L=\C [L'/L]$.
The weight
\begin{equation}\label{wt def}
k(f) = \tilde{c}(0,0)
\end{equation}
is the value of $\tilde{c}(0)$ at the trivial coset in $L'/L$.
\end{remark}
The space of all forms \eqref{input form} is an infinite dimensional $$-vector space.
As evidenced by the following theorem, this gives us a great deal of freedom to choose a Borcherds product $ψ(f)$ with prescribed properties.
\begin{theorem}\label{thm:good borcherds}
Assume $n>2$.
Given any infinite subset $\mathscr{A} \subset \Z^+$, there is a weakly holomorphic form \eqref{input form} satisfying
\begin{enumerate}
\item $c(m) \in \Z$ for all $m\in \Z$,
\item $c(-m) =0$ for all positive integers $m \not\in \mathscr{A}$,
\item $k(f) \neq 0$,
\item
up to a vertical divisor supported in characteristics $p\mid D$,
the divisor of $\psi(f)$ is equal to \eqref{naive Bdiv}.
\end{enumerate}
In particular, the divisor of $\psi(f)$ contains no irreducible components supported on the boundary.
\end{theorem}
The proof will occupy the entirety of the next subsection.
\subsection{Borcherds products with prescribed properties}
As $L' ⊂V$ is the dual lattice of $L$ with respect to the quadratic form \ref{Q quadratic}, any $μ∈L'/L$ determines a coset $ -Q(μ) ⊂$.
Denote by $P(ω_L)$ the space of finite Fourier polynomials
\[
\sum_{\mu\in L'/L}\sum_{\substack{m \in \Z-Q(\mu)\\ m\leq 0}} c(m ,\mu ) \varphi_\mu \cdot q^m
\]
valued in $S_L=[L'/L]$, whose coefficients satisfy $c(m,μ)=c(m,-μ)$.
As $n≥2$, we may view
H_2-n(ω_L) ⊂P(ω_L)
by sending a harmonic Maass form \eqref{eq:fourierf} to its principal part
\[
\sum_{m\le 0} c^+(m) \cdot q^m
\sum_{\mu\in L'/L} \sum_{m\le 0} c^+(m,\mu) \varphi_\mu \cdot q^m .
\]
In particular, this allows us to view
M^!_2-n(ω_L) ⊂P(ω_L).
Denote by $S_L[[q^1/D]]$ the space of $S_L$-valued formal power series in the variable $q^1/D$, and denote by
\[
\vartheta \define q\frac{d}{dq} : S_L[[q^{1/D}]] \to S_L[[q^{1/D}]]
\]
the Ramanujan theta operator.
For any $k∈$, taking $q$-expansions allows us to view $M_k(ω̅_L) ⊂S_L[[q^1/D]]$.
Following [7] we consider the $$-bilinear pairing
\[
\{ - , - \} : P(\omega_L)\times S_L[[q^{1/D}]] \to \C
\]
defined by
\[
\{p,g\} = \sum_{ \substack{ m\ge 0 \\ \mu\in L'/L } } c(-m,\mu) b(m,\mu) ,
\]
where $c(m,μ)$ and $b(m,μ)$ denote the coefficients of $p$ and $g$, respectively.
\begin{proposition}
\label{prop:crit}
For any
\[
p=\sum_{\mu\in L'/L}\sum_{m<0} c(m,\mu)\varphi_\mu\, q^{m}\in P(\omega_L),
\]
the following are equivalent:
\begin{enumerate}
\item
There exists a weakly holomorphic modular form $f\in M_{2-n}^!(\omega_L)$
whose principal part agrees with $p$ up to a constant in $S_L$.
\item
We have $\{p, g\}=0$ for every $g\in S_{n}(\bar\omega_L)$.
\end{enumerate}
When these conditions holds, the constant term
$c(0,0)$ of $f$ is related to the value of the Eisenstein series
\begin{equation}\label{E_L}
E_L=E_L(\tau,s_0,n)\in M_n(\bar\omega_L)
\end{equation}
of \ref{eq:fouriereis} at $s_0=(n-1)/2$ by
\begin{align}
\label{eq:ctcond}
-c(0,0)= \{p,E_L\}=\sum_{\mu\in L'/L}\sum_{m>0} c(-m,\mu)\, B(m,\mu, s_0) .
\end{align}
\end{proposition}
\begin{proof}
See [3], or Corollary 3.9 of [7].
\end{proof}
\begin{proposition}
\label{prop:keyprop}
Let $\mathscr{A} \subset \Z^+$ be any infinite subset.
If $n>2$, there exists a weakly holomorphic form $f\in M_{2-n}^!(\omega_L)$ whose Fourier coefficients $c(m,\mu)$ are integers satisfying the following properties:
\begin{itemize}
\item[(i)]
if $c(-m,\mu)\neq 0$ with $m >0$, then $\mu=0$ and $m\in \mathscr{A}$,
\item[(ii)]
$c(0,0)\neq 0$,
\item[(iii)] $\{f,\vartheta(g)\}= 0$ for all $g\in M_{n-2}(\bar \omega_L)$.
\end{itemize}
\end{proposition}
%The weakly holomorphic form $f\in M^!_{2-n}(\bar\omega_L)$ of the proposition can be obtained as the lift of an element of $M_{2-n}^{!,\infty}(D,\chi)$ as in \eqref{eq:lift}.
\begin{proof}
We generalize the argument of \cite[Proposition~3.1]{Br2}.
It follows from the main result of [42] that
the space $M_{2-n}^!(\omega_L)$ has a basis of weakly holomorphic modular forms with integral coefficients. Hence, it suffices to show the existence of an $f\in M_{2-n}^!(\omega_L)$ with {\em rational} coefficients satisfying the stated properties.
Write $M_n(\bar \omega_L,\Q)$ for the $\Q$-vector space of modular forms in $M_\kappa(\bar\omega_L)$ with rational coefficients, and $S_n(\bar \omega_L,\Q)$ for the subspace of cusp forms with rational coefficients.
To lighten notation, throughout the proof we denote by $M$ the finite dimensional $\Q$-vector space
\[
M=M_n(\bar \omega_L,\Q)\oplus \vartheta M_{n-2}(\bar \omega_L,\Q)\subset
\]
and by $S$ the subspace
\[
S= S_n(\bar \omega_L,\Q)\oplus \vartheta M_{n-2}(\bar \omega_L,\Q)\subset M.
\]
The $\Q$-duals are denoted $M^\vee$ and $S^\vee$, and we denote by
\[
\pr: M^\vee\to S^\vee .
\]
the surjection induced by $S\subset M$.
For $\mu\in L'/L$ and $m\in \Z+Q(\mu)$, denote by
$a_{m,\mu} \in M^\vee$ the linear functional sending
\[
g=\sum_{\nu \in L'/L} \sum_{\ell \ge 0} b(\ell,\nu) \varphi_\nu \cdot q^\ell \in M
\]
to the Fourier coefficient $a_{m,\mu}(g)= b(m,\mu).$
Let $M_{\mathscr{A}}^\vee\subset M^\vee$ be the subspace generated by all functionals $a_{m,0}$ with $m\in \mathscr{A}$, and fix $e_1,\dots,e_d\in M_{\mathscr{A}}^\vee$ such that
$\pr(e_1),\dots,\pr(e_d)$ is a basis of the subspace $\pr(M_{\mathscr{A}}^\vee)\subset S^\vee$.
For every $m\in \mathscr{A}$ there is a unique tuple
\[
r(m) = ( r_1(m),\dots,r_d(m) ) \in \Q^d
\]
such that
\[
\pr(a_{m,0}) = r_1(m)\cdot \pr(e_1)+\ldots+r_d(m)\cdot \pr(e_d).
\]
The linear combination
\[
\tilde a_{m,0} \define a_{m,0} - ( r_1(m)\cdot e_1+\cdots+r_d(m)\cdot e_d) \in M_{\mathscr{A}}^\vee
\]
clearly lies in the kernel of $\pr$.
\begin{lemma}
There is an $m\in \mathscr{A}$ such that the Eisenstein series \eqref{E_L} satisfies $\tilde a_{m,0}(E_{L}) \neq 0$.
\end{lemma}
\begin{proof}
We assume on the contrary that $\tilde a_{m,0}(E_{L})=0$ for all $m\in \mathscr{A}$.
In other words, that the coefficients \eqref{eq:coeff} satisfy
\begin{align}
\label{eq:eisa}
B(m,0,s_0) = r_1(m)\cdot e_1( E_{L})+\ldots+r_d(m)\cdot e_d(E_{L})
\end{align}
for all such $m$.
Let $\|r\|$ be the euclidian norm of a vector $r\in \R^d$, and denote by $\|\cdot\|$ a norm on $S^\vee\otimes_\Q \R$, say the operator norm with respect to a fixed norm on the finite dimensional vector space $S\otimes_\Q \R$.
Since $\pr(e_1),\dots,\pr(e_d)$ are linearly independent, there exists an $\eps>0$ such that
\[
\eps \|r\|\leq \| r_1 \pr(e_1)+\ldots+r_d \pr(e_d)\|
\]
for all $r=(r_1,\dots,r_d)\in \R^d$.
Taking $r=r(m)$, we obtain
\[
\eps \cdot \|r(m)\| \leq \| \pr (a_{m,0}) \| .
\]
On the other hand, \eqref{eq:eisa} implies that there is a constant $c>0$ such that
\[
|B(m,0,s_0)|\leq c\cdot \|r(m)\|.
\]
Combining these last two inequalities, we find
\begin{align}
\label{eq:3}
|B(m,0,s_0)| \leq \frac{c}{\eps}\cdot \| \pr (a_{m,0}) \|
\end{align}
for all $m\in \mathscr{A}$.
The Hecke bound for the coefficients of (scalar valued) cusp forms of weight $n$ for $\Gamma(D)$ implies that
\[
|\pr (a_{m,0})(g)|= O(m^{n/2}),
\]
as $m\to \infty$ for any $g\in S_n(\bar \omega_L)$.
On the other hand, an elementary estimate shows that
\[
|\pr (a_{m,0})(g)|= O(m^{n-2+\delta}),
\]
as $m\to \infty$ for any $\delta>0$ and any $g\in\vartheta ( M_{n-2}(\bar \omega_L))$.
As $n>2$, these bounds imply
$\| \pr (a_{m,0}) \| = O(m^{n-3/2})$.
Combining this with \eqref{eq:3} shows that
\[
|B(m,0,s_0)| = O(m^{n-3/2})
\]
for $m\in \mathscr{A}$ and $m\to \infty$, contradicting Corollary \ref{cor:eisgrowth}.
\end{proof}
We now complete the proof of Proposition \ref{prop:keyprop}.
The lemma provides us with an $a=\tilde a_{m,0}\in M_{\mathscr{A}}^\vee$ satisfying $\pr(a)=0$ and $a(E_L)\neq 0$. By definition of $M_{\mathscr{A}}^\vee$, we may expand $a$ as a finite linear combination
\[
a = \sum_{m \in \mathscr{A}} c(m,0) a_{m,0}
\]
with $c(m,0) \in \Q$,
and then form the Fourier polynomial
\[
p = \sum_{m \in \mathscr{A}} c(m,0) \varphi_0 \cdot q^m \in P(\omega_L).
\]
The condition $\pr(a)=0$ implies that $\{ p,g\}=0$ for all $g\in S_{n}(\bar\omega_L)$, and
$\{ p, \vartheta(g)\}=0$ for all $g\in S_{n-2}(\bar\omega_L)$.
In particular, Proposition \ref{prop:crit} provides us with a form $f\in M^!_{2-n}(\omega_L)$
whose principal part agrees with $p$ up to a constant, satisfies
\[
\{ f, \vartheta(g) \} = \{ p,\vartheta(g)\} =0
\]
for all $g\in S_{n-2}(\bar\omega_L)$, and has constant term $ \{ p,E_L\} =a(E_L) \neq 0$.
\end{proof}
As a special case of the following proposition, the form $f$ of Proposition \ref{prop:keyprop} lies in the image of the lifting map
\begin{equation}\label{holomorphic lift}
M^{!,\infty}_{2-n}(D,\varepsilon^n) \map{h\mapsto \tilde{h}} M_{2-n}^!(\omega_L)
\end{equation}
defined by \eqref{eq:lift}.
\begin{proposition}
\label{prop:keypropscal}
Assume $n>2$, and let $f\in M_{2-n}^!(\omega_L)$ be a form whose Fourier coefficients $c(m,\mu)$ satisfy $c(m,\mu)=0$ for all $(m,\mu)$ with $m<0$ and $\mu\neq 0$.
In the notation of \eqref{eq:lift}, there exists an $h\in M^{!,\infty}_{2-n}(D,\varepsilon^n)$
with principal part
\begin{align}
\label{eq:prinh}
\sum_{m<0} c(m,0) q^m + \mathrm{constant},
\end{align}
and such that $\tilde h=f$.
\end{proposition}
\begin{proof}
Using the basis $\{ \varphi_\mu\}_{\mu\in L'/L}$ of $S_L$, any form $g(\tau) \in S_n(\bar\omega_L)$ can be written as $g(\tau) =\sum_{\mu\in L'/L} g_\mu(\tau) \varphi_\mu$.
Taking the component corresponding to $\mu=0$ defines a linear map
\[
S_n(\bar\omega_L)\map{ g\mapsto g_0} S_{n}(D,\varepsilon^n).
\]
We claim that the map $g\mapsto g_0$ is surjective. Indeed, this is equivalent to the injectivity of the adjoint map $ S_{n}(D,\varepsilon^n)\to S_n(\bar\omega_L)$, which is just the map $g\mapsto \tilde{g}$ of \eqref{eq:lift}.
This injectivity follows from the explicit formula for the Fourier expansion of $\tilde g $ found in Proposition 3.3.2 of [14], along with the fact that the $\Q/\Z$-valued quadratic form on $L'/L$ represents all elements of $\frac{1}{D}\Z/\Z$ primitively.
Now suppose we are given a cusp form
\begin{equation}\label{dual cusp}
\sum_{m>0} b(m) q^m \in S_{n}(D,\varepsilon^n).
\end{equation}
If we choose a $g\in S_n(\bar\omega_L)$ such that $g_0 = \sum_m b(m) q^m$, then
\[
\sum_{m>0} c(-m,0)b(m)= \{ f,g\}=0,
\]
where the first equality follows from our hypotheses on the coefficients of $f$, and the second follows from the residue theorem.
As this vanishing holds for all forms \eqref{dual cusp}, it follows from
Serre duality on the modular curve $X_0(D)$ that there exists an $h\in M^{!,\infty}_{2-n}(D,\varepsilon^n)$ with principal part \eqref{eq:prinh}.
In particular, it follows from \eqref{eq:coeffrel} that $\tilde{h} -f$ is holomorphic at $\infty$. As $\tilde{h} -f$ has negative weight, it is identically $0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:good borcherds}]
First pick a vector-valued form $\tilde{f}\in M_{2-n}^!(\omega_L)$ as in Proposition \ref{prop:keyprop}, and then apply Proposition \ref{prop:keypropscal} to pick a scalar-valued form
\[
f(\tau) = \sum_{ m \gg -\infty } c(m) q^m \in M^{!,\infty}_{2-n}(D,\varepsilon^n)
\]
that maps to it under \eqref{holomorphic lift}.
It follows from the relation \eqref{eq:coeffrel} between the coefficients of $f$ and $\tilde{f}$ that
for all $m>0$ the coefficient $c(-m)$ is an integer, and vanishes unless $m\in \mathscr{A}$.
Now we use the fact that $M^{!,\infty}_{2-n}(D,\varepsilon^n)$ has a basis of forms with integer coefficients\footnote{One can deduce this from the corresponding statement for holomorphic modular forms, by multiplying weakly holomorphic modular forms by powers of Ramanujan's discriminant to kill the poles at $\infty$}. For any $\sigma \in \Aut(\C/\Q)$, it follows that the formal $q$-expansion $f^\sigma$ again lies in $M^{!,\infty}_{2-n}(D,\varepsilon^n)$. The difference $f(\tau)-f^\sigma(\tau)$ is then holomorphic at every cusp with weight $n-2<0$, and so vanishes identically.
Thus all coefficients of $f(\tau)$ are rational, and we may replace $f(\tau)$ by a positive integer multiple to assume that $c(m)\in \Z$ for all $m\in \Z$.
As we chose $\tilde{f}$ so that its constant term $\tilde{c}(0,0)$ is nonzero, the associated Borcherds product $\psi(f)$ has nonzero weight by Remark \ref{rem:lifted weight}.
It only remains to verify property (4) in Theorem \ref{thm:good borcherds}.
For this we appeal to Theorem 5.3.3 of [13], which tells us that
\begin{equation}\label{divisor boundary}
\mathrm{div}(\psi(f)) = \bar{\mathcal{Z}} _V(f)
+ \sum_{m>0} c(-m) \mathcal{B}(m) ,
\end{equation}
up to a linear combination of divisors supported in characteristics $p\mid D$.
Here, as in (5.3.3) of [13],
\begin{equation}\label{boundary mult}
\mathcal{B}(m) = \frac{m}{n-2} \sum_\Phi \rho_{L_0}(m) \cdot \mathcal{S}_V(\Phi),
\end{equation}
where the sum is over a finite set $\{ \Phi \}$ indexing the irreducible boundary components $\mathcal{S}_V(\Phi) \subset \partial \bar{\mathcal{S}}_V$, each of which is connected and smooth over $\co_\kk$. Inside the sum, each $L_0$ is a self-dual hermitian $\co_\kk$-module (which depends on $\Phi$) of signature $(n-2,0)$, and
\[
\rho_{L_0}(m) = \# \{ x\in L_0 : \langle x,x\rangle =m \}
\]
is the number of times that lattice represents $m$.
As explained in \S 3.1 of [13], each $\Phi$ is an equivalence class of pairs $(I , g)$ in which $I \subset V$ is an isotropic $\kk$-line, and $g\in G(\A_f)$. This data determines a filtration
\[
\mathfrak{a} \subset \mathfrak{a}^\perp \subset gL
\]
by $\co_\kk$-module direct summands, with $\mathfrak{a} = I \cap gL$ isotropic of rank one and $\mathfrak{a}^\perp = \{ x\in gL : \langle x, \mathfrak{a} \rangle =0 \}$.
The quotient $L_0 = \mathfrak{a}^\perp / \mathfrak{a}$ inherits a self-dual hermitian form from that on $gL \subset V$, and the filtration admits a (non-canonical) splitting
\[
gL = \mathfrak{a} \oplus L_0 \oplus \mathfrak{b}
\]
in which $\mathfrak{b}$ is rank one and isotropic, and $\mathfrak{a} \oplus \mathfrak{b}$ is orthogonal to $L_0$.
As $L_0$ and $gL$ are themselves self-dual hermitian $\co_\kk$-modules, we may form modular forms valued in the finite dimensional vector spaces $S_{L_0}=\C [L_0'/L_0]$ and $S_{gL} = \C[(gL)'/(gL)]$, exactly we did for $L$. The action of $g\in G(\A_f)$ defines a canonical bijection
L' / L \iso (gL)'/(gL),
which induces an isomorphism $S_L \iso S_{gL}$ respecting the Weil representations on source and target.
To each $\Phi$, we may attach the $S_{L_0}$-valued theta series
\[
\Theta_\Phi(\tau) = \sum_{ \mu \in L_0' / L_0 } \Theta_{\Phi,\mu} (\tau) \varphi_\mu \in M_{n-2}(\bar{\omega}_{L_0} )
\]
\Theta_{\Phi,\mu} (\tau) = \sum_{ x\in \mu+L_0 } q^{\langle x,x\rangle} .
As in Theorem 4.1 of [48], there is an induced $S_{gL}$-valued modular form
\[
\ind_L(\Theta_\Phi) = \sum_{\substack{\mu\in L_0'/L_0\\
\beta \in \mathfrak{d}_\kk^{-1} \mathfrak{a}/\mathfrak{a}}} \Theta_{E,\mu}(\tau)\varphi_{\mu+\beta}
\in M_{n-2}(\bar\omega_{gL}) ,
\]
and we use $S_L \iso S_{gL}$ to view $\ind_L( \Theta_\Phi )\in M_{n-2}(\bar\omega_{L})$.
Using the relation \eqref{eq:coeffrel} between the coefficients of $\tilde{f}$ and $f$, we see that
\[
\sum_{m>0} m c(-m) \rho_{L_0}(m) = \{ \tilde{f} , \vartheta(\ind_L( \Theta_\Phi )) \} =0,
\]
where the second equality follows from condition (iii) in Proposition \ref{prop:keyprop}. Comparing with \eqref{boundary mult} shows that
\sum_{m>0} c(-m) \mathcal{B}(m) = 0,
and property (4) of Theorem \ref{thm:good borcherds} follows by comparison with \eqref{divisor boundary}.
\end{proof}
%We now prove a second existence result for Borcherds products. It can be used to determine the arithmetic volumes of the special divisors $Z(p,0)$ for almost all primes $p\equiv 1\pmod{D}$. \note{I don't think we need this. There is a cleaner way to get at volume of $Z(p,0)$}
%Assume that $n\geq 3$ and let $\kk$ and $L$ be as above. Let $\mathscr{A}$ be an infinite set of positive integers.
%There exists a positive integer $t_0$ such that for all $m_0\in \mathscr{A}$ with $m_0\geq t_0$ there is a Borcherds product $\Psi$ of weight $0$ on $X^*$ with
%\dv_{X^*}(\Psi) = \sum_{\substack{m\in \mathscr{A}\\m\geq t_0}} c(m) Z^*(m,0),
%where $c(m_0)\neq 0$ and only finitely many of the other coefficients $c(m)$ are non-zero.
%In particular, the divisor of $\Psi$ does not include any boundary components.
%In view of \cite[Theorem 8.1]{Hof} (see also \cite[Theorem 13.3]{Bo1}) and Corollary \ref{cor:boundary}, Theorem \ref{thm:bpdiv2} is a consequence of the following proposition.
%%Assume that $n\geq 3$.
%There exists a positive integer $t_0$ such that for all $m_0\in \mathscr{A}$ with $m_0\geq t_0$ there is a
%weakly holomorphic modular form $f\in M_{2-n}^!(\omega_L)$ with integral Fourier coefficients $c(m,\mu)$ with the properties:
%if $(m,\mu)\in \Q^+\times L'/L$ with $c(-m,\mu)\neq 0$, then $\mu=0$ and $m\in \mathscr{A}$,
%\item[(ii)] $c(-m_0,0)\neq 0$ and
%$c(0,0)= 0$,
%\item[(iii)] $\{f,\vartheta(g)\}= 0$ for all $g\in M_{n-2}(\bar \omega_L)$.
%We use the notation of the proof of
%Proposition \ref{prop:keyprop}. For $l\in \Z^+$ we let
%$A_l\subset M^\vee$ be the linear span of all functionals
%a_{m,0}\in M^\vee \quad \text{with $m\in \mathscr{A}$ and $m\geq l$.}
%We obtain a descending chain of subspaces
%M^\vee \supset A_1 \supset A_2 \supset A_3 \supset\dots .
%Since $M^\vee$ has finite dimension this chain becomes stationary, i.e., there exists a positive integer $t_0$ such that $A_l=A_{t_0}$ for all $l\geq t_0$.
%Hence for every $m_0\in \mathscr{A}$ with $m_0\geq t_0$ and for every $l\geq m_0$ we have
%a_{m_0,0}\in A_l.
%Consequently, there exist $m_1,\dots, m_d\in A_l$ and coefficients $c(m_1),\dots ,c(m_d)\in \Q$ such that
%-a_{m_0,0}= \sum_{j=1}^d c(m_j) a_{m_j,0}.
%According to Proposition~\ref{prop:crit}, there exists an $f\in M_{2-n}^!(\omega_L)$ with principal part
%q^{-m_0}\varphi_0+\sum_{j=1}^d c(m_j)q^{-m_j}\varphi_0
%that satisfies (i)--(iii).
\subsection{A carefully chosen Borcherds product}
Recalling the functions $𝐚_k(s)$ of \eqref{a_k}, set
𝐛_V,1(s) =𝐚_1(s).
For $k ≥2$ even, set
\[
\mathbf{b}_{V,k}(s)
= \mathbf{a}_k(s)
\prod_{\ell \mid D} \left( 1+ \leg{-1}{\ell}^{ \frac{k}{2} } \mathrm{inv}_\ell(V) \ell^{ -s- \frac{k}{2} } \right).
\]
For $k ≥3$ odd, set
\[
\mathbf{b}_{V,k}(s)
= \mathbf{a}_k(s)
\prod_{\ell \mid D} \left( 1+ \leg{-1}{\ell}^{ \frac{k-1}{2} } \mathrm{inv}_\ell(V) \ell^{ -s +\frac{1-k}{2}} \right)^{-1}.
\]
When $k>1$ we have
$𝐛_V,k(s) 𝐛_V,k+1(s) = 𝐚_k(s) 𝐚_k+1 (s)$,
which implies that the function \eqref{A_V} factors as
\begin{equation}\label{A_V factorization}
\mathbf{A}_V(s) = \mathbf{b}_{V,1}(s) \cdots \mathbf{b}_{V,n}(s) .
\end{equation}
Now suppose $n>2$, abbreviate
\[
\mathscr{A} = \{ \mbox{primes } p \equiv 1\pmod{D} \},
\]
and assume that the weakly modular form
\[
f(\tau) = \sum_{m \gg -\infty} c(m) q^m \in M_{2-n}^{ ! , \infty}( D , \eps^n )
\]
of \eqref{input form} is chosen as in Theorem \ref{thm:good borcherds}.
In particular, the divisor of the Borcherds product $ψ(f)$ contains no components of the boundary, and so
\begin{equation}\label{borcherds divisor}
\mathrm{div} ( \psi(f) ) = \bar{\mathcal{Z}}_V(f) +
\vertical(f)
\end{equation}
in which
\[
\bar{\mathcal{Z}}_V(f)
= \sum_{ p \in \mathscr{A} } c(-p) \bar{\mathcal{Z}}_V(p),
\]
and $ (f)$ is supported in characteristics dividing $D$.
\begin{remark}
The notation $\vertical(f)$ is slightly misleading, as $ \bar{\mathcal{Z}}_V(f)$ may itself have vertical components.
Any such components are supported on the exceptional divisor $\mathrm{Exc}_V \subset \bar{\mathcal{S}}_V$, by
Corollary 3.7.3 of [13].
\end{remark}
\begin{proposition}\label{prop:awesome B}
The Borcherds product $\psi(f)$ has weight
\[
k(f) = \frac{1}{\mathbf{b}_{V,n}(0)} \sum_{p \in \mathscr{A} } c(-p) ( p^{n-1} + 1 ) ,
\]
and, recalling the notation \eqref{constant metrics}, satisfies
\[
( \vertical (f) , 0 ) = (E,0) - k(f) \sum_{\ell \mid D} \left( 0 , \frac{ \log(\ell) }{1+ \beta_\ell } \right) \in \widehat{\Pic}(\mathcal{S}_V)_\Q,
\]
where $E$ is a divisor (which may depend on $f$) supported on the exceptional divisor
of Definition \ref{def:special exceptional}, and
\begin{equation}\label{bang}
\beta_\ell = (-1)^{n+1} \cdot \begin{cases}
\leg{-1}{\ell}^{\frac{n}{2} } \inv_\ell(V) \ell^{ \frac{n}{2} } & \mbox{if $n$ is even} \\[2ex]
\leg{-1}{\ell}^{\frac{n-1}{2} } \inv_\ell(V) \ell ^{ \frac{n-1}{2} } & \mbox{if $n$ is odd.}
\end{cases}.
\end{equation}
\end{proposition}
\begin{proof}
The proof requires a short digression on Eisenstein series.
For any divisor $r\mid D$ set $r'=D/r$. Our assumption that $D$ is odd implies that the quadratic character $\eps : (\Z/D\Z)^\times \to \{ \pm 1\}$ determined by $\kk$ is
\[
\eps(a) = \leg{a}{D}.
\]
Hence we may factor $\eps = \eps_r \cdot \eps_{r'}$ with
\[
\eps_r(a) = \left( \frac{a}{r} \right) \quad \mbox{and} \quad \eps_{r'} (a) = \left( \frac{a}{r'} \right).
\]
Define the quadratic Gauss sum
\[
\tau(\eps_r) = \sum_{ a \in (\Z/r\Z)^\times } \eps_r(a) e^{2\pi i a/r} = \begin{cases}
\sqrt{r} & \mbox{if } r\equiv 1\pmod{4} \\
i\sqrt{r} & \mbox{if } r \equiv 3\pmod{4},
\end{cases}
\]
and similarly with $r$ replaced by $r'$.
\begin{lemma}\label{lem:eisenstein formulas}
For every divisor $r\mid D$ there is an Eisenstein series
\[
E_r = \sum_{m \ge 0} e_r(m) \cdot q^m \in M_n(\Gamma_0(D),\eps^n)
\]
whose Fourier coefficients are as follows.
The constant term is
\[
e_r(0)= \begin{cases}
1 & \mbox{if }r=1 \\
0 & \mbox{otherwise.}
\end{cases}
\]
If $n$ is even the coefficients indexed by $m>0$ are
\[
e_r(m) =
\frac{ r^{n/2} (-2\pi i )^n }{ D^n \Gamma(n) L_D( n ,\eps^n ) }
\sum_{ \substack{ c\mid m \\ c>0 \\ \gcd( m /c, r) =1 } }
c^{n-1} \sum_{ d \mid \gcd( c , r' ) } d \mu( r' / d) .
\]
If $n$ is odd the coefficients indexed by $m>0$ are
\[
e_r(m) =
\eps_r(r') \frac{ r^{n/2} (-2\pi i )^n \tau(\eps_{r'}) }{ D^n \Gamma(n) L_D( n ,\eps^n ) }
\sum_{ \substack{ c\mid m \\ c>0 \\ \gcd( m /c, r) =1 } }
\eps_r( m/c) \eps_{r'}(c) \cdot c^{n-1}.
\]
In both formulas $L_D(s,\eps^n)$ is the Dirichlet $L$-function with Euler factors at all $\ell\mid D$ removed.
\end{lemma}
\begin{proof}
% Choose a matrix
% \[
% R_r = \left(\begin{matrix} \alpha & \beta \\ \gamma r' & \delta r \end{matrix}\right) \in \Gamma_0(D/r)
% \]
% with $\alpha,\beta,\gamma,\delta\in \Z$, set
% $
% W_r = R_r \cdot \left( \begin{smallmatrix} r \\ & 1 \end{smallmatrix}\right),
% $
%and define
%E (z)
% =
% \sum_{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \Gamma_\infty \backslash \Gamma_0(D) } \frac{ \eps^n (d) } { (c z + d)^n } \in M_n( \Gamma_0(D) , \eps^n).
% \]
% Here $\Gamma_\infty \subset \mathrm{SL}_2(\Z)$ is the subgroup of upper triangular matrices.
%The claim is that
% \[
%E_r = \eps_r(-\beta) \eps_{r'}(\alpha r) \cdot ( E \mid_n W_r ) \in M_n(\Gamma_0(D),\eps^n).
% has the stated Fourier expansion.
For each $s,t\in \Z/D\Z$ define an Eisenstein series
\[
G_{(s,t)}(z) = \sum_{ \substack{ c,d \in \Z \\ (c,d) \neq (0,0) \\ (c,d) \equiv (s,t) \pmod{D} } } ( cz + d)^{-n} .
\]
Theorem 7.1.3 of [43] shows that
\[
E_r(z) =
\frac{ \eps^n_r(r') }{ 2 r^{n/2} L_D( n ,\eps^n ) }
\sum_{ s,t \in \Z/D\Z } \eps_r^n( s ) \eps_{r'}^n( t) G_{ ( s,t ) } ( r' z )
\]
has the desired Fourier expansion.
% Indeed, one first checks that [warning:there was a bad search and replace in the formulas below. Some $\alpha$ should be $\beta$]
% \[
%L_D( n ,\eps^n ) \cdot E (z)
% = \frac{1}{2} \sum_{ s , t \in \Z/D\Z } \eps^n(t) G_{(s,t)} (Dz) .
%Applying $|_nW_r$ to both sides and using
% \[
% W_r
% =
% \left( \begin{matrix} D^{-1} & \\ & 1 \end{matrix} \right)
% \left(\begin{matrix} \alpha r & \alpha r' \\ \gamma & \delta \end{matrix}\right)
% \left( \begin{matrix} D & \\ & r \end{matrix} \right),
% \]
%one then finds
% \begin{align*}
% L_D( n ,\eps^n ) \cdot E_r(z)
% & = \frac{ \eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2 D^{n/2} } \sum_{ s ,t\in \Z/D\Z } \eps^n(t) G_{(s,t)} \big|_n \left( \begin{matrix} D \\ & 1 \end{matrix} \right) W_r \\
%& =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r)}{ 2 D^{n/2}} \sum_{ s , t \in \Z/D\Z } \eps^n(t)
% G_{ ( \alpha rs + \gamma t , \alpha r' s + \delta t ) } \big|_n
% \left( \begin{matrix} D & \\ & r \end{matrix} \right) \\
% & =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2r^{n/2} } \sum_{ s,t \in \Z/D\Z } \eps^n(t)
% G_{ ( \alpha rs + \gamma t , \alpha r' s + \delta t ) } ( r' z ) \\
% & =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2 r^{n/2} } \sum_{ s,t \in \Z/D\Z } \eps^n( -\alpha r' s + \alpha rt)
% G_{ ( s,t ) } ( r' z ) \\
% & =
% \frac{\eps^n_r(r') }{ 2 r^{n/2} }
% \sum_{ s,t \in \Z/D\Z } \eps_r^n( s ) \eps_{r'}^n( t) G_{ ( s,t ) } ( r' z )
% \end{align*}
%where the second to last formula is obtained by
%(s,t) \mapsto ( \delta s - \gamma t , -\alpha r' s + \alpha rt)
%= (s,t) \cdot \left( \begin{matrix} \delta & -\alpha r' \\ -\gamma & r \end{matrix} \right) .
%First suppose that $n$ is even, so that $\eps_r^n$ and $\eps_{r'}^n$ are the trivial Dirichlet characters modulo $r$ and $r'$, respectively.
%The constant term is
%g_r(0) = \begin{cases}
%2 \zeta_D(n ) & \mbox{if } r=1 \\
%0 & \mbox{otherwise,}
%and when $m>0$ we have
%g_r (m) =
% \frac{2 r^n (-2\pi i )^n }{ D^n \Gamma(n) } \sum_{ \substack{ c\mid m \\ c>0 \\ \gcd( m/c, r) =1 } }
% c^{n-1} \sum_{ d \mid \gcd( c , r' ) } d \mu( r' / d) .
%Now suppose that $n$ is odd, so that $\eps_r^n=\eps_r$ and $\eps_{r'}^n=\eps_{r'}$ are primitive Dirichlet characters modulo $r$ and $r'$. The constant term is
%g_r(0) = \begin{cases}
%2 L (n, \eps ) & \mbox{if } r=1 \\
%0 & \mbox{otherwise,}
%and when $m>0$ we have
%g_r (m) =
% \frac{2 r^n (-2\pi i )^n \tau(\eps_{r'}) }{ D^n \Gamma(n) }
% \sum_{ \substack{ c\mid m \\ c>0 \\ \gcd( m/c, r) =1 } }
% \eps_r( m/c) \eps_{r'}(c) \cdot c^{n-1} .
\end{proof}
%The above formulas can be rewritten using the relations
% \frac{ 2 \zeta(n) } { \zeta(1-n) } = \frac{ (-2\pi i )^n }{ \Gamma(n) }
% \frac{ 2 L(n,\eps_{r'}) }{ L(1-n,\eps_{r'}) } = \frac{ (-2\pi i )^n \tau (\eps_{r'}) r^n } { D^n \Gamma(n) }
%for $n$ even and odd (respectively).
Suppose $r\mid D$, and set $r'=D/r$.
If $n$ is even, set
\[
a = \frac{ (-2\pi i )^n \mu(D) }{ D^n \Gamma(n) L_D(n , \eps^n) } \quad\mbox{and} \quad
b_r = r^{n/2} \mu( r ) .
\]
If $n$ is odd, set
\[
a =
\frac{ (-2\pi i )^n \tau(\eps) }{ D^n \Gamma(n) L_D(n , \eps^n) } \quad\mbox{and} \quad
b_r = \frac{ r^{n/2} \eps_r( r') \tau(\eps_{r'}) }{ \tau(\eps) }.
\]
One may check directly that
b_r=\prod_{\ell\mid r} b_\ell,
where the product is over all primes $\ell\mid r$, and that when $p \in \mathscr{A}$ the formulas of Lemma \ref{lem:eisenstein formulas} simplify to
\begin{equation}\label{simple miyake}
e_r(p) = a b_r \cdot ( p^{n-1}+1 ) .
\end{equation}
|
# RF-Diffusion: Radio Signal Generation via Time-Frequency Diffusion
Guoxuan Chi1, Zheng Yang1🖂, Chenshu Wu2, Jingao Xu1, Yuchong Gao3,
Yunhao Liu1, Tony Xiao Han4 1Tsinghua University 2The University of Hong Kong
3Beijing University of Posts and Telecommunications 4Huawei Technologies Co.,
Ltd chiguoxuan, hmilyyz, wucs32, xujingao13, gaoyc01<EMAIL_ADDRESS><EMAIL_ADDRESS>
(2024)
###### Abstract.
Along with AIGC shines in CV and NLP, its potential in the wireless domain has
also emerged in recent years. Yet, existing RF-oriented generative solutions
are ill-suited for generating high-quality, time-series RF data due to limited
representation capabilities. In this work, inspired by the stellar
achievements of the diffusion model in CV and NLP, we adapt it to the RF
domain and propose RF-Diffusion. To accommodate the unique characteristics of
RF signals, we first introduce a novel Time-Frequency Diffusion theory to
enhance the original diffusion model, enabling it to tap into the information
within the time, frequency, and complex-valued domains of RF signals. On this
basis, we propose a Hierarchical Diffusion Transformer to translate the theory
into a practical generative DNN through elaborated design spanning network
architecture, functional block, and complex-valued operator, making RF-
Diffusion a versatile solution to generate diverse, high-quality, and time-
series RF data. Performance comparison with three prevalent generative models
demonstrates the RF-Diffusion’s superior performance in synthesizing Wi-Fi and
FMCW signals. We also showcase the versatility of RF-Diffusion in boosting Wi-
Fi sensing systems and performing channel estimation in 5G networks.
RF Signal, Generative Model, Time-Frequency Diffusion, Wireless Sensing,
Channel Estimation
🖂 Zheng Yang is the corresponding author. Our project is available here.
††ccs: Human-centered computing Ubiquitous and mobile computing††ccs: Networks
Mobile networks††journalyear: 2024††conference: International Conference On
Mobile Computing And Networking; September 30–October 4, 2024; Washington
D.C., DC, USA††booktitle: International Conference On Mobile Computing And
Networking (ACM MobiCom ’24), September 30–October 4, 2024, Washington D.C.,
DC, USA††doi: 10.1145/3636534.3649348††isbn: 979-8-4007-0489-5/24/09
## 1\. Introduction
Artificial intelligence generated content (AIGC) has catalyzed a revolutionary
impact in both industrial and academic frontier, birthing a constellation of
cutting-edge products founded on deep neural networks (DNNs). Remarkable
odysseys include Stable Diffusion (Rombach et al., 2022), Midjourney
(Midjourney, 2023), DALL-E (Ramesh et al., 2022) for image creation, and
ChatGPT (OpenAI, 2023) for text generation.
Nowadays, AIGC is gradually knocking on the door of the radio-frequency (RF)
domain. Current practice offers initial proof of its potential to boost
wireless systems in terms of data augmentation (Rizk et al., 2019), signal
denoising (Bando et al., 2020) and time-series prediction (Hamdan and Hamdi,
2020). In downstream tasks like device localization (Zhao et al., 2023), human
motion sensing (Yang et al., 2022a), and channel estimation (Liu et al.,
2021), such progress not only enhances system performance but also cuts down
the cumbersome ground truth annotation costs for application-layer DNN
training.
Existing RF data generation models can be broadly divided into two main
categories:
$\bullet$ Environment modeling based generative model. This approach exploits
LiDAR point clouds or video footage to craft a detailed 3D model of the
environment. It then employs physical models, like ray tracing (McKown and
Hamilton, 1991), to simulate how RF signals interact with surroundings, which
eventually aids in forecasting the signals a receiver might capture. However,
one notable limitation is the method’s insufficient consideration of how the
materials and properties of targets can affect RF signal propagation.
Additionally, obtaining a 3D model with accuracy compatible with RF signal
wavelengths (e.g., 1-10 mm) remains a challenge and will significantly raise
system expenses. While the recent study uses the neural radiance field for
implicit modeling of RF-complex environments to estimate signal propagation
(Zhao et al., 2023), it requires a stationary receiver (Rx), which complicates
generating essential time-series data for wireless communication systems or
tasks like human motion recognition.
$\bullet$ Data-driven probabilistic generative model. Current innovations
leverage models like generative adversarial network (GAN) and variational
autoencoder (VAE) to augment RF datasets (Ha et al., 2020). Essentially, these
models learn the distribution within the training data and then generate new
RF data that follow this distribution. However, these models mainly focus on
expanding feature-level distributions and struggle to precisely generate raw
RF signals due to their constrained representation capabilities (Yang et al.,
2022a). Additionally, most of them are designed for specific tasks with
dedicated loss functions and DNN architectures, limiting their versatility. On
the other hand, GAN’s training is notoriously fickle due to the tug-of-war
between the generator and discriminator (Xia et al., 2022).
Table 1. Illustrative examples. Methods | Examples | | SSIM
---|---|---|---
| Wi-Fi | FMCW | | Wi-Fi | FMCW
Ground Truth | | | | N/A | N/A
Ours | | | | 0.81 | 0.75
DDPM (Ho et al., 2020) | | | | 0.65 | 0.58
DCGAN (Radford et al., 2015) | | | | 0.68 | 0.61
CVAE (Sohn et al., 2015a) | | | | 0.47 | 0.4
Remark. Albeit inspiring, there still lacks a versatile generative model for
generating accurate and time-series raw RF signals suitable for diverse
downstream applications.
Recently, Diffusion Model has emerged as a luminous star in the visual AIGC
cosmos, underpinning a variety of innovative DNNs for a range of prominent
image/video applications such as Stable Diffusion, Midjourny, and DALL-E.
Compared to the aforementioned generative models, its unique iterative process
of noise addition (i.e., noising) and removal (i.e., denoising) allows for
precise capture of intricate raw data distributions (Yang et al., 2022b).
Moreover, its training is straightforward and avoids typical problems like
mode collapse or convergence troubles, since it doesn’t juggle competing parts
or require delicate fine-tuning (Dhariwal and Nichol, 2021).
These compelling advantages inspire us to embrace the diffusion model for
synthesizing RF data. However, transferring existing diffusion models (Ho et
al., 2020) to the RF domain faces significant challenges arising from RF
signal’s unique characteristics beyond images, as summarized below.
$(i)$ Time series. RF signals capture dynamic details like target movement and
environment/channel changes over time, unlike static snapshots. Diffusion
models designed for single-image generation struggle to synthesize RF signal
sequences.
$(ii)$ Frequency domain. Essential RF details (e.g., Doppler shift, chirp) are
embedded in the frequency domain. While recent video diffusion models can
create time series, they mainly focus on the spatial domain (e.g., pixel-wise
brightness), discarding the rich information in the frequency domain.
$(iii)$ Number field. RF data is complex-valued with both amplitude and phase
readings. While existing diffusion models only focus on amplitude (e.g., light
strength), the phase data can’t be ignored due to its crucial role in wireless
systems.
In summary, while diffusion models hold great promise, there is a need to
upgrade current models to suit the unique traits of RF signals and tap into
the underlying information in the time series, frequency, and complex-valued
domains.
Our Work. We propose RF-Diffusion, the first versatile generative model for RF
signals based on Diffusion model. To overcome the above challenges, we expand
existing denoising-based diffusion model to the time-frequency domain by
revisiting its theoretical foundation, overall DNN architecture, and detailed
operator design, enabling RF-Diffusion to generate diverse, high-quality, and
time-series RF data.
$\bullet$ Time-Frequency Diffusion Theory. We first propose the time-frequency
diffusion (TFD) theory as a novel paradigm to guide diffusion models in
extracting and leveraging characteristics of RF signals across both temporal
and frequency domains. Specifically, we demonstrate a diffusion model could
effectively destruct and restore high-quality RF signals by alternating
between adding noise in the time domain and blurring in the frequency domain
(§3).
$\bullet$ Hierarchical Diffusion Transformer Design. We further re-design the
DNNs of existing denoising-based diffusion model to be compatible with TFD.
The derived DNN, designated as the hierarchical diffusion transformer (HDT),
from a top-down perspective, incorporates $(i)$ a hierarchical architecture to
fully uncover time-frequency details by decoupling spatio-temporal dimensions
of RF data; $(ii)$ attention-based diffusion blocks leveraging enhanced
Transformers to extract RF features; and $(iii)$ a complex-valued design to
encode both signal strength and phase information. The three key designs work
hand-in-hand to enable RF-Diffusion to generate high-quality RF data (§4).
We implement RF-Diffusion and conduct extensive experiments that include
synthesis of both Wi-Fi and FMCW signals. To provide a clear understanding of
its performance, an intuitive comparison of the time-frequency spectrograms
generated by RF-Diffusion and those from related works is presented in Table
1. Evaluation results demonstrate that RF-Diffusion generates RF signals with
high fidelity, achieving an average structural similarity of $81\%$ relative
to the ground truth. This performance surpasses prevalent generative models
such as DDPM, DCGAN, and CVAE by over $18.6\%$. We also demonstrate the
performance of RF-Diffusion in two case studies: augmented Wi-Fi gesture
recognition and 5G FDD channel estimation. By employing RF-Diffusion as a data
augmentor, existing wireless gesture recognition systems experience a
significant accuracy improvement ranging from $4\%$ to $11\%$. When applied to
the channel estimation task, RF-Diffusion showcases a substantial $5.97$ dB
improvement in SNR compared to state-of-the-arts.
In summary, this paper makes the following contributions.
(1) We propose RF-Diffusion, the first generative diffusion model tailored for
RF signal. RF-Diffusion is versatile and can be leveraged in a wide spectrum
of fundamental wireless tasks such as RF data augmentation, channel
estimation, and signal denoising, propelling AIGC to shine in the RF domain.
(2) We present the Time-Frequency Diffusion theory, an advanced evolution
beyond traditional denoising-based diffusion methods. The integration of TFD
with its bespoke Hierarchical Diffusion Transformer (HDT) enables enhanced
precision in time-series sampling and a balanced focus on spectral details of
the data.
(3) We fully implement RF-Diffusion. Extensive evaluation results from case
studies show RF-Diffusion’s efficacy.
Community Contribution. RF-Diffusion’s code and pre-trained model are publicly
available. Our solution, in part or in all, could provide a collection of
tools for both industry and academia to push forward AIGC in RF domain.
Moreover, its ability to handle time-series sampling while highlighting the
spectral nuances of the data has potential benefits beyond the wireless
community, offering value to video, audio processing, and other time-series-
dependent modalities.
## 2\. Overview
Figure 1. RF-Diffusion overview.
We propose RF-Diffusion, a pioneering probabilistic generative model for RF
data that leverages the diffusion model framework, as detailed in Fig. 1. At
its core, RF-Diffusion aligns with the principle of denoising-based diffusion
models by employing a dual-process approach: a forward process of integrating
noise into the data, and a reverse process of generating data from noise.
However, RF-Diffusion distinguishes itself through two innovative features:
$(i)$ RF-Diffusion incorporates the proposed Time-Frequency Diffusion (§3)
theory to direct each stage of state transition in both forward (i.e.,
$q(\boldsymbol{x}_{t}|\boldsymbol{x}_{t-1})$) and reverse (i.e.,
$p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})$) processes, enabling RF-
Diffusion to harness the RF signal information across both the time and
frequency domain.
$(ii)$ RF-Diffusion introduces the Hierarchical Diffusion Transformer (§4),
which is a restructured DNN model for the reverse generation process, to align
with the Time-Frequency Diffusion theory and the characteristics of RF
signals.
As for the specific data flow, RF-Diffusion gradually introduces Gaussian
noise in the time domain and blurs the spectrum in the frequency domain at
each stage in the forward direction. As the diffusion step $t$ advances, the
original RF signal $\boldsymbol{x}_{0}$ diminishes, eventually degrading into
noise. In TFD theory, we demonstrate any destructed signal
$\boldsymbol{x}_{t}$ can be restored to its original form $\boldsymbol{x}_{0}$
using a parameterized reverse process. Guided by the destruction process
alternating in time-frequency domain, the reverse restoration process
emphasizes both time-domain amplitude accuracy and frequency-domain continuity
to achieve time-frequency high-fidelity signal generation.
In the reverse direction, HDT are served as the parameterized model for
learning the restoration process. It decouples the Gaussian noise and the
spectral blur, effectively addressing them in the spatial denoise and time-
frequency deblur stages, respectively. During its training, HDT takes
destructed signal $\boldsymbol{x}_{t}$ as the model input, and uses the signal
of previous diffusion step $\boldsymbol{x}_{t-1}$ to supervise the output.
Once trained, RF-Diffusion is capable of iteratively transforming fully
degraded noise back into a specific type of signal.
## 3\. Time-Frequency Diffusion
In this section, we introduce the proposed Time-Frequency Diffusion (TFD)
process. Unlike prevailing denoising diffusion models, the time-frequency
diffusion process comprehensively addresses two potential distortions in
wireless signal data: 1) amplitude distortion due to additive Gaussian noise;
2) spectral aliasing resulting from insufficient time resolution. Therefore,
the learned reverse process focuses not only on precisely reconstructing the
amplitude of individual samples but also on preserving spectral resolution in
time-series signals. In what follows, we first introduce the forward
destruction process (§3.1) which jointly eliminates the original data
distribution in the time and the frequency domain. On this basis, we describe
how to reverse the process (§3.2) and fit it through a parameterized model,
which is the basis of our conditional generation (§3.3) task.
### 3.1. Forward Destruction Process
The time-frequency diffusion model is proposed for the RF signal, which can be
treated as the complex-valued time-series data. Therefore, we take the signal
as a two-dimensional complex tensor $\boldsymbol{x}\in\mathbb{C}^{M\times N}$,
where $M$ represents the spatial dimension of each sample, while $N$
represents the temporal dimension of the times series.
Given a signal that follows a specific distribution $\boldsymbol{x}_{0}\sim
q(\boldsymbol{x}_{0})$, the forward destruction process yields a progression
of random variables
$\boldsymbol{x}_{1},\boldsymbol{x}_{2},\dots,\boldsymbol{x}_{T}$. Each
diffusion step in this process disrupts the original distribution from both
the time and frequency domains. Specifically, the forward diffusion process
from step $t-1$ to $t$ is described as follows:
* •
Frequency Blur. To dissipate the spectral details of the original signal, the
Fourier transform $\mathfrak{F}(\cdot)$ is first performed to the temporal
dimension. Subsequently, with the predefined Gaussian convolution kernel
$\mathbf{G}_{t}$, a cyclic convolution $*$ operation is performed on the
spectrum, resulting in a blurred spectrum
$\mathbf{G}_{t}*\mathfrak{F}(\boldsymbol{x}_{t-1})$.
* •
Time-series Noise. To drown out the amplitude details of the signal, complex
standard Gaussian noise $\boldsymbol{\epsilon}\sim\mathcal{CN}(0,\mathbf{I})$
is introduced, and a weighted summation is performed with a predefined
parameter $\sqrt{\alpha_{t}}$, where $\alpha_{t}\in(0,1)$.
By combining the above two steps, we get:
(1)
$\boldsymbol{x}_{t}=\sqrt{\alpha_{t}}\mathfrak{F}^{-1}(\mathbf{G}_{t}*\mathfrak{F}(\boldsymbol{x}_{t-1}))+\sqrt{1-\alpha_{t}}\boldsymbol{\epsilon},$
where $\mathfrak{F}^{-1}(\cdot)$ indicates the inverse Fourier transform.
To ensure the practical feasibility of the time-frequency diffusion process,
it is essential that the transition from $\boldsymbol{x}_{0}$ to
$\boldsymbol{x}_{t}$ for any given step $t\in[1,T]$ can be executed with an
acceptable time complexity, instead of involving an iteration of $t$ steps. To
simplify this process, certain advantageous characteristics of the Fourier
transform and the Gaussian function are leveraged. Based on the convolution
theorem (Weisstein, 2023), we have
$\mathfrak{F}^{-1}(\mathbf{G}_{t}*\mathfrak{F}(\boldsymbol{x}_{t-1}))=\mathfrak{F}^{-1}(\mathbf{G}_{t})\boldsymbol{x}_{t-1}$
111Vector multiplications in this paper default to element-wise products..
Therefore, the operation in Eqn. 1 can be expressed as:
(2) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\sqrt{\alpha_{t}}\boldsymbol{g}_{t}\boldsymbol{x}_{t-1}+\sqrt{1-\alpha_{t}}\boldsymbol{\epsilon},$
where $\boldsymbol{g}_{t}=\mathfrak{F}^{-1}(\boldsymbol{G}_{t})$ is still a
Gaussian kernel, which means the convolution of the signal with the Gaussian
kernel in the frequency domain can be equivalently transformed into the
multiplication of the signal with another Gaussian kernel in the time domain.
For ease of notion, let
$\boldsymbol{\gamma}_{t}=\sqrt{\alpha_{t}}\boldsymbol{g}_{t}$, and
$\sigma_{t}=\sqrt{1-\alpha_{t}}$, indicating the weight of the signal
$\boldsymbol{x}_{t-1}$ and the standard deviation of the added noise at step
$t$.
Since the forward process is a Markov chain, by recursively applying Eqn. 2
and incorporating with the reparametrization trick (Kingma and Welling, 2013),
the relationship between the original signal $\boldsymbol{x}_{0}$ and the
degraded signal $\boldsymbol{x}_{t}$ can be obtained:
(3) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\bar{\boldsymbol{\gamma}}_{t}\boldsymbol{x}_{0}+\sum_{s=1}^{t}{(\sqrt{1-\alpha_{s}}\frac{\bar{\boldsymbol{\gamma}}_{t}}{\bar{\boldsymbol{\gamma}}_{s}})\boldsymbol{\epsilon}}=\bar{\boldsymbol{\gamma}}_{t}\boldsymbol{x}_{0}+\bar{\boldsymbol{\sigma}}_{t}\boldsymbol{\epsilon},$
where
$\bar{\boldsymbol{\gamma}}_{t}=\prod_{s=1}^{t}\boldsymbol{\gamma}_{s}=\boldsymbol{\gamma}_{t}\cdots\boldsymbol{\gamma}_{1}$.
As $\alpha_{t}$ and $\boldsymbol{g}_{t}$ are predefined hyperparameters
corresponding to the noise and blur scheduling strategy, any
$\bar{\boldsymbol{\gamma}}_{t}$ and $\bar{\boldsymbol{\sigma}}_{t}$ are
constant coefficients, representing the weight of the original signal and the
standard deviation of the added noise. Thus, the forward destruction process
to any step $t$ can be quickly completed without iteration. Stated in
probabilistic terms, essentially $\boldsymbol{x}_{t}$ follows an non-isotropic
Gaussian distribution conditioned on $\boldsymbol{x}_{0}$:
(4)
$q(\boldsymbol{x}_{t}|\boldsymbol{x}_{0})=\mathcal{CN}(\boldsymbol{x}_{0};\bar{\boldsymbol{\mu}}_{t},\bar{\boldsymbol{\sigma}}_{t}^{2}\mathbf{I}),$
where
$\bar{\boldsymbol{\mu}}_{t}=\bar{\boldsymbol{\gamma}}_{t}\boldsymbol{x}_{0}$
and
$\bar{\boldsymbol{\sigma}}_{t}=\sum_{s=1}^{t}{(\sqrt{1-\alpha_{s}}\frac{\bar{\boldsymbol{\gamma}}_{t}}{\bar{\boldsymbol{\gamma}}_{s}}})$.
Specifically, the vector $\bar{\boldsymbol{\gamma}}_{t}$ consists of distinct
weighting coefficients, each applied multiplicatively across the temporal
dimension of the original signal to perform weighting adjustments.
It is proven in Appendix A that as the diffusion step $t$ increases, the
original signal is gradually eliminated, and $\boldsymbol{x}_{t}$ eventually
converges to a closed-form noise distribution:
(5)
$\lim_{T\to\infty}\boldsymbol{x}_{T}=\lim_{T\to\infty}\sum_{t=1}^{T}{(\sqrt{1-\alpha_{t}}\frac{\bar{\boldsymbol{\gamma}}_{T}}{\bar{\boldsymbol{\gamma}}_{t}}})\boldsymbol{\epsilon}=\lim_{T\to\infty}\bar{\boldsymbol{\sigma}}_{T}\boldsymbol{\epsilon},$
where
$\bar{\boldsymbol{\sigma}}_{T}=\sum_{t=1}^{T}{(\sqrt{1-\alpha_{t}}\frac{\bar{\boldsymbol{\gamma}}_{T}}{\bar{\boldsymbol{\gamma}}_{t}}})\boldsymbol{\epsilon}$
is determined by predefined noise scheduling strategy in practical
implementation.
### 3.2. Reverse Restoration Process
The restoration process is the reversal of the destruction, which gradually
eliminates the noise and restores the original data distribution.
To learn a parameterized distribution $p_{\theta}(\boldsymbol{x}_{0})$ which
approximates the original distribution $q(\boldsymbol{x}_{0})$, an effective
approach is to minimize their Kullback-Leibler (KL) divergence:
(6) $\displaystyle\theta$
$\displaystyle=\operatorname*{arg\,min}_{\theta}D_{\mathrm{KL}}(q(\boldsymbol{x}_{0})\|p_{\theta}(\boldsymbol{x}_{0}))$
$\displaystyle=\operatorname*{arg\,min}_{\theta}(\mathbb{E}_{q(\boldsymbol{x}_{0})}[-\log
p_{\theta}(\boldsymbol{x}_{0})]+\mathbb{E}_{q(\boldsymbol{x}_{0})}[\log
q(\boldsymbol{x}_{0})])$
$\displaystyle=\operatorname*{arg\,max}_{\theta}\mathbb{E}_{q(\boldsymbol{x}_{0})}[\log
p_{\theta}(\boldsymbol{x}_{0})].$
Unfortunately, $q(\boldsymbol{x}_{0})$ is intractable to calculate in general
(Sohl-Dickstein et al., 2015; Kong et al., 2020), therefore
$\mathbb{E}_{q(\boldsymbol{x}_{0})}[\log p_{\theta}(\boldsymbol{x}_{0})]$
cannot be expressed explicitly. Building on the concepts of prior works (Song
and Ermon, 2019; Ho et al., 2020), we approximate the distribution by
maximizing the variational lower bound. As established in (Ho et al., 2020),
the optimization problem in Eqn. 6 can be approximated as:
(7)
$\theta=\operatorname*{arg\,min}_{\theta}D_{\mathrm{KL}}(q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})\|{p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})}),$
where $q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})$
represents the actual reverse process conditioned on $\boldsymbol{x}_{0}$,
while $p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})$ denotes the
reverse process fitted by our model. Eqn. 7 shows the problem of
reconstructing the original data distribution can be transformed into a
problem of fitting the reverse process. Rewrite
$q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})$ based on the
Bayesian theorem (Appendix B), and we prove it follows a Gaussian distribution
over $\boldsymbol{x}_{t-1}$:
(8) $\displaystyle
q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})\sim\mathcal{CN}(\boldsymbol{x}_{t-1};\tilde{\boldsymbol{\mu}}_{t-1},\tilde{\boldsymbol{\sigma}}_{t-1}^{2}\mathbf{I}),$
$\displaystyle\tilde{\boldsymbol{\mu}}_{t-1}$
$\displaystyle=\frac{1}{\bar{\boldsymbol{\sigma}}_{t}^{2}}(\boldsymbol{\gamma}_{t}\bar{\boldsymbol{\sigma}}_{t-1}^{2}\boldsymbol{x}_{t}+\bar{\boldsymbol{\gamma}}_{t-1}\boldsymbol{\sigma}^{2}_{t}\boldsymbol{x}_{0}),\
\
\tilde{\boldsymbol{\sigma}}_{t-1}=\frac{\bar{\boldsymbol{\sigma}}_{t-1}}{\bar{\boldsymbol{\sigma}}_{t}}\boldsymbol{\sigma}_{t}.$
Let’s assume that $p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})$ is a
Gaussian Markov process:
(9)
$p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})\sim\mathcal{CN}(\boldsymbol{x}_{t-1};\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t}),\boldsymbol{\sigma}_{\theta}^{2}(\boldsymbol{x}_{t})\mathbf{I}).$
Therefore, the KL divergence of two Gaussian distributions in Eqn. 7 can be
simplified as follows:
(10) $\displaystyle
D_{\mathrm{KL}}(q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})\|{p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})})$
$\displaystyle=$
$\displaystyle\mathbb{E}_{q(\boldsymbol{x}_{0})}[\frac{1}{2\tilde{\boldsymbol{\sigma}}_{t}^{2}}\lVert\tilde{\boldsymbol{\mu}}_{t-1}-\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t})\rVert^{2}]+C.$
In summary, the optimization of the the parameterized model
$p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})$ can be achieved by
minimizing the mean square error (MSE) between $\boldsymbol{\mu}_{\theta}$ and
$\tilde{\boldsymbol{\mu}}_{t-1}$. In other words, if a model can infer the
mean value $\tilde{\boldsymbol{\mu}}_{t-1}$ of the previous step from the
input $\boldsymbol{x}_{t}$ of the current diffusion step, then it is competent
for the data generation task.
Figure 2. Illustration of the conditional forward and reverse trajectories.
### 3.3. Conditional Generation
In most practical applications, the generation process is expected to be
guided by the condition label $\boldsymbol{c}$, which indicates a specific
type of the generated signal (e.g., the signal corresponding to a specific
device location or human activity).
Incorporating the conditional generation mechanism into RF-Diffusion offers
significant advantages: $(i)$ Enhanced practicality. The conditional
generation mechanism enables RF-Diffusion system to generate signals of
different categories based on various condition combinations. This eliminates
the need for training separate models for each signal type, significantly
improving the model’s utility in practical applications. $(ii)$ Increased
signal diversity. A well-trained conditional generation model creates diverse
samples featuring any conceivable combination of characteristics within the
condition-label space of the training dataset, which extends the model’s
generalizability beyond the initial scope of the training set, ensuring that
data augmentation contributes to performance improvements in downstream tasks.
In this context, the condition input $\boldsymbol{c}$ defines specific
scenarios, including various rooms, Tx-Rx deployments, human activity types,
and signal bandwidths. This input guides the generation process to produce
data that aligns with the conditional distribution
$p_{\theta}(\boldsymbol{x}|\boldsymbol{c})$. An illustration of the
conditional forward and reverse processes is presented in Fig. 2.
Following the conclusion of previous work (Sohn et al., 2015a; Dhariwal and
Nichol, 2021; Ho and Salimans, 2021), we directly incorporate the condition
$\boldsymbol{c}$ in both the forward process Eqn. 4 and the reverse process
Eqn. LABEL:eqn:reverse-prob, and get
$q(\boldsymbol{x}_{t}|\boldsymbol{x}_{0},\boldsymbol{c})$ and
$q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0},\boldsymbol{c})$
respectively. Then, by combining Eqn. 7 and Eqn. LABEL:eqn:kl, the
optimization can be written as:
(11) $\displaystyle\theta$
$\displaystyle=\operatorname*{arg\,min}_{\theta}D_{\mathrm{KL}}(q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0},\boldsymbol{c})\|{p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t}),\boldsymbol{c}})$
$\displaystyle=\operatorname*{arg\,min}_{\theta}\mathbb{E}_{q(\boldsymbol{x}_{0})}[\lVert\tilde{\boldsymbol{\mu}}_{t-1}-\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t}(\boldsymbol{x}_{0},t,\boldsymbol{\epsilon}),\boldsymbol{c})\rVert^{2}].$
Algorithm 1 RF-Diffusion Training.
1:Dataset following $\boldsymbol{x}\sim q(\boldsymbol{x})$ with condition
$\boldsymbol{c}$
2:Trained model $\mu_{\theta}$
3:Set hyperparameters $T$, $\alpha_{t}$ and $\boldsymbol{g}_{t}$
4:while $\mu_{\theta}$ not converged do
5: Sample $\boldsymbol{x}_{0}\sim q(\boldsymbol{x}_{0})$ with condition
$\boldsymbol{c}$ from dataset
6: Sample diffusion step $t\in\mathrm{Uniform}(1,\dots,T)$
7: Sample noise $\boldsymbol{\epsilon}\sim\mathcal{CN}(0,\mathbf{I})$
8: Get
$\boldsymbol{x}_{t}=\bar{\boldsymbol{\gamma}}_{t}\boldsymbol{x}_{0}+\bar{\boldsymbol{\sigma}}_{t}\boldsymbol{\epsilon}$
$\triangleright$ Eqn. 3
9: Calculate $\tilde{\boldsymbol{\mu}}_{t-1}$ based on $\boldsymbol{x}_{0}$
and $\boldsymbol{x}_{t}$ $\triangleright$ Eqn. LABEL:eqn:reverse-prob
10: Minimize
$\lVert\tilde{\boldsymbol{\mu}}_{t-1}-\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t}(\boldsymbol{x}_{0},t,\boldsymbol{\epsilon}),\boldsymbol{c})\rVert^{2}$
$\triangleright$ Eqn. 11
11:end while
Algorithm 2 RF-Diffusion Sampling.
1:Trained model $\mu_{\theta}$, condition $\boldsymbol{c}$
2:Generated sample $\boldsymbol{x}_{0}$
3:Set hyperparameters $T$, $\alpha_{t}$ and $\boldsymbol{g}_{t}$
4:Sample noise $\boldsymbol{\epsilon}\sim\mathcal{CN}(0,\mathbf{I})$
5:Let $\boldsymbol{x}_{T}=\bar{\boldsymbol{\sigma}}_{T}\boldsymbol{\epsilon}$
$\triangleright$ Eqn. 5
6:for $t=T,\dots,1$ do
7: Get model output
$\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t},\boldsymbol{c})$, and let
$\boldsymbol{\sigma}_{\theta}=\tilde{\boldsymbol{\sigma}}_{t-1}$
8: Sample $\boldsymbol{x}_{t-1}\sim
p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t})$ with
$\boldsymbol{\mu}_{\theta}$ and $\boldsymbol{\sigma}_{\theta}$, which means
let
$\boldsymbol{x}_{t-1}=\boldsymbol{\mu}_{\theta}(\boldsymbol{x}_{t},\boldsymbol{c})+\boldsymbol{\sigma}_{\theta}\boldsymbol{\epsilon}$
$\triangleright$ Eqn. 9
9:end for
10:return $\boldsymbol{x}_{0}$
The training process of the parameterized model used for restoration is
summarized Algorithm 1. By incorporating the desired signal type as a
conditional input, the trained model can iteratively synthesize the original
signal from a sampled noise. The generative process is illustrated in
Algorithm 2.
Figure 3. Hierarchical Diffusion Transformer design.
## 4\. Hierarchical Diffusion Transformer
To bridge the gap between time-frequency theory and a practical generative
model, we introduce a hierarchical diffusion transformer (HDT). Our proposed
HDT incorporates many innovative designs, aligning it with the underlying
time-frequency diffusion theory and making it adept for RF signal generation.
We first introduce the overarching hierarchical design (§4.1), followed by the
detailed design of our proposed attention-based diffusion block (ADB) (§4.2).
Addressing the challenge of complex-valued signal generation, we extend the
core design of the classic transformer block (Vaswani et al., 2017) into the
complex domain (§4.3). Moreover, we propose phase modulation encoding (PME)
(§4.4), a novel positional encoding approach tailored for complex-valued
neural networks.
### 4.1. Hierarchical Architecture
From the top perspective, HDT adopts a hierarchical architecture to
efficiently decouples the estimation of non-isotropic noise. As shown in Fig.
3, HDT is divided into two stages: spatial denoising and time-frequency
deblurring.
The diffusion step, denoted as $t$, is encoded, thereby informing the model
about the current input’s diffusion level. The conditional vector $c$
undergoes encoding as well. In conjunction with the input
$\boldsymbol{x}_{t}^{(n)}$, these components engage in computations, striving
to discern the latent correlation between the input and its pertinent
condition.
Our observation is that the non-isotropic noise can be dissected into two
components: 1) Independent Gaussian noise $\boldsymbol{\epsilon}$ across both
the spatial dimension $M$ and the temporal dimension $N$. 2) Different
information and noise weights (i.e., $\bar{\gamma}_{t}^{(n)}$ and
$\bar{\sigma}_{t}^{(n)}$) along the temporal dimension $N$. Therefore, by
splitting the time-series data into separate samples, we get
$\boldsymbol{x}_{t}^{(n)}=\bar{\gamma}_{t}^{(n)}\boldsymbol{x}_{0}^{(n)}+\bar{\sigma}_{t}^{(n)}\boldsymbol{\epsilon}^{(n)}$.
Herein, within each sample, both signal and noise weights remain constant.
Therefore, each spatial denoising module processes a single sample
$\boldsymbol{x}_{t}^{(n)}$ of the input sequence independently. During this
stage, denoising circumvents the temporal domain weighting induced by spectral
blurring, focusing exclusively on the Gaussian noise
$\boldsymbol{\epsilon}^{(n)}$ introduced into the original information. This
approach resonates with the principles of denoising diffusion (Ho et al.,
2020).
Although the spatial denoising module effectively mitigates the impact of
noise $\boldsymbol{\epsilon}$, its individual treatment for each sample
disregards the temporal weighting effects originating from spectral blurring.
Therefore, the processed results are concatenated as
$\hat{\boldsymbol{\mu}}=[\hat{\boldsymbol{\mu}}^{(1)},\cdots,\hat{\boldsymbol{\mu}}^{(N)}]$
and serve as sequence input for the time-frequency deblurring module, aiming
to estimate the mean value $\tilde{\boldsymbol{\mu}}_{t-1}$.
### 4.2. Attention-based Diffusion Block
As shown in Fig. 3, the input data are process by a sequence of transformer
blocks in both the denoising and deblur stage. We introduce an innovative
attention-based diffusion block to jointly analyze the noisy input
$\boldsymbol{x}_{t}$, condition $\boldsymbol{c}$, and step $t$.
Self-attention for feature extraction. The multi-head self-attention module
captures autocorrelation feature from the noisy input and extracts the high-
level representations implicit in the signal. Compared to convolutional layers
with translation invariance, attention layers are sensitive to the positional
information of each sample in the sequence, thus enabling more effective
restoration of the original signal.
Cross-attention for conditioning. To enhance the conditional generation
capability, RF-Diffusion incorporates a cross-attention module to learn the
latent associations between the inputs and their corresponding conditions.
This module is designed to directly capture the intricate dynamics between the
inputs and specified conditions, thereby improving the diversity and fidelity
of generated signals.
Adaptive layer normalization for diffusion embedding. Inspired by the
widespread usage of adaptive normalization layer (adaLN) (Perez et al., 2018)
in existing conditional generative models (Brock et al., 2018; Dhariwal and
Nichol, 2021), we explore replacing standard layer normalization with adaLN.
Rather than directly learn dimension-wise scale $a$ and shift parameters $b$,
we regress them from the $t$, embedding the diffusion step information into
our model.
### 4.3. Complex-Valued Module Design
In order to work effectively with complex-valued wireless signals, the RF-
Diffusion model is designed as a complex-valued neural network. Several
adaptations have been made to HDT to facilitate complex-valued operations.
Complex-valued attention module. Two key improvements have been implemented in
the dot-product attention mechanism to accommodate complex number computation:
1) The dot product of the query and key vectors is extended to the hermitian
inner product, $\boldsymbol{q}^{\mathrm{H}}\boldsymbol{k}$, which captures the
correlation of two vectors in the complex space. This preserves the effective
information of both the real and imaginary parts to the fullest extent. 2)
Given that the softmax function operates on real numbers, adjustments have
been made to make it compatible with complex vectors. Specifically, softmax is
applied to the magnitude of the dot product, while the phase information
remains unchanged. This modification maintains the probabilistic
interpretation of vector relevance. In mathematical terms, the complex-valued
attention computation for complex vectors $\boldsymbol{q}$ and
$\boldsymbol{k}$ can be expressed as:
(12)
$\mathrm{softmax}(\lvert\boldsymbol{q}^{\mathrm{H}}\boldsymbol{k}\rvert)\exp(j\angle{(\boldsymbol{q}^{\mathrm{H}}\boldsymbol{k})}).$
Complex-valued feed-forward module. Feed-forward module consists of two main
of operations: linear transformation and non-linear activation. A complex-
valued linear transformation can be decomposed into real-valued ones (Trabelsi
et al., 2018). Specifically, for complex-valued input
$\boldsymbol{x}=\boldsymbol{x}_{r}+j\boldsymbol{x}_{i}$, the transformation
with complex weight $\boldsymbol{w}=\boldsymbol{w}_{r}+j\boldsymbol{w}_{i}$
and bias $\boldsymbol{b}=\boldsymbol{b}_{r}+j\boldsymbol{b}_{i}$ can be
written as follows:
(13)
$\displaystyle\boldsymbol{w}\boldsymbol{x}+\boldsymbol{b}=\left[\begin{array}[]{l}\Re(\boldsymbol{w}\boldsymbol{x}+\boldsymbol{b})\\\
\Im(\boldsymbol{w}\boldsymbol{x}+\boldsymbol{b})\end{array}\right]=\left[\begin{array}[]{rr}\boldsymbol{w}_{r}&\
-\boldsymbol{w}_{i}\\\ \boldsymbol{w}_{r}&\
\boldsymbol{w}_{i}\end{array}\right]\left[\begin{array}[]{l}\boldsymbol{x}_{r}\\\
\boldsymbol{x}_{i}\end{array}\right]+\left[\begin{array}[]{l}\boldsymbol{b}_{r}\\\
\boldsymbol{b}_{i}\end{array}\right].$
Furthermore, applying an activation function $g(\cdot)$ to a complex value can
be seen as activating the real and imaginary parts separately:
$g({\boldsymbol{x}})=g({\boldsymbol{x}_{r}})+jg({\boldsymbol{x}_{i}})$.
Figure 4. Illustration of phase modulation encoding.
### 4.4. Phase Modulation Encoding
Leveraging the attention mechanism, a Transformer network parallelly processes
the entire sequence. Yet, it lacks inherent capability to discern the
positional information of the input. To address this, we introduce an
innovative phase modulation encoding (PME) strategy tailored for complex
spaces, serving as the positional encoding scheme for HDT.
As illustrated in Fig. 4, suppose the maximum dimension of each vector in the
sequence is $d$. For the $i$-th element of the $n$-th vector in the sequence,
PME operates as follows:
(14) $\displaystyle\mathrm{PME}(\boldsymbol{x}^{(n)}(i),n)$
$\displaystyle=\boldsymbol{x}_{i}^{(n)}\exp{(jn\theta_{i})},$
where $\theta_{i}$ is given by $\theta_{i}=10000^{-\frac{i}{d}}$. This
procedure can be conceptualized as a phase modulation process—essentially
imparting a specific phase offset to the original data based on the position
$n$ in the sequence.
The PME inherently decodes the relative position during computation,
establishing its essential role in position encoding. Specifically, when
executing the complex-domain Attention operation on the encoded key vector
$\boldsymbol{k}$ and query vector $\boldsymbol{q}$, it is equivalent to:
(15)
$\displaystyle\mathrm{PME}(\boldsymbol{q},n)^{\mathrm{H}}\mathrm{PME}(\boldsymbol{k},m)=\mathrm{PME}(\boldsymbol{q}^{\mathrm{H}}\boldsymbol{k},m-n).$
Therefore, the relative position information $m-n$ can be derived. This
enables our model to learn more proficiently by integrating the positional
details of the sequence.
## 5\. Implementation
We implement RF-Diffusion based on PyTorch and train our model on 8 NVIDIA
GeForce 3090 GPUs, incorporating essential implementation techniques outlined
below.
Exponential moving average. Following common practice in most generative
models, we adopt the exponential moving average (EMA) mechanism with a decay
rate of $0.999$. EMA calculates a sliding average of the model’s weights
during training, which improves model robustness.
Weight initialization. We zero-initialize each final layer before the residual
to accelerate large-scale training (Goyal et al., 2017), and apply Xavier
uniform initialization (Glorot and Bengio, 2010) to other layers, which is a
standard weight initialization technique in transformer-based models
(Dosovitskiy et al., 2021).
Hyperparameters. We train our model using AdamW optimizer (Kinga et al., 2015;
Loshchilov and Hutter, 2019) with an initial learning rate of $1\times
10^{-3}$. A step learning rate scheduler with a decay factor of $0.5$ is
adopted to improve training efficiency. In the training process, we apply a
dropout rate of $0.1$ to mitigate overfitting.
Noise scheduling strategy. In our implementation, the rate of data destruction
is designed to increase incrementally from a lower to a higher intensity as
the diffusion process progresses. This is aimed at achieving a balance between
the model complexity and the generation quality (Kong et al., 2020).
Specifically, we configure the diffusion process with a maximum of $T=300$
steps. The noise coefficient, $\beta_{t}=\sqrt{1-\alpha_{t}}$, is set to
linearly increase from $10^{-4}$ to $0.03$, i.e., $\beta_{t}=10^{-4}t$. In
parallel, the standard deviation of the Gaussian convolution kernel in the
frequency domain, denoted as $\boldsymbol{G}_{t}$, is adjusted to linearly
escalate from $10^{-3}$ to $0.3$, facilitating the controlled amplification of
noise across the diffusion steps.
Data preprocessing. Each signal sequence from the dataset is either
interpolated or downsampled to a consistent length of 512. This guarantees
uniformity in the model’s input length. Prior to input into our model, each
sample within the input sequence is normalized by average signal power, which
means each sample is divided by the average L2-norm of all the samples in the
sequence.
(a) Classroom
(b) Hall
(c) Office
Figure 5. Experimental scenarios.
## 6\. Evaluation
### 6.1. Experiment Design
#### 6.1.1. Data Collection.
As shown in Fig. 5, our dataset comprises wireless signals collected under
three distinct scenarios, featuring variations in room selection, device
location, and human factors, including their location, orientation, and
activities. We compile condition labels for each sequence into a conditional
vector $\boldsymbol{c}$, guiding both training and sampling phases. Our
research evaluates RF-Diffusion’s proficiency in producing signals across
different modulation modes, focusing on Wi-Fi and FMCW radar signals as two
primary types of wireless sensing and communications.
* •
Wi-Fi. We collect Wi-Fi signal based on the commercial NIC IWL5300 working in
5.825 GHz with 40 MHz bandwidth. The transmitter injects Wi-Fi packets to 3
receivers to extract the channel state information (CSI) corresponding to the
environment.
* •
FMCW. FMCW signals are recorded using the mmWave radar IWR1443 (Instruments,
2022). This radar device can be placed at either one of two different
locations in each scene, working at a frequency band from 77 GHz to 81 GHz.
More than 20,000 Wi-Fi sequences and 13,000 FMCW sequences are collected. Each
sequence has an associated condition label indicating the room, device
placement, human ID, location, orientation and activity type. All experiments
conducted in this paper conform to the IRB policies.
#### 6.1.2. Comparative Methods.
We compare RF-Diffusion with three representative data generation model:
* •
DDPM (Ho et al., 2020). The denoising diffusion probabilistic model (DDPM)
introduces Gaussian noise to original data and subsequently learns to reverse
this process, thereby generating raw data from the noise.
* •
DCGAN (Radford et al., 2015). The deep convolutional generative adversarial
network (DCGAN) stands as a widely recognized GAN. In DCGAN, two models (i.e.,
generator and discriminator) are simultaneously trained in an adversarial
manner. Once trained, the generator can produce data that convincingly
bypasses the discriminator’s scrutiny.
* •
CVAE (Sohn et al., 2015b). The conditional variational autoencoder (CVAE)
learns the Gaussian implicit representation of the data, thereby enabling data
generation. This method is widly adopted in both sensing (Ha et al., 2020) and
communication (Liu et al., 2021) systems to synthesize of wireless features.
To adapt them for RF signal, we have re-implemented the model using complex-
valued neural networks (Trabelsi et al., 2018).
#### 6.1.3. Evaluation Metrics.
For a comprehensive evaluation, we adopt two metrics, both of which are
commonly used in previous research for evaluating data-driven generative
models (Jiang et al., 2021; Nichol and Dhariwal, 2021; Ho et al., 2020;
Dhariwal and Nichol, 2021; Allen-Zhu and Li, 2023). Recognizing that a
definitive “gold standard” for generative models has not been established,
these metrics are among the most authoritative.
* •
SSIM (Wang et al., 2004): The Structural Similarity Index Measure (SSIM) is a
prominent criterion for gauging the similarity between two samples by
analyzing their means and covariances. We’ve adapted SSIM for the complex
domain, making it suitable for assessing complex-valued signals.
* •
FID (Heusel et al., 2017): The Fréchet Inception Distance (FID) evaluates
generative models by measuring the Fréchet distance between the high-level
features of real and synthesized data. We adopt a pretrained STFNets (Yao et
al., 2019) as the feature extractor to better fit the property of wireless
signals.
### 6.2. Overall Generation Quality
The evaluation result for RF-Diffusion on Wi-Fi and FMCW signal are
illustrated in Fig. 6 and Fig. 7 respectively. As shown, our proposed RF-
Diffusion has proved the superiority over comparative methods on both two
metrics.
Specifically, as shown in Fig. 6, RF-Diffusion generates Wi-Fi signal with an
average SSIM of $0.81$, exceeding DDPM, DCGAN, and CVAE by $25.4\%$, $18.6\%$
and $71.3\%$ respectively. RF-Diffusion achieves an FID of $4.42$,
outperforming the above comparative methods by $42.4\%$, $63.0\%$, and
$57.3\%$.
RF-Diffusion also outperforms the comparative methods in terms of generating
high-fidelity FMCW signals. As shown in Fig. 7, the FMCW signal generated by
RF-Diffusion achieves an average SSIM of $0.75$ and an average FID of $6.10$.
The impressive performance of RF-Diffusion can be attributed to several key
factors: 1) Our proposed time-frequency diffusion adopted by RF-Diffusion
emphasizes refining the frequency spectrum of the RF signal, thereby
preserving finer spectral details in the generated signals, which is difficult
to be captured by other methods. 2) Through its iterative generation approach,
RF-Diffusion attains precise reconstruction of data details via multi-step
approximations, leading to a superior quality of generated data. 3) In
contrast to DCGAN, which optimizes two models concurrently, RF-Diffusion’s
loss function is more streamlined and its training process more stable,
ensuring a richer diversity in the generated signal and contributing to a
commendable FID score.
(a) SSIM
(b) FID
Figure 6. Wi-Fi signal generation quality.
(a) SSIM
(b) FID
Figure 7. FMCW signal generation quality.
Figure 8. Impact of diffusion method.
Figure 9. Impact of network design.
Figure 10. Scalability analysis.
### 6.3. Micro-benchmarks
#### 6.3.1. Impact of Diffusion Methods.
To validate the efficacy of our proposed time-frequency diffusion theory, we
retained the network model architecture of RF-Diffusion but replaced the time-
frequency diffusion process with two alternate schemes: 1) Gaussian diffusion,
which is similar to DDPM and by only introduces Gaussian noise to the signal
amplitude, and 2) blur diffusion which only performs spectral blurring. As
depicted in Fig. 10, our time-frequency diffusion theory consistently
outperforms both in terms of the SSIM and FID metrics. Specifically, the SSIM
values for time-frequency diffusion, Gaussian diffusion, and blur diffusion
stand at $0.81$, $0.71$, and $0.45$, respectively. This translates to the
time-frequency diffusion offering an SSIM improvement of $13.9\%$ over
Gaussian diffusion and a notable $79.2\%$ over blur diffusion. In terms of the
FID, the time-frequency diffusion surpasses the other two methods by margins
of $41.3\%$ and $83.5\%$, respectively. The results indicates that the time-
frequency diffusion theory successfully incorporates two diffusion methods on
orthogonal spaces, and thus achieving complementary benefits.
#### 6.3.2. Impact of Network Design.
To demonstrate the advantages of our proposed hierarchical diffusion
transformer (HDT), we compare it against: 1) single-stage diffusion
transformer (SDT), which is a simplified form of HDT, with only one stage for
end-to-end data restoration, and 2) U-Net (Ronneberger et al., 2015), a
popular choice in prevalent diffusion models. As shown in Fig. 10, our
proposed HDT outperforms the SDT and U-Net. Specifically, the SSIM for HDT,
SDT, and U-Net are $0.81$, $0.75$, and $0.68$ respectively. This indicate that
HDT achieves a SSIM boost of $7.7\%$ over SDT and a significant $18.9\%$
increment compared to U-Net. When assessed using the FID metric, HDT continues
to lead by margins of $48.2\%$ and $49.3\%$ against SDT and U-Net,
respectively. The outstanding performance benefits from the follows aspects:
1) Compared with SDT, HDT can efficiently decouple the non-isotropic noise
introduced in the diffusion process and eliminate it through two sequential
stages; 2) Compared with translation-invariant U-Net, HDT’s transformer
architecture can effectively distinguish the signal characteristics at
different times, thereby achieving more accurate signal generation.
#### 6.3.3. Scalability Analysis.
Scalability refers to a model’s ability to enhance its performance with
increasing size, which is critical for large generative models like RF-
Diffusion. To verify the scalability of RF-Diffusion, we trained 9 models of
different sizes, exploring different numbers of attention-based diffusion
blocks (16B, 32B, 64B) and hidden dimensions (64, 128, 256). Fig. 10
illustrates that the FID performance of RF-Diffusion is strongly correlated
with model parameters and GFLOPs, indicating that scaling up model parameters
and additional model computation is the critical ingredient for better
performance. Increasing the model size is anticipated to further enhance RF-
Diffusion’s performance.
## 7\. Case Study
This section showcases how RF-Diffusion benefits wireless researches in two
distinct downstream tasks: Wi-Fi-based gesture recognition and 5G FDD channel
estimation.
### 7.1. Wi-Fi Gesture Recognition
Wireless sensing (Chi et al., 2022; Yang et al., 2023; Zhang et al., 2023; Gao
et al., 2023) has emerged as a major research focus. By serving as a data
augmentor, RF-Diffusion can boost the performance of existing wireless sensing
systems, all while preserving the original model structure without any
modifications. In particular, our approach involves initially training RF-
Diffusion using a real-world dataset. Subsequently, RF-Diffusion generates
synthetic RF signals of the designated type, guided by condition labels. These
synthetic samples are then integrated with the original dataset, collectively
employed to train the wireless sensing model. Both RF-Diffusion-augmented
solution and baseline are fundamentally based on the same real-world dataset,
ensuring a fair comparison, as RF-Diffusion itself is trained on this real-
world dataset and no extra data is ever involved.
We illustrate this approach through the case of Wi-Fi-based gesture
recognition and evaluate the performance gains achieved by integrating RF-
Diffusion into established gesture recognition models.
(a) Cross-domain
(b) In-domain
Figure 11. Performance of augmented Wi-Fi sensing.
#### 7.1.1. Experiment Design.
We select two different types of Wi-Fi-based model for a comprehensive
evaluation:
* •
Widar3.0 (Zheng et al., 2019) is a gesture recognition model founded on
physical principles. It initially extracts features from raw signals and
subsequently conducts recognition through a deep neural network.
* •
EI (Jiang et al., 2018) is a data-driven end-to-end human activity recognition
model that takes raw signal as input.
We utilize the publicly available dataset from Widar3.0 (Zheng et al., 2019)
to assess performance. This evaluation encompasses scenarios where RF-
Diffusion and comparative methods (§6.1.2) were employed as data augmentors.
Figure 12. Impact of synthetic data volume.
#### 7.1.2. Cross-domain Evaluation.
We first evaluate the sensing performance when the training and testing set
are from different domains (i.e., room, device placement, human location,
orientation, etc.), a common scenario in real-world wireless sensing system
deployments. We synthesize an equivalent volume of data as the real-world
dataset using the pre-trained RF-Diffusion. Subsequently, both synthesized and
authentic datasets are used for training. As shown in Fig. 11a, integrating
RF-Diffusion brings performance improvements of $4.7\%$ and $11.5\%$ for
Widar3.0 and EI, respectively. Integrating DDPM can bring relatively limited
performance gains of only $1.8\%$ and $7.3\%$, respectively. Additionally, the
integration of DCGAN or CVAE may result in a degradation of recognition
accuracy due to deviations in the synthetic data distribution from the
original data distribution.
Compared with Widar 3.0, the EI model obtains a more significant improvement
since: 1) EI is more sensitive to the data volume and data diversity as an
end-to-end DNN; 2) the information in the wireless signals generated in a
data-driven manner cannot be fully exploited in Widar3.0 when being converted
into physical features.
In conclusion, RF-Diffusion enhances the cross-domain performance of wireless
sensing systems in two aspects:
* •
Enhanced data diversity. Synthetic training data with higher diversity avoid
model overfitting and thus implicitly improves the model’s domain
generalization ability.
* •
Feature distillation. The generative model RF-Diffusion implicitly imparts its
learned signal features to the recognition model through synthetic training
data, contributing to improved performance.
#### 7.1.3. In-domain Evaluation.
In the in-domain scenario, the training and testing data are from the same
domain. As shown in Fig. 11b, the integration of RF-Diffusion yields
performance improvements of $1.8\%$ and $8.7\%$ for Widar3.0 and EI,
respectively. Overall, compared with the cross-domain scenario, the
performance gain in the in-domain case is relatively modest. This is
attributed to the limited impact of diverse synthetic training data on
enhancing the model’s performance within the same domain. For in-domain
testing, even with a less diverse synthetic data generated by DCGAN, an
obvious performance gain can be achieved.
#### 7.1.4. Impact of Synthesized Data Ratio.
We further investigate the impact of the synthesized data ratio used for
training to provide more insights. We evaluate Widar 3.0 and EI in both cross-
domain (CD) and in-domain (ID) cases.
As shown in the Fig. 12, we introduce varying quantities of synthetic data
(from $+25\%$ to $+200\%$) to the real-world dataset for joint training of the
recognition model. Notably, as the volume of synthetic data increases, the
trend in recognition accuracy exhibits an ascent to a peak followed by a
decline. Specifically, in the cross-domain case, Widar3.0 reaches the highest
accuracy of $92.7\%$ with $+100\%$ synthetic data, while EI reaches the
highest accuracy of $83.8\%$ with $+125\%$ synthetic data. In the in-domain
case, Widar3.0 achieved the highest accuracy of $93.4\%$ with $+50\%$
synthetic data, while EI achieved the highest accuracy of $89.1\%$ with
$+75\%$ synthetic data. Drawing from these statistical findings, we deduce the
following insights: 1) For most wireless recognition models, judicious
incorporation of synthetic data into the training set can effectively enhance
model performance. 2) Excessive introduction of synthetic data can potentially
shift the training data distribution away from the original, consequently
diminishing recognition accuracy. 3) The cross-domain scenario requires a
greater infusion of synthetic data into the training set to achieve optimal
model performance compared to the in-domain scenario. 4) Data-driven end-to-
end models (e.g., EI) reap more substantial benefits from data augmentation
facilitated by RF-Diffusion.
### 7.2. 5G FDD Channel Estimation
(a) Channel amplitude and phase
(b) SNR
Figure 13. Performance of channel estimation.
In this section, we discuss how RF-Diffusion enables channel estimation of the
Frequency Domain Duplex (FDD) system in 5G, where the uplink and downlink
transmissions operate at different frequency bands. Therefore, the principle
of reciprocity that two link channels are equal no longer holds (Liu et al.,
2021). To estimate the downlink channel state, client devices must receive
additional symbols from a base station with a massive antenna array and send
back the estimated results, causing unsustainable overheads. To solve this
problem, substantial research is devoted to predicting the downlink channel by
observing the uplink channel state information. For example, FNN (Bakshi et
al., 2019) and FIRE (Liu et al., 2021) make use of a fully connected network
and a VAE to transfer the estimated CSI from the uplink to the downlink,
respectively.
We discover that by employing the uplink CSI as conditional input, RF-
Diffusion demonstrates the capacity to estimate downlink channel CSI in a
generative manner. Specifically, in RF-Diffusion, the downlink CSI
$\boldsymbol{x}_{\mathrm{down}}$ serves as the target data for generation,
while the uplink CSI is encoded as the condition
$\boldsymbol{c}_{\mathrm{up}}$ and input into the model. The trained RF-
Diffusion learns the correlation between $\boldsymbol{c}_{\mathrm{up}}$ and
$\boldsymbol{x}_{\mathrm{down}}$, thereby accomplishing the channel estimation
task. This efficacy is rooted in the assumption of shared propagation paths,
positing that both link channels are shaped by the same underlying physical
environment (Huang et al., 2019; Vasisht et al., 2016).
#### 7.2.1. Experiment Design.
Our evaluation is based on the publicly available dataset Argos (Shepard et
al., 2016), which is a real-world MIMO dataset collected in a complex
environment with a large number of non-line-of-sight (NLoS) propagation. Each
CSI frame contains 52 subcarriers. Similar to previous works (Zhao et al.,
2023; Liu et al., 2021), we designate the initial 26 subcarriers for the
uplink channel, while the remaining 26 are allocated to the downlink channel.
For a comprehensive evaluation, we compare our approach against three
different types of channel estimation solutions:
* •
NeRF2 (Zhao et al., 2023) implicitly learns the signal propagation environment
from the uplink channel state based on a neural network, and then estimate the
downlink channel.
* •
FIRE (Liu et al., 2021) is a channel estimation system based on VAE, which
compresses the uplink channel CSI into a latent space representation and
further transforms it into a downlink channel estimation.
* •
Codebook (Kaltenberger et al., 2008), commonly used in standard
implementations per the 3GPP physical channel standard(3rd Generation
Partnership Project, 3GPP), requires both base stations and clients to
maintain a codebook of vectors created using predefined rules. Clients measure
the channel locally, select the closest codebook vectors, and send the
corresponding indices back to the base station.
#### 7.2.2. Channel Estimation Accuracy.
As shown in Fig. 13a, when we input the blue uplink CSI as a condition into
the trained RF-Diffusion, the red downlink estimate will be output, which
closely aligns with the ground truth downlink channel state. Assessment of
channel estimation accuracy employs the Signal-to-Noise Ratio (SNR) metric
(Zhao et al., 2023; Liu et al., 2021). This metric gauges the congruence
between the estimated downlink channel $\boldsymbol{x}_{\mathrm{est}}$ and the
ground truth $\boldsymbol{x}_{\mathrm{down}}$ through the following
formulation:
(16)
$\mathrm{SNR}=-10\log_{10}\left(\frac{\lVert\boldsymbol{x}_{\mathrm{down}}-\boldsymbol{x}_{\mathrm{est}}\rVert^{2}}{\lVert\boldsymbol{x}_{\mathrm{down}}\rVert^{2}}\right).$
A higher positive SNR corresponds to enhanced proximity between the predicted
channel and the ground truth. As shown in Fig. 13b, RF-Diffusion achieves the
highest SNR among all comparative methods with an average SNR of $27.01$,
outperforming NeRF2 and FIRE by $34.6\%$ and $77.5\%$ respectively, and
achieves more than $5\times$ performance gain compared with the standard
implementation based on codebook.
The observed underperformance of NeRF2 can be attributed to its treatment of
the signal propagation space as a time-invariant system, a characterization
that may not hold in practical scenarios. VAE-based FIRE and codebook-based
methods fall short in the fine-grained characterization of the underlying
distribution of channel states. In contrast, RF-Diffusion adeptly learns the
intricate correlation between the uplink and downlink channels, leveraging its
robust modeling capacity to achieve highly accurate channel estimation.
## 8\. Related Work
We briefly review the related works in the following.
Diffusion probabilistic models. Diffusion probabilistic models (Yang et al.,
2022b; Sohl-Dickstein et al., 2015) have emerged as a powerful new family of
deep generative models with record-breaking performance in many applications
(Dhariwal and Nichol, 2021), including image synthesis, point cloud
completion, and natural language processing, etc. One of the best-known
diffusion model is the DDPM (Sohl-Dickstein et al., 2015), which progressively
destruct data by injecting gaussian noise, then learn to reverse this process
for high-fidelity sample generation. On this basis, DDIM (Nichol and Dhariwal,
2021) expedites reverse sampling, while LDM (Rombach et al., 2022) conducts
diffusion in latent space to curtail computational overhead. The above schemes
have been widely used in a wide range of tasks such as image super-resolution
(Saharia et al., 2022b; Ho et al., 2022), inpainting (Lugmayr et al., 2022),
and style transfer (Saharia et al., 2022a). The most recent studies (Lee et
al., 2022; Rissanen et al., 2023) have successfully applied a combination of
blurring and additive noise to the image, yielding satisfactory results.
Although first proposed for image generation, the diffusion model’s
versatility extends to other domains including point cloud completion (Zhou et
al., 2021; Lyu et al., 2021), text generation (Austin et al., 2021; Gong et
al., 2023), audio synthesis (Kong et al., 2020; Chen et al., 2021), and
beyond. In addition, diffusion model has a great potential for multi-modal
generation. By integrating pre-trained language model (Radford et al., 2021),
the diffusion models achieve impressive performance in text-to-image (Nichol
et al., 2022; Ramesh et al., 2022) and text-to-audio (Popov et al., 2021)
tasks.
RF-Diffusion, in contrast, stands as the pioneering diffusion model tailored
for wireless signal generation. It introduces an innovative time-frequency
diffusion process, which regulates noise and blurring across two orthogonal
domains, thus encompassing both temporal and spectral intricacies of wireless
signals. By generating high-fidelity signals, RF-Diffusion benefits wireless
applications like Wi-Fi sensing and 5G channel estimation.
Signal generation in wireless systems. Conventional wireless signal generation
schemes are mainly based on modeling and simulation. In particular, these
methods involve utilizing LiDAR-scanned 3D models, and employing
electromagnetic (EM) ray tracing techniques (McKown and Hamilton, 1991) to
simulate the distribution of wireless signals. Recent studies (Korany et al.,
2019; Cai et al., 2020; Zhang et al., 2022) have integrated vision-based human
reconstruction techniques with signal propagation models, enabling the
generation of wireless signals that interact with the human body.
Unfortunately, the above schemes fails to model the structure material and
physical characteristics, which constraints their performance in real-world
applications. The recently proposed NeRF2 (Zhao et al., 2023) learns the
properties of the signal propagation space based on a deep neural network and
then accomplishes the signal generation task. However, NeRF2 is limited to
specific static scenarios and degrades for dynamic real-world scenarios. RF-
EATS (Ha et al., 2020) and FallDar (Yang et al., 2022a) employ Variational
Autoencoders (VAEs) to extract environment-independent features, thereby
enhancing the generalizability of wireless sensing models. Additionally, other
studies have utilized Generative Adversarial Networks (GANs) to generate
Doppler spectrum (Erol et al., 2019). Other research endeavors have addressed
channel estimation in wireless communication systems using either GANs (Doshi
et al., 2022; Balevi and Andrews, 2021) or VAEs (Baur et al., 2022; Liu et
al., 2021). Nonetheless, due to their limited representation capabilities,
solutions based on GANs and VAEs struggle to faithfully characterize the
intrinsic properties of original wireless signals. Consequently, the
aforementioned systems are suitable solely for specific tasks, lacking the
competence for general-purpose wireless data generation.
In contrast, RF-Diffusion, as a versatile generative model for wireless
signals, can proficiently generate fine-grained signals in high-fidelity, even
within dynamic scenarios.
## 9\. Discussion and Future Work
RF-Diffusion is a pioneering attempt towards diffusion-based RF signal
generation, and there is room for continued research in various perspectives.
$\bullet$ RF-Diffusion for data-driven downstream tasks. Extensive practices
(He et al., 2023; Shivashankar and Miller, 2023; Azizi et al., 2023; Zheng et
al., 2023) indicate that synthetic data from generative models significantly
enhances data-driven downstream tasks. As a conditional generative model, RF-
Diffusion effectively captures the representative features and their novel
combinations, while randomizing non-essential details. This approach allows
for the generation of innovative data samples that extend beyond the initial
scope of the dataset, thus improving the generalization ability of downstream
models. This paper specifically explores and experiments with applying RF-
Diffusion to augment Wi-Fi gesture recognition, demonstrating its potential.
However, the applicability of RF-Diffusion extends to any data-driven task in
wireless communication and sensing.
$\bullet$ RF-Diffusion as a simulation tool. As a probabilistic generative
model, RF-Diffusion operates independently of any signal propagation
assumptions and does not require pre-modeling of the environment. This
flexibility implies that, while RF-Diffusion offers novel opportunities for
signal synthesis, it may not achieve the same level of stability and precision
as traditional signal simulation tools in all scenarios. RF-Diffusion is not
designed to supplant simulation tools but rather to introduce a novel data-
driven approach for signal synthesis, which is particularly valuable in
complex and dynamic environments, such as indoor spaces with human activity,
where accurate modeling poses challenges.
$\bullet$ Autoregressive signal generation. RF-Diffusion, a non-autoregressive
generative model, processes time series as a unified entity, necessitating
downsampling and interpolation for variable-length sequences, which limits its
versatility. The advent of autoregressive models like GPT introduces
alternative methods for time-series signal generation, which improves
adaptability for sequences of differing lengths and enable effective
exploration of temporal correlation features.
## 10\. Conclusion
This paper presents RF-Diffusion, the pioneering generative diffusion model
designed for RF signals. RF-Diffusion excels in generating high-fidelity time-
series signals by employing a novel time-frequency diffusion process. This
process captures the intricate characteristics of RF signals across spatial,
temporal, and frequency domains. This theoretical framework is then translated
into a practical generative model based on the hierarchical diffusion
transformer. RF-Diffusion exhibits remarkable versatility. It holds
significant potential for essential wireless tasks, ranging from boosting the
accuracy of wireless sensing systems, to estimating channel states in
communication systems, shedding light on the applications of AIGC in wireless
research.
## 11\. Acknowledgments
We sincerely thank the MobiSense Group, the anonymous reviewers, and our
shepherd for their constructive comments and feedback in improving this work.
This paper is supported in part by the NSFC under grant No. 62372265,
No.62302254, and No. 62222216, and by the Hong Kong RGC ECS under grant
27204522 and RIF under grant R5060-19.
## Appendix A Convergence of Forward Destruction Process
As $T\to\infty$, the forward process converges to a distribution independent
of the original signal. The above proposition is equivalent to the following
two conditions: (1) $\lim_{T\to\infty}\bar{\boldsymbol{\mu}}_{T}=\mathbf{0}$,
(2) $\lim_{T\to\infty}\bar{\boldsymbol{\sigma}}_{T}<\infty$. We find a
sufficient condition for the above to hold true: all element in
$\boldsymbol{\gamma}_{t}=\sqrt{\alpha_{t}}\boldsymbol{g}_{t}$ should be less
than 1, i.e., $\forall n,\gamma_{t}^{(n)}<1$. Under this condition, according
to Eqn. 3,
$\lim_{T\to\infty}\bar{\boldsymbol{\mu}}_{T}=\lim_{T\to\infty}\bar{\boldsymbol{\gamma}}_{T}\boldsymbol{x}_{0}=\mathbf{0}$
holds. Let $\alpha_{\mathrm{min}}=\min(\alpha_{t}),t\in[1,T]$ and
${\gamma}^{(n)}_{\mathrm{max}}=\max(\gamma_{t}^{(n)})$ and
$\boldsymbol{\gamma}_{\mathrm{max}}=({\gamma}^{(1)}_{\mathrm{max}},\dots,{\gamma}^{(N)}_{\mathrm{max}})$.
It can be proven that:
(17)
$\displaystyle\lim_{T\to\infty}\bar{\boldsymbol{\sigma}}_{T}=\lim_{T\to\infty}\sum_{t=1}^{T}{(\sqrt{1-\alpha_{t}}\frac{\bar{\boldsymbol{\gamma}}_{T}}{\bar{\boldsymbol{\gamma}}_{t}})}$
$\displaystyle\leq\sqrt{1-\alpha_{\mathrm{min}}}\lim_{T\to\infty}\sum_{t=1}^{T}{(\boldsymbol{\gamma}_{\mathrm{max}})^{t-1}}=\frac{\sqrt{1-\alpha_{\mathrm{min}}}}{1-\boldsymbol{\gamma}_{\mathrm{max}}}<\infty$
## Appendix B Reverse Process Distribution
Based on the Bayes’ theorem, we get:
(18) $\displaystyle
q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{t},\boldsymbol{x}_{0})=q(\boldsymbol{x}_{t}|\boldsymbol{x}_{t-1},\boldsymbol{x}_{0})\frac{q(\boldsymbol{x}_{t-1}|\boldsymbol{x}_{0})}{q(\boldsymbol{x}_{t}|\boldsymbol{x}_{0})}$
$\displaystyle\propto$
$\displaystyle\exp(-\frac{1}{2}(\frac{(\boldsymbol{x}_{t}-\boldsymbol{\gamma}_{t}\boldsymbol{x}_{t-1})^{2}}{\boldsymbol{\sigma}_{t}^{2}}+\frac{(\boldsymbol{x}_{t-1}-\bar{\boldsymbol{\gamma}}_{t-1}\boldsymbol{x}_{0})^{2}}{\bar{\boldsymbol{\sigma}}_{t-1}^{2}}-\frac{(\boldsymbol{x}_{t}-\bar{\boldsymbol{\gamma}}_{t}\boldsymbol{x}_{0})^{2}}{\bar{\boldsymbol{\sigma}}_{t}^{2}}))$
$\displaystyle=$
$\displaystyle\exp(((\frac{\boldsymbol{\gamma}_{t}}{\boldsymbol{\sigma}_{t}})^{2}+(\frac{1}{\bar{\boldsymbol{\sigma}}_{t-1}})^{2})\boldsymbol{x}_{t-1}^{2}-(\frac{2\boldsymbol{\gamma}_{t}}{\boldsymbol{\sigma}_{t}^{2}}\boldsymbol{x}_{t}+\frac{2\bar{\boldsymbol{\gamma}}_{t-1}}{\bar{\boldsymbol{\sigma}}_{t-1}^{2}}\boldsymbol{x}_{0})\boldsymbol{x}_{t-1}+C(\boldsymbol{x}_{t},\boldsymbol{x}_{0})),$
in which the recursive relationship is used:
$\bar{\boldsymbol{\sigma}}_{t}^{2}=\boldsymbol{\gamma}_{t}^{2}\bar{\boldsymbol{\sigma}}_{t-1}^{2}+\boldsymbol{\sigma}_{t}^{2}$,
which can be inferred by combining Eqn. 2 and Eqn. 3.
## References
* (1)
* 3rd Generation Partnership Project (3GPP) 3rd Generation Partnership Project (3GPP). 2023\. _5G; NR; Physical channels and modulation._ Technical Specification (TS) 38.211. 3rd Generation Partnership Project (3GPP). Version 17.5.0.
* Allen-Zhu and Li (2023) Zeyuan Allen-Zhu and Yuanzhi Li. 2023. Forward Super-Resolution: How Can GANs Learn Hierarchical Generative Models for Real-World Distributions. In _The Eleventh International Conference on Learning Representations_.
* Austin et al. (2021) Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. 2021\. Structured denoising diffusion models in discrete state-spaces. _Advances in Neural Information Processing Systems_ 34 (2021), 17981–17993.
* Azizi et al. (2023) Shekoofeh Azizi, Simon Kornblith, Chitwan Saharia, Mohammad Norouzi, and David J Fleet. 2023\. Synthetic data from diffusion models improves imagenet classification. _arXiv preprint arXiv:2304.08466_ (2023).
* Bakshi et al. (2019) Arjun Bakshi, Yifan Mao, Kannan Srinivasan, and Srinivasan Parthasarathy. 2019. Fast and efficient cross band channel prediction using machine learning. In _The 25th Annual International Conference on Mobile Computing and Networking_. 1–16.
* Balevi and Andrews (2021) Eren Balevi and Jeffrey G Andrews. 2021. Wideband channel estimation with a generative adversarial network. _IEEE Transactions on Wireless Communications_ 20, 5 (2021), 3049–3060.
* Bando et al. (2020) Yoshiaki Bando, Kouhei Sekiguchi, and Kazuyoshi Yoshii. 2020\. Adaptive Neural Speech Enhancement with a Denoising Variational Autoencoder.. In _INTERSPEECH_.
* Baur et al. (2022) Michael Baur, Benedikt Fesl, Michael Koller, and Wolfgang Utschick. 2022. Variational autoencoder leveraged mmse channel estimation. In _2022 56th Asilomar Conference on Signals, Systems, and Computers_. IEEE, 527–532.
* Brock et al. (2018) Andrew Brock, Jeff Donahue, and Karen Simonyan. 2018\. Large Scale GAN Training for High Fidelity Natural Image Synthesis. In _International Conference on Learning Representations_.
* Cai et al. (2020) Hong Cai, Belal Korany, Chitra R Karanam, and Yasamin Mostofi. 2020\. Teaching rf to sense without rf training measurements. _Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies_ 4, 4 (2020), 1–22.
* Chen et al. (2021) Nanxin Chen, Yu Zhang, Heiga Zen, Ron J Weiss, Mohammad Norouzi, and William Chan. 2021\. WaveGrad: Estimating Gradients for Waveform Generation. In _International Conference on Learning Representations_.
* Chi et al. (2022) Guoxuan Chi, Zheng Yang, Jingao Xu, Chenshu Wu, Jialin Zhang, Jianzhe Liang, and Yunhao Liu. 2022. Wi-drone: wi-fi-based 6-DoF tracking for indoor drone flight control. In _Proceedings of the 20th Annual International Conference on Mobile Systems, Applications and Services_. 56–68.
* Dhariwal and Nichol (2021) Prafulla Dhariwal and Alexander Nichol. 2021. Diffusion models beat gans on image synthesis. _Advances in neural information processing systems_ 34 (2021), 8780–8794.
* Doshi et al. (2022) Akash S Doshi, Manan Gupta, and Jeffrey G Andrews. 2022\. Over-the-Air Design of GAN Training for mmWave MIMO Channel Estimation. _IEEE Journal on Selected Areas in Information Theory_ 3, 3 (2022), 557–573.
* Dosovitskiy et al. (2021) Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. 2021\. An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. In _International Conference on Learning Representations_.
* Erol et al. (2019) Baris Erol, Sevgi Z Gurbuz, and Moeness G Amin. 2019\. GAN-based synthetic radar micro-Doppler augmentations for improved human activity recognition. In _2019 IEEE Radar Conference (RadarConf)_. IEEE, 1–5.
* Gao et al. (2023) Yuchong Gao, Guoxuan Chi, Guidong Zhang, and Zheng Yang. 2023\. Wi-Prox: Proximity Estimation of Non-directly Connected Devices via Sim2Real Transfer Learning. In _GLOBECOM 2023-2023 IEEE Global Communications Conference_. IEEE.
* Glorot and Bengio (2010) Xavier Glorot and Yoshua Bengio. 2010. Understanding the difficulty of training deep feedforward neural networks. In _Proceedings of the thirteenth international conference on artificial intelligence and statistics_. JMLR Workshop and Conference Proceedings, 249–256.
* Gong et al. (2023) Shansan Gong, Mukai Li, Jiangtao Feng, Zhiyong Wu, and Lingpeng Kong. 2023. DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models. In _The Eleventh International Conference on Learning Representations_.
* Goyal et al. (2017) Priya Goyal, Piotr Dollár, Ross Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, Andrew Tulloch, Yangqing Jia, and Kaiming He. 2017. Accurate, large minibatch sgd: Training imagenet in 1 hour. _arXiv preprint arXiv:1706.02677_ (2017).
* Ha et al. (2020) Unsoo Ha, Junshan Leng, Alaa Khaddaj, and Fadel Adib. 2020\. Food and liquid sensing in practical environments using $\\{$RFIDs$\\}$. In _17th USENIX Symposium on Networked Systems Design and Implementation (NSDI 20)_. 1083–1100.
* Hamdan and Hamdi (2020) Mutasem Q Hamdan and Khairi A Hamdi. 2020. Variational Auto-encoders application in wireless Vehicle-to-Everything communications. In _2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring)_. IEEE.
* He et al. (2023) Ruifei He, Shuyang Sun, Xin Yu, Chuhui Xue, Wenqing Zhang, Philip Torr, Song Bai, and Xiaojuan Qi. 2023\. Is synthetic data from generative models ready for image recognition?. In _The Eleventh International Conference on Learning Representations_.
* Heusel et al. (2017) Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. 2017\. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In _Advances in Neural Information Processing Systems_ , Vol. 30. Curran Associates, Inc.
* Ho et al. (2020) Jonathan Ho, Ajay Jain, and Pieter Abbeel. 2020. Denoising diffusion probabilistic models. _Advances in neural information processing systems_ 33 (2020), 6840–6851.
* Ho et al. (2022) Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. 2022\. Cascaded diffusion models for high fidelity image generation. _The Journal of Machine Learning Research_ 23, 1 (2022), 2249–2281.
* Ho and Salimans (2021) Jonathan Ho and Tim Salimans. 2021. Classifier-Free Diffusion Guidance. In _NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications_.
* Huang et al. (2019) Chongwen Huang, George C Alexandropoulos, Alessio Zappone, Chau Yuen, and Mérouane Debbah. 2019\. Deep learning for UL/DL channel calibration in generic massive MIMO systems. In _ICC 2019-2019 IEEE International Conference on Communications (ICC)_. IEEE, 1–6.
* Instruments (2022) Texas Instruments. 2022\. Texas Instruments IWR1443BOOST. https://www.ti.com/tool/IWR1443BOOST
* Jiang et al. (2021) Liming Jiang, Bo Dai, Wayne Wu, and Chen Change Loy. 2021\. Focal frequency loss for image reconstruction and synthesis. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_. 13919–13929.
* Jiang et al. (2018) Wenjun Jiang, Chenglin Miao, Fenglong Ma, Shuochao Yao, Yaqing Wang, Ye Yuan, Hongfei Xue, Chen Song, Xin Ma, Dimitrios Koutsonikolas, et al. 2018\. Towards environment independent device free human activity recognition. In _Proceedings of the 24th annual international conference on mobile computing and networking_. 289–304.
* Kaltenberger et al. (2008) Florian Kaltenberger, David Gesbert, Raymond Knopp, and Marios Kountouris. 2008. Performance of multi-user MIMO precoding with limited feedback over measured channels. In _IEEE GLOBECOM 2008-2008 IEEE Global Telecommunications Conference_. IEEE, 1–5.
* Kinga et al. (2015) D Kinga, Jimmy Ba Adam, et al. 2015\. Adam: A method for stochastic optimization. In _International conference on learning representations (ICLR)_ , Vol. 5. San Diego, California;, 6\.
* Kingma and Welling (2013) Diederik P Kingma and Max Welling. 2013. Auto-encoding variational bayes. _arXiv preprint arXiv:1312.6114_ (2013).
* Kong et al. (2020) Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. 2020. DiffWave: A Versatile Diffusion Model for Audio Synthesis. In _International Conference on Learning Representations_.
* Korany et al. (2019) Belal Korany, Chitra R Karanam, Hong Cai, and Yasamin Mostofi. 2019. XModal-ID: Using WiFi for through-wall person identification from candidate video footage. In _The 25th Annual International Conference on Mobile Computing and Networking_. 1–15.
* Lee et al. (2022) Sangyun Lee, Hyungjin Chung, Jaehyeon Kim, and Jong Chul Ye. 2022. Progressive Deblurring of Diffusion Models for Coarse-to-Fine Image Synthesis. In _NeurIPS 2022 Workshop on Score-Based Methods_. https://openreview.net/forum?id=KP8BrpZBbv
* Liu et al. (2021) Zikun Liu, Gagandeep Singh, Chenren Xu, and Deepak Vasisht. 2021. FIRE: enabling reciprocity for FDD MIMO systems. In _Proceedings of the 27th Annual International Conference on Mobile Computing and Networking_. 628–641.
* Loshchilov and Hutter (2019) Ilya Loshchilov and Frank Hutter. 2019. Decoupled Weight Decay Regularization. In _International Conference on Learning Representations_.
* Lugmayr et al. (2022) Andreas Lugmayr, Martin Danelljan, Andres Romero, Fisher Yu, Radu Timofte, and Luc Van Gool. 2022\. Repaint: Inpainting using denoising diffusion probabilistic models. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_. 11461–11471.
* Lyu et al. (2021) Zhaoyang Lyu, Zhifeng Kong, Xudong Xu, Liang Pan, and Dahua Lin. 2021. A conditional point diffusion-refinement paradigm for 3d point cloud completion. _arXiv preprint arXiv:2112.03530_ (2021).
* McKown and Hamilton (1991) John W McKown and R Lee Hamilton. 1991. Ray tracing as a design tool for radio networks. _IEEE Network_ 5, 6 (1991), 27–30.
* Midjourney (2023) Midjourney. 2023\. Midjourney. https://www.midjourney.com/
* Nichol and Dhariwal (2021) Alexander Quinn Nichol and Prafulla Dhariwal. 2021. Improved denoising diffusion probabilistic models. In _International Conference on Machine Learning_. PMLR, 8162–8171.
* Nichol et al. (2022) Alexander Quinn Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob Mcgrew, Ilya Sutskever, and Mark Chen. 2022. GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models. In _Proceedings of the 39th International Conference on Machine Learning_.
* OpenAI (2023) OpenAI. 2023. GPT-4 Technical Report. arXiv:2303.08774 [cs.CL]
* Perez et al. (2018) Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. 2018. Film: Visual reasoning with a general conditioning layer. In _Proceedings of the AAAI conference on artificial intelligence_ , Vol. 32.
* Popov et al. (2021) Vadim Popov, Ivan Vovk, Vladimir Gogoryan, Tasnima Sadekova, and Mikhail Kudinov. 2021. Grad-tts: A diffusion probabilistic model for text-to-speech. In _International Conference on Machine Learning_. PMLR, 8599–8608.
* Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. 2021\. Learning transferable visual models from natural language supervision. In _International conference on machine learning_. PMLR, 8748–8763.
* Radford et al. (2015) Alec Radford, Luke Metz, and Soumith Chintala. 2015. Unsupervised representation learning with deep convolutional generative adversarial networks. _arXiv preprint arXiv:1511.06434_ (2015).
* Ramesh et al. (2022) Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. 2022. Hierarchical text-conditional image generation with clip latents. _arXiv preprint arXiv:2204.06125_ 1, 2 (2022), 3\.
* Rissanen et al. (2023) Severi Rissanen, Markus Heinonen, and Arno Solin. 2023\. Generative Modelling with Inverse Heat Dissipation. In _The Eleventh International Conference on Learning Representations_. https://openreview.net/forum?id=4PJUBT9f2Ol
* Rizk et al. (2019) Hamada Rizk, Ahmed Shokry, and Moustafa Youssef. 2019\. Effectiveness of data augmentation in cellular-based localization using deep learning. In _2019 IEEE Wireless Communications and Networking Conference (WCNC)_. IEEE.
* Rombach et al. (2022) Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. 2022\. High-resolution image synthesis with latent diffusion models. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_. 10684–10695.
* Ronneberger et al. (2015) Olaf Ronneberger, Philipp Fischer, and Thomas Brox. 2015\. U-net: Convolutional networks for biomedical image segmentation. In _Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18_. Springer, 234–241.
* Saharia et al. (2022a) Chitwan Saharia, William Chan, Huiwen Chang, Chris Lee, Jonathan Ho, Tim Salimans, David Fleet, and Mohammad Norouzi. 2022a. Palette: Image-to-image diffusion models. In _ACM SIGGRAPH 2022 Conference Proceedings_. 1–10.
* Saharia et al. (2022b) Chitwan Saharia, Jonathan Ho, William Chan, Tim Salimans, David J Fleet, and Mohammad Norouzi. 2022b. Image super-resolution via iterative refinement. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ 45, 4 (2022), 4713–4726.
* Shepard et al. (2016) Clayton Shepard, Jian Ding, Ryan E Guerra, and Lin Zhong. 2016\. Understanding real many-antenna MU-MIMO channels. In _2016 50th Asilomar Conference on Signals, Systems and Computers_. IEEE, 461–467.
* Shivashankar and Miller (2023) C Shivashankar and Shane Miller. 2023. Semantic Data Augmentation With Generative Models. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_. 863–873.
* Sohl-Dickstein et al. (2015) Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. 2015. Deep unsupervised learning using nonequilibrium thermodynamics. In _International conference on machine learning_. PMLR, 2256–2265.
* Sohn et al. (2015a) Kihyuk Sohn, Honglak Lee, and Xinchen Yan. 2015a. Learning structured output representation using deep conditional generative models. _Advances in neural information processing systems_ 28 (2015).
* Sohn et al. (2015b) Kihyuk Sohn, Honglak Lee, and Xinchen Yan. 2015b. Learning Structured Output Representation using Deep Conditional Generative Models. In _Advances in Neural Information Processing Systems_ , C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett (Eds.), Vol. 28. Curran Associates, Inc.
* Song and Ermon (2019) Yang Song and Stefano Ermon. 2019. Generative modeling by estimating gradients of the data distribution. _Advances in neural information processing systems_ 32 (2019).
* Trabelsi et al. (2018) Chiheb Trabelsi, Olexa Bilaniuk, Ying Zhang, Dmitriy Serdyuk, Sandeep Subramanian, Joao Felipe Santos, Soroush Mehri, Negar Rostamzadeh, Yoshua Bengio, and Christopher J Pal. 2018\. Deep Complex Networks. In _International Conference on Learning Representations_.
* Vasisht et al. (2016) Deepak Vasisht, Swarun Kumar, Hariharan Rahul, and Dina Katabi. 2016. Eliminating channel feedback in next-generation cellular networks. In _Proceedings of the 2016 ACM SIGCOMM Conference_. 398–411.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. _Advances in neural information processing systems_ 30 (2017).
* Wang et al. (2004) Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. 2004\. Image quality assessment: from error visibility to structural similarity. _IEEE transactions on image processing_ 13, 4 (2004), 600–612.
* Weisstein (2023) Eric W Weisstein. 2023\. Convolution Theorem. https://mathworld.wolfram.com/ConvolutionTheorem.html
* Xia et al. (2022) Weihao Xia, Yulun Zhang, Yujiu Yang, Jing-Hao Xue, Bolei Zhou, and Ming-Hsuan Yang. 2022\. Gan inversion: A survey. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ (2022).
* Yang et al. (2022b) Ling Yang, Zhilong Zhang, Yang Song, Shenda Hong, Runsheng Xu, Yue Zhao, Yingxia Shao, Wentao Zhang, Bin Cui, and Ming-Hsuan Yang. 2022b. Diffusion models: A comprehensive survey of methods and applications. _arXiv preprint arXiv:2209.00796_ (2022).
* Yang et al. (2023) Zheng Yang, Yi Zhang, Kun Qian, and Chenshu Wu. 2023\. SLNet: A Spectrogram Learning Neural Network for Deep Wireless Sensing. In _20th USENIX Symposium on Networked Systems Design and Implementation (NSDI 23)_. 1221–1236.
* Yang et al. (2022a) Zheng Yang, Yi Zhang, and Qian Zhang. 2022a. Rethinking fall detection with Wi-Fi. _IEEE Transactions on Mobile Computing_ (2022).
* Yao et al. (2019) Shuochao Yao, Ailing Piao, Wenjun Jiang, Yiran Zhao, Huajie Shao, Shengzhong Liu, Dongxin Liu, Jinyang Li, Tianshi Wang, Shaohan Hu, et al. 2019\. Stfnets: Learning sensing signals from the time-frequency perspective with short-time fourier neural networks. In _The World Wide Web Conference_. 2192–2202.
* Zhang et al. (2023) Guidong Zhang, Guoxuan Chi, Yi Zhang, Xuan Ding, and Zheng Yang. 2023. Push the Limit of Millimeter-wave Radar Localization. _ACM Transactions on Sensor Networks_ 19, 3 (2023), 1–21.
* Zhang et al. (2022) Xiaotong Zhang, Zhenjiang Li, and Jin Zhang. 2022. Synthesized Millimeter-Waves for Human Motion Sensing. In _Proceedings of the 20th ACM Conference on Embedded Networked Sensor Systems_. 377–390.
* Zhao et al. (2023) Xiaopeng Zhao, Zhenlin An, Qingrui Pan, and Lei Yang. 2023\. NeRF2: Neural Radio-Frequency Radiance Fields. _arXiv preprint arXiv:2305.06118_ (2023).
* Zheng et al. (2023) Chenyu Zheng, Guoqiang Wu, and Chongxuan Li. 2023. Toward Understanding Generative Data Augmentation. _arXiv preprint arXiv:2305.17476_ (2023).
* Zheng et al. (2019) Yue Zheng, Yi Zhang, Kun Qian, Guidong Zhang, Yunhao Liu, Chenshu Wu, and Zheng Yang. 2019. Zero-effort cross-domain gesture recognition with Wi-Fi. In _Proceedings of the ACM MobiSys_.
* Zhou et al. (2021) Linqi Zhou, Yilun Du, and Jiajun Wu. 2021. 3d shape generation and completion through point-voxel diffusion. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_. 5826–5835.
|
# Collisionless Pattern Discovery in Robot Swarms Using Deep Reinforcement
Learning
Nelson Sharma1, Aswini Ghosh1, Rajiv Misra1, Supratik Mukhopadhyay2, and
Gokarna Sharma3
3Department of Computer Science, Kent State University, Kent, Ohio, USA
E-mail: {nelson_2121cs07, aswini_2121cs02<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>2Department of Environmental Sciences,
Louisiana State University, Baton Rouge, Louisiana, USA 1Department of
Computer Science and Engineering, Indian Institute of Technology, Patna, India
###### Abstract
We present a deep reinforcement learning-based framework for automatically
discovering patterns available in any given initial configuration of fat robot
swarms. In particular, we model the problem of collision-less gathering and
mutual visibility in fat robot swarms and discover patterns for solving them
using our framework. We show that by shaping reward signals based on certain
constraints like mutual visibility and safe proximity, the robots can discover
collision-less trajectories leading to “well-formed” gathering and visibility
patterns.
###### Index Terms:
Swarm robots, Pattern formation, Gathering, Collision avoidance, Deep
reinforcement learning
## I Introduction
In recent years, swarm robotics [1] has gained a lot of attention from the
research community. In a swarm, a number of robots work together collectively
for performing a certain task, such as disaster recovery, foraging, mapping,
etc [2]. In many cases, it is assumed that there is no explicit means of
communication between the robots (e.g., in a bandwidth-limited disaster zone);
a robot can only see others through a camera that provides a $360^{\circ}$
view. The robots are anonymous (no unique identifiers), disoriented (no
agreement on coordinate systems or units of distance), autonomous (no external
control), and indistinguishable (no external identifiers). In the classical
oblivious swarm model [4, 52], the position of every robot is considered as a
point on 2D plane and has a local co-ordinate system. Each robot senses other
robots using sensory equipment installed on it, such as a camera. However, the
robots are considered silent since it is assumed that there is no direct
communication among them [4, 52]. Each robot executes the same algorithm.
Additionally, each robot executes Look-Compute-Move (LCM) cycle – when a robot
is activated, it takes a snapshot of its surroundings from its local
perspective (Look), then it uses the snapshot and the algorithm it is
executing to determine an action to take (Compute), and finally it performs
the action (Move). A swarm of robots can operate asynchronously, semi-
synchronously or fully synchronously. In the fully synchronous operating
model, every robot in the swarm becomes active in every LCM cycle and cycles
of every robot are synchronized (i.e., there is a global clock). In the semi-
synchronous operating model, not all swarm robots necessarily become active in
every LCM cycle, but LCM cycles are synchronized (i.e., there is still a
global clock). In the asynchronous operating model, not every robot is
necessarily active all the time, the duration of each LCM cycle is arbitrary,
and the LCM cycles of different robots are not synchronized (i.e., no global
clock). In the classical model, robots are oblivious because they do not keep
memory of what they did in any past LCM cycle.
There has been a lot of research recently in forming patterns with maximum
visibility and gathering without collisions for swarm robots in the classical
model [7, 8, 9, 10, 11, 12, 13, 14]. Pattern formation concerns arranging $n$
robots in a given pattern on the plane. A configuration of robots matches the
pattern if rotation, translation, and scaling of the configuration can produce
the pattern explicitly [7, 8, 9, 10, 11, 12, 13, 14]. An algorithm solves a
pattern formation problem, if given any configuration of $n$ robots located
initially at distinct points on a plane, it enables them to reach a
configuration that matches the given pattern and remain stationary thereafter
[39]. Specific patterns like circle formation have been considered in [23, 24,
25, 26, 27]. Convex hull and cubic spline formations as solutions to the
mutual visibility problem – i.e., each robot sees all others in the swarm,
have also been studied in the classical model [17, 41]. Pattern formation has
been studied in [42] for UAVs. Pattern Formation via Deep Reinforcement
Learning (DRL) has been studied in [43]. The objective in the gathering
problem is to arrange the robots as close together as possible around a
gathering point (that may or may not be known a priori) [38].
Existing pattern formation algorithms belong to two broad classes:
combinatorial algorithms and machine learning-based approaches. In the
combinatorial paradigm [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37], it is assumed that the target pattern to be formed is given as input to
the robots and one designs a sophisticated algorithm to arrange robots to be
positioned on the target pattern positions, starting from a given initial
configuration. Designing such algorithms is labor-intensive, error-prone, and
time consuming. Additionally, the designed combinatorial algorithm may not
work for any initial configuration, for example, many combinatorial algorithms
on the literature assume that the initial configurations are asymmetric [3, 4,
5, 6]. The same kind of assumption of asymmetric initial configuration extends
also to the combinatorial algorithms for gathering in the literature. Existing
machine learning based approaches [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51] attempt to solve pattern formation for a given target pattern,
i.e., similarly as in the combinatorial algorithms, the target pattern needs
to be given to the machine learning algorithm.
It would be ideal, given any initial configuration (symmetric/asymmetric), if
the valid target patterns could be identified and discovered, even when the
target pattern is not given as input a priori. In this paper, we precisely aim
to do that for the first time in the context of swarm robotics. Particularly,
we seek a _single algorithm_ that can identify and _discover_ patterns that
provide solutions to problems such as mutual visibility or gathering, rather
than giving as input the solution pattern beforehand and the goal of the
algorithm is to reach to the given target from the given initial
configuration.
We present a DRL-based framework for automatically discovering patterns
available in any given initial configuration of robot swarms. We depart from
the literature and consider robot swarms which are not points but fat – and
opaque robots with an extent and hence they occupy certain area. In
particular, we model the problem of collision-less gathering and mutual
visibility in fat opaque robot swarms and discover patterns for solving them
using our framework. Since our proposed approach is DRL-based, it is based on
designing good reward functions. We show that designing reward signals based
on constraints like mutual visibility and safe proximity, the robots can
discover collision-less trajectories leading to “well-formed” gathering and
visibility patterns.
In summary, we make the following two major contributions.
* •
We model the problem of automatic pattern discovery for collision-less
gathering in robot swarms and formulate it in terms of DRL framework.
* •
We propose DRL based approach for the collision-less gathering and pattern
discovery using multiple policies to discover “well-formed” gathering
patterns.
## II RELATED WORK
Specific target pattern formation for swarm robots have been extensively
studied in [40, 41, 27, 28, 29, 30, 31]. Optimal convex hull pattern formation
assuming obstructed visibility has been studied in [40]. In [41], the authors
studied proposed self organised cubic spline based pattern formation method.
Several researchers have studied circular pattern formation [27] problem. In
[29], authors have investigated the plane formation problem that requires a
swarm of robots moving in the three-dimensional Euclidean space to land on a
common plane and considered the fully synchronous setting and where robots are
endowed with only visual perception.Most works that studied pattern formation
problem considered the classic oblivious point swarm model [30, 39]. In [39],
the authors presented a combinatorial framework for the pattern formation
problem and used a randomized procedure for leader election in their proposed
algorithm. They measured the time complexity of their algorithms in “epochs”
(i.e., time interval in which every robot performs at least one look-compute-
move step).
For the fundamental problem of gathering, the authors in [16] considered
gathering of $N$ autonomous robots on a plane, which requires all robots to
meet at a single point under limited visibility. Here the point of gathering
is not known beforehand. The authors in [38] significantly improved the time
complexity of gathering under limited visibility model with the assumption
that the point of gathering is known beforehand.
The authors in [37] considered the state machine representation to formulate
the gathering problem and developed a distributed algorithm that solves the
problem of gathering for any number of fat robots. In [43], the authors
proposed a Deep Reinforcement Learning (DRL)-based method that generates
general-purpose pattern formation strategies for any target pattern. A bio-
inspired algorithm was proposed in [47] to control robot swarms for pattern
formation. The authors in [49] have formulated reward functions to ensure
drones arrive at their destinations following predefined routes avoiding
collisions. The authors in [50] presented the idea of guided policy learning
and investigated how to learn to control a group of cooperative agents with
limited sensing capabilities such as robot swarms. The authors in [51]
presented a framework for collision-less flying of drones over a directed
circulant digraph.
However, none of the above discussed approaches have attempted discovering
collision-less patterns where mutual visibility is maximized with or without
predefined gathering points. In this paper, we consider a DRL-based framework
for discovering “well-formed” patterns for mutual visibility and gathering
where gathering point may not be known beforehand.
## III System Model and Preliminaries
Robot Swarm. A robot swarm is a collection on $n$ robots moving autonomously
inside a rectangular region of size $(2X_{w},2Y_{w})$ on a 2D Cartesian plane
with origin being at the center of the rectangle. The robots are _fat_ and
_opaque_ , meaning that a robot $(n\in N)$ is represented as a circular opaque
disk of radius $R_{bot}$. As shown in Fig. 1 (left), a robot has a center and
a circular body of fixed size. A robot’s position vector $(P_{n})$ and
velocity vector $(V_{n})$ are assumed to be known at all times, recorded using
sensors internal to robots. It is assumed that all robots are fitted with
sensors that can detect the positions of neighboring robots within a fixed
distance from their centers, called the _scan-radius_ ($R_{scan}$). The
sensors’ readings for a robot provide: (i) the number of neighbors
($G^{all}_{n}$), (ii) the number of fully visible neighbors ($G^{vis}_{n}$),
and (iii) the number of occluded neighbors ($G^{occ}_{n}$). Additionally, a
_safe distance_ ($\delta_{s}$) is defined, which can be used to calculate a
measure of how unsafe is the current positioning of the robots.
Figure 1: (left) Fat robot model (right) Visibility model
Since robots are fat and opaque, a robot $B_{i}$ is said to be occluded by
robot $B_{j}$ from the view of robot $B_{k}$ if any part of $B_{i}$ lies in
the shadow formed by $B_{j}$ assuming that a point source of light placed at
the center of $B_{k}$, as shown in Fig. 1 (right). Two robots are _mutually
visible_ to each other if and only if none of them are occluded by any robot
from the other’s view. The robots are allowed to move in any direction by
changing their velocity vector $(V_{n}=[V_{n}^{x},V_{n}^{y}])$ within a fixed
range ($V^{min}\leq V_{n}^{x},V_{n}^{y}\leq V^{max}$ ) at each time cycle.
Problem Formulation. The goal of robot swarm is to discover appropriate
patterns for gathering such that the robots do not collide with each other and
maintain maximum visibility among one another during pattern formation. We
consider two gathering problem variations as follows:
Gathering at a Predefined Point. In this case, the robots are initially
scattered far apart, i.e., mostly outside the scan-radius of each other. It is
assumed that the gathering point is at the origin of the environment.
Gathering without a Predefined Point. We assume a very large scan-radius and
all robots are within the scan-radius of each other. Gathering point is not
known a priori and the robot swarm has to decide the appropriate gathering
place.
Markov Decision Process. We formulate the above variations of the gathering
problem as an appropriate _Markov Decision Process (MDP)_ and solve it using a
DRL-based approach that tries to maximize its underlying reward function. The
MDP is formulated as follows:
* •
State Space ($\mathcal{S}$): The State Space contains the position coordinates
of all the robots.
$\mathcal{S}=\\{P_{n}\\}:\forall n\in\\{1...N\\}$ (1)
* •
Action Space ($\mathcal{A}$): The Action Space contains the velocity
components that can be set at a given time-step, for all the robots.
$\mathcal{A}=\\{[V_{n}^{x},V_{n}^{y}]\\}:\forall n\in\\{1...N\\}$ (2)
* •
Reward Function ($\mathcal{R}$): The reward for each time-step is based on the
_Reward Signal_ ($\mathcal{C}$) from the current state ($S_{t}$) and the next
state ($S_{t+1}$).
$\mathcal{R}(S_{t+1}|S_{t},A_{t})=\mathcal{C}(S_{t+1})-\mathcal{C}(S_{t})$ (3)
where, the Reward Signal for a given state is a weighted sum of multiple
signals. We define two reward signals, $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$
as follows:
$\mathcal{C}_{1}(S_{t})=\sum_{C\in C_{1}}C(S_{t})$ (4)
$\mathcal{C}_{2}(S_{t})=\sum_{C\in C_{2}}C(S_{t})$ (5)
where, $C_{1}$ and $C_{2}$ are sets of signals defined as:
$C_{1}=\\{C_{close},C_{safety},C_{neighbors},C_{visible}\\}$ (6)
$C_{2}=\\{C_{nclose},C_{safety},C_{neighbors},C_{visible}\\}$ (7)
where, the members are defined as follows:
* –
$C_{close}$ : measure of closeness of robots from the origin. This signal
causes robots to move closer to the gathering point, i.e., the origin.
$C_{close}(S_{t})=1-W_{close}\sum_{n\in N}||P_{n}(t)||$ (8)
* –
$C_{safety}$ : is the sum of safety measure ($F_{safe}$) for all robots, which
helps robots to avoid collisions maintaining appropriate distance between each
other.
$C_{safety}(S_{t})=W_{safety}\sum_{i,j\in N,i\neq j}F_{safe}(i,j,t)$ (9)
where, $F_{safe}$ is defined between robots $i$ and $j$ at time-step $t$,
based on a known constant _safe-distance_ ($\delta_{s}$), as follows:
$F_{d}=||P_{i}(t)-P_{j}(t)||-\delta_{s}$
$F_{safe}(i,j,t)=\begin{cases}\sqrt{\delta_{s}+R_{bot}-F_{d}},&F_{d}\geq 0,\\\
\text{0},&F_{d}<0,\\\ \end{cases}$
* –
$C_{neighbors}$ : denotes the number of neighbors and encourages the robots to
have more robots in their neighborhood.
$C_{neighbors}(S_{t})=W_{neighbors}\sum_{n\in N}G_{n}^{all}(t)$ (10)
* –
$C_{visible}$ : denotes total number of neighbors that are currently fully
visible for each robot and encourages maximum visibility within the robot
swarm.
$C_{visible}(S_{t})=W_{visible}\sum_{n\in N}G_{n}^{vis}(t)$ (11)
* –
$C_{nclose}$ : measures the average distances from all neighbors for each
robot. In the absence of a priori gathering point, this signal encourages
robots to move closer to each other for gathering.
$C_{nclose}(S_{t})=1-W_{nclose}\sum_{i,j\in N,i\neq j}||P_{i}(t)-P_{j}(t)||$
(12)
The weights assigned to each signal ($W_{i}$) are chosen so as to normalize
the values to the range $[0,1]$.
* •
Initial States ($\mathcal{S}_{0}$): We consider three types of initial state
configurations:
* –
_Scattered_ : robots are uniformly scattered along a roughly circular shape
with radius $\min(X_{w},Y_{w})$.
* –
_Distributed_ : robots are equally scattered near any two of the corners of
the environment.
* –
_Packed_ : robots are closely placed in a line (or multiple lines).
An initial state configuration $(\mathcal{S}_{0},\epsilon)$ is chosen at the
start of each episode where $\mathcal{S}_{0}$ is a set of $n$ coordinates, one
for each robot, and $\epsilon$ is the noise radius. A robot $(B_{n})$ is
placed at a location ($P_{n}(0)$) given by:
$P_{n}(0)=P_{n}^{0}+\mathcal{U}(-\epsilon,\epsilon)$ (13)
where $P_{n}^{0}$ is an predefined point generated based on the initial state
$(\mathcal{S}_{0})$. Initial position of a robot is determined by adding
uniformly sampled noise ($\epsilon$) to each dimension of $P_{n}^{0}$.
For gathering at a predefined point, we use the reward signal
$\mathcal{C}_{1}$, which tries to maximize the $C_{close}$ signal causing
robots to move closer to a pre-defined gathering point. In contrast, for
gathering without a predefined point, we shall use the reward signal
$\mathcal{C}_{2}$ in our MDP which tries to maximize the $C_{nclose}$ signal
causing robots to move closer to each other. For both the problems, the common
reward signal is $\mathcal{C}_{safe}$ which encourages collision-less movement
at a safe distance; and $\mathcal{C}_{visible}$ which encourages maximum
mutual visibility.
Proximal Policy Optimization (PPO). PPO [53] is a policy optimization
algorithm where the parameterized policy is updated using one of two objective
functions. The first is _clipped objective_ function ($L_{C}$) defined as
follows:
$L_{C}=\mathbb{E}_{t}\left[\min(r_{t}\hat{A}_{t},clip(r_{t},1-\epsilon_{c},1+\epsilon_{c})\hat{A}_{t})\right]$
(14)
where, $\hat{A}_{t}$ represents the advantage estimate, $\epsilon_{c}$
represents the _clipping hyper-parameter_ and $r_{t}$ is the ratio of
probabilities given by the current policy ($\pi_{i}$) and an older version of
it ($\pi_{i-1}$), defined as:
$r_{t}=\frac{\pi_{i}(A_{t}|S_{t})}{\pi_{i-1}(A_{t}|S_{t})}$ (15)
The clipped objective can be maximized using a mini-batch ($\mathcal{D}$) of
trajectories, in which case the objective becomes:
$L_{C}=\frac{1}{|\mathcal{D}|T}\sum_{\mathcal{D}}\sum_{t=0}^{T}\min(r_{t}\hat{A}_{t},clip(r_{t},1-\epsilon_{c},1+\epsilon_{c})\hat{A}_{t})$
(16)
where, $T$ represents the trajectory length. The advantage is estimated using
a truncated version of Generalized Advantage Estimation (GAE)[54] defined as:
$\delta_{t}=\mathcal{R}_{t}+\gamma V(S_{t+1})-V(S_{t})$ (17)
$\hat{A}_{i}(\gamma,\lambda)=\sum_{l=0}^{T-t-1}(\gamma\lambda)^{l}\delta_{l+t}$
(18)
where, $\lambda$ is a hyper-parameter, $\gamma$ is the discount factor, and
$V$ represents the value function.
Notice that PPO is an on-policy algorithm, i.e., it uses the latest policy to
sample actions during the learning process. The exploration is dependent on
the randomness in the stochastic policy. As the policy is tuned over time, the
randomness in sampled action decreases which often causes the agent to explore
less, leading to the policy settling in a local optima. We shall take
advantage of this fact and use multiple policies to move around local optima
until we find a satisfactory one. This can be achieved, in one way, by
altering the value of discount factor during multiple runs of the learning
algorithm.
## IV Algorithm
To discover a gathering pattern, we use Algorithm 2 to generate two policies
sequentially using an augmented version of the Clipped Proximal Policy
Optimization (Clipped-PPO) algorithm developed in [53]. The first policy
($\Pi_{0}$) is called the _Base_ policy and the second one ($\Pi_{1}$) is
called the _Auxiliary_ policy. The base policy finds near-optimal patterns in
most cases but may sometimes get struck on sub-optimal patterns (i.e., local
optima). The Auxiliary policy is used to break out of local optima and improve
the reward further. The base policy uses a larger value of the discount factor
($\gamma$) and trains on longer horizons. On the other hand, the auxiliary
policy uses a smaller value of discount factor and trains on short horizons
using only the sub-optimal patterns formed by the base policy as the initial
states. The pattern can be sub-optimal, in the sense that the robots do
satisfy the gathering criteria only to a certain degree. In other words, it
might be possible that the reward can be further maximized to obtain a more
optimal pattern. In such a case, the auxiliary policy is trained starting at
various sub-optimal patterns formed by the base policy and is expected to
improve the pattern further using a few more steps. The algorithm additionally
uses hyper-parameters which are described in Table I.
Table I: Hyper-parameters used in proposed algorithms Hyper-Parameter | Description
---|---
$\gamma$ | Discount Factor
$\lambda$ | GAE Lambda
$\epsilon_{c}$ | Clipping Range for Objective
$\alpha$ | Learning Rate for policy updates
$\beta$ | Learning Rate for value updates
$Z$ | No. of Learning Epochs
Algorithm 1 Gathering using Clipped PPO
0: Set of Initial States $({\sigma})$
0: Hyper-Parameters $(\gamma,\lambda,\epsilon_{c},\alpha,\beta,Z)$
1: Initialize Policy-Network $(\pi_{\theta})$ and Value-Network $(V_{\phi})$
with parameters $\theta_{0}$ and $\phi_{0}$ respectively
2: Initialize Learning Environment with a uniform Initial State Distribution
on set $\sigma$.
3: Initialize an empty set $\sigma^{*}$
4: for $i=1$ to $Z$ do
5: Collect batch of trajectories
$\mathcal{D}_{i}=\\{S_{t},A_{t},R_{t},S_{t+1}\\}$ from the environment using
current policy $\pi_{\theta_{i}}$ and compute _Rewards-to-go_ ($\hat{R}_{t}$)
at each time-step $t$
$\hat{R}_{t}=\sum_{l=t}^{T}\mathcal{R}(S_{l+1}|(S_{l},A_{l}))$
6: Compute Clipped Objective ($L_{C}$) for batch $\mathcal{D}_{i}$ using
equation (16)
7: Update Policy Network using:
$\theta_{i+1}=\theta_{i}+\alpha\cdot\nabla_{\theta}L_{C}$
8: Update Value Network using:
$E(\phi_{i})=\frac{1}{|\mathcal{D}_{i}|T}\sum_{\mathcal{D}_{i}}\sum_{t=0}^{T}(V_{\phi_{i}}(S_{t})-\hat{R}_{t})^{2}$
$\phi_{i+1}=\phi_{i}-\beta\cdot\nabla_{\phi}E(\phi_{i})$
9: end for
10: Evaluate Policy $\pi_{\theta}$ in the Environment using initial states
from $\sigma$. For each $S_{i}\in\sigma$, add the final state $S^{*}_{i}$ to
set $\sigma^{*}$ only if $\mathcal{S}^{*}_{i}$ is not a terminal state.
11: Output policy $\pi_{\theta}$ and set of states $\sigma^{*}$
The clipping parameter ($\epsilon_{c}$) is gradually increased with each epoch
of Algorithm 1. The Discount Factor Schedule ($F_{\gamma}$) is a function
which provides appropriate value of discount factor on each successive run of
Algorithm 2. We train only two successive policies where the value of Discount
Factor is high for the first run and reduced during the second run. The
auxiliary policy is responsible for improving patterns discovered by the base
policy and can be trained on customized reward signals that give rise to
various patterns while satisfying given gathering criteria.
Algorithm 2 Pattern Discovery for robot swarm
0: Initial State Configuration $(\mathcal{S}_{0},\epsilon)$
0: Hyper-Parameters $(\lambda,\epsilon_{c},\alpha,\beta,Z)$
0: Discount Factor Schedule $(F_{\gamma})$
1: Initialize set of initial states
$\sigma\leftarrow(\mathcal{S}_{0},\epsilon)$
2: Initialize $t\leftarrow 0$
3: Initialize empty list of policies ($\Pi$)
4: while $|\sigma|>0$ do
5: Run Algorithm[1] with Set of Initial States ($\sigma$) and hyper-parameters
$(F_{\gamma}(t),\lambda,\epsilon_{c},\alpha,\beta,Z)$ to obtain $\pi^{t}$ and
$\sigma^{t}$
6: Append $\pi^{t}$ to list of policies $\Pi$
7: Update set of initial states $\sigma\leftarrow\sigma^{t}$
8: $t\leftarrow t+1$
9: Evaluate policy list ($\Pi$) with given Initial State Configuration
$(\mathcal{S}_{0},\epsilon)$ to obtain desired trajectory set $\mathcal{T}$
10: If $\mathcal{T}$ forms valid collision-less patterns then Break
11: end while
12: if $|\sigma|=0$ then
13: No valid pattern formation trajectory was discovered
14: else
15: Output Policy List $\Pi$
16: end if
## V Experiments and Results
To test the pattern discovery algorithm (Algorithm 2), we conducted a total of
six experiments under three different swarm compositions with two different
gathering criteria (reward signals) starting in one of the three initial state
configurations (Scattered, Distributed, Packed). The experimental settings are
described in Table II.
Table II: Experiment Settings Experiment | Swarm Size | Gathering Target | Initial States
---|---|---|---
A | $6$ | Origin | Packed
B | $8$ | Origin | Scattered
C | $10$ | Origin | Distributed
D | 6 | Undefined | Packed
E | 8 | Undefined | Scattered
F | 10 | Undefined | Distributed
The training results for swarm size of 10 and 8 (in experiments C and E) are
shown in Figs. 2 and 3, respectively. Note that, in both cases, the auxiliary
policy improves the patterns formed by the base policy which is evident from
the improvement seen in the reward signal during auxiliary training.
Figure 2: [Exp C] Training result for Base Policy and Auxiliary Policy for
swarm of size 10 with pre-defined gathering point Figure 3: [Exp E] Training
result for Base Policy and Auxiliary Policy for swarm of size 8 with undefined
gathering point
Fig. 4 shows the reward obtained by robot during a single episode of pattern
formation. When the _step-reward_ received by the agent becomes zero, it
indicates that the base policy cannot improve the pattern any further, and
hence, has converged to a stable pattern. It can be seen that the base policy
leads to a sub-optimal pattern which is then improved by the auxiliary policy.
Figure 4: [Exp C] Cumulative Reward obtained during single episode: (a) Reward
Obtained by Base Policy, (b) Reward Obtained by Auxiliary Policy.
The patterns discovered by the trained policies are shown in Figs. 5–8, which
show the (a) initial states, (b) the final pattern formed by base policy, and
(c) the improved pattern formed by auxiliary policy. It should be noted that
the patterns discovered are quite different in both the cases and the proposed
algorithm is able to gather robots even when gathering without a predefined
point. Similar results can be seen for swarm size of 8 (in experiments B and
E) in Figs. 6 and 8 which also show that the auxiliary policy is able to
improve the base pattern even though the improved pattern is only slightly
different in case of experiment E as seen in Fig. 8.
Figure 5: [Exp C] Pattern formation for swarm of size 10 in distributed initial state with pre-defined gathering point Figure 6: [Exp F] Pattern formation for swarm of size 10 in distributed initial state with undefined gathering point Figure 7: [Exp B] Pattern formation for swarm of size 8 in scattered initial state with pre-defined gathering point Figure 8: [Exp E] Pattern formation for swarm of size 8 in scattered initial state with undefined gathering point Figure 9: Swarm size vs average pattern formation steps for distributed and scattered initial state configurations Table III: Average Steps taken for pattern formation Experiment | Swarm Size | Base Steps | Auxiliary Steps | Total Steps
---|---|---|---|---
A | 6 | 113.63 | 4.01 | 117.64
B | 8 | 728.37 | 18.22 | 746.59
C | 10 | 1730.20 | 162.38 | 1892.58
D | 6 | 269.88 | 2.89 | 272.77
E | 8 | 521.71 | 223.56 | 1168.83
F | 10 | 1550.26 | 486.97 | 2037.23
Table III shows the average number of time-steps taken for robots to form a
collision-less gathering pattern under different experimental settings and
standard initial state configurations with uniform noise
$\sim\mathcal{U}(-2R_{bot},2R_{bot})$. A total of $200$ random initial states
were generated to test pattern formation steps under the corresponding initial
state configuration. To study the relationship between swarm size and number
of steps needed for pattern formation, we conducted similar experiments with
robot swarms of sizes ranging from $4$ to $10$ under distributed and scattered
initial state configurations; see Fig. 9. Note that the number of steps varies
almost linearly with the swarm size.
## VI Conclusions and Future Work
In this paper, we proposed a DRL-based framework for automatic pattern
discovery and gathering for swarms of oblivious fat opaque robots. Using
state-of-the-art DRL techniques for policy optimization and proper reward
system design, the agents trained using the proposed methods were shown to be
able to discover collision-less trajectories for gathering in desirable
patterns. With the use of multiple policies at different stages of gathering,
it was possible for the robots to improve their patterns successively instead
of being struck at local optima.
In the future, we shall work upon improving the robustness and scale of our
models by making them fully distributed and autonomous while also being able
to handle environmental uncertainty like unseen obstacles, actuator failures,
and recovery mechanisms. We look forward to explore trajectory planing and
environmental mapping as well.
## References
* [1] M. Dorigo, M. Birattari, and M. Brambilla, “Swarm robotics,” _Scholarpedia_ , vol. 9, no. 1, p. 1463, 2014.
* [2] C. Blum and D. Merkle, _Swarm intelligence: introduction and applications_. Springer Science & Business Media, 2008.
* [3] R. Vaidyanathan, G. Sharma, J.L. Trahan, On fast pattern formation by autonomous robots, in: SSS, 2018, pp.203–220.
* [4] P. Flocchini, G. Prencipe, N. Santoro, Distributed Computing by Oblivious Mobile Robots, Synthesis Lectures on Distributed Computing Theory, Morgan and Claypool Publishers, 2012.
* [5] S. Das, P. Flocchini, G. Prencipe, N. Santoro, M. Yamashita, Autonomous mobile robots with lights, Theor. Comput. Sci. 609(P1) (2016) 171–184.
* [6] D. Peleg, Distributed coordination algorithms for mobile robot swarms: new directions and challenges, in: IWDC, 2005, pp.1–12.
* [7] Q. Bramas, S. Tixeuil, Probabilistic asynchronous arbitrary pattern formation (short paper), in: SSS, 2016, pp.88–93.
* [8] Y. Dieudonné, F. Petit, V. Villain, Leader election problem versus pattern formation problem, in: DISC, 2010, pp.267–281.
* [9] P. Flocchini, G. Prencipe, N. Santoro, P. Widmayer, Arbitrary pattern formation by asynchronous, anonymous, oblivious robots, Theor. Comput. Sci. 407(1–3) (2008) 412–447.
* [10] N. Fujinaga, Y. Yamauchi, H. Ono, S. Kijima, M. Yamashita, Pattern formation by oblivious asynchronous mobile robots, SIAM J. Comput. 44(3) (2015) 740–785.
* [11] I. Suzuki, M. Yamashita, Distributed anonymous mobile robots: formation of geometric patterns, SIAM J. Comput. 28(4) (1999) 1347–1363.
* [12] M. Yamashita, I. Suzuki, Characterizing geometric patterns formable by oblivious anonymous mobile robots, Theor. Comput. Sci. 411(26–28) (2010) 2433–2453.
* [13] Y. Yamauchi, M. Yamashita, Pattern formation by mobile robots with limited visibility, in: SIROCCO, 2013, pp.201–212.
* [14] Y. Yamauchi, M. Yamashita, Randomized pattern formation algorithm for asynchronous oblivious mobile robots, in: DISC, 2014, pp.137–151.
* [15] R. Vaidyanathan, C. Busch, J.L. Trahan, G. Sharma, S. Rai, Logarithmic-time complete visibility for robots with lights, in: IPDPS, 2015, pp.375–384.
* [16] P. Poudel, G. Sharma, Universally optimal gathering under limited visibility, in: SSS, 2017, pp.323–340.
* [17] G. Sharma, C. Busch, S. Mukhopadhyay, Brief announcement: complete visibility for oblivious robots in linear time, in: SPAA, 2017, pp.325–327.
* [18] G. Sharma, R. Vaidyanathan, J.L. Trahan, Constant-time complete visibility for asynchronous robots with lights, in: SSS, 2017, pp.265–281.
* [19] G. Sharma, R. Vaidyanathan, J.L. Trahan, C. Busch, S. Rai, Logarithmic-time complete visibility for asynchronous robots with lights, in: IPDPS, 2017, pp.513–522.
* [20] S. Cicerone, G.D. Stefano, A. Navarra, Asynchronous arbitrary pattern formation: the effects of a rigorous approach, Distrib. Comput. 32(2) (2019) 91–132.
* [21] S. Cicerone, G.D. Stefano, A. Navarra, Asynchronous embedded pattern formation without orientation, in: DISC, 2016, pp.85–98.
* [22] S. Das, P. Flocchini, N. Santoro, M. Yamashita, Forming sequences of geometric patterns with oblivious mobile robots, Distrib. Comput. 28(2) (2015) 131–145.
* [23] X. Défago, A. Konagaya, Circle formation for oblivious anonymous mobile robots with no common sense of orientation, in: FOMC, 2002, pp.97–104.
* [24] X. Défago, S. Souissi, Non-uniform circle formation algorithm for oblivious mobile robots with convergence toward uniformity, Theor. Comput. Sci. 396(1–3) (2008) 97–112.
* [25] Y. Dieudonné, O. Labbani-Igbida, F. Petit, Circle formation of weak mobile robots, in: SSS, 2006, pp.262–275.
* [26] A. Dutta, S.G. Chaudhuri, S. Datta, K. Mukhopadhyaya, Circle formation by asynchronous fat robots with limited visibility, in: ICDCIT, 2012, pp.83–93.
* [27] B. Katreniak, Biangular circle formation by asynchronous mobile robots, in: SIROCCO, 2005, pp.185–199.
* [28] P. Flocchini, G. Prencipe, N. Santoro, G. Viglietta, Distributed computing by mobile robots: uniform circle formation, Distrib. Comput. 30(6) (2017) 413–457.
* [29] Y. Yamauchi, T. Uehara, S. Kijima, M. Yamashita, Plane formation by synchronous mobile robots in the three-dimensional Euclidean space, J. ACM 64(3) (2017) 16.
* [30] S. Das, P. Flocchini, G. Prencipe, N. Santoro, Synchronized dancing of oblivious chameleons, in: FUN, 2014, pp.113–124.
* [31] Z. Derakhshandeh, R. Gmyr, A.W. Richa, C. Scheideler, T. Strothmann, Universal shape formation for programmable matter, in: SPAA, 2016, pp.289–299.
* [32] [30]G.A. Di Luna, P. Flocchini, N. Santoro, G. Viglietta, Y. Yamauchi, Brief announcement: shape formation by programmable particles, in: DISC, 2017, pp.48:1–48:3.
* [33] A. Cord-Landwehr, B. Degener, M. Fischer, M. Hüllmann, B. Kempkes, A. Klaas, P. Kling, S. Kurras, M. Märtens, F. Meyer auf der Heide, C. Raupach, K. Swierkot, D. Warner, C. Weddemann, D. Wonisch, A new approach for analyzing convergence algorithms for mobile robots, in: ICALP, 2011, pp.650–661.
* [34] J.D. Lawrence, A Catalog of Special Plane Curves, Dover Books on Mathematics, Dover, New York, NY, 2014.
* [35] H. Attiya, J. Welch, Distributed Computing: Fundamentals, Simulations and Advanced Topics, John Wiley & Sons, 2004.
* [36] H.M. El-Boghdadi, R. Vaidyanathan, J.L. Trahan, S. Rai, On the communication capability of the self-reconfigurable gate array architecture, in: IPDPS, 2002, pp.1–8.
* [37] C. Agathangelou, C. Georgiou, M. Mavronicolas, A distributed algorithm for gathering many fat mobile robots in the plane, in: PODC, 2013, pp.250–259.
* [38] GOKARNA SHARMA,COSTAS BUSCH, SUPRATIK MUKHOPADHYAY, and CHARLES MALVEAUX, Tight Analysis of a Collisionless Robot Gathering Algorithm ACM Transactions on Autonomous and Adaptive SystemsVolume 12 Issue 1 March 2017 Article No.: 3pp 1–20https://doi.org/10.1145/3056460
* [39] Ramachandran ,Vaidyanathana,GokarnaSharma ,bJerryTrahana, On fast pattern formation by autonomous robots,https://doi.org/10.1016/j.ic.2021.104699
* [40] Rory Hector; Ramachandran Vaidyanathan; Gokarna Sharma; Jerry L. Trahan ,Optimal Convex Hull Formation on a Grid by Asynchronous Robots With Lights,IEEE Transactions on Parallel and Distributed Systems ( Volume: 33, Issue: 12, 01 December 2022)
* [41] Belkacem Khaldi , Fouzi Harrou , Foudil Cherif , and Ying Sun,Toward Emerging Cubic-Spline Patterns With a Mobile Robotics Swarm System, IEEE TRANSACTIONS ON COGNITIVE AND DEVELOPMENTAL SYSTEMS, VOL. 14, NO. 2, JUNE 2022
* [42] Gunasekaran Raja, Saran V S, Sudha Anbalagan, Ali Kashif Bashir,Collisionless Fast Pattern Formation Mechanism for Dynamic Number of UAVs ,GLOBECOM 2020 - 2020 IEEE Global Communications Conference.
* [43] Jia Wang; Jiannong Cao; Milos Stojmenovic; Miao Zhao; Jinlin Chen; Shan Jiang,Pattern-RL: Multi-robot Cooperative Pattern Formation via Deep Reinforcement Learning.
* [44] Abhik Singla, Sindhu Padakandla , and Shalabh Bhatnagar, Memory-Based Deep Reinforcement Learning for Obstacle Avoidance in UAV With Limited Environment Knowledge, IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 22, NO. 1, JANUARY 2021.
* [45] Linbo Luo , Xinyu Wang, Jianfeng Ma , Member, IEEE, and Yew-Soon Ong , GrpAvoid: Multigroup Collision-Avoidance Control and Optimization for UAV Swarm, IEEE TRANSACTIONS ON CYBERNETICS
* [46] Haitao Zhao ,Hai Liu, Yiu-Wing Leung,and Xiaowen Chu, Self-Adaptive Collective Motion of Swarm Robots
* [47] Seongin Na , Hanlin Niu , Barry Lennox ,and Farshad Arvin ,Bio-Inspired Collision Avoidance in Swarm Systems via Deep Reinforcement Learning IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 71, NO. 3, MARCH 2022.
* [48] E. Sahin; T.H. Labella; V. Trianni; J.-L. Deneubourg; P. Rasse; D. Floreano; L. Gambardella; F. Mondada; S. Nolfi; M. Dorigo SWARM-BOT: pattern formation in a swarm of self-assembling mobile robots ,IEEE International Conference on Systems, Man and Cybernetics (SMC)
* [49] Sirui Song, Yuanhang Zhang, Xi Qin, Kirk Saunders, Jundong Liu , Vision-guided Collision Avoidance Through Deep Reinforcement Learning, NAECON 2021 - IEEE National Aerospace and Electronics Conference
* [50] Maximilian Huttenrauch, Adrian sosic,and Gerhard Neumann, Guided Deep Reinforcement Learning for Swarm Systems, https://doi.org/10.48550/arXiv.1709.06011
* [51] ZBIGNIEW R. BOGDANOWICZ , Flying Swarm of Drones Over Circulant Digraph,Armament Research, Development and Engineering Center, Picatinny Arsenal, NJ USA
* [52] Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, Lecture Notes in Computer Science 11340, Springer, 2019
* [53] John Schumann, Fillip Wolski, Prafulla Dhariwal, Alec Radford, Oleg Klimov ,Proximal Policy Optimization Algorithms,arXiv:1707.06347v2 [cs.LG] 28 Aug 2017
* [54] John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, Pieter Abbeel,High-Dimensional Continuous Control Using Generalized Advantage Estimation,arXiv:1506.02438v6 [cs.LG] 20 Oct 2018
|
# The multiplexed light storage of Orbital Angular Momentum based on atomic
ensembles
Xin Yang Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an, 710049, China Hong Chang Key Laboratory of Time
and Frequency Primary Standards, National Time Service Center, Chinese Academy
of Sciences, Xi’an 710600, China Jinwen Wang Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an,
710049, China Yan Ma Key Laboratory of Time and Frequency Primary Standards,
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China
Yun Chen Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an, 710049, China Shuwei Qiu Ministry of Education
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed
Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an,
710049, China Zehao Shen Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an, 710049, China Chengyuan
Wang Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an, 710049, China Quan Quan Ministry of Education
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed
Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an,
710049, China Dong Wei Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an, 710049, China Haixia
Chen Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an, 710049, China Mingtao Cao Key Laboratory of Time
and Frequency Primary Standards, National Time Service Center, Chinese Academy
of Sciences, Xi’an 710600, China Hong Gao Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an,
710049, China Corresponding<EMAIL_ADDRESS>Fuli Li Ministry of
Education Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an, 710049, China
###### Abstract
The improvement of the multi-mode capability of quantum memory can further
improve the utilization efficiency of the quantum memory and reduce the
requirement of quantum communication for storage units. In this letter, we
experimentally investigate the multi-mode light multiplexing storage of
orbital angular momentum (OAM) mode based on rubidium vapor, and
demultiplexing by a photonic OAM mode splitter which combines a Sagnac loop
with two dove prisms. Our results show a mode extinction ratio higher than
80$\%$ at 1 $\mu$s of storage time. Meanwhile, two OAM modes have been
multiplexing stored and demultiplexed in our experimental configuration. We
believe the experimental scheme may provide a possibility for high channel
capacity and multi-mode quantum multiplexed quantum storage based on atomic
ensembles.
††journal: journal
Qauntum memories, enabling the storage of an input photonic qubit and
retrieval on controllable time, which can beat any classical device and
constitute essential components in quantum repeaters and optical quantum
information processing[1, 2]. Over the past years, various memory protocols
have been proposed, such as electromagnetically induced transparency (EIT)[3],
gradient echo storage[4], optical frequency comb [5], Raman scheme [6] and the
Duan–Lukin–Cirac–Zoller protocol [7]. The EIT storage protocol is widely used
among these storage protocols because of its simple configuration and easy
implementation.
Improving the multi-mode storage capability in the spatial domain can
effectively improve the working efficiency of quantum repeaters and reduce the
requirement for quantum communication on the storage unit. Orbital angular
momentum (OAM) beam is an attractive light field mode, and its topological
charge can be taken as an arbitrary integer[8]. In principle, an infinite-
dimensional Hilbert space can be constructed to realize higher dimensional
spatial mode encoding of photon OAM. Therefore, using the OAM mode for storage
is an effective way to realize efficient quantum repeaters. In recent years,
research on the storage of photon OAM light modes has been carried out [9, 10,
11]. These studies mainly focus on storing low-order vector beams and finite-
dimensional photon OAM quantum states [12, 13, 11], but the high-dimensional
properties of OAM have not been fully utilized in any quantum memory
experiment and multi-OAM mode storage simultaneously based on atomic ensembles
has not been studied.
It is well known that effectively identifying the OAM mode is significant in
multi-mode storage. To date, there are many traditional measurement methods
for the photon OAM quantum state, including the round-hole diffraction
measurement [14], optical transformation method [15] and interference method
[16, 17]. However, the schemes of state identification of OAM quantum states
after storage are all based on the projection measurement scheme[18]; The
projection measurement scheme can only be used for the identification of the
OAM quantum states but cannot achieve the separation of the OAM quantum
states.An applicable quantum memory with multimode capacity should be achieved
multiplexed storage and demultiplexed retrieval simultaneously. Therefore, if
photons encoded with high-dimensional OAM in a quantum memory, not only the
multimode storage shall be realized but also the stored OAM quantum states
need to be efficiently separated after storage.
Figure 1: (a)The experimental setup of Multiplexing quantum memory. PBS,
polarization beam splitter; BS, beam splitter; HWP, half-wave plate; QWP,
quarter-wave plate; SLM, spatial light modulator. (b) The setup of
demultiplexing; DP, Dove prism; ICCD, intensified charge coupled device (c)
Schematic energy diagram of 87Rb and light coupling.
In this letter, we experimentally investigate multi-mode quantum multiplexing
storage and demultiplexing based on rubidium vapor. Specifically, OAM beams
with different topological charges have been stored in atomic ensembles
through the use of electromagnetic induced transparency storage protocol and
demultiplexed into different output ports by using the photon OAM mode
splitter which combines a Sagnac loop with two dove prisms. Our results show a
mode extinction ratio higher than 80 $\%$ at 1 $\mu$s of storage time.
Meanwhile, two OAM modes have been multiplexed storage and demultiplexed
output in different ports by using an OAM mode splitter. The results show that
our experimental system can absolutely separate these two OAM modes after
storage. We believe that the experimental scheme has applications in high
channel capacity multi-mode quantum multiplexing storage and demultiplexing in
atomic ensembles.
The experimental setup is shown in Fig.1. The output of a 795 nm external
cavity diode laser is divided into two parts. One part is applied for
frequency locking, and the other one is sent through a single-mode fiber (SMF)
to improve the spatial mode. After the fiber, the beam passes through a half-
wave plate and polarization beam splitter to control the intensity of the
control and probe beams. The transmitted part is chosen as the probe beam with
the waist diameter of 2 mm (shown in Fig.1(a)), while the reflected part is
selected as the control beam, whose size is expanded three times larger than
the probe beam by a telescope. The acousto-optic modulations(AOM) are inserted
into these two parts for pulse modulation. The spatial light
modulator(HAMAMATSU-X10468-07) is used for loading grating to generate the OAM
light in the probe beam part. For effective imaging, the probe beam with OAM
passed through the 4f imaging system (not shown in the Fig.1) and propagates
collinearly with the control beam. Finally, Both beams incident into the
Rubidium cell and with the corresponding Zeeman sublevels of 87Rb. The energy-
level configuration is shown in Fig.1(c). In our experiment, the laser
frequency is locked to the $5S_{1/2},F=2\to 5P_{1/2},F^{\prime}=1$ transition
of the 87Rb D1 line. The Rb cell has a length of 50 mm. A three-layer $\mu$
-metal magnetic shield isolates the cell from ambient magnetic fields. In our
experiment, the power of the control and probe beam are The temperature of the
cell is set to 65°C with a controller.
The control and the probe beams are separated into the reflected and
transmitted parts using a QWP and a PBS after the Rb cell, and the reflected
part is the probe beam carrying OAM. Then a Sagnac loop with two Dove prisms
is applied to demultiplex OAM beams after storage. As shown in Fig.1(b), the
OAM probe beam is injected into the Sagnac loop and divided into two parts.
These two parts go through the same path (clockwise and counterclockwise
paths) and inject into the Dove prism with different rotation angles. The Dove
prisms in the two paths produce different phases, and finally, these two parts
interfere with each other in the beam splitter passes through the Sagnac loop.
The spatial intensity distribution of the retrieval signal is recorded by an
intensified charge-coupled device camera with a shutter time of 2 nanoseconds
(ICCD,ANDOR TECHNOLOGY), as shown in Fig.1(b).
Now let us focus on the experimental procedure shown in Fig.1(d). First, the
probe beam pulse with 5$\mu$s is input to the spatial light modulator(SLM)
before the rubidium cell, and an OAM light pulse is generated to load a fork-
shaped grating. In one-third of the back edge of the probe light pulse, the
control light is turned off and then switched on again by using an AOM after a
controllable time. The retrieved probe beam can be demultiplexed after the
sagnac loop and detected by ICCD1 or ICCD2, depending on the topological
charge of the probe light pulse. Secondly, our system realizes the
multiplexing memory and demultiplexing of OAM states using the OAM mode
splitter. To clearly understand the principle of demultiplexing based on the
Sagnac loop, we assume that the quantum state of the probe beam is
$\left|l\right\rangle$($l$ can be any integer). The transmitted and reflected
states become $\frac{1}{2}\left|l\right\rangle$ and
$\frac{i}{2}\left|l\right\rangle$. The outgoing light at the ICCD1 and the
ICCD2 ports are expressed as
$\displaystyle
T_{ICCD1}=\frac{i}{2}\exp(i(2l\alpha+\pi))\left|l\right\rangle+\frac{i}{2}\exp(i\pi)\left|l\right\rangle$
(1a) $\displaystyle
T_{ICCD2}=-\frac{1}{2}\exp(i(2l\alpha+\pi))\left|l\right\rangle+\frac{1}{2}\exp(i\pi)\left|l\right\rangle$
(1b)
The first term of Eq.1 is the quantum state that undergoes a counterclockwise
path, and the second term describes the quantum state that undergoes a
clockwise path. $\alpha$ is the rotation angle of dove prism. Actually, we set
the angle of Dove Prism1 (DP1) to 90 degrees experimentally. Thus, the
expression above becomes
$\displaystyle
T_{ICCD1}=-\frac{i}{2}\exp(il\pi)\left|l\right\rangle-\frac{i}{2}\left|l\right\rangle$
(2a) $\displaystyle
T_{ICCD2}=\frac{1}{2}\exp(il\pi))\left|l\right\rangle-\frac{1}{2}\left|l\right\rangle$
(2b)
According to the Eq.2, when the topological charge of the incident probe beam
is an odd number, we can obtain the $T_{ICCD1}=0$ and
$T_{ICCD2}=\left|l\right\rangle$, which means that the probe beam with quantum
state $\left|l\right\rangle$ constructive interference in the ICCD2 port and
destructive interference at the ICCD1 port. The output is reversed when the
topological charge is an even number.
## 1 Results and analysis
Figure 2: (a) The demultiplexing of the probe beam after 0.25 $\mu$s of
memory. (b) The demultiplexing of the probe beam after 1 $\mu$s of memory.
After checking the system reliability again, mainly including light leakage
and ICCD normal trigger. We shown Our experimental results in Fig. 2(a).
Clearly, the retrieval signal when carrying topological charge of 0, 2, 4 is
captured by ICCD1, while, the signal carrying topological charge of 1 and 3
are captured by ICCD2 after 0.25$\mu$s storage time, respectively. The
demultiplexing system still has good characteristics after 1 $\mu$s storage
time as shown in Fig. 2(b). It can be seen that our demultiplexing system can
separate the OAM mode very well and has a good mode contrast. In addition, we
use oblique lens to realize self-interference of retrieval signal in
experiment. The interference pattern is shown in the upper right corner of
Fig. 2. As the results show that the OAM states remain unchanged after
storage, and different OAM states are separated in different output channels.
In order to study the quantum multiplexing memory and demultiplexing
characteristics of OAM probe beams with different topological charges at
different storage times. For effectively compare the retrieval mode quality
under specific storage time, the mode extinction ratio is defined as
$V_{T_{ICCD1}/T_{ICCD2}}=|\frac{T_{ICCD1}-T_{ICCD2}}{T_{ICCD1}+T_{ICCD2}}|$
(3)
We experimentally changed the clock switch of the control beam and the
grating, and all results are shown in Fig. 3.
Figure 3: The separation contrast of different OAM probe beams at different
storage times. The black, red, blue, pink and green lines express the OAM
probe beam with $l=0$, $l=1$, $l=2$, $l=3$ and $l=4$, respectively.
We can obtain several differences by comparing the mode extinction ratio
obtained from our demultiplexing system. It is observed that the Gauss pulse
($l=0$) maintains high mode extinction ratio about 80 $\%$ after 2.5 $\mu$s of
quantum memory, as the black line shows in Fig. 3. The OAM pulse with $l=1$
remains an extinction ratio higher than 80 $\%$ after 1.5 $\mu$s of storage,
as shown in the red line. However, the optical mode extinction ratio drops
dramatically when the storage time exceeds 1.5 $\mu$s. The reason is due to
the motion of the atomic spin wave that intensifies the diffusion of the
readout optical mode when the optical mode carrying OAM is recorded into the
atomic spin wave, which leads to a decrease in the extinction ratio after
demultiplexing in our system. This influence becomes more pronounced when the
optical mode carries higher-order topological charges, as shown in the blue
line, pink and green lines of Fig. 3.
In order to verify the multi-mode capabilities of multiplexing storage and
demultiplexing in our experimental setup. Two OAM modes expressed as
$\left|1\right\rangle+\left|2\right\rangle$,
$\left|2\right\rangle+\left|3\right\rangle$ and
$\left|3\right\rangle+\left|4\right\rangle$ have been generated for multi-mode
storage, and the intensity distribution as shown in the left column of Fig. 4.
The figure shows that, two OAM modes injected into the memory system interfere
to form a moon-like intensity distribution. After 0.25 $\mu$s stored, this two
OAM modes with different topological charges are separated into different
paths, as shown in the two right columns of Fig. 4. All odd OAM modes have
been recorded by ICCD1, while ICCD2 has recorded the even OAM modes, and the
separated OAM modes have a good intensity distribution. An effective OAM mode
separator can not only effectively separate the OAM modes in different
channels, but also maintain the OAM modes characteristic. Therefore, oblique
lenses are applied again to achieve OAM mode self-interference. The experiment
results are shown in the upper right corner of Fig. 4, the two OAM states
remain unchanged after demultiplexing. At this point, our experimental system
can not only realize the storage of the single OAM state and separated it in
different ports by using an OAM modes splitter but can also store and separate
two OAM modes simultaneously.
Figure 4: Multiplexing memory and demultiplexing output of two OAM modes at
0.25 $\mu$s.
In conclusion, we have demonstrated the experimental realization of multi-mode
multiplexed storage and demultiplexed of OAM beams at room temperature based
on atomic ensembles. With the Sagnac interferometer embedding two dove prisms,
we obtained a retrieval signal of the OAM mode with an odder or even
topological charge in different interfering ports and a mode extinction ratio
higher than 80 $\%$ at the specific storage time. Meanwhile, two OAM
superimposed modes have been stored and demultiplexed successfully, and the
results show that our experimental scheme can effectively separate the two OAM
modes after quantum memory. The OAM demultiplexing devices based on the Sagnac
loop structure not only have higher separation contrast but can also be
extended to a higher dimensional multimode multiplexing storage and
demultiplexing of OAM modes. It is worth noting that quantum memory with
multimode capacity not only needs to maintain the storage modes, but also
needs to maintain the high fidelity. Our results may have applications in high
channel capacity multi-mode quantum multiplexing storage and demultiplexing
based on atomic ensembles.
Funding National Natural Science Foundation of China (NSFC) (11374238,
11534008, 11574247, 11604257, 11604258, 11774286); National Science Foundation
(NSF) (1602755);
Disclosures
Disclosures The authors declare no conflicts of interest.
Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.
## References
* [1] A. I. Lvovsky, B. C. Sanders, and W. Tittel, Nature photonics 3, 706 (2009).
* [2] R. Zhao, Y. Dudin, S. Jenkins, C. Campbell, D. Matsukevich, T. Kennedy, and A. Kuzmich, Nature Physics 5, 100 (2009).
* [3] J. P. Marangos, Journal of modern optics 45, 471 (1998).
* [4] D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Physical Review A 86, 023801 (2012).
* [5] T. Fortier and E. Baumann, Communications Physics 2, 1 (2019).
* [6] D. Saunders, J. Munns, T. Champion, C. Qiu, K. Kaczmarek, E. Poem, P. Ledingham, I. Walmsley, and J. Nunn, Physical review letters 116, 090501 (2016).
* [7] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001).
* [8] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Light: Science & Applications 8, 1 (2019).
* [9] A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nature Photonics 8, 234 (2014).
* [10] T.-S. Yang, Z.-Q. Zhou, Y.-L. Hua, X. Liu, Z.-F. Li, P.-Y. Li, Y. Ma, C. Liu, P.-J. Liang, X. Li _et al._ , Nature communications 9, 1 (2018).
* [11] C. Wang, Y. Yu, Y. Chen, M. Cao, J. Wang, X. Yang, S. Qiu, D. Wei, H. Gao, and F. Li, Quantum Science and Technology 6, 045008 (2021).
* [12] V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, Nature communications 6, 1 (2015).
* [13] Y.-H. Ye, M.-X. Dong, Y.-C. Yu, D.-S. Ding, and B.-S. Shi, Optics letters 44, 1528 (2019).
* [14] G. C. Berkhout and M. W. Beijersbergen, Physical review letters 101, 100801 (2008).
* [15] G. C. Berkhout, M. P. Lavery, M. J. Padgett, and M. W. Beijersbergen, Optics letters 36, 1863 (2011).
* [16] H. Huang, Y. Ren, Y. Yan, N. Ahmed, Y. Yue, A. Bozovich, B. I. Erkmen, K. Birnbaum, S. Dolinar, M. Tur _et al._ , Optics letters 38, 2348 (2013).
* [17] M. Padgett, J. Arlt, N. Simpson, and L. Allen, American Journal of Physics 64, 77 (1996).
* [18] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
sample
|
# VL-InterpreT: An Interactive Visualization Tool for Interpreting Vision-
Language Transformers
Estelle Aflalo111Equal Contributions
Intel Labs
<EMAIL_ADDRESS>Meng Du111Equal Contributions
Intel Labs, UCLA
<EMAIL_ADDRESS>Shao-Yen Tseng
Intel Labs
<EMAIL_ADDRESS>Yongfei Liu
Microsoft Research
<EMAIL_ADDRESS>Chenfei Wu
Microsoft Research
<EMAIL_ADDRESS>Nan Duan
Microsoft Research
<EMAIL_ADDRESS>Vasudev Lal
Intel Labs
<EMAIL_ADDRESS>
###### Abstract
Breakthroughs in transformer-based models have revolutionized not only the NLP
field, but also vision and multimodal systems. However, although visualization
and interpretability tools have become available for NLP models, internal
mechanisms of vision and multimodal transformers remain largely opaque. With
the success of these transformers, it is increasingly critical to understand
their inner workings, as unraveling these black-boxes will lead to more
capable and trustworthy models. To contribute to this quest, we propose VL-
InterpreT, which provides novel interactive visualizations for interpreting
the attentions and hidden representations in multimodal transformers. VL-
InterpreT is a task agnostic and integrated tool that (1) tracks a variety of
statistics in attention heads throughout all layers for both vision and
language components, (2) visualizes cross-modal and intra-modal attentions
through easily readable heatmaps, and (3) plots the hidden representations of
vision and language tokens as they pass through the transformer layers. In
this paper, we demonstrate the functionalities of VL-InterpreT through the
analysis of KD-VLP, an end-to-end pretraining vision-language multimodal
transformer-based model, in the tasks of Visual Commonsense Reasoning (VCR)
and WebQA, two visual question answering benchmarks. Furthermore, we also
present a few interesting findings about multimodal transformer behaviors that
were learned through our tool.
## 1 Introduction
Since transformers were introduced in Vaswani _et al_. [30], not only have
they seen massive success in NLP applications, their impact on computer vision
and multimodal problems has also become increasingly disruptive. However, the
internal mechanisms of transformers that lead to such successes are not well
understood. Although efforts have been made to interpret the attentions [8]
and hidden states [22] of transformers for NLP, such as BERT [12],
investigations in the mechanisms of vision and multimodal transformers are
relatively scarce, and tools for probing such transformers are also limited.
Given the fast-growing number of successful vision and multimodal transformers
(_e.g_., ViT [13] and CLIP [25]), enhanced interpretability of these models is
needed to guide better designs in the future.
Past research has shown the importance of interpreting the inner mechanisms of
transformers. For example, Clark _et al_. [8] found certain BERT attention
heads specialized in handling certain syntactic relations, as well as
interesting ways in which BERT attention utilizes special tokens and
punctuation marks. Additionally, Lin _et al_. [21] showed that the linguistic
information encoded in BERT becomes increasingly abstract and hierarchical in
later layers. These studies provide valuable insights into the functions of
various elements in transformer architecture for NLP, and shed light on their
limitations.
This paper presents VL-InterpreT111A screencast of our application is
available at https://www.youtube.com/watch?v=4Rj15Hi_Pdo. Source code and a
link to a live demo: https://github.com/IntelLabs/VL-InterpreT, which is an
interactive visualization tool for interpreting the attentions and hidden
representations of vision-language (VL) transformers. Importantly, it is a
single system that analyzes and visualizes several aspects of multimodal
transformers: first, it tracks behaviors of both vision and language attention
components in attention heads throughout all layers, as well as the
interactions across the two modalities. Second, it also visualizes the hidden
representations of vision and language tokens as they pass through transformer
layers.
The main contributions of our work are:
* •
Our tool allows interactive visualization for probing hidden representations
of tokens in VL transformers.
* •
Our tool allows systematic analysis, interpretation, and interactive
visualization of cross- and intra-modal components of attention in VL
transformers.
* •
As an application of VL-InterpreT, we demonstrate multimodal coreference in
two analyses: 1) how contextualized tokens in different modalities referring
to the same concept are mapped to proximate representations, and 2) how
attention components capture the conceptual alignment within and across
modalities.
## 2 Related Work
As deep learning models flourish, many tools and methods have been proposed to
offer insight into their inner workings. Some methods are general-purpose [24,
32], while others are nuanced for specific models such as CNNs [35, 36] or
RNNs [16, 27, 17]. In transformers, the introduction of attention not only
helped improve performance, but also served as an additional component towards
interpretability.
Interpretability of NLP transformers was initially approached through the
analysis of attention to capture its alignments with syntactic or semantic
relationships [31, 8]. Following this, subsequent works introduced additional
functionalities including visualizations of hidden representations, task
matching of attention heads, aggregate metrics, and interactive datapoint
analysis [28, 15, 18, 20]. While common in allowing a user to understand the
inner workings of transformers, each tool introduces novel applications. For
example, LIT [28] enables probing for bias through examination of coreferences
in counterfactuals. InterpreT [18] allows tracking of token representations
through the layers and offers users the ability to define new metrics to
identify coreference relationships in attention heads. Additionally, T3-Vis
[20] focused on allowing users to improve transformer training by integrating
the training dynamics in their visualization tool.
Interpreting vision transformers, such as those for object detection [4, 13]
or image captioning [10, 19], has also become increasingly popular. Cordonnier
_et al_. [9] showed that the first few layers in transformers can learn to
behave similarly to convolutional layers, and demonstrated the filter patterns
through visualization of image-to-image attention. As illustrated in this
paper, the attention mechanism in transformers is a natural gateway to
understanding vision models, as the heatmaps of attention can be used to
highlight salient image regions. Furthermore, Chefer _et al_. [7] proposed a
method for visualizing self-attention models by calculating a LRP [1]-based
relevancy score for each attention head in each layer, and propagating
relevancies through the network. The end result is a class-specific
visualization of image regions that led to the classification outcome.
Multimodal interpretability has, up until now, mainly entailed using probing
tasks to study the impact of each modality on the responses generated by the
model. These probing tasks aim to quantify the information captured in the
hidden representations by training classifiers, or applying metrics to
embeddings at different points in a model. For instance, Cao _et al_. [3]
proposed various probing tasks to analyze VL transformers, where the authors
observed modality importance during inference, and identified attention heads
tailored for cross-modal interactions as well as alignments between image and
text representations. Additionally, other works have proposed probing tasks to
interpret VL transformers for aspects such as visual-semantics [11], verb
understanding [14], and other concepts such as shape and size [26]. However, a
disadvantage of probing tasks is the amount of work: additional training of
the classifiers is often required, and specific task objectives must be
defined to capture different embedded concepts. Finally, most aforementioned
works require image-caption pairs as input, and are therefore not best suited
for interpreting multimodal transformers in tasks such as visual question
answering.
Recently, a first attempt at explaining predictions by a VL transformer was
proposed in [6]. There the authors constructed a relevancy map using the
model’s attention layers to track the interactions between modalities. The
relevancy map is updated by a set of update rules that back-propagates
relevancies of the prediction result back to the input. This map is very
useful in understanding how model decisions are formed, but a more
comprehensive interpretation for other aspects of VL transformers is still
needed.
Our proposed tool, VL-InterpreT, differs from previous works in that it
interprets various aspects of multimodal transformers in a single interface.
This interactive interface allows users to explore interactions between tokens
in each modality from a bottom-up perspective, without tying to task-specific
inputs and outcomes. To the best of our knowledge, this is the first
interactive tool for interpreting multimodal transformers.
## 3 System Design
### 3.1 Workflow
Figure 1: VL-InterpreT workflow. Data shown in this figure are for
illustration purposes only.
VL-InterpreT is designed as a two-stage workflow: First, the attentions and
hidden states of a given multimodal model are generated and saved for a set of
examples (in this case, 100 examples). Next, the saved data, along with the
metadata of the corresponding examples, are loaded into our tool to enable
visualizations of the inner workings of the model. The workflow of VL-
InterpreT is shown in Figure 1, and the user interface is shown in Figure 3.
Different from interpretability tools for NLP transformers, VL-InterpreT
addresses the analysis and interpretation of the following properties of
multimodal transformers:
* •
Input: In general, multimodal transformers are able to process inputs
originating from different modalities, _e.g_., video, audio, or language.
Here, we only consider VL transformers where the input is composed of visual
and textual tokens. These tokens are mapped into a shared space, allowing for
concept-level alignment between the two modalities.
* •
Attention: Because of the bi-modal nature of the input, the resulting
attention can be split into four components: language-to-language, vision-to-
vision, vision-to-language, and language-to-vision. An illustration of these
components is shown in Figure 1.
* •
Hidden states: In each layer, the transformer produces as many hidden states
as the number of input tokens. Each input token is embedded as a _d_
-dimension vector after processed by each layer.
Figure 2: The VL-InterpreT user interface (rearranged for print).
Figure 3: The Attention Head Summary plot colored by the metrics selected from
the dropdown menu
### 3.2 Visualizations
#### 3.2.1 Attention heads components
In this section, we describe each attention component and how VL-InterpreT
visualizes them interactively. The attention matrix in a VL transformer, as
they are loaded in VL-InterpreT, is of size
$(N_{layers},N_{heads},L_{v}+L_{l},L_{v}+L_{l})$, where $N_{layers}$ and
$N_{heads}$ correspond to the number of layers and heads, respectively,
$L_{v}$ to the number of visual tokens, and $L_{l}$ to the number of text
tokens.
* •
The Language-to-Vision attention component (L2V) of size
$(N_{layers},N_{heads},L_{l},L_{v})$ reflects the text tokens’ dependency on
the visual tokens. This partition of the attention contains the attention
scores calculated from the dot product of the query vector based on the
selected image patch and the key vectors from text tokens. These attention
scores are the weights given to value vectors of text tokens when summing for
the updated representation of a specific image patch in the next layer. A user
can select any attention head and image patch in the interface of our tool,
and the corresponding L2V attention weights are displayed as a heatmap
overlaid on the input text, to visualize how much each text token contributes
to the updated image patch representation (see Figure 7).
* •
The Vision-to-Language attention component (V2L) of size
$(N_{layers},N_{heads},L_{v},L_{l})$ reflects the visual tokens’ dependency on
the text tokens. This component of attention in a VL transformer arises
through the query-key dot product where the query vectors are computed from
text token embeddings, and the key vectors are computed from the image token
embeddings. A user can select a specific attention head and a text token, and
the corresponding attention scores will be overlaid onto the image as a
heatmap (see Figure 6). This visualization helps users understand the relative
contributions of various parts of the image to the updated representations of
the text tokens. The interactive application also allows users to play an
animation, in which the heatmap over the image is automatically displayed in a
sequence for each word.
* •
The Language-to-Language attention component (L2L) of size
$(N_{layers},N_{heads},L_{l},L_{l})$ corresponds to the attention mapping in
NLP Transformers. This component visualizes how all text tokens attends to
each individual token of the input sentence (see Figure 6).
* •
The Vision-to-Vision attention component (V2V) of size
$(N_{layers},N_{heads},L_{v},L_{v})$ is analogous to language-to-language
attention, but in the visual space. It represents the attention between one
visual tokens and all visual tokens, including itself. Similar to the V2L
component, an attention vector (of size $(L_{v},1)$) here in a given head can
also be translated into a heatmap and overlaid onto the image. This
visualization is useful for identifying the contributions of different image
patches to the updated representation of the image patch selected by a user.
#### 3.2.2 Attention head summary
This functionality allows users to visualize a head summary plot containing
statistics of the attentions calculated for all heads and layers. For an
attention matrix of size $(N_{layers},N_{heads},L_{v}+L_{l},L_{v}+L_{l})$, the
head summary computes statistical metrics over the last two dimensions,
resulting in a plot of size $(N_{layers},N_{heads})$.
* •
The mean attention in an attention head is generated by calculating the
average of the corresponding attention matrix, while some tokens can be
excluded from this calculation. A user can select a summary metric from the
list of options. Each metric can be restricted to different components of the
attention. For example, instead of computing the mean of all input tokens
regardless of modality, the calculation can be limited to the vision-to-
language part or the vision-to-vision part of the attention. Additionally,
users may also focus on both cross-modal components (i.e., V2L and L2V
averaged) or both intra-modal ones (i.e., V2V and V2L averaged). See the drop
down menu in Figure 3.
* •
Custom metrics: Based on users’ interests, custom metrics can be integrated in
this plot to show relevant attention heads. For example, we created a custom
metric to look for the attention heads responsible for aligning the same
person between vision and language modalities, based on the V2L component.
This metric, labeled Spearman correlation b/w person attention and segment
(see Figure 3), is the Spearman correlation between the panoptic segmentation
mask for a person in the image (generated by Maskformer model [2]), and the
attention heatmap to the corresponding person token in the Vision-to-Language
attention component.
This metric allows a user to identify attention heads that perform a function
similar to panoptic segmentation for people (see Figure 4).
Figure 4: Custom metric: Average correlation between the V2L attention to the
[PERSON] tokens (bottom right) and the person’s panoptic segmentation mask
(top right).
Finally, for each metric, a mean is automatically computed for each layer and
shown in the rightmost column of the attention head summary (see Figures 3 and
4). This column visualizes the general behavior in each layer for the selected
metrics, and shows its evolution throughout the layers of the transformer.
#### 3.2.3 Hidden state representation
For each input token, a _d-_ dimensional hidden representation is produced by
the transformer after each layer (in our setup, $d=768$). This pool of hidden
representations is then filtered by two criteria: (1) if the related text is a
stop word and (2) if the related image patch comes from a part of the
background (e.g., wall, ground, etc.). In order to visualize the remaining
hidden representations in a readable form, t-distributed Stochastic Neighbor
Embedding (t-SNE) [29] was applied to reduce dimensions and create disjoint
t-SNE spaces for different layers. This way, given a selected example, VL-
InterpreT tracks the hidden representations both before the first layer and
after each subsequent layer, and plots them in two-dimensional spaces.
Figure 8(a) shows the data points representing the visual (in blue) and
textual (in red) tokens from a given example. When hovering on a data point
from language, the corresponding text is displayed. When hovering on a data
point representing visual tokens, the image is shown with a highlighted blue
patch corresponding to the visual token. Further observations on the hidden
states often reveal the concept-level vision-text alignments that are learnt
in this multimodal setup (see Section 4.2.3). To help further understand this
alignment, VL-InterpreT allows a user to select a token (text or image patch)
from a given example, and shows the nearest token in the other modality from
the whole subset of examples, marked with a green star.
## 4 Case Studies
To demonstrate the functionalities of VL-InterpreT, we analyze an end-to-end
VL transformer model, KD-VLP [23], on two benchmarks: Visual Commonsense
Reasoning (VCR) [34] and WebQA [5]. Nonetheless, our tool is generally
applicable to a variety of multimodal transformer configurations and types of
VL datasets.
### 4.1 Model
The KD-VLP model used in our case study is a transformer for end-to-end
vision-language processing. This model utilizes a ResNet backbone for visual
inputs, and is pretrained using text-oriented, image-oriented, and cross-modal
tasks in the form of masked language modeling, object-guided masked vision
modeling, and phrase-region alignment, respectively. Depending on the
application, the KD-VLP model can be fine-tuned for classification or
generation tasks given bi-modal input of image and text.
### 4.2 Analysis on VCR
The VCR benchmark consists of 290K multiple choice QA problems derived from
110K movie scenes. This dataset is uniquely valuable in that it requires
higher-order cognition and commonsense reasoning about the world. Given an
image and a question, the objective is to select an appropriate answer from
four possible choices, and then provide the rationale. To predict the correct
answer and rationale, the KD-VLP model is fed with the image, the question,
and each answer or rationale individually. The predicted answer $a_{p}$ is the
answer/rationale choice that receives the highest probability score, _i.e_.,
$a_{p}=\underset{a_{j}}{\arg\\!\max}~{}f(v,q,a_{j})$ (1)
where $f$ is the KD-VLP model, $v$ is the image, $q$ the question, and
$\\{a_{j}|~{}j\in[1,2,3,4]\\}$ are the possible answer choices.
The functionalities of VL-InterpreT are demonstrated using example val-445
from the VCR validation set, or example ID 89 in the VL-InterpreT live demo.
This data sample, shown in Figure 6, comprises an image showing a little girl
running to a man and a woman in a garden. The question is:
Where is [PERSON_0] running to?
where [PERSON_0] corresponds to the little girl on the right side of the image
(the locations of persons are provided in the VCR dataset and also passed to
the model). We also analyze the answer predicted by the model (which in this
case is correct):
[PERSON_0] is running to help [PERSON_1] and [PERSON_2] with the plants.
where [PERSON_1] and [PERSON_2] refer to the couple.
The following analyses on this example will highlight the visualization
capabilities that our application provides.
#### 4.2.1 Attention head summary
This functionality allows a user to identify interesting heads based on
various metrics. By selecting Mean cross-modal attention from the dropdown
menu (see Figure 3), a user can identify attention heads specialized in cross-
modal attention. For instance, in Figure 5 the eighth attention head in layer
11 (denoted as (11, 8)) has, on average, the highest attention across
modalities. Thus, we focus on this specific head and plot its cross-modal (V2L
and L2V) attention. Apart from cross-modal attention, specific heads could
also been identified through Mean image-to-text attention for analyzing V2L
attentions, or Mean text-to-image attention for L2V attentions. Analogous
procedure also applies to L2L and V2V attention components – for instance, the
metric Mean image-to-image attention (without self) shows the heads’
attentions from every image patch to the every other image patches, excluding
each patch itself and its neighbors. As described in section 3.2.2, a custom
metric was used to identify attention heads with V2L components for [PERSON]
tokens highly correlated with their panoptic segmentations. As shown in Figure
4, head (8, 9) scores particularly high in this metric. With head (8, 9)
selected, the V2L attention attention heatmap for the person token (Figure 4,
bottom right) indeed aligns well with the panoptic segmentation of the
corresponding woman (Figure 4, top right). Furthermore, this correlation
metric exhibits a trend where middle and later layers tend to have higher
correlation coefficients on average (above 0.5) than early layers (less than
0.1), showing the evolution of attention patterns as the transformer layers
grow deeper.
Figure 5: Attention Head Summary
#### 4.2.2 Attention components
As described in Section 3.2.1, Figure 6 shows the heatmaps over the image and
the text generated by the Language-to-Language and Vision-to-Language
attention components of head (11, 8). This figure shows how attentions differ
for two selected tokens: [PERSON_0] and plants. The V2L components are
represented as heatmaps over the images at the bottom right of Figures 6(a)
and 6(b). It can be observed that the attention is concentrated on regions
corresponding to the text, namely the little girl for the attention to
[PERSON_0] (Figure 6(a)) and the plants for the attention to the plants token
(Figure 6(b)). The L2L components are on the top right of these figures. For
both selected text tokens, related text tokens (including themselves) are
highlighted in the heatmaps. For the example in Figure 6(a), the attention to
[PERSON_0] is mostly from [PERSON_0], running to, and running to help
[PERSON_1]. In the other example in Figure 6(b), the attention to plants is
mostly from the plants token itself.
Figure 7 visualizes the attentions to vision tokens, i.e., the Language-to-
Vision and the Vision-to-Vision attentions. In this example, we select an
image patch that is a part of the plants on the left image for analysis.
Accordingly, the L2V component (top right of Figure 7) shows that the
attention to this image patch is mainly from the [CLS] token as well as the
plants token, which aligns with the concept behind the selected region. As for
the V2V component (bottom right of the figure), it is also interesting that
most of the regions containing plants in the image attend to this specific
patch of plants, which again shows a conceptual alignment. In summary, this
example shows evidence of a unified concept of “plants”, where the attention
has a consistent pattern between intra- and cross-modal components.
(a) Attention to the text token [PERSON_0]
(b) Attention to the text token plants
Figure 6: Two selected text tokens and the corresponding L2L (top right) and
V2L (bottom right) attention to them. Figure 7: Attentions to vision for a
selected image patch in the left image. The heatmaps show the attention from
the text (top right) as well as other patches in the image (bottom right).
#### 4.2.3 Hidden states
Figure 8 demonstrates the capabilities of the VL-InterpreT for visualizing
hidden states. When selecting the text token plants (marked in orange) from
the previous example val-445 (ex 89), Figure 8(b) shows that in layer 11, the
closest image patch from the whole pool of examples is the [IMG_52] (marked
with a green star) from val-495 (ex 99). It is interesting that even when it
comes from a very different example, this token also refers to the plants in
the image. Other than layer 11, users can select different layers on the right
to see other closest image tokens to plants throughout layers.
(a) Hovering over a data point representing a visual token displays the
corresponding image patch.
(b) Clicking on the text token plants marks it orange, and displays the
closest image token from the whole dataset, marked as a green star.
Figure 8: t-SNE plot from the hidden representations of the selected example.
(a) Predicted: A cross appears at the top of the dome of Saint Peter’s
Basilica.
(b) Predicted: There are 6 pillars in front of the Façade of the Kurhaus.
(c) Predicted: The center of both the Aster amellus and the Wood Anemone is
yellow.
(d) Predicted: The ring around the eye of Trogon surrucura is red.
(e) Predicted: The Traditional wedding attire of the Yoruba culture in Nigeria
is white.
Correct color: maroon.
(f) Predicted: The olive that goes in Moroccan Tagine is yellow.
Correct color: green.
Figure 9: V2L attention heatmaps for WebQA examples and generated answers.
Figures have been rearranged for print. (a)-(d) are correct predictions and
(e) and (f) are incorrect.
Sections 4.2.2 and 4.2.3 show evidence of alignment between visual and textual
concepts of plants. We see that the concept of ”plants” in both modalities and
across examples is captured by proximate representations. As such, VL-
InterpreT allows studying how such sense of objectness emerges by probing the
attentions and hidden states across all layers of the transformer.
### 4.3 Analysis on WebQA
The WebQA benchmark focuses on multimodal, multihop reasoning for open-domain
question answering. This benchmark emulates a knowledge-seeking query to a
search engine for information which may be contained in either text-based
articles or images. Given a query, the goal is to identify which information
is relevant across modalities, and to generate a full natural language answer
based on the selected sources. The dataset contains 50k QA pairs, half of
which are text-based and the other half are image-based.
We use the KD-VLP model to first select relevant sources using a
classification head. Then, by adding a decoder in a Fusion-in-Decoder manner
as in [33], a predicted answer is generated based on the retrieved sources.
Similar behaviors can be studied for WebQA as in the previous section. The
following analyses will focus on the L2V attention in the KD-VLP encoder, as
these visualizations helps in understanding how the model generates answers.
For this analysis, attention head (11, 5), identified in the same way as
previous analyses, was selected. Figure 9 shows the V2L attention from image
to the highlighted word above the pictures. These heatmaps show that the model
attends more to the regions that help answer the question. For instance, when
the question is about pillars, Figure 9(b) shows that the model attention
comes particularly from the columns in the picture. In order to determine the
color of the center of the flower in Figure 9(c), the model exhibits attention
from the flower center in the image and generates the correct color (yellow).
Furthermore, these visualizations can also be generated for incorrectly
answered questions, providing insights into the reason why incorrect answers
were generated. For example, one may imagine why the model answered
incorrectly in Figure 9(e) and 9(f): In 9(e), the attire that the question
asks about was misidentified. That is, instead of getting attention from
patches of the maroon dress, the model focuses on the white feathers. A
similar behavior is also seen in 9(f), where the model fails to locate the
olives but focuses on the yellow vegetable in the tagine. In both cases, the
model answers according to the identified object color (i.e., ”white” feathers
and ”yellow” olive). These interpretations of attention helps users identify
why a model fails in certain cases, and provides guidance for future efforts
in improving model accuracy.
## 5 Conclusions and Future Directions
In this paper we presented VL-InterpreT, an interactive visualization tool for
interpreting vision-language transformers. This tool allows for interactive
analysis of attention and hidden representations in each layer of any VL
transformer. VL-InterpreT can be used to freely explore the interactions
between and within different modalities to better understand the inner
mechanisms of a transformer model, and to obtain insight into why certain
predictions are made. Through case studies, we demonstrated how VL-InterpreT
can be used to validate the learning of cross-modal concepts, as well as to
“explain” cases of failure.
In the latest version, VL-Interpret is able to run a live model to process
user-generated examples in real time. This allows interactive manipulations of
inputs, including both text and image, to study their effects on the attention
and hidden representations.
For future work, we would like to include aggregated metrics and
visualizations over multiple samples to obtain a more comprehensive
understanding of model operation. In addition, we hope to experiment with any
additional functionalities that will assist users in interpreting multimodal
transformers, and continue to enhance this interpretability tool.
## References
* [1] Sebastian Bach, Alexander Binder, Grégoire Montavon, Frederick Klauschen, Klaus-Robert Müller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PloS one, 10(7):e0130140, 2015.
* [2] Alexander Kirillov Bowen Cheng, Alexander G. Schwing. Per-pixel classification is not all you need for semantic segmentation. 2021\.
* [3] Jize Cao, Zhe Gan, Yu Cheng, Licheng Yu, Yen-Chun Chen, and Jingjing Liu. Behind the scene: Revealing the secrets of pre-trained vision-and-language models. In European Conference on Computer Vision, pages 565–580. Springer, 2020.
* [4] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In European Conference on Computer Vision, pages 213–229. Springer, 2020.
* [5] Yingshan Chang, Mridu Narang, Hisami Suzuki, Guihong Cao, Jianfeng Gao, and Yonatan Bisk. Webqa: Multihop and multimodal qa. arXiv preprint arXiv:2109.00590, 2021.
* [6] Hila Chefer, Shir Gur, and Lior Wolf. Generic attention-model explainability for interpreting bi-modal and encoder-decoder transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 397–406, October 2021.
* [7] Hila Chefer, Shir Gur, and Lior Wolf. Transformer interpretability beyond attention visualization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 782–791, 2021.
* [8] Kevin Clark, Urvashi Khandelwal, Omer Levy, and Christopher D. Manning. What does BERT look at? An analysis of BERT’s attention. In BlackboxNLP@ACL, 2019.
* [9] Jean-Baptiste Cordonnier, Andreas Loukas, and Martin Jaggi. On the relationship between self-attention and convolutional layers. In International Conference on Learning Representations, 2020.
* [10] Marcella Cornia, Matteo Stefanini, Lorenzo Baraldi, and Rita Cucchiara. Meshed-memory transformer for image captioning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10578–10587, 2020.
* [11] Adam Dahlgren Lindström, Johanna Björklund, Suna Bensch, and Frank Drewes. Probing multimodal embeddings for linguistic properties: the visual-semantic case. In Proceedings of the 28th International Conference on Computational Linguistics, pages 730–744, Barcelona, Spain (Online), Dec. 2020\. International Committee on Computational Linguistics.
* [12] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171–4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.
* [13] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021.
* [14] Lisa Anne Hendricks and Aida Nematzadeh. Probing image-language transformers for verb understanding. In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 3635–3644, Online, Aug. 2021. Association for Computational Linguistics.
* [15] Benjamin Hoover, Hendrik Strobelt, and Sebastian Gehrmann. exBERT: A Visual Analysis Tool to Explore Learned Representations in Transformer Models. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: System Demonstrations, pages 187–196, Online, July 2020. Association for Computational Linguistics.
* [16] Bo-Jian Hou and Zhi-Hua Zhou. Learning with interpretable structure from gated rnn. IEEE transactions on neural networks and learning systems, 31(7):2267–2279, 2020.
* [17] Andrej Karpathy, Justin Johnson, and Li Fei-Fei. Visualizing and understanding recurrent networks. arXiv preprint arXiv:1506.02078, 2015.
* [18] Vasudev Lal, Arden Ma, Estelle Aflalo, Phillip Howard, Ana Simoes, Daniel Korat, Oren Pereg, Gadi Singer, and Moshe Wasserblat. InterpreT: An interactive visualization tool for interpreting transformers. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: System Demonstrations, pages 135–142, Online, Apr. 2021. Association for Computational Linguistics.
* [19] Guang Li, Linchao Zhu, Ping Liu, and Yi Yang. Entangled transformer for image captioning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 8928–8937, 2019.
* [20] Raymond Li, Wen Xiao, Lanjun Wang, Hyeju Jang, and Giuseppe Carenini. T3-vis: visual analytic for training and fine-tuning transformers in NLP. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pages 220–230, Online and Punta Cana, Dominican Republic, Nov. 2021. Association for Computational Linguistics.
* [21] Yongjie Lin, Yi Chern Tan, and Robert Frank. Open sesame: Getting inside BERT’s linguistic knowledge. In Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP, pages 241–253, Florence, Italy, Aug. 2019. Association for Computational Linguistics.
* [22] Nelson F. Liu, Matt Gardner, Yonatan Belinkov, Matthew E. Peters, and Noah A. Smith. Linguistic knowledge and transferability of contextual representations. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 1073–1094, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.
* [23] Yongfei Liu, Chenfei Wu, Shao-Yen Tseng, Vasudev Lal, Xuming He, and Nan Duan. KD-VLP: Improving end-to-end vision-and-language pretraining with object knowledge distillation, 2021.
* [24] Harsha Nori, Samuel Jenkins, Paul Koch, and Rich Caruana. Interpretml: A unified framework for machine learning interpretability. arXiv preprint arXiv:1909.09223, 2019.
* [25] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision. In ICML, 2021.
* [26] Emmanuelle Salin, Badreddine Farah, Stéphane Ayache, and Benoit Favre. Are vision-language transformers learning multimodal representations? A probing perspective. In AAAI 2022, 2022.
* [27] Hendrik Strobelt, Sebastian Gehrmann, Hanspeter Pfister, and Alexander M Rush. Lstmvis: A tool for visual analysis of hidden state dynamics in recurrent neural networks. IEEE transactions on visualization and computer graphics, 24(1):667–676, 2017.
* [28] Ian Tenney, James Wexler, Jasmijn Bastings, Tolga Bolukbasi, Andy Coenen, Sebastian Gehrmann, Ellen Jiang, Mahima Pushkarna, Carey Radebaugh, Emily Reif, and Ann Yuan. The language interpretability tool: Extensible, interactive visualizations and analysis for NLP models. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pages 107–118, Online, Oct. 2020. Association for Computational Linguistics.
* [29] Laurens van der Maaten. Learning a parametric embedding by preserving local structure. In Proceeding of AISTATS 2009., 2009.
* [30] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
* [31] Jesse Vig and Yonatan Belinkov. Analyzing the structure of attention in a transformer language model. In Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP, pages 63–76, 2019.
* [32] James Wexler, Mahima Pushkarna, Tolga Bolukbasi, Martin Wattenberg, Fernanda Viégas, and Jimbo Wilson. The what-if tool: Interactive probing of machine learning models. IEEE transactions on visualization and computer graphics, 26(1):56–65, 2019.
* [33] Donghan Yu, Chenguang Zhu, Yuwei Fang, Wenhao Yu, Shuohang Wang, Yichong Xu, Xiang Ren, Yiming Yang, and Michael Zeng. KG-fid: Infusing knowledge graph in fusion-in-decoder for open-domain question answering, 2022.
* [34] Rowan Zellers, Yonatan Bisk, Ali Farhadi, and Yejin Choi. From recognition to cognition: Visual commonsense reasoning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 6720–6731, 2019.
* [35] Quanshi Zhang, Ruiming Cao, Feng Shi, Ying Nian Wu, and Song-Chun Zhu. Interpreting cnn knowledge via an explanatory graph. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018\.
* [36] Quanshi Zhang, Yu Yang, Haotian Ma, and Ying Nian Wu. Interpreting cnns via decision trees. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6261–6270, 2019.
|
space in a linear form, and in $\text{rank}_{1,w}$ it is essentially done in
the original space of the variables through nonlinear inequalities.
In the first experiment we set
$\lambda=10^{-2},\mu=10^{-4},p=50,d=1325,\pi=0.1$ and
$N\in\\{10,50,100,200\\}$. In Figure 1 we report the optimality gap and
solution time for different convex relaxations. In determining the optimality
gap, we compare the objective value of the given relaxation against the best
known feasible solution for the instance (among the ones found by the B&B
method applied to any of the formulations of (35) presented in (41), (47) and
(56). This solution corresponds to the optimal solution if the time limit has
not been reached in the B&B algorithm, or the best feasible solution reported
by MOSEK if the time limit has been reached.
The results in Figure 1 suggest that the qualities of the convex relaxations
based on the (56) formulation are the best. In particular,
$\text{rank}_{1}^{+}\\!$ relaxation attains an average gap smaller than $5\%$.
Moreover, adding valid inequalities to the sets $\Delta_{b}$ and $\Omega_{b}$
significantly improve the quality of the $\text{rank}_{1}\\!$ and
$\text{rank}_{1}^{+}\\!$ relaxations. For example, the
$\text{rank}_{1,v}^{+}\\!$ relaxation attains the average gap smaller than
$1\%$, which is $5$ times smaller than the gap attained by
$\text{rank}_{1}^{+}\\!$. However, adding these valid inequalities comes with
a computational downside. Specifically, the optimization problems now involve
more constraints, which result in longer solution times. It is also worth to
note that the optimality gap of the $\text{rank}_{1}\\!$ relaxations is
significantly superior to that of the separable and natural relaxations,
albeit at the expense of relatively longer solution times.
Finally, we highlight that the optimality gap of the continuous relaxations
from $\text{rank}_{1,v}\\!$ and $\text{rank}_{1,w}\\!$, with the latter being
inspired by (Wei et al., 2022b, Theorem 1), are identical. This is due to the
fact that when the variables ${\bm{w}}$ are relaxed to be continuous the
projection of $\text{rank}_{1,v}\\!$ in the original space leads to precisely
the same inequalities as in $\text{rank}_{1,w}\\!$. Nonetheless, it takes
roughly twice as long to solve the $\text{rank}_{1,w}\\!$ relaxation than the
$\text{rank}_{1,v}\\!$ relaxation. This is expected as the
$\text{rank}_{1,w}\\!$ relaxation introduces a considerably larger number of
constraints and variables compared to the $\text{rank}_{1,v}\\!$ relaxation.
It is also important to note that a complete implementation of (Wei et al.,
2022b, Theorem 1) requires using a characterization of
$\operatorname{conv}({\cal Z}\backslash\\{\bm{0}\\})$ which may possibly
involve an exponential number of constraints. In contrast,
$\text{rank}_{1,v}\\!$ relaxation handles this complexity through the use of
binary variables ${\bm{w}}$, making the $\text{rank}_{1,v}\\!$ relaxation much
more applicable in practice.
Figure 1: Comparison of different continuous relaxations for $p=50$ and
$\pi=0.1$ as $N$ varies.
We next examine the B&B performance of these alternative formulations of (35)
in which we always keep the variables ${\bm{z}}$ as binary but we create two
variants for each of the $\text{rank}_{1}\\!$ and $\text{rank}_{1}^{+}\\!$
formulations based on whether the variables ${\bm{w}}$ are kept as binary or
${\bm{w}}$ are relaxed to be continuous:
* •
In _separable reformulation_ , we consider (41).
* •
In _$\text{rank}_{1}\\!$ reformulation_, we consider (47).
* •
In _$\text{rank}_{1,r}\\!$ reformulation_, we replace $\Delta_{b}$ in (47)
with $\operatorname{relax}_{{\bm{w}}}(\Delta_{b})$.
* •
In _$\text{rank}_{1,v}\\!$ reformulation_, we replace $\Delta_{b}$ in (47)
with $\Delta_{b}\cap\Delta_{v}$.
* •
In _$\text{rank}_{1,r,v}\\!$ reformulation_, we replace $\Delta_{b}$ in (47)
with $\operatorname{relax}_{{\bm{w}}}(\Delta_{b}\cap\Delta_{v})$.
* •
In _$\text{rank}_{1,w}\\!$ reformulation_, we consider (4).
* •
In _$\text{rank}_{1}^{+}\\!$ reformulation_, we consider (56).
* •
In _$\text{rank}_{1,r}^{+}\\!$ reformulation_, we replace $\Omega_{b}$ in (56)
with $\operatorname{relax}_{{\bm{w}}}(\Omega_{b})$.
* •
In _$\text{rank}_{1,v}^{+}\\!$ reformulation_, we replace $\Omega_{b}$ in (56)
with $\Omega_{b}\cap\Omega_{v}$.
* •
In _$\text{rank}_{1,r,v}^{+}\\!$ reformulation_, we replace $\Omega_{b}$ in
(56) with $\operatorname{relax}_{{\bm{w}}}(\Omega_{b}\cap\Omega_{c})$.
Figure 2 reports the true optimality gap computed using the best known
heuristic solution (in our experiments this was usually obtained by a variant
of (47) formulation) as discussed earlier, the optimality gap reported by the
solver, the number of B&B nodes, and the solution time.
Figure 2: The B&B performance for $p=50$ and $\pi=0.1$ as $N$ varies.
We start by analyzing the true optimality gap and solver gap in Figure 2. As
also seen in Figure 1, in terms of true optimality gap, the formulations based
on (56) consistently outperform those based on (47) in terms of the true
optimality gap. This is primarily because the relaxations based on (56) can
produce high-quality lower bounds, even though these formulations take longer
to solve and thus result in significantly fewer number of nodes explored in
B&B. Among $\text{rank}_{1}\\!$ and $\text{rank}^{+}_{1}\\!$ variants, the
ones where the variables ${\bm{w}}$ are kept as binary perform better in terms
of the optimality gaps than the ones where ${\bm{w}}$ are relaxed to be
continuous even though having ${\bm{w}}$ binary results in fewer B&B nodes.
This is because when ${\bm{w}}$ are binary, the solver is able to leverage the
structure of the sets $\Delta_{b}$ and $\Omega_{b}$ to generate further cuts
and results in higher quality relaxations. Among $\text{rank}_{1}\\!$
variants, the performance of $\text{rank}_{1,v}\\!$ seems to be best and
$\text{rank}_{1,w}\\!$ seems to be the worst. As the continuous relaxation of
$\text{rank}_{1,w}$ takes longer to solve compared to those based on (47), its
B&B can explore only a smaller number of nodes, and therefore, results in a
worse optimality gap.
When we examine the gaps reported by the solver, we still observe that the
same phenomena, but this time the associated gaps reported by the solver are
considerably larger than the true optimality gaps. This is because for this
class of problems, the heuristic methods utilized by the solver are not very
advanced, and the good quality feasible solutions of a formulation are often
found at integral nodes in the B&B tree, so essentially by chance. Therefore,
the B&B procedure for the formulations that admit quick to solve node
relaxations often results in high quality feasible solutions. Consequently, in
the case of expensive lifted formulations such as (56) while the actual
optimality gaps are very close to zero, the solver is unable to report this
gap due to the inferior quality of the feasible solution found in the
associated B&B tree. To address this issue, BARON introduces the concept of
“relaxation-only constraints” (Sahinidis, 1996, accessed November 13, 2023).
These constraints are active during the relaxation step, but are disregarded
during the local search step. To the best of our knowledge, MOSEK 10 does not
support this feature, and thus in our figures we report both the true
optimality gap and the solver reported gap for the formulations tested.
In the second experiment we set $\lambda=10^{-2},\mu=10^{-4},N=100,\pi=0.1$
and $p\in\\{10,20,30,40,50\\}$, which translates to
$d\in\\{65,230,495,860,1325\\}$. Figure B.1 and Figure B.2 in Appendix B
compare the quality of different continuous relaxations and also report the
performance of the B&B algorithm, respectively. The observations from Figure
B.1 are very similar to the ones in Figure 1; thus we omit this discussion for
brevity. Despite the similarity between the observations in Figure B.2 and
Figure 2, it is worth noting that when $p=10$, the B&B performance is slightly
different. In particular, when $p=10$, all methods except
$\text{rank}_{1,r}^{+}\\!$ and $\text{rank}_{1,r,v}^{+}\\!$ can solve the
optimization problem in less than $20$ seconds. This implies that the integer
programs may be relatively simple to solve when the dimension is small. As a
result, stronger relaxations may not be needed when dealing with small scale
instances.
In the last experiment we set $\lambda=10^{-2},\mu=10^{-4},p=50,d=1325,N=100$
and $\pi\in\\{0.1,0.3,0.5,0.7,0.9\\}$; see Figure B.3 and Figure B.4 in
Appendix B for the quality of different continuous relaxations and the B&B
performance. In all of these instances time limit was reached in B&B, so the
solution time is not reported in Figure B.4. As the value of $\pi$ increases,
we notice that the optimality gap of the $\text{rank}_{1}\\!$ relaxation gets
closer to that of separable relaxation. This is expected as the binary
variable $w_{j}$ models whether ${\bm{a}}_{j}^{\top}{\bm{x}}=0$ or not. When
$\pi$ is large, the probability of such event is low. As a result, $w_{j}$ is
assigned a value of $1$ with high probability, which make the
$\text{rank}_{1}\\!$ relaxations much less effective. As a final observation,
we note that the value of $\pi$ seems to not affect the quality of the
$\text{rank}_{1}^{+}\\!$ relaxations; in particular these relaxations continue
to be of high quality even for high values of $\pi$.
##### Acknowledgements
This research is supported by Early Postdoc Mobility Fellowship SNSF grant
P2ELP2_195149 and AFOSR grant FA9550-22-1-0365.
## References
* Aktürk et al. (2009) M. S. Aktürk, A. Atamtürk, and S. Gürel. A strong conic quadratic reformulation for machine-job assignment with controllable processing times. _Operations Research Letters_ , 37(3):187–191, 2009.
* Atamtürk and Gómez (2018) A. Atamtürk and A. Gómez. Strong formulations for quadratic optimization with M-matrices and indicator variables. _Mathematical Programming_ , 170(1):141–176, 2018\.
* Atamtürk and Gómez (2019) A. Atamtürk and A. Gómez. Rank-one convexification for sparse regression. _arXiv:1901.10334_ , 2019.
* Atamtürk and Gómez (2020) A. Atamtürk and A. Gómez. Safe screening rules for $\ell_{0}$-regression from perspective relaxations. In _International Conference on Machine Learning_ , pages 421–430, 2020.
* Atamtürk and Gómez (2022) A. Atamtürk and A. Gómez. Supermodularity and valid inequalities for quadratic optimization with indicators. _Mathematical Programming (Forthcoming)_ , pages 1–44, 2022.
* Atamtürk et al. (2021) A. Atamtürk, A. Gómez, and S. Han. Sparse and smooth signal estimation: Convexification of $\ell_{0}$-formulations. _Journal of Machine Learning Research_ , 22:52–1, 2021\.
* Bacci et al. (2019) T. Bacci, A. Frangioni, C. Gentile, and K. Tavlaridis-Gyparakis. New MINLP formulations for the unit commitment problems with ramping constraints. _Optimization Online_ , 2019.
* Behdin and Mazumder (2021) K. Behdin and R. Mazumder. Archetypal analysis for sparse nonnegative matrix factorization: Robustness under misspecification. _arXiv:2104.03527_ , 2021.
* Ben-Tal and Nemirovski (2001) A. Ben-Tal and A. Nemirovski. _Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications_. SIAM, 2001.
* Bertsimas et al. (2021a) D. Bertsimas, R. Cory-Wright, and J. Pauphilet. A new perspective on low-rank optimization. _arXiv:2105.05947_ , 2021a.
* Bertsimas and King (2016) D. Bertsimas and A. King. OR forum—an algorithmic approach to linear regression. _Operations Research_ , 64(1):2–16, 2016.
* Bertsimas et al. (2016) D. Bertsimas, A. King, and R. Mazumder. Best subset selection via a modern optimization lens. _Annals of Statistics_ , 44(2):813–852, 2016\.
* Bertsimas et al. (2021b) D. Bertsimas, J. Pauphilet, and B. Van Parys. Sparse classification: a scalable discrete optimization perspective. _Machine Learning_ , 110(11):3177–3209, 2021b.
* Bertsimas and Van Parys (2020) D. Bertsimas and B. Van Parys. Sparse high-dimensional regression: Exact scalable algorithms and phase transitions. _Annals of Statistics_ , 48(1):300–323, 2020\.
* Bien et al. (2013) J. Bien, J. Taylor, and R. Tibshirani. A LASSO for hierarchical interactions. _Annals of Statistics_ , 41(3):1111, 2013.
* Bienstock (1996) D. Bienstock. Computational study of a family of mixed-integer quadratic programming problems. _Mathematical Programming_ , 74(2):121–140, 1996\.
* Ceria and Soares (1999) S. Ceria and J. Soares. Convex programming for disjunctive convex optimization. _Mathematical Programming_ , 86(3):595–614, 1999\.
* Combettes (2018) P. L. Combettes. Perspective functions: Properties, constructions, and examples. _Set-Valued and Variational Analysis_ , 26(2):247–264, 2018.
* Cozad et al. (2014) A. Cozad, N. V. Sahinidis, and D. C. Miller. Learning surrogate models for simulation-based optimization. _AIChE Journal_ , 60(6):2211–2227, 2014.
* Cozad et al. (2015) A. Cozad, N. V. Sahinidis, and D. C. Miller. A combined first-principles and data-driven approach to model building. _Computers & Chemical Engineering_, 73:116–127, 2015\.
* Dantzig and Eaves (1973) G. B. Dantzig and B. C. Eaves. Fourier-Motzkin elimination and its dual. _Journal of Combinatorial Theory_ , 14(3):288–297, 1973.
* Deza and Atamtürk (2022) A. Deza and A. Atamtürk. Safe screening for logistic regression with $\ell_{0}$-$\ell_{2}$ regularization. _arXiv:2202.00467_ , 2022.
* Frangioni and Gentile (2006) A. Frangioni and C. Gentile. Perspective cuts for a class of convex 0–1 mixed integer programs. _Mathematical Programming_ , 106(2):225–236, 2006\.
* Frangioni et al. (2020) A. Frangioni, C. Gentile, and J. Hungerford. Decompositions of semidefinite matrices and the perspective reformulation of nonseparable quadratic programs. _Mathematics of Operations Research_ , 45(1):15–33, 2020.
* Gómez (2021) A. Gómez. Outlier detection in time series via mixed-integer conic quadratic optimization. _SIAM Journal on Optimization_ , 31(3):1897–1925, 2021.
* Günlük and Linderoth (2010) O. Günlük and J. Linderoth. Perspective reformulations of mixed integer nonlinear programs with indicator variables. _Mathematical Programming_ , 124(1):183–205, 2010\.
* Han and Gómez (2021) S. Han and A. Gómez. Compact extended formulations for low-rank functions with indicator variables. _arXiv:2110.14884_ , 2021.
* Hastie et al. (2015) T. Hastie, R. Tibshirani, and M. Wainwright. _Statistical learning with sparsity: the lasso and generalizations_. CRC Press, 2015.
* Hazimeh and Mazumder (2020a) H. Hazimeh and R. Mazumder. Fast best subset selection: Coordinate descent and local combinatorial optimization algorithms. _Operations Research_ , 68(5):1517–1537, 2020a.
* Hazimeh and Mazumder (2020b) H. Hazimeh and R. Mazumder. Learning hierarchical interactions at scale: A convex optimization approach. In _International Conference on Artificial Intelligence and Statistics_ , pages 1833–1843, 2020b.
* Hazimeh et al. (2023) H. Hazimeh, R. Mazumder, and P. Radchenko. Grouped variable selection with discrete optimization: Computational and statistical perspectives. _Annals of Statistics_ , 51(1):1–32, 2023.
* Hazimeh et al. (2021) H. Hazimeh, R. Mazumder, and A. Saab. Sparse regression at scale: Branch-and-bound rooted in first-order optimization. _Mathematical Programming_ , pages 1–42, 2021.
* Heller and Tompkins (1956) I. Heller and C. B. Tompkins. An extension of a theorem of Dantzig’s. In H. W. Kuhn and A. W. Tucker, editors, _Linear Inequalities and Related Systems_ , pages 247–254. Princeton University Press, 1956.
* Hiriart-Urruty and Lemaréchal (2004) J.-B. Hiriart-Urruty and C. Lemaréchal. _Fundamentals of Convex Analysis_. Springer, 2004.
* Huang et al. (2012) J. Huang, P. Breheny, and S. Ma. A selective review of group selection in high-dimensional models. _Statistical Science_ , 27(4), 2012.
* Jeon et al. (2017) H. Jeon, J. Linderoth, and A. Miller. Quadratic cone cutting surfaces for quadratic programs with on–off constraints. _Discrete Optimization_ , 24:32–50, 2017.
* Kucukyavuz et al. (2020) S. Kucukyavuz, A. Shojaie, H. Manzour, L. Wei, and H.-H. Wu. Consistent second-order conic integer programming for learning Bayesian networks. _arXiv:2005.14346_ , 2020.
* Liu et al. (2022) P. Liu, S. Fattahi, A. Gómez, and S. Küçükyavuz. A graph-based decomposition method for convex quadratic optimization with indicators. _Mathematical Programming (Forthcoming)_ , 2022.
* Lubin and Dunning (2015) M. Lubin and I. Dunning. Computing in operations research using Julia. _INFORMS Journal on Computing_ , 27(2):238–248, 2015.
* Manzour et al. (2021) H. Manzour, S. Küçükyavuz, H.-H. Wu, and A. Shojaie. Integer programming for learning directed acyclic graphs from continuous data. _INFORMS Journal on Optimization_ , 3(1):46–73, 2021.
* Natarajan (1995) B. K. Natarajan. Sparse approximate solutions to linear systems. _SIAM Journal on Computing_ , 24(2):227–234, 1995\.
* Ramachandra et al. (2021) A. Ramachandra, N. Rujeerapaiboon, and M. Sim. Robust conic satisficing. _arXiv:2107.06714_ , 2021.
* Rockafellar (1970) R. T. Rockafellar. _Convex Analysis_. Princeton University Press, 1970.
* Rudin and Ustun (2018) C. Rudin and B. Ustun. Optimized scoring systems: Toward trust in machine learning for healthcare and criminal justice. _Interfaces_ , 48(5):449–466, 2018.
* Sahinidis (1996) N. V. Sahinidis. BARON: A general purpose global optimization software package. _Journal of Global Optimization_ , 8:201–205, 1996.
* Sahinidis (accessed November 13, 2023) N. V. Sahinidis. BARON user manual v. 2023.11.10. https://minlp.com/downloads/docs/baron%20manual.pdf, accessed November 13, 2023.
* Tibshirani (1996) R. Tibshirani. Regression shrinkage and selection via the lasso. _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 58(1):267–288, 1996.
* Wei et al. (2022a) L. Wei, A. Atamtürk, A. Gómez, and S. Küçükyavuz. On the convex hull of convex quadratic optimization problems with indicators. _arXiv:2201.00387_ , 2022a.
* Wei et al. (2020) L. Wei, A. Gómez, and S. Küçükyavuz. On the convexification of constrained quadratic optimization problems with indicator variables. In _International Conference on Integer Programming and Combinatorial Optimization_ , pages 433–447, 2020.
* Wei et al. (2022b) L. Wei, A. Gómez, and S. Küçükyavuz. Ideal formulations for constrained convex optimization problems with indicator variables. _Mathematical Programming_ , 192(1):57–88, 2022b.
* Wolsey (1989) L. A. Wolsey. Submodularity and valid inequalities in capacitated fixed charge networks. _Operations Research Letters_ , 8(3):119–124, 1989.
* Xie and Deng (2020) W. Xie and X. Deng. Scalable algorithms for the sparse ridge regression. _SIAM Journal on Optimization_ , 30(4):3359–3386, 2020.
## A Additional Proofs
###### Proof
_of Lemma 4 _ As for assertion (i), let $\overline{\cal
U}\\!:=\\!{\mathbb{R}}\times{\cal U}$ and $\overline{\cal
V}:=\\{(\tau,{\bm{\mu}},{\bm{t}}\hskip
1.00006pt):{\bm{1}}^{\top}{\bm{t}}=\tau\\}$. By construction, ${\cal V}$ is
the projection of $\overline{\cal U}\cap\overline{\cal V}$ onto
$(\tau,{\bm{\mu}})-$space. Hence, $\operatorname{conv}({\cal
V})=\operatorname{conv}(\operatorname{Proj}_{\tau,{\bm{\mu}}}(\overline{\cal
U}\cap\overline{\cal
V}))=\operatorname{Proj}_{\tau,{\bm{\mu}}}(\operatorname{conv}(\overline{\cal
U}\cap\overline{\cal V}))$. Therefore, in the remainder of the proof we
characterize convex hull of $\overline{\cal U}\cap\overline{\cal V}$. As
$\overline{\cal V}$ is convex, to prove $\operatorname{conv}(\overline{\cal
U}\cap\overline{\cal V})=\operatorname{conv}(\overline{\cal
U})\cap\overline{\cal V}$ it suffices to show that
$\operatorname{conv}(\overline{\cal U})\cap\overline{\cal
V}\subseteq\operatorname{conv}(\overline{\cal U}\cap\overline{\cal V})$.
Take a point $(\bar{\tau},\bar{\bm{\mu}},\bar{\bm{t}}\hskip 0.43057pt)$ from
$\operatorname{conv}(\overline{\cal U})\cap\overline{\cal V}$. Since
$(\bar{\tau},\bar{\bm{\mu}},\bar{\bm{t}}\hskip 0.43057pt)\in\overline{\cal
V}$, we have $\bar{\tau}={\bm{1}}^{\top}\bar{\bm{t}}$. On the other hand,
since $(\bar{\tau},\bar{\bm{\mu}},\bar{\bm{t}}\hskip
0.43057pt)\in\operatorname{conv}(\overline{\cal U})$, we can always express it
as a convex combination of a finite number of points in $\overline{\cal U}$,
that is, $(\bar{\tau},\bar{\bm{\mu}},\bar{\bm{t}}\hskip
0.43057pt)=\sum_{k\in[q]}\lambda_{k}(\tau^{k},{\bm{\mu}}^{k},{\bm{t}}^{k})$
for some finite $q$, $(\tau^{k},{\bm{\mu}}^{k},{\bm{t}}^{k})\in\overline{\cal
U}$, and $\lambda_{k}>0$ for all $k\in[q]$ satisfy
$\sum_{k\in[q]}\lambda_{k}=1$. Notice that there is no restriction on $\tau$
in the definition of $\overline{\cal U}$. Hence, we can take
$\tau^{k}:={\bm{1}}^{\top}{\bm{t}}^{k}$ for all $k\in[q]$. Therefore, we have
$(\tau^{k},{\bm{\mu}}^{k},{\bm{t}}^{k})\in\overline{\cal V}$ for all
$k\in[q]$. We thus deduce
$(\tau^{k},{\bm{\mu}}^{k},{\bm{t}}^{k})\in\overline{\cal U}\cap\overline{\cal
V}$ for all $k\in[q]$. Moreover, because
$\bar{\bm{t}}=\sum_{k\in[q]}\lambda_{k}{\bm{t}}^{k}$, we have
$\sum_{k\in[q]}\lambda_{k}\tau^{k}=\sum_{k\in[q]}\lambda_{k}({\bm{1}}^{\top}t^{k})={\bm{1}}^{\top}(\sum_{k\in[q]}\lambda_{k}t^{k})={\bm{1}}^{\top}\bar{\bm{t}}=\bar{\tau}$.
Hence, this proves that any point
$(\bar{\tau},\bar{\bm{\mu}},\bar{\bm{t}}\hskip 0.43057pt)$ can be written as a
convex combination of points from $\overline{\cal U}\cap\overline{\cal V}$ as
desired. Therefore, $\operatorname{conv}(\overline{\cal U}\cap\overline{\cal
V})=\operatorname{conv}(\overline{\cal U})\cap\overline{\cal V}$, and the
claim follows by projecting onto $(\tau,{\bm{\mu}})-$space.
As for assertion (ii), note that the set ${\cal V}$ is the affine
transformation of the convex set $\widetilde{\cal U}=\\{\bm{\eta}\in{\cal
U}:{\bm{B}}\bm{\eta}=\bm{b}\\}$, that is, ${\cal V}={\bm{A}}\,\widetilde{\cal
U}$. We then have
$\displaystyle\operatorname{cl}({\cal
V})=\operatorname{cl}(\operatorname{rint}({\cal
V}))=\operatorname{cl}({\bm{A}}(\operatorname{rint}(\widetilde{\cal U})))$
$\displaystyle=\operatorname{cl}({\bm{A}}(\operatorname{rint}(\operatorname{cl}(\widetilde{\cal
U}))))$
$\displaystyle=\operatorname{cl}(\operatorname{rint}({\bm{A}}(\operatorname{cl}(\widetilde{\cal
U}))))=\operatorname{cl}({\bm{A}}(\operatorname{cl}(\widetilde{\cal U}))).$
The first equality holds since a convex set and its relative interior have the
same closure. The second equality holds as the linear transformation and the
relative interior operators are interchangeable for convex sets (see
(Rockafellar, 1970, Theorem 6.6)); thus, we have $\operatorname{rint}({\cal
V})=\operatorname{rint}({\bm{A}}\,\widetilde{\cal
U})={\bm{A}}(\operatorname{rint}(\widetilde{\cal U}))$. The third equality
holds as the relative interior of a convex set and the relative interior of
its closure are the same. The fourth equality follows from the convexity of
$\operatorname{cl}(\widetilde{\cal U})$, which allows us to interchange the
linear transformation and the relative interior operators. Finally, the last
inequality holds as the closure of the relative interior of a convex set
equals the closure of the set. Since we assumed that there exists a point
$\bm{\eta}^{\star}\in\operatorname{rint}({\cal U})$ satisfying the condition
${\bm{B}}\bm{\eta}^{\star}=\bm{b}$, we have $\operatorname{cl}(\widetilde{\cal
U})=\\{\bm{\eta}\in\operatorname{cl}({\cal U}):{\bm{B}}\bm{\eta}=\bm{b}\\}$
thanks to (Rockafellar, 1970, Corrolay 6.5.1). Thus, we showed that
$\displaystyle\operatorname{cl}({\cal
V})=\operatorname{cl}({\bm{A}}(\operatorname{cl}(\widetilde{\cal
V})))=\operatorname{cl}(\\{{\bm{\mu}}:\exists\bm{\eta}\in\operatorname{cl}({\cal
U})\text{ s.t. }{\bm{A}}\bm{\eta}={\bm{\mu}},{\bm{B}}\bm{\eta}=\bm{b}\\}).$
As we assumed that $\bm{\eta}=\bm{0}$ is the only
$\bm{\eta}\in\operatorname{rec}(\operatorname{cl}(\widetilde{\cal V}))$ with
${\bm{A}}\bm{\eta}=\bm{0}$, by (Rockafellar, 1970, Theorem 9.1), we have
$\operatorname{cl}({\bm{A}}(\operatorname{cl}(\widetilde{\cal
V})))\\!=\\!{\bm{A}}(\operatorname{cl}(\widetilde{\cal V}))$. This completes
the proof. ∎
###### Proof
_of Lemma 6 _ Note that the matrix
$\displaystyle{\bm{A}}=\begin{bmatrix}-1&~{}{\bm{1}}^{\top}\\\\[2.15277pt]
0&~{}{\bm{1}}^{\top}\end{bmatrix}$
is totally unimodular. Recall that if $A$ is totally unimodular, then the
matrix $[{\bm{A}}^{\top},~{}{\bm{I}},~{}-{\bm{I}}]^{\top}$ is also totally
unimodular. By definition,
$\Delta_{1}=\\{(w,{\bm{z}})\in\\{0,1\\}^{1+d}:~{}w\leq{\bm{1}}^{\top}{\bm{z}},~{}{\bm{1}}^{\top}{\bm{z}}\leq\kappa\\}$.
As the feasible set of $\Delta_{1}$ will be represented by the matrix
$[{\bm{A}}^{\top},~{}{\bm{I}},~{}-{\bm{I}}]^{\top}$ and an integer vector, the
set $\Delta_{1}$ is totally unimodular, and $\operatorname{conv}(\Delta_{1})$
is thus given by the continuous relaxation of $\Delta_{1}$. ∎
###### Proof
_of Lemma 7 _ Note that the matrix
$\displaystyle{\bm{A}}=\begin{bmatrix}1&-{\bm{1}}_{d-1}^{\top}&0\\\\[2.15277pt]
0&-{\bm{1}}_{d-1}^{\top}&1\end{bmatrix}$
is totally unimodular as every square submatrix of $A$ has determinant $0$,
$+1$, or $-1$. Moreover, it is easy to see that
$\displaystyle\Delta_{1}$
$\displaystyle\textstyle=\\{(w,{\bm{z}})\in\\{0,1\\}^{1+d}:~{}w\leq{\bm{1}}^{\top}{\bm{z}},~{}z_{d}\leq\sum_{i\in[d-1]}z_{i}\\}$
$\displaystyle\textstyle=\\{(w,{\bm{z}})\in\\{0,1\\}^{1+d}:~{}w\leq\sum_{i\in[d-1]}z_{i},~{}z_{d}\leq\sum_{i\in[d-1]}z_{i}\\},$
which implies that the feasible set of $\Delta_{1}$ can be represented by the
matrix $[{\bm{A}}^{\top},~{}{\bm{I}},~{}-{\bm{I}}]^{\top}$ and an integer
vector. Thus, the new representation of $\Delta_{1}$ is totally unimodular,
and $\operatorname{conv}(\Delta_{1})$ is given by its continuous relaxation. ∎
###### Proof
_of Lemma 8 _ The set $\Delta_{1}$ can be written as
$\displaystyle\Delta_{1}$
$\displaystyle=\left(\left\\{0\right\\}\times\left\\{\bm{0}\right\\}\right)\cup\left(\left\\{0,1\right\\}\times{\cal
Z}\backslash\left\\{\bm{0}\right\\}\right).$
Since $\operatorname{conv}(A\times
B)=\operatorname{conv}(A)\times\operatorname{conv}(B)$ and
$\operatorname{conv}(A\cup
B)=\operatorname{conv}(\operatorname{conv}(A)\cup\operatorname{conv}(B))$, we
arrive at
$\displaystyle\operatorname{conv}(\Delta_{1})$
$\displaystyle=\operatorname{conv}\left(\left(\left\\{0\right\\}\times\left\\{\bm{0}\right\\}\right)\cup\left([0,1]\times\operatorname{conv}({\cal
Z}\backslash\\{\bm{0}\\})\right)\right)$
$\displaystyle=\left\\{(w,{\bm{z}})\in{\mathbb{R}}^{1+d}:\exists\lambda\in[0,1]\text{~{}s.t.~{}}\begin{array}[]{l}{\bm{F}}^{0}{\bm{z}}\geq\bm{0},~{}0\leq
w\leq\lambda\\\\[4.30554pt]
{\bm{z}}^{\top}\bm{f}^{+}_{k}\geq\lambda,~{}\forall k\in{\cal
K},\\\\[4.30554pt] {\bm{z}}^{\top}\bm{f}_{l}^{-}\leq\lambda,~{}\forall
l\in{\cal L}\end{array}\right\\}$
The proof concludes by projecting out the variable $\lambda$ using the
Fourier-Motzkin elimination approach. ∎
###### Proof
_of Lemma 9 _ By (Wei et al., 2022b, Lemma 3), we have
$\displaystyle\textstyle\operatorname{conv}({\cal
Z}\backslash\left\\{\bm{0}\right\\})=\left\\{{\bm{z}}\in[0,1]^{d}:~{}\begin{array}[]{l}1\leq\sum_{i\in[d-1]}z_{i}-(d-2)z_{d},\\\\[4.30554pt]
z_{d}\leq z_{i},~{}\forall i\in[d-1]\end{array}\right\\}.$ (A.3)
The proof concludes by applying Lemma 8. ∎
###### Proof
_of Theorem 3.3 _ Recall the definition of $\widetilde{\cal T}$ and
$\Delta_{p}$. By letting
$\displaystyle{\bm{\beta}}_{i}:=(s_{i},{\bm{x}}_{i}),\,\bm{\gamma}:={\bm{t}},\,{\bm{\delta}}:=({\bm{w}},{\bm{z}}),\,\Delta:=\Delta_{p},\,{\bm{C}}_{i}=[-1,{\bm{a}}_{i}^{\top}],\,{\mathbb{C}}_{i}=\left\\{0\right\\},\,\forall
i\in[p],$
we can represent the set $\widetilde{\cal T}$ as an instance of the set ${\cal
W}$ defined as in (11). Then, Proposition 2 yields
$\displaystyle\operatorname{cl}\operatorname{conv}(\widetilde{\cal
T})\\!=\\!\left\\{({\bm{x}},{\bm{z}},{\bm{s}},{\bm{w}},{\bm{t}})\\!\in\\!{\mathbb{R}}^{d+d+p+p+p}:\begin{array}[]{l}h^{\pi}(s_{i},w_{i})\leq
t_{i},~{}\forall i\in[p]\\\\[4.30554pt]
{\bm{a}}_{i}^{\top}{\bm{x}}_{i}=s_{i},~{}\forall i\in[p]\\\\[4.30554pt]
({\bm{w}},{\bm{z}})\in\operatorname{conv}(\Delta_{p})\end{array}\right\\}.$
In the following we characterize $\operatorname{cl}\operatorname{conv}({\cal
T})$ in terms of $\operatorname{cl}\operatorname{conv}(\widetilde{\cal T})$.
Let ${\cal
T}^{\prime}:=\\{(\tau,{\bm{x}},{\bm{z}}):\exists{\bm{s}},{\bm{w}},{\bm{t}}\text{~{}s.t.~{}}({\bm{x}},{\bm{z}},{\bm{s}},{\bm{w}}.{\bm{t}})\in\widetilde{\cal
T},{\bm{1}}^{\top}{\bm{t}}=\tau\\}$. By Proposition 3, we have
$\displaystyle\operatorname{cl}\operatorname{conv}({\cal
T})=\operatorname{cl}\big{(}\operatorname{conv}(\operatorname{Proj}_{\tau,{\bm{x}},{\bm{z}}}({\cal
T}^{\prime}))\big{)}=\operatorname{cl}\big{(}\operatorname{Proj}_{\tau,{\bm{x}},{\bm{z}}}(\operatorname{conv}({\cal
T}^{\prime}))\big{)}.$
Moreover, applying Lemma 4 (i) yields
$\operatorname{conv}({\cal
T}^{\prime})\\!=\\!\\{(\tau,{\bm{x}},{\bm{z}}):\exists{\bm{s}},{\bm{w}},{\bm{t}}\text{
s.t.
}({\bm{x}},{\bm{z}},{\bm{s}},{\bm{w}},{\bm{t}})\in\operatorname{conv}(\widetilde{\cal
T}),\,{\bm{1}}^{\top}{\bm{t}}=\tau\\}.$
By letting
$\displaystyle\begin{array}[]{c}{\bm{\mu}}:=(\tau,{\bm{x}},{\bm{z}}),~{}\bm{\eta}:=(\tau,{\bm{x}},{\bm{z}},{\bm{s}},{\bm{w}},{\bm{t}}),~{}{\cal
U}:={\mathbb{R}}\times\operatorname{conv}(\widetilde{\cal T}),\\\\[4.30554pt]
{\bm{A}}:=[{\bm{I}}_{1+2d},~{}\bm{0}],~{}{\bm{B}}:=(-1,\bm{0},\bm{0},\bm{0},\bm{0},{\bm{1}}_{p})^{\top},~{}\bm{b}:=0,\end{array}$
we observe that
$\operatorname{Proj}_{\tau,{\bm{x}},{\bm{z}}}(\operatorname{conv}({\cal
T}^{\prime}))=\\{{\bm{\mu}}:\exists\bm{\eta}\in{\cal U}\text{ s.t.
}{\bm{A}}\bm{\eta}={\bm{\mu}},~{}{\bm{B}}\bm{\eta}=\bm{b}\\}$ as in Lemma 4
(ii). Moreover, the first requirement of Lemma 4 (ii) is trivially satisfied
as the variable $\tau$ is free in the set ${\cal U}$. In addition, we have the
set
$\displaystyle\left\\{\bm{\eta}\in\operatorname{cl}({\cal
U}):~{}{\bm{A}}\bm{\eta}=\bm{0},~{}{\bm{B}}\bm{\eta}=\bm{b}\right\\}$
$\displaystyle=$
$\displaystyle\left\\{(0,\bm{0},\bm{0},{\bm{s}},{\bm{w}},{\bm{t}}):~{}\begin{array}[]{l}({\bm{w}},\bm{0})\in\operatorname{conv}(\Delta_{p}),~{}{\bm{1}}^{\top}{\bm{t}}=0\\\\[4.30554pt]
s_{i}={\bm{a}}_{i}^{\top}{\bm{x}}_{i},\,h^{\pi}(s_{i},w_{i})\leq
t_{i},\,\forall i\in[p]\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{(0,\bm{0},\bm{0},{\bm{s}},{\bm{w}},{\bm{t}}):~{}{\bm{w}}=\bm{0},~{}{\bm{s}}=\bm{0},~{}h^{\pi}(0,0)\leq
t_{i},\,\forall
i\in[p],~{}{\bm{1}}^{\top}{\bm{t}}=0\right\\}=\left\\{\bm{0}\right\\},$
where the first equation holds by the definition of
$\operatorname{cl}\operatorname{conv}(\widetilde{\cal T})$ and the fact that
${\bm{A}}\bm{\eta}=\bm{0}$ implies
$(\tau,{\bm{x}},{\bm{z}})=(0,\bm{0},\bm{0})$, and the second equation holds
because, by definition of $\Delta_{p}$ we have $0\leq
w_{i}\leq{\bm{1}}^{\top}{\bm{z}}_{i}$ which enforces $w_{i}=0$ as
${\bm{z}}_{i}=\bm{0}$ for all $i\in[p]$, and the last equation holds as the
function $h$ satisfies $h(0)=0$, which along with ${\bm{1}}^{\top}{\bm{t}}=0$
implies ${\bm{t}}=\bm{0}$. Thus, we can apply Lemma 4 (ii) to conclude that
$\displaystyle\operatorname{cl}\operatorname{conv}({\cal
T})=\\{(\tau,{\bm{x}},{\bm{z}}):\exists{\bm{s}},{\bm{w}},{\bm{t}}\text{ s.t.
}({\bm{x}},{\bm{z}},{\bm{s}},{\bm{w}},{\bm{t}})\in\operatorname{cl}\operatorname{conv}(\widetilde{\cal
T}),~{}{\bm{1}}^{\top}{\bm{t}}=\tau\\}.$
The proof concludes by projecting out the variable ${\bm{t}}$ using the
Fourier-Motzkin elimination approach. ∎
The convex hull of $\Delta_{p}$ relies on the set
${\cal
Z}_{i}:=\\{{\bm{z}}_{i}\in{\mathbb{R}}^{d_{i}}:\exists\tilde{\bm{z}}\in{\cal
Z}~{}\text{s.t.}~{}\tilde{\bm{z}}_{i}={\bm{z}}_{i},~{}\tilde{\bm{z}}_{j}=\bm{0},\forall
j\neq i\\}$
for every $i\in[p]$. In particular, given any $i\in[p]$, we assume that
$\displaystyle\operatorname{conv}({\cal
Z}_{i}\backslash\left\\{\bm{0}\right\\})=\left\\{{\bm{z}}_{i}\in{\mathbb{R}}^{d_{i}}:{\bm{F}}_{i}^{0}{\bm{z}}_{i}\geq\bm{0},~{}\begin{array}[]{l}{\bm{z}}_{i}^{\top}\bm{f}^{+}_{ik}\geq
1,~{}\forall k\in{\cal K}_{i},\\\\[4.30554pt]
{\bm{z}}_{i}^{\top}\bm{f}_{il}^{-}\leq 1,~{}\forall l\in{\cal
L}_{i}\end{array}\right\\}.$ (A.6)
###### Lemma A.1
Given the representation of $\operatorname{conv}({\cal
Z}_{i}\backslash\left\\{\bm{0}\right\\})$ as in (A.6) for any $i\in[p]$, then
$\displaystyle\operatorname{conv}(\Delta_{p})=\left\\{({\bm{w}},{\bm{z}})\in{\mathbb{R}}^{p+d}:\begin{array}[]{l}\bm{0}\leq{\bm{w}}\leq\bm{1},~{}{\bm{1}}^{\top}{\bm{w}}\leq
1,~{}{\bm{F}}_{i}^{0}{\bm{z}}_{i}\geq\bm{0},~{}\forall i\in[p],\\\\[4.30554pt]
{\bm{z}}_{i}^{\top}\bm{f}^{+}_{ik}\geq w_{i},~{}\forall i\in[p],\forall
k\in{\cal K}_{i},\\\\[4.30554pt] {\bm{z}}_{i}^{\top}\bm{f}_{il}^{-}\leq
1,~{}\forall i\in[p],\forall l\in{\cal L}_{i},\\\\[4.30554pt]
{\bm{z}}_{i}^{\top}\bm{f}_{il}^{-}\leq{\bm{z}}_{i}^{\top}\bm{f}^{+}_{ik},~{}\forall
i\in[p],\forall k\in{\cal K},\forall l\in{\cal L}\end{array}\right\\}.$
###### Proof
_of Lemma A.1 _ The set $\Delta_{p}$ can be written as the union of $p+1$
sets. Namely,
$\displaystyle\Delta_{p}$
$\displaystyle=\left(\left\\{\bm{0}\right\\}\times\left\\{\bm{0}\right\\}\right)\bigcup\limits_{i\in[p]}\left(\left\\{\bm{0},{\bm{e}}_{i}\right\\}\times\left(\times_{k\in[i-1]}\left\\{\bm{0}\right\\}\times{\cal
Z}_{i}\backslash\left\\{\bm{0}\right\\}\times_{k\in\\{i+1,\dots,p\\}}\left\\{\bm{0}\right\\}\right)\right).$
Since $\operatorname{conv}(A\times
B)=\operatorname{conv}(A)\times\operatorname{conv}(B)$ and
$\operatorname{conv}(A\cup
B)=\operatorname{conv}(\operatorname{conv}(A)\cup\operatorname{conv}(B))$, we
arrive at
$\displaystyle\operatorname{conv}(\Delta_{p})=\left\\{({\bm{w}},{\bm{z}})\in{\mathbb{R}}^{p+d}:\exists\bm{\lambda}\in{\mathbb{R}}^{p}\text{~{}s.t.~{}}\begin{array}[]{l}\bm{0}\leq\bm{\lambda}\leq{\bm{1}},~{}{\bm{1}}^{\top}\bm{\lambda}\leq
1,~{}{\bm{w}}\leq\bm{\lambda},\\\\[4.30554pt]
{\bm{F}}_{i}^{0}{\bm{z}}_{i}\geq\bm{0},~{}\forall i\in[p]\\\\[4.30554pt]
{\bm{z}}_{i}^{\top}\bm{f}^{+}_{ik}\geq\lambda_{i},~{}\forall i\in[p],\forall
k\in{\cal K}_{i},\\\\[4.30554pt]
{\bm{z}}_{i}^{\top}\bm{f}_{il}^{-}\leq\lambda_{i},~{}\forall i\in[p],\forall
l\in{\cal L}_{i}\end{array}\right\\}$
The proof concludes by projecting out the variable $\bm{\lambda}$ using the
Fourier-Motzkin elimination approach. ∎
###### Proof
_of Lemma 13 _ Note that the matrix
$\displaystyle{\bm{A}}=\begin{bmatrix}{\bm{1}}^{\top}&~{}\phantom{-}\bm{0}^{\top}\\\
\\!\\!{\bm{I}}&~{}-{\bm{I}}\\\
\bm{0}^{\top}&\phantom{-}{\bm{1}}^{\top}\end{bmatrix}$
is totally unimodular. To see this, we partition the rows of ${\bm{A}}$ into
the two disjoint sets $\\{1\\}$ and $\\{2,\dots,d+2\\}$. As such, ${\bm{A}}$
is totally unimodular because every entry in ${\bm{A}}$ is $0$, $+1$, or $-1$,
every column of ${\bm{A}}$ contains at most two non-zero entries, and two
nonzero entries in a column of ${\bm{A}}$ with the same signs belong to either
$\\{1\\}$ or $\\{2,\dots,d+2\\}$, whereas two nonzero entries in a column of
${\bm{A}}$ with the opposite signs belong to $\\{2,\dots,d+2\\}$. Hence, by
(Heller and Tompkins, 1956, Theorem 2), the matrix ${\bm{A}}$ is totally
unimodular. As the feasible set of $\Omega$ is represented by the matrix
$[{\bm{A}}^{\top},~{}{\bm{I}},~{}-{\bm{I}}]^{\top}$ and an integer vector, the
set $\Omega$ is totally unimodular, and $\operatorname{conv}(\Omega)$ is given
by its continuous relaxation. ∎
We next provide ideal and lifted descriptions of $\operatorname{conv}(\Omega)$
for general ${\cal Z}$. Lemma A.2 (i) is particularly useful when the set
$\Omega\backslash\left\\{\bm{0}\right\\}$ is totally unimodular. In this case,
we can provide an ideal description for $\operatorname{conv}(\Omega)$. On the
other hand, Lemma A.2 (ii) is useful when the set ${\cal Z}$ is totally
unimodular.
###### Lemma A.2
The following holds.
1. (i)
Suppose that $\operatorname{conv}(\Omega\backslash\left\\{\bm{0}\right\\})$
admits the following ideal representation
$\displaystyle\operatorname{conv}(\Omega\backslash\left\\{\bm{0}\right\\})=\left\\{\bm{\delta}\in{\mathbb{R}}^{2d}:{\bm{F}}^{0}\bm{\delta}\geq\bm{0},~{}\begin{array}[]{l}\bm{\delta}^{\top}\bm{f}^{+}_{k}\geq
1,~{}\forall k\in{\cal K},\\\\[4.30554pt] \bm{\delta}^{\top}\bm{f}_{l}^{-}\leq
1,~{}\forall l\in{\cal L}\end{array}\right\\}.$
Then, we have
$\displaystyle\operatorname{conv}(\Omega)\\!=\\!\left\\{\bm{\delta}\in{\mathbb{R}}^{d+d}:{\bm{F}}^{0}\bm{\delta}\geq\bm{0},~{}\begin{array}[]{l}\bm{\delta}^{\top}\bm{f}^{+}_{k}\geq
0,~{}\forall k\in{\cal K},\\\\[4.30554pt] \bm{\delta}^{\top}\bm{f}_{l}^{-}\leq
1,~{}\forall l\in{\cal L},\\\\[4.30554pt]
\bm{\delta}^{\top}\bm{f}_{l}^{-}\leq\bm{\delta}^{\top}\bm{f}^{+}_{k},~{}\forall
k\in{\cal K},\forall l\in{\cal L}\end{array}\right\\}.$
2. (ii)
Let ${\cal O}_{i}=\\{{\bm{z}}\in{\mathbb{R}}^{d}:z_{i}=0\\}$. Then, we have
$\displaystyle\operatorname{conv}(\Omega)=\left\\{({\bm{w}},{\bm{z}})\\!:\\!\exists\tilde{\bm{z}}^{0},\dots,\tilde{\bm{z}}^{d}\text{
s.t.
}\begin{array}[]{l}{\bm{w}}\in{\mathbb{R}}_{+}^{d},~{}{\bm{1}}^{\top}{\bm{w}}\leq
1,\\\\[4.30554pt]
{\bm{z}}=\tilde{\bm{z}}^{0}+\sum_{i\in[d]}\tilde{\bm{z}}^{i}\\\\[4.30554pt]
\tilde{\bm{z}}^{0}\in(1-{\bm{1}}^{\top}{\bm{w}})\cdot\operatorname{conv}({\cal
Z})\\\\[4.30554pt] \tilde{\bm{z}}^{i}\in w_{i}\cdot\operatorname{conv}({\cal
Z}\backslash{\cal O}_{i}),~{}\forall i\in[d]\end{array}\right\\}.$
###### Proof
For case (i), notice that the set $\Omega$ can be written as the union of two
sets
$\Omega=\left(\left\\{\bm{0}\right\\}\times\left\\{\bm{0}\right\\}\right)\cup\left(\Omega\backslash\left\\{\bm{0}\right\\}\right)$.
We thus have
$\displaystyle\operatorname{conv}(\Omega)$
$\displaystyle=\operatorname{conv}\left(\left(\left\\{\bm{0}\right\\}\times\left\\{\bm{0}\right\\}\right)\cup\left(\operatorname{conv}(\Omega\backslash\\{\bm{0}\\})\right)\right)$
$\displaystyle=\left\\{\bm{\delta}\in{\mathbb{R}}^{d+d}:\exists\lambda\in[0,1]\text{~{}s.t.~{}}{\bm{F}}^{0}\bm{\delta}\geq\bm{0},~{}\begin{array}[]{l}\bm{\delta}^{\top}\bm{f}^{+}_{k}\geq\lambda,~{}\forall
k\in{\cal K},\\\\[4.30554pt]
\bm{\delta}^{\top}\bm{f}_{l}^{-}\leq\lambda,~{}\forall l\in{\cal
L}\end{array}\right\\}.$
The proof concludes by projecting out the variable $\lambda$ using the
Fourier-Motzkin elimination approach.
For case (ii), note that the set $\Omega$ can be written as the union of $d+1$
sets, that is, $\Omega=\left(\left\\{\bm{0}\right\\}\times{\cal
Z}\right)\bigcup\left(\cup_{i\in[d]}\left(\left\\{{\bm{e}}_{i}\right\\}\times({\cal
Z}\backslash{\cal O}_{i})\right)\right)$. We thus have
$\displaystyle\operatorname{conv}(\Omega)$ $\displaystyle=$
$\displaystyle\operatorname{conv}\left(\left(\left\\{\bm{0}\right\\}\times\operatorname{conv}({\cal
Z})\right)\bigcup\left(\cup_{i\in[d]}\left\\{{\bm{e}}_{i}\right\\}\times\operatorname{conv}({\cal
Z}\backslash{\cal O}_{i})\right)\right)$ $\displaystyle=$
$\displaystyle\left\\{({\bm{w}},{\bm{z}}):\begin{array}[]{l}\exists{\bm{w}}^{0},\dots,{\bm{w}}^{d}\\\
\exists{\bm{z}}^{0},\dots,{\bm{z}}^{d}\\\
\exists\bm{\lambda}\in{\mathbb{R}}_{+}^{d}\end{array}\text{ s.t.
}\begin{array}[]{l}{\bm{1}}^{\top}\bm{\lambda}\leq 1,\\\\[4.30554pt]
{\bm{w}}=(1-{\bm{1}}^{\top}\bm{\lambda}){\bm{w}}^{0}+\sum_{i\in[d]}\lambda_{i}{\bm{w}}^{i}\\\\[4.30554pt]
{\bm{z}}=(1-{\bm{1}}^{\top}\bm{\lambda}){\bm{z}}^{0}+\sum_{i\in[d]}\lambda_{i}{\bm{z}}^{i}\\\\[4.30554pt]
({\bm{w}}^{0},{\bm{z}}^{0})\in\left\\{\bm{0}\right\\}\times\operatorname{conv}({\cal
Z})\\\\[4.30554pt]
({\bm{w}}^{i},{\bm{z}}^{i})\in\left\\{{\bm{e}}_{i}\right\\}\times\operatorname{conv}({\cal
Z}\backslash{\cal O}_{i}),~{}\forall i\in[d]\end{array}\right\\}.$
The proof concludes by introducing the new variables
$\tilde{\bm{z}}^{0}=(1-{\bm{1}}^{\top}\bm{\lambda}){\bm{z}}^{0}$ and
$\tilde{\bm{z}}^{i}=\lambda_{i}{\bm{z}}^{i}$ for every $i\in[d]$, and then
projecting out the variable $\bm{\lambda}$ using the observation that
${\bm{w}}=\bm{\lambda}$. ∎
Armed with Lemma A.2, we are ready to provide characterizations for
$\operatorname{conv}(\Omega)$ in the cases of weak and strong hierarchy
constraints.
###### Proof
_of Lemma 14 _ First, note that
$\displaystyle\Omega\backslash\\{\bm{0}\\}$
$\displaystyle\textstyle=\\{({\bm{w}},{\bm{z}})\in\\{0,1\\}^{d+d}:~{}{\bm{w}}\leq{\bm{z}},~{}{\bm{1}}^{\top}{\bm{w}}\leq
1,~{}\sum_{i\in[d-1]}z_{i}\geq 1\\}.$
Based on this let us consider the matrix
$\displaystyle{\bm{A}}=\begin{bmatrix}{\bm{I}}_{d}\quad&-{\bm{I}}_{d}\\\\[2.15277pt]
{\bm{1}}_{d}^{\top}\quad&\bm{0}_{d}^{\top}\\\\[2.15277pt]
\bm{0}_{d}^{\top}\quad&(-{\bm{1}}_{d-1},0)^{\top}\end{bmatrix}.$
This matrix is totally unimodular as we can partition rows of ${\bm{A}}$ into
the subsets $[d]$ and $\\{d+1,d+2\\}$ such that they satisfy the requirements
of (Heller and Tompkins, 1956, Theorem 2). Thus, the set
$\Omega\backslash\\{\bm{0}\\}$ is represented by constraints based on the
matrix $[{\bm{A}}^{\top},~{}{\bm{I}},~{}-{\bm{I}}]^{\top}$ and an integer
vector. Thus, $\Omega\backslash\\{\bm{0}\\}$ admits a totally unimodular
representation, and $\operatorname{conv}(\Omega\backslash\\{\bm{0}\\})$ is
given by its continuous relaxation. The proof concludes by applying Lemma A.2
(i). ∎
###### Proof
_of Lemma 15 _ Note that ${\cal Z}$ is totally unimodular. Hence, the
continuous relaxation of ${\cal Z}$ gives $\operatorname{conv}({\cal Z})$. For
any $i\in[d-1]$, we also have ${\cal Z}\backslash{\cal
O}_{i}=\\{{\bm{z}}\in\\{0,1\\}^{d}:z_{i}=1,~{}z_{d}\leq z_{j},~{}\forall
j\in[d-1]\\}.$ It is easy to see that the continuous relaxation of ${\cal
Z}\backslash{\cal O}_{i}$ gives its convex hull. Besides, we have ${\cal
Z}\backslash{\cal O}_{d}=\\{\bm{1}\\}$. The proof follows by applying Lemma
A.2 (ii). ∎
## B Additional Numerical Results
Figure B.1: Comparison of different continuous relaxations for $N=100$,
$\pi=0.1$ as $p$ (and thus $d$) varies.
Figure B.2: Performance of the B&B algorithm for $N=100$, $\pi=0.1$ as $p$
(and thus $d$) varies.
Figure B.3: Comparison of different continuous relaxations for
$(p,N)\\!=\\!(50,100)$ as $\pi$ varies.
Figure B.4: Performance of the B&B algorithm for $(p,N)\\!=\\!(50,100)$ as
$\pi$ varies.
|
capbtabboxtable[][]
# Orthogonal calibration via posterior projections with applications to the
Schwarzschild model
Antik Chakraborty<EMAIL_ADDRESS>Department of Statistics, Purdue
University Jonelle L. Walsh<EMAIL_ADDRESS>George P. and Cynthia
Woods Mitchell Institute for Fundamental Physics and Astronomy, Department of
Physics and Astronomy, Texas A&M University Louis Strigari<EMAIL_ADDRESS>George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy, Texas A&M University Bani K.
Mallick<EMAIL_ADDRESS>Department of Statistics, Texas A&M University
Anirban Bhattacharya<EMAIL_ADDRESS>Department of Statistics, Texas
A&M University
###### Abstract
The orbital superposition method originally developed by Schwarzschild (1979)
is used to study the dynamics of growth of a black hole and its host galaxy,
and has uncovered new relationships between the galaxy’s global
characteristics. Scientists are specifically interested in finding optimal
parameter choices for this model that best match physical measurements along
with quantifying the uncertainty of such procedures. This renders a
statistical calibration problem with multivariate outcomes. In this article,
we develop a Bayesian method for calibration with _multivariate outcomes_
using orthogonal bias functions thus ensuring parameter identifiability. Our
approach is based on projecting the posterior to an appropriate space which
allows the user to choose any nonparametric prior on the bias function(s)
instead of having to model it (them) with Gaussian processes. We develop a
functional projection approach using the theory of Hilbert spaces. A finite-
dimensional analogue of the projection problem is also considered. We
illustrate the proposed approach using a BART prior and apply it to calibrate
the Schwarzschild model illustrating how a multivariate approach may resolve
discrepancies resulting from a univariate calibration.
Keywords: Bayesian; Computer model; Emulator; Scientific modeling; Regression
trees.
## 1 Introduction
Scientific endeavors typically aim to describe a natural physical phenomenon.
In modern applications, these descriptions increasingly involve complex
mathematical equations typically solved using expensive computer codes. For
example, the MIT2D climate model
(http://web.mit.edu/globalchange/www/climate.html) simulates probability
distributions of ocean, surface and upper atmospheric temperatures. In our
motivating application, Schwarzschild’s orbital integral method
(Schwarzschild, 1979) is used to simulate mass distributions of galaxies.
These orbit-based models are the main method for measuring the masses of
supermassive black holes at the centers of nearby galaxies (Kormendy and Ho,
2013), and have led to the establishment of empirical relationships between
the masses of black holes and large-scale galaxy properties, which
surprisingly suggest that black holes and their host galaxies grow and evolve
together over time (McConnell and Ma, 2013; Kormendy and Ho, 2013; Saglia et
al., 2016). Besides determining black hole masses, Schwarzschild modeling is a
powerful tool for measuring a galaxy’s mass-to-light ratio, dark matter halo
properties, stellar orbital distribution, viewing orientation, and intrinsic
three-dimensional shape, allowing for the further study of galaxy assembly
histories (Mehrgan et al., 2019). These scientific models involve parameters
which carry physical meaning. Inferring about the parameters while addressing
the uncertainty of the data observed from the the physical process was first
formulated as a statistical problem in Sacks et al. (1989), and was further
studied from a Bayesian point of view in Kennedy and O’Hagan (2001), who
popularized the term computer code calibration. Since then statistical
calibration methods have been used in many scientific disciplines; e.g.
climatology (Forest et al., 2008; Salter et al., 2018), cell biology (Xie and
Xu, 2021), mechanical engineering (Gattiker et al., 2006) etc.
Parallel to the widening applications of statistical calibration methods,
substantial effort has also been dedicated to the development of more robust
calibration methods within the statistics community, especially after the
seminal work of Kennedy and O’Hagan (2001). In their work, the authors
proposed a framework that simultaneously models the data observed from a
physical process and data generated from computer code implementations of
mathematical models emulating the physical process. The fundamental
contribution of their work was to include a discrepancy/bias function in their
model, which the authors interpreted as a unknown function that captures the
inability of the mathematical model to explain variations in the observed
data. A Bayesian nonparametric regression approach based on Gaussian processes
(Rasmussen and Williams, 2005) was then used to model the unknown bias
function as well as the computer code simulating the physical process. This
enabled fast emulation of the computer code simulators and produced rapid
estimates of input parameters of the computer code that best approximated the
observed data. Since then several authors have proposed statistical methods
for computer code calibration. Higdon et al. (2008) developed a basis vector
approach to calibration when the computer code output is very high-dimensional
using principal components; Bayarri et al. (2007) proposed a general framework
for Bayesian calibration where a modular Bayes approach was advocated to
overcome significant computational and inferential challenges associated with
a fully Bayesian analysis of calibration problems; Chakraborty et al. (2013)
modeled the computer code using multivariate adaptive regression splines
(Friedman, 1991; Denison et al., 1998) to avoid overfitting the computer
model; Pratola and Higdon (2016) proposed to model the bias and computer model
using Bayesian additive regression trees (BART; Chipman et al. (2010)) which
has gained immense popularity in the machine learning community recently. Tuo
and Wu (2015, 2016) studied the general calibration model of Kennedy and
O’Hagan (2001) from a theoretical point-of-view and showed that the
calibration parameters are not identifiable. As a remedy, Tuo and Wu (2015)
defined the calibration parameter as the point in the input parameter space
that minimizes the $L^{2}$ loss function between the physical process and the
computer code. Plumlee (2017) extended it to more general losses and then
proposed a modified covariance function for Gaussian process priors used for
the bias function; Xie and Xu (2021) treated the calibration parameter as a
functional of the bias function and used a projected Bayesian approach to
avoid identifiability issues.
In this article, we propose a generalized Bayesian approach for computer code
calibration with multivariate output which takes into account identifiability
issues pointed out by Tuo and Wu (2016). We start from the definition of the
optimal calibration parameter as the point in the input space of the computer
code that minimizes a certain loss function. Following Plumlee (2017), this
identifiability issue can be mitigated by imposing a orthogonality constraint
on the bias function with respect to the derivatives of the computer code. We
first generalize the restrictions on the bias function to the situation where
there are multiple potentially correlated outcomes. In the univariate outcome
case, Plumlee (2017) developed a modified Gaussian process prior on the bias
function that respects the orthogonality constraint. However, as indicated by
other authors (Pratola and Higdon, 2016) Gaussian process priors often suffer
from scalability issues to big data sets and high-dimensional input spaces.
The scalability issues are only exacerbated when one has to deal with
multivariate covariance functions. On the other hand, while nonparametric
priors such as the BART are highly scalable, introducing a priori constraints
such as the orthogonality constraint indicated above is not immediate. To
overcome these issues, we propose to posit a unconstrained prior on the bias
function and project the obtained posterior on the relevant space. For
inference based on the projected posterior we develop two algorithms for
posterior sampling. The first one is essentially a Hilbert projection on the
space of square-integrable functions and the second one borrows inspiration
from a projection interpretation of the multivariate Gaussian distribution.
Numerical experiments in Section 5 show superiority of the proposed method
both in terms of computational scalability and modeling flexibility. Indeed,
the method provides a more general framework for orthogonal calibration
whereby practitioners are free to choose a nonparametric prior of their own
choice; for example deep Gaussian process priors used recently in Marmin and
Filippone (2022). We also note that our approach is significantly different
from a related projection based method by Xie and Xu (2021) where the authors
consider a Gaussian process prior on the bias function. The calibration
parameter, defined as the minimizer of an appropriate loss function between
the observed data and the computer model is then treated as a functional of
the bias function. Hence, a prior is induced on the calibration parameter
through the Gaussian process prior on the bias function. Although this serves
the statistical issue of identifiability, properties of the induced prior is
less understood.
The rest of the paper is organized as follows. In Section 2, we briefly
describe our motivating example which is then followed by the development of
the orthogonal posterior projection approach in Section 3. We then compare the
efficiency of the proposed method with other state-of-the-art methods in terms
of estimating the calibration parameter and computational scalability. We end
with an application on the computer code data from Schwazschild model. An R
package for implementing the proposed method is available to download from
https://github.com/antik015/OCal.
## 2 Schwarzschild’s model
The study of the mass distribution within galaxies is central in the quest of
understanding of black holes, stellar components, dark matter, and the growth
of galaxies. Schwarzschild (1979) presented a orbit superposition method for
constructing a self-consistent mass model of galaxies. It consists of
“integrating a representative set of orbits in a static triaxial gravitational
potential, and finding weights for these orbits such that their superposition
reproduces the assumed mass distribution” (Quenneville et al., 2021). Several
modifications to the initial method have since been proposed among which van
den Bosch et al. (2008)’s triaxial orbit superposition has become very
popular. In standard applications of the method, the model is compared with
kinematic and photometric data to determine best-fit parameters such as black
hole mass $(\theta_{1})$, stellar mass-to-light ratio $(\theta_{2})$, and
fraction of dark matter halo $(\theta_{2})$.
(a) Left: $v_{S}$; Right: $v_{F}$
(b) Left: $\tau_{S}$; Right: $\tau_{F}$
(c) Left: $h_{3,S}$; Right: $h_{3,F}$
(d) Left: $h_{4,S}$; Right: $h_{4,F}$
Figure 1: Orbital superposition output versus observed data on four moments of
the velocity distribution for
$\theta=(10.22,10,1)^{\mathrm{\scriptscriptstyle{T}}}$. The superposition
method is implemented by a code originally developed by van den Bosch et al.
(2008).
In it’s current version (van den Bosch et al., 2008), for a given input
parameter combination
$t=(t_{1},t_{2},t_{3})^{\mathrm{\scriptscriptstyle{T}}}$, the code outputs the
first four moments of the line-of-sight velocity distribution
$f_{S}(x;t)=(v_{S}(x;t),\tau_{S}(x;t),h_{3,S}(x;t),h_{4,S}(x;t))^{\mathrm{\scriptscriptstyle{T}}}$
for points $x$ in a 2-dimensional spatial grid $\mathcal{X}$. The points $x$
are typically chosen to match the locations of the physical photometric data,
providing further information about the mass distribution. Specifically,
physical data
$y_{F}(x_{i})=(v_{F}(x_{i}),\tau_{F}(x_{i}),h_{3,F}(x_{i}),h_{4,F}(x_{i}))^{\mathrm{\scriptscriptstyle{T}}}$
is available for $i=1,\ldots,n,\,n=105$ locations of $\mathcal{X}=[-1,1]^{2}$.
In Figure 1, color coded output from the code and the observed data are shown
when $t=(10.22,10,1)^{\mathrm{\scriptscriptstyle{T}}}$ at the $n$ locations.
This superposition technique is implemented by a computer code (FORTRAN) which
is extremely computationally intensive; e.g. for one input of $\theta$,
approximately 3 hours is needed to generate the output. The code output is
available for three different values of $t_{3}=1,2,3$, and for each value
$t_{3}$, the output is available over a two-dimensional grid of values for
$(t_{1},t_{2})$ where $t_{1}$ ranges from 10.22 to 10.31 and $t_{2}$ ranges
from 10 to 10.25. Overall, the computer code output is available for $N=476$
different values of $t$. Naturally, an exhaustive search over the input
parameter space that compares the code output to the observed data to find the
best-fit parameter combination is prohibitive. However, from limited
experiments performed separately on each of the 4 outputs of the model,
scientists have seen that for the point $\tilde{\theta}=(9.92,9.29,1.27)$ the
model outputs best approximated the observed data up to slight variations for
each of the outcome. It is, however, unclear how a joint analysis of the model
outputs and the physical data can be carried out that adequately addresses the
uncertainty in the observed data, and whether such an analysis would reveal
regions in the parameter space that perform relatively well. This is important
for designing future experiments where scientists plan to include other
parameters that could potentially lead to better predictions of the physical
process on previously unobserved points - a task typically suited for a
multivariate analysis when outcomes are correlated. Inspired by this
application, in Section 3, we first develop a general methodology for
orthogonal calibration of multivariate computer models where the key focus is
on parameter identifiability and modeling flexibility. The validity of the
proposed methodology is then investigated in a series of numerical experiments
in Section 5, followed by an application on the Schwarzschild’s model in
Section 6.
## 3 Method
### 3.1 Orthogonal calibration
We begin with a quick review of orthogonal calibration, and introduce notation
necessary to develop our method. Suppose we are interested in a real physical
system $y_{R}(\cdot)\in\mathbb{R}^{q},\,q\geq 1$ with controllable inputs
$x\in\mathcal{X}$ where $\mathcal{X}$ is some closed, bounded subset of
$\mathbb{R}^{d}$. We do not directly observe the system. Instead, we observe
stochastic field observations $y_{F}(\cdot)$ at $n$ input points called the
design $\mathcal{D}=\\{x_{1},\ldots,x_{n}\\}$, for which a natural stochastic
model is
$y_{F}(x_{i})=y_{R}(x_{i})+\epsilon(x_{i}),\,\epsilon(x_{i})\overset{ind.}{\sim}{\mathrm{N}}_{q}(0,\Sigma_{F}),\,i=1,\ldots,n.$
(1)
Suppose in addition to the field observations, we also have a simulator
$f:\mathcal{X}\times\Theta\to\mathbb{R}^{q}$ for the physical system
(depending on the controllable input $x$ as well as other parameters
$t\in\Theta\subset\mathbb{R}^{p}$) which typically comes in the form of a
computer code/model. The parameter $t$ represents our understanding of the
physical system, and we hope that for some unknown $\theta^{*}\in\Theta$, the
computer model closely approximates the physical process $y_{R}(\cdot)$.
Formally, suppose $L\\{y_{R}(\cdot),f(\cdot,\cdot)\\}$ is a loss function that
can distinguish between the best possible parameter $\theta^{*}$ and all other
parameter values $t$. If there exists a unique $t^{*}$ such that
$y_{R}(x)=f(x,t^{*})$ for all $x\in\mathcal{X}$ and the loss $L$ is strictly
proper following Gneiting and Raftery (2007), then $\theta^{*}=t^{*}$.
However, existence of such a $t^{*}$ is not always guaranteed for most
practical computer models and loss functions. In the absence of such a
parameter, the computer model is biased and $\theta^{*}$ is defined as the
parameter combination at which the loss $L\\{y_{R}(\cdot),f(\cdot,\cdot)\\}$
is minimized , i.e.
$\theta^{*}=\operatorname*{arg\,min}_{t\in\Theta}L\\{y_{R}(\cdot),f(\cdot,t)\\}$.
To account for such bias, Kennedy and O’Hagan (2001) proposed the following
model for the observed data on the system
$y_{F}(x_{i})=f(x_{i},\theta)+b_{\theta}(x_{i})+\epsilon(x_{i}),\,\epsilon(x_{i})\overset{ind.}{\sim}{\mathrm{N}}(0,\Sigma_{F}),\,i=1,\ldots,n,$
(2)
where $\theta$ represents the model parameter that targets the population
parameter $\theta^{*}$ defined above, and $b_{\theta}(x)$ is interpreted as
the discrepancy or bias between the physical process and the assumed computer
model, i.e. $b_{\theta}(x)=y_{R}(x)-f(x,\theta)$. Without added restrictions
in (2), $\theta$ is not identifiable in (2). We provide a sufficient condition
for identifiablity under the loss
$L\\{y_{R}(\cdot),f(\cdot,t)\\}=\sum_{k=1}^{q}\int_{\mathcal{X}}\\{y_{R,k}(x)-f_{k}(x,t)\\}^{2}dx,$
(3)
in the following proposition, extending the result of Plumlee (2017) to
multivariate outcomes.
###### Proposition 3.1.
Consider the loss $L\\{y_{R}(\cdot),f(\cdot,t)\\}$ defined above. Let
$g_{j,k}(x,t)=\frac{\partial}{\partial t_{j}}f_{k}(x,t_{j})$, the partial
derivative with respect to $t_{j}$ of the $k$-th component $f_{k}(x,t)$ of the
computer model $f(x,t)$, which is assumed to exist for all $j,k$. A sufficient
condition for $\frac{\partial}{\partial
t}L\\{y_{R}(\cdot),f(\cdot,t)\\}\rvert_{t=\theta^{*}}=0$ is
$\int_{\mathcal{X}}\sum_{k=1}^{q}g_{j,k}(x,\theta^{*})b_{\theta^{*},k}(x)dx=0,\quad\text{for
all }j=1,\ldots,p.$ (4)
###### Proof.
Write $y_{R,k}(x)=f_{k,\theta}(x)+b_{\theta,k}(x)$ to obtain that the $j$-th
element of
$\frac{\partial}{\partial
t}L\\{y_{R}(\cdot),f(\cdot,t)\\}\rvert_{t=\theta}=\\\
-2\int_{\mathcal{X}}\sum_{k=1}^{q}g_{j,k}(x)b_{\theta,k}(x)dx,$
and recall
$\theta^{*}=\operatorname*{arg\,min}_{t\in\Theta}L\\{y_{R}(\cdot),f(\cdot,t)\\}$
which completes the proof. ∎
In other words, if the bias functions satisfy a linear constraint with respect
to the gradients of the computer model at $\theta$, then $\theta$ is
identifiable.
### 3.2 Bayesian inference on model (2)
The influential paper Kennedy and O’Hagan (2001) originally proposed the model
(2) for $q=1$, and considered Gaussian process priors (Rasmussen and Williams,
2005) as a prior distribution over the unknown function $b_{\theta}(x)$
together with a Normal/Uniform prior over the parameter $\theta$. For now, we
assume that the functional form of the computer model $f(x,t)$ is explicitly
known for every $x\in\mathcal{X},t\in\Theta$ and defer the extension of the
proposed method to unavailable $f$ later. A Gaussian process prior over an
unknown function $h(\cdot)$ is typically elicited by a mean function
$m(\cdot)$ and a covariance function $C(\cdot,\cdot)$. Under this prior, for
any finite collection of points $\\{u_{1},\ldots,u_{N}\\}$, the prior
distribution over the vector
$\\{h(u_{1}),\ldots,h(u_{N})\\}^{\mathrm{\scriptscriptstyle{T}}}$ has a
multivariate Gaussian distribution with mean $\\{m(u_{1}),\ldots,m(u_{N})\\}$
and covariance matrix $K=(K_{ij})$ and $K_{ij}=C(u_{i},u_{j}),\,1\leq i,j\leq
N$. Let $\Pi(\theta)$ be the prior distribution over $\theta$ and suppose
$b_{\theta}$ is endowed with a Gaussian process prior with a zero mean
function and a covariance kernel $C(\cdot,\cdot)$. Then with $n$ observations
on $y_{F}(\cdot)$ on the design $\mathcal{D}$, the posterior distribution of
$(\theta,\mathbf{b}),\,\mathbf{b}=\\{b_{\theta}(x_{1}),\ldots,b_{\theta}(x_{n})\\}^{\mathrm{\scriptscriptstyle{T}}}$
can be obtained using Bayes theorem as
$\Pi(\theta,\mathbf{b}\mid
y^{(n)})\propto\ell(y^{(n)}\mid\theta,\mathbf{b})\Pi(\theta)\Pi(\mathbf{b}),$
(5)
where given $\theta$,
$y^{(n)}=\\{y_{F}(x_{1})-f(x_{1},\theta),\ldots,y_{F}(x_{n})-f(x_{n},\theta)\\}^{\mathrm{\scriptscriptstyle{T}}}$
and $\ell(y^{(n)}\mid\theta,\mathbf{b})$ is the likelihood corresponding to a
Gaussian distribution with mean $\mathbf{b}$ and covariance matrix
$\sigma^{2}_{F}\mathrm{I}_{n}$. The marginal posterior distribution of
$\theta$, $\Pi(\theta\mid y^{(n)})$ can be obtained by observing that
$y^{(n)}\mid\theta\sim{\mathrm{N}}(0,K+\sigma_{F}^{2}\mathrm{I}_{n})$.
Tuo and Wu (2016) proved that vanilla Gaussian process priors over the bias
$b_{\theta}(\cdot)$ may lead to inaccurate inference on $\theta^{*}$, defined
as the minimizer of the loss function discussed above, even when observations
are available for a large number of points $n$. This is mainly due to the fact
that a generic Gaussian process on the bias function is not guaranteed to
satisfy the orthogonality condition (4). In light of this, Plumlee (2017)
advocated a modified covariance kernel for $b_{\theta}(\cdot)$ such that
realizations of such a Gaussian process prior satisfy (4) almost surely, see
also Plumlee and Joseph (2018). This modified covariance kernel is
$C_{\theta}(x,x^{\prime})=C(x,x^{\prime})-h_{\theta}(x)^{\mathrm{\scriptscriptstyle{T}}}H_{\theta}^{-1}h_{\theta}(x^{\prime}),$
(6)
where the $j$-th element of the vector $h_{\theta}(x)\in\mathbb{R}^{p}$ is
given by $\int_{\mathcal{X}}g_{j}(x,\theta)C(x,u)du$ and the $(j,k)$-th
element of the matrix $H_{\theta_{0}}\in\mathbb{R}^{p\times p}$ is
$\int_{\mathcal{X}}\int_{\mathcal{X}}g_{j}(u,\theta)g_{k}(u^{\prime},\theta)C(u,u^{\prime})dudu^{\prime}$
with $x,x^{\prime}\in\mathcal{X}$ for $1\leq j,k\leq p$. This covariance
function is obtained by observing that under a Gaussian process prior on
$b_{\theta_{0}}$, the prior on
$(b_{\theta},\int_{\mathcal{X}}g_{1}(x,\theta)b_{\theta}(x)dx,\ldots,\int_{\mathcal{X}}g_{p}(x,\theta_{0})b_{\theta}(x)dx)$
is again a Gaussian process and one then simply finds the conditional
covariance of $b_{\theta}$ given that the last $p$ elements of the above
vector are 0, and thus the orthogonality condition in (4) is enforced. Of
course, apriori we do not have knowledge on $\theta$. However, assuming the
knowledge of the system $y_{R}(x)$ and the computer model $f(x,t)$ for all
$t\in\Theta$, a consistent estimator of $\theta$ can be used for implementing
orthogonal covariance function in practice (Plumlee, 2017).
### 3.3 Posterior projections
Although the modified covariance kernel seemingly resolves idenitifiability
issue of $\theta$, it hinges on the assumption that the user chooses to model
$b_{\theta}(x)$ with a Gaussian process. This leaves out a plethora of other
nonparametric priors for functions that have been widely popular in the
literature, for example Bayesian additive regression trees (BART) (Chipman et
al., 2010), Bayesian multivariate adaptive regression splines (Denison et al.,
1998), Bayesian neural networks (Neal, 2012) etc. Furthermore, these methods
have been successfully applied in computer code calibration problems (Pratola
and Higdon, 2016; Higdon et al., 2008; Chakraborty et al., 2013). Many of
these priors have been shown to have optimal theoretical properties along with
sufficient computational scalability (Ročková and van der Pas, 2020; Linero
and Yang, 2018). In this article, our aim is to develop a procedure where a
user is free to choose any nonparametric prior for $b_{\theta}(x)$ and the
procedure automatically takes care of orthogonality constraints discussed
above.
Let $L^{2}_{q}(\mathcal{X})$ be the space of $q$-dimensional square-integrable
functions on $\mathcal{X}$ equipped with the inner-product $\langle
f_{1},f_{2}\rangle=\sum_{k=1}^{q}\int f_{1,k}(x)f_{2,k}(x)dx$. Then
$L^{2}_{q}(\mathcal{X})$ is a tensor Hilbert space with norm
$\|\cdot\|_{L^{2}_{q}}$ induced by the inner product:
$\|f\|_{L^{2}_{q}}=\langle f,f\rangle=\sum_{k=1}^{q}\int f_{k}^{2}(x)dx$. We
assume that for every $t\in\Theta$,
$g_{j}(\cdot,t)=(g_{j,1}(\cdot,t),\ldots,g_{j,q}(\cdot,t))^{\mathrm{\scriptscriptstyle{T}}}\in
L^{2}_{q}(\mathcal{X})$ for all $j=1,\ldots,p$ and the corresponding bias
$b_{t}(\cdot)\in L^{2}_{q}(\mathcal{X})$. Define
$\mathcal{F}_{\tilde{\theta}}=\\{b_{\tilde{\theta}}:\int\sum_{k=1}^{q}g_{j,k}(x,\tilde{\theta})b_{\tilde{\theta},k}(x)dx=0,j=1,\ldots,p\\}$
for a fixed $\tilde{\theta}$. Here $\tilde{\theta}$ is our guess about the
parameter combination at which the computer model best approximates the system
given the observed data and the computer model. Such a data-dependent choice
of $\tilde{\theta}$ involves minimizing the corresponding empirical risk. Let
$\tilde{\theta}$ be defined as
$\tilde{\theta}=\operatorname*{arg\,min}_{t\in\Theta}\frac{1}{nq}\sum_{k=1}^{q}\sum_{i=1}^{n}\\{y_{F,k}(x_{i})-f_{k}(x_{i},t)\\}^{2}.$
(7)
Validity of this definition is provided in the following result where we first
establish $\theta^{*}$ as the minimizer of population risk and then leverage
on empirical risk minimization results to show that $\tilde{\theta}$ converges
to $\theta^{*}$ for large $n$.
###### Proposition 3.2.
Let $P=P_{y\mid x}P_{x}$ denote the data generating distribution where $P_{x}$
is the uniform measure on $\mathcal{X}$. Suppose that $\Theta$ is compact and
that the $k$-th computer model is Lipshitz in $t\in\Theta$ uniformly over
$x\in\mathcal{X}$, that is there exists $L_{k}>0$ such that
$|f_{k}(x,t_{1})-f_{k}(x,t_{2})|\leq L_{k}\|t_{1}-t_{2}\|_{2}^{2}$ for
$k=1,\ldots,q$. Then $\tilde{\theta}\overset{P}{\to}\theta^{*}$.
A proof is provided in the appendix. Our definition of $\tilde{\theta}$ is
different from the definition used in (Plumlee, 2017, Proposition 1) where the
author assumes the knowledge of $y_{R}(\cdot)$ along with the computer model
$f(\cdot,t)$ for all $t$. We argue that explicit functional form of
$y_{R}(\cdot)$ may not be available in many practical scenarios, and hence
define $\tilde{\theta}$ as detailed above. The compactness of $\Theta$ is a
technical assumption which nevertheless is often practical in many calibration
problems.
Having defined the point at which we want the bias function $b(x)$ to satisfy
the orthogonality condition (4), we now proceed to define the projection
posterior. Instead of defining a prior distribution on
$b_{\tilde{\theta}}\in\mathcal{F}_{\tilde{\theta}}$, our strategy is to start
with a generic prior and then project the posterior distribution to
$\mathcal{F}_{\tilde{\theta}}$. Standard Hilbert space theory implies that
$\mathcal{F}_{\tilde{\theta}}$ is a non-empty, closed and convex subset of
$L^{2}_{q}(\mathcal{X})$. Then by the Hilbert projection theorem (Tsiatis,
2006), there exists a unique projection $b_{\tilde{\theta}}^{*}$ of any
$b_{\tilde{\theta}}\in L^{2}_{q}(\mathcal{X})$. Define the projection as
$T_{\mathcal{F}_{\tilde{\theta}}}(b_{\tilde{\theta}})=\\{b_{\tilde{\theta}}^{*}\in\mathcal{F}_{\tilde{\theta}}:\|b_{\tilde{\theta}}^{*}-b_{\tilde{\theta}}\|_{L^{2}_{q}}=\inf_{b\in\mathcal{F}_{\tilde{\theta}}}\|b-b_{\tilde{\theta}}\|_{L^{2}_{q}}\\}.$
(8)
We now describe how $T_{{\mathcal{F}}_{\tilde{\theta}}}$ is used to define the
projection posterior. Suppose a prior distribution
$\Pi(\theta,b)=\Pi(\theta)\Pi(b)$ is elicited on $\Theta\times
L^{2}_{q}(\mathcal{X})$. Let
$\widetilde{B}=\widetilde{B}_{1}\times\widetilde{B}_{2}$ be a measurable
subset of the Borel $\sigma$-algebra of
$\Theta\times{\mathcal{F}}_{\tilde{\theta}}$ . Then given $y^{(n)}$, we define
the posterior probability $\Pi_{\text{proj}}(\widetilde{B}\mid y^{(n)})$ of
$\widetilde{B}$ under the prior $\Pi(\theta,b)$ as $\Pi(B\mid y^{(n)})$ where
$B=\widetilde{B}_{1}\times
T_{\mathcal{F}_{\tilde{\theta}}}^{-1}(\widetilde{B}_{2})$ where
$T_{\widetilde{\mathcal{F}}_{\theta}}^{-1}(\widetilde{B}_{2})=\\{b_{\tilde{\theta}}:T_{\mathcal{F}_{\tilde{\theta}}}(b_{\theta})\in\widetilde{B}_{2}\\}$,
that is,
$\Pi_{\text{proj}}(\widetilde{B}\mid y^{(n)})=\Pi(B\mid
y^{(n)})=\dfrac{\int_{B}\ell(y^{(n)}\mid\theta,b)d\Pi(\theta,b)}{\int\ell(y^{(n)}\mid\theta,b)d\Pi(\theta,b)}.$
(9)
Measurability of $T$ is guaranteed since $\mathcal{F}_{\tilde{\theta}}$ is
non-empty, closed and convex; see also Sen et al. (2018). Although (8) is
defined as a solution to an optimization problem, in this particular case, it
has an explicit form which we call functional projection.
###### Lemma 3.3.
Fix $b\in L^{2}_{q}(\mathcal{X})$ and let $\tilde{\theta}$ be defined as
above. Let the functions $g_{j,k}(x),\,j=1,\ldots,p$ satisfy the following:
$\sum_{j=1}^{p}\alpha_{j}g_{j,k}(x)=0$ for all $x\in\mathcal{X}$ and for all
$k=1,\ldots,q$ iff $\alpha_{j}=0,\,j=1,\ldots,p$. Then
$T_{\mathcal{F}_{\tilde{\theta}}}(b)=b^{*}(x)=b(x)-\sum_{j=1}^{p}\lambda_{j}^{*}g_{j}(x)$
where the vector
$\lambda=(\lambda_{1}^{*},\ldots,\lambda_{p}^{*})^{\mathrm{\scriptscriptstyle{T}}}$
satisfies $Q\lambda=\eta$ with $\eta=(\sum_{k=1}^{q}\langle
b_{k},g_{1,k}\rangle,\ldots,\sum_{k=1}^{q}\langle
b_{k},g_{p,k}\rangle)^{\mathrm{\scriptscriptstyle{T}}}$ and $Q$ is a $p\times
p$ matrix with elements $Q_{jj^{\prime}}=\sum_{k=1}^{q}\langle
g_{j,k},g_{j^{\prime},k}\rangle$.
The proof is provided in the Appendix. Having defined the projection for our
purpose, we can then devise an MCMC algorithm to sample from
$\Pi_{\text{proj}}(\cdot\mid y^{(n)})$ defined in (9).
1\. Update $b_{\tilde{\theta}}\sim\Pi(b_{\tilde{\theta}}\mid\theta,y^{(n)})$
2\. Project $b_{\tilde{\theta}}$ onto $\mathcal{F}_{\tilde{\theta}}$ following
the Proposition 3.3 to obtain $b^{*}_{\tilde{\theta}}$.
3\. Update $\theta\sim\Pi(\theta\mid b^{*}_{\tilde{\theta}},y^{(n)})$.
Algorithm 1 Projection sampler 1
As an example, consider the case when $b_{\tilde{\theta}}$ is endowed with a
zero mean Gaussian process prior with covariance kernel $C(\cdot,\cdot)$. Then
following Algorithm 1, we update
$b_{\tilde{\theta}}\sim\Pi(b\mid\tilde{\theta},y^{(n)})$ which is a
multivariate Gaussian distribution. Next, we compute the corresponding
projection $b^{*}_{\tilde{\theta}}$ using Lemma 3.3. Finally, we sample
$\theta\sim\Pi(\theta\mid b^{*}_{\tilde{\theta}},y^{(n)})$. This strategy also
works for other priors such as BART, where in the first step we update the
parameters of the BART parameters using their respective full conditionals.
The next steps are exactly the same. An alternative finite-dimensional
projection method is described in the supplementary document.
For a full Bayesian inference on (2) under the projection posterior framework
described above, one should ideally elicit a prior $\Pi(\Sigma_{F})$ on the
unknown covariance matrix $\Sigma_{F}$. However, this often adds to
computational challenges already present in a calibration problem and may not
influence the uncertainty observed in $\theta$ \- the central goal of these
problems (Bayarri et al., 2007). As an alternative, a modular Bayes approach
is considered here where we use a plug-in estimate $\hat{\Sigma}_{F}$. This
estimate is constructed by fitting a nonparametric regression model, i.e.
$y_{F,k}(x)=y_{R,k}(x)+\epsilon_{k}(x)$ to each outcome separately with
idiosyncratic variance parameters $\sigma_{F,k}^{2}$ with conjugate Inverse
Gamma priors and $y_{R,k}\sim\Pi_{k}$ for some nonparametric prior on
$y_{R,k}$. We then set $\hat{\Sigma}_{F}=\text{Cov}(E)$ where $E$ is the error
matrix with columns $E_{k}=y_{F,k}-\bar{y}_{R}$ where $\bar{y}_{R}$ is the
posterior mean of predictions on the training set. Given $\hat{\Sigma}_{F}$,
it can be assumed without loss of generality that
$\epsilon(x)\sim{\mathrm{N}}(0,\mathrm{I}_{q})$ in (2). This also makes prior
elicitation on the multivariate bias $b$ simpler; a natural choice is
$\Pi(b)=\prod_{k=1}^{q}\Pi(b_{k})$.
## 4 Explicitly unavailable computer model
Until now we have assumed that the computer model $f(x,t)$ is either
explicitly specified or can be evaluated cheaply using a code for every
$(x,t)\in\mathcal{X}\times\Theta$. However, in many calibration problems,
including ours, this is not the case; the computer code can be computationally
very expensive; see also Section 2. The standard approach in the literature
has been to approximate the computer model using a nonparametric method,
typically the Gaussian process (Santner et al., 2003). While the original
calibration framework developed by Kennedy and O’Hagan (2001) modeled the
observed data and the computer model simultaneously, a modular approach to
inference has been advocated by many authors, e.g. Bayarri et al. (2007);
Plumlee (2017) wherein the modeling of the explicitly defined computer model
is done separately from the modeling of the observed data. Here, we adopt this
modular approach with one crucial difference with Plumlee (2017) where the
author uses a probabilistic definition of the computer model using a Gaussian
process. We, on the other hand, use a deterministic definition $\hat{f}$ of
the computer model obtained using any nonparametric approximator such as the
posterior mean of a BART fit, a neural net obtained by minimizing the least
square error between code evaluations and the neural net output, or a random
forest fit; such an approach was also adopted by Xie and Xu (2021). Our
motivation for doing so is to allow for more computational and modeling
flexibility than a Gaussian process framework. Specifically, we work with the
posterior mean of a BART fit which proved to be superior among all other
choices of models for $f(x,t)$ in our numerical experiments as well as in our
motivating example. In particular, we compared the squared error loss on a
held-out test data of size $n_{t}=50$ for these methods:
$(n_{t})^{-1}\sum_{i=1}^{n_{t}}\\{f(x,t_{i})-\hat{f}(x,t_{i})\\}^{2}$ for each
location $x$ in the observed data $x\in\mathcal{X}$ and each output of the
computer model. In the best case scenario, the ratio of the squared error loss
for the posterior predictive mean improved over the second best method (Random
Forest) by a factor of almost 5.
## 5 Simulations and univariate applications
### 5.1 Univariate simulations
In this section, we compare our method to the GP orthogonal calibration by
Plumlee (2017) (OGP) and the projected calibration (PCAL) method by Xie and Xu
(2021) in several simulation experiments. Since these methods were originally
developed for one outcome, we restrict here to the case $q=1$. We focus on
each of these methods’ ability to estimate $\theta^{*}$, uncertainty
quantification, and their associated computing time. For generating the data,
we consider two situations when the definition of the computer model is
available explicitly:
1. 1.
Model 1: $y_{R}(x)=4x+x\sin 5x$, $f(x,t)=tx$ where $x\in[0,1]$ (Plumlee, 2017,
Example 5.1),
2. 2.
Model 2: $f(x,t)=7\\{\sin(2\pi t_{1}-\pi)\\}^{2}+2\\{(2\pi
t_{2}-\pi)^{2}\sin(2\pi x-\pi)\\}$, $y_{R}(x)=f(x,\theta^{*})$ where
$\theta^{*}=(0.2,0.3)$, $x\in[0,1]$ (Xie and Xu, 2021, Configuration 1).
In Model 1, $\theta^{*}=3.56$. We additionally consider one more situation
where we do not assume the knowledge of $f(x,t)$ in Model 2 and approximate it
by the posterior predictive mean of a BART when $N=7^{2}$ code outputs are
available on the observed values of $x$, meaning we have for each value
$\\{\theta_{1},\ldots,\theta_{N}\\}$ the computer code output is available for
$f(\theta_{k},x_{i}),\,k=1,\ldots,N,i=1,\ldots,n$.
* •
Model 3: $f(x,t)$ is same as Model 2 but only outputs $f_{1},\ldots,f_{N}$ are
available. Everything else is same as Model 2.
In this case, we set $\tilde{\theta}$ to be the parameter value at which the
sample $L^{2}$-distance is minimized between the observed data and the
posterior predictive mean of $f$.
For all cases we simulate $n=100$ field observations from the model
$y^{F}(x_{i})=y^{R}(x_{i})+\epsilon_{i},\,\epsilon_{i}\sim{\mathrm{N}}(0,0.2^{2})$
where we sample the $x_{i}$’s uniformly over $[0,1]$. For the proposed method,
we consider two priors on the bias function - GP and BART. For the GP prior,
we use the covariance kernel
$C(x,x^{\prime})=\sigma^{2}(1+|x-x^{\prime}|/\psi)\exp\\{-|x-x^{\prime}|/\psi\\}$.
We set $\psi=1/2$ and use a plug-in estimate of $\sigma^{2}$ which is obtained
by fitting a non-parametric BART regression model to the observed field data
$(y_{F,i},x_{i})_{i=1}^{n}$. The corresponding projected versions are
abbreviated as PGP and PBART. We also consider the functional and finite
dimensional projections for PGP and PBART, with the finite-dimensional version
abbreviated as PGP(Fn) and PBART(Fn). For Model 3, we only consider the
infinite dimensional projections. In Tables 1, 2 and 3 we report the results
of our experiments from a replicated simulation study. All numbers are
averaged over 100 replications and for each model we draw 5000 posterior
samples using MCMC with 1000 burn in samples. In terms of estimation quality
of the calibration parameters, all the different methods perform similarly.
Indeed, the posterior mean and standard deviation in Table 1 and Table 2 are
almost identical. However, the key difference is in the respective runtimes of
these methods. The runtimes reported in the tables are comparable up to coding
language and efficiencies of these implementations. Nonetheless, PBART
outperforms all other methods by quite a large margin providing at least an
order of magnitude improvement over the next fastest method. We note here, the
runtimes reported for PCAL correspond to the standard implementation of the
method and not the approximated calibration method proposed in Xie and Xu
(2021); the approximated calibration method is much faster compared to its
standard version although even for that the average runtime for Model 2 is
36.33 seconds which is almost 4 times the time of PBART. In terms of
uncertainty quantification, all the methods in consideration provide roughly
the nominal coverage although it seems that PGP and PBART is biased downwards
(but only slightly). This is also apparent in the relatively sharper posterior
distributions of $\theta$ plotted in Figure 2.
| Model 1 ($\theta^{*}=3.56$)
---|---
| PGP | PGP(Fn) | PBART | PBART(Fn) | OGP | PCAL
Mean | 3.56 | 3.59 | 3.57 | 3.58 | 3.57 | 3.57
Std. Dev. | 0.008 | 0.007 | 0.02 | 0.02 | 0.003 | 0.003
Coverage | 0.93 | 0.93 | 0.92 | 0.93 | 0.95 | 0.95
Runtime | 3.78 sec | 30.72 sec | 1.86 sec | 17 min | 1.88 | 13.92 min
Table 1: Simulation results for Model 1. We report the posterior mean,
posterior standard deviation, coverage of the 95% credible intervals and the
average runtime.
| Model 2 ($\theta^{*}=(0.2,0.3)$) |
---|---|---
| PGP | PGP(Fn) | PBART | PBART(Fn) | OGP | PCAL
Mean | (0.2, 0.3) | (0.2, 0.31) | (0.19, 0.3) | (0.2, 0.31) | (0.2, 0.3) | (0.2, 0.3)
Std. Dev. | (0.0002, 0.0002) | (0.0003, 0.0004) | (0.0008, 0.0007) | (0.0008, 0.0008) | (0.001, 0.002) | (0.003, 0.0006)
Coverage | 0.91 | 0.91 | 0.92 | 0.92 | 0.96 | 0.96
Runtime | 18.18 sec | 2.65 min | 9.38 sec | 1.35 hrs | 16.58 min |
Table 2: Simulation results for Model 2. We report the posterior mean,
posterior variance, coverage of the 95% credible intervals and the average
runtime.
| Model 3 ($\theta_{0}=(0.2,0.3)$)
---|---
| PGP | PBART
Mean | (0.21, 0.33) | (0.22, 0.34)
Std. Dev. | (0.03, 0.03) | (0.04, 0.03)
Runtime | 24 min | 20 min
Table 3: Simulation results for Model 3. We report the posterior mean,
posterior variance and the average runtime.
(a) Model 1: $\pi(\theta\mid\text{data})$
(b) Model 2: $\pi(\theta_{1}\mid\text{data})$
(c) Model 2: $\pi(\theta_{2}\mid\text{data})$
Figure 2: Posterior distribution of the calibration parameter $\theta$.
### 5.2 Univariate analysis on Schwarzschild model outputs
We apply the proposed posterior projection technique on each output of the
Schwarzschild model separately. We illustrate the specifics of our application
with the first output of the Schwarzschild model, which is the mean velocity.
Similar to the Kennedy and O’Hagan (2001) setup, we have the following model
for the observed velocities $v_{F}(x)$ and the corresponding code output
$v_{S}(x)$ at location $x\in\mathcal{X}$:
$v_{F}(x_{i})=v_{R}(x_{i})+\epsilon_{i}=v_{S}(x_{i};\tilde{\theta})+b_{v,\tilde{\theta}}(x_{i})+\epsilon_{i},\quad\epsilon_{i}\sim{\mathrm{N}}(0,\sigma_{v}^{2}),$
(10)
where we let $\tilde{\theta}$ to be the parameter value where the $L^{2}$ loss
defined in previous sections is minimized and $b_{v,\tilde{\theta}}(\cdot)$ is
the bias function corresponding to mean velocity at $\tilde{\theta}$. Since
the explicit form of the model is not available, we use the posterior mean of
a BART fit as the definition of $v_{S}(x;t)$ based on the model outputs only.
We fix $\sigma_{v}$ to the posterior mean of $\sigma$ of a BART regression fit
to $(v_{F,i},x_{i})_{i=1}^{n}$. We set
$\theta_{j}\sim{\mathrm{N}}(0,\gamma^{2})$ with $\gamma=10$ as priors for
$\theta_{j}$, and $\Pi(\theta)=\prod_{j=1}^{p}\Pi(\theta_{j})$. We implement
PGP and PBART under these settings where for implementing Step 3 of Algorithm
1, we use an adaptive Metropolis-Hastings sampler Haario et al. (2001) which
is tuned to maintain an acceptance rate of approximately 0.3.
In Table 4, we summarize the posterior distribution of the calibration
parameters for all the four different outputs of the Schwarzschild model.
Specifically, we report the posterior mean, standard deviation and the 95%
credible interval for all the three calibration parameters
$(\theta_{1},\theta_{2},\theta_{3})$. As expected, there are some differences
in the posterior means for different velocity moments being fit. For example,
the posterior mean for $\theta_{1}$ differs when using $h_{4}$ compared to
when using $\tau$. The difference may be due to the physical degeneracy
between the orbital anisotropy and black hole mass, such that large $\tau$ at
the center of the 2-dimensional spatial grid can imply a large black hole mass
but so too can a smaller $\tau$ with radial anisotropy, which is reflected in
the values of $h_{4}$. Or, it could be due simply to the model inadequacy of a
univariate treatment of the problem. In addition, there seems to a broad
agreement between the posterior means for $\theta_{1}$ and $\theta_{2}$ under
the two priors. However, the key difference is in the posterior mean of
$\theta_{3}$. For the GP prior, the posterior mean for the 4 outcomes is
roughly around 1, whereas for BART, the posterior mean is around 1.4. One
possible explanation for this discrepancy is that for $\theta_{3}$, the
computer code output is only available for $\theta_{3}=1,2,3$. The scarcity of
data along this dimension may potentially lead to unstable estimates. A joint
model accounting for the covariance between the measurements can reconcile
this apparent discrepancy in the posterior distribution and the problems
induced by sparse data. Numerical experiments in the next subsection
illustrates the benefits of a joint model when outcomes are correlated.
| | PGP | PBART
---|---|---|---
| | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$
$v$ | Mean | 9.46 | 9.06 | 0.82 | 9.59 | 9.16 | 1.22
Std. Dev. | 0.14 | 0.08 | 0.09 | 0.27 | 0.15 | 0.07
Interval | (9.21, 9.80) | (8.93, 9.18) | (0.65, 1.00) | (9.20, 9.97) | (8.93, 9.50) | (1.12, 1.32)
$\tau$ | Mean | 9.98 | 9.12 | 1.02 | 9.59 | 9.03 | 1.42
Std. Dev. | 0.09 | 0.07 | 0.10 | 0.10 | 0.13 | 0.18
Interval | (9.81, 10.14) | (8.99, 9.26) | (0.87, 1.23) | (9.41, 9.76) | (8.84, 9.31) | (1.15, 1.76)
$h_{3}$ | Mean | 9.96 | 9.36 | 1.04 | 10.09 | 9.27 | 1.41
Std. Dev. | 0.07 | 0.08 | 0.12 | 0.07 | 0.08 | 0.06
Interval | (9.86, 10.11) | (9.15, 9.49) | (0.87, 1.25) | (9.97, 10.23) | (9.12, 9.41) | (1.25 , 1.41)
$h_{4}$ | Mean | 10.18 | 9.07 | 1.09 | 9.73 | 9.09 | 1.53
Std. Dev. | 0.11 | 0.06 | 0.06 | 0.06 | 0.12 | 0.09
Interval | (9.89, 10.62) | (8.97, 9.20) | (0.97, 1.19) | (9.61, 9.82) | (8.85, 9.28) | (1.35, 1.69)
Table 4: Posterior summaries for the calibration parameters $\theta_{1}$ =
logarithm of mass of the black hole, $\theta_{2}$ = stellar mass-to-light
ratio and $\theta_{3}$ = fraction of dark matter. The posterior mean, standard
deviation (Std. Dev.) and the 95% credible interval is reported here.
### 5.3 Multivariate simulations
Here we consider a simulation scenario where more than outcome is measured,
i.e. $q>1$. We illustrate this with the case $q=2$. For $x\in[0,1]$, consider
the model
$\begin{pmatrix}y_{F,1}(x)\\\
y_{F,2}(x)\end{pmatrix}=\begin{pmatrix}f_{1}(x,\theta)\\\
f_{2}(x,\theta)\end{pmatrix}+\begin{pmatrix}b_{\theta,1}(x)\\\
b_{\theta,2}(x)\end{pmatrix}+\begin{pmatrix}\epsilon_{1}\\\
\epsilon_{2}\end{pmatrix}$ (11)
where $y_{R,1}(x)=4x+x\sin 5x$, $y_{R,2}(x)=[1+\exp\\{-6(x-0.5)\\}]^{-1}$ are
the real data-generating processes. The respective computer models are
$f_{1}(x,t)=tx$, $f_{2}(x,t)=\Phi\\{t(x-0.5)\\}$ with $\Phi(\cdot)$ being the
cdf of $Z\sim{\mathrm{N}}(0,1)$. The errors are assumed to follow
${\mathrm{N}}(0,\Sigma)$. The loss
$L\\{y_{R}(\cdot),f(\cdot,t)\\}=\sum_{k=1}^{2}\int_{\mathcal{X}}\\{y_{R,k}(x)-f_{k}(x,t)\\}^{2}$
is minimized at $\theta^{*}=3.56$ approximately. However, the behavior of the
$L_{2}$-loss in these two cases is very different, especially near the
minimum. Indeed, as shown in the left panel of Figure 3, for the pair
$(y_{R,1},f_{1})$, the loss increases quite sharply around 3.56, whereas for
$(y_{R,2},f_{2})$, the increase in loss for $t>3.56$ is much slower. As such,
when data from model (11) is fitted separately in the univariate framework,
one should expect more uncertainty in $\theta$ for $(y_{R,2},f_{2})$. On the
other hand, a joint multivariate fit should reflect the fact when the two real
processes and their corresponding computer models are compared together, the
posterior distribution of $\theta$ is much more centered around 3.56.
(a) $L(\theta)=\int_{0}^{1}\left\\{y_{R,1}(x)-f_{1}(x,t)\right\\}^{2}$
(b) $L(\theta)=\int_{0}^{1}\left\\{y_{R,2}(x)-f_{2}(x,t)\right\\}^{2}$
Figure 3: Loss comparison
For data generation, we consider $n=100$ and a covariance matrix $\Sigma_{0}$
such that the diagonal elements are $0.2^{2}$ and the off-diagonal elements
are $0.012$. We consider PGP and PBART for this setup and use the functional
projection. We fit this data separately in the univariate framework and then a
joint model is fitted. For a fair comparison all hyperparameter values were
the same. We plot the posterior distribution of $\theta$ obtained using the
PBART method in Figure 4. In the first and second panel, the posterior
distribution of $\theta$ is shown when two separate univariate models are
fitted, and in the third panel we show that for a multivariate fit. As
mentioned earlier, the posterior distribution of $\theta$ when data for the
second outcome is fitted separately, is more dispersed and has relatively
large mass in the interval [3, 4]. This is not the case for the first outcome,
as almost the entire mass is within [3.5, 3.65]. However, the posterior
distribution of $\theta$ form a multivariate fit is able to borrow information
across the two outcomes so that most of the mass of the distribution is within
[3.4,3.7].
Figure 4: Posterior distribution of $\theta$ for univariate and multivariate
PBART fits for model 11. Left two panels show posteriors for two separate
univariate fits, and the third panel shows posterior from a combined
multivariate fit.
## 6 Calibrating the Schwarzschild model
Here we carry out a joint calibration of the Schwarzschild model. For this, we
first estimate the covariance matrix as $\Sigma=\text{Cov}(E)$, where $E$ is a
$n\times q$ error matrix obtained by fitting univariate BART regressions to
the field data of each outcome. Here we fit the PBART method with functional
projection. The marginal posterior distribution of $\theta$ is shown in Figure
5. The 95% symmetric posterior (marginal) credible intervals are [9.54, 9.87],
[8.78, 9.13] and [0.97, 1.38] for $\theta_{1},\theta_{2},\theta_{3}$,
respectively. Intervals for $\theta_{1},\theta_{2}$ from the joint analysis
clearly seem to lie at the intersection of the intervals from the univariate
model, but the posterior of $\theta_{3}$ exhibits slight bimodality. Overall,
by modeling the four outcomes simultaneously uncertainty in $\theta$ have
reduced which may significantly cut down computation time for future
applications with an expanded parameter space. We also show the marginal
posterior predictive mean of the outcomes. In the first panel of Figure 6, we
plot the posterior predictive mean at the observed locations for
$v_{R}(\cdot)$. For reference, the observed data is plotted on the third panel
of Figure 6. Similar plots for $\tau_{R}(\cdot),h_{3,R}(\cdot)$ and
$h_{4,R}(\cdot)$ are provided in Figures 7, 8 and 9, respectively. As
mentioned earlier, the outcomes in this context represent the first four
moments of velocity, and the model performance decreases as the moments
increase. We also observed a similar phenomenon when we looked at the relative
squared error loss for a held-out data set of size 20. For $v_{F}(x)$, We
define the relative squared error loss as
$n_{t}^{-1}\sum_{i=1}^{n_{t}}(1-\hat{v}_{F}(x_{i})/v_{F}(x_{i}))^{2}$ where
$\hat{v_{F}}(x_{i})$ represents the posterior mean of samples of
$[v_{S}(x_{i},\theta)+b_{v,\theta}(x_{i})]$. Here $n_{t}$ is the test data
size. The definition is similar for other outcomes. The values we obtained are
0.17, 0.003, 3.92 and 30.42 for $v,\tau,h_{3},h_{4}$, respectively.
Figure 5: Posterior distribution of $\theta$ from multivariate PBART fit for
the Schwarzschild model.
Figure 6: Posterior predictive mean of PBART is plotted for $v_{R}(\cdot)$ in
the first panel. In the second panel, the observed values $v_{F}(\cdot)$ are
also shown for reference.
Figure 7: Posterior predictive mean of PBART is plotted for $\tau_{R}(\cdot)$
in the first panel. In the second panel, the observed values $\tau_{F}(\cdot)$
are also shown for reference.
Figure 8: Posterior predictive mean of PBART is plotted for $h_{3,R}(\cdot)$
in the first panel. In the second panel, the observed values $h_{3,F}(\cdot)$
are also shown for reference.
Figure 9: Posterior predictive mean of PBART is plotted for $h_{4,R}(\cdot)$
in the first panel. In the second panel, the observed values $h_{4,R}(\cdot)$
are also shown for reference.
## 7 Discussion
The Schwarzschild model, although computationally intensive, is an important
tool in understanding the dynamics of a black hole and its host galaxy.
Motivated by this application we developed a multivariate calibration method
that ensures parameter identifiablity under the squared error loss. The key
benefits of the proposed projection posterior approach is its flexibility in
accommodating user-specified prior distributions on the multivariate bias
function, and the fact that such a projection is available analytically.
Benefits of a multivariate analysis is demonstrated through numerical
experiments and it seems to impact the analysis of Schwarzschild model
positively as well. Conceptually, the projection approach can be extended to
alternative loss functions that lead to different sufficiency conditions for
identifiability and high-dimensional outcomes where low-rank models for the
error covariance might be warranted.
## References
* Bayarri et al. (2007) Bayarri, M., D. Walsh, J. Berger, J. Cafeo, G. Garcia-Donato, F. Liu, J. Palomo, R. Parthasarathy, R. Paulo, and J. Sacks (2007). Computer model validation with functional output. Annals of Statistics 35(5), 1874–1906.
* Bayarri et al. (2007) Bayarri, M. J., J. O. Berger, R. Paulo, J. Sacks, J. A. Cafeo, J. Cavendish, C.-H. Lin, and J. Tu (2007). A framework for validation of computer models. Technometrics 49(2), 138–154.
* Chakraborty et al. (2013) Chakraborty, A., B. K. Mallick, R. G. Mcclarren, C. C. Kuranz, D. Bingham, M. J. Grosskopf, E. M. Rutter, H. F. Stripling, and R. P. Drake (2013). Spline-based emulators for radiative shock experiments with measurement error. Journal of the American Statistical Association 108(502), 411–428.
* Chipman et al. (2010) Chipman, H. A., E. I. George, and R. E. McCulloch (2010). BART: Bayesian additive regression trees. The Annals of Applied Statistics 4(1), 266–298.
* Denison et al. (1998) Denison, D. G., B. K. Mallick, and A. F. Smith (1998). Bayesian mars. Statistics and Computing 8(4), 337–346.
* Forest et al. (2008) Forest, C. E., B. Sansó, and D. Zantedeschi (2008). Inferring climate system properties using a computer model. Bayesian Analysis 3(1), 1–37.
* Friedman (1991) Friedman, J. H. (1991). Multivariate adaptive regression splines. The annals of statistics 19(1), 1–67.
* Gattiker et al. (2006) Gattiker, J., D. Higdon, S. Keller-McNulty, M. McKay, L. Moore, and B. Williams (2006). Combining experimental data and computer simulations, with an application to flyer plate experiments. Bayesian Analysis 1(4), 765–792.
* Gneiting and Raftery (2007) Gneiting, T. and A. E. Raftery (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association 102(477), 359–378.
* Haario et al. (2001) Haario, H., E. Saksman, J. Tamminen, et al. (2001). An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242.
* Higdon et al. (2008) Higdon, D., J. Gattiker, B. Williams, and M. Rightley (2008). Computer model calibration using high-dimensional output. Journal of the American Statistical Association 103(482), 570–583.
* Kennedy and O’Hagan (2001) Kennedy, M. C. and A. O’Hagan (2001). Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63(3), 425–464.
* Kormendy and Ho (2013) Kormendy, J. and L. C. Ho (2013, August). Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies. Annual Review of Astronomy and Astrophysics 51(1), 511–653.
* Linero and Yang (2018) Linero, A. R. and Y. Yang (2018). Bayesian regression tree ensembles that adapt to smoothness and sparsity. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80(5), 1087–1110.
* Marmin and Filippone (2022) Marmin, S. and M. Filippone (2022). Deep gaussian processes for calibration of computer models. Bayesian Analysis 1(1), 1–30.
* McConnell and Ma (2013) McConnell, N. J. and C.-P. Ma (2013, February). Revisiting the Scaling Relations of Black Hole Masses and Host Galaxy Properties. The Astrophysical Journal 764(2), 184.
* Mehrgan et al. (2019) Mehrgan, K., J. Thomas, R. Saglia, X. Mazzalay, P. Erwin, R. Bender, M. Kluge, and M. Fabricius (2019, December). A 40 Billion Solar-mass Black Hole in the Extreme Core of Holm 15A, the Central Galaxy of Abell 85. The Astrophysical Journal 887(2), 195.
* Neal (2012) Neal, R. M. (2012). Bayesian learning for neural networks, Volume 118. Springer Science & Business Media.
* Plumlee (2017) Plumlee, M. (2017). Bayesian calibration of inexact computer models. Journal of the American Statistical Association 112(519), 1274–1285.
* Plumlee and Joseph (2018) Plumlee, M. and V. R. Joseph (2018). Orthogonal gaussian process models. Statistica Sinica, 601–619.
* Pollard and Radchenko (2006) Pollard, D. and P. Radchenko (2006). Nonlinear least-squares estimation. Journal of Multivariate Analysis 97(2), 548–562.
* Pratola and Higdon (2016) Pratola, M. T. and D. M. Higdon (2016). Bayesian additive regression tree calibration of complex high-dimensional computer models. Technometrics 58(2), 166–179.
* Quenneville et al. (2021) Quenneville, M. E., C. M. Liepold, and C.-P. Ma (2021). Dynamical modeling of galaxies and supermassive black holes: Axisymmetry in triaxial schwarzschild orbit superposition models. The Astrophysical Journal Supplement Series 254(2), 25\.
* Rasmussen and Williams (2005) Rasmussen, C. E. and C. K. I. Williams (2005). Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press.
* Ročková and van der Pas (2020) Ročková, V. and S. van der Pas (2020). Posterior concentration for bayesian regression trees and forests. The Annals of Statistics 48(4), 2108–2131.
* Sacks et al. (1989) Sacks, J., W. J. Welch, T. J. Mitchell, and H. P. Wynn (1989). Design and analysis of computer experiments. Statistical science 4(4), 409–423.
* Saglia et al. (2016) Saglia, R. P., M. Opitsch, P. Erwin, J. Thomas, A. Beifiori, M. Fabricius, X. Mazzalay, N. Nowak, S. P. Rusli, and R. Bender (2016, February). The SINFONI Black Hole Survey: The Black Hole Fundamental Plane Revisited and the Paths of (Co)evolution of Supermassive Black Holes and Bulges. The Astrophysical Journal 818(1), 47.
* Salter et al. (2018) Salter, J. M., D. B. Williamson, J. Scinocca, and V. Kharin (2018). Uncertainty quantification for spatio-temporal computer models with calibration-optimal bases. arXiv preprint arXiv:1801.08184 12.
* Santner et al. (2003) Santner, T. J., B. J. Williams, and W. I. Notz (2003). The Design and Analysis of Computer Experiments. Springer series in statistics. Springer.
* Schwarzschild (1979) Schwarzschild, M. (1979). A numerical model for a triaxial stellar system in dynamical equilibrium. The Astrophysical Journal 232, 236–247.
* Sen et al. (2018) Sen, D., S. Patra, and D. Dunson (2018). Constrained inference through posterior projections. arXiv preprint arXiv:1812.05741.
* Tsiatis (2006) Tsiatis, A. A. (2006). Semiparametric theory and missing data, Volume 4. Springer.
* Tuo and Wu (2015) Tuo, R. and C. J. Wu (2015). Efficient calibration for imperfect computer models. Annals of Statistics 43(6), 2331–2352.
* Tuo and Wu (2016) Tuo, R. and C. J. Wu (2016). A theoretical framework for calibration in computer models: parametrization, estimation and convergence properties. SIAM/ASA Journal on Uncertainty Quantification 4(1), 767–795.
* van den Bosch et al. (2008) van den Bosch, R., G. Van De Ven, E. Verolme, M. Cappellari, and P. De Zeeuw (2008). Triaxial orbit based galaxy models with an application to the (apparent) decoupled core galaxy ngc 4365. Monthly Notices of the Royal Astronomical Society 385(2), 647–666.
* Xie and Xu (2021) Xie, F. and Y. Xu (2021). Bayesian projected calibration of computer models. Journal of the American Statistical Association 116(536), 1965–1982.
* Yang et al. (2017) Yang, Y., A. Bhattacharya, and D. Pati (2017). Frequentist convergence and sup-norm convergence rate in gaussian process regression. arxiv preprint arXiv:1708.04753.
## Appendix A Proofs
### A.1 Proof of Proposition 3.2
###### Proof.
We first characterize $\theta$ in terms of the expected loss. Recall that for
$Q=\mathrm{I}_{q}$,
$\theta=\operatorname*{arg\,min}_{t\in\Theta}\int_{x}\sum_{k=1}^{q}\\{y_{R,k}(x)-f_{k}(x,t)\\}^{2}dx$.
For any fixed $t\in\Theta$, let $e(x)=\\{y^{F}(x)-f(x,t)\\}$, then
$\displaystyle
E_{P}\\{e(x)^{\mathrm{\scriptscriptstyle{T}}}e(x)\\}=(\text{Vol}(\mathcal{X}))^{-1}\int_{x}\left[\sum_{k=1}^{q}E_{P_{y\mid
x}}\\{e^{2}_{k}(x)\\}\right]dx$
$\displaystyle=(\text{Vol}(\mathcal{X}))^{-1}\int_{x}\sum_{k=1}^{q}E_{P_{y\mid
x}}\\{y_{F,k}-y_{R,k}(x)\\}^{2}dx+(\text{Vol}(\mathcal{X}))^{-1}\int_{x}\sum_{k=1}^{q}\\{y_{R,k}(x)-f_{k}(x,t)\\}^{2}dx$
$\displaystyle=\text{tr}(\Sigma_{F})+(\text{Vol}(\mathcal{X}))^{-1}\int_{x}e(x)^{\mathrm{\scriptscriptstyle{T}}}e(x)dx,$
where $\text{tr}(A)$ denotes the trace of a matrix $A$. Hence, $\theta$ can be
seen as
$\operatorname*{arg\,min}_{t\in\Theta}\int_{x}\sum_{k=1}^{q}\\{y_{R,k}(x)-f_{k}(x,t)\\}^{2}dx=\operatorname*{arg\,min}_{t\in\Theta}E_{P}[e(x)^{\mathrm{\scriptscriptstyle{T}}}e(x)].$
Now define
$Q_{n,k}(t)=n^{-1}\sum_{i=1}^{n}\\{y_{F,k}(x_{i})-f_{k}(x_{i},t)\\}^{2}$ and
$Q_{n}(t)=q^{-1}\sum_{k=1}^{q}Q_{n,k}(t)$ which is simply
$E_{P_{n}}[e(x)^{\mathrm{\scriptscriptstyle{T}}}e(x)]$ where $P_{n}$ is the
corresponding joint empirical measure. Invoking Theorem 3 of Pollard and
Radchenko (2006) and the compactness of $\Theta$, we get that
$\sup_{f}|Q_{n}(t)-E_{P}(Q_{n}(t))|\overset{P}{\to}0$ which implies that
$\theta_{n}^{*}\overset{P}{\to}\theta$. ∎
### A.2 Proof of Lemma 3.3
###### Proof.
The projection operator is the solution to the optimization problem
$\displaystyle\min\|b_{\theta}-b_{\theta}^{*}\|_{L^{2}_{q}}\quad\text{subject
to }\sum_{k=1}^{q}\langle g_{j,k},b_{\theta,k}^{*}\rangle=0,\,j=1,\ldots,p.$
Hence, the Lagrangian dual of the optimization problem is
$\displaystyle\min\sum_{k=1}^{q}\|b_{\theta,k}-b_{\theta,k}^{*}\|_{L}^{2}+\sum_{j=1}^{p}\lambda_{j}\left[\sum_{k=1}^{q}\langle
g_{j,k},b_{\theta,k}^{*}\rangle\right].$
Solving for $b_{\theta,k}^{*}$ gives
$b_{\theta,k}^{*}=b_{\theta,k}+\sum_{j=1}^{p}\lambda_{j}g_{j,k}$. Thus,
$\|b_{\theta,k}-b_{\theta,k}^{*}\|_{L^{2}_{q}}=\lambda^{\mathrm{\scriptscriptstyle{T}}}(\sum_{k=1}^{q}Q_{k})\lambda$
where
$\lambda=(\lambda_{1},\ldots,\lambda_{p})^{\mathrm{\scriptscriptstyle{T}}}$
and $Q_{k}^{p\times p}$ is a matrix with $(j,j^{\prime})$-th element $\langle
g_{j,k},g_{j^{\prime},k}\rangle$. Also, $\sum_{k=1}^{q}\langle
g_{j,k},b_{\theta,k}^{*}\rangle=\sum_{k=1}^{q}\langle
g_{j,k},b_{\theta,k}\rangle+\sum_{j^{\prime}=1}^{p}\lambda_{j^{\prime}}Q_{k,jj^{\prime}}$.
This implies that $Q\lambda=-\eta$ where $\eta\in\mathbb{R}^{p\times 1}$ with
$\eta_{j}=\sum_{k=1}^{q}\langle g_{j,k},b_{\theta,k}\rangle$ which proves the
result. ∎
Supplementary materials for “Orthogonal calibration via posterior projections
with applications to the Schwarzschild model”
## Appendix S.1 Finite-dimensional projection
When the central focus of analysis is assessing uncertainty in $\theta$, and
$\Pi(b_{k})$ is a Gaussian process, then the projection posterior approach can
be implemented within a finite dimensional setting by ensuring
$\sum_{k=1}^{q}\mathbf{b}_{k}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{g}_{j,k}=0$
where
$\mathbf{b}_{k}=(b_{k}(x_{1}),\ldots,b_{k}(x_{n}))^{\mathrm{\scriptscriptstyle{T}}}$
and $\mathbf{g}_{j,k}=(g_{j,k}(x_{1}),\ldots,g_{j,k}(x_{n}))$. This is a
consequence of posteriors of $\mathbf{b}_{k}$ being Gaussian under a Gaussian
process prior and projections of multivariate Gaussian random vectors to
linear subspaces. For example, suppose $X^{d\times
1}\sim{\mathrm{N}}(\mu,\Sigma)$ and consider the set
$S=\\{x:A^{\mathrm{\scriptscriptstyle{T}}}x=0\\}\subset\mathbb{R}^{n}$ where
$A^{d\times p}$ is matrix of rank $p$. The conditional distribution of $X\mid
X\in S$ is again a multivariate Gaussian with mean $\mu-\Sigma
A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})^{-1}A\mu$ and covariance matrix
$\Sigma-\Sigma A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})^{-1}A\Sigma$. Next, define the following
projection for any $x\in\mathbb{R}^{d}$
$P_{S}(x)=\Sigma^{1/2}\operatorname*{arg\,min}_{y\in S}\|y-\Sigma^{-1/2}x\|.$
(S.1)
Standard linear algebra yields that $P_{S}(x)$ has the form
$\Sigma^{1/2}(I-P_{A})\Sigma^{-1/2}y$ where
$P_{A}=A(A^{\mathrm{\scriptscriptstyle{T}}}A)^{-1}A^{\mathrm{\scriptscriptstyle{T}}}$
is the projection matrix of $A$. This leads to the following interpretation of
the multivariate Gaussian distribution conditional on linear equality
constraints.
###### Proposition S.1.1.
Suppose $X\sim{\mathrm{N}}(\mu,\Sigma)$ and consider the set
$S=\\{x:A^{\mathrm{\scriptscriptstyle{T}}}x=0\\}\subset\mathbb{R}^{d}$ where
$A^{d\times p}$ is matrix of rank $p$. Then the random variable
$P_{S}(X)\sim{\mathrm{N}}(\mu-\Sigma
A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})^{-1}A\mu,\Sigma-\Sigma
A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})^{-1}A\Sigma).$
###### Proof.
We have
$\displaystyle\text{Cov}(X\mid X\in S)$ $\displaystyle=\Sigma-\Sigma
A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})^{-1}A\Sigma$
$\displaystyle=\Sigma^{1/2}(I-\Sigma^{1/2}A^{\mathrm{\scriptscriptstyle{T}}}(A\Sigma
A^{\mathrm{\scriptscriptstyle{T}}})A\Sigma^{1/2})\Sigma^{1/2}$
$\displaystyle=\Sigma^{1/2}(I-P_{B})\Sigma^{1/2}=\Sigma^{1/2}(I-P_{B})^{2}\Sigma^{1/2}=\text{Cov}(P_{S}(X)),$
where $B=A\Sigma^{1/2}$ and $P_{B}=P_{A}$ since $A$ is full rank and $\Sigma$
is non-singular by assumption. A similar analysis shows that the mean also
agrees. ∎
The proof is provided in the appendix. We illustrate this within the
orthogonal calibration context. For the sake of simplicity, consider $q=1$. We
set $A^{n\times p}=(a_{1},\ldots,a_{p})$ where
$a_{j}=(g_{j}(x_{1},\tilde{\theta}),\ldots,g_{j}(x_{n},\tilde{\theta}))^{\mathrm{\scriptscriptstyle{T}}}$.
Let $b=(b(x_{1}),\ldots,b(x_{n}))^{\mathrm{\scriptscriptstyle{T}}}$ denote the
vector of bias function evaluated at $x_{1},\ldots,x_{n}$. Then the finite-
dimensional analogue of the orthogonality condition is
$A^{\mathrm{\scriptscriptstyle{T}}}b=0$. When $b_{\tilde{\theta}}(\cdot)$ is
endowed with a zero mean Gaussian process prior with covariance kernel
$C(\cdot,\cdot)$, then apriori the vector
$b_{\tilde{\theta}}\sim{\mathrm{N}}(0,K)$ with $K_{ij}=C(x_{i},x_{j}),\,1\leq
i,j\leq n$. Also,
$b_{\tilde{\theta}}\mid\theta,y^{(n)}\sim{\mathrm{N}}((K+\sigma^{2}_{F})^{-1}z^{(n)},(K+\sigma^{2}_{F})^{-1})$
where
$z^{(n)}=(y(x_{1})-f(x_{1},\theta),\ldots,y(x_{2})-f(x_{n},\theta))^{\mathrm{\scriptscriptstyle{T}}}$.
Using Proposition S.1.1, we can then obtain $b^{*}$. Hence, for Gaussian
process priors, only the second step in Algorithm 1 from the main draft will
change under the finite-dimensional projection setup.
We now demonstrate how the finite-dimensional projection can be executed when
$b_{\tilde{\theta}}$ is specified a non-Gaussian process prior. Here the
posterior of $\mathbf{b}_{k}$ is also non-Gaussian. Suppose the prior is
parameterized by $\eta$. For instance, with a BART prior, $\eta$ contains the
trees and leaf node parameters. A prior on $b_{\tilde{\theta}}$ is specified
by a prior distribution on $\eta$ which then produces a full conditional
distribution of $\eta\mid\theta,y^{(n)}$. A posterior sample of
$b_{\tilde{\theta}}\in\mathbb{R}^{n}$ can be drawn by sampling from
$\eta\mid\theta,y^{(n)}$. For many priors, the joint distribution can be well
approximated by a multivariate Gaussian with some mean and covariance
$(\beta,\Phi)$ (Yang et al., 2017). To find the approximating mean and
covariance, we simply draw $M$ independent draws of $b_{\tilde{\theta}}$ by
sampling $\\{\eta_{1},\ldots,\eta_{M}\\}$ independently from
$\eta\mid\theta,y^{(n)}$ and set
$\beta=M^{-1}\sum_{m=1}^{M}b_{\tilde{\theta},m}$ and
$\phi=(M-1)^{-1}(b_{\tilde{\theta},m}-\beta)(b_{\tilde{\theta},m}-\beta)^{\mathrm{\scriptscriptstyle{T}}}$
where $b_{\tilde{\theta},m}$ is the draw of $b_{\tilde{\theta}}$ corresponding
to $\eta_{m},\,m=1,\ldots,M$. We note here, that $M$ can be made arbitrarily
large compared to the dimension $n$ of $b_{\tilde{\theta}}$ so that the sample
mean and covariance provide reasonable approximations to their population
counterparts. This technique is summarized in the following algorithm.
1\. Sample $M$ independent copies of parameters $\eta\mid\theta,y^{(n)}$ and
consider the corresponding draws of $b_{\tilde{\theta}}$ as
$\\{b_{\tilde{\theta},1},\ldots,b_{\tilde{\theta,M}}\\}$.
2\. Compute the quantities $\beta=M^{-1}\sum_{m=1}^{M}b_{\tilde{\theta},m}$
and
$\Phi=(M-1)^{-1}(b_{\tilde{\theta},m}-\beta)(b_{\tilde{\theta},m}-\beta)^{\mathrm{\scriptscriptstyle{T}}}$.
2\. Set the projection as
$b_{\tilde{\theta}}^{*}=\Phi^{1/2}(I-P_{G})\Phi^{-1/2}b_{\tilde{\theta}}$.
3\. Update $\theta\sim\Pi(\theta\mid b^{*}_{\tilde{\theta}},y^{(n)})$.
Algorithm 2 Projection sampler 2: non-GP priors
Clearly, for non-Gaussian priors for $b_{k}$, the finite-dimensional
projection approach is computationally much more expensive. However, if the
prior is a GP, this approach is relatively less expensive to implement and
yields similar results to the functional projection approach described in the
main draft.
|
# Linear mixed modelling of federated data when only the mean, covariance, and
sample size are available
Marie Analiz April Limpoco Interuniversity Institute for Biostatistics and
Statistical Bioinformatics (I-BioStat), Data Science Institute (DSI), Hasselt
University, Hasselt, Belgium Christel Faes Interuniversity Institute for
Biostatistics and Statistical Bioinformatics (I-BioStat), Data Science
Institute (DSI), Hasselt University, Hasselt, Belgium Niel Hens
Interuniversity Institute for Biostatistics and Statistical Bioinformatics
(I-BioStat), Data Science Institute (DSI), Hasselt University, Hasselt,
Belgium Centre for Health Economic Research and Modelling Infectious Diseases
(CHERMID), Vaccine & Infectious Disease Institute, Antwerp University,
Antwerp, Belgium
###### Abstract
In medical research, individual-level patient data provide invaluable
information, but the patients’ right to confidentiality remains of utmost
priority. This poses a huge challenge when estimating statistical models such
as linear mixed models, which is an extension of linear regression models that
can account for potential heterogeneity whenever data come from different data
providers. Federated learning algorithms tackle this hurdle by estimating
parameters without retrieving individual-level data. Instead, iterative
communication of parameter estimate updates between the data providers and
analyst is required. In this paper, we propose an alternative framework to
federated learning algorithms for fitting linear mixed models. Specifically,
our approach only requires the mean, covariance, and sample size of multiple
covariates from different data providers once. Using the principle of
statistical sufficiency within the framework of likelihood as theoretical
support, this proposed framework achieves estimates identical to those derived
from actual individual-level data. We demonstrate this approach through real
data on 15 068 patient records from 70 clinics at the Children’s Hospital of
Pennsylvania (CHOP). Assuming that each clinic only shares summary statistics
once, we model the COVID-19 PCR test cycle threshold as a function of patient
information. Simplicity, communication efficiency, and wider scope of
implementation in any statistical software distinguish our approach from
existing strategies in the literature.
Keywords: linear mixed model, federated learning, sufficiency principle, data
privacy, aggregated data
Correspondence: Marie Analiz April Limpoco
Interuniversity Institute for Biostatistics and Statistical Bioinformatics
(I-BioStat),
Data Science Institute (DSI), Hasselt University, Hasselt, Belgium
Email<EMAIL_ADDRESS>
## 1 Introduction
Understanding and extracting useful information from data are some of the
shared goals between data providers and data analysts. However, both parties
must also respect the right to data privacy of individuals from whom the data
were collected. This imposes restrictions on how much and which kind of data
can be disclosed by the data providers to the data analysts.1 Consequently,
this adds to the challenge of estimating statistical models. For example, in
health research involving patient data from different hospitals, estimating a
linear mixed model to account for potential heterogeneity across hospitals
requires individual-level patient records. In the interest of data
confidentiality, hospitals might be reluctant to share the full data unless
the data analyst goes through a series of processes and paperwork, which may
take considerable time.2
A possible compromise is to employ federated learning algorithms.3 In this
setting, only parameter estimate updates and not the individual-level data are
sent by data providers to a centralized server to build a global model.4
Prevalent models for which a federated learning algorithm was developed
include linear regression,5 generalized linear mixed models,6, 7 logistic
regression, support vector machine (SVM), K-Means, neural network, Bayesian
network, and random forest.8, 9 To implement this strategy, a network of
computer connections that enable iterative communication between the data
providers and data analysts has to be set up. In practice, at least in the
healthcare context, it is not easy to set up such networks that fulfill the
requirements conducive to federated learning.10, 11, 12
Instead of an iterative communication, data providers may be more willing to
share summary data once. For a linear regression model, if these summary data
contain sufficient statistics, then model estimation is possible even when the
individual-level data are unavailable.13 For a linear mixed model involving
random effects per data provider, Luo et al14 expressed the likelihood in
terms of aggregate matrices, which in turn are composed of sufficient
statistics. They did this by applying the Woodbury matrix identity and some
linear algebra concepts. Since most of the existing functionalities in
statistical softwares require the individual observations as input, model
estimation using summaries requires the development of new functionalities.
Luo et al14 developed an R package called pda 15 to implement their proposed
method, which they referred to as distributed linear mixed model (DLMM), but
its structure is in a distributed model estimation context. Specifically, in
practice, data providers must send their aggregate matrices to a central
online server.
In the meta-analysis setting where the “data providers" are the relevant
studies, there are advantages of using individual participant data (IPD) over
aggregate data (AD) especially when the number of studies is small.16, 17
However, IPD is seldom available from studies and so constructing a substitute
for the unavailable IPD, called pseudo-IPD, was an alternative. One strategy
is to require the specification of the joint distribution and correlation
structure from which several sets of pseudo-IPD will be simulated to obtain
parameter estimates per set.16 These will then be aggregated to arrive at a
single estimate per parameter of interest. Since this method involves
simulating from an assumed distribution whose parameters are based on the
available AD, a point to consider in practice is how many sets should be
produced to achieve the desired accuracy.
Another approach requires only one set of pseudo-IPD to yield estimates
exactly equal to the actual IPD estimates.17 In this approach, pseudo-IPD are
constructed by first generating random numbers from any distribution and then
transforming them such that the mean and standard deviation of the resulting
data are exactly equal to those indicated in the studies. Unlike DLMM which
also yields exactly the same estimates as the actual IPD estimates, this
pseudo-IPD approach can be implemented in any statistical software with
existing linear mixed model functionalities. A limitation of both pseudo-IPD
methods though, in the context of meta-analysis, is that the covariance
structure of the relevant variables is rarely available from studies, thus
making it difficult to include more covariates in the model. In particular,
their scope only includes studies with two treatments or exposure groups. In
the context of federated data, this is not a problem since data providers like
hospitals may be willing to supply the covariance structure of the variables,
aside from the mean vector.
In this paper, we apply the pseudo-IPD meta-analysis framework of
Papadimitropoulou et al 17 to the federated data setting to estimate a linear
mixed model with random effects per data provider when only the mean,
covariance, and sample size are made available once. More importantly, we
extend it so that multiple covariates can be included, distinguishing our
approach from theirs. This principle of using pseudo-data in the context of
federated data analysis is novel. To summarize the difference of our approach
from those that exist in the literature, we present Table 1. Among the
existing methods, DLMM and our proposed strategy employ the concept of
sufficient statistics and are thus expected to yield theoretically the same
results. The major difference lies in how the sufficient statistics are
utilized: DLMM uses them directly in the re-expressed form of the likelihood,
while we use them to generate pseudo-data before feeding these into the
classical form of the likelihood. Hence, our proposed strategy may be
considered as an alternative and is not necessarily superior over DLMM; though
it has some advantages in terms of generalisability and implementation. The
following points distinguish our approach from DLMM:
1. 1.
Extension to more complex models. The principle of generating pseudo-data
facilitates model building, performing post hoc procedures, and extension to
more complex models such as generalized linear models (GLMs) while keeping the
communication exchange with the data providers to just once. Although the idea
of DLMM has already been extended to GLMs, at least one communication exchange
is needed. In addition, in their current framework, only linear models have
the option to include random effects.18 On the contrary, we have found in our
current research work that it is possible to apply a similar principle of
pseudo-data generation from one-time shared summary statistics to the other
members of the GLM family. While in this paper we focus on linear models, we
will discuss the extension towards GLMs in the discussion of the paper.
2. 2.
Practical implementation. Our proposed method makes use of the classical form
of the likelihood which feeds on individual observations. This is the basis
for existing LMM functionalities in various statistical softwares. This means
that after the pseudo-data are generated, we can use any of the existing
software programs available to fit the mixed model. This is advantageous due
to the strength of existing functionalities for linear mixed modelling which
can be found not only in R 15 but also in other statistical softwares. This
implies that compared to pda,19 current versions of existing software packages
(e.g. lmer in lme4 20) have been optimized in terms of numerical stability,
computational efficiency, optimization algorithms, and bugs fixing.20
Furthermore, user support is more available online for possible problems that
a user may encounter when using these existing functionalities. Thus,
utilizing these existing functionalities may be preferred by practitioners, if
possible.
Table 1: Difference of proposed strategy from existing methods
Aspect | Our | Federated | DLMM | pseudo-IPD
---|---|---|---|---
| approach | learning | | in meta-analysis
Input | one set of | parameter estimate | aggregate | at least one set
| pseudo-data | updates until | data | of pseudo-data
| with identical | convergence | matrices | with similar
| sufficient statistics | | | sufficient statistics
| as actual data | | | as actual data
Number of | multiple | multiple | multiple | one/limited
covariates | | | |
Estimation | use pseudo-data | global model is | likelihood is rewritten | same as our
strategy | as substitute to | estimated iteratively | in terms of aggregate | approach but
| actual unavailable | from local parameter | matrices instead of | repeated multiple
| data | estimates of | individual observations | times; estimates
| | data providers | | aggregated
| | | | into one
Theoretical estimates | identical to | close to | identical to | close to
produced | estimates from | estimates from | estimates from | estimates from
| actual data | actual data | actual data | actual data
Communication rounds | once | more than once | once | once
between data providers | | | |
and data analyst | | | |
Infrastructure | none | central server must be | online server where | none
requirements | | connected to data | data providers send |
| | providers’ databases | aggregate matrices |
Software | any statistical | TensorFlow Federated, | R package pda 15 | any statistical
implementation | software with | OpenFL, PySyft | | software with
| LMM functionality | | | LMM functionality
In the next section (2), we present the details of our proposed framework. We
then demonstrate it in section 3 on deidentified publicly available real data
consisting of the results of COVID-19 testing at the Children’s Hospital of
Pennsylvania (CHOP) in 2020 which can be found in the R15 package
medicaldata.21 Finally, we discuss implications as well as potential research
directions in section 4, before we close with a conclusion.
## 2 Methods
### 2.1 Principles of data reduction
This section briefly revisits the principles of data reduction, which are
thoroughly discussed by Casella and Berger.22 We aim to draw insights about
dealing with parameter estimation given limited data. We begin with the
sufficiency principle, which guarantees that the entire sample need not be
available, as inferences about a parameter $\theta$ can be derived from the
sufficient statistic $T(\mathbf{X})$, if it exists. In other words, even if
the only information known is $T(\mathbf{x})$, inference about the parameter
of interest $\theta$ can still be made. In connection with our objective, if
the data providers supply sufficient statistics for the model of interest,
which in this case is the linear mixed model, then parameter estimation is
still possible even without disclosing the individual-level data.
Once sufficient statistics are identified, if they exist, parameter estimation
can then proceed by directly plugging in the sufficient statistics instead of
the individual observations into the log-likelihood of the model. An
alternative strategy is to generate a sample $\mathbf{x_{2}}$ which is
different from the original sample $\mathbf{x_{1}}$, but such that
$T(\mathbf{x_{1}})=T(\mathbf{x_{2}})$, and use $\mathbf{x_{2}}$ as input to
the existing functionalities that estimate a linear mixed model. The
sufficiency principle will still guarantee that the same conclusion can be
drawn even though $\mathbf{x_{1}}\neq\mathbf{x_{2}}$. Casella and Berger22
explicitly mentioned that the conditional probability distribution given a
sufficient statistic $T(\mathbf{x_{1}})$ can be used to draw a sample
$\mathbf{x_{2}}$ and generate equivalent information about $\theta$. However,
they did not discuss how to obtain this probability distribution in practice.
To supplement the sufficiency principle, we use the likelihood principle
(Appendix A.1). Specifically, if we aim to exactly estimate a parameter
through its likelihood based on $\mathbf{x_{1}}$ but only its sufficient
statistic $T(\mathbf{x_{1}})$ is available, we can generate a sample
$\mathbf{x_{2}}$ such that $T(\mathbf{x_{1}})=T(\mathbf{x_{2}})$ and use the
likelihood based on $\mathbf{x_{2}}$ to estimate the parameter of interest. To
this end, the distribution that generated $\mathbf{x_{2}}$ becomes immaterial.
In the succeeding sections, we implement this idea. Specifically, we first
identify the sufficient statistics and then generate another sample which we
refer to as pseudo-data such that the sufficient statistics of the actual and
pseudo-data are equal. We start with sufficient statistics for a linear
regression model and extend the idea to a linear mixed model.
### 2.2 Sufficient statistics for a linear regression model
The concepts presented in section 2.1 were demonstrated by Lee, Brown, and
Ryan13 for a linear regression model. In particular, given $n$ observations,
an intercept, and $p-1$ predictors, if $\mathbf{X}$ denotes the $n\times p$
design matrix and $\mathbf{y}$ is the $n\times 1$ vector of continuous
responses, the linear regression coefficients $\bm{\beta}$ are estimated as
$\displaystyle\hat{\bm{\beta}}$
$\displaystyle=(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T}\mathbf{y},$
where knowing the aggregate information $\mathbf{X}^{T}\mathbf{X}$ and
$\mathbf{X}^{T}\mathbf{y}$ instead of the individual-level data $\mathbf{X}$
and $\mathbf{y}$ will still yield exactly the same regression coefficient
estimates $\hat{\bm{\beta}}$. We expand on this and explicitly show that the
sample mean, sample covariance matrix, and sample size are sufficient to
produce the same parameter estimates as with using the individual-level data.
In addition to the work of Lee, Brown, and Ryan13, we include estimation of
the variance
$\displaystyle\hat{\sigma}^{2}_{MLE}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}\hat{\bm{\beta}})^{2},\text{
or}$ $\displaystyle\hat{\sigma}^{2}_{OLS}$
$\displaystyle=\frac{1}{n-p}\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}\hat{\bm{\beta}})^{2}.$
For more specific details regarding the derivations, see Appendix A.2.
We begin by looking at the log-likelihood
$\displaystyle l(\bm{\beta},\sigma^{2};\mathbf{y},\mathbf{X})$
$\displaystyle=-\frac{n}{2}\text{ln}(2\pi)-\frac{n}{2}\text{ln}(\sigma^{2})-\frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2},$
where $\mathbf{x}_{i}$ is a vector representing the $i$th row in the design
matrix $\mathbf{X}$. We see here that information from the sample is required
only in the last term. Moreover, the sum of squares of this term can be
expressed as
$\displaystyle\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$
$\displaystyle=\sum_{i=1}^{n}y_{i}^{2}-2\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}+\sum_{i=1}^{n}\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}.$
From this, we find that knowing $n$, $\sum_{i=1}^{n}y_{i}^{2}$,
$\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$, and
$\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ is sufficient to construct
the log-likelihood and estimate the parameters, even in the absence of
individual-level data. In particular, $\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$
and $\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ are sufficient to
estimate the coefficients $\bm{\beta}$ while the variance $\sigma^{2}$ also
requires $\sum_{i=1}^{n}y_{i}^{2}$ in addition to the other two. Furthermore,
these values can be obtained from the vector of sample means and sample
covariance matrix of the response variable and the predictors. Specifically,
performing some algebraic manipulations will show that
$\sum_{i=1}^{n}y_{i}^{2}$ can be derived from the sample variance
$s^{2}_{\mathbf{y}}$, the sample mean $\bar{\mathbf{y}}$, and the sample size
$n$
$\displaystyle\sum_{i=1}^{n}y_{i}^{2}$
$\displaystyle=s^{2}_{\mathbf{y}}(n-1)+n\bar{\mathbf{y}}^{2}.$
For $\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$, we note that it is a $1\times p$
matrix
$\displaystyle\begin{bmatrix}\sum_{i=1}^{n}y_{i}&\sum_{i=1}^{n}y_{i}x_{i1}&\sum_{i=1}^{n}y_{i}x_{i2}&...&\sum_{i=1}^{n}y_{i}x_{ij}&...&\sum_{i=1}^{n}y_{i}x_{i(p-1)}\end{bmatrix},$
such that the first element can be obtained from the sample mean
$\bar{\mathbf{y}}$ while the rest of the elements needs the sample covariance
between $\mathbf{y}$ and each of the predictors ($s_{\mathbf{yx}_{j}}$).
Lastly, since $\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$is a $p\times p$
matrix
$\displaystyle\begin{bmatrix}n&\sum_{i}{x_{i1}}&\ldots&\sum_{i}{x_{i(p-1)}}\\\
\sum_{i}{x_{i1}}&\sum_{i}{x_{i1}^{2}}&\ldots&\sum_{i}{x_{i1}x_{i(p-1)}}\\\
\vdots&\vdots&\ddots&\vdots\\\
\sum_{i}{x_{i(p-1)}}&\sum_{i}{x_{i(p-1)}x_{i1}}&\ldots&\sum_{i}{x_{i(p-1)}^{2}}\end{bmatrix},$
performing similar derivations as above reveals that computing
$\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ only requires the sample mean
($\bar{\mathbf{x}}_{j}$), sample variance ($s^{2}_{\mathbf{x}_{j}}$), and
sample covariances among predictors $j$ and $k$
($s_{\mathbf{x}_{j}\mathbf{x}_{k}}$).
### 2.3 Sufficient statistics for a linear mixed model
A more realistic assumption when handling federated data is that the
observations from the same data provider are more similar than observations
from different sources, violating the independence assumption of a linear
regression model. To account for this, a linear mixed model is more
appropriate. Assuming that there are $m$ data providers, let $y_{hi}$ be the
continuous response of individual $i$ from data provider $h$;
$\mathbf{x}_{hi}$ be a $p$-dimensional vector consisting of an intercept and
$p-1$ predictors; $\bm{\beta}$ be the p-dimensional vector of fixed effects;
$\mathbf{z}_{hi}$ be the $q$-dimensional vector corresponding to the $q$
random effects; ${\mathbf{u}}_{h}$ be the $q$-dimensional random effects
vector, which represents the deviation of data provider $h$ from the overall
pattern; and $\epsilon_{hi}\sim N(0,\sigma^{2})$ be the random error. For a
linear mixed model with random effects for each data provider, the model
structure will be
$\displaystyle y_{hi}$
$\displaystyle=\mathbf{x}_{hi}^{T}\bm{\beta}+{\mathbf{z}}_{hi}^{T}{\mathbf{u}}_{h}+\epsilon_{hi}.$
For a model with a random intercept and slope, $q=2$,
$\mathbf{z}_{hi}=[1,z_{hi}]$ and $\mathbf{u}_{h}\sim N(\bm{0},\mathbf{G})$
where $\mathbf{G}$ is the $2\times 2$ random effects covariance matrix. To
estimate parameters, the marginal log-likelihood used is
$\displaystyle l(\bm{\beta},\sigma^{2},\mathbf{G};\mathbf{y},\mathbf{X})$
$\displaystyle=-\frac{1}{2}\sum_{h=1}^{m}\\{\text{log}|\bm{\Sigma}_{h}|+(\mathbf{y}_{h}-\mathbf{X}_{h}\bm{\beta})^{T}\bm{\Sigma}_{h}^{-1}(\mathbf{y}_{h}-\mathbf{X}_{h}\bm{\beta})\\},$
where $\mathbf{X}_{h}$ and $\mathbf{y}_{h}$ are the design matrix and response
vector, respectively, of data provider $h$, $|.|$ is the matrix determinant
and
$\bm{\Sigma}_{h}=\bm{\Sigma}_{h}(\sigma^{2},\mathbf{G})=\mathbf{Z}_{h}\mathbf{G}\mathbf{Z}_{h}^{T}+\sigma^{2}I_{n_{h}}$.
Due to the seemingly entangled data and parameter matrices, it is not
straightforward to identify the aggregate statistics that can be used when the
individual-level data are not available. Luo et al14 showed that by utilizing
the Woodbury matrix identity and some linear algebra concepts, the data can be
disentangled from the parameters to reconstruct the profile log-likelihood. In
their approach, only aggregate matrices $\mathbf{X}_{h}^{T}\mathbf{X}_{h}$,
$\mathbf{X}_{h}^{T}\mathbf{y}_{h}$, $\mathbf{y}_{h}^{T}\mathbf{y}_{h}$, and
$n_{h}$ from each data provider $h$ are required to produce identical
estimates as those produced with the individual-level data. We have shown in
the previous section that these aggregate data can actually be derived from
the sample mean, sample covariance matrix, and sample size of each data set.
Therefore, these summary statistics per data provider (e.g. per hospital) are
also sufficient to estimate a linear mixed model in the absence of individual-
level data.
### 2.4 Proposed method
In this section, we provide details for creating the pseudo-data for a single
variable and then extend it to a set of variables. As mentioned earlier, we
opt to use these pseudo-data as input to the model estimation process rather
than directly utilizing the sufficient statistics so that we can still use the
existing linear regression or linear mixed model functionality in any
statistical software such as the lmer function in the R package lme4.20
#### 2.4.1 Single variable
The goal of constructing pseudo-data for linear models is to have exactly the
same sample mean and variance as the original unavailable individual-level
data. We do this by performing a linear transformation. For instance, suppose
the original univariate sample $\mathbf{x}_{d}$ is unknown, but its sample
mean $\bar{x}_{d}$ and sample standard deviation $s_{d}$ are available. We
consider a linear transformation of a randomly generated data set
$\mathbf{x}_{r}$ into pseudo-data $\mathbf{x}_{\pi}$ which has equal mean and
standard deviation as $\mathbf{x}_{d}$. To this end, we let
$\displaystyle x_{\pi_{i}}$ $\displaystyle=a+bx_{r_{i}},$
where $\mathbf{x}_{r}$ can come from any distribution and has sample mean
$\bar{x}_{r}$ and sample standard deviation $s_{r}$. Examining the
relationship between the mean of the randomly generated data $\bar{x}_{r}$ and
that of the transformed data $\bar{x}_{\pi}$, we find that
$\displaystyle\bar{x}_{\pi}=a+b\bar{x}_{r},$
while for the variances:
$\displaystyle s_{\pi}^{2}=b^{2}s_{r}^{2},$
from which we find that to obtain identical means $\bar{x}_{d}=\bar{x}_{\pi}$
and standard deviations $s_{d}=s_{\pi}$ between the original unknown data
$\mathbf{x}_{d}$ and the pseudo-data $\mathbf{x}_{\pi}$, we should let
$\displaystyle b$
$\displaystyle=\frac{s_{\pi}}{s_{r}}=\frac{s_{d}}{s_{r}},\text{ and}$
$\displaystyle a$
$\displaystyle=\bar{x}_{\pi}-\frac{s_{\pi}}{s_{r}}\bar{x}_{r}=\bar{x}_{d}-\frac{s_{d}}{s_{r}}\bar{x}_{r},$
so that
$\displaystyle x_{\pi_{i}}$
$\displaystyle=\bar{x}_{d}+s_{d}\frac{x_{r_{i}}-\bar{x}_{r}}{s_{r}}.$
In summary, the algorithm to generate pseudo-data for a single variable is:
1. 1.
Generate $n$ random numbers $\mathbf{x}_{r}$ from any distribution.
2. 2.
Compute the sample mean $\bar{x}_{r}$ and sample standard deviation $s_{r}$ of
$\mathbf{x}_{r}$.
3. 3.
Transform $\mathbf{x}_{r}$ into the desired pseudo-data $\mathbf{x}_{\pi}$
using:
$\displaystyle x_{\pi_{i}}$
$\displaystyle=\bar{x}_{d}+s_{d}\frac{x_{r_{i}}-\bar{x}_{r}}{s_{r}},$
where $\bar{x}_{d}$ and $s_{d}$ are the sample mean and sample standard
deviation, respectively, of the original unavailable data.
#### 2.4.2 Set of variables
The strategy to construct pseudo-data for more than one variable is analogous
to that for a single variable. Ripley 23 presents an approach for multivariate
normal distribution, but for our case, we do not have to impose normality on
the pseudo-data nor on the initial set of random numbers. We just have to
ensure that the resulting pseudo-data has exactly the same sample mean vector
and sample covariance matrix as the individual-level data. Thus, from the
algorithm for a single variable, we replace $\bar{x}_{d}$ with the sample mean
vector for all variables ($\hat{\bm{\mu}}_{d}$). As for $s_{d}$, we take the
Cholesky decomposition of the covariance matrix of all variables
($\hat{\bm{\Sigma}}_{d}$). Note that for a multiple linear regression or a
linear mixed model, the set of variables comprises both the predictors and the
response variable. The following algorithm provides a summary to generate
pseudo-data with sample size $n$ for $p$ variables consisting of $p-1$
predictors and a continuous response variable:
1. 1.
Generate random numbers
$\mathbf{R}=[\mathbf{r}_{1},...,\mathbf{r}_{i},...\mathbf{r}_{n}]^{T}$ which
is an $n\times p$ matrix where each column is generated independently from any
distribution.
2. 2.
Compute the mean vector $\hat{\bm{\mu}}_{r}$ and the covariance matrix
$\hat{\bm{\Sigma}}_{r}$ of $\mathbf{R}$.
3. 3.
Generate the $i$th pseudo-data point as
$\bm{\pi}_{i}=\hat{\bm{\mu}}_{d}+L_{\hat{\bm{\Sigma}}_{d}}(L_{\hat{\bm{\Sigma}}_{r}})^{-1}(\mathbf{r}_{i}-\hat{\bm{\mu}}_{r}),$
where $L_{\hat{\bm{\Sigma}}_{d}}$ and $L_{\hat{\bm{\Sigma}}_{r}}$ are the
lower triangular matrices of the Cholesky decomposition of
$\hat{\bm{\Sigma}}_{d}$ and $\hat{\bm{\Sigma}}_{r}$. Here, $\bm{\pi}_{i}$ is
the pseudo-data vector consisting of the response variable and the predictors;
that is,
$[y_{\pi_{i}},x_{\pi_{i1}},...,x_{\pi_{ij}},...,x_{\pi_{i(p-1)}}]^{T}$.
An issue that arises with using the Cholesky decomposition is the need to have
a positive definite covariance matrix. In practical settings, especially when
there are binary variables involved, this may not be true. An alternative
procedure is to perform a singular value decomposition (SVD) on the centered
values of $\mathbf{R}$ to obtain the matrix of right singular vectors denoted
as $\mathbf{V}$. SVD can always be performed on any rectangular matrix such as
this $n\times p$ matrix $\mathbf{R}$. The product of $\mathbf{R}$ and
$\mathbf{V}$ is then computed and the resulting matrix elements are then
divided by the root mean square $\sqrt{\sum_{i}x_{rv_{ij}}^{2}/(n-1)}$ of each
column $j$. Denoting this as $\mathbf{R_{V}}$, the $i$th pseudo-data point is
then generated as
$\displaystyle\bm{\pi}_{i}$
$\displaystyle=\hat{\bm{\mu}}_{d}+\mathbf{U}\bm{\Lambda}^{1/2}\mathbf{r_{v}}_{i}\hskip
2.84526pt,$
where $\mathbf{U}$ is the matrix of eigenvectors of $\hat{\bm{\Sigma}}_{d}$
and $\bm{\Lambda}^{1/2}$ is a diagonal matrix whose elements are the square
root of the eigenvalues of $\hat{\bm{\Sigma}}_{d}$. Eigendecomposition only
requires that the matrix is diagonalized. Since a covariance matrix is always
symmetric, it follows that it is always diagonalized and thus eigenvectors and
eigenvalues can always be computed.
This alternative procedure is implemented by the function mvrnorm in the R
package MASS24 where each column of $\mathbf{R}$ is generated from a standard
normal distribution. The user may wish to explore using a different
distribution such as a uniform distribution to generate $\mathbf{R}$ by
slightly altering the code for this function. Note that mvrnorm returns an
error whenever the sample size of the pseudo-data $n$ is smaller than the
number of variables $p$. The reason behind this is that the function svd used
in mvrnorm returns a reduced matrix $\mathbf{V}$, affecting the dimension of
the matrix $\mathbf{R_{V}}$ and making the matrix multiplication incompatible.
We modified this setting so that the full SVD is returned. This R code is
provided in Appendix LABEL:modified_mvrnorm.
## 3 An illustrative example: COVID-19 testing at CHOP
We demonstrate the proposed approach on a real dataset: the COVID-19 testing
results from different clinics at the Children’s Hospital of Pennsylvania
(CHOP). It is a publicly available and deidentified dataset consisting of all
patients who got tested at the hospital, and can be accessed from the R
package medicaldata 21. This dataset provides information about patients at
CHOP who got tested for COVID-19 from days four to 107 of the pandemic in
2020. A total of 88 clinics from this hospital provided 15 524 patient records
which were anonymized, time-shifted, and permuted. The COVID-19 test was
performed via PCR. For a description of the variables, the reader may refer to
the documentation of the medicaldata 21 package which is available online.
Since we have access to the entire individual-level data, we will demonstrate
how data providers can preprocess their raw data to achieve the expected
summary data for our proposed method. In general, the data provider must
supply the name and a brief description of each variable, the number of
observations and mean per variable, and the covariance matrix. For model
selection purposes, they must also provide the summary statistics for the
standardized version, log- and squared transformations of the numeric
variables, as well as for the two-way interactions among the variables. We
illustrate these using R software, but these results can also be implemented
using any other statistical software.
Suppose we are interested in how factors such as gender, age, and whether the
specimen was collected in a drive-thru site affect the cycle at which
threshold reached during PCR (numeric variable from 14 to 45). For this data
set, the cycle threshold is the response variable while gender, age, drive
thru indicator, and an interaction term for age and gender are the predictors.
The cycle at which threshold reached during PCR is a measure of how much
amplification is necessary to detect the target viral gene, and is inversely
proportional to viral load.25 This means that if more cycles are needed to
detect a viral gene, then the presence of the viral gene is less likely. Some
studies found a negative correlation between the cycle threshold and disease
severity 26 and mortality among patients.27 In this illustrative example, we
will model how the aforementioned regressors influence this measure to
hypothesize about the clinical outcomes among patients at CHOP.
### 3.1 Preliminary analysis by the data provider
Each data provider should supply the data analyst with a good overview of all
the available variables. The data provider may use the function skim from the
R package skimr28 to accomplish this. Since the CHOP data from R is already
composed of the pooled individual-level records from all 88 clinics, applying
the aforementioned function covers all clinics already. Table 2 displays the
metadata for all 88 clinics having a total of 15 524 patient records during
the COVID-19 pandemic in 2020.
Table 2: Metadata of all clinics at CHOP
Variable | Type | Number of | Complete | Number of | Number of |
---|---|---|---|---|---|---
name | | missing observations | rate | empty cells | unique values |
Fake first name | char | 0 | 1.00 | 0 | 832 |
Fake last name | char | 0 | 1.00 | 0 | 27 |
Gender | char | 0 | 1.00 | 0 | 2 |
Test ID | char | 0 | 1.00 | 0 | 2 |
Clinic name | char | 0 | 1.00 | 0 | 88 |
Result | char | 0 | 1.00 | 0 | 3 |
Demographic group | char | 0 | 1.00 | 0 | 5 |
Payor group | char | 7087 | 0.54 | 0 | 7 |
Patient class | char | 7077 | 0.54 | 0 | 9 |
Subject ID | num | 0 | 1.00 | - | - |
Day of | | | | | |
pandemic | num | 0 | 1.00 | - | - |
Age | num | 0 | 1.00 | - | - |
Drive thru | | | | | |
indicator | num | 0 | 1.00 | - | - |
Cycle threshold | | | | | |
result | num | 209 | 0.99 | - | - |
Orderset | num | 0 | 1.00 | - | - |
Collection to | | | | | |
receive time | num | 0 | 1.00 | - | - |
Receive to | | | | | |
verification time | num | 0 | 1.00 | - | - |
Our proposed method assumes that the summary data per data provider were
computed from complete observations. Additionally, for categorical variables,
the data provider must also indicate the levels. For instance, for binary
variables such as gender, the variable name in the summary data must reflect
the non-reference category (e.g. gendermale). For this dataset, of the 88
clinics, only 70 clinics with a total of 15 068 observations were included in
the analysis after filtering out incomplete observations and invalid values
such as NA. Those clinics with only one observation were removed as well
because in a federated data setting, it is not so common for a data provider
(e.g. a clinic) to have only one patient. Moreover, even if a clinic with only
one patient exists, summarizing the data does not make sense and neither does
handing over the single patient record to the data analyst because it goes
against that patient’s right to data confidentiality. Should the clinic do so
after fulfilling some legal requirements to ensure privacy, this observation
itself can be combined with the generated pseudo-data. In Appendix
LABEL:chop_app, we provide an R code for preprocessing this data set. As an
example, Table 3 displays the summary statistics for the Inpatient Ward A
clinic at CHOP as well as the covariance matrix for the variables we are
including in our model.
Table 3: Summary statistics for the Inpatient Ward A clinic at CHOP
Variable | n | Mean | Variance-Covariance
---|---|---|---
name | | | log of Cycle | Gendermale | Age | Drive thru | Gendermale $\times$
| | | threshold | | | | Age
log of Cycle threshold | 208 | 3.803 | 0.001 | 0.001 | 0.001 | 0.000 | -0.001
Gendermale | 208 | 0.529 | 0.001 | 0.250 | 0.053 | 0.002 | 0.369
Age | 208 | 1.373 | 0.001 | 0.053 | 10.506 | 0.003 | 6.621
Drive thru | 208 | 0.005 | 0.000 | 0.002 | 0.003 | 0.005 | 0.006
Gendermale $\times$ Age | 208 | 0.779 | -0.001 | 0.369 | 6.621 | 0.006 | 7.085
### 3.2 LMM estimation by the data analyst
After receiving the sufficient statistics from the data providers, the data
analyst formulates the linear mixed model. To study the relationship between
COVID-19 PCR test cycle threshold and patient information namely gender, age,
their interaction, and the drive thru indicator, we fit two linear mixed
models: one with only a random intercept per clinic and another with a random
intercept and a random slope for age. This allows variations in mean cycle
threshold and age effects across the clinics. We use the logarithm of the
cycle threshold to ensure nonnegative values for this variable and we
standardize age to avoid numerical problems during the optimization of the
log-likelihood. One set of pseudo-data was generated for each clinic using the
proposed method. Using the lmer function in the R package lme4 on the pseudo-
data and on the actual data, we estimate the parameters of the model. Table 4
displays the results for the model with only a random intercept while Table 5
presents the estimates for an LMM with an additional random slope for age. For
both models, we find that only the scaled residuals are different between the
LMM using pseudo-data and using actual data. This is expected since residuals
are computed from individual observations and thus cannot be reproduced from a
different set of values such as the pseudo-data. Additionally, we can
reproduce the AIC and confidence intervals, as shown in the same tables. We
select the model with both random intercept and random slope since it has
lower AIC and BIC values.
From the estimates of the selected model, we observe that among the fixed
effects, the interaction of gender and age significantly affect the cycle
threshold, whereas drive-thru testing does not. This significant interaction
effect indicate that the overall impact of a patient’s gender on the log cycle
threshold also depends on the age group of the patient, and vice versa.
Moreover, since the effects of age are allowed to vary across clinics, the
interpretation also varies per clinic. For instance, for a male patient who
belongs to clinic $h$ and is one standard deviation or around 16 years older,
the log cycle threshold changes by $-0.0057+b_{age_{h}}$. If he belongs to the
Inpatient Ward A, the log cycle threshold decreases by $0.008$. On the other
hand, a female patient belonging to the same clinic is expected to have
$0.003$ decrease in log cycle threshold for every one standard deviation
increase in age. This suggests that for this clinic, older patients tend to
have more viral load, although the difference across age groups is slightly
more pronounced among males and among females. Note that this per clinic
interpretation is also supported by our proposed approach since the random
effects prediction are exactly equal to those derived from the actual data..
Lastly, the estimated variance components of the random effects suggest that
there is not much variation across the clinics.
Table 4: Comparison of LMMs with random intercept only based on pseudo-data
and based on actual CHOP data.
| LMM estimates
---|---
| pseudo-data | actual data
REML criterion at convergence | -20473 | -20473
Scaled residuals: | |
Min | -4.5919 | -9.3287
$Q_{1}$ | -0.5773 | 0.1482
Median | 0.0249 | 0.2174
$Q_{3}$ | 0.5760 | 0.2538
Max | 3.7464 | 1.1855
(Intercept) | 3.7871 (0.0039)*** | 3.7871 (0.0039)***
Gendermale | 0.0021 (0.002)*** | 0.0021 (0.002)***
Std. Age | -0.0046 (0.0015)*** | -0.0046 (0.0015)***
Drive thru | -0.0043 (0.0058)*** | -0.0043 (0.0058)***
Gendermale $\times$ Std. Age | -0.0061 (0.0020)*** | -0.0061 (0.0020)***
AIC | -20459.04 | -20459.04
BIC | -20405.7 | -20405.7
n | 15068 | 15068
number of clinics | 70 | 70
$\sigma_{Int}$ | 0.0216 | 0.0216
$\sigma$ | 0.1222 | 0.1222
| 95% confidence bounds
---|---
| pseudo-data | actual data
| 2.5 % | 97.5 % | 2.5 % | 97.5 %
$\sigma_{Int}$ | 0.0160 | 0.0282 | 0.0160 | 0.0282
$\sigma$ | 0.1208 | 0.1235 | 0.1208 | 0.1235
(Intercept) | 3.7793 | 3.7949 | 3.7793 | 3.7949
Gendermale | -0.0018 | 0.006 | -0.0018 | 0.006
Std. Age | -0.0076 | -0.0015 | -0.0076 | -0.0015
Drive thru | -0.0156 | 0.0071 | -0.0156 | 0.0071
Gendermale $\times$ Std. Age | -0.0100 | -0.0022 | -0.0100 | -0.0022
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$ | |
Table 5: Comparison of LMMs with random intercept and random slope for age
based on pseudo-data and based on actual CHOP data.
| LMM estimates
---|---
| pseudo-data | actual data
REML criterion at convergence | -20513.2 | -20513.2
Scaled residuals: | |
Min | -4.7364 | -9.3604
$Q_{1}$ | -0.5789 | 0.1229
Median | 0.0241 | 0.2030
$Q_{3}$ | 0.5777 | 0.2560
Max | 3.7682 | 1.8698
(Intercept) | 3.7851 (0.0045)*** | 3.7851 (0.0045)***
Gendermale | 0.0021 (0.0020)*** | 0.0021 (0.0020)***
Std. Age | -0.0005 (0.0037)*** | -0.0005 (0.0037)***
Drive thru | -0.0038 (0.0059)*** | -0.0038 (0.0059)***
Gendermale $\times$ Std. Age | -0.0052 (0.0020)*** | -0.0052 (0.0020)***
AIC | -20495.15 | -20495.15
BIC | -20426.57 | -20426.57
n | 15068 | 15068
number of clinics | 70 | 70
$\sigma_{Int}$ | 0.0249 | 0.0249
$\sigma_{Age}$ | 0.0128 | 0.0128
$\sigma_{\text{Int}\times\text{Age}}$ | -0.10 | -0.10
$\sigma$ | 0.1219 | 0.1219
| 95% confidence bounds
---|---
| pseudo-data | actual data
| 2.5 % | 97.5 % | 2.5 % | 97.5 %
$\sigma_{Int}$ | 0.0185 | 0.0323 | 0.0185 | 0.0323
$\sigma_{\text{Int}\times\text{Age}}$ | -0.5797 | 0.3917 | -0.5797 | 0.3917
$\sigma_{Age}$ | 0.008 | 0.019 | 0.008 | 0.019
$\sigma$ | 0.1205 | 0.1233 | 0.1205 | 0.1233
(Intercept) | 3.7762 | 3.7942 | 3.7762 | 3.7942
Gendermale | -0.0018 | 0.006 | -0.0018 | 0.006
Age | -0.0081 | 0.0074 | -0.0081 | 0.0074
Drive thru | -0.0152 | 0.0077 | -0.0152 | 0.0077
Gendermale $\times$ Age | -0.0092 | -0.0013 | -0.0092 | -0.0013
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$ | |
## 4 Discussion
In this paper, we have demonstrated that data privacy in a federated data
setting can be achieved not only through machine learning algorithms but even
through age-old statistical concepts such as sufficiency and likelihood. These
principles enable data reduction without losing important information about
the parameter of interest. Specifically, since the sufficient statistics
already contain the information required to estimate the parameters of a
statistical model, data providers do not have to share individual observations
anymore. This approach is very useful and applicable to settings where
individual-level data are too sensitive to be shared, such as patient data
from hospitals. For a linear regression model and a linear mixed model, this
is true since their sufficient statistics exist. For other more complex models
such as generalized linear models with and without random effects, this may
not be as straightforward.
Since our approach produces identical estimates as those obtained from pooling
the individual-level records across multiple data providers, we have
confidence that our estimates are as good as the estimates that the
established estimation techniques claim. Among the methods proposed in the
literature, maximum likelihood (ML) and residual or restricted maximum
likelihood (REML) have become the standard methods for estimating the
parameters of a linear mixed model.29 However, between these two, ML
estimators do not account for the degrees of freedom lost when estimating the
fixed effects, resulting in biased variance parameter estimates towards the
null, especially for small samples.30, 31 For this reason, REML estimation of
the variance parameters is preferred over ML. Moreover, REML estimators have
shown improved properties whenever the number of clusters is small,32, 33
which suffers from finite sample bias more than models involving small cluster
sizes.34 Despite these desirable properties, REML does not completely solve
the issues related to inflated Type I error rates for fixed effects.35 To
address this, the Kenward-Roger correction has been recommended as best
practice in the literature since it has been shown to maintain nominal Type I
error rates.32 Due to the nature of our proposed strategy, versatility in
specifying the estimation procedure (ML or REML) and applying corrections
(e.g. Kenward-Roger) whenever the sample size is small can be easily
implemented with identical results as with the actual observations.
In contrast to the study of Luo et al 14 which also yields exactly the same
estimates for LMM, our approach is a simple one and thus can more easily be
implemented in practice. Another advantage of our proposed framework is that
we do not need to specify a distribution from which the pseudo-data come from,
unlike the method proposed by Song et al.16 Additionally, the concept can be
applied using any statistical software that can estimate LMM, thus enabling a
wider scope of implementation. Another edge we have is the computational
efficiency of generating only one set of pseudo-data compared to methodologies
that simulate data multiple times and aggregate the estimates to form a single
parameter estimate. As a consequence, we are spared from the question of how
many simulations to run and which aggregation method to best implement.
Lastly, in contrast to federated learning algorithms in the literature, our
approach does not require more than one communication iteration between the
data providers and data analyst, nor do we need to set up a network among the
databases. Hence, we are significantly minimizing, if not totally eliminating,
the risk of disclosing sensitive data.
A limitation of our proposed approach is the inability to compute residuals,
which require individual response values from the original data. The pseudo-
data we generate, although similar in some characteristics to the original,
cannot be used to compute residuals. Thus, model diagnostics through residual
plots cannot be performed. In general, even when the individual-level data are
available, model checking through residual analysis is a non-trivial task.36,
37 This is especially true for visual assessment because of the element of
subjectivity and the difficulty in discerning patterns when the sample size is
very large, which is important when deciding possible corrective measures.38
Formal tests on residuals, on the other hand, can become very sensitive to
large samples, which may lead to falsely concluding violations of the
assumption. Hence, a good recourse is to be aware of (1) the consequences of
potential violations of the model assumptions and (2) possible remedies to
mitigate these consequences.
To begin with, the normality assumption regarding the response variable has
the least impact on tests and inference derived from linear regression 39, 40,
41 and linear mixed models 42 as long as outliers are handled properly.
Specifically for linear mixed models, inference on the fixed effects remain
valid even when the random effects do not follow a normal distribution.43
Gelman and Hill41 do not recommend checking the residuals for normality while
Galecki and Burzykowski 43 note that normality is not important for ordinary
least squares although it is for maximum likelihood estimation. On the
contrary, heteroscedasticity affects the standard errors of the parameter
estimates of a classical linear regression model even though the point
estimates remain unbiased.44 This affects confidence interval construction as
well as inference about the covariate effects. One way to address this without
altering the interpretation of effects through variable transformation 45, 46
is by using robust standard errors.47 We can show that with additional summary
statistics namely the third and fourth joint sample moments, our approach can
also achieve identical robust variance estimates as the ones derived from
actual data (Appendix A.3). Analogously, a robust or empirical variance
estimator has been proposed for linear mixed models and has been shown to be
consistent under misspecification of the correlation structure as long as the
mean is correctly specified.48
Validity, additivity and linearity, and independence of errors are the top
most important assumptions when utilizing linear regression models.41 Although
these cannot be evaluated in our proposed framework, model selection
procedures may help mitigate the impact of violations to these assumptions, if
they exist. Harrell38 proposed fitting a flexible parametric model that allows
for most departures from the assumptions as an alternative to residual
analysis. In light of our proposed strategy, selecting from candidate models
that consider potential violations is recommended in practice (e.g.
considering different combination of regressors, polynomial terms). The
independence assumption may be relaxed by considering a random effects model
instead of the classical linear regression model.
Another consequence of our approach is the inability to perform training and
testing since partitioning the original data is not possible. As a result,
model validation would not be an option, and predictive accuracy of the
resulting model might be difficult to assess. A potential remedy is to
generate pseudo-data which embodies the statistical properties of the original
data more closely than just having identical summary statistics such as the
mean vector and covariance matrix. Several studies present different
strategies to generate and analyze synthetic data from a statistical
disclosure control perspective as well as from a machine learning perspective
49, 50, 51, 52, but these would of course not be as simple anymore and would
require more data processing from the data providers’ end.
An interesting point to consider is the impact of rounding off sufficient
statistics. In the current implementation of the proposed framework, we assume
that the mean vector and covariance matrix from each data provider contain the
exact values and not the rounded off values. In practice, these sufficient
statistics may be rounded off to a few decimal places. To examine this, we
implemented the proposed approach on sufficient statistics that were rounded
off to two decimal places. We observed that there were only slight differences
in the model estimates when compared to using the exact values of the
sufficient statistics (e.g. the difference starts from the third decimal
place). The direction of effect and the overall inference also remained the
same. This suggests that the method is robust to rounded values of the
sufficient statistics in this setting, although a more thorough sensitivity
analysis is encouraged to draw more conclusive findings.
A field related to federated learning is meta-analysis. Like federated
learning, meta-analysis aims to build a global model that synthesizes
information from multiple studies. Since meta-analysis dates back to as early
as 1976 53 while federated learning is fairly recent,54 it is worthwhile to
explore meta-analysis techniques in addressing the challenges of a federated
data setting. Traditionally, meta-analysis directly utilizes the aggregated
information from studies. These techniques are called aggregate data meta-
analysis (ADMA). However, individual participant data meta-analysis (IPDMA) is
now regarded as the “gold standard", but accessing individual-level data
remains an obstacle.55 The work of Papadimitropoulou et al17 involving pseudo-
IPD thus provides a good solution to performing IPDMA even when studies only
include aggregate information. Because of its similarity to a meta-analysis
setting, the federated data setting also benefits from this framework. In
contrast to a meta-analysis setting though, multiple variables can be included
in a federated data setting because data providers may be more willing to
share the covariance structure of the variables.
Some future research avenues include dealing with missing data and generating
sufficient statistics for interaction terms and transformed variables e.g.
log- and quadratic-transformed variables, from the sufficient statistics of
the main variables. Presently, our approach assumes that the data providers
hand over summary data for complete observations. They must also supply the
sufficient statistics for the transformed variables and for possible
interaction terms on top of those for the main variables. Including random
slopes and more complex hierarchical models is also a viable direction.
Currently, the authors are working on generalized linear models with and
without random intercept wherein we find the potential of extending the idea
of pseudo-data generation that matches the summary statistics of the actual
data.
## 5 Conclusion
In this paper, we have demonstrated that parameter estimation of a linear
mixed model can be performed on federated data by generating pseudo-data from
the sample size, mean vector, and covariance matrix supplied by each data
provider. The principles of statistical sufficiency and likelihood provide a
good theoretical support to the validity of the proposed framework. Estimates
achieved from this approach are identical to those obtained from the actual
individual-level data, which are difficult to access due to privacy reasons.
Simplicity, computational and communication efficiency, and potentially wider
scope of implementation through any statistical software distinguish our
approach from the existing strategies in the literature. Extending this
approach to generalized linear mixed models is a current work in progress.
## 6 Conflict of Interest
The author(s) declare(s) no potential conflicts of interest with respect to
the research, authorship and/or publication of this article.
## 7 Funding
The author(s) disclosed receipt of the following financial support for the
research, authorship, and/or publication of this article: The author(s)
acknowledge(s) support from the Methusalem financement program of the Flemish
Government.
## References
* 1 Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation) (Text with EEA relevance). European Commission. https://eur-lex.europa.eu/eli/reg/2016/679/oj. Updated 2020. Accessed September 13, 2023.
* 2 Chiruvella V, Guddati AK. Ethical issues in patient data ownership. Interact J Med Res. 2021; 10(2): e22269. https://doi.org/10.2196/22269.
* 3 Sadilek A, Liu L, Nguyen D, et al. Privacy-first health research with federated learning. npj Digit Med. 2021; 4(132). https://doi.org/10.1038/s41746-021-00489-2.
* 4 Banabilah S, Aloqaily M, Alsayed E, Malik N, Jararweh Y. Federated learning review: Fundamentals, enabling technologies, and future applications. Inf Process Manag. 2022; 59(6): 103061. https://doi.org/10.1016/j.ipm.2022.103061.
* 5 Yue X, Kontar RA, Gómez AME. Federated data analytics: A study on linear models. IISE Trans. 2023; 0(0): 1-13. https://doi.org/10.1080/24725854.2022.2157912.
* 6 Li W, Tong J, Anjum MM, Mohammed N, Chen Y, Jiang X. Federated learning algorithms for generalized mixed-effects model (GLMM) on horizontally partitioned data from distributed sources. BMC Med Inform Decis Mak. 2022; 22(1): 269. https://doi.org/10.1186/s12911-022-02014-1.
* 7 Yan Z, Zachrison KS, Schwamm LH, Estrada JJ, Duan R. A privacy-preserving and computation-efficient federated algorithm for generalized linear mixed models to analyze correlated electronic health records data. PLoS One. 2023; 18(1): e0280192. https://doi.org/10.1371/journal.pone.0280192.
* 8 Crowson MG, Moukheiber D, Arévalo AR, et al. A systematic review of federated learning applications for biomedical data. PLOS Digit Health. 2022; 1(5): e0000033. https://doi.org/10.1371/journal.pdig.0000033.
* 9 Antunes RS, CostaA. d C, Küderle A, Yari IA, Eskofier B. Federated Learning for Healthcare: Systematic Review and Architecture Proposal. ACM Trans Intell Syst Technol. 2022; 13(4). https://doi.org/10.1145/3501813.
* 10 Wen J, Zhang Z, Lan Y, Cui Z, Cai J, Zhang W. A survey on federated learning: challenges and applications. Int J Mach Learn Cybern. 2023; 14(2): 513-535. https://doi.org/10.1007/s13042-022-01647-y.
* 11 Li T, Sahu AK, Talwalkar A, Smith V. Federated Learning: Challenges, Methods, and Future Directions. IEEE Signal Process Mag. 2020; 37(3): 50-60. https://doi.org/10.1109/MSP.2020.2975749.
* 12 Rieke N, Hancox J, Li W, et al. The future of digital health with federated learning. npj Digit Med. 2020; 3(1): 119. https://doi.org/10.1038/s41746-020-00323-1.
* 13 Lee JYL, Brown JJ, Ryan LM. Sufficiency Revisited: Rethinking Statistical Algorithms in the Big Data Era. Am Stat. 2017; 71(3): 202-208. https://doi.org/10.1080/00031305.2016.1255659.
* 14 Luo C, Islam M, Sheils N, et al. DLMM as a lossless one-shot algorithm for collaborative multi-site distributed linear mixed models. Nat Commun. 2022; 13(1). https://doi.org/10.1038/s41467-022-29160-4.
* 15 R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2024.
* 16 Song Y, Sun F, Redline S, Wang R. Random-effects meta-analysis of combined outcomes based on reconstructions of individual patient data. Res Synth Methods. 2020; 11(5): 594–616. https://doi.org/10.1002/jrsm.1406.
* 17 Papadimitropoulou K, Stijnen T, Dekkers O, Le Cessie S. One-stage random effects meta-analysis using linear mixed models for aggregate continuous outcome data. Res Synth Methods. 2018; 10. https://doi.org/10.1002/jrsm.1331.
* 18 Privacy-preserving distributed algorithms: A solution for next generation data sharing for collaborative modelling. Penn Computing, Inference and Learning Lab. https://pdamethods.org. Accessed July 16, 2024.
* 19 Luo C, Duan R, Edmondson M, et al. R package, Version 1.2.7: Privacy-Preserving Distributed Algorithms. Penn Computing, Inference and Learning Lab; 2024.
* 20 Bates D, Mächler M, Bolker B, Walker S. Fitting Linear Mixed-Effects Models Using lme4. J Stat Softw. 2015; 67(1): 1–48. https://doi.org/10.18637/jss.v067.i01.
* 21 Higgins P. R package, Version 0.2.0: Data Package for Medical Datasets. 2021\.
* 22 Casella G, Berger R. Statistical Inference. 2nd ed. California: Duxbury Resource Center; 2001.
* 23 Ripley BD. Stochastic Simulation. New York: John Wiley & Sons, Inc.; 1987.
* 24 Venables WN, Ripley BD. Modern Applied Statistics with S. 4th ed. New York: Springer; 2002. ISBN 0-387-95457-0.
* 25 Rao SN, Manissero D, Steele VR, Pareja J. A Systematic Review of the Clinical Utility of Cycle Threshold Values in the Context of COVID-19. Infect Dis Ther. 2020; 9(3): 573-586. https://doi.org/10.1007/s40121-020-00328-z.
* 26 Shah VP, Farah WH, Hill JC, et al. Association Between SARS-CoV-2 Cycle Threshold Values and Clinical Outcomes in Patients With COVID-19: A Systematic Review and Meta-analysis. Open Forum Infect Dis. 2021; 8(9): ofab453. https://doi.org/10.1093/ofid/ofab453.
* 27 Choudhuri J, Carter J, Nelson R, et al. SARS-CoV-2 PCR cycle threshold at hospital admission associated with patient mortality. PLoS One. 2021; 15(12): 1-14. https://doi.org/10.1371/journal.pone.0244777.
* 28 Waring E, Quinn M, McNamara A, Arino de la Rubia E, Zhu H, Ellis S. R package, Version 2.1.5: Compact and Flexible Summaries of Data. 2022.
* 29 Gumedze F, Dunne T. Parameter estimation and inference in the linear mixed model. Linear Algebra Appl. 2011; 435(8): 1920-1944. https://doi.org/10.1016/j.laa.2011.04.015.
* 30 Lin C, McAllister A. Monte Carlo Comparison of Four Methods for Estimation of Genetic Parameters in the Univariate Case1. J Dairy Sci. 1984; 67(10): 2389-2398. https://doi.org/10.3168/jds.S0022-0302(84)81587-8.
* 31 Swallow WH, Monahan JF. Monte Carlo Comparison of ANOVA, MIVQUE, REML, and ML estimators of variance components. Technometrics. 1984; 26(1): 47. https://doi.org/10.1080/00401706.1984.10487921.
* 32 Ferron JM, Bell BA, Hess MR, Rendina-Gobioff G, Hibbard ST. Making treatment effect inferences from multiple-baseline data: The utility of multilevel modeling approaches. Behav Res Methods. 2009; 41(2): 372-384. https://doi.org/10.3758/BRM.41.2.372.
* 33 McNeish D, Stapleton LM. Modeling clustered data with very few clusters. Multivariate Behav Res. 2016; 51(4): 495–518. https://doi.org/10.1080/00273171.2016.1167008.
* 34 Snijders TAB, Bosker RJ. Standard Errors and Sample Sizes for Two-Level Research. J Educ Behav Stat. 1993; 18(3): 237–259. https://doi.org/10.3102/10769986018003237.
* 35 McNeish D. Small Sample Methods for Multilevel Modeling: A Colloquial Elucidation of REML and the Kenward-Roger Correction. Multivariate Behav Res 2017; 52(5): 661–670. https://doi.org/10.1080/00273171.2017.1344538.
* 36 Bradburn MJ, Clark TG, Love SB, Altman DG. Survival Analysis Part III: Multivariate data analysis – choosing a model and assessing its adequacy and fit. Br J Cancer. 2003; 89(4): 605-611. https://doi.org/10.1038/sj.bjc.6601120.
* 37 Lindsey J. Nonlinear Models in Medical Statistics. Oxford statistical science series. New York, USA: Oxford University Press; 2001.
* 38 Harrell F. Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis. Springer Series in Statistics. Switzerland, AG: Springer International Publishing; 2015.
* 39 Ali MM, Sharma SC. Robustness to nonnormality of regression F-tests. J Econom. 1996; 71(1): 175-205. https://doi.org/10.1016/0304-4076(94)01700-X.
* 40 Box GEP, Watson GS. Robustness to non-normality of regression tests. Biometrika 1962; 49(1/2): 93–106. https://doi.org/10.2307/2333470.
* 41 Gelman A, Hill J. Data Analysis Using Regression and Multilevel/Hierarchical Models. Analytical Methods for Social Research. New York, USA: Cambridge University Press; 2006.
* 42 Schielzeth H, Dingemanse NJ, Nakagawa S, et al. Robustness of linear mixed-effects models to violations of distributional assumptions. Methods Ecol Evol. 2020; 11(9): 1141-1152. https://doi.org/10.1111/2041-210X.13434.
* 43 Galecki A, Burzykowski T. Linear Mixed-Effects Models Using R: A Step-by-Step Approach. Springer Texts in Statistics. Springer New York; 2013.
* 44 Long JS, Ervin LH. Using heteroscedasticity consistent standard errors in the linear regression model. Am Stat. 2000; 54(3): 217–224. https://doi.org/10.2307/2685594.
* 45 Weisberg S. Applied Linear Regression. Wiley Series in Probability and Statistics. 3rd ed. New Jersey, USA: John Wiley & Sons, Inc.; 2005.
* 46 Carroll R, Ruppert D. Transformation and Weighting in Regression. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Taylor & Francis; 1988.
* 47 White H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 1980; 48(4): 817–838. https://doi.org/10.2307/1912934.
* 48 Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika 1986; 73(1): 13-22. https://doi.org/10.1093/biomet/73.1.13.
* 49 Raghunathan TE. Synthetic Data. Annu Rev Stat Appl. 2021; 8(1): 129-140. https://doi.org/10.1146/annurev-statistics-040720-031848.
* 50 Hernandez M, Epelde G, Alberdi A, Cilla R, Rankin D. Synthetic data generation for tabular health records: A systematic review. Neurocomputing (Amst). 2022; 493: 28-45. https://doi.org/10.1016/j.neucom.2022.04.053.
* 51 Figueira A, Vaz B. Survey on Synthetic Data Generation, Evaluation Methods and GANs. Mathematics. 2022; 10(15): 2733. https://doi.org/10.3390/math10152733.
* 52 Murtaza H, Ahmed M, Khan NF, Murtaza G, Zafar S, Bano A. Synthetic data generation: State of the art in health care domain. Comput Sci Rev. 2023; 48: 100546. https://doi.org/10.1016/j.cosrev.2023.100546.
* 53 Glass GV. Primary, Secondary, and Meta-Analysis of Research. Educ Res. 1976; 5(10): 3-8. https://doi.org/10.3102/0013189X005010003.
* 54 McMahan HB, Moore E, Ramage D, Hampson S, Arcasy BA. Communication-efficient learning of deep networks from decentralized data. Talk presented at: 20th International conference on artificial intelligence and statistics (AISTATS); April 20-22, 2017; Florida, USA. https://proceedings.mlr.press/v54/mcmahan17a/mcmahan17a.pdf.
* 55 Nevitt SJ, Marson AG, Davie B, Reynolds S, Williams L, Smith CT. Exploring changes over time and characteristics associated with data retrieval across individual participant data meta-analyses: systematic review. BMJ. 2017; 357: j1390. https://doi.org/10.1136/bmj.j1390.
## Appendix A Appendix
### A.1 Likelihood principle
Another principle of data reduction discussed by 22 is the likelihood
principle. Stated more formally,
> “If $\mathbf{x_{1}}$ and $\mathbf{x_{2}}$ are two samples such that the
> likelihood $L(\theta|\mathbf{x_{1}})$ is proportional to
> $L(\theta|\mathbf{x_{2}})$, that is, there exists a constant
> $C(\mathbf{x_{1}},\mathbf{x_{2}})$ such that
>
>
> $L(\theta|\mathbf{x_{1}})=C(\mathbf{x_{1}},\mathbf{x_{2}})L(\theta|\mathbf{x_{2}})$
>
> then the conclusions drawn from $\mathbf{x_{1}}$ and $\mathbf{x_{2}}$ should
> be identical.”
They demonstrated this principle for the case of a normal distribution and
showed that the likelihood of a parameter $\mu$ given a sample
$\mathbf{x_{1}}$ ($L(\mu|\mathbf{x_{1}})$) can be exactly equal to the
likelihood of the same parameter given another sample $\mathbf{x_{2}}$
($L(\mu|\mathbf{x_{2}})$) if their sample means are equal
($\mathbf{\bar{x}}_{1}=\mathbf{\bar{x}}_{2}$).
### A.2 Showing the sufficient statistics for a linear regression model
Given $n$ observations, an intercept, and $p-1$ predictors, if $\mathbf{X}$
denotes the $n\times p$ design matrix and $\mathbf{y}$ is the $n\times 1$
vector of continuous responses, the linear regression coefficients
$\bm{\beta}$ and the variance $\sigma^{2}$ can be estimated through the log-
likelihood
$\displaystyle l(\bm{\beta},\sigma^{2};\mathbf{y},\mathbf{X})$
$\displaystyle=-\frac{n}{2}\text{ln}(2\pi)-\frac{n}{2}\text{ln}(\sigma^{2})-\frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$
where $\mathbf{x}_{i}$ is a vector containing the $i$th row in the design
matrix. We see here that information from the sample is required only in the
last term. Moreover, the sum of squares of this term can be expressed as
$\displaystyle\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$
$\displaystyle=\sum_{i=1}^{n}(y_{i}^{2}-2y_{i}\mathbf{x}_{i}^{T}\bm{\beta}+(\mathbf{x}_{i}^{T}\bm{\beta})(\mathbf{x}_{i}^{T}\bm{\beta})^{T}).$
Since $\mathbf{x}_{i}^{T}\bm{\beta}$ is just the dot product of two vectors
$\mathbf{x}_{i}$ and $\bm{\beta}$, commutativity applies such that the
equation above can also be written as
$\displaystyle\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$
$\displaystyle=\sum_{i=1}^{n}(y_{i}^{2}-2y_{i}\mathbf{x}_{i}^{T}\bm{\beta}+(\bm{\beta}^{T}\mathbf{x}_{i})(\bm{\beta}^{T}\mathbf{x}_{i})^{T})$
which when simplified further yields
$\displaystyle\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$
$\displaystyle=\sum_{i=1}^{n}(y_{i}^{2}-2y_{i}\mathbf{x}_{i}^{T}\bm{\beta}+\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta})$
$\displaystyle=\sum_{i=1}^{n}y_{i}^{2}-2\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}+\sum_{i=1}^{n}\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}$
From this, we find that knowing $n$, $\sum_{i=1}^{n}y_{i}^{2}$,
$\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$, and
$\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ is sufficient to construct
the log-likelihood and estimate the parameters, even in the absence of
individual-level data. In particular, $\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$
and $\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ are sufficient to
estimate the coefficients $\bm{\beta}$ while the variance $\sigma^{2}$ also
requires $\sum_{i=1}^{n}y_{i}^{2}$ in addition to the other two. Furthermore,
these values can be obtained from the vector of sample means and sample
covariance matrix of the response variable and the predictors. Specifically,
since the sample variance $s_{\mathbf{y}}^{2}$ is computed as
$\displaystyle s^{2}_{\mathbf{y}}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(y_{i}-\bar{\mathbf{y}})^{2}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(y_{i}^{2}-2y_{i}\bar{\mathbf{y}}+\bar{\mathbf{y}}^{2})$
$\displaystyle=\frac{1}{n-1}\left(\sum_{i=1}^{n}y_{i}^{2}-2\bar{\mathbf{y}}\sum_{i=1}^{n}y_{i}+n\bar{\mathbf{y}}^{2}\right)$
$\displaystyle=\frac{1}{n-1}\left(\sum_{i=1}^{n}y_{i}^{2}-n\bar{\mathbf{y}}^{2}\right),$
performing some algebraic manipulations will show that
$\sum_{i=1}^{n}y_{i}^{2}$ can be derived from the sample variance
$s^{2}_{\mathbf{y}}$, the sample mean $\bar{\mathbf{y}}$, and the sample size
$n$
$\displaystyle\sum_{i=1}^{n}y_{i}^{2}$
$\displaystyle=s^{2}_{\mathbf{y}}(n-1)+n\bar{\mathbf{y}}^{2}.$
For $\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$, we note that
$y_{i}\mathbf{x}_{i}^{T}$ is a $1\times p$ matrix
$\displaystyle\begin{bmatrix}y_{i}&y_{i}x_{i1}&y_{i}x_{i2}&...&y_{i}x_{ij}&...&y_{i}x_{i(p-1)}\end{bmatrix}$
where the first element of $\mathbf{x}_{i}$ is 1 corresponding to the
intercept. Thus, $\sum_{i=1}^{n}y_{i}\mathbf{x}_{i}^{T}$ is a $1\times p$
matrix
$\displaystyle\begin{bmatrix}\sum_{i=1}^{n}y_{i}&\sum_{i=1}^{n}y_{i}x_{i1}&\sum_{i=1}^{n}y_{i}x_{i2}&...&\sum_{i=1}^{n}y_{i}x_{ij}&...&\sum_{i=1}^{n}y_{i}x_{i(p-1)}\end{bmatrix}.$
The first element can be obtained from the sample mean $\bar{\mathbf{y}}$
while the rest of the elements needs the sample covariance between
$\mathbf{y}$ and each of the predictors:
$\displaystyle s_{\mathbf{yx}_{j}}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(y_{i}-\bar{\mathbf{y}})(x_{ij}-\bar{\mathbf{x}}_{j})$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(y_{i}x_{ij}-\bar{\mathbf{y}}x_{ij}-y_{i}\bar{\mathbf{x}}_{j}+\bar{\mathbf{y}}\bar{\mathbf{x}}_{j})$
$\displaystyle=\frac{1}{n-1}\left(\sum_{i=1}^{n}y_{i}x_{ij}-\bar{\mathbf{y}}\sum_{i=1}^{n}x_{ij}-\bar{\mathbf{x}}_{j}\sum_{i=1}^{n}y_{i}+n\bar{\mathbf{y}}\bar{\mathbf{x}}_{j}\right)$
$\displaystyle=\frac{1}{n-1}\left(\sum_{i=1}^{n}y_{i}x_{ij}-\bar{\mathbf{y}}\sum_{i=1}^{n}x_{ij}\right),$
and thus,
$\displaystyle\sum_{i=1}^{n}{y_{i}x_{ij}}$
$\displaystyle=s_{\mathbf{yx}_{j}}(n-1)+\bar{\mathbf{y}}\sum_{i=1}^{n}x_{ij}.$
Lastly, for $\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$, each summand
$\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ yields a $p\times p$ matrix
$\displaystyle\begin{bmatrix}1&x_{i1}&\ldots&x_{i(p-1)}\\\
x_{i1}&x_{i1}^{2}&\ldots&x_{i1}x_{i(p-1)}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{i(p-1)}&x_{i(p-1)}x_{i1}&\ldots&x_{i(p-1)}^{2}\end{bmatrix}$
which when summated over $n$ observations yields
$\displaystyle\begin{bmatrix}n&\sum_{i}{x_{i1}}&\ldots&\sum_{i}{x_{i(p-1)}}\\\
\sum_{i}{x_{i1}}&\sum_{i}{x_{i1}^{2}}&\ldots&\sum_{i}{x_{i1}x_{i(p-1)}}\\\
\vdots&\vdots&\ddots&\vdots\\\
\sum_{i}{x_{i(p-1)}}&\sum_{i}{x_{i(p-1)}x_{i1}}&\ldots&\sum_{i}{x_{i(p-1)}^{2}}\end{bmatrix}.$
Performing similar derivations as above reveals that computing
$\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ only requires the sample mean
($\bar{\mathbf{x}}_{j}$), variance ($s^{2}_{\mathbf{x}_{j}}$), and covariances
among predictors $j$ and $k$ ($s_{\mathbf{x}_{j}\mathbf{x}_{k}}$), namely
$\displaystyle\sum_{i=1}^{n}x_{ij}^{2}$
$\displaystyle=s^{2}_{\mathbf{x}_{j}}(n-1)+n\bar{\mathbf{x}}_{j}^{2},$
$\displaystyle\sum_{i=1}^{n}{x_{ij}x_{ik}}$
$\displaystyle=s_{\mathbf{x}_{j}\mathbf{x}_{k}}(n-1)+\bar{\mathbf{x}}_{j}\sum_{i=1}^{n}x_{ik}.$
### A.3 Robust variance estimation from summary statistics
When estimating the robust variance of linear regression coefficients, the
following estimator is used when the individual observations are available:
$\displaystyle\hat{V}(\hat{\bm{\beta}})$
$\displaystyle=(\mathbf{X}^{T}\mathbf{X})^{-1}(\mathbf{X}^{T}\mathbf{W}\mathbf{X})^{-1}(\mathbf{X}^{T}\mathbf{X})^{-1}$
where $\mathbf{X}$ is the design matrix and $\mathbf{W}$ is an $n\times n$
diagonal matrix whose elements consist of the squared residuals
$\hat{e}_{i}^{2}=(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}$.
$\mathbf{X}^{T}\mathbf{X}$ is a $p\times p$ matrix that is equivalent to
$\sum_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$, which we have shown
(Appendix A.2) can be computed from the mean vector and covariance matrix of
the variables. Recall that $\mathbf{x}_{i},i=1,..,n$ denotes the vector
representing the $i$th row of $\mathbf{X}$. On the other hand,
$(\mathbf{X}^{T}\mathbf{W}\mathbf{X})^{-1}$ can be shown to be composed of
summary statistics involving the third and fourth joint sample moments.
Specifically,
$\displaystyle(\mathbf{X}^{T}\mathbf{W}\mathbf{X})^{-1}=$
$\displaystyle\sum_{i=1}^{n}\hat{e}_{i}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(y_{i}-\mathbf{x}_{i}^{T}\bm{\beta})^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(y_{i}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}-2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})\mathbf{x}_{i}\mathbf{x}_{i}^{T}+(\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta})\mathbf{x}_{i}\mathbf{x}_{i}^{T})$
The first term $y_{i}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ is just a product
of a scalar $y_{i}^{2}$ and the matrix $\mathbf{x}_{i}\mathbf{x}_{i}^{T}$,
which results in the matrix
$\displaystyle\begin{bmatrix}y_{i}^{2}&y_{i}^{2}x_{i1}&\ldots&y_{i}^{2}x_{i(p-1)}\\\
y_{i}^{2}x_{i1}&y_{i}^{2}x_{i1}^{2}&\ldots&y_{i}^{2}x_{i1}x_{i(p-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
y_{i}^{2}x_{i(p-1)}&y_{i}^{2}x_{i(p-1)}x_{i1}&\ldots&y_{i}^{2}x_{i(p-1)}^{2}\end{bmatrix}$
whose summation over all observations $i$ becomes
$\displaystyle\begin{bmatrix}\sum_{i}y_{i}^{2}&\sum_{i}y_{i}^{2}x_{i1}&\ldots&\sum_{i}y_{i}^{2}x_{i(p-1)}\\\
\sum_{i}y_{i}^{2}x_{i1}&\sum_{i}y_{i}^{2}x_{i1}^{2}&\ldots&\sum_{i}y_{i}^{2}x_{i1}x_{i(p-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\sum_{i}y_{i}^{2}x_{i(p-1)}&\sum_{i}y_{i}^{2}x_{i(p-1)}x_{i1}&\ldots&\sum_{i}y_{i}^{2}x_{i(p-1)}^{2}\end{bmatrix},$
where we find that availability of the third and fourth joint sample moments
involving the response variable makes it possible for this matrix to be
computed.
Similarly, the second term
$2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ is also
the product of a scalar $2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})$ and the matrix
$\mathbf{x}_{i}\mathbf{x}_{i}^{T}$ resulting in
$\displaystyle\begin{bmatrix}2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i1}&\ldots&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i(p-1)}\\\
2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i1}&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i1}^{2}&\ldots&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i1}x_{i(p-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i(p-1)}&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i(p-1)}x_{i1}&\ldots&2(y_{i}\mathbf{x}_{i}^{T}\bm{\beta})x_{i(p-1)}^{2}\end{bmatrix}.$
Summing over all observations leads to the matrix
$\displaystyle\begin{bmatrix}2\sum_{i}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&2\sum_{i}x_{i1}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&2\sum_{i}x_{i(p-1)}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}\\\
2\sum_{i}x_{i1}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&2\sum_{i}x_{i1}^{2}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&2\sum_{i}x_{i1}x_{i(p-1)}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}\\\
\vdots&\vdots&\ddots&\vdots\\\
2\sum_{i}x_{i(p-1)}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&2\sum_{i}x_{i(p-1)}x_{i1}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&2\sum_{i}x_{i(p-1)}^{2}y_{i}\mathbf{x}_{i}^{T}\bm{\beta}\end{bmatrix}$
which can be computed from the third and fourth joint sample moments.
Likewise, the final term
$(\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta})\mathbf{x}_{i}\mathbf{x}_{i}^{T}$
summed over all observations becomes
$\displaystyle\begin{bmatrix}\sum_{i}\bm{\beta}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\sum_{i}\bm{\beta}^{T}x_{i1}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&\sum_{i}\bm{\beta}^{T}x_{i(p-1)}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}\\\
\sum_{i}\bm{\beta}^{T}x_{i1}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\sum_{i}\bm{\beta}^{T}x_{i1}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&\sum_{i}\bm{\beta}^{T}x_{i1}x_{i(p-1)}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}\\\
\vdots&\vdots&\ddots&\vdots\\\
\sum_{i}\bm{\beta}^{T}x_{i(p-1)}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\sum_{i}\bm{\beta}^{T}x_{i(p-1)}x_{i1}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}&\ldots&\sum_{i}\bm{\beta}^{T}x_{i(p-1)}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\bm{\beta}\end{bmatrix}$
and is also composed of the third and fourth joint sample moments.
|
# On a Side Condition for
Wronskian-Involving Differential Equations
Nicoleta Bîlă 111Department of Mathematics and Computer Science, Fayetteville
State University, 1200 Murchison Road, Fayetteville, NC 28301, E-mail:
<EMAIL_ADDRESS>
###### Abstract
The purpose of this paper is to make a few connections among specific concepts
occurring in differential geometry and the theory of differential equations
with the aim of identifying an intriguing class of undetermined nonlinear
ordinary differential equations whose solutions satisfy a specific side
condition consisting in a homogeneous third-order linear ordinary differential
equation. A method for solving this class of Wronskian-involving differential
equations based on the proposed side condition is presented. The Tzitzeica
curve equation arising in the theory of space curves is considered as an
example, and new closed and integral-form solutions for this equation are
obtained.
MSC 2000: 34A05, 34A30, 34A34, 53A04, 53A15.
Keywords: Nonlinear Ordinary Differential Equations; Linear Ordinary
Differential Equations; Wronskian; Tzitzeica Curves.
## 1 Introduction
Since there is no general theory for integrating nonlinear ordinary
differential equations (ODEs), it will always be a challenge to explore
methods involving intriguing side conditions that would allow one “to undo”
the nonlinearity encapsulated among the unknown functions and their
derivatives and obtain particular solutions to the studied equation. The
classical Lie method [7] is one of the most well-known techniques that may be
applied in this situation and, for instance, would allow one to reduce the
order of an ODE by one whenever the equation is invariant with respect to a
particular one-parameter Lie group of transformations. However, this reduction
may become more complicated in the case of underdetermined nonlinear ODEs of
higher order that have fewer equations than unknowns and contain arbitrary
constants or functions. In this paper, the focus is on a specific class of
nonlinear differential equations (4) that expresses a relation between the
Wronskian (1) of the unknown functions and the Wronskian (2) of their first
derivatives. This class of autonomous ODEs is analyzed in the particular case
when its solutions satisfy the auxiliary equation (9). For instance, the
Tzitzeica curve equation (23) belongs to this class. It is important to point
out that the side condition (9) is not found by using the classical Lie method
but rather is related to how the nonlinearity among the unknown functions and
their derivatives has occurred while deriving the equation.
Introduced in 1812 by Józef Maria Hoëné-Wroński and later on mentioned by
Thomas Muir in [6], the Wronskian determinant (or, simply, the Wronskian) of a
set of $n$ smooth functions is the determinant of the $n\times n$ matrix whose
entries are the functions and their derivatives up to the $(n-1)$st order.
Namely, the functions are listed in the first row, their first derivatives are
placed on the second row, and so on, and their $(n-1)$st derivatives are
written on the last row ([4], p. 221). In particular, for $n=3$, the Wronskian
$W(x,y,z)(t)$ of the smooth functions $x(t)$, $y(t)$, and $z(t)$ is given by
(1). If the functions represent the set of fundamental solutions of a $n$th-
order homogeneous linear ODE, then their Wronskian does not vanish and
satisfies the Abel-Liouville-Ostrogradski formula (see, e.g., [4], p. 239).
Interestingly, this formula allows us to determine the Wronskian associated
with the solutions without effectively integrating the linear differential
equation.
On the other hand, the Wronskian also occurs in the theory of space curves.
For a smooth regular space curve (20), the Wronskian $W(x,y,z)(t)$ of the
functions defining parametrically the curve may arise, for instance, in the
calculation of the distance $d(t)$ from the origin of the system of
coordinates to the osculating plane at an arbitrary point of the curve while
the Wronskian $W(x^{\prime},y^{\prime},z^{\prime})(t)$ of their first
derivatives (2) is used in the computation of the curve’s torsion $\tau(t)$ as
in (21). Therefore, for example, any condition imposed upon on $d(t)$ and
$\tau(t)$ yields a Wronskian-involving underdetermined nonlinear differential
equation for the curve’s defining functions.
In this paper, the class of nonlinear ODEs (4) involving the Wronskians (1)
and (2) of the unknown functions $x(t)$, $y(t)$, and $z(t)$ and, respectively,
of their first derivatives $x^{\prime}(t)$, $y^{\prime}(t)$, and
$z^{\prime}(t)$ is introduced and analyzed in the context of the family of
solutions satisfying the auxiliary linear differential equation (9) such that
the condition (3) holds. The motivation of considering differential equations
of this kind lies in the study of the condition (22) that was introduced for
specific curves by the Romanian mathematician Gheorghe Tzitzeica in 1911
during his work on affine invariants [11]. Nowadays, a space curve satisfying
the relation (22) that may be written in an equivalent form as (23) is called
a Tzitzeica curve. Along with Tzitzeica curves, Tzitzeica surfaces are affine
invariants. A Tzitzeica surface is a surface for which the ratio of its
Gaussian curvature and the fourth power of the distance from the origin to the
tangent plane at any arbitrary point of the surface is constant [10]. It may
be shown that the asymptotic curves on a Tzitzeica surface with negative
Gaussian curvature are Tzitzeica curves (see, e.g., [2]). Although during the
past years the Tzitzeica curves have been of interest to many geometers, their
related nonlinear ODE has not been studied in detail so far, and, hence, there
are known only a few examples of Tzitzeica curves defined explicitly in
algebraic, transcendental, or integral forms (see [2], [3], [5], and [13]). In
[2], Agnew et al. used the Mathematica software to find and illustrate the
asymptotic curves on the Tzitzeica surface of revolution $z(x^{2}+y^{2})=1$
and expressed them in cylindrical coordinates in terms of logarithmic and
exponential functions. In [5], Crâşmăreanu determined elliptic and hyperbolic
cylindrical Tzitzeica curves written in integral form. In the paper by
Williams [13], the nonlinear Tzitzeica curve equation has been derived
explicitly, and new closed-form solutions have been presented. Bîlă and Eni
[3] showed that the nonlinear ODE (23) admits particular solutions obtained by
augmenting it with a side condition consisting in a third-order homogeneous
linear ODE with constant coefficients. In this paper, it is shown that the
Tzitzeica curve equation also admits solutions that satisfy the side condition
(9). Although this is a slight generalization of the auxiliary condition
introduced in [3], by Proposition 3.1, the new attached equation yields a
larger family of solutions to the Tzitzeica curve equation. The new
interesting solutions (16) and (26) are expressed in closed-form or in terms
of Airy’s Ai and Bi functions. In Fig. 1, the software Geogebra has been used
to visualize the Tzitzeica curve (26) on the Tzitzeica surface of equation
$yz=1-4x^{2}$, surface that was found in [12] by applying the classical Lie
method to the Tzitzeica surface partial differential equation. Additionally,
for various functions $\gamma(t)$ in (9), new Tzitzeica curves expressed in
terms of other special functions [1] such as Bessel or generalized
hypergeometric functions may be investigated. Theorem 2.1 gives a
generalization of these results for the class of nonlinear ODEs (4).
The structure of the paper is the following. In Section 2, the side condition
(9) is explored in the case of the class of underdetermined nonlinear
Wronskian-involving ODEs (4) for which (3) holds. A new method for solving
these types of nonlinear differential equations with the help of the auxiliary
condition (9) is introduced. The novel approach is exemplified in Section 3
for the Tzitzeica curve equation (23) for which intriguing solutions are
obtained. In the last section, conclusions of this work are presented.
## 2 A class of Wronskian-Involving Equations
### 2.1 On a related side condition
If $x(t)$, $y(t)$, and $z(t)$ are smooth functions defined on an open interval
$I\subset\mathbb{R}$, then their Wronskian is defined as
$W(x,y,z)(t)=\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|.$
(1)
Similarly, the Wronskian of the functions’ first derivatives is given by
$W(x^{\prime},y^{\prime},z^{\prime})(t)=\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|.$
(2)
In what follows, assume that
$W(x,y,z)(t)\neq 0\quad\text{and}\quad
W(x^{\prime},y^{\prime},z^{\prime})(t)\neq 0,$ (3)
for all $t\in I$. This supposition implies that the functions $x(t)$, $y(t)$,
and $z(t)$ and, respectively, their first derivatives $x^{\prime}(t)$,
$y^{\prime}(t)$, and $z^{\prime}(t)$ are linearly independent (see [4], p.
221). Consider the differential equation
${\cal{F}}\left(W(x,y,z)(t),W(x^{\prime},y^{\prime},z^{\prime})(t)\right)=0$
(4)
involving the Wronskians (1) and (2), where ${\cal{F}}$ is a smooth real-
valued function in two variables. The equation above is a third-order
underdetermined differential equation in the unknown functions $x(t)$, $y(t)$,
and $z(t)$. Next, the problem of finding solutions to (4) is explored in the
context of a specific side condition that would allow the Wronskian (2) to be
expressed in terms of the Wronskian (1). Suppose that the functions $x(t)$,
$y(t)$, and $z(t)$ satisfy the following homogeneous third-order linear ODE
$u^{\prime\prime\prime}+\beta(t)u^{\prime\prime}+\gamma(t)u^{\prime}+\delta
u=0,$ (5)
where $\beta(t)$ and $\gamma(t)$ are smooth functions defined on the interval
$I$, and $\delta$ is a nonzero constant. For simplicity, in this paper,
$\delta$ is considered a nonzero constant. However, the method explained below
may be modified to accommodate $\delta$ variable too. Since the functions
$x(t)$, $y(t)$, and $z(t)$ are linearly independent, they form a fundamental
set of solutions to (5). Therefore, the general solution to this equation may
be written as $u(t)=C_{1}x(t)+C_{2}y(t)+C_{3}z(t)$, where $C_{i}$ are
arbitrary real constants. If the leading derivatives
$x^{\prime\prime\prime}=-\beta(t)x^{\prime\prime}-\gamma(t)x^{\prime}-\delta
x$,
$y^{\prime\prime\prime}=-\beta(t)y^{\prime\prime}-\gamma(t)y^{\prime}-\delta
y$, and
$z^{\prime\prime\prime}=-\beta(t)z^{\prime\prime}-\gamma(t)z^{\prime}-\delta
z$ are substituted into (2), then after applying a few properties of
determinants, this Wronskian may be expressed as
$W(x^{\prime},y^{\prime},z^{\prime})(t)=-\delta W(x,y,z)(t),\quad t\in I.$ (6)
Replacing the above relation into the differential equation (4) yields
${\cal{G}}\left(W(x,y,z)(t);\delta\right)=0,$ (7)
where ${\cal{G}}$ denotes the resulting function that depends on $W(x,y,z)(t)$
(the notation used here shows that $\delta$ occurs as well in the reduced
equation due to (6)). On the other hand, according to Abel-Liouville-
Ostrogradski formula (see, e.g., [4], p. 239), the Wronskian of the set of
fundamental solutions of (5) satisfies the first-order linear ODE
$W^{\prime}(x,y,z)(t)=-\beta(t)W(x,y,z)(t).$ (8)
The differentiation of (7) with respect to $t$ leads to
$W^{\prime}(x,y,z)(t)=0$ for any $t\in I$. By (3), the Wronskian $W(x,y,z)(t)$
does not vanish, and, thus, the equation (8) implies $\beta(t)=0$ for all
$t\in I$. In conclusion, the equation (5) reduces to
$u^{\prime\prime\prime}+\gamma(t)u^{\prime}+\delta u=0,$ (9)
where $\gamma(t)$ is a smooth function and $\delta$ is a nonzero constant. In
this way, the equation (4) has been reduced to the homogeneous third-order
linear ODE (9) along with the equation (7) provided that (3) holds. Observe
that now the Wronskian $W(x,y,z)(t)=C_{0}$ is constant, and the equation (7)
turns into the compatibility condition
${\cal{G}}\left(C_{0};\delta\right)=0.$ (10)
In conclusion, the following result has been proven.
###### Theorem 2.1.
Any three linearly independent solutions $x(t)$, $y(t)$, and $z(t)$ of the
homogeneous third-order linear ODE (9) conditioned to (3) satisfy the
nonlinear differential equation (4) provided that (10) holds.
### 2.2 Special side conditions
In what follows, the ODE (9) is integrated in a few particular cases that
yield to explicit or integral form solutions. Recall that in each of these
cases the Wronskian of the solutions is constant. As it has been shown in the
previous section, the equation (4) has been reduced to the side condition (9)
and compatibility condition (10).
###### Remark 2.1.
If $\gamma(t)=0$, then the equation (9) becomes $u^{\prime\prime\prime}+\delta
u=0$, and its set of fundamental solutions is
$\displaystyle x(t)$ $\displaystyle=\exp\left(-\delta^{1/3}t\right),$
$\displaystyle y(t)$
$\displaystyle=\exp\left(\frac{1}{2}\delta^{1/3}t\right)\cos\left(\frac{\sqrt{3}}{2}\delta^{1/3}t\right),$
$\displaystyle z(t)$
$\displaystyle=\exp\left(\frac{1}{2}\delta^{1/3}t\right)\sin\left(\frac{\sqrt{3}}{2}\delta^{1/3}t\right),$
(11)
with $t\in\mathbb{R}$. In this case, $W(x,y,z)(t)=-3\delta\sqrt{3}/2$.
###### Remark 2.2.
For $\gamma(t)=\tilde{\gamma}$, where $\tilde{\gamma}$ is a nonzero real
constant, the equation (9) is reduced to the homogeneous linear ODE with
constant coefficients
$u^{\prime\prime\prime}+\tilde{\gamma}u^{\prime}+\delta u=0$ (12)
whose solutions have been discussed in detail in the context of the Tzitzeica
curve equation in [3]. The characteristic equation related to (12) is the
depressed cubic equation $v^{3}+\tilde{\gamma}v+\delta=0$. The character of
the solutions depends on the sign of the associated determinant
$D=-4\gamma^{3}-27\delta^{2}.$
a) If $D>0$, the depressed cubic equation has three real distinct solutions
$v_{i}\neq 0$, $i=1,2,3$ whose sum is zero. The conditions $v_{3}\neq v_{i}$,
with $i=1,2$, imply $v_{2}\neq-2v_{1}$ and $v_{2}\neq-v_{1}/2$. In this case,
the fundamental set of solutions of (12) is
$x(t)=\exp\left(v_{1}t\right),\quad y(t)=\exp\left(v_{2}t\right),\quad
z(t)=\exp\left(-\left(v_{1}+v_{2}\right)t\right),$ (13)
with $t\in\mathbb{R}$, for which
$W(x,y,z)(t)=(v_{2}-v_{1})(2v_{1}+v_{2})(v_{1}+2v_{2})$.
b) For $D=0$, the characteristic equation has three real solutions out of
which two are equal, i.e., $v_{1}=v_{2}\neq 0$ and $v_{3}=-2v_{1}$. Therefore,
$x(t)=\exp(v_{1}t),\quad y(t)=t\exp(v_{1}t),\quad z(t)=\exp(-2v_{1}t),$ (14)
with $t\in\mathbb{R}$, form a fundamental solution set to (12). In this case,
their Wronskian is given by $W(x,y,z)(t)=9v_{1}^{2}$.
c) If $D<0$, the depressed cubic equation has two nonreal complex conjugate
solutions, $v_{1,2}=m\pm in$ and one real nonzero solution $v_{3}=-2m$, where
$m,n\neq 0$ are real numbers. The integration of (12) yields
$x(t)=\exp(mt)\cos(nt),\;\;y(t)=\exp(mt)\sin(nt),\;\;z(t)=\exp(-2mt),$ (15)
where $t\in\mathbb{R}$, and for which $W(x,y,z)(t)=n(9m^{2}+n^{2})$.
###### Remark 2.3.
If $\gamma(t)=\delta t$, then (9) may be integrated once and becomes
$u^{\prime\prime}+\delta tu=C$, where $C\in\mathbb{R}$ (here $C$ is nonzero
whenever three linearly independent solutions to (9) are needed). The
solutions of the latter ODE are given in integral form in terms of the Airy’s
Ai and Bi functions of the first and second kind, respectively ([8], p. 214),
that is,
$\displaystyle x(t)$ $\displaystyle=\text{Ai}\left(-\delta^{1/3}t\right),$
$\displaystyle y(t)$ $\displaystyle=\text{Bi}\left(-\delta^{1/3}t\right),$
$\displaystyle z(t)$ $\displaystyle=\pi\delta^{-1/3}\left(x(t)\int
y(s)ds-y(t)\int x(s)ds\right),$ (16)
with $t\in\mathbb{R}$, and their Wronskian is $W(x,y,z)(t)=-\delta^{1/3}/\pi$.
### 2.3 On an integration method involving a side condition
Assume that $x_{0}(t)$ is a known solution to ODE (9). In this situation, a
method for integrating the ODE (9) may be introduced as follows.
Step 1. Substitute $u=x_{0}(t)$ into the ODE (9) and solve the equation for
$\gamma(t)$. Denote $\gamma_{0}(t;\delta)$ the resulting solution that depends
on $\delta$ too.
Step 2. Replace $\gamma_{0}(t;\delta)$ in (9) and obtain
$u^{\prime\prime\prime}+\gamma_{0}(t;\delta)u^{\prime}+\delta u=0$. By using
the change of function
$u(t)=x_{0}(t)\int w(s)ds,$ (17)
the previous equation is reduced to
$x_{0}(t)w^{\prime\prime}+3x^{\prime}_{0}(t)w^{\prime}+\left[3x^{\prime\prime}_{0}(t)+\gamma_{0}(\delta;t)x_{0}(t)\right]w=0$
(18)
which is a homogeneous linear second-order ODE for the function $w(t)$.
Step 3. Integrate (18) and determine its general solution
$w(t)=C_{1}w_{1}(t)+C_{2}w_{2}(t)$, where $w_{1}(t)$ and $w_{2}(t)$ represent
its fundamental solution set.
Step 4. Replace $w(t)$ into (17) to have
$u(t)=x_{0}(t)\int w(s)ds=C_{1}x_{0}(t)\int w_{1}(s)ds+C_{2}x_{0}(t)\int
w_{2}(s)ds+C_{3}x_{0}(t),$
where $t\in I$. In conclusion,
$x(t)=x_{0}(t),\quad y(t)=x_{0}(t)\int w_{1}(s)ds,\quad z(t)=x_{0}(t)\int
w_{2}(s)ds,$ (19)
with $t\in I$, form the fundamental set of solutions of the equation of (9).
## 3 Solutions to Tzitzeica Curve Equation
### 3.1 Tzitzeica Curve Equation
In this section, the compatibility of the side condition (9) for the
underdetermined nonlinear ODE (4) is presented in detail in the case of the
Tzitzeica curve equation (23). Consider
$\mathbf{r}(t)=\left(x(t),y(t),z(t)\right),\quad t\in I$ (20)
a smooth, regular space curve, where $I\subset\mathbf{R}$ is an interval.
Assume that the curve’s curvature
$k(t)=\frac{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||}{||\mathbf{r^{\prime}}(t)||^{3}},\quad
t\in I$
and torsion
$\tau(t)=\frac{\langle\mathbf{r^{\prime}}(t),\mathbf{r^{\prime\prime}}(t),\mathbf{r^{\prime\prime\prime}}(t)\rangle}{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||^{2}},\quad
t\in I$ (21)
do not vanish on $I$. Here
$||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||$ is the
magnitude of the cross product of the tangent vector $\mathbf{r^{\prime}}(t)$
and the acceleration vector $\mathbf{r^{\prime\prime}}(t)$,
$\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)$,
$||\mathbf{r^{\prime}}(t)||$ is the magnitude of $\mathbf{r^{\prime}}(t)$, and
$\langle\mathbf{r^{\prime}}(t),\mathbf{r^{\prime\prime}}(t),\mathbf{r^{\prime\prime\prime}}(t)\rangle=\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|$
is the mixed product of vectors $\mathbf{r^{\prime}}(t)$,
$\mathbf{r^{\prime\prime}}(t)$, and $\mathbf{r^{\prime\prime\prime}}(t)$ (see,
for instance, [9], p. 48). The curvature shows the amount by which a curve
deviates from being a line, and its torsion shows how sharply the curve is
twisting out of its osculating plane. If the torsion of the curve is nonzero
then $W(x^{\prime},y^{\prime},z^{\prime})(t)\neq 0$ and, hence, the functions
$x^{\prime}(t)$, $y^{\prime}(t)$, and $z^{\prime}(t)$ are linearly
independent. On the other hand, the osculating plane at an arbitrary point of
the curve (20) is the plane generated by the vectors $\mathbf{r^{\prime}}(t)$
and $\mathbf{r^{\prime\prime}}(t)$. Therefore, its equation in the determinant
form is given by
$\left|\begin{array}[]{ccc}x-x(t)&y-y(t)&z-z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|=0.$
If $d(t)$ denotes the distance from the origin to the osculating plane of the
curve, it can be shown that
$d(t)=\frac{1}{||\mathbf{r^{\prime}}(t)\times\mathbf{r^{\prime\prime}}(t)||}\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|.$
The distance $d(t)$ does not vanish iff $W(x,y,z)(t)\neq 0$. In what follows,
in addition to the condition that the curvature and the torsion of the curve
(20) are nonzero, the distance $d(t)$ is assumed nonzero on $I$ as well.
Therefore, the curve’s defining functions $x(t)$, $y(t)$, and $z(t)$ are
assumed to satisfy the condition (3).
A Tzitzeica curve is defined as a space curve with the property that the ratio
of its torsion $\tau(t)$ and the square of the distance $d(t)$ from the origin
to the osculating plane at an arbitrary point of the curve is constant, i.e,
$\frac{\tau(t)}{d^{2}(t)}=\alpha,$ (22)
for all $t\in I$, where $\alpha\neq 0$ is a nonzero real number called here
the curve’s constant. Substituting $\tau(t)$ and $d(t)$ into (22) yields
$\left|\begin{array}[]{ccc}x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\\\
x^{\prime\prime\prime}(t)&y^{\prime\prime\prime}(t)&z^{\prime\prime\prime}(t)\end{array}\right|=\alpha\left|\begin{array}[]{ccc}x(t)&y(t)&z(t)\\\
x^{\prime}(t)&y^{\prime}(t)&z^{\prime}(t)\\\
x^{\prime\prime}(t)&y^{\prime\prime}(t)&z^{\prime\prime}(t)\end{array}\right|^{2}.$
The above equation can be rewritten in the terms of the Wronskians (1) and (2)
as follows
$W(x^{\prime},y^{\prime},z^{\prime})(t)=\alpha\left[W(x,y,z)(t)\right]^{2}.$
(23)
Based on Theorem 2.1, the Wronskian-involving equation (23) admits particular
solutions satisfying the side condition (9) provided that (3) holds.
Assume that the functions $x(t)$, $y(t)$, and $z(t)$ are solutions to the ODE
(5). In this case, the Wronskian of their first derivatives (2) satisfies the
relation (6) which may be used to rewrite the equation (23) in the form
$-\delta W(x,y,z)(t)=\alpha\left[W(x,y,z)(t)\right]^{2},$ (24)
or, equivalently, as follows
$W(x,y,z)(t)\left[\alpha W(x,y,z)(t)-\delta)\right]=0.$
Thus, in this case, the above relation represents the compatibility condition
(10) . Taking into account that $W(x,y,z)(t)$ does not vanish on $I$, it
follows
$\alpha=-\frac{\delta}{W(x,y,z)(t)}.$ (25)
Since the Tzitzeica curve equation (24) is a particular case of the Wronskian-
involving equation (4), according to Theorem 2.1, the following result has
been proven.
###### Proposition 3.1.
Any three linearly independent solutions $x$, $y$, and $z$ of the homogeneous
third-order linear ODE (9) for which the condition (3) holds satisfy the
Tzitzeica curve equation (24), where the curve’s constant is given by (25).
### 3.2 Examples of Tzitzeica Curves
A few examples of Tzitzeica curves that derive from the side condition (9) are
presented below. However, more examples may be given by exploring other
options for the function $\gamma(t)$ in this auxiliary condition.
Example 1. According to Remark 2.1, the functions (11) represent a fundamental
set of solutions for the side condition (9) in the case $\gamma(t)=0$. Since
the Tzitzeica curve’s equation (24) reduces to (25), then (11) represents a
Tzitzeica curve with $\alpha=-2\sqrt{3}/9$ that lies on the surface
$S:x(y^{2}+z^{2})=1$. After applying the affine transformation
$T:\tilde{x}=z,\tilde{y}=x,\tilde{z}=y$ to the surface $S$, this becomes
$\tilde{S}:z(x^{2}+y^{2})=1$. It may be shown that $\tilde{S}$ is a Tzitzeica
surface with negative Gaussian curvature, and, therefore, its asymptotic
curves are Tzitzeica curves. Agnew et al [2] have found the asymptotic curves
(expressed in cylindrical coordinates) for $\tilde{S}$ by using the software
Mathematica. It may be shown that the transformed Tzitzeica curve (11) via the
transformation $T$ is an asymptotic curve on the surface $\tilde{S}$.
Example 2. Consider the side condition (12) in Remark 2.2. Then there are
three classes of transcendental Tzitzeica curves that are classified in terms
of the sign of the determinant $D$ of the depressed equation. The solutions
(13), (14), and (15) represent Tzitzeica curves for which their associated
curve’s constants are given, respectively, by (25). These curves have been
found and discussed in detailed in [3].
Example 3. The solution (16) represents a new Tzitzeica curve for which, by
(25), the curve’s constant is $\alpha=\pi\delta^{2/3}$. Interestingly, this
curve is expressed in terms of Airy Ai and Bi functions.
Example 4. This example refers to the method introduced in Subsection 2.3.
Assume that $x_{0}(t)=t^{-3/2}$ is a known solution to (9). At Step 1, after
substituting $x_{0}(t)$ back in the equation and solving for $\gamma(t)$, it
follows
$\gamma(t)=\gamma_{0}(t;\delta)=\frac{8\delta t^{3}-105}{12t^{2}}.$
In particular, for $\delta=-27/8$, the ODE (9) becomes
$u^{\prime\prime\prime}-\frac{9t^{3}+35}{4t^{2}}u^{\prime}-\frac{27}{8}u=0.$
At Step 2, the change of variable (17) reduces the above equation to
$w^{\prime\prime}-\frac{9}{2t}w^{\prime}+\left(-\frac{9t}{4}+\frac{5}{2t^{2}}\right)w=0.$
The set of fundamental solutions is found at Step 3 to be
$w_{1}(t)=\frac{3}{2}\left(t^{2}-\sqrt{t}\right)\exp\left({t^{3/2}}\right),\quad
w_{2}(t)=-\frac{3}{2}\left(t^{2}+\sqrt{t}\right)\exp\left(-{t^{3/2}}\right),$
for $t>1$. Therefore, at Step 4, the following Tzitzeica curve is found
$\displaystyle x(t)$ $\displaystyle=t^{-3/2},$ $\displaystyle y(t)$
$\displaystyle=\left(1-2t^{-3/2}\right)\exp\left({t^{3/2}}\right),$
$\displaystyle z(t)$
$\displaystyle=\left(1+2t^{-3/2}\right)\exp\left({-t^{3/2}}\right),$ (26)
where $t>1$. The Wronskian (1) becomes $W(x,y,z)(t)=27/4$, and, from (25), the
curve’s constant takes the value $\alpha=1/2$. The new Tzitzeica curve (26)
lies on the Tzitzeica surface $M:\;yz=1-4x^{2}$ with negative Gaussian
curvature, surface that was found by using the symmetry analysis theory in
[12] (see Fig. 1). The curve (26) is not an asymptotic curve on $M$ as this is
an hyperboloid with one sheet whose asymptotic curves are lines.
Figure 1: The Tzitzeica curve defined by (26) for $t\in[1.001,8]$ represented
on the Tzitzeica surface of equation $yz=1-4x^{2}$.
## 4 Discussion and Conclusions
The motivation of this work is based on the study of the Tzitzeica curve
equation (24) which is an underdetermined nonlinear autonomous ordinary
differential equation arising from the condition stating that the ratio of the
curve’s torsion $\tau(t)$ and the square of the distance $d(t)$ from the
origin to its osculating plane at an arbitrary point of the curve is constant.
This paper is a continuation of author’s work on closed-form solutions for the
Tzitzeica curves that she has initiated with her students (see [3] and [13]).
It is shown that the auxiliary equation (9) provides a large family of
solutions for the class of Wronskian-involving equations (4) which includes
the Tzitzeica curve nonlinear ODE (24). The side condition (9) has been
derived by using a linear combination of derivatives of the unknown functions
that would allow specific properties of determinants to be applied. Another
key observation is the relation (6) that shows that the Wronskian (2) may be
expressed in terms of the Wronskian (1). In Section 2, a method based on the
side condition (9) is introduced and allows one to find a Tzitzeica curve if a
solution $x_{0}(t)$ to (9) is known. As a consequence, new solutions for (4)
and, in particular, for (24) are obtained. It is intriguing how the auxiliary
condition (9) may provide a large variety of solutions to the nonlinear ODEs
(4) and, hence, to (24). In the future, it will be interesting to make the
connection of these results with the classical Lie symmetries associated with
the Tzitzeica curve equation and with other particular Wronskian-type
equations.
Acknowledgments. Part of these results have been presented to the online XVth
International Conference on Differential Geometry and Dynamical Systems, held
on August 26–29, 2021, Bucharest, Romania, and, hence, the author would like
to thank the organizers, in particular, Prof. Dr. C. Udrişte and Prof. Dr. V.
Bălan. The author would also like to thank Prof. Dr. M. Crâşmăreanu for
reading the manuscript and making interesting comments that have improved the
presentation of the paper.
## References
* [1] Abramowitz, M. and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover (1972)
* [2] A. F. Agnew, A. Bobe, W. G. Boskoff and B. D. Suceava, Tzitzeica curves and surfaces, The Mathematica Journal, 12, 1-18 (2010)
* [3] N. Bîlă and M. Eni, Particular solutions to the Tzitzeica curve equation, Differential Geometry - Dynamical Systems, 24, 38-47 (2022)
* [4] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 8th edition, John Wiley & Sons, Inc. (1986)
* [5] M. Crâşmăreanu, Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Appl.,7, No. 1, 37-42 (2002)
* [6] T. Muir, A treatise on the theorie of determinants, London. Macmillan (1882)
* [7] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York (1986)
* [8] A. D. Polyanin, V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations Second Edition, Chapman & Hall/CRC Press, Boca Raton (2003)
* [9] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited (2012)
* [10] G. Tzitzeica, Sur une nouvelle classes de surfaces, Comptes Rendus, Acad. Sci. Paris, Paris, 144, 1257-1259 (1907)
* [11] G. Tzitzeica, Sur certaines courbes gauches, Ann. de l’Ec. Normale Sup., 28, 9-32 (1911)
* [12] C. Udrişte, N. Bîlă, Symmetry group of Tzitzeica surfaces PDE, Balkan J. Geom. Appl., 4(2), 123-140 (1999)
* [13] L. R. Williams, On The Tzitzeica Curve Equation, Explorations: The Undergraduate Research and Creative Activities Journal for the State of North Carolina, VIII, 105-115 (2013)
|
# Study of semi-boosted top quark reconstruction performance on the line shape
of a $t\bar{t}$ resonance
J. Pácalt J. Kvita
###### Abstract
We study the top quark pair events production in $pp$ collisions in the
$\ell$+jets channel at the energy of $\sqrt{s}=14$ TeV for Standard Model as
well as new physics processes. We explore the usage of semi-boosted topologies
where the top quark decays into a high-transverse momentum (boosted) hadronic
$W$-jet and an isolated $b$-jet and study their performance in the $t\bar{t}$
events kinematic reconstruction. An important event fraction is recovered and
the correlation of selected kinematic variables between the detector and
particle level is studied. Quality of the reconstructed mass line shape of a
hypothetical scalar resonance decaying into $t\bar{t}$ is evaluated and
compared for regimes of a different degree of the transverse boost. Unfolding
performance is checked in terms of comparing the excess of events in spectra
before and after the unfolding, concluding with the proof of a signal
significance loss after the unfolding procedure for both energy and angle
related observables, with possible applications in current LHC experiments.
## 1 Introduction
This work studies the top quarks pair kinematic reconstruction in a collider
detector close to that of the ATLAS detector [1] at the Large Hadron Collider
(LHC) [2] at CERN using a parameterized detector simulation provided by
Delphes [3]. The LHC nominally collides protons at four interaction points
where the detectors are located. The two main multi-purpose detector
facilities are the ATLAS and CMS [4] detectors which are versatile particle
detectors with ability to discern all kinds of particles with the exception of
neutrinos. Both are of similar phase-space coverage, resolution and detection
and identification capabilities based on different experimental technologies.
Quarks and gluons originating in collisions are not detected directly because
they become confined in hadrons or, in case of the top quark, decay before
their arrival to the detector. The process of hadronization forms showers of
particles collimated in the direction of the original particle, resulting in a
hadronic jet reaching the detector. The degree of collimation is proportional
to the momentum of the parent particle and this results in particular
perpetual positions of hadronic showers in the detector. If the primarily
particle has a large momentum with respect to the beam (transverse momentum,
${p_{\rm T}}$), the corresponding particle shower is more collimated leading
to a reconstructed hadronic final state with an imprint of the parent particle
four-vector, including its mass. Also, stable particles of different origin
can overlap in a jet.
The energy of particles used in colliders increases with the advance of the
experimental technology. This leads to enrichment of events with particles of
higher transverse momenta. This paper studies the process of top and anti-top
quark pair($t\bar{t}$) production $pp\rightarrow t\bar{t}$ at the LHC at CERN
at the center-of-mass energy $\sqrt{s}=14$ TeV. This paper also considers a
process with a hypothetical massive heavy scalar particle $y_{0}$ as a
mediator for the process $pp\rightarrow y_{0}\rightarrow t\bar{t}$ through a
triangle loop for the enhancement of events in the phase space of large
transverse momenta.
## 2 The ${t\bar{t}}$ final states topologies
The $t\bar{t}$ events are categorized into three channels, according to the
decaying products of the top quarks. The top quark decay is described by the
following process: $t\rightarrow W^{+}q$ $(q=b,s,d)$. The rest of allowed
decay processes are weak neutral currents which are heavily suppressed and
their contribution is negligible. Furthermore, the decay of the top quark is
mainly to the bottom quark thanks to the large value of the CKM mixing matrix
element between the bottom and top quarks. The $W$ boson has two main decay
modes; hadronic (68%) and leptonic (32%) [5]. There are two $W$ boson decays
in each $t\bar{t}$ event and the $t\bar{t}$ decay channels can thus be
categorized based on the combination of $W$ decay modes to all-hadronic, semi-
leptonic and dilepton channels. This analysis focuses on the semi-leptonic
channel.
The degree of collimation of the produced particle showers and their angular
separation in the detector defines the topology of an event. In the resolved
topology, $t\bar{t}$ decay products are reconstructed as individual jets and a
lepton, see Fig. 1a). Events in this topology are usually produced at lower
invariant masses of the $t\bar{t}$ pair. In the semi-boosted topology, decay
products on the side where the $W$ boson is decaying hadronically are
collimated enough to form one jet in the detector with exception of the jet
from the $b$ quark, see Fig. 1b). In the semi-boosted mixed topology, which is
a special case of semi-boosted topology, the angularly isolated jet is one of
the $W$ boson hadronic decay products, see Fig. 1c). In the boosted topology,
all products from the hadronically decaying top quark are collimated and form
one large jet in the detector, see Fig. 1d). The fractions of the topologies
are correlated to the energy spectrum of the $t\bar{t}$ pair forming a gradual
transition from the resolved to boosted topologies. The number of events of
the resolved topology drops significantly with increasing energy of the
process. In an intermediate energy regime, the number of events in the boosted
topology is not yet large enough to fill the gad after the resolved topology.
Finding ways to improve the events reconstruction efficiency by adding the
semi-boosted topologies, which reside in the aforementioned transition energy
region, is one of the aims of this paper. This can help gain statistics in
$t\bar{t}$ analyses. All these four topologies mentioned are explored in this
paper. In addition, we study the resolution of the mass peak of an
hypothetical scalar resonance decaying to ${t\bar{t}}$ and we check the
performance of the unfolding procedure in terms of its ability to retain an
excess of a possible new physics signal.
|
---|---
a) | b)
|
c) | d)
Figure 1: A schematic of $pp\rightarrow y_{0}\rightarrow t\bar{t}$ decays in
the resolved a), the semi-boosted mixed b), the semi-boosted c) and the
boosted d) topologies in the $\ell$+jets channel. Red (green) cones represent
small-$R$ (large-$R$) jets.
The semiboosted topologies have been used in analyses at the LHC as _e.g._ by
CMS [6, 7, 8] or ATLAS [9, 10, 11, 12], although mostly at lower energies or
in cases where the boosted $W$-tagged hadronic jet plays a key rôle in the
analysis. We argue that their potential is still worth exploring in the
${t\bar{t}}$ final states also at the highest energies of the LHC to come. We
employ the semiboosted topologies in the ${t\bar{t}}\rightarrow\ell$+jets
final states and emphasize their ability to recover a non-negligible event
fraction as well as explore their usage in unfolding differential
distributions, i.e. in precision measurements as well in searches for new
physics.
## 3 Samples
Events were generated for the processes $pp\rightarrow{t\bar{t}}{}$ (SM) and
$pp\rightarrow y_{0}^{\prime}\rightarrow{t\bar{t}}{}$ with the addition of a
$y_{0}$ scalar particle to the Standard Model [13, 14, 15, 16, 17, 18, 19, 20]
using the MadGraph5 version 2.6.4 simulation toolkit [21]. The spin-0 model
[19] contains an s-channel color-singlet scalar mediator with purely flavour-
diagonal couplings proportional to the masses of the SM particles, therefore
leaving top quark as the only relevant SM fermion coupling to the new
hypothetical scalar. The parton shower and the hadronization processes were
simulated using Pythia8 [22]. Masses of the hypothetical $y_{0}$ particle,
which serves effectively as a source of semi-boosted and boosted top quarks,
were selected as 500, 600, 700, 800, 900 and 1000 GeV to sample through the
region where the number of events in the resolved topology starts to decline
rapidly (500 GeV) to where the number of the boosted topology events is
becoming dominant (1000 GeV). The decay width of $y_{0}$ of 1%, 10% and 30% of
its mass for each sample were studied, results and values in the tables and
plots are shown for the decay width of 10%.
The SM $t\bar{t}$-sample without the hypothetical $y_{0}$ particle ensures the
correspondence with data measured in LHC experiments and is used as a
background to $y_{0}$ and for corrections for the unfolding procedure. The top
quark mass in simulation was set to 173 GeV (MadGraph5 default).
To ensure the strength of the evidence from the contribution of different
samples, all samples were weighted to the same luminosity ($\sim
12~{}\mathrm{fb}^{-1}$) for stacking and unfolding purposes, and which
corresponds to the luminosity of the $t\bar{t}$ sample.
The cross-sections of the samples used are summarized in Table 3. The numbers
of events in the table are presented for one statistically independent sample
and the generated samples also include charge conjugated processes in the
decay, _i.e._ the top and anti-top quark decays were swapped. All samples were
generated at the next-to-leading order (NLO) accuracy in perturbative quantum
chromodynamics (QCD), allowing also hard process with the additional
high-${p_{\rm T}}$ jet production.
The samples for the description of the $W+$jets and $WW+$jets backgrounds were
prepared within the same framework as the signal samples. The addition of the
associated production of one $W$ boson and two $b$ quarks
($Wbb\rightarrow\ell\nu bb$ \+ jets) background and two $W$ bosons and two $b$
quarks ($WWbb\rightarrow\ell\nu bb+$jets) background brings the analysis close
to those over data while the $y_{0}$ sample represents a signal of new
physics.
The ATLAS-like detector was simulated using the Delphes version 3.4.1 package
[3] with a modified ATLAS card111The modification is the addition of
information about $B$-hadrons and in the reconstruction of both small as well
as large-$R$ jets.. This simulation is able to perform particle propagation
through the magnetic field as well as hadronic and electromagnetic calorimeter
simulation including the response of the detector, muon identification system
and missing energy. The Delphes package has its own reconstruction procedure
leading to detector-level jets with a simulated realistic energy response
based on the performance of the ATLAS detector at LHC. Jets with two distance
parameters 0.4 and 1.0 were reconstructed using the anti-$k_{t}$ algorithm
[23] to form small-$R$ jets (small jets) and large-$R$ jets (large jets),
using the FastJet algorithm [24]. The Delphes package has its built-in jet
energy scale correction for jets, which is in our case used for small jets
only as pre-correction, with a private jet energy scale correction applied on
top of it for small-$R$ and as a fully private correction for the large-$R$
jets. More details can be found in Appendix A and in the selection section.
For the large-$R$ jets see Section 4.3 while for small-$R$ jets see Section
4.4.
The cross-section, processes details and the generated number of events for
samples generated by the MadGraph5 package; c.c. stands for the charge
conjugation and $\ell$ for an electron or muon. A cut of
$p_{\mathrm{T,SJ}}>20$ GeV is applied at the generator level. In the left
column, values on the $y_{0}$ lines indicate its generated mass.
Sample | Cross-section [pb] | Generated process | Events
---|---|---|---
$Wbb$+jets | 153.40000 | $pp\rightarrow W^{+}+j,W^{+}\rightarrow\ell^{+}+\nu_{\ell}$+c.c. | 655,855
$WWbb$+jets | 180.30000 | $pp\rightarrow W^{+}W^{-}b\bar{b},W^{+}\rightarrow\ell^{+}+\nu_{\ell},W^{-}\rightarrow jj$+c.c. | 1,000,000
$y_{0}$ 1000 GeV | 0.031 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 900 GeV | 0.053 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 800 GeV | 0.091 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 700 GeV | 0.160 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 600 GeV | 0.270 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$y_{0}$ 500 GeV | 0.410 | $pp\rightarrow y_{0}\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 500,000
$t\bar{t}+$jets | 178.60000 | $pp\rightarrow t\bar{t},t\rightarrow bjj,\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}$+c.c. | 2,934,961
## 4 Object and event selection
Events considered in the analysis are reconstructed at two levels; once with
the Delphes ATLAS-like detector simulation, forming detector level spectra,
and at the particle level. The event selection and the requirements differ
slightly for the reconstruction level and for the boosted, semi-boosted, semi-
boosted mixed and resolved topologies and are described below. Unless stated
otherwise, the same object and event selection applies to the particle-level
objects and selections.
### 4.1 Missing transverse energy requirement
The missing transverse energy ($E_{\mathrm{T,miss}}$) is a measure of energy
imbalance in the plane transverse to the beam and is equal to the value of
negative vector sum in the transverse plane of energies of all objects leaving
a calorimeter deposit. By definition this should equal to zero thanks to the
law of energy conservation, but the energy taken away by the undetected
neutrinos is not counted for in the detector and their contribution to
$\ell/\mu$ channels via leptonic $\tau$ decays is small. The magnitude of the
missing transverse energy is required to be $E_{\mathrm{T,miss}}>25$ GeV for
all topologies as well as for both the detector and the particle levels. This
ensures that only events in which neutrinos carry away a considerable amount
of energy are chosen for the analysis. This is a standard requirement for the
missing energy in most of top quark analyses in channels involving a charged
lepton.
### 4.2 Lepton selection
A requirement on the lepton (muon or electron) transverse momentum ensures the
selected lepton comes from the hard process and a cut of
$p_{\mathrm{T},\ell}>25$ GeV is used, a typical value in real experiment also
due to trigger requirements. Tau leptons are not considered in this analysis
as they decay before they enter the detector. In case more leptons fulfilling
the ${p_{\rm T}}$ requirement, only the electron or muon with the highest
transverse momentum is taken into account. This requirement is the same for
all topologies.
Charged leptons may radiate low energy photons which are highly collimated. E.
g. for electrons the separation of such photons and the lepton is below the
resolution of the detector and thus the photon energy is included by
construction at the detector level. The lepton dressing procedure is performed
at the particle level reconstruction to correct for this phenomenon, in which
the photon four-vectors, fulfilling the condition of the angular separation
threshold $\Delta
R_{\mathrm{\gamma,\ell}}=\sqrt{\Delta\eta_{\mathrm{\gamma,\ell}}^{2}+\Delta\phi_{\mathrm{\gamma,\ell}}^{2}}<0.1$,
are added to the lepton four-vector.
### 4.3 Large jet selection
Jets are the experimental signatures of hadronic final states of quarks and
gluons, which form particle showers entering the detector. In a typical
collider detector, jet constituents are clustered energy deposits in
calorimeters, or stable particles at the particle level. The jet four-vector
is the result of the reconstruction with the anti-$k_{t}$ algorithm.
We call jets reconstructed with a distance parameter $\Delta R=1$ as large
jets or large-$R$ jets. A private jet energy scale correction is derived on
the $t\bar{t}$ sample and applied to the detector level large jet before the
selection in order to correct jet energies to the particle level. The
magnitude of the jet energy scale correction is about 5% depending on jet
pseudorapidity $\eta$222The pseudorapidity is defined as a function of the
polar angle $\theta$ as $\eta\equiv-\ln\tan\frac{\theta}{2}$. and
$p_{\mathrm{T}}$. In the event selection, the transverse momentum of large
jets is required to be $p_{\mathrm{T,LJ}}>100$ GeV. This condition helps to
reduce the number of events with jets not coming from top quark or $W$ decays.
Furthermore, all large jets are considered in the pseudorapidity range
$|\eta|<2.5$. This constraint ensures in practice better jet identification as
the forward (large $|\eta|$) region is not well instrumented for tracking and
has a worse energy resolution. The isolation criterion of jets to be isolated
from the lepton ensures that the selected lepton is not contained within the
large jet by following the requirement of $\Delta
R_{\mathrm{LJ,\ell}}=\sqrt{\Delta\eta_{\mathrm{LJ,\ell}}^{2}+\Delta\phi_{\mathrm{LJ,\ell}}^{2}}>1$.
All these requirements are applied to all three topologies333There is no large
jet in the resolved topology. and both the detector and the particle levels.
Each large jet is then probed for the top quark and $W$ boson tagging (see
Appendix B for tagging and mistag efficiencies), first for the hypothesis as
coming from the top quark decay, then, in the semi-boosted topology, as coming
from the $W$ boson decay and in case none of the tagging was successful, the
event is then considered as a candidate for the semi-boosted mixed or the
resolved topology.
Tagging for the boosted topology is based on the constraint on the mass of the
large jet $110$ GeV$<M_{\mathrm{LJ}}<240$ GeV and a constraint combining the
large jet mass and a jet substructure variable $\tau_{3,2}$ [25], roughly a
consistency measure of finding three sub-jets inside the studied large jet
rather than two sub-jets, as $M_{\mathrm{LJ}}/\tau_{3,2}>256$ GeV. The value
for the second constraint was added to avoid background events, e.g. a large
jet from the $W$ boson. The selection is depicted in Fig. 2 (right) by the
area inside the dotted lines. The jet substructure variable
$\tau_{\mathrm{N}}$ is defined as
$\tau_{\mathrm{N}}=\frac{1}{d_{\mathrm{0}}}\sum_{k}p_{\mathrm{T},k}\min{R_{1,k},R_{2,k},...,R_{N,k}},$
(1)
where $d_{0}$ is a normalization parameter computed as
$d_{\mathrm{0}}=\Delta R\,\sum_{k}p_{\mathrm{T},k}$ (2)
with $\Delta R$ being the jet distance parameter used (here 1.0). The
subjettinesses are then combined into a ratio, such as
$\tau_{2,1}=\frac{\tau_{2}}{\tau_{1}}$, which has the ability to distinguish
the compatibility of the jet substructure with the one subjet hypothesis
(ratio closer to unity) in comparison to the scenario with two subjets (ratio
closer to zero). The substructure variable $\tau_{3,2}$ is defined in a
similar manner and effectively compares the jet substructure consistency with
two or three subjets.
|
---|---
Figure 2: Distribution of the jet substructure variable $\tau_{2,1}$ (left)
and $\tau_{3,2}$ (right) in the dependence on the large jet mass
($M_{\mathrm{LJ}}$) for the sample with $M_{\mathrm{y_{0}}}=700$ GeV at the
detector level. The large jets in the red dotted area are selected for the
reconstruction of the boosted $W$ boson and the top quark. The mass window is
set around the expected masses of the $W$ boson and top quark, respectively.
Tagging of large jets for the semi-boosted topology is based on the large jet
mass window $60$ GeV $<M_{\mathrm{LJ}}<120$ GeV and the jet substructure
variable $\tau_{2,1}$, reflecting the consistency of the jet to contain two
sub-jets inside the studied large jets rather than a single sub-jet, as
$\tau_{2,1}<0.6$. The selection is shown in Fig. 2 (left).
The large jet in the semi-boosted topology can be tagged as coming from the
$W$ boson, but there is no expected peak in the large jet mass spectrum in the
semi-boosted mixed topology, and thus it cannot be tagged based on its mass.
The case of the semi-boosted mixed topology is also considered, where the
isolated small jet does not originate from the $b$-quark but from a light
quark from the $W$ boson decay. To ensure the consistency with a $b$-quark
being a part of the large jet, there is a requirement on the large jet to
overlap with a small jet originating from a $b$-quark by requiring the
condition $\Delta
R_{\mathrm{LJ,SJ}}=\sqrt{\Delta\eta_{\mathrm{LJ,SJ}}^{2}+\Delta\phi_{\mathrm{LJ,SJ}}^{2}}<0.5$
at the detector level, see Section 4.4 for further information about small
jets. A similar condition is set at the particle level for the large jet to
contain a $B$-hadron within $\Delta R_{\mathrm{LJ,B-had}}<0.5$.
### 4.4 Small jets selection
The general requirements for small jets (small-$R$ jets), which are jets
reconstructed with a distance parameter $\Delta R=0.4$, are on the transverse
momentum $p_{T}>25$ GeV, pseudorapidity $|\eta|<2.5$ and the isolation from
the selected lepton $\Delta R_{\mathrm{SJ,\ell}}>0.5$. The jet energy scale
correction is applied on the detector level small jet objects before the
selection, which was derived on the $t\bar{t}$ sample on top of the Delphes
default jet energy scale. The magnitude of this residual jet energy scale
correction is about 2%. The identification of the small jets as coming from
the $b$-quark, $b$-tagging, is done by the Delphes simulation at the detector
level using the efficiency parameterization taken from [26], leading to the
$b$-tagging efficiency of 67.4% (72.8%) for jets of ${p_{\rm T}}=50$ (100)
GeV. At the particle level, $b$-tagging is done by the requirement of
containing a $B$-hadron within the jet as $\Delta R_{\mathrm{SJ,B-had}}<0.2$
for $B$ hadrons of ${p_{\rm T}}>5$ GeV as recommended by the LHC Top WG [27].
#### 4.4.1 Small jet for the reconstruction of the leptonically decaying top
quark
The small jet for the reconstruction of the leptonically decaying top quark
has to fulfill the angular condition $\Delta R_{\mathrm{SJ,\ell}}<2$, which
ensures that it lies in the vicinity of the selected lepton, and must be
$b$-tagged. The condition of a large jet isolation from the lepton $\Delta
R_{\mathrm{SJ,LJ}}>1.5$ applies to all topologies with the exception of the
resolved topology, where there is no large jet. Such a selected small jet is
then removed from the jet collection and from further consideration. This is
the only selected small jet in case of the boosted topology.
#### 4.4.2 Small jet for the hadronically decaying top quark, semi-boosted
topology
For the reconstruction of the hadronically decaying top quark in the semi-
boosted topology a small $b$-tagged jet is required in the vicinity to the
selected large jet $1<\Delta R_{\mathrm{SJ,LJ}}<1.5$. Thus a partial overlap
between the selected large jet and the considered small jet is allowed, _i.e._
the selected $b$-tagged small jet should be partially contained in the
selected large jet.
#### 4.4.3 Small jet for the hadronically decaying top quark, semi-boosted
mixed topology
The conditions for the semi-boosted mixed topology are similar to the
conditions for the semi-boosted topology. The vicinity condition to the large
jet remains unchanged but the small jet is required not to be $b$-tagged while
a $b$-tag is required for the selected large jet.
#### 4.4.4 Small jet for the hadronically decaying top quark, resolved
topology
The resolved topology selection is tried as the last option before the event
is discarded. The reconstruction of the hadronically decaying top quark in the
resolved topology requires three small jets, one of them $b$-tagged. The
algorithm first takes two small non-$b$-tagged jets with the highest
transverse momentum and tests their invariant mass $M_{\mathrm{SJ,SJ}}<120$
GeV to avoid dijets not corresponding to the mass of the $W$ boson. Then it
adds the four-vector of the remaining $b$-tagged jet444One $b$-tagged jet is
used in the reconstruction of the leptonically decaying top quark.. If all
such three jets are found, the event is accepted.
## 5 Reconstruction
The events passing the selection described in the previous chapter are
entering the top anti-top quark pair four-vector reconstruction described in
this section and illustrated on the $y_{0}\rightarrow t\bar{t}$ sample with
$M_{\mathrm{y_{0}}}=700$ GeV, although all samples were processed the same
way.
### 5.1 Leptonically decaying top quark
The reconstruction of the leptonically decaying top quark is the same for all
four studied topologies, starting with setting the transverse momentum of the
neutrino ($p_{\mathrm{T,\nu}}$) with the missing transverse energy
$E_{\mathrm{T,miss}}$. The missing energy together with the four-vector of the
selected lepton is used to calculate the longitudinal component of the
neutrino momentum ($p_{z,\nu}$) from the $W$ boson mass constraint
$M_{W}=M_{\ell\nu}$, which leads to a quadratic equation with two solutions in
general. The solution which leads to the more central neutrino in the rapidity
is accepted in the reconstruction. If the solution leads to a complex number
result, the imaginary part is discarded. This procedure is often used in top
quark analyses, e.g. in the ATLAS experiment [1]. The $W$ boson is
reconstructed as the sum of four-vectors of the lepton and the reconstructed
neutrino. Finally, the top quark four-vector is formed from the reconstructed
$W$ boson four-vector and the selected $b$-tagged small jet as described in
Section 4.4.1. The mass of the reconstructed leptonically decaying top quark
in the studied topologies is shown in Fig. 3 at both the detector and
particles levels.
|
---|---
Figure 3: Comparison between topologies for the shapes of the reconstructed
leptonically decaying top quark mass ($M_{\mathrm{t,lep}}$) for the sample
with $M_{\mathrm{y_{0}}}=700$ GeV at the particle (left) and the detector
(right) levels. The vertical dashed lines indicate the position of the maximum
value in the spectrum for each of the topologies, which happens to be the same
bin for all topologies at the particle level.
### 5.2 Hadronically decaying top quark, boosted topology
The recognition of the boosted event is done by selecting a large jet
fulfilling conditions specified in Section 4.3. Since there is no
reconstructed $W$ boson candidate in this case, the top-tagged large jet is
considered to contain most of the products coming from the top quark decay. To
verify this, the mass of the large jet corresponding to the reconstructed
hadronically decaying top quark is shown in Fig. 4 at both the detector and
the particle levels, showing a peak around the top quark mass.
### 5.3 Hadronically decaying top quark, semi-boosted topology
The reconstruction of the hadronically decaying top quark in the semi-boosted
topology uses the selected $W$-tagged large jet and one small $b$-tagged jet,
fulfilling the conditions mentioned in Section 4.4.2. The large jet is
considered as the hadronically decaying $W$ boson, with its mass shown in Fig.
5. The reconstructed top quark is formed by adding the selected $b$-tagged
small jet four-vector and its mass is shown in Fig. 4, again exhibiting the
expected peak which is sharper at the particle level due to finite detector
resolution.
### 5.4 Hadronically decaying top quark, semi-boosted mixed topology
The reconstruction of the hadronically decaying top quark in the semi-boosted
mixed topology is performed by summing one large jet and one non-$b$-tagged
small jet four-vectors, see Section 4.4.3 for details. The invariant mass of
such a large jet does not produce a peak but in the combination with the
selected small jet the resulting invariant mass peak should correspond to the
one of the top quark as shown in Fig. 4.
### 5.5 Hadronically decaying top quark, resolved topology
The resolved topology has the largest combinatorial ambiguity as it involves
largest multiplicity of objects for the reconstruction, starting with the $W$
boson reconstruction from two highest transverse momentum small jets which are
not tagged as $b$-jets. The reconstructed $W$ boson candidate mass, shown in
Fig. 5, is required to be 60–120 GeV, otherwise the event is discarded. The
third selected small jet for the reconstruction in the resolved topology is
required to be $b$-tagged and is added to the reconstructed $W$ boson forming
finally the four-vector of the hadronically decaying top quark with its mass
shown in Fig. 4.
|
---|---
Figure 4: Comparison between topologies for the hadronically decaying top quark mass ($M_{\mathrm{t,had}}$) for the sample with $M_{\mathrm{y_{0}}}=700$ GeV at the particle (left) and the detector (right) level. The vertical dashed lines indicate the position of the maximum value in the spectrum for each of the topologies. |
---|---
Figure 5: Comparison between topologies for the shapes of the reconstructed
hadronically decaying $W$ boson mass ($M_{\mathrm{W,had}}$) for the sample
with $M_{\mathrm{y_{0}}}=700$ GeV at the particle level (left) and the
detector level (right). The vertical dashed lines indicate the position of the
maximum value in the spectrum for each of the topologies.
### 5.6 Top anti-top quark pair system
The four-vector of the top anti-top quark (${t\bar{t}}{}$) pair system is
reconstructed as the combination of the leptonically and hadronically decaying
top quarks. Its mass and the contributing fractions of events from the
particular topologies are shown in Fig. 6. The fractions depend on the mass of
the hypothetical $y_{0}$ particle as shown in Fig. 7. This plot illustrates
the existence of the transition region between the resolved and the boosted
topology which was mentioned in Section 1 and which benefits from the
implementation of the semi-boosted and semi-boosted mixed topologies via their
non-negligible event fractions.
|
---|---
Figure 6: Contributions of different topologies to the reconstruction of the
${t\bar{t}}$ invariant mass ($M_{\mathrm{t\bar{t}}}$) for the sample with
$M_{\mathrm{y_{0}}}=700$ GeV in descendant order: boosted (blue), semi-boosted
(red), semi-boosted mixed (purple), and resolved (green) at the particle
(left) and the detector (right) level. The corresponding percentage is
presented in the legend for each topology.
---
Figure 7: The fraction of events contributing to the $t\bar{t}$ reconstruction
from each topology over samples with various masses of the hypothetical
$y_{0}$ particle ($M_{y_{0}}$) at the detector (solid lines, full markers) and
particle (dotted lines, open markers) levels.
### 5.7 Migration of events between topologies
The kinematic reconstruction at the detector and particle levels is
accomplished in parallel under the same selection requirements but the
resulting event topologies are not necessarily the same at the two
reconstruction levels. This is described by a migration matrix between the two
levels in terms of the topologies as illustrated in Fig. 8 (left). Similar
migrations further apply to values and bins of any studied observable. The
example for the reconstructed ${t\bar{t}}{}$ pair transverse momentum
($p_{\mathrm{T,t\bar{t}}}$) is shown in Fig. 8 (right), this plot also shows
migration of events between different bins. A matching condition, requiring
the same topology at the detector and particle levels, is applied for the
purpose of unfolding and only those events are taken into account to study
migration between bins in selected spectra in the subsequent unfolding
procedure.
|
---|---
Figure 8: The migration of events between the detector and the particle levels
between the topologies (left) and an example of the migration of events for
the transverse momentum of the $t\bar{t}$ system $p_{\mathrm{T,t\bar{t}}}$
between the resolved (R), semi-boosted mixed (SBM), semi-boosted (SB) and
boosted (B) topologies (right) for the sample with $M_{\mathrm{y_{0}}}=700$
GeV. The bin range for each of the sub-spectrum is 0–1000 GeV.
## 6 Results
The results are summarized in this section consisting of the resolution of
different samples, the performance of the unfolding process for selected
variables in dedicated stacked samples; and the significance studies of a
Beyond the Standard Model (BSM) signal over the Standard Model background
before and after unfolding.
### 6.1 Resolution of the ${t\bar{t}}{}$ resonance mass peak
As expected for the $y_{\mathrm{0}}\rightarrow t\bar{t}$ BSM process, the
reconstructed $t\bar{t}$ mass ($M_{\mathrm{t\bar{t}}}$) spectrum peaks around
the value of the mass of the studied hypothetical particle $y_{0}$ by
construction. The width of the distribution is a measure of the resolution of
the $t\bar{t}$ resonance mass in given topology. A Gaussian curve was used to
determine its width at both the detector and the particle levels and is shown
in Fig. 9 as absolute (top) as well as relative (bottom), i.e. when divided by
the $y_{0}$ mass in the corresponding sample.
|
---|---
Figure 9: The comparison of the $t\bar{t}$ system mass resolution for the
samples with different masses of the hypothetical particle $y_{0}$ in the
particular topologies the absolute (left) and relative (right) resolution with
respect to $M_{\mathrm{y_{0}}}$. The horizontal shift in the position of
markers around each mass point in the right plot is on the purpose to avoid
the loss of information due to their overlap.
The relative resolution is comparable between all the topologies with the
exception for the semi-boosted mixed topology at the detector level where the
resolution is slightly worse, being approximately 15% while the resolution of
the other topologies is 9–13%.
### 6.2 The unfolding procedure and significance tests
The unfolding procedure corrects for finite detector resolution effects in the
measured spectra. The Fully Bayesian Unfolding (FBU) [28] was chosen for the
purposes of this procedure as implemented with the PyMC3 package [29]. In
short, FBU uses the Bayesian theorem to estimate the truth (here particle)
level spectrum from the measured detector-level spectrum using the migration
matrix. As such, the matrix must be evaluated using simulated events and is
normalized so that it describes the migration of events in a selected spectrum
from particle-level to detector level bins. FBU returns a binned posterior
corresponding to the estimated probability distribution of the variable in
each particle-level bin. Its maximum can be chosen as the truth-level estimate
for the observable in given bin. Among the main advantages of this method are
that the migration matrix does not need to be inverted (which is a numerically
unstable task) nor the problems is regularized and therefore becoming modified
as e.g. in the singular value decomposition method [30]. Further advantages
are the absence of iterations (and a need to terminate them at some point) as
in the iterative Bayesian unfolding [31]; and the full control over the result
as the full probability density is revealed in each bin.
In general, the unfolding process can be described by the following formula
$\hat{T}_{\mathrm{i}}=\frac{1}{f_{\mathrm{i,eff}}}M_{\mathrm{ij}}^{-1}f_{\mathrm{j,acc}}(D_{\mathrm{j}}-B_{\mathrm{j}})\,,$
(3)
where $\hat{T}_{i}$ is the estimate of the particle level spectrum in bin $i$,
$f_{\mathrm{i,eff}}$ is the efficiency correction, $M_{\mathrm{ij}}^{-1}$
stands for the main unfolding procedure using the migration matrix555Fully
Bayesian Unfolding does not use the inverted migration matrix for the
estimation of the particle level spectra, the notation in the formula stands
for a shorthand of the unfolding procedure in general. $M_{\mathrm{ij}}$,
$f_{\mathrm{j,acc}}$ is the acceptance correction in detector level bin $j$,
$D_{\mathrm{j}}$ is the measured detector level spectrum and $B_{\mathrm{j}}$
is the estimated background. The efficiency and acceptance correction factors
are defined as
$f_{\mathrm{i,eff}}=\frac{P_{\mathrm{t\bar{t},i}}^{\mathrm{match}}}{P_{\mathrm{t\bar{t},i}}}\quad\mathrm{and}\quad
f_{\mathrm{j,acc}}=\frac{D_{\mathrm{t\bar{t},j}}^{\mathrm{match}}}{D_{\mathrm{t\bar{t},j}}}\,,$
(4)
where, in the context of this study, $P_{\mathrm{t\bar{t},i}}$ is the
particle-level spectrum in bin $i$ using the $t\bar{t}$ sample as the model
process, while $P_{\mathrm{t\bar{t},i}}^{\mathrm{match}}$ is the particle
level spectrum in bin $i$ for events matched to the detector level events,
_i.e._ only events reconstructed at both particle and detector levels in the
same topology contribute to this spectrum. Similarly,
$D_{\mathrm{t\bar{t},j}}$ is the detector-level spectrum in bin $j$ and
$D_{\mathrm{t\bar{t},j}}^{\mathrm{match}}$ is the detector level spectrum in
bin $j$ for events matched to the particle level. Both correction factors are
in the range between 0 and 1 by definition and were prepared using a
statistically independent $t\bar{t}$ sample w.r.t. the ${t\bar{t}}{}$ sample
used to compose the spectra to be unfolded.
The test spectra entering the unfolding procedure have the addition of the
$y_{0}$ signal sample with an amplified cross section by an ad hoc number of
$10^{3}$ ($5\cdot 10^{3}$ in case of resolved topology) to study the impact of
the unfolding on the strength of a well-present signal of similar significance
in all topologies.
### 6.3 Unfolding selected spectra
Unfolding of three selected spectra is described in this section on the
example in the semi-boosted topology, namely the transverse momentum of the
hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$), the
invariant mass of the reconstructed top anti-top quark pair
($M_{\mathrm{t\bar{t}}}$) and the production angle of the hadronically
decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$).
The simulated samples were divided into two statistically independent halves.
The sum of first such halves is taken as the pseudo-data serving as input into
the unfolding procedure while their statistically independent counterparts are
stacked to form the expected total prediction.
The comparison between the pseudo-data (black markers) and stacked histograms
of the prediction (filled) for the transverse momentum of the hadronically
decaying top quark in the semi-boosted topology is shown in Fig. 10 (left).
The histogram and the stacked spectra agree well, their difference is within
the statistical uncertainties. The binning was selected to respect the
resolution, falling statistics, and so that the unfolding proceeds fast enough
but still delivers useful information about spectra shape. The $t\bar{t}$
system invariant mass spectrum was chosen due to the possibility to see the
hints of events from the sample with the hypothetical particle $y_{0}$. The
spectrum entering the unfolding procedure and its statistically independent
counterpart is shown in Fig. 10 (right).
The production angle of the hadronically decaying top quark was chosen to
study the performance of this variable against the different topologies as an
example of an angular variable. The general assumption is that boosted
topology has jets preferably in smaller $|\eta|$ range due to the usually
larger transverse momentum of the particles. This trend of the central $\eta$
preference drops in semi-boosted topologies and diminishes in the resolved
topology. The spectrum used in unfolding and its statistically independent
counterpart is also shown in Fig. 10 (bottom).
The corrections used in the unfolding procedure were evaluated with the
statistically independent sample from the one forming the pseudo-data using
the $t\bar{t}$ sample. The corresponding acceptance and efficiency corrections
are shown in Fig. 11 and the migration matrices are presented in Fig. 12.
The unfolding results for the three spectra are shown in Fig. 13. A $\chi^{2}$
test was computed between the unfolded and the $t\bar{t}$ particle level
spectra with the $y_{0}$ signal included, resulting in
$\chi^{2}_{\mathrm{t\bar{t}+y_{0}}}$. As the input pseudo-data do contain the
$y_{0}$ signal, this comparison is a closure test of the unfolding procedure
ability to recover the particle level spectrum which consists of
${t\bar{t}}{}$ as well as the $y_{0}$ signal sample. Another $\chi^{2}$ test
was performed between the unfolded spectrum and the $t\bar{t}$-only particle
level spectrum, resulting in $\chi^{2}_{\mathrm{t\bar{t}}}$. This tests
evaluates the incompatibility of the unfolded pseudo-data with the
${t\bar{t}}$-only hypothesis. The middle panels of Fig. 13 show the ratio of
the unfolded spectra over the particle level spectra from the $t\bar{t}$
sample (black full markers)where the disagreement with the $t\bar{t}$-only
particle level spectrum is caused by the presence of the $y_{0}$ sample in the
pseudo-data.
|
---|---
Figure 10: Comparison of the detector level spectra from two statistically independent parts (full markers and filled stack) for the $t\bar{t}$ sample with the addition of $Wbb$ and $WWbb$ backgrounds and an admixture of events from the sample with $M_{\mathrm{y_{0}}}=700$ GeV for the transverse momentum of the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left), for the top anti-top quark pair invariant mass ($M_{\mathrm{t\bar{t}}}$, right) and for the crossing angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom), all spectra are reconstructed in the semi-boosted topology. The hatched bands in the top panel represent the statistical uncertainty in each sample. |
---|---
Figure 11: The acceptance and the efficiency for the two statistically independent $t\bar{t}$ samples for the reconstructed hadronically decaying top quark transverse momentum ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left), the invariant mass of the reconstructed $t\bar{t}$ system ($M_{\mathrm{t\bar{t}}}$, right) and for the production angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom). All spectra are reconstructed in the semi-boosted topology. Indices 1 and 2 denote the two statistically independent samples. |
---|---
Figure 12: The migration matrices for the reconstructed transverse momentum of the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, left) for the reconstructed invariant mass of $t\bar{t}$ system ($M_{\mathrm{t\bar{t}}}$, right) and for the crossing angle of the hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom), all in the semi-boosted topology. All matrices were derived from the $t\bar{t}$ sample and used in the unfolding procedure. The correlation factor $\rho$ is calculated between the detector and particle levels. |
---|---
Figure 13: The comparison between the unfolded spectrum (black full markers),
the detector level spectrum (blue) with the acceptance and efficiency
correction applied to be comparable to the particle level one; and the
$t\bar{t}$-only particle level spectrum (red) for the transverse momentum of
the hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, top
left), for the invariant mass of the reconstructed $t\bar{t}$ spectrum
($M_{\mathrm{t\bar{t}}}$, top right) and for the production angle of the
hadronically decaying top quark ($\cos\theta^{*}_{\mathrm{t,had}}$, bottom),
all in the semi-boosted topology. The $\chi^{2}_{\mathrm{t\bar{t}}}$ test is
performed between the unfolded and $t\bar{t}$ particle level spectra while
$\chi^{2}_{\mathrm{t\bar{t}+y_{0}}}$ between the unfolded and the $t\bar{t}$
particle level spectra with the $y_{0}$ signal included (closure test). The
middle panels show the ratio of the unfolded spectrum over the particle level
spectrum of the $t\bar{t}$ sample (full markers). Here the disagreement with
the $t\bar{t}$-only particle level spectrum is caused by the addition of the
$y_{0}$ signal before unfolding, the yellow band shows the statistical
uncertainty in the particle level spectrum from the $t\bar{t}$ sample. The
bottom panels show the detector (open markers, dashed line) and unfolded (full
markers, solid line) $y_{0}$ signal significances in each bin. The detector-
level $y_{0}$ signal significance is calculated using the original detector-
level spectrum, _i.e._ without applying the acceptance and efficiency
correction.
### 6.4 Significance at the detector and unfolded levels
The strength of the $y_{0}$ signal is quantified by the significance which
considers the total statistical uncertainties in samples used in given bin.
The significance $S$ in bin $i$ before unfolding is defined as
$S_{\mathrm{i,det}}\equiv(P_{\mathrm{i}}^{t\bar{t}+y_{0}+B}-T_{\mathrm{i}}^{t\bar{t}}-B_{\mathrm{i},1}-\ldots-
B_{\mathrm{i},k})/\sqrt{\sigma_{P_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}+\sigma_{B_{\mathrm{i},1}}^{2}+\ldots}\,,$
(5)
where $P_{\mathrm{i}}$ is the number of the the pseudo data events consisting
from the signal and the background added to the expected $t\bar{t}$ sample in
bin $i$; $T_{\mathrm{i}}$ is the detector level spectrum from the
statistically independent $t\bar{t}$ sample, $B_{\mathrm{i},k}$ is the
background contribution to the studied spectra from the $k$-th background
sample; and $\sigma_{P_{\mathrm{i}}}$, $\sigma_{T_{\mathrm{i}}}$ and
$\sigma_{B_{\mathrm{i},k}}$ are the statistical uncertainties in the pseudo
data, $t\bar{t}$ and the $k$-th background samples, respectively (all at the
detector level). The composition of the sample is denoted in the upper index,
_e.g._ $t\bar{t}+y_{0}+B$ stands for the sum of the background, $t\bar{t}$ and
the $y_{0}$ samples. Systematic uncertainties are not part of this study as
their effect would be largely coherent across topologies, thus not changing
the conclusions nor hierarchy of the observed patterns.
A similar significance is defined after the unfolding procedure, before which
the background was subtracted, as
$S_{\mathrm{i,unf}}\equiv(U_{\mathrm{i}}^{t\bar{t}+y_{0}}-T_{\mathrm{i}}^{t\bar{t}})/\sqrt{\sigma_{U_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}}\,,$
(6)
where $U_{\mathrm{i}}$ is the number of the unfolded pseudo-data events in bin
$i$, $T_{\mathrm{i}}$ is the particle level spectrum from the statistically
independent $t\bar{t}$ sample; and $\sigma_{U_{\mathrm{i}}}$ and
$\sigma_{T_{\mathrm{i}}}$ are the statistical uncertainties in the unfolded
spectrum bin $i$ and in the statistically independent $t\bar{t}$ sample at the
particle level, respectively. The binned detector and unfolded significances
are shown under the ratio plots of the unfolded spectra in Fig. 13. The
integral significance, strength of the signal over the whole spectrum, is
defined similarly for both the detector and the unfolded level. The detector
level integral significance is defined as
$S_{\mathrm{I,det}}\equiv\sum_{i=0}^{m}(P_{\mathrm{i}}^{t\bar{t}+y_{0}+B}-T_{\mathrm{i}}^{t\bar{t}}-B_{\mathrm{i},1}-\ldots-
B_{\mathrm{i},k})/\sqrt{\sum_{i=0}^{m}(\sigma_{P_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}})\,,$
(7)
where $m$ is number of bins in given spectrum. The detector level integral
significance is the same for all variables.
The integral significance at the unfolded level is defined as
$S_{\mathrm{I,unf}}\equiv\sum_{i=0}^{m}(U_{\mathrm{i}}^{t\bar{t}+y_{0}}-T_{\mathrm{i}}^{t\bar{t}})/\sqrt{\sum_{i=0}^{m}(\sigma_{U_{\mathrm{i}}}^{2}+\sigma_{T_{\mathrm{i}}}^{2}})\,.$
(8)
The unfolded integral significance varies slightly over spectra as in the
unfolding procedure the integral of the spectrum may not be preserved. The
values of both the detector and the unfolded integral significance are
presented in the legend in Fig. 14.
The significances were calculated for the three selected spectra in all
topologies and at both detector and particle levels. The comparison between
significances before and after the unfolding procedure over the studied
topologies are shown in Fig. 14 for the spectra of
$p_{\mathrm{T}}^{\mathrm{t,had}}$, $M_{\mathrm{t\bar{t}}}$ and
$\cos\theta^{*}_{\mathrm{t,had}}$.
The significance uncertainties were estimated by using 100 pseudo experiments
for each spectrum with a smeared content in each detector-level bin. The
smearing was performed by drawing a random number from the Gaussian
distribution within the $\sigma$ parameter equal to the statistical
uncertainty in the total detector-level spectrum in the given bin and with the
mean parameter set to zero. Each such spectrum was unfolded using the same
procedure and corrections and the binned significances were evaluated. The
resulting standard deviation of significances in each bin is considered as the
statistical uncertainty in the unfolded significance. The statistical
uncertainty of the significances is already presented as error bars in Fig.
14.
|
---|---
Figure 14: The detector (open markers, dashed line) and unfolded (full
markers, solid line) significances for the transverse momentum of the
hadronically decaying top quark ($p_{\mathrm{T}}^{\mathrm{t,had}}$, top left),
invariant mass of the reconstructed $t\bar{t}$ pair ($M_{\mathrm{t\bar{t}}}$,
top right) and the crossing angle of the hadronically decaying top quark
($\cos\theta^{*}_{\mathrm{t,had}}$, bottom) plotted for all the topologies in
each bin. The orange band defines the area where the absolute value of the
significance is below three, corresponding to the 3-$\sigma$ interval. The
lower pads present the ratios of the unfolded over the detector level
significances, without uncertainties which are highly correlated.
The signal significances in the $p_{\mathrm{T}}^{\mathrm{t,had}}$ spectrum
peaks at different values of $p_{\mathrm{T}}^{\mathrm{t,had}}$ depending on
the topology as the event selection in each topology biases the spectrum and
effectively selects different ranges in $M_{\mathrm{t\bar{t}}}$, too. On the
other hand, significance in the $M_{\mathrm{t\bar{t}}}$ spectrum peaks around
the value of the generated $y_{0}$ mass of 700 GeV as expected, with a slight
tail to lower values in the resolved topology which is the least suitable one
to reconstruct a resonance of such a large mass. In contrast, the
$\cos\theta^{*}_{\mathrm{t,had}}$ is very flat also for the signal sample and
there is no clear isolated excess of signal events in this spectrum, with the
exception of the boosted topology which selects, by construction,
high-${p_{\rm T}}$ large jets and thus also top quarks, consequently more
localized in the central rapidity region, producing a pear around zero in
$\cos\theta^{*}_{\mathrm{t,had}}$.
The three selected spectra are thus good candidate observables to illustrate
different behavior and spread of significances over bins, also presenting a
selection of observables of a dimension of energy as well as dimensionless
(angular). The binned significances are in general lower after the unfolding,
for which an explicit proof is delivered by this study. The cause of this is
as follows.
While a sharper spectrum may be recovered by unfolding, the procedure in
general correlates information among bins by maximizing a likelihood function
in case of FBU, or minimizing (possibly modified and regularized) $\chi^{2}$
or iterating and sequentially improving the result for the case of other
methods. An increase in the correlation across bins of the unfolded spectrum
is a known and important fact and a correlation matrix should preferably be
published along with unfolded spectra from real experiments, as done _e.g._ in
[32, 33] where a correlation matrix between the observables was also
evaluated. We observe that in case of the FBU method the posteriors variance
usually increases, leading to larger absolute as well as relative
uncertainties of the unfolded spectrum w.r.t. the particle level one. This
effectively decreases the significance of the observed signal excess. An
increase of the statistical uncertainties with the number of iterations in
case of the Iterative Bayesian Unfolding [31] was also reported by other
analyses [34]. We note that in our case the BSM signal significances decrease
by 20–40%. Other more explicit regularization methods like the SVD [30]
provide a spectrum with a smaller statistical uncertainty by definition
(effectively ditching small regularized response matrix eigenvalues which
would lead to large variations), but are prone to unfolding biases towards the
underlying simulation spectrum. Also the FBU extension with a regularization
term leads to more narrow posteriors (smaller statistical uncertainty) [35].
This places the standard FBU (without an explicit regularization term) among
high-level unfolding methods with realistic statistical uncertainties.
## 7 Conclusions
The results of the semi-boosted and semi-boosted mixed reconstruction
algorithm show potential to enhance the number of events in the $t\bar{t}$
analyses in the semi-leptonic decay channel. The estimates show the enrichment
in events between 20% and 50% in the $t\bar{t}$ pair mass region ranging from
500 GeV to 1000 GeV. The resolution in the semi-boosted topology and the
resolved or boosted topology is comparable, only the semi-boosted mixed has a
worse resolution roughly by factor of 1.5. The performance of the unfolding
procedure including a simple background model shows results corresponding well
to the particle level and the significance of the enhanced signal of the
hypothetical $y_{0}$ particle is still visible after the unfolding. Values of
the detector and the unfolded integral significance are comparable, yet there
is 20–30% decrease in significance between the detector and the unfolding
levels caused by the unfolding in the reconstructed $t\bar{t}$ mass spectrum
and 5–20% in the reconstructed transverse momentum of the hadronically
decaying top quark spectrum and the production angle of the top quarks, i.e.
both for energy-dependent as well as angular variables.
The concrete proof of diminishing a BSM signal significance by the unfolding
procedure (using FBU) is, to our knowledge, shown explicitly for the first
time in this study. Our findings thus support why the model-dependent searches
for new physics at the LHC are mostly done at the detector level, using
concrete detector-level (fully-simulated) BSM signals. We attribute the
decrease of the significance to the increase of the statistical uncertainty in
the unfolded spectrum, _i.e._ to the posterior widening in the case of the FBU
method. Due to this, although the $M_{{t\bar{t}}}$ spectrum becomes sharper
after unfolding, revealing a narrower peak of the $y_{0}$ resonance, the
binned significance does not increase due to larger statistical uncertainties
of the unfolded spectrum caused by unfolding-induced correlation across bins.
On the other hand, the significance stays substantial even after unfolding,
opening doors to a comparison of any theory prediction at the particle-level
to the unfolded data.
The described algorithm proves that semi-boosted and semi-boosted mixed
topologies are sensitive to the possible presence of BSM signals. The
selection criteria chosen close to those in real analyses make the studied
algorithms applicable also in current LHC experiments.
## 8 Acknowledgments
The authors gratefully acknowledge the support from the Czech Science
Foundation project GAČR 19-21484S and project IGA_PrF_2021_004 of the Faculty
of Science of the Palacky University Olomouc, Czech Republic. This work was
performed as a part of fulfillment of doctoral studies of J. Pacalt at the
Applied physics programme at the Faculty of Science of the Palacky University.
## References
* [1] G. Aad et al. The ATLAS Experiment at the CERN Large Hadron Collider. JINST, 3:S08003, 2008.
* [2] Lyndon R Evans and Philip Bryant. LHC Machine. JINST, 3:S08001. 164 p, 2008. This report is an abridged version of the LHC Design Report (CERN-2004-003).
* [3] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi. DELPHES 3, A modular framework for fast simulation of a generic collider experiment. JHEP, 02:057, 2014.
* [4] S. Chatrchyan et al. The CMS Experiment at the CERN LHC. JINST, 3:S08004, 2008.
* [5] Particle Data Group. Review of Particle Physics. Progress of Theoretical and Experimental Physics, 2020(8), 08 2020\. 083C01.
* [6] Vardan Khachatryan et al. Identification techniques for highly boosted W bosons that decay into hadrons. JHEP, 12:017, 2014.
* [7] Serguei Chatrchyan et al. Search for Anomalous $t\bar{t}$ Production in the Highly-Boosted All-Hadronic Final State. JHEP, 09:029, 2012. [Erratum: JHEP 03, 132 (2014)].
* [8] Serguei Chatrchyan et al. Search for Resonant $t\bar{t}$ Production in Lepton+Jets Events in $pp$ Collisions at $\sqrt{s}=7$ TeV. JHEP, 12:015, 2012.
* [9] Georges Aad et al. Identification of boosted, hadronically decaying W bosons and comparisons with ATLAS data taken at $\sqrt{s}=8$ TeV. Eur. Phys. J. C, 76(3):154, 2016.
* [10] Morad Aaboud et al. Measurement of jet-substructure observables in top quark, $W$ boson and light jet production in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector. JHEP, 08:033, 2019.
* [11] Georges Aad et al. A new method to distinguish hadronically decaying boosted $Z$ bosons from $W$ bosons using the ATLAS detector. Eur. Phys. J. C, 76(5):238, 2016.
* [12] Morad Aaboud et al. Performance of top-quark and $W$-boson tagging with ATLAS in Run 2 of the LHC. Eur. Phys. J. C, 79(5):375, 2019.
* [13] Y. Afik, F. Maltoni, K. Mawatari, P. Pani, G. Polesello, Y. Rozen, and M. Zaro. DM+$b\bar{b}$ simulations with DMSimp: an update. In Dark Matter at the LHC 2018: Experimental and theoretical workshop, 11 2018.
* [14] Chiara Arina, Mihailo Backović, Jan Heisig, and Michele Lucente. Solar $\gamma$ rays as a complementary probe of dark matter. Phys. Rev. D, 96(6):063010, 2017.
* [15] Andreas Albert et al. Recommendations of the LHC Dark Matter Working Group: Comparing LHC searches for dark matter mediators in visible and invisible decay channels and calculations of the thermal relic density. Phys. Dark Univ., 26:100377, 2019.
* [16] Sabine Kraml, Ursula Laa, Kentarou Mawatari, and Kimiko Yamashita. Simplified dark matter models with a spin-2 mediator at the LHC. Eur. Phys. J. C, 77(5):326, 2017.
* [17] Goutam Das, Celine Degrande, Valentin Hirschi, Fabio Maltoni, and Hua-Sheng Shao. NLO predictions for the production of a spin-two particle at the LHC. Phys. Lett. B, 770:507–513, 2017.
* [18] Matthias Neubert, Jian Wang, and Cen Zhang. Higher-Order QCD Predictions for Dark Matter Production in Mono-$Z$ Searches at the LHC. JHEP, 02:082, 2016.
* [19] Mihailo Backović, Michael Krämer, Fabio Maltoni, Antony Martini, Kentarou Mawatari, and Mathieu Pellen. Higher-order QCD predictions for dark matter production at the LHC in simplified models with s-channel mediators. Eur. Phys. J. C, 75(10):482, 2015.
* [20] Olivier Mattelaer and Eleni Vryonidou. Dark matter production through loop-induced processes at the LHC: the s-channel mediator case. Eur. Phys. J. C, 75(9):436, 2015.
* [21] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro. The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. JHEP, 07:079, 2014.
* [22] Torbjörn Sjöstrand, Stefan Ask, Jesper R. Christiansen, Richard Corke, Nishita Desai, Philip Ilten, Stephen Mrenna, Stefan Prestel, Christine O. Rasmussen, and Peter Z. Skands. An introduction to pythia 8.2. Computer Physics Communications, 191:159–177, Jun 2015.
* [23] Matteo Cacciari, Gavin P Salam, and Gregory Soyez. The anti-ktjet clustering algorithm. Journal of High Energy Physics, 2008(04):063–063, Apr 2008.
* [24] Matteo Cacciari, Gavin P. Salam, and Gregory Soyez. FastJet User Manual. Eur. Phys. J., C72:1896, 2012.
* [25] Benjamin Nachman, Pascal Nef, Ariel Schwartzman, Maximilian Swiatlowski, and Chaowaroj Wanotayaroj. Jets from Jets: Re-clustering as a tool for large radius jet reconstruction and grooming at the LHC. JHEP, 02:075, 2015.
* [26] Expected performance of the ATLAS $b$-tagging algorithms in Run-2. Technical report, CERN, Geneva, Jul 2015. All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2015-022.
* [27] LHC Top WG. Particle level objects and pseudo-top-quark definitions. https://twiki.cern.ch/twiki/bin/view/LHCPhysics/ParticleLevelTopDefinitions, 2014\.
* [28] Georgios Choudalakis. Fully bayesian unfolding. https://arxiv.org/abs/1201.4612v4, 2012.
* [29] John Salvatier, Thomas V Wiecki, and Christopher Fonnesbeck. Probabilistic programming in python using pymc3. PeerJ Computer Science, 2:e55, 2016.
* [30] Andreas Höcker and Vakhtang Kartvelishvili. Svd approach to data unfolding. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 372(3):469–481, Apr 1996.
* [31] G. D’Agostini. Improved iterative bayesian unfolding. https://arxiv.org/abs/1010.0632, 2010.
* [32] Georges Aad et al. Measurements of normalized differential cross sections for $t\bar{t}$ production in pp collisions at $\sqrt{s}=7$ TeV using the ATLAS detector. Phys. Rev. D, 90(7):072004, 2014.
* [33] M. Aaboud et al. Measurements of top-quark pair differential cross-sections in the lepton+jets channel in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector. JHEP, 11:191, 2017.
* [34] Georges Aad et al. Differential top-antitop cross-section measurements as a function of observables constructed from final-state particles using pp collisions at $\sqrt{s}=7$ TeV in the ATLAS detector. JHEP, 06:100, 2015.
* [35] P. Baron and J. Kvita. Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method. Symmetry, 12:2100, 2020.
## Appendix A Jet energy scale derivation and closure tests
The jet energy scale (JES) procedure corrects for a finite energy response of
the detector to hadronic final states (jets) as function of their angle in the
detector and energy. The goal is to correct detector-level jet energies to the
particle level. The method to calculate the jet response thus uses information
from Monte Carlo simulation and forms a ratio between value of the jet energy
measured at the simulated detector $E_{\mathrm{det}}$ to the energy of a
angularly matched the particle level jet ($E_{\mathrm{ptcl}}$). In detail,
first a jet at the detector level is chosen, then the particle level jet
candidate with the smallest distance parameter defined as
$\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<R_{\mathrm{cut}}\,.$ (9)
is chosen as the matching jet, with the $R_{\mathrm{cut}}$ parameter set to
0.2 for small ($R=0.4$) jets and 0.3 for large ($R=1$) jets.
The correction is binned in the detector-level jet pseudorapidity and
transverse momentum
$\mathrm{JES}(\eta,p_{\mathrm{{T}}})=\left\langle\frac{E_{\mathrm{ptcl}}^{\mathrm{jet}}}{E_{\mathrm{det}}^{\mathrm{jet}}}\right\rangle,$
(10)
and so the inverse value of the JES correction is the response of the
detector-level jet. Histograms of the jet response corresponding to different
energy intervals are filled, each fitted by a Gaussian function. The mean of
the fit is plotted against the reconstructed energy and fitted by a polynomial
logarithmic function which is used to interpolate the JES correction to any
energy. The visualization of the derived JES correction in dependence on the
jet ${p_{\rm T}}$ and $\eta$ is in Fig. 15 for small jets (left) and large
jets (right).
|
---|---
Figure 15: The visualization of derived JES correction functions for large
jets (left) and for small jets (right).
A closure test was performed, in which the JES factors are applied to jets on
a statistically independent sample to the one used to derive the JES
corrections, but otherwise generated under the same settings. The correction
is re-derived on this already pre-corrected sample, thus the corrected
detector level jet energy over the matched particle level jet energy is
expected to be around unity by construction.
The closure test results as function of the transverse momentum and
pseudorapidity $\eta$ of jets are shown for both small jets and large jets in
Fig. 16 where the mean deviation from unity is well within 5%.
|
---|---
|
Figure 16: Jet energy correction closure tests for the large jets (right) and
small jets (left) as function of the transverse momentum (top) and $\eta$
(bottom) of jets.
## Appendix B Top quark and $W$ boson tagging efficiencies
The tagging of the large-$R$ jets originating from the top quark or the $W$
boson is a common practice in high energy physics and was used also in this
paper. In order to evaluate the tagging efficiencies, a comparison to the
generator level information is needed to define a truth jet label as top, $W$
or light otherwise. The angularly closest large-$R$ jet to the direction of
the original top quark was used as a probe for the truth top tagging
efficiency $\varepsilon_{\mathrm{top}}$ which is defined as follows
$\varepsilon_{\mathrm{top}}=\frac{N_{\mathrm{jet,top}}^{\mathrm{match\&tag}}}{N_{\mathrm{jet,top}}^{\mathrm{match}}}\,,$
(11)
where $N_{\mathrm{jet,top}}^{\mathrm{match\&tag}}$ is the number of jets
matched to the original top quark and top-tagged by the tagging technique; and
$N_{\mathrm{jet,top}}^{\mathrm{match}}$ is number of all jets tagged as
possibly originating from top quark, as marked by the algorithm. The truth $W$
boson tagging efficiency is defined in a similar manner. The mistag (fake)
efficiency was also evaluated, which describes the false positivity of the
tagger on jets not originating from the top quark or the $W$ boson. It is
defined by the following formula in case of the top tagging
$\varepsilon_{\mathrm{mis,top}}=\frac{N_{\mathrm{jet,top}}^{\mathrm{\neg
match\&tag}}}{N_{\mathrm{jet,top}}^{\mathrm{\neg match}}}\,,$ (12)
where $\varepsilon_{\mathrm{mis,top}}$ is the mistag efficiency of the top
quark tagger, $N_{\mathrm{jet,top}}^{\mathrm{\neg match\&tag}}$ is the number
of large jets which are not matched to the generator level top quark but are
top-tagged by the tagger; and $N_{\mathrm{jet,top}}^{\mathrm{\neg match}}$ is
the number of all large jets not matched to the generator level top quark. The
$W$ mistag efficiency is calculated in a similar way for the both detector and
particle levels.
Both tag and mistag efficiencies are shown in Fig 17 for the $W$ (left) and
top (right) tagger, studied at both detector and particle levels in dependence
on the transverse momentum of the large jet. The sample for the mistag
efficiency was the production of $2j2b$ events and contained 32.5M events,
with events generated in exclusive ${p_{\rm T}}$ ranges of the leading and
sub-leading jets in order to populate the phase space of higher transverse
momenta.
|
---|---
Figure 17: Tagging (blue) and mistag (red) efficiencies for the $W$ boson
(left) and for the top quark (right) in dependence on the transverse momentum
of the large jet studied on the $t\bar{t}$ sample at both particle and
detector levels.
|
# Language Models can Infer Action Semantics for Classical Planners from
Environment Feedback
Wang Zhu Ishika Singh Robin Jia Jesse Thomason
Department of Computer Science
University of Southern California
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Classical planning approaches guarantee finding a set of actions that can
achieve a given goal state when possible, but require an expert to specify
logical action semantics that govern the dynamics of the environment.
Researchers have shown that Large Language Models (LLMs) can be used to
directly infer planning steps based on commonsense knowledge and minimal
domain information alone, but such plans often fail on execution. We bring
together the strengths of classical planning and LLM commonsense inference to
perform domain induction, learning and validating action pre- and post-
conditions based on closed-loop interactions with the environment itself. We
propose PSALM, which leverages LLM inference to heuristically complete partial
plans emitted by a classical planner given partial domain knowledge, as well
as to infer the semantic rules of the domain in a logical language based on
environment feedback after execution. Our analysis on 7 environments shows
that with just one expert-curated example plans, using LLMs as heuristic
planners and rule predictors achieves lower environment execution steps and
environment resets than random exploration while simultaneously recovering the
underlying ground truth action semantics of the domain.
## 1 Introduction
Classical planning requires extensive domain knowledge to produce a sequence
of actions that achieve a specified goal. The domain contains expert-annotated
action semantics that govern the dynamics of the environment. For example,
traditional symbolic methods, like Planning Domain Description Language (PDDL)
solvers, have action semantics annotated in a domain file. Classical planning
algorithms systematically explore the state space based on actions that can be
executed in any given state as per the domain definition. Therefore, the
resultant plan is guaranteed to succeed, if the specified goal is achievable.
However, it is tedious to exhaustively define the domain to enable classical
planning in an environment, and requires a human expert for every new
environment.
We propose a novel domain induction task setting in PDDL, in which an agent
must automatically infer the action semantics of an environment without manual
annotation or error correction. In this setting, the domain information, such
as object properties and action functions headers, are given to an agent. The
agent is then asked to infer the action semantics through interacting with the
environment and learning from resulting feedback. Our proposed setting is
motivated by the longer-term goal of building real-world robots that can
explore a new environment (_e.g_., a kitchen) and learn to perform new tasks
in the environment (_e.g_., cleaning) without human specification of the
environment.
To solve this domain induction problem, we draw inspiration from prior work
that prompts Large Language Models (LLMs) to perform robotic planning tasks.
These methods do not require explicit domain specification, as they assume
implicit encoding of world knowledge in LLMs for generating unseen plans. For
instance, ProgPrompt Singh et al. (2023) generates program-like high-level
plans from LLM. However, LLM generated plans suffer from two major issues: the
plan is not guaranteed to execute successfully, and the environment-specific
prompt makes it hard to transfer similar performance to new environments.
Hence, instead of using the LLM directly as a planner, we use the LLM in
conjunction with a symbolic solver to iteratively explore the environment and
predict the action semantics based on environment feedback.
We propose Predicting Semantics of Actions with Language Models (PSALM), a
novel method that combines language models with symbolic solvers to induce
domain world models. At a high level, our method maintains a probabilistic
memory of learned action semantics, which it iteratively improves by
interacting with the environment. We use LLMs to sample (possibly incorrect)
candidate trajectories and infer action semantics based on the result of
executing those trajectories in the environment. This leverages LLMs’ strong
commonsense reasoning abilities, as well as their ability to generate
syntactically valid formal semantics. Meanwhile, we use a symbolic solver to
search for ways to achieve the final goal state based on our current belief
about the action semantics; this compensates for the LLMs’ comparatively poor
ability to efficiently search through a large state space.
Across 7 diverse environments, we demonstrate the effectiveness and efficiency
of leveraging LLMs for domain induction. PSALM consistently induces correct
domain files, and does so with substantially fewer total execution steps and
numbers of resets than multiple baseline approaches. Overall, this integration
of LLMs and symbolic solvers presents a promising avenue for advancing domain
induction capabilities in robotics.
## 2 Related Works
### 2.1 Classical Planning
Classical task planning algorithms have been widely applied in autonomous
spacecrafts, military logistics, manufacturing, games, and robotics. The
automated STRIPS planner was the first algorithm that operated the Shakey
robot Fikes and Nilsson (1971) in 1970. Classical planners classically require
finite, deterministic, and full state information to generate guaranteed plans
when a path from the initial to the goal state is possible. Some other
frameworks were also shown to be useful for robot planning Carbonell et al.
(1991); Nau et al. (2003). Planning domain description language (PDDL) Jiang
et al. (2019); Fox and Long (2003) and answer set programming (ASP) Brewka et
al. (2011); Lifschitz (2002) are commonly used specification languages for
classical planning domains.
### 2.2 Application of LLMs for Task Planning
Most works assume access to training data knowledge contained in the LLM as a
source of open-domain world understanding, and apply LLMs directly for task
planning. While this method bypasses explicit domain definition as required in
classical planning for plan generation, the plan is not guaranteed to succeed.
Several works have shown that LLMs can act as planners Huang et al. (2022);
Ichter et al. (2022), but such stochastic, generative approaches lose the
success guarantees of classical planners. API-based programmatic plan
generation Singh et al. (2023); Liang et al. (2023), which introduces some
symbolic structure and constraints, has been shown to perform better but still
does not ensure success. Program synthesis for planning has been previously
proposed in LEAPS Trivedi et al. (2021), which generates programs using a
learned latent space. In recent developments Silver et al. (2023); Liu et al.
(2023), PDDL has been used as a translation medium, with LLMs used to generate
either a PDDL plan or goal. Subsequently, a classical planner is used to plan
for the PDDL goal. Generating a PDDL goal eliminates the need to track
execution state, required when generating a plan using LLMs instead. However,
using a classical planner necessitates specification of the full domain. In
this paper, we propose a new problem of domain induction guided by LLMs. We
use LLMs to guide preliminary environment interactions and planning, which
exposes the domain functioning based on execution feedback.
### 2.3 Domain Induction and Memory Augmentation
Knowledge acquisition for task planning through dialog and web access has been
studied in the past. Some works Perera et al. (2015); Amiri et al. (2019)
construct a knowledge base in an open domain to refer to for grounding user
utterance. Recent works have shown LLM based memory augmentation to be
effective. Sarch et al. (2023) builds a memory of language instruction and
corresponding plan to retrieve from for prompting the LLM with a new
instruction, where the retrieved interaction might inform planning. Oswald et
al. (2024) utilizes LLM to generate PDDL actions given detailed text action
descriptions. Closer to our paper, Smirnov et al. (2024) attempts to generate
syntactically correct but not semantically guaranteed PDDL domain using LLMs,
by exhaustively listing syntax errors in the prompt to correct the generated
domains. Han et al. (2024) leverages language feedback from human during
embodied interaction for domain predicates and action semantics prediction.
In this work, we propose a simple and general approach for retrieving domain
via LLM guided interactions with the environment. Our method, PSALM, aims to
accurately recover the correct domain file without requiring human feedback.
Figure 1: An example from the Blocksworld domain: PDDL domain file (left),
PDDL problem file (mid) and PDDL plan (right).
## 3 Problem Formulation and Background
We define the classical planning problem, the planing language PDDL we use,
and the novel domain induction problem in PDDL we study.
### 3.1 Classical planning problem
The basic framework for classical planning is the goal-directed deterministic
planning problem. A classical planning problem $\mathrm{P}$ is formally
defined as a tuple
$\langle\mathcal{D},\mathcal{O},s^{i},\mathcal{S}^{g}\rangle$, where
$\mathcal{D}$ represents the domain (environment), $\mathcal{O}$ represents a
set of objects in the domain, $s^{i}$ denotes the initial state, and
$\mathcal{S}^{g}$ denotes the goal specification, _i.e_., a set of goal states
satisfying the goal conditions. A plan solution to $\mathrm{P}$ is a $T$-step
sequence of actions $a_{1..T}$ (see Fig. 1 right for an example), which once
executed at $s^{i}$ would lead to a state in $\mathcal{S}^{g}$.
### 3.2 PDDL
Planning Definition and Domain Language (PDDL, Aeronautiques et al., 1998), is
a standard encoding language to represent the planning problems. Fig. 1 lists
an example of the PDDL files from the classical Blocksworld domain. The PDDL
domain file (Fig 1 left) defines $\mathcal{D}$, which contains the predicate
definition representing objects properties in the domain, and the symbolic
action set $\mathcal{A}$ in the environment. Each action has an action name
(_e.g_., Put-down), a list of parameters (_e.g_., ?ob), and action semantics.
The action semantics $\Phi_{a}$ of an action $a\in\mathcal{A}$ includes the
preconditions that describe when an action is valid to execute, and the
postconditions (effects) that describe what happens when an action is
executed. Fig. 2 depicts $\Phi_{a}$ as two lists of statements. The PDDL
problem file (Fig 1 mid) defines the objects $\mathcal{O}$, the initial state
$s^{i}$, and the goal states $\mathcal{S}_{g}$.
### 3.3 Domain induction in PDDL
When performing classical planning in a new environment in PDDL, we have to
define the domain file for the new environment. Human experts must carefully
annotate the action semantics to enable the symbolic solver to find solutions.
We propose a novel domain induction task, where agents must automatically find
the action semantics for a new domain without human annotation or correction.
In the domain induction task, the agent knows
$\mathcal{D}\setminus\bigcup_{a\in\mathcal{A}}{\Phi_{a}}$, i.e., everything
about the domain except the action semantics. In addition, the agent has
access to one problem file $\langle\mathcal{O},s^{i},\mathcal{S}^{g}\rangle$,
The goal of the agent is to learn the correct action semantics $\Phi_{a}$ for
each action $a$. During the learning process, the agent interacts with the
environment, but only starting at $s^{i}$, and can reset to $s^{i}$ as many
times as it wants. Once the action semantics is learned, the ground truth
domain file can be constructed and used for by the symbolic solver for
efficient and robust task solving in the domain.
We evaluate the domain induction learning process on three measures. (1) The
accuracy (Acc):
$\sum_{a\in\mathcal{A}}|\hat{\Phi}_{a}\cap\Phi_{a}|/\sum_{a\in\mathcal{A}}|\Phi_{a}|$,
where $\hat{\Phi}_{a}$ is the predicted action semantics for action $a$. We
use Acc to measure how close the agent is towards finding a solution of the
given problem. Though a solution not ensures 100% Acc, matching the ground
truth action semantics is guaranteed for a solution from the symbolic solver.
We care about the follow two measures once we find a solution. (2) The number
of resets (NR): how many times we put the robot back to $s_{i}$; (3) The
number of executed steps (NES): how many total actions we take during the
learning, including the failed ones.
Figure 2: The pipeline of PSALM in four steps: (1) sample trajectories from a
trajectory sampler; (2) execute the trajectories in the environment to get
feedbacks (3) predict action semantics for each action with environment
feedback, and update the memory based on the prediction; (4) sample action
semantics from the memory to construct the domain file for the symbolic solver
to check the success.
## 4 Proposed Approach: PSALM
We propose the framework Predicting Semantics of Actions with Language Models
(PSALM), to learn action semantics through LLM and solver’s interactions with
the environment.
### 4.1 Overview
As shown in Fig. 2, given the task we want to solve, we first use an LLM as
the trajectory sampler to sample trajectories, and execute the trajectories in
the environment to get feedbacks. We then use the LLM again together with a
rule-based parser, as the action semantics predictor to predict action
semantics, _i.e_., preconditions and postconditions, for each action, based on
this environment feedback. We update the memory of the learned action
semantics and finally sample the current belief of the action semantics to the
symbolic solver to check the success.
If the symbolic solver finds a solution, we execute the plan in the
environment to check if the plan succeeds. If the plan reaches the goal in the
environment, we finish the loop; otherwise, we will pass the result of the
failed plan to the LLM to predict the action semantics again. If the symbolic
solver does not find a solution, we will provide some partial candidate
trajectories from the symbolic solver to the trajectory sampler, to seed the
sampled trajectory.
We elaborate on three components in the following section: the trajectory
sampler, the action semantics predictor, and the memory of learned action
semantics.
### 4.2 Trajectory sampler
We sample trajectories by prompting an LLM to generate new trajectories. The
LLM trajectory sampler takes in the domain information and problem information
as input in the prompt, as described in the problem setup. Besides, the memory
information from the sampled preconditions and postconditions is directly
converted to text in the prompt. For example, the postconditions (on-table
?ob) (arm-empty) of the action put-down are converted to natural language The
effects are (on-table ?ob), (arm-empty). As the symbolic solver is performing
a tree-based search algorithm given a time-limit of $W$, if the solver finds
any candidate trajectories, i.e., not a complete solution but a partial
solution stopped by the timer or a dead end, we will leverage the
deterministic search from the symbolic solver, and include the $k$ longest
candidate trajectories as input to the trajectory sampler. The trajectory
sampler will be asked to pick one of the candidate trajectories, and generate
a trajectory starting from it. Additionally, we store previous failed
trajectories, which are used to (1) filter out invalid candidate trajectories
to avoid the trajectory sampler stuck at the same failed ones (2) list up to
$g$ past failed trajectories in the prompt for higher quality sampling from
the language model. We sample $l$ trajectories from the trajectory sampler.
When $l>1$, we predict action semantics for each trajectory separately, and
update the memory with all predictions.
#### Prospection.
To avoid unnecessary steps taken in the environment execution, and improve
trajectory quality, we perform trajectory prospection. For a generated
trajectory $\mathbf{\tau}=a_{1:T}$, we check if is aligned with the current
belief of the environment action semantics for $v$ steps. Concretely, assuming
$v\leq T$, if any action in $a_{1:v}$ does not satisfy the preconditions in
the sampled action semantics, starting from that action, we will keep randomly
sampling one action until the sampled action is valid in the current action
semantics, and repeat this until we have $v$ total actions. Otherwise, if
$a_{1:v}$ is all valid, we execute the original $a_{1:T}$ in the environment.
#### Random sampler.
To compare the LLM sampler with a random one, we assess the random sampler’s
performance, which samples $v$ actions per trajectory. The random sampler also
takes the solver information as input, i.e., it starts from the longest
candidate trajectory if available. Random sampler can also take advantage of
prospection.
### 4.3 Action semantics predictor
We combine LLM-based and rule-based action semantics prediction in PSALM’s
action semantics predictor.
#### LLM-based predictor.
We prompt the LLM to generate the action semantics of each action separately.
Apart from the domain, problem, memory information, for a trajectory
$\mathbf{\tau}=a_{1:T}$ failed at step $t+1$, the LLM action semantics
predictor takes in valid actions $a_{1:t}$, state transitions
$s_{j+1}-s_{j},j=1..t,s_{1}=s^{i}$, and error message for action $a_{t+1}$ as
input in the prompt. We assume the error message is provided by the
environment. When an action fails, the error message points out which
precondition is not satisfied. When no action fails but the plan is still
invalid, the error message points out the goal is not reached. We show in the
analysis that the error message is necessary for predicting action
preconditions. For each action, we prompt the LLM once to predict the
preconditions and once for the postconditions.
#### Rule-based predictor.
As the error message is given, we can parse the output from the environment
feedbacks for rule-based action semantics prediction. The rule-based predictor
parses the error message from the environment output to infer one or more
missing preconditions suggested. It also parses the state transition
descriptions to derive guaranteed postconditions. For each loop, we add the
rule-based action semantics to the memory.
### 4.4 Memory of action semantics
We keep a memory of the learned action semantics. For each action’s action
semantics, we store two lists of predicted statements for preconditions and
postconditions. Each statement $\phi$ has a probability
$p(\phi\in\Phi_{a}|a)$, in short $p(\phi|a)$. We use this probability to
sample the current belief of action semantics, and then construct the current
belief of the domain file for the symbolic solver.
The first time a statement $\phi$ is predicted as part of the action semantics
for $a$, the probability will be assigned to $1$. Afterwards, this probability
will be updated following an exponential forgetting rule. Suppose at time step
$t$, the predicted action semantics of $a$ is $\hat{\Phi}_{a,t}$, the memory
updating rule of statement $\phi$ is
$\displaystyle
p_{t+1}(\phi|a)=\begin{cases}\mathbbm{1}[\phi\in\hat{\Phi}_{a,t+1}],&p_{t}(\phi|a)=0\\\
\gamma_{\phi}p_{t}(\phi|a)+(1-\gamma_{\phi})\mathbbm{1}[(\phi)\in\hat{\Phi}_{a,t+1}],&\text{otherwise}\\\
\end{cases}$
where $\gamma_{\phi}$ is the forgetting factor. If a statement $\phi$ has only
been predicted by LLM, $\gamma_{\phi}=0.8$. Once a statement $\phi$ is
predicted by rule-based predictor, $\gamma_{\phi}=1$, meaning the statement
keeps at probability $1$ in the memory and never decays. Notice that all the
statements predicted in the current time step and in the memory
$\phi\in\bigcup_{t^{\prime}\in\\{1..t+1\\}}\hat{\Phi}_{a,t^{\prime}}$ will be
updated following the rule.
Figure 3: We compare PSALM with multiple variations over 7 domains. We report
on NES and the results suggest (1) LLM as a trajectory sampler greatly reduces
the execution steps; (2) LLM and rule-based action semantics predictors have
complementary benefits; (3) Prospection is helpful overall. TS is short for
trajectory sampler and ASP is short for action semantics predictor.
## 5 Experiments
We discuss the experimental setups, the results, and the ablation studies,
which show that language models can efficiently learn action semantics from
environment feedback.
### 5.1 Experimental setups
All experiments use Fast-Downward
111https://github.com/aibasel/downward/tree/release-22.12.0 planner with a
search time limit of $W=30$ seconds. We use VAL 222https://github.com/KCL-
Planning/VAL as simulator for plan validation and extracting state condition
changes, error messages. We set the maximum number of loops to 1k for pure
random baselines and to 100 for any method involving LLMs, considering the
cost. We report the average over 3 runs for all the methods.
For the main results, we use GPT-4
API333https://platform.openai.com/docs/models/gpt-4 as the language model
agent, with temperature 0, one-shot prompting following Liu et al. (2023). We
list the prompts in Appendix A. We use $v=10$ prospection steps, $l=1$ sampled
trajectory, $k=3$ candidate trajectories, and $g=5$ failed trajectories per
loop. We perform ablation studies over the hyperparameters in the analysis.
The total cost of the API call is around $500.
We experiment on 7 symbolic domains from International Planning Competitions;
each defines 20 tasks that vary in number of the environment objects and
optimal plan length. (1) Barman: The robot is a bartender with 2 hands
preparing cocktails for a customer’s order, using the specified ingredients
and appropriate tools; (2) Blocksworld: The robot reorganizes a collection of
block piles arranged on a table, into a specified configuration while adhering
to the simple physics principles; (3) Floortile: A set of robots painting
color patterns on floor tiles, allowed to move around but not to step on
painted tiles; (4) Grippers: A set of robots with 2 grippers each are given a
task to move objects among different rooms; (5) Storage: The robot lifts and
drops crates initially stored in different areas, into a depot, using a given
set of hoists; (6) Termes: The robot constructs complex structures by
transporting and positioning blocks, as well as using them as a means to move
adjacent blocks; (7) Tyreworld: The robot is assigned with changing flat
tires, which involves tasks such as removing flat tires, inflating the intact
tires, tightening nuts, and returning tools to the boot.
### 5.2 Main results
Fig. 3 compares PSALM with multiple baselines over 7 domains. Everything in
the figure gets 100% accuracy, so we focus on NES. We do not put the pure
random baselines in the figure, since none of the pure random baselines
complete the task within 1k resets.
#### LLM as a sampler greatly reduces the execution steps.
Comparing the blue bars and the purple bars (w/o LLM-TS), we show that the LLM
trajectory sampler greatly reduces the execution steps over all domains. The
random sampler, even with prospection, lacks the commonsense reasoning
knowledge for choosing trajectories likely be solutions of the problem.
#### LLM and rule-based action semantics predictions have complementary
benefits.
Comparing the orange bars (w/o rule-ASP) and the green bars (w/o LLM-ASP),
rule-based action semantics predictor is more important in Floortile, Storage
and Tyreworld, while LLM action semantics predictor is more important in the
other four domains. With the commonsense prior knowledge provided by the LLM,
and the exactness of the information from the rule-based parser, combining
both predictions is always a better solution over all domains.
#### Prospection induces redundant actions sometimes, but is necessary
overall.
For certain domains with a small number of actions $|\mathcal{A}|$, such as
Grippers ($|\mathcal{A}|=3$) and Termes ($|\mathcal{A}|=7$), a large number of
prospection steps causes unnecessary exploration, comparing to the red bars
(w/o prospect.). However, prospection is overall helpful, as the red bar shows
around 1.4 times execution steps on average compared to our method.
Especially, in the Barman domain, prospection reduces more than a half of the
execution steps. On the other hand, notice that all the exploration during
prospection time is simulated by not executed, a large number of prospection
steps is not time-consuming during the trajectory sampling time.
Figure 4: Additional analysis for PSALM. (Left) We vary the type of LLM and
show that PSALM works with GPT-3.5 and Mistral-7B on the Termes domain.
(Middle) Using the LLM prior before trajectory sampling (darker bars) enables
the random baselines to work better compared to not having the prior (lighter
bars), though it can adversely affect the full PSALM method. (Right)
Experiments where we remove the error message from input to the LLM action
semantics predictor. Without error messages, PSALM works only on easy domains.
For the experiments that fail to find a solution of the problem, we put the
accuracy number on top of their bars.
### 5.3 Analysis
We first show PSALM works on some weaker or even public LLMs. We then discuss
the language model prior can effectively enable the random baselines to work,
as the starting point of the dynamics model. Finally, we show that the error
message is important for the action semantics prediction, without which the
domain induction is not possible with the current LLMs for some domains.
#### PSALM can work on less powerful and public LLMs.
Fig. 4 left shows PSALM can work on GPT-3.5 and Mistral-7B, though with more
number of resets (NR) and number of execution steps (NES). Prospection reduces
the gap between the different LLMs, perhaps because it offsets differences in
commonsense reasoning ability.
#### LLM prior enables random baselines.
We try to use LLM to predict the action semantics $\hat{\Phi}_{0,a}$ for each
action $a$, before having any sampled trajectory, and then start the iteration
with the memory of the dynamics model initialized from $\hat{\Phi}_{0,a}$.
This prediction is solely based on LLM’s commonsense reasoning capability.
Fig. 4 middle shows on the Termes domain the LLM prior can enable the random
baseline without prospection to work within 1k resets, and greatly reduce NR
and NES for the random baseline with prospection. However, on more complex
domains such as Barman, the LLM prior does not enable the random baselines,
_i.e_., failing to find a solution within 1k resets, with and without
prospection. The finding implies we can perform domain induction on some
easier domains with limited access, or even one call, of language models. On
the other hand, LLM prior is harmful to PSALM, as it inserts noisy predictions
of action semantics which requires additional resets to erase.
#### Error messages are crucial for domain induction.
For real-world use, error messages may be hard to obtain from the environment.
So, we study whether PSALM can succeed without receiving error messages about
failed actions. Note that without the error message, the action semantics
predictor can only be LLM-based, not rule-based. Fig. 4 right shows that on
slightly complex domains like Termes, PSALM cannot predict the correct
dynamics model, _i.e_., the accuracy is not 100. PSALM works on very simple
domains like Grippers, but it requires 7x more NES to learn. Moreover,
prospection is essential when we have no error message.
### 5.4 Ablation studies
We perform ablation studies on prospection steps, number of sampled
trajectories, number of candidate trajectories, and number of failed
trajectories per run, to show their influence on the PSALM framework. The
results validate our choice of the hyperparameters in the main results.
#### Few prospection steps is enough for PSALM.
We vary prospection steps $v$ from $0,1,5,10$ in Fig, 5(a). Notice the random
baseline does not find a solution when $v=0,1,5$, and reaches an Acc of 78.5.
We conclude the number of prospection steps matters more when the trajectory
sampler is random, while a small number of prospection is enough for language
model trajectory sampler.
(a)
(b)
(c)
(d)
Figure 5: Ablation studies for PSALM. (a). Varying the number of prospection
steps $v$. Few prospection steps is enough for PSALM. (b). Varying the number
of sampled trajectories $l$. One sampled trajectory per run is enough. (c).
Varying the number of candidate trajectories $k$ passed from the symbolic
solver to the LLM trajectory sampler. More candidate trajectories help when
prospection is used. (d). Varying the number of failed trajectories $g$ shown
to the LLM trajectory sampler. A certain amount of failed trajectories is
required for the LLM.
#### One sampled trajectory per run is enough.
We vary the number of sampled trajectories $l$ from $1,3,5$ in Fig. 5(b). For
both LLM and random samplers, We see the total number of steps grows linearly
with number of sampled trajectory, which means LLMs are hard to learn more
information from just sampling more each run, and one sampled trajectory per
run is enough.
#### More candidate trajectories help on prospection.
We vary the number of candidate trajectories $k$ from $0,1,3$ in the LLM
prompt. The results in Fig. 5(c) suggest including candidate trajectory in the
prompt is beneficial, while one candidate trajectory is enough for pure LLM
prediction, more candidate trajectories help LLM with prospection. On the
other hand, as the candidate trajectories can be very long for certain domain
like Barman, and thus result in a very long prompt, we do not experiment with
more than 3 candidate trajectories in the prompt.
#### Choose the number of failed trajectories with care.
We vary the number of failed trajectories $g$ from $0,1,5$ in the LLM prompt.
To select $g$ failed trajectories, we first filter trajectories with no less
than 3 steps, and then sample $g$ trajectories from the filtered ones for the
LLM prompt to avoid generating those trajectories as prefix. The results in
Fig. 5(d) suggest that multiple failed trajectories help LLM action semantics
prediction, but only one failed trajectory could be harmful. We hypothesize
the one failed trajectory in the input might distract the model from following
the candidate trajectories.
## 6 Conclusion and Discussions
In conclusion, we propose a novel domain induction problem in PDDL, where
agents automatically infer the action semantics for a new domain without human
annotation. We introduce a simple but strong framework PSALM, which combines
the commonsense reasoning ability of large language models with the precision
of symbolic solvers for domain induction. To update the action semantics in a
memory, PSALM uses LLMs as agents for trajectory sampling and action semantics
prediction. We demonstrate the effectiveness and efficiency of leveraging LLMs
for domain induction over 7 domains. This integration of LLMs and symbolic
solvers presents a promising avenue for advancing domain induction
capabilities in robotics.
## 7 Limitations
Though PSALM shows to be relatively robust across multiple LLMs, we are not
claim that PSALM can work with any LLM. Besides, as discussed in the analysis,
the error message, which might be hard to get from some real-world
environment, is crucial in the PSALM framework, given the current LLM’s
reasoning ability. Future works may explore methods without the error message
as the input. On the other hand, to apply the domain induction setup to the
real world, the environment predicates and object types have to be annotated
or derived from the environment. The domain induction problem without these
annotations can be a harder. We will leave this extension for future studies.
## Acknowledgments and Disclosure of Funding
WZ and RJ were supported in part by a grant from Open Philanthropy. IS and JT
were supported in part by a grant from the Army Research Lab (ARL) Army AI
Innovations Institute (A2I2), award number W911NF-23-2-0010.
## References
* Aeronautiques et al. [1998] Constructions Aeronautiques, Adele Howe, Craig Knoblock, ISI Drew McDermott, Ashwin Ram, Manuela Veloso, Daniel Weld, David Wilkins Sri, Anthony Barrett, Dave Christianson, et al. Pddl| the planning domain definition language. _Technical Report, Tech. Rep._ , 1998.
* Amiri et al. [2019] Saeid Amiri, Sujay Bajracharya, Cihangir Goktolgal, Jesse Thomason, and Shiqi Zhang. Augmenting knowledge through statistical, goal-oriented human-robot dialog. In _2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2019.
* Brewka et al. [2011] Gerhard Brewka, Thomas Eiter, and Mirosław Truszczyński. Answer set programming at a glance. _Commun. ACM_ , 2011.
* Carbonell et al. [1991] Jaime Carbonell, Oren Etzioni, Yolanda Gil, Robert Joseph, Craig Knoblock, Steve Minton, and Manuela Veloso. Prodigy: An integrated architecture for planning and learning. _SIGART Bull._ , 1991.
* Fikes and Nilsson [1971] Richard E. Fikes and Nils J. Nilsson. Strips: A new approach to the application of theorem proving to problem solving. _Artificial Intelligence_ , 1971.
* Fox and Long [2003] Maria Fox and Derek Long. Pddl2.1: An extension to pddl for expressing temporal planning domains. _ArXiv_ , 2003.
* Han et al. [2024] Muzhi Han, Yifeng Zhu, Song-Chun Zhu, Ying Nian Wu, and Yuke Zhu. Interpret: Interactive predicate learning from language feedback for generalizable task planning. In _Robotics: Science and Systems (RSS)_ , 2024.
* Huang et al. [2022] Wenlong Huang, Pieter Abbeel, Deepak Pathak, and Igor Mordatch. Language models as zero-shot planners: Extracting actionable knowledge for embodied agents. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvári, Gang Niu, and Sivan Sabato, editors, _International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA_ , Proceedings of Machine Learning Research, 2022.
* Ichter et al. [2022] Brian Ichter, Anthony Brohan, Yevgen Chebotar, Chelsea Finn, Karol Hausman, Alexander Herzog, Daniel Ho, Julian Ibarz, Alex Irpan, Eric Jang, Ryan Julian, Dmitry Kalashnikov, Sergey Levine, Yao Lu, Carolina Parada, Kanishka Rao, Pierre Sermanet, Alexander T Toshev, Vincent Vanhoucke, Fei Xia, Ted Xiao, Peng Xu, Mengyuan Yan, Noah Brown, Michael Ahn, Omar Cortes, Nicolas Sievers, Clayton Tan, Sichun Xu, Diego Reyes, Jarek Rettinghouse, Jornell Quiambao, Peter Pastor, Linda Luu, Kuang-Huei Lee, Yuheng Kuang, Sally Jesmonth, Kyle Jeffrey, Rosario Jauregui Ruano, Jasmine Hsu, Keerthana Gopalakrishnan, Byron David, Andy Zeng, and Chuyuan Kelly Fu. Do as i can, not as i say: Grounding language in robotic affordances. In _6th Annual Conference on Robot Learning_ , 2022.
* Jiang et al. [2019] Yu-qian Jiang, Shi-qi Zhang, Piyush Khandelwal, and Peter Stone. Task planning in robotics: an empirical comparison of pddl- and asp-based systems. _Frontiers of Information Technology & Electronic Engineering_, 2019.
* Liang et al. [2023] Jacky Liang, Wenlong Huang, Fei Xia, Peng Xu, Karol Hausman, Brian Ichter, Pete Florence, and Andy Zeng. Code as policies: Language model programs for embodied control. In _2023 IEEE International Conference on Robotics and Automation (ICRA)_ , 2023.
* Lifschitz [2002] Vladimir Lifschitz. Answer set programming and plan generation. _Artificial Intelligence_ , 2002.
* Liu et al. [2023] Bo Liu, Yuqian Jiang, Xiaohan Zhang, Qiang Liu, Shiqi Zhang, Joydeep Biswas, and Peter Stone. LLM+P: Empowering large language models with optimal planning proficiency. _ArXiv preprint_ , 2023.
* Nau et al. [2003] Dana Nau, Tsz-Chiu Au, Okhtay Ilghami, Ugur Kuter, J William Murdock, Dan Wu, and Fusun Yaman. Shop2: An htn planning system. _J. Artif. Intell. Res. (JAIR)_ , 2003.
* Oswald et al. [2024] James Oswald, Kavitha Srinivas, Harsha Kokel, Junkyu Lee, Michael Katz, and Shirin Sohrabi. Large language models as planning domain generators. In _Proceedings of the International Conference on Automated Planning and Scheduling_ , volume 34, pages 423–431, 2024.
* Perera et al. [2015] Vittorio Perera, Robin Soetens, Thomas Kollar, Mehdi Samadi, Yichao Sun, Daniele Nardi, René van de Molengraft, and Manuela Veloso. Learning task knowledge from dialog and web access. _Robotics_ , 2015.
* Sarch et al. [2023] Gabriel Sarch, Yue Wu, Michael Tarr, and Katerina Fragkiadaki. Open-ended instructable embodied agents with memory-augmented large language models. In _Findings of the Association for Computational Linguistics: EMNLP 2023_ , 2023.
* Silver et al. [2023] Tom Silver, Soham Dan, Kavitha Srinivas, Joshua B. Tenenbaum, Leslie Pack Kaelbling, and Michael Katz. Generalized planning in pddl domains with pretrained large language models. _ArXiv preprint_ , 2023.
* Singh et al. [2023] Ishika Singh, Valts Blukis, Arsalan Mousavian, Ankit Goyal, Danfei Xu, Jonathan Tremblay, Dieter Fox, Jesse Thomason, and Animesh Garg. Progprompt: Generating situated robot task plans using large language models. In _2023 IEEE International Conference on Robotics and Automation (ICRA)_ , 2023.
* Smirnov et al. [2024] Pavel Smirnov, Frank Joublin, Antonello Ceravola, and Michael Gienger. Generating consistent pddl domains with large language models. _ArXiv preprint_ , 2024.
* Trivedi et al. [2021] Dweep Trivedi, Jesse Zhang, Shao-Hua Sun, and Joseph J. Lim. Learning to synthesize programs as interpretable and generalizable policies. In Marc’Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan, editors, _Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual_ , 2021.
## Appendix A LLM Prompts
We list the prompt template for LLM trajectory sampler (Fig. 6) and LLM action
semantics predictor (Fig. 7), and provide one complete example for each. Fig.
8, 9 is the example for trajectory sampler, and Fig. 10, 11 is the example for
action semantics predictor.
Figure 6: LLM trajectory sampler prompt template Figure 7: LLM action
semantics predictor prompt template Figure 8: LLM trajectory sampler prompt
example Figure 9: LLM trajectory sampler prompt example (cont’) Figure 10: LLM
action semantics predictor prompt example Figure 11: LLM action semantics
predictor prompt example (cont’)
|
# OFA: Unifying Architectures, Tasks, and Modalities Through a Simple
Sequence-to-Sequence Learning Framework
Peng Wang, An Yang, Rui Men, Junyang Lin, Shuai Bai
Zhikang Li, Jianxin Ma, Chang Zhou, Jingren Zhou, Hongxia Yang
DAMO Academy, Alibaba Group
{zheluo.wp, ya235025, menrui.mr, junyang.ljy, baishuai.bs,
zhikang.lzk, jason.mjx, ericzhou.zc, jingren.zhou<EMAIL_ADDRESS>
Correspondence to: Chang<EMAIL_ADDRESS>
###### Abstract
In this work, we pursue a unified paradigm for multimodal pretraining to break
the scaffolds of complex task/modality-specific customization. We propose OFA,
a Task-Agnostic and Modality-Agnostic framework that supports Task
Comprehensiveness. OFA unifies a diverse set of cross-modal and unimodal
tasks, including image generation, visual grounding, image captioning, image
classification, language modeling, etc., in a simple sequence-to-sequence
learning framework. OFA follows the instruction-based learning in both
pretraining and finetuning stages, requiring no extra task-specific layers for
downstream tasks. In comparison with the recent state-of-the-art vision &
language models that rely on extremely large cross-modal datasets, OFA is
pretrained on only $20$M publicly available image-text pairs. Despite its
simplicity and relatively small-scale training data, OFA achieves new SOTAs in
a series of cross-modal tasks while attaining highly competitive performances
on uni-modal tasks. Our further analysis indicates that OFA can also
effectively transfer to unseen tasks and unseen domains. Our code and models
are publicly available at https://github.com/OFA-Sys/OFA.
Figure 1: Examples of various tasks supported by OFA.
_K_ eywords Unified frameworks $\cdot$ Multimodal pretraining $\cdot$
Multitask learning $\cdot$ Zero-shot learning
## 1 Introduction
Building an omnipotent model that handles as many tasks and modalities as
human beings is an attractive goal in the AI community. The possibilities of
achieving this goal may largely depend on whether massive varieties of
modalities, tasks and training regimes can be represented with only a few
forms that can be unified and managed by a single model or system.
Recent developments of the Transformer [1] architecture have shown its
potential for being a universal computation engine [2, 3, 4, 5, 6, 7, 8]. In
the settings of supervised learning, the “pretrain-finetune” paradigm achieves
excellent success in many domains. In the regimes of few-/zero-shot learning,
language models with prompt / instruction tuning prove powerful zero-/few-shot
learners [3, 9, 10]. These advances have provided more significant than ever
opportunities for the emergence of an omni-model.
To support better generalization for open-ended problems while maintaining
multitask performance and ease of use, we advocate that an omnipotent model
should have the following three properties: 1\. Task-Agnostic (TA): unified
task representation to support different types of tasks, including
classification, generation, self-supervised pretext tasks, etc., and to be
agnostic to either pretraining or finetuning. 2\. Modality-Agnostic (MA):
unified input and output representation shared among all tasks to handle
different modalities. 3\. Task Comprehensiveness (TC): enough task variety to
accumulate generalization ability robustly.
However, it is challenging to satisfy these properties while maintaining
superior performance in downstream tasks. Current language and multimodal
pretrained models readily fail at parts of these properties, due to their
following design choices. 1\. Extra learnable components for finetuning, e.g.,
task-specific heads [2], adapters [11], soft prompts [12]. This makes the
model structure task-specific and poses discrepancy between pretraining and
finetuning. Such designs are also not friendly to supporting unseen tasks in a
zero-shot manner. 2\. Task-specific formulation. For most current methods,
pretraining, finetuning and zero-shot tasks usually differ in task formulation
and training objectives. This violates TA and it is burdensome to scale up the
task population to achieve TC. 3\. Entangling modality representation with
downstream tasks. It is a common practice for Vision-Language models to take
the detected objects as part of the image input features [8, 13, 14, 15, 16,
17]. Though it demonstrates better downstream task performance on some closed-
domain datasets, it depends on an extra object detector which usually fails at
open-domain data.
Therefore, we explore an omni-model for multimodal pretraining and propose
OFA, hopefully “One For All”, which achieves the objectives of unifying
architectures, tasks, and modalities, and supports the three properties
above.111This work is the latest one of our M6 series [18, 19, 20, 21]. We
formulate both pretraining and finetuning tasks in a unified sequence-to-
sequence abstraction via handcrafted instructions [9, 10] to achieve Task-
Agnostic. A Transformer is adopted as the Modality-Agnostic compute engine,
with a constraint that no learnable task- or modality-specific components will
be added to downstream tasks. It is available to represent information from
different modalities within a globally shared multimodal vocabulary across all
tasks. We then support Task Comprehensiveness by pretraining on varieties of
uni-modal and cross-modal tasks.
To summarize:
* •
We propose OFA, a Task-Agnostic and Modality-Agnostic framework that supports
Task Comprehensiveness. OFA is the first attempt to unify the following vision
& language, vision-only and language-only tasks, including understanding and
generation, e.g., text-to-image generation, visual grounding, visual question
answering (VQA), image captioning, image classification, language modeling,
etc., via a simple sequence-to-sequence learning framework with a unified
instruction-based task representation.
* •
OFA is pretrained on the publicly available datasets of $20$M image-text
pairs, in comparison with recent models that rely on paired data of a much
larger scale [22, 23]. OFA achieves state-of-the-art performances in a series
of vision & language downstream tasks, including image captioning, visual
question answering, visual entailment, referring expression comprehension,
etc.
* •
OFA, as a multimodal pretrained model, achieves comparable performances on
unimodal tasks with SOTA pretrained models in language or vision, e.g.,
RoBERTa, ELECTRA and DeBERTa for natural language understanding, UniLM,
Pegasus and ProphetNet for natural language generation, and MoCo-v3, BEiT and
MAE for image classification.
* •
We verify that OFA achieves competitive performance in zero-shot learning.
Also, it can transfer to unseen tasks with new task instructions and adapt to
out-of-domain information without finetuning.
## 2 Related Work
Figure 2: A demonstration of the pretraining tasks, including visual
grounding, grounded captioning, image-text matching, image captioning, VQA,
object detection, image infilling as well as text infilling.
#### Language Pretraining & Vision Pretraining
Natural language pretraining has revolutionized the whole NLP research
community. A representation of this track is the birth of BERT [2] and GPT
[24]. A number of studies have been progressively advancing pretraining by
improving pretraining tasks and designing more sophisticated model
architectures [25, 26, 27, 28, 29, 30, 31]. Having witnessed the success of
natural language pretraining, researchers have promoted self-supervised
learning (SSL) in computer vision [32, 33, 34, 35]. Recently, mirroring masked
language modeling (MLM) in language pretraining, generative pretraining [36,
37] with ViT architecture [6] further boosts downstream performance.
#### Multimodal Pretraining
Multimodal pretraining has been developing rapidly [38, 13, 39, 40, 14, 41,
42, 43, 44, 15, 16, 17, 45, 46, 47]. Researchers have applied the masking
strategies and the encoder-decoder architecture to adapt models to generation
tasks [15, 17, 18, 22]. Besides, to simplify preprocessing, patch projection
has received attention and helped Transformer achieve SOTA performance in
downstream tasks [22, 48]. To make full use of large-scale weakly supervised
data, [49] trains a bi-encoder on $400$ million pairs and demonstrates
excellent performance in retrieval tasks. Another line of work is text-to-
image synthesis. A bunch of works [50, 51, 18, 52] incorporate Transformer
with VQVAE [53] or VQGAN [54] to generate high-quality images with high
resolution. However, the previously mentioned methods are limited in
processing a single type of data, such as cross-modal data only or limited in
their capabilities. Also, the discrepancy between pretraining and finetuning
behaviors limits the transferability to open-ended data.
#### Unified Frameworks
To pursue the unified models, [55] demonstrate a uniform format to represent
tasks. In NLP, recent studies unify diverse tasks covering natural language
understanding and generation to text-to-text transfer [30] or language
modeling [3]. Following this idea, [56] and [57] demonstrate text-generation-
based multimodal pretrained models. [7] and [58] propose a simple framework
that can process information from multiple modalities with a uniform byte-
sequence representation. [59] and [60] unify tasks of different modalities by
designing various task-specific layers. [61] explores to employ a retrieval-
based unified paradigm. However, these multimodal pretrained models suffer
from performance degradation in downstream tasks, e.g., VQA, image captioning,
etc., and they have no image generation capability.
## 3 OFA
In this work, we propose OFA, a unified Seq2Seq framework for the unification
of I/O & architectures, tasks, and modalities. The overall framework is
illustrated in Figure 2.
### 3.1 I/O & Architecture
#### I/O
The most common practice of multimodal pretraining is the pretraining of
Transformer models on image-text pair corpus at scale. This requires data
preprocessing or modality-specific adaptors to enable the joint training of
both visual and linguistic information with the Transformer architecture.
Compared with the complex, resource&time-consuming object feature extraction,
we aim for simplicity and directly use ResNet modules to convolve
$\textrm{x}_{v}\in\mathbb{R}^{H\times W\times C}$ to $P$ patch features of the
hidden size, following [62] and [22]. As to processing the linguistic
information, we follow the practice of GPT [24] and BART [31] that we apply
byte-pair encoding (BPE) [63] to the given text sequence to transform it into
a subword sequence and then embed them to features.
To process different modalities without task-specific output schema, it is
essential to represent data of various modalities in a unified space. A
possible solution is to discretize text, image, and object and represent them
with tokens in a unified vocabulary. Recent advances in image quantization
[53, 54] have demonstrated effectiveness in text-to-image synthesis [50, 18,
51, 19], and thus we utilize this strategy for the target-side image
representations. Sparse coding is effective in reducing the sequence length of
image representation. For example, an image of the resolution of $256\times
256$ is represented as a code sequence of the length of $16\times 16$. Each
discrete code strongly correlates with the corresponding patch [36].
Apart from representing images, it is also essential to represent objects
within images as there are a series of region-related tasks. Following [64],
we represent objects as a sequence of discrete tokens. To be more specific,
for each object, we extract its label and its bounding box. The continuous
corner coordinates (the top left and the bottom right) of the bounding box are
uniformly discretized to integers as location tokens $\langle
x_{1},y_{1},x_{2},y_{2}\rangle$. As to the object labels, they are
intrisically words and thus can be represented with BPE tokens.
Finally, we use a unified vocabulary for all the linguistic and visual tokens,
including subwords, image codes, and location tokens.
#### Architecture
Following the previous successful practices in multimodal pretraining [14, 17,
22], we choose Transformer as the backbone architecture, and we adopt the
encoder-decoder framework as the unified architecture for all the pretraining,
finetuning, and zero-shot tasks. Specifically, both the encoder and the
decoder are stacks of Transformer layers. A Transformer encoder layer consists
of a self attention and a feed-forward network (FFN), while a Transformer
decoder layer consists of a self attention, an FFN and a cross attention for
building the connection between the decoder and the encoder output
representations. To stabilize training and accelerate convergence, we add head
scaling to self attention, a post-attention layer normalization (LN) [65], and
an LN following the first layer of FFN [66]. For positional information, we
use two absolute position embeddings for text and images, respectively.
Instead of simply adding the position embeddings, we decoupling the position
correlation from token embeddings and patch embeddings [67]. In addition, we
also use 1D relative position bias for text [30] and 2D relative position bias
for image [22, 62].
### 3.2 Tasks & Modalities
A unified framework is designed to provide architecture compatibility across
different modalities and downstream tasks so that opportunities can arise to
generalize to unseen tasks within the same model. Then we have to represent
the possible downstream tasks concerning different modalities in a unified
paradigm. Therefore, an essential point for the design of pretraining tasks is
the consideration of multitask and multimodality.
To unify tasks and modalities, we design a unified sequence-to-sequence
learning paradigm for pretraining, finetuning, and inference on all tasks
concerning different modalities. Both pretraining tasks and downstream tasks
of cross-modal and uni-modal understanding and generation are all formed as
Seq2Seq generation. It is available to perform multitask pretraining on
multimodal and uni-modal data to endow the model with comprehensive
capabilities. Specifically, we share the identical schema across all tasks,
while we specify handcrafted instructions for discrimination [9].
For cross-modal representation learning, we design $5$ tasks, including visual
grounding (VG), grounded captioning (GC), image-text matching (ITM), image
captioning (IC), and visual question answering (VQA). For VG, the model learns
to generate location tokens specifying the region position $\langle
x_{1},y_{1},x_{2},y_{2}\rangle$ based on the input of the image $x^{i}$ and
the instruction “Which region does the text $x^{t}$ describe?” where $x^{t}$
refers to the region caption. GC is an inverse task of VG. The model learns to
generate a description based on the input image $x^{i}$ and the instruction
“What does the region describe? region: $\langle
x_{1},y_{1},x_{2},y_{2}\rangle$”. For ITM, we use each original image-text
pair as the positive sample and construct a new one as the negative by pairing
the image with a randomly substituted caption. The model learns to
discriminate whether the given image and text are paired by learning to
generate “Yes” or “No” based on the input image $x^{i}$ and the instruction
“Does the image describe $x^{t}$?”. As to image captioning, this task can
naturally adapt to the sequence-to-sequence format. The model learns to
generate the caption based on the given image and the instruction “What does
the image describe?”. For VQA, we send the image and the question as the input
and require the model to learn to generate correct answers.
For uni-modal representation learning, we design $2$ tasks for vision and $1$
task for language, respectively. The model is pretrained with image infilling
and object detection for vision representation learning. Recent advances in
generative self-supervised learning for computer vision show that masked image
modeling is an effective pretraining task [36, 37]. In practice, we mask the
middle part of the images as the input. The model learns to generate the
sparse codes for the central part of the image based on the corrupted input
and the specified instruction “What is the image in the middle part?”. We
additionally add object detection to pretraining following [44]. The model
learns to generate human-annotated object representations, i.e., the sequence
of object position and label, based on the input image and the text “What are
the objects in the image?” as the instruction. Both tasks strengthen the
representation learning on both pixel and object levels. For language
representation learning, following the practice of [31], we pretrain the
unified model on plain text data with text infilling.
In this way, we unify multiple modalities and multiple tasks to a single model
and pretraining paradigm. OFA is pretrained jointly with those tasks and data.
Thus, it can perform different tasks concerning natural language, vision, and
cross-modality.
### 3.3 Pretraining Datasets
We construct pretraining datasets by incorporating Vision & Language data
(i.e., image-text pairs), Vision data (i.e., raw image data, object-labeled
data), and Language data (i.e., plain texts). For replication, we only use
datasets that are publicly available. We carefully filter our pretraining data
and exclude images that appear in the validation and test sets of downstream
tasks to avoid data leakage. We provide more details about pretraining
datasets in Appendix A.1.
### 3.4 Training & Inference
We optimize the model with the cross-entropy loss. Given an input $x$, an
instruction $s$ and an output $y$, we train OFA by minimizing
$\mathcal{L}=-\sum_{i=1}^{|y|}{\rm log}P_{\theta}(y_{i}|y_{<i},x,s)$, where
$\theta$ refers to the model parameters. For inference, we apply the decoding
strategies, e.g., beam search, to enhance the quality of generation. However,
this paradigm has several problems in classification tasks: 1\. optimizing on
the entire vocabulary is unnecessary and inefficient; 2\. the model may
generate invalid labels out of the closed label set during inference. To
overcome these issues, we introduce a search strategy based on prefix tree
(Trie, [68]). Experimental results show that the Trie-based search can enhance
the performance of OFA on classification tasks. See Appendix B for more
details.
### 3.5 Scaling Models
Table 1: Detailed hyperparameters of OFA model configuration. We list the
configuration for OFA of $5$ different sizes.
Model #Param. Backbone Hidden size Intermediate Size #Head #Enc. Layers #Dec.
Layers $\text{OFA}\rm_{Tiny}$ 33M ResNet50 256 1024 4 4 4
$\text{OFA}\rm_{Medium}$ 93M ResNet101 512 2048 8 4 4 $\text{OFA}\rm_{Base}$
182M ResNet101 768 3072 12 6 6 $\text{OFA}\rm_{Large}$ 472M ResNet152 1024
4096 16 12 12 $\text{OFA}\rm_{Huge}$ 930M ResNet152 1280 5120 16 24 12
In order to investigate how OFA of different model sizes perform in downstream
tasks, we have developed $5$ versions of OFA models, scaling from $33$M to
$940$M parameters, and we list their detailed hyperparameters in Table 1.
To be more specific, we have built basic models of $\rm Base$ and $\rm Large$
sizes, $\text{OFA}\rm_{Base}$ and $\text{OFA}\rm_{Large}$. As our network
configuration is similar to BART [31], their sizes are similar to those of
$\text{BART}\rm_{Base}$ and $\text{BART}\rm_{Large}$. Additionally, we have
developed OFA of a larger size, which we name it $\text{OFA}\rm_{Huge}$, or
OFA without specific mentioning in the tables. Its size is comparable to that
of $\text{SimVLM}\rm_{Huge}$ or $\text{ViT}\rm_{Huge}$. To investigate whether
smaller OFA can still reach satisfactory performance, we have developed
$\text{OFA}\rm_{Medium}$ and $\text{OFA}\rm_{Tiny}$, which are solely around
half and less than $20\%$ as large as $\text{OFA}\rm_{Base}$.
## 4 Experiments
This section provides experimental details and analyses to demonstrate our
model’s effectiveness. See Appendix A for implementation details.
### 4.1 Results on Cross-modal Tasks
Table 2: Experimental results on cross-modal understanding tasks including VQA
and visual entailment. Note that we report the best results from the previous
SOTAs, and specifically SimVLM is a huge-size model comparable to ViT-Huge
pretrained on 1.8B image-text pairs, and Florence is built with CoSwin-H and
RoBERTa and it is pretrained on 900M image-text pairs.
Model VQA SNLI-VE test-dev test-std dev test UNITER [14] 73.8 74.0 79.4 79.4
OSCAR [15] 73.6 73.8 - - VILLA [16] 74.7 74.9 80.2 80.0 VL-T5 [56] - 70.3 - -
VinVL [17] 76.5 76.6 - - UNIMO [46] 75.0 75.3 81.1 80.6 ALBEF [69] 75.8 76.0
80.8 80.9 METER [70] 77.7 77.6 80.9 81.2 VLMo [48] 79.9 80.0 - - SimVLM [22]
80.0 80.3 86.2 86.3 Florence [23] 80.2 80.4 - - $\text{OFA}\rm_{Tiny}$ 70.3
70.4 85.3 85.2 $\text{OFA}\rm_{Medium}$ 75.4 75.5 86.6 87.0
$\text{OFA}\rm_{Base}$ 78.0 78.1 89.3 89.2 $\text{OFA}\rm_{Large}$ 80.3 80.5
90.3 90.2 OFA 82.0 82.0 91.0 91.2
Table 3: Experimental results on MSCOCO Image Captioning. We report the
results on the Karpathy test split. Note that SimVLM and LEMON are huge-size
models.
Model Cross-Entropy Optimization CIDEr Optimization BLEU@4 METEOR CIDEr SPICE
BLEU@4 METEOR CIDEr SPICE VL-T5 [56] 34.5 28.7 116.5 21.9 - - - - OSCAR [15]
37.4 30.7 127.8 23.5 41.7 30.6 140.0 24.5 UNICORN [57] 35.8 28.4 119.1 21.5 -
- - - VinVL [17] 38.5 30.4 130.8 23.4 41.0 31.1 140.9 25.2 UNIMO [46] 39.6 -
127.7 - - - - - LEMON [71] 41.5 30.8 139.1 24.1 42.6 31.4 145.5 25.5 SimVLM
[22] 40.6 33.7 143.3 25.4 - - - - $\text{OFA}\rm_{Tiny}$ 35.9 28.1 119.0 21.6
38.1 29.2 128.7 23.1 $\text{OFA}\rm_{Medium}$ 39.1 30.0 130.4 23.2 41.4 30.8
140.7 24.8 $\text{OFA}\rm_{Base}$ 41.0 30.9 138.2 24.2 42.8 31.7 146.7 25.8
$\text{OFA}\rm_{Large}$ 42.4 31.5 142.2 24.5 43.6 32.2 150.7 26.2 OFA 43.9
31.8 145.3 24.8 44.9 32.5 154.9 26.6
Table 4: Experimental results on the $3$ datasets of referring expression
comprehension, namely RefCOCO, RefCOCO+, and RefCOCOg. We report the<EMAIL_ADDRESS>on different test splits of the datasets.
Model RefCOCO RefCOCO+ RefCOCOg val testA testB val testA testB val-u test-u
VL-T5 [56] - - - - - - - 71.3 UNITER [14] 81.41 87.04 74.17 75.90 81.45 66.70
74.86 75.77 VILLA [16] 82.39 87.48 74.84 76.17 81.54 66.84 76.18 76.71 MDETR
[72] 86.75 89.58 81.41 79.52 84.09 70.62 81.64 80.89 UNICORN [57] 88.29 90.42
83.06 80.30 85.05 71.88 83.44 83.93 $\text{OFA}\rm_{Tiny}$ 80.20 84.07 75.00
68.22 75.13 57.66 72.02 69.74 $\text{OFA}\rm_{Medium}$ 85.34 87.68 77.92 76.09
83.04 66.25 78.76 78.58 $\text{OFA}\rm_{Base}$ 88.48 90.67 83.30 81.39 87.15
74.29 82.29 82.31 $\text{OFA}\rm_{Large}$ 90.05 92.93 85.26 85.80 89.87 79.22
85.89 86.55 OFA 92.04 94.03 88.44 87.86 91.70 80.71 88.07 88.78
We evaluate our models on different cross-modal downstream tasks, covering
cross-modal understanding and generation. Specifically, we implement
experiments on multimodal understanding datasets including VQAv2 for visual
question answering and SNLI-VE [73] for visual entailment, and multimodal
generation including MSCOCO Image Caption [74] for image captioning, RefCOCO /
RefCOCO+ / RefCOCOg [75, 76] for referring expression comprehension as this
task can be viewed as bounding box generation, and MSCOCO Image Caption for
text-to-image generation. More details are provided in Appendix A.3.
Table 2 presents the performance of OFA and baseline models on VQA and SNLI-
VE. In general, OFA achieves the best performance in both tasks with $82.0$ on
the VQA test-std set and $91.2$ on the SNLI-VE test set. For smaller-size
models, $\text{OFA}\rm_{Large}$ can outperform the recent SOTAs, e.g., VLMo
and SimVLM, and $\text{OFA}\rm_{Base}$ can beat the SOTAs before the
aforementioned two models in both tasks. This demonstrates that OFA can
achieve superior performance on cross-modal understanding tasks and scaling up
OFA can bring significant improvements, reflecting the strong potential of
large-scale pretrained models.
Table 3 presents the performance of OFA and baseline models on the MSCOCO
image captioning dataset. We report the results on the Karpathy test split,
and we demonstrate the performance of models trained with Cross-Entropy
optimization and additionally with CIDEr optimization based on reinforcement
learning. In comparison with the previous SOTA $\text{SimVLM}\rm_{Huge}$ for
Cross-Entropy optimization, OFA outperforms it by around $2$ points in CIDEr
evaluation. For CIDEr optimization, OFA of the $3$ sizes all outperform the
huge-size LEMON, and OFA demonstrates a new SOTA of $154.9$ CIDEr score. By
May 31 2022, the single-model OFA had topped the MSCOCO Image Caption
Leaderboard.222https://competitions.codalab.org/competitions/3221#results
To evaluate the capability of visual grounding, we conduct experiments on
RefCOCO, RefCOCO+, and RefCOCOg. While we unify locations to the vocabulary,
visual grounding can be viewed as a sequence generation task. As there is only
one target for each query, we limit the generation length to $4$ in order to
generate a bounding box by $<x_{1},y_{1},x_{2},y_{2}>$. Experimental results
in Table 4 show that OFA reaches the SOTA performance on the $3$ datasets.
Compared with the previous SOTA UNICORN [57], OFA achieves significant
improvement with a gain of $3.61$, $6.65$ and $4.85$ points on the testA sets
of RefCOCO and RefCOCO+ as well as the test-u set of RefCOCOg.
Figure 3: Qualitative comparison with state-of-the-art models for text-to-
image generation task. We present more qualitative examples of text-to-image
generation for better demonstration in Appendix C.
Text-to-image generation is a challenging task even for pretrained model. As
we pretrain OFA with the task “image-infilling”, i.e., recovering masked
patches by generating the corresponding codes [36], and thus OFA is able to
generate code. We thus directly finetune OFA on the MSCOCO Image Caption
dataset for text-to-code generation. At the inference stage, we additionally
transform the generated codes to an image with the code decoder. Specifically,
we use the codes from VQGAN [54] following [52]. Experimental results show
that OFA outperforms the baselines in all the metrics. Note that increasing
the sampling size during inference is expected to bring clear improvements on
FID and IS. Compared with DALLE [50], CogView [51] and NÜWA [52], whose
sampling sizes are $512$, $60$ and $60$, respectively, OFA outperforms these
SOTA methods on FID and IS with a much smaller sampling size $24$. This
illustrates that OFA has learned better correspondence among the query text,
the image and the image codes.
Table 5: Experimental results on text-to-image generation. Models are
evaluated on FID, CLIPSIM, and IS scores. OFA outperforms the baselines,
including the concurrent SOTA NÜWA. We report the results of $\text{OFA}_{\rm
Large}$. Note that GLIDE additionally has $1.5B$ parameters for upsampling
except for the $3.5B$ parameters.
Model FID$\downarrow$ CLIPSIM$\uparrow$ IS$\uparrow$ DALLE [50] 27.5 - 17.9
CogView [51] 27.1 33.3 18.2 GLIDE [77] 12.2 - - Unifying [78] 29.9 30.9 - NÜWA
[52] 12.9 34.3 27.2 OFA 10.5 34.4 31.1
We compare OFA with CogView and GLIDE on generation quality with normal and
counterfactual queries.333For more implementation details, please refer to
Appendix A.3 Normal queries describe existing things in the real world, while
counterfactual queries refer to those describing things that could only exist
in our imagination. For normal queries, both CogView and OFA generate images
semantically consistent with the given texts, in comparison with GLIDE. The
generated examples from our model can provide more sophisticated details of
objects, say the horse and the double-decker bus. For counterfactual queries,
we find that OFA is the only one that can generate the three imaginary scenes,
which indicates its imaginative power based on its strong capability to align
text to the image. See Appendix C for more qualitative examples.
### 4.2 Results on Uni-modal Tasks
As the design of OFA unifies different modalities, we evaluate its performance
on unimodal tasks, namely tasks of natural language and computer vision. For
natural language tasks, we evaluate OFA on $6$ tasks of the GLUE benchmark
[79] for natural language understanding and Gigaword abstractive summarization
[80] for natural language generation. For computer vision, we evaluate OFA on
the classic ImageNet-1K [81] dataset for image classification. More details
are provided in Appendix A.3.
Table 6: Experimental results on the GLUE benchmark datasets [79]. For
comparison, we list the performance of multimodal pretrained models as well
the recent SOTA models that were pretrained on natural language data only.
Following [28], we finetune RTE and MRPC starting from the checkpoint
finetuned on MNLI.
Model SST-2 RTE MRPC QQP MNLI QNLI Multimodal Pretrained Baseline Models
VisualBERT [38] 89.4 56.6 71.9 89.4 81.6 87.0 UNITER [14] 89.7 55.6 69.3 89.2
80.9 86.0 VL-BERT [8] 89.8 55.7 70.6 89.0 81.2 86.3 VilBERT [13] 90.4 53.7
69.0 88.6 79.9 83.8 LXMERT [40] 90.2 57.2 69.8 75.3 80.4 84.2 Uni-Perceiver
[61] 90.2 64.3 86.6 87.1 81.7 89.9 SimVLM [22] 90.9 63.9 75.2 90.4 83.4 88.6
FLAVA [60] 90.9 57.8 81.4 90.4 80.3 87.3 UNIMO [46] 96.8 - - - 89.8 - Natural-
Language-Pretrained SOTA Models BERT [2] 93.2 70.4 88.0 91.3 86.6 92.3 RoBERTa
[28] 96.4 86.6 90.9 92.2 90.2 93.9 XLNet [25] 97.0 85.9 90.8 92.3 90.8 94.9
ELECTRA [82] 96.9 88.0 90.8 92.4 90.9 95.0 DeBERTa [83] 96.8 88.3 91.9 92.3
91.1 95.3 Ours OFA 96.6 91.0 91.7 92.5 90.2 94.8
Table 7: Experimental results on Gigaword abstractive summarization. We report
performance on the ROUGE evaluation [84]
.
Model Gigaword ROUGE-1 ROUGE-2 ROUGE-L BERTSHARE [85] 38.13 19.81 35.62 MASS
[86] 38.73 19.71 35.96 UniLM [29] 38.45 19.45 35.75 PEGASUS [87] 39.12 19.86
36.24 ProphetNet [88] 39.55 20.27 36.57 UNIMO [46] 39.71 20.37 36.88 OFA 39.81
20.66 37.11
As OFA has been pretrained on plain text data, it can be directly transferred
to natural language downstream tasks. For natural language generation, it is
essentially a sequence-to-sequence generation task, and for natural language
understanding, typically text classification, we regard them as generation
tasks where labels are essentially word sequences. Additionally, for each
task, we design a manual instruction to indicate the model what types of
questions it should answer. We list our instruction design in Appendix A.3.
We demonstrate that even a unified multimodal pretrained model can achieve
highly competitive performance in natural language tasks. Specifically, in the
evaluation of natural language understanding, OFA surpasses multimodal
pretrained models by large margins in all tasks. In comparison with the state-
of-the-art natural language pretrained models, including RoBERTa [28], XLNET
[25], ELECTRA [82], and DeBERTa [83], OFA reaches a comparable performance. In
the evaluation of natural language generation, OFA even reaches a new state-
of-the-art performance on the Gigaword dataset.
Also, OFA can reach a competitive performance in image classification. Table 8
shows the performance of OFA on image classification. $\text{OFA}\rm_{Large}$
achieves higher accuracy than previous backbone models such as EfficientNet-B7
[89] and ViT-L [6]. We also compare OFA with self-supervised pretraining
models based on contrastive learning and masked image modeling. OFA
outperforms contrastive-based models such as SimCLR [32] and MoCo-v3 [33, 35]
with similar parameters. Compared with pretrained models based on masked image
modeling, e.g., BEiT-L [36] and MAE-L [37], OFA can achieve similar
performance.
These aforementioned results in both natural language and vision tasks
indicate that a unified multimodal pretrained model is not only effective in
multimodal tasks but also capable of tackling unimodal tasks, and in the
future, it might be sufficient for such a model to solve complex tasks
concerning different modality combinations.
Table 8: ImageNet-1K finetuning results. All the listed models do not use
extra labeled image classification samples during training for fair
comparison. We report the results of $\text{OFA}\rm_{Large}$.
Model Top-1 Acc. EfficientNet-B7 [89] 84.3 ViT-L/16 [6] 82.5 DINO [90] 82.8
SimCLR v2 [32] 82.9 MoCo v3 [35] 84.1 BEiT384-L/16 [36] 86.3 MAE-L/16 [37]
85.9 OFA 85.6
### 4.3 Zero-shot Learning & Task Transfer
The instruction-guided pretraining enables OFA to perform zero-shot inference.
Following Uni-Perceiver [61], we evaluate our model on the $6$ tasks of the
GLUE benchmark, including single-sentence classification and sentence pair
classification. Table 9 demonstrates that OFA generally outperforms Uni-
Perceiver. However, both models do not achieve satisfactory performance in
sentence-pair classification (with $\text{Acc}.<60\%$). We hypothesize that
the missing sentence-pair data in the pretraining dataset attributes to the
performance.
Also, we find that the model performance is highly sensitive to the design of
instructions. To obtain the best result, one should search a proper
instruction template possibly from a large pool of candidates. A slight change
to manual prompts or model parameters may drastically influence the model
performance, which is not robust. We leave this issue to the future work.
We observe that the model can transfer to unseen tasks well with new task
instructions. We design a new task called grounded question answering and
present examples in Figure 4. In this scenario, given a question about a
certain region on the image, the model should provide a correct answer. We
find that the model can achieve a satisfactory performance in this new task,
which reflects its strong transferability. Besides, OFA can solve tasks with
the out-of-domain input data. For example, OFA without finetuning achieves
satisfactory performance in VQA for the out-of-domain images. Examples are
demonstrated in Figure 5. OFA can also perform accurate visual grounding on
the out-of-domain images, e.g., anime pictures, synthetic images, etc., and we
demonstrate more examples on Figure 11 in Appendix C.
Table 9: Zero-shot performance on $6$ GLUE tasks and SNLI-VE.
Model SST-2 RTE MRPC QQP QNLI MNLI SNLI-VE Acc. Acc. F1 F1 Acc. Acc. Acc.
(dev/test) Uni-Perceiver 70.6 55.6 76.1 53.6 51.0 49.6 - $\text{OFA}_{\rm
Base}$ 71.6 56.7 79.5 54.0 51.4 37.3 49.71 / 49.18
Figure 4: Qualitative results on an unseen task grounded QA. We design a new
task called grounded question answering, where the model should answer a
question about a certain region in the image. More samples are provided in
Figure 10 in Appendix C. Figure 5: Qualitative results on unseen domain VQA.
During pretraining, only real-world photographs are used for VQA. We present
cases of VQA on out-of-domain images, i.e., the iconic and sci-fi images, and
demonstrate their capability of transferring to unseen domains. More samples
are provided in Figure 9 in Appendix C.
### 4.4 Ablation on Multitask Pretraining
Thanks to the unified framework, OFA has been pretrained on multiple tasks and
thus endowed with comprehensive capabilities. However, the effects of each
task are still undiscovered. We verify their effects on multiple downstream
tasks, including image captioning, VQA, image classification, and text-to-
image generation.
We first evaluate how uni-modal pretraining tasks influence the performance in
both cross-modal and uni-modal tasks. Table 10 demonstrates our experimental
results. We observe some interesting phenomena about the effects of uni-modal
pretraining tasks. Text infilling brings improvement on image caption ($+0.8$
CIDEr) and VQA ($+0.46$ Acc.). Natural language pretraining betters the
contextualized representation of language and thus enhances performance in
cross-modal tasks. However, it is noticed that the language pretraining task
may degrade the performance in image classification, leading to the decrease
in ImageNet-1K ($-1.0$ Acc.). Also, it is interesting to find that it does not
encourage improvement in text-to-image generation ($-0.1$ CLIPSIM). It may
attribute to the simplicity of text in this task, which indicates that
improved representation of language does not affect the performance. As to
image infilling, it significantly improves the performance in image
classification ($+1.0$ Acc.) and text-to-image generation ($+0.6$ CLIPSIM).
Learning to recover images is an effective self-supervised task for image
representation, and it also encourages the decoder’s ability to generate image
codes. However, it hurts the performance in image captioning and VQA. Both
tasks require a strong capability in generating texts, and the decoder’s
learning of image generation naturally brings performance degradation in
captioning ($-0.7$ CIDEr) and VQA ($-0.3$ Acc.).
Table 10: Ablation results of OFA. All models are pretrained for $250k$ steps.
w/o ground. represents the removal of both visual grounding and grounded
captioning tasks. Note that all models are only finetuned with the cross-
entropy loss in image captioning.
Model Caption VQA ImageNet Image Generation CIDEr Test-dev Top-1 Acc. FID /
CLIPSIM / IS $\text{OFA}\rm_{Base}$ 135.6 76.0 82.2 20.8 / 31.6 / 21.5 w/o
text infill. 134.8 75.6 83.2 20.3 / 31.7 / 21.8 w/o image infill. 136.3 76.3
81.8 23.2 / 31.0 / 20.0 w/o det. 133.3 75.4 81.4 20.9 / 31.5 / 21.6 w/o
ground. 134.2 75.5 82.0 21.2 / 31.5 / 21.5
Furthermore, we evaluate how multimodal tasks impact the performance. Previous
studies have provided evidence of the contribution of conventional pretraining
tasks, e.g., MLM, MOC, ITM, VQA, image captioning, etc. [14, 17]. However,
they miss other tasks, e.g., detection and visual grounding & grounded
captioning. We conduct experiments on these tasks and find that tasks
predicting regions are crucial to multimodal tasks, with a performance
increase in image captioning ($+2.3$ CIDEr & $+1.4$ CIDEr) and VQA ($+0.6$
Acc. & $+0.5$ Acc.). It suggests that detection and visual grounding &
grounded captioning help the model grasp fined-grained alignments between
vision and language. Region information contributes little to text-to-image
generation ($+0.1$ CLIPSIM & $+0.1$ CLIPSIM), as this task requires far less
text-region alignment information. We surprisingly find that detection can
encourage the performance in visual understanding ($+0.8$ Acc.). It indicates
that incorporating region information might be essential to visual
understanding, especially on images with complex objects.
## 5 Conclusion
In this work, we propose OFA, a Task-Agnostic and Modality-Agnostic framework
supporting Task Comprehensiveness. OFA achieves the unification in
architecture, tasks and modalities, and thus is capable of multimodal & uni-
modal understanding and generation, without specification in additional layers
or tasks. Our experiments show that OFA creates new SOTAs in a series of
tasks, including image captioning, VQA, visual entailment, and referring
expression comprehension. OFA also demonstrates a comparable performance with
language / vision pretrained SOTA models in uni-modal understanding and
generation tasks, e.g., GLUE, abstractive summarization, and image
classification. We provide a further analysis to demonstrate its capability in
zero-shot learning and domain & task transfer, and we also verify the
effectiveness of pretraining tasks.
In the future, we will continue exploring the issues discovered in this work.
Also, we endeavor to figure out a reasonable solution to building an omni-
model essentially generalizable to the complex real world.
## Acknowledgments
We would like to thank Jie Zhang, Yong Li, Jiamang Wang, Shao Yuan, and Zheng
Cao for their support to this project, and we would like to thank Guangxiang
Zhao and Fei Sun for their insightful comments to our paper.
## References
* [1] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NeurIPS 2017, pages 5998–6008, 2017.
* [2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: pre-training of deep bidirectional transformers for language understanding. In Jill Burstein, Christy Doran, and Thamar Solorio, editors, NAACL-HLT 2019, pages 4171–4186. Association for Computational Linguistics, 2019.
* [3] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. arXiv preprint arXiv:2005.14165, 2020.
* [4] Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168, 2021.
* [5] Steffen Schneider, Alexei Baevski, Ronan Collobert, and Michael Auli. wav2vec: Unsupervised pre-training for speech recognition. arXiv preprint arXiv:1904.05862, 2019.
* [6] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.
* [7] Andrew Jaegle, Felix Gimeno, Andrew Brock, Andrew Zisserman, Oriol Vinyals, and Joao Carreira. Perceiver: General perception with iterative attention. arXiv preprint arXiv:2103.03206, 2021.
* [8] Weijie Su, Xizhou Zhu, Yue Cao, Bin Li, Lewei Lu, Furu Wei, and Jifeng Dai. Vl-bert: Pre-training of generic visual-linguistic representations. In International Conference on Learning Representations, 2019.
* [9] Jason Wei, Maarten Bosma, Vincent Y Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M Dai, and Quoc V Le. Finetuned language models are zero-shot learners. arXiv preprint arXiv:2109.01652, 2021.
* [10] Victor Sanh, Albert Webson, Colin Raffel, Stephen H Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Teven Le Scao, Arun Raja, et al. Multitask prompted training enables zero-shot task generalization. arXiv preprint arXiv:2110.08207, 2021.
* [11] Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin De Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. Parameter-efficient transfer learning for nlp. In International Conference on Machine Learning, pages 2790–2799. PMLR, 2019.
* [12] Brian Lester, Rami Al-Rfou, and Noah Constant. The power of scale for parameter-efficient prompt tuning. arXiv preprint arXiv:2104.08691, 2021.
* [13] Jiasen Lu, Dhruv Batra, Devi Parikh, and Stefan Lee. Vilbert: Pretraining task-agnostic visiolinguistic representations for vision-and-language tasks. In NeurIPS, 2019.
* [14] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed El Kholy, Faisal Ahmed, Zhe Gan, Yu Cheng, and Jingjing Liu. Uniter: Universal image-text representation learning. In ECCV, 2020.
* [15] Xiujun Li, Xi Yin, Chunyuan Li, Xiaowei Hu, Pengchuan Zhang, Lei Zhang, Lijuan Wang, Houdong Hu, Li Dong, Furu Wei, Yejin Choi, and Jianfeng Gao. Oscar: Object-semantics aligned pre-training for vision-language tasks. In ECCV, 2020.
* [16] Zhe Gan, Yen-Chun Chen, Linjie Li, Chen Zhu, Yu Cheng, and Jingjing Liu. Large-scale adversarial training for vision-and-language representation learning. ArXiv, abs/2006.06195, 2020.
* [17] Pengchuan Zhang, Xiujun Li, Xiaowei Hu, Jianwei Yang, Lei Zhang, Lijuan Wang, Yejin Choi, and Jianfeng Gao. Vinvl: Revisiting visual representations in vision-language models. 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5575–5584, 2021.
* [18] Junyang Lin, Rui Men, An Yang, Chang Zhou, Ming Ding, Yichang Zhang, Peng Wang, Ang Wang, Le Jiang, Xianyan Jia, et al. M6: A chinese multimodal pretrainer. arXiv preprint arXiv:2103.00823, 2021.
* [19] Zhu Zhang, Jianxin Ma, Chang Zhou, Rui Men, Zhikang Li, Ming Ding, Jie Tang, Jingren Zhou, and Hongxia Yang. M6-ufc: Unifying multi-modal controls for conditional image synthesis. arXiv preprint arXiv:2105.14211, 2021.
* [20] An Yang, Junyang Lin, Rui Men, Chang Zhou, Le Jiang, Xianyan Jia, Ang Wang, Jie Zhang, Jiamang Wang, Yong Li, et al. Exploring sparse expert models and beyond. arXiv preprint arXiv:2105.15082, 2021.
* [21] Junyang Lin, An Yang, Jinze Bai, Chang Zhou, Le Jiang, Xianyan Jia, Ang Wang, Jie Zhang, Yong Li, Wei Lin, et al. M6-10t: A sharing-delinking paradigm for efficient multi-trillion parameter pretraining. arXiv preprint arXiv:2110.03888, 2021.
* [22] Zirui Wang, Jiahui Yu, Adams Wei Yu, Zihang Dai, Yulia Tsvetkov, and Yuan Cao. Simvlm: Simple visual language model pretraining with weak supervision. ArXiv, abs/2108.10904, 2021.
* [23] Lu Yuan, Dongdong Chen, Yi-Ling Chen, Noel C. F. Codella, Xiyang Dai, Jianfeng Gao, Houdong Hu, Xuedong Huang, Boxin Li, Chunyuan Li, Ce Liu, Mengchen Liu, Zicheng Liu, Yumao Lu, Yu Shi, Lijuan Wang, Jianfeng Wang, Bin Xiao, Zhen Xiao, Jianwei Yang, Michael Zeng, Luowei Zhou, and Pengchuan Zhang. Florence: A new foundation model for computer vision. ArXiv, abs/2111.11432, 2021.
* [24] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. URL https://s3-us-west-2.amazonaws.com/openai-assets/researchcovers/ languageunsupervised/language understanding paper. pdf, 2018.
* [25] Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G. Carbonell, Ruslan Salakhutdinov, and Quoc V. Le. Xlnet: Generalized autoregressive pretraining for language understanding. In NeurIPS 2019, pages 5754–5764, 2019.
* [26] Yu Sun, Shuohuan Wang, Yu-Kun Li, Shikun Feng, Xuyi Chen, Han Zhang, Xin Tian, Danxiang Zhu, Hao Tian, and Hua Wu. ERNIE: enhanced representation through knowledge integration. CoRR, abs/1904.09223, 2019.
* [27] Yu Sun, Shuohuan Wang, Yu-Kun Li, Shikun Feng, Hao Tian, Hua Wu, and Haifeng Wang. ERNIE 2.0: A continual pre-training framework for language understanding. CoRR, abs/1907.12412, 2019.
* [28] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized BERT pretraining approach. CoRR, abs/1907.11692, 2019.
* [29] Li Dong, Nan Yang, Wenhui Wang, Furu Wei, Xiaodong Liu, Yu Wang, Jianfeng Gao, Ming Zhou, and Hsiao-Wuen Hon. Unified language model pre-training for natural language understanding and generation. In NeurIPS 2019, pages 13042–13054, 2019.
* [30] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of Machine Learning Research, 21(140):1–67, 2020.
* [31] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. BART: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension. In ACL 2020, July 2020.
* [32] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597–1607. PMLR, 2020.
* [33] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020.
* [34] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. arXiv preprint arXiv:2006.07733, 2020.
* [35] Xinlei Chen and Kaiming He. Exploring simple siamese representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15750–15758, 2021.
* [36] Hangbo Bao, Li Dong, and Furu Wei. Beit: Bert pre-training of image transformers. arXiv preprint arXiv:2106.08254, 2021.
* [37] Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. Masked autoencoders are scalable vision learners. arXiv preprint arXiv:2111.06377, 2021.
* [38] Liunian Harold Li, Mark Yatskar, Da Yin, Cho-Jui Hsieh, and Kai-Wei Chang. Visualbert: A simple and performant baseline for vision and language. ArXiv, abs/1908.03557, 2019.
* [39] Luowei Zhou, Hamid Palangi, Lei Zhang, Houdong Hu, Jason J. Corso, and Jianfeng Gao. Unified vision-language pre-training for image captioning and VQA. In AAAI 2020, pages 13041–13049, 2020.
* [40] Hao Tan and Mohit Bansal. Lxmert: Learning cross-modality encoder representations from transformers. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 5100–5111, 2019.
* [41] Gen Li, Nan Duan, Yuejian Fang, Daxin Jiang, and Ming Zhou. Unicoder-vl: A universal encoder for vision and language by cross-modal pre-training. CoRR, abs/1908.06066, 2019.
* [42] Junyang Lin, An Yang, Yichang Zhang, Jie Liu, Jingren Zhou, and Hongxia Yang. Interbert: Vision-and-language interaction for multi-modal pretraining. arXiv preprint arXiv:2003.13198, 2020.
* [43] Jiasen Lu, Vedanuj Goswami, Marcus Rohrbach, Devi Parikh, and Stefan Lee. 12-in-1: Multi-task vision and language representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10437–10446, 2020.
* [44] Haiyang Xu, Ming Yan, Chenliang Li, Bin Bi, Songfang Huang, Wenming Xiao, and Fei Huang. E2e-vlp: End-to-end vision-language pre-training enhanced by visual learning. arXiv preprint arXiv:2106.01804, 2021.
* [45] Fei Yu, Jiji Tang, Weichong Yin, Yu Sun, Hao Tian, Hua Wu, and Haifeng Wang. Ernie-vil: Knowledge enhanced vision-language representations through scene graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 3208–3216, 2021.
* [46] Wei Li, Can Gao, Guocheng Niu, Xinyan Xiao, Hao Liu, Jiachen Liu, Hua Wu, and Haifeng Wang. UNIMO: towards unified-modal understanding and generation via cross-modal contrastive learning. In Chengqing Zong, Fei Xia, Wenjie Li, and Roberto Navigli, editors, ACL/IJCNLP 2021, pages 2592–2607. Association for Computational Linguistics, 2021.
* [47] Zhicheng Huang, Zhaoyang Zeng, Bei Liu, Dongmei Fu, and Jianlong Fu. Pixel-bert: Aligning image pixels with text by deep multi-modal transformers. ArXiv, abs/2004.00849, 2020.
* [48] Wenhui Wang, Hangbo Bao, Li Dong, and Furu Wei. Vlmo: Unified vision-language pre-training with mixture-of-modality-experts. ArXiv, abs/2111.02358, 2021.
* [49] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision. In Marina Meila and Tong Zhang, editors, ICML 2021, volume 139 of Proceedings of Machine Learning Research, pages 8748–8763. PMLR, 2021.
* [50] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. arXiv preprint arXiv:2102.12092, 2021.
* [51] Ming Ding, Zhuoyi Yang, Wenyi Hong, Wendi Zheng, Chang Zhou, Da Yin, Junyang Lin, Xu Zou, Zhou Shao, Hongxia Yang, et al. Cogview: Mastering text-to-image generation via transformers. arXiv preprint arXiv:2105.13290, 2021.
* [52] Chenfei Wu, Jian Liang, Lei Ji, Fan Yang, Yuejian Fang, Daxin Jiang, and Nan Duan. N$\backslash$" uwa: Visual synthesis pre-training for neural visual world creation. arXiv preprint arXiv:2111.12417, 2021.
* [53] Aäron van den Oord, Oriol Vinyals, and Koray Kavukcuoglu. Neural discrete representation learning. In NIPS, 2017.
* [54] Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12873–12883, 2021.
* [55] Lukasz Kaiser, Aidan N Gomez, Noam Shazeer, Ashish Vaswani, Niki Parmar, Llion Jones, and Jakob Uszkoreit. One model to learn them all. arXiv preprint arXiv:1706.05137, 2017.
* [56] Jaemin Cho, Jie Lei, Haochen Tan, and Mohit Bansal. Unifying vision-and-language tasks via text generation. In ICML, 2021.
* [57] Zhengyuan Yang, Zhe Gan, Jianfeng Wang, Xiaowei Hu, Faisal Ahmed, Zicheng Liu, Yumao Lu, and Lijuan Wang. Crossing the format boundary of text and boxes: Towards unified vision-language modeling. ArXiv, abs/2111.12085, 2021.
* [58] Andrew Jaegle, Sebastian Borgeaud, Jean-Baptiste Alayrac, Carl Doersch, Catalin Ionescu, David Ding, Skanda Koppula, Daniel Zoran, Andrew Brock, Evan Shelhamer, et al. Perceiver io: A general architecture for structured inputs & outputs. arXiv preprint arXiv:2107.14795, 2021.
* [59] Ronghang Hu and Amanpreet Singh. Unit: Multimodal multitask learning with a unified transformer. arXiv preprint arXiv:2102.10772, 2021.
* [60] Amanpreet Singh, Ronghang Hu, Vedanuj Goswami, Guillaume Couairon, Wojciech Galuba, Marcus Rohrbach, and Douwe Kiela. Flava: A foundational language and vision alignment model. arXiv preprint arXiv:2112.04482, 2021.
* [61] Xizhou Zhu, Jinguo Zhu, Hao Li, Xiaoshi Wu, Xiaogang Wang, Hongsheng Li, Xiaohua Wang, and Jifeng Dai. Uni-perceiver: Pre-training unified architecture for generic perception for zero-shot and few-shot tasks. arXiv preprint arXiv:2112.01522, 2021.
* [62] Zihang Dai, Hanxiao Liu, Quoc V Le, and Mingxing Tan. Coatnet: Marrying convolution and attention for all data sizes. arXiv preprint arXiv:2106.04803, 2021.
* [63] Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words with subword units. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1715–1725, 2016.
* [64] Ting Chen, Saurabh Saxena, Lala Li, David J Fleet, and Geoffrey Hinton. Pix2seq: A language modeling framework for object detection. arXiv preprint arXiv:2109.10852, 2021.
* [65] Lei Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. CoRR, abs/1607.06450, 2016.
* [66] Sam Shleifer, Jason Weston, and Myle Ott. Normformer: Improved transformer pretraining with extra normalization. arXiv preprint arXiv:2110.09456, 2021.
* [67] Guolin Ke, Di He, and Tie-Yan Liu. Rethinking positional encoding in language pre-training. In International Conference on Learning Representations, 2020.
* [68] Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009.
* [69] Junnan Li, Ramprasaath R Selvaraju, Akhilesh Deepak Gotmare, Shafiq Joty, Caiming Xiong, and Steven Hoi. Align before fuse: Vision and language representation learning with momentum distillation. In Thirty-Fifth Conference on Neural Information Processing Systems, 2021.
* [70] Zi-Yi Dou, Yichong Xu, Zhe Gan, Jianfeng Wang, Shuohang Wang, Lijuan Wang, Chenguang Zhu, Nanyun Peng, Zicheng Liu, and Michael Zeng. An empirical study of training end-to-end vision-and-language transformers. ArXiv, abs/2111.02387, 2021.
* [71] Xiaowei Hu, Zhe Gan, Jianfeng Wang, Zhengyuan Yang, Zicheng Liu, Yumao Lu, and Lijuan Wang. Scaling up vision-language pre-training for image captioning. CoRR, abs/2111.12233, 2021.
* [72] Aishwarya Kamath, Mannat Singh, Yann LeCun, Ishan Misra, Gabriel Synnaeve, and Nicolas Carion. Mdetr - modulated detection for end-to-end multi-modal understanding. ArXiv, abs/2104.12763, 2021.
* [73] Ning Xie, Farley Lai, Derek Doran, and Asim Kadav. Visual entailment: A novel task for fine-grained image understanding. arXiv preprint arXiv:1901.06706, 2019.
* [74] Xinlei Chen, Hao Fang, Tsung-Yi Lin, Ramakrishna Vedantam, Saurabh Gupta, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco captions: Data collection and evaluation server. arXiv preprint arXiv:1504.00325, 2015.
* [75] Licheng Yu, Patrick Poirson, Shan Yang, Alexander C Berg, and Tamara L Berg. Modeling context in referring expressions. In European Conference on Computer Vision, pages 69–85. Springer, 2016.
* [76] Junhua Mao, Jonathan Huang, Alexander Toshev, Oana Camburu, Alan L Yuille, and Kevin Murphy. Generation and comprehension of unambiguous object descriptions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 11–20, 2016.
* [77] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021.
* [78] Yupan Huang, Hongwei Xue, Bei Liu, and Yutong Lu. Unifying multimodal transformer for bi-directional image and text generation. In Proceedings of the 29th ACM International Conference on Multimedia, pages 1138–1147, 2021.
* [79] Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R Bowman. Glue: A multi-task benchmark and analysis platform for natural language understanding. arXiv preprint arXiv:1804.07461, 2018.
* [80] Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 379–389, 2015.
* [81] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
* [82] Kevin Clark, Minh-Thang Luong, Quoc V. Le, and Christopher D. Manning. ELECTRA: pre-training text encoders as discriminators rather than generators. In 8th International Conference on Learning Representations, ICLR 2020. OpenReview.net, 2020.
* [83] Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. Deberta: decoding-enhanced bert with disentangled attention. In 9th International Conference on Learning Representations, ICLR 2021. OpenReview.net, 2021.
* [84] Chin-Yew Lin. ROUGE: A package for automatic evaluation of summaries. In Text Summarization Branches Out, Barcelona, Spain, July 2004\. Association for Computational Linguistics.
* [85] Sascha Rothe, Shashi Narayan, and Aliaksei Severyn. Leveraging pre-trained checkpoints for sequence generation tasks. Transactions of the Association for Computational Linguistics, 8:264–280, 2020.
* [86] Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu. MASS: masked sequence to sequence pre-training for language generation. In ICML 2019, pages 5926–5936, 2019.
* [87] Jingqing Zhang, Yao Zhao, Mohammad Saleh, and Peter Liu. Pegasus: Pre-training with extracted gap-sentences for abstractive summarization. In International Conference on Machine Learning, pages 11328–11339. PMLR, 2020.
* [88] Weizhen Qi, Yu Yan, Yeyun Gong, Dayiheng Liu, Nan Duan, Jiusheng Chen, Ruofei Zhang, and Ming Zhou. Prophetnet: Predicting future n-gram for sequence-to-sequence pre-training. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: Findings, pages 2401–2410, 2020.
* [89] Mingxing Tan and Quoc Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning, pages 6105–6114. PMLR, 2019.
* [90] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. arXiv preprint arXiv:2104.14294, 2021.
* [91] Soravit Changpinyo, Piyush Sharma, Nan Ding, and Radu Soricut. Conceptual 12m: Pushing web-scale image-text pre-training to recognize long-tail visual concepts. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3558–3568, 2021.
* [92] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu Soricut. Conceptual captions: A cleaned, hypernymed, image alt-text dataset for automatic image captioning. In ACL 2018, pages 2556–2565, 2018.
* [93] Vicente Ordonez, Girish Kulkarni, and Tamara L. Berg. Im2text: Describing images using 1 million captioned photographs. In NeurIPS 2011, pages 1143–1151, 2011.
* [94] Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalantidis, Li-Jia Li, David A. Shamma, Michael S. Bernstein, and Li Fei-Fei. Visual genome: Connecting language and vision using crowdsourced dense image annotations. International Journal of Computer Vision, 123(1):32–73, 2017.
* [95] Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. Making the v in vqa matter: Elevating the role of image understanding in visual question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 6904–6913, 2017.
* [96] Drew A Hudson and Christopher D Manning. Gqa: A new dataset for real-world visual reasoning and compositional question answering. In CVPR 2019, pages 6700–6709, 2019.
* [97] Bart Thomee, David A Shamma, Gerald Friedland, Benjamin Elizalde, Karl Ni, Douglas Poland, Damian Borth, and Li-Jia Li. Yfcc100m: The new data in multimedia research. Communications of the ACM, 59(2):64–73, 2016.
* [98] Alina Kuznetsova, Hassan Rom, Neil Alldrin, Jasper Uijlings, Ivan Krasin, Jordi Pont-Tuset, Shahab Kamali, Stefan Popov, Matteo Malloci, Alexander Kolesnikov, et al. The open images dataset v4. International Journal of Computer Vision, 128(7):1956–1981, 2020\.
* [99] Shuai Shao, Zeming Li, Tianyuan Zhang, Chao Peng, Gang Yu, Xiangyu Zhang, Jing Li, and Jian Sun. Objects365: A large-scale, high-quality dataset for object detection. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 8430–8439, 2019.
* [100] Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al. The pile: An 800gb dataset of diverse text for language modeling. arXiv preprint arXiv:2101.00027, 2020.
* [101] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR 2016, pages 770–778, 2016.
* [102] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In ICLR 2019, 2019.
* [103] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q. Weinberger. Deep networks with stochastic depth. In ECCV, 2016.
* [104] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pages 311–318, 2002.
* [105] Satanjeev Banerjee and Alon Lavie. Meteor: An automatic metric for mt evaluation with improved correlation with human judgments. In Proceedings of the acl workshop on intrinsic and extrinsic evaluation measures for machine translation and/or summarization, pages 65–72, 2005.
* [106] Ramakrishna Vedantam, C Lawrence Zitnick, and Devi Parikh. Cider: Consensus-based image description evaluation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4566–4575, 2015.
* [107] Peter Anderson, Basura Fernando, Mark Johnson, and Stephen Gould. Spice: Semantic propositional image caption evaluation. In European conference on computer vision, pages 382–398. Springer, 2016.
* [108] Andrej Karpathy and Li Fei-Fei. Deep visual-semantic alignments for generating image descriptions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3128–3137, 2015.
* [109] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.
* [110] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. Advances in neural information processing systems, 29:2234–2242, 2016.
* [111] Steven J Rennie, Etienne Marcheret, Youssef Mroueh, Jerret Ross, and Vaibhava Goel. Self-critical sequence training for image captioning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7008–7024, 2017.
* [112] Guangxiang Zhao, Wenkai Yang, Xuancheng Ren, Lei Li, and Xu Sun. Well-classified examples are underestimated in classification with deep neural networks. CoRR, abs/2110.06537, 2021.
* [113] Ekin D Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V Le. Randaugment: Practical automated data augmentation with a reduced search space. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 702–703, 2020.
* [114] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data augmentation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 13001–13008, 2020.
* [115] Hongyi Zhang, Moustapha Cissé, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings. OpenReview.net, 2018.
* [116] Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, and Junsuk Choe. Cutmix: Regularization strategy to train strong classifiers with localizable features. In 2019 IEEE/CVF International Conference on Computer Vision, ICCV 2019, Seoul, Korea (South), October 27 - November 2, 2019, pages 6022–6031. IEEE, 2019.
* [117] Christoph Schuhmann, Richard Vencu, Romain Beaumont, Robert Kaczmarczyk, Clayton Mullis, Aarush Katta, Theo Coombes, Jenia Jitsev, and Aran Komatsuzaki. Laion-400m: Open dataset of clip-filtered 400 million image-text pairs. arXiv preprint arXiv:2111.02114, 2021.
## Appendix A Implementation Details
### A.1 Pretraining Datasets
We construct pretraining datasets by incorporating Vision & Language data
(i.e., image-text pairs), Vision data (i.e., raw image data, object-labeled
data), and Language data (i.e., plain texts). For replication, the pretraining
datasets are publicly available. We carefully filter our pretraining data and
exclude images that appear in the validation and test sets of downstream tasks
to avoid data leakage. The statistics on the pretraining datasets are listed
in Table 11.
#### Cross-modal Data
For vision & language pretraining, we mainly apply image-text pairs, including
image-caption pairs, image-QA pairs, and image-region pairs, as the
pretraining data. For the pretraining tasks of image captioning and image-text
matching, we collect Conceptual Caption 12M (CC12M) [91], Conceptual Captions
(CC3M) [92], SBU [93], MSCOCO image captions (COCO) [74], and Visual Genome
Captions (VG Captions) [94]. Specifically, the part of data from VG requires
some additional processing. As texts in VG captions describe local regions on
the images, we retrieve regions with area larger than $16,384$ pixels and
construct region-caption pairs. For visual question answering, we collect
VQAv2 [95], VG-QA [94], as well as GQA [96]. VQAv2 is a visual question
answering dataset with real-world photographs from COCO. VG-QA is also a
visual question answering dataset with real-world photographs from VG. The
questions of VG-QA are related to specific regions on the images. GQA is a
large VQA dataset featuring compositional questions. The images of GQA are
also collected from VG. For visual grounding and grounded captioning, we
collect data from RefCOCO [75], RefCOCO+ [75], RefCOCOg [76] and VG captions.
Additional processing is applied to VG Captions for this task. Specifically,
we use the data of VG that contains regions with area smaller than $16,384$
pixels for Visual Grounding, in order to encourage model to grasp fine-grained
alignments between vision and language.
#### Uni-modal Data
Uni-modal data includes vision and language data. Vision data consists of raw
images for image infilling and object-labeled images for object detection. For
image infilling, we collect raw images from OpenImages, YFCC100M [97] and
ImageNet-21K [81], and exclude annotations. Thus the model is unable to access
labels in the pretraining stage. For object detection, we collect OpenImages
[98], Object365 [99], VG and COCO for object detection. Language data consists
of plain texts, i.e., passages consisting of sentences. We use around 140GB of
data from Pile [100] to leverage its diversity. Specifically, we extract
natural language data and implement preprocessing methods, including
truncation to the length of $512$.
Table 11: Statistics on the datasets of pretraining tasks. “#Image” denotes
the number of distinct images, and “#Sample” denotes the number of samples.
*For language data, we report its storage following the previous studies [2,
28].
Type Pretraining Task Source #Image #Sample Vision & Language Image Captioning
CC12M, CC3M, SBU, COCO, VG-Cap 14.78M 15.25M Image-Text Matching Visual
Question Answering VQAv2, VG-QA, GQA 178K 2.92M Visual Grounding RefCOCO,
RefCOCO+, RefCOCOg, VG-Cap 131K 3.20M Grounded Captioning Vision Detection
OpenImages, Object365, VG, COCO 2.98M 3.00M Image Infilling OpenImages,
YFCC100M, ImageNet-21K 36.27M - Language Masked Language Modeling Pile
(Filtered) - 140GB*
### A.2 Pretraining Details
For the image processing, we first resize and crop the images into different
resolutions, $256\times 256$ for $\text{OFA}\rm_{Tiny}$ and
$\text{OFA}\rm_{Medium}$, $384\times 384$ for $\text{OFA}\rm_{Base}$,
$480\times 480$ for $\text{OFA}\rm_{Large}$ and $\text{OFA}\rm_{Huge}$, with a
fixed patch size of $16\times 16$. Note that training $\text{OFA}\rm_{Large}$
and $\text{OFA}\rm_{Huge}$ are time and computation consuming, we first train
them with images of the resolution of $384\times 384$ and $256\times 256$, and
continue pretraining with images of the resolution of $480\times 480$.
For each patch, we obtain its feature vector with the first three blocks of
ResNet [101]. The ResNet module is jointly trained along with the transformer
module. Note that through extensive experiments we find that random sampling
patches [47] does not bring additional benefits in our scenario. For the text
processing, we tokenize the texts with the same BPE Tokenizer [63] as BART
[31]. The maximum text sequence length of both encoder and decoder is set to
$256$. We share parameters between the embedding and the decoder softmax
output layer.
From our preliminary experiments, we find that the initialization for
Transformer plays an important role. For $\text{OFA}\rm_{Base}$ and
$\text{OFA}\rm_{Large}$, we initialize the transformer with most of the
weights of $\text{BART}\rm_{Base}$ and $\text{BART}\rm_{Large}$ considering
the slight difference between OFA Transformer and BART as described in Sec
3.1. For OFA of the other sizes, we pretrain language models with the same
pretraining strategy with BART and use the pretrained weights to initialize
the Transformer in OFA.
We use the AdamW [102] optimizer with $(\beta_{1},\beta_{2})=(0.9,0.999)$ and
$\epsilon=1e\text{-}8$ to pretrain our models. We set the peak learning rate
to $2e\text{-}4$, and apply a scheduler with linear decay with a warmup ratio
of $0.01$ to control the learning rate. For regulation, we set dropout to
$0.1$ and use weight decay with $0.01$. We employ stochastic depth [103] with
a $0.1$ rate (applied to encoder and decoder except for convolution blocks).
We mix all the pretraining data within each batch, which contains $2,048$
vision&language samples, $256$ object detection samples, $256$ image-only
samples and $512$ text-only samples. All models are pretrained for at least
$300K$ steps except the models used for ablation study.
### A.3 Details of Downstream Tasks
We verify the capability of OFA on various downstream tasks in both finetuning
and zero-shot settings. We design various task-specific instructions to
transfer the knowledge learned from pretraining to downstream tasks
effectively. The instructions of different tasks are listed in Table 12. For
finetuning, if not specified, the input image resolution is set to $480\times
480$, and the other hyper-parameters remain the same as for pretraining. The
experimental details of different downstream tasks, including both multimodal
and uni-modal tasks, are listed below:
#### Image Captioning
Image captioning is a standard vision&language task that requires models to
generate an appropriate and fluent caption for an image. We adopt the most
widely used MSCOCO Image Caption dataset [74] to evaluate the multi-modal
generation capability of OFA. We report BLEU-4 [104], METEOR [105], CIDEr
[106], and SPICE [107] scores on the Karpathy test split [108]. Following the
previous standard practice, we first finetune OFA with cross-entropy loss for
$2$ epochs with a batch size of $128$ and a learning rate of $1e-5$, and label
smoothing is set to $0.1$. We then finetune the model with CIDEr optimization
for $3$ epochs with a batch size of $64$, and disable dropout and stochastic
depth. We report both scores at the two stages.
#### Visual Question Answering
Visual question answering (VQA) is a cross-modal task that requires the models
to answer the question given an image. Previous works such as VLMo [48] or
SimVLM [22] define VQA as a classification task. They use a linear output
layer to predict the probability of each candidate answer on a given set. In
contrast with these studies, to adapt the generative OFA model to VQA
benchmark, we use the Trie-based search strategy mentioned in Sec. 3.4 to
ensure that the answer generated by OFA is constrained in the candidate set.
We evaluate our model with other baselines on the commonly used VQAv2 dataset
[95]. Accuracy scores on both test-dev and test-std sets are reported. The OFA
models of all the reported sizes are finetuned for $40,000$ steps with a batch
size of $512$. The learning rate is $5e-5$ with the label smoothing of $0.1$.
When finetuning $\text{OFA}\rm_{Large}$ and $\text{OFA}\rm_{Huge}$, we
increase the image resolution from $480$ to $640$. Linear interpolation of the
image absolute positional embedding proposed in [6] is employed when
transferring the pretrained OFA to VQA finetuning. During Trie-based
searching, we constrain the generated answers over the most frequent $3,129$
answer candidates. Exponential moving average (EMA) with decay rate $0.9999$
is employed in finetuning.
Table 12: Instructions for downstream tasks.
Task Dataset Instruction Target Image Captioning COCO
$\rm\left[\textbf{Image}\right]$ What does the image describe? {Caption}
Visual Question Answering VQA $\rm\left[\textbf{Image}\right]$ {Question}
{Answer} Visual Entailment SNLI-VE $\rm\left[\textbf{Image}\right]$ Can image
and text1 “{Text1}" imply text2 “{Text2}"? Yes/No/Maybe Referring Expression
Comprehension RefCOCO, RefCOCO+, RefCOCOg $\rm\left[\textbf{Image}\right]$
Which region does the text “{Text}" describe? {Location} Image Generation COCO
What is the complete image? caption: {Caption} {Image} Image Classification
ImageNet-1K $\rm\left[\textbf{Image}\right]$ What does the image describe?
{Label} Single-Sentence Classification SST-2 Is the sentiment of text “{Text}"
positive or negative? Positive/Negative Sentence-Pair Classification RTE Can
text1 “{Text1}" imply text2 “{Text2}"? Yes/No MRPC Does text1 “{Text1}" and
text2 “{Text2}" have the same semantics? Yes/No QQP Is question “{Question1}"
and question “{Question2}" equivalent? Yes/No MNLI Can text1 “{Text1}" imply
text2 “{Text2}"? Yes/No/Maybe QNLI Does “{Text}" contain the answer to
question “{Question}"? Yes/No Text Summarization Gigaword What is the summary
of article “{Article}"? {Summary}
#### Visual Entailment
Visual entailment requires the model to evaluate how the given image and text
are semantically correlated, i.e., entailment, neutral, or contradiction. We
perform experiments on the SNLI-VE dataset [73]. The image premise, text
premise and text hypothesis are fed to the encoder, and the decoder generates
appropriate labels. To transfer the knowledge learned by pretraining to this
task, we convert the labels entailment/neutral/contradiction to yes/maybe/no.
We also use the Trie-based search strategy to constrain the generated labels
over the candidate set. We report accuracy on both dev and test sets. The OFA
model is finetuned for $6$ epochs with a learning rate of $2e-5$ and a batch
size of $256$.
#### Referring Expression Comprehension
Referring expression comprehension requires models to locate an image region
described by a language query. Different from the approach taken by most
previous methods [13, 14] which ranks a set of candidate bounding boxes
detected by a pretrained object detector, our method directly predicts the
best matching bounding box without any proposals. We perform experiments on
RefCOCO [75], RefCOCO+ [75], and RefCOCOg [76]. Consistent with other
downstream tasks, we formulate referring expression comprehension as a
conditional sequence generation task. In detail, given an image and a language
query, OFA generates the box sequence (e.g., $\langle
x_{1},y_{1},x_{2},y_{2}\rangle$) in an autoregressive manner. We report the
standard metric<EMAIL_ADDRESS>on the validation and test sets. For finetuning, the
input image resolution is set to $512\times 512$. We finetune the OFA model on
each dataset for about $10$ epochs with a batch size of $128$. The learning
rate is $3e-5$ with the label smoothing of $0.1$. Each query only corresponds
to an image region, so we limit the maximum generated length to $4$ during
inference.
#### Image Generation
Following the same setting with [52], we train our model on the MS COCO train
split and evaluate our model on the validation split by randomly sampling
$30,000$ images. We use Fréchet Inception Distance (FID) [109] and Inception
Score (IS) [110] to evaluate the quality of the images. Following the previous
studies [78, 52], we also compute CLIP Similarity Score (CLIPSIM) to evaluate
the semantic similarity between the query text and the generated images.
During finetuning, OFA learns to generate the image code sequence according to
the given text query only. The model is first finetuned with cross-entropy and
then with CLIPSIM optimization following [111, 78]. In the first stage, we
finetune the OFA model for about $50$ epochs with a batch size of $512$ and a
learning rate of $1e-3$. In the second stage, the model is finetuned for extra
$5000$ steps with a batch size of $32$ and a learning rate of $1e-6$. During
the evaluation, we sample $24$ images with the resolution of $256\times 256$
for each query and choose the best one using the pretrained CLIP model [49].
For case study, we compare OFA with CogView and GLIDE. CogView provides an API
website 444https://wudao.aminer.cn/CogView/index.html. Note that this API
samples 8 images of resolution of $512\times 512$ for each query. We select
the first one of generated images and resize it to the resolution of
$256\times 256$. GLIDE provides a Colab
notebook.555https://colab.research.google.com/drive/1q6tJ58UKod1eCOkbaUNGzF3K5BbXlB5m.
Note that the only publicly available GLIDE model is of base size
($\sim$385M).
#### Image Classification
We provide finetuning results on ImageNet-1K [81] following recent studies in
self-supervised learning for computer vision. During finetuning and inference,
a Trie-based search strategy is employed to constrain the generated text into
the set of $1,000$ candidate labels. We finetune OFA for $32$ epochs and a
batch size of $256$. The learning rate is $5e-5$. The ratio for label
smoothing is $0.1$. The encouraging loss proposed in [112] is employed with
the hyperparameter LE set to $0.75$. Following [36], we use the same random
resize cropping, random flipping, RandAug [113] and random erasing [114]
transformations as data augmentation strategies. Mixup [115] and CutMix [116]
are used with overall $0.5$ probability to be performed on each batch and
alpha is $0.8$ and $1.0$, respectively. To adapt the mixed soft target of
Mixup and CutMix into generation paradigm during finetuning, we run the
decoder twice each with one of the target sequences to be mixed and sum the
loss weighted by the mixing ratio.
#### Natural Language Understanding
To verify the natural language understanding ability of OFA, we select $6$
language understanding tasks from GLUE benchmark [79], including both single-
sentence classification tasks and sentence-pair classification tasks. To adapt
to sentence-pair classification, previous models [2, 28] usually use segment
embeddings to distinguish different sentences. Unlike those models, OFA can
apply the model to sentence-pair classification tasks by constructing
appropriate instructions without introducing additional segment embeddings.
For the hyper-parameters of finetuning, we tune the training epochs among
$\\{5,7,10\\}$, learning rate among $\\{3e-5,5e-5,6e-5,7e-5,1e-4\\}$, batch
size among $\\{32,64,128\\}$, weight decay among $\\{0.01,0.05\\}$, and
dropout rate among $\\{0.0,0.1\\}$. We report the best performance on the
development set for each task.
#### Natural Language Generation
We verify the natural language generation ability of OFA in the Gigaword
dataset [80]. We report ROUGE-1/ROUGE-2/ROUGE-L to evaluate the generation
results following [80]. We finetune the OFA models for $6$ epochs with a batch
size of $512$. The learning rate is $1e-4$ with the label smoothing of $0.1$,
and the maximum input text sequence length is set to $512$. During inference,
we set the length penalty to $0.7$ and beam size to $6$, and limit the maximum
generated length to 32.
## Appendix B Trie-based Search
This section describes how to use Trie-based search to improve model
performance on downstream classification tasks. When dealing with
classification tasks, we first construct a Trie where nodes are annotated with
tokens from the candidate label-set. During finetuning, the model computes the
log-probabilities of the target tokens based on their positions on the Trie.
As shown in Figure 6, when computing the log-probabilities of the target token
“sky”, we only consider tokens in {“sky”, “ocean”} and forcefully set the
logits for all invalid tokens to $-\infty$. During inference, we constrain the
generated labels over the candidate set. As shown in Table 13, Trie-based
search strategy can boost the performance of OFA in various downstream
classification tasks.
Figure 6: Example of Trie-based search where the constraint labels are “blue
sky”, “blue ocean” and “green”. When computing the log-prob of token “sky”, we
only consider tokens in {“sky”, “ocean”} and forcefully set the logits for all
invalid tokens to $-\infty$. Table 13: Ablation results of Trie. The removal
of Trie-based search degenerates the performance on downstream tasks. Note
that the baseline $\text{OFA}\rm_{Base}$ is only pre-trained for 250k steps,
which is also used in Table 10.
Model VQA SNLI-VE ImageNet MRPC QQP Test-dev Acc. Dev Acc. Top-1 Acc. F1 F1
$\text{OFA}\rm_{Base}$ 76.03 89.2 82.2 90.6 88.4 w/o Trie 75.86(-0.17)
89.0(-0.2) 81.9(-0.3) 90.1(-0.5) 88.2(-0.2)
## Appendix C Qualitative Examples
This section provides more qualitative examples of multiple tasks, including
text-to-image generation, open-domain VQA, grounded question answering, and
open-domain visual grounding, from the generation of OFA. By reading this
section, we hope that readers can better perceive OFA.
Figure 7: Examples of text-to-image generation. For better demonstration, we
continue finetuning OFA on a subset of LAION-400M [117]. Figure 8: Examples of
text-to-image generation. Figure 9: More samples of VQA task on unseen
domains. The answers are generated by pretrained OFA without finetuning. The
datasets used in VQA pretraining task only contain real-world photographs. We
present more cases of VQA task on out-of-domain (non-photographic) images and
demonstrate the capability of transferring OFA to these unseen domains. Figure
10: Samples of the unseen grounded question answering task. In this task, the
model should answer a question about a particular region in the image. This
task is unseen in pretraining. We demonstrate that directly transferring
pretrained OFA to this new task without finetuning works well. Figure 11:
Samples of visual grounding task generated by OFA for various unseen domains:
(a) anime (the corresponding animations are Pokemon and One Piece); (b)
synthetic images with attribute combinations.
|
# CenterFusion: Center-based Radar and Camera Fusion for 3D Object Detection
Ramin Nabati, Hairong Qi
University of Tennessee Knoxville
{rnabati<EMAIL_ADDRESS>
###### Abstract
The perception system in autonomous vehicles is responsible for detecting and
tracking the surrounding objects. This is usually done by taking advantage of
several sensing modalities to increase robustness and accuracy, which makes
sensor fusion a crucial part of the perception system. In this paper, we focus
on the problem of radar and camera sensor fusion and propose a middle-fusion
approach to exploit both radar and camera data for 3D object detection. Our
approach, called CenterFusion, first uses a center point detection network to
detect objects by identifying their center points on the image. It then solves
the key data association problem using a novel frustum-based method to
associate the radar detections to their corresponding object’s center point.
The associated radar detections are used to generate radar-based feature maps
to complement the image features, and regress to object properties such as
depth, rotation and velocity. We evaluate CenterFusion on the challenging
nuScenes dataset, where it improves the overall nuScenes Detection Score (NDS)
of the state-of-the-art camera-based algorithm by more than 12%. We further
show that CenterFusion significantly improves the velocity estimation accuracy
without using any additional temporal information. The code is available at
https://github.com/mrnabati/CenterFusion.
## 1 Introduction
Autonomous vehicles are usually equipped with different types of sensors to
take advantage of their complimentary characteristics. Using multiple sensor
modalities increases robustness and accuracy, but also introduces new
challenges in designing the perception system. Sensor fusion is one of these
challenges, which has motivated many studies in 2D and 3D object detection [4,
10, 14, 19], semantic segmentation [33, 16] and object tracking [1, 7] in
recent years.
Most of the recent sensor fusion methods focus on exploiting LiDAR and camera
for 3D object detection. LiDARs use the time of flight of laser light pulses
to calculate distance to surrounding objects. LiDARs provide accurate 3D
measurement at close range, but the resulting point cloud becomes sparse at
long range, reducing the system’s ability to accurately detect far away
objects. Cameras provide rich appearance features, but are not a good source
of information for depth estimation. These complementary features have made
LiDAR-camera sensor fusion a topic of interest in recent years. This
combination has been proven to achieve high accuracy in 3D object detection
for many applications including autonomous driving, but it has its
limitations. Cameras and LiDARs are both sensitive to adverse weather
conditions (e.g. snow, fog, rain), which can significantly reduce their field
of view and sensing capabilities. Additionally, LiDARs and cameras are not
capable of detecting objects’ velocity without using temporal information.
Estimating objects’ velocity is a crucial requirement for collision avoidance
in many scenarios, and relying on the temporal information might not be a
feasible solution in time-critical situations.
Figure 1: CenterFusion network architecture. Preliminary 3D boxes are first
obtained using the image features extracted by the backbone. The frustum
association module uses the preliminary boxes to associate radar detections to
objects and generate radar feature maps. The image and radar features maps are
then concatenated and used to refine the preliminary detections by
recalculating depth and rotation as well as estimating objects’ velocity and
attributes.
For many years, radars have been used in vehicles for Advanced Driving
Assistance System (ADAS) applications such as collision avoidance and Adaptive
Cruise Control (ACC). Compared to LiDARs and cameras, radars are very robust
to adverse weather conditions and are capable of detecting objects at very
long range (up to 200 meters for automotive radars). Radars use the Doppler
effect to accurately estimate the velocities of all detected objects, without
requiring any temporal information. Additionally, compared to LiDARs, Radar
point clouds require less processing before they can be used as object
detection results. These features and their lower cost compared to LiDARs,
have made radars a popular sensor in autonomous driving applications.
Despite radar’s popularity in the automotive industry, few studies have
focused on fusing radar data with other sensors. One reason for this is the
fact that there are not many datasets containing radar data for autonomous
driving applications, which makes conducting research in this area difficult.
Additionally, due to inherent differences between LiDAR and radar point
clouds, applying or adapting existing LiDAR-based algorithms to radar point
cloud proves to be extremely difficult. Radar point clouds are significantly
more sparse than their LiDAR counter parts, making it unfeasible to use for
extracting objects’ geometry information. Aggregating multiple radar sweeps
increases the density of the points, but also introduces delay into the
system. Moreover, although radar point clouds are usually represented as
points in the 3D coordinate system, the reported vertical measurement of the
points are usually not accurate or even non-existent, as most automotive
radars only report the distance and azimuth angle to objects.
In order to effectively combine multiple sensing modalities, a variety of
sensor fusion schemes have been developed [8] taking advantage of the
hierarchical feature representation in neural networks. In an early fusion
approach, the raw or pre-processed sensory data from different sensor
modalities are fused together. With this approach, the network learns a joint
representation from the sensing modalities. Early fusion methods are usually
sensitive to spatial or temporal misalignment of the data [8]. On the other
hand, a late fusion approach combines the data from different modalities at
the decision level, and provides more flexibility for introducing new sensing
modalities to the network. However, a late fusion approach does not exploit
the full potential of the available sensing modalities, as it does not acquire
the intermediate features obtained by learning a joint representation. A
compromise between the early and late fusion approaches is referred to as
middle fusion. It extracts features from different modalities individually and
combines them at an intermediate stage, enabling the network to learn joint
representations and creating a balance between sensitivity and flexibility.
We propose CenterFusion, a middle-fusion approach to exploit radar and camera
data for 3D object detection. CenterFusion focuses on associating radar
detections to preliminary detection results obtained from the image, then
generates radar feature maps and uses it in addition to image features to
accurately estimate 3D bounding boxes for objects. Particularly, we generate
preliminary 3D detections using a key point detection network, and propose a
novel frustum-based radar association method to accurately associate radar
detections to their corresponding objects in the 3D space. These radar
detections are then mapped to the image plane and used to create feature maps
to complement the image-based features. Finally, the fused features are used
to accurately estimate objects’ 3D properties such as depth, rotation and
velocity. The network architecture for CenterFusion is shown in Fig. 1.
We evaluate CenterFusion on the challenging nuScenes [2] dataset, where it
outperforms all previous camera-based object detection methods in the 3D
object detection benchmark. We also show that exploiting radar information
significantly improves velocity estimation for objects without using any
temporal information.
## 2 Related Work
### 2.1 Single-modality Methods
Monocular 3D object detection methods use a single camera to estimate 3D
bounding boxes for objects. Many studies have been reported, taking different
approaches to extract the depth information from monocular images. 3D RCNN
[11] uses Fast R-CNN [9] with an additional head and 3D projection. It also
uses a collection of CAD models to learn class-specific shape priors for
objects. Deep3DBox [17] regresses a set of 3D object properties using a
convolutional neural network first, then uses the geometric constraints of 2D
bounding boxes to produce a 3D bounding box for the object. CenterNet [34]
takes a different approach and uses a keypoint detection network to find
objects’ center point on the image. Other object properties such as 3D
dimension and location are obtained by regression using only the image
features at the object’s center point.
LiDARs have been widely used for 3D object detection and tracking in
autonomous driving applications in recent years. The majority of LiDAR-based
methods either use 3D voxels [12, 35] or 2D projections [13, 5, 29, 31] for
point cloud representation. Voxel-based methods are usually slow as a result
of the voxel grid’s high dimensionality, and projection-based methods might
suffer from large variances in object shapes and sizes depending on the
projection plane. PointRCNN [25] directly operates on raw point clouds and
generates 3D object proposals in a bottom-up manner using point cloud
segmentation. These proposals are refined in the second stage to generate the
final detection boxes.
Figure 2: Difference between actual and radial velocity. For target A,
velocity in the vehicle coordinate system and the radial velocity are the same
($v^{A}$). For target B on the other hand, radial velocity ($v_{r}$) as
reported by the radar is different from the actual velocity of the object
($v^{B}$) in the vehicle coordinate system.
### 2.2 Fusion-based Methods
Most existing sensor fusion methods focus on the LiDAR and camera fusion
problem. MV3D [4] extracts features from the front view and Bird’s Eye View
(BEV) representations of the LiDAR data, in addition to the RGB image. The
features obtained from the LiDAR’s BEV are then used to generate 3D object
proposals, and a deep fusion network is used to combine features from each
view and predict the object class and box orientations. PointFusion [28]
processes the image and LiDAR data using a CNN and a PointNet model
respectively, and then generate 3D object proposals using the extracted
features. Frustum PointNet [23] directly operates on the raw point clouds
obtained from an RGB-D camera and uses the RGB image and a 2D object detector
to localize objects in the point cloud.
Few studies have focused on fusing radars with other sensors for autonomous
driving applications. RadarNet [30] fuses radar and LiDAR data for 3D object
detection. It uses an early fusion mechanism to learn joint representations
from the two sensors, and a late-fusion mechanism to exploit radar’s radial
velocity evidence and improve the estimated object velocity. In [3], Chadwick
_et al_. project radar detections to the image plane and use them to boost the
object detection accuracy for distant objects. In [20] authors use radar
detections to generate 3D object proposals first, then project them to the
image plane to perform joint 2D object detection and depth estimation. CRF-Net
[22] also projects radar detections to the image plane, but represents them as
vertical lines where the pixel values correspond to the depth of each
detection point. The image data is then augmented with the radar information
and used in a convolutional network to perform 2D object detection.
Figure 3: Frustum association. An object detected using the image features
(left), generating the ROI frustum based on object’s 3D bounding box (middle),
and the BEV of the ROI frustum showing radar detections inside the frustum
(right). $\delta$ is used to increase the frustum size in the testing phase.
$\hat{d}$ is the ground truth depth in the training phase and the estimated
object depth in the testing phase. Figure 4: Expanding radar points to 3D
pillars (top image). Directly mapping the pillars to the image and replacing
with radar depth information results in poor association with objects’ center
and many overlapping depth values (middle image). Frustum association
accurately maps the radar detections to the center of objects and minimizes
overlapping (bottom image). Radar detections are only associated to objects
with a valid ground truth or detection box, and only if all or part of the
radar detection pillar is inside the box. Frustum association also prevents
associating radar detections caused by background objects such as buildings to
foreground objects, as seen in the case of pedestrians on the right hand side
of the image.
## 3 Preliminary
### 3.1 Radar Point Cloud
Radars are active sensors that transmit radio waves to sense the environment
and measure the reflected waves to determine the location and velocity of
objects. Automotive radars usually report the detected objects as 2D points in
BEV, providing the azimuth angle and radial distance to the object. For every
detection, the radar also reports the instantaneous velocity of the object in
the radial direction. This radial velocity does not necessarily match the
object’s actual velocity vector in it’s direction of movement. Fig. 2
illustrates the difference between the radial as reported by the radar, and
actual velocity of the object in the vehicle’s coordinate system.
We represent each radar detection as a 3D point in the egocentric coordinate
system, and parameterize it as $P=(x,y,z,v_{x},v_{y}$) where $(x,y,z)$ is the
position and $(v_{x},v_{y})$ is the reported radial velocity of the object in
the $x$ and $y$ directions. The radial velocity is compensated by the ego
vehicle’s motion. For every scene, we aggregate 3 sweeps of the radar point
cloud (detections within the past 0.25 seconds). The nuScenes dataset provides
the calibration parameters needed for mapping the radar point clouds from the
radar coordinates system to the egocentric and camera coordinate systems.
### 3.2 CenterNet
CenterNet [34] represents the state-of-the-art in 3D object detection using
single camera. It takes an image $I\in\mathbb{R}^{W\times H\times 3}$ as input
and generates a keypoint heatmap
$\hat{Y}\in[0,1]^{\frac{W}{R}\times\frac{H}{R}\times C}$ as output where $W$
and $H$ are the image width and height, $R$ is the downsampling ratio and $C$
is the number of object categories. A prediction of $\hat{Y}_{x,y,c}=1$ as the
output indicates a detected object of class $c$ centered at position $(x,y)$
on the image. The ground-truth heatmap
$Y\in[0,1]^{\frac{W}{R}\times\frac{H}{R}\times C}$ is generated from the
ground-truth 2D bounding boxes using a Gaussian kernel. For each bounding box
center point $p_{i}\in\mathcal{R}^{2}$ of class $c$ in the image, a Gaussian
heatmap is generated on $Y_{:,:,c}$. The final value of $Y$ for class $c$ at
position $q\in\mathcal{R}^{2}$ is defined as [34]:
$Y_{qc}=\max_{i}\exp(-\frac{(p_{i}-q)^{2}}{2\sigma_{i}^{2}})$ (1)
where $\sigma_{i}$ is a size-adaptive standard deviation, controlling the size
of the heatmap for every object based on its size. A fully convolutional
encode-decoder network is used to predict $\hat{Y}$.
To generate 3D bounding boxes, separate network heads are used to regress
object’s depth, dimensions and orientation directly from the detected center
points. Depth is calculated as an additional output channel
$\hat{D}\in[0,1]^{\frac{W}{R}\times\frac{H}{R}}$ after applying the inverse
sigmoidal transformation used in Eigen _et al_. [6] to the original depth
domain. The object dimensions are directly regressed to their absolute values
in meter as three output channels
$\hat{\Gamma}\in[0,1]^{\frac{W}{R}\times\frac{H}{R}\times 3}$. Orientation is
encoded as two bins with 4 scalars in each bin, following the orientation
representation in Mousavian _et al_. [18]. For each center point, a local
offset is also predicted to compensate for the discretization error caused by
the output strides in the backbone network [34].
Given the annotated objects ${p_{0},p_{1},...}$ in an image, the training
objective is defined as below based on the focal loss [15]:
$L_{k}=\frac{1}{N}\sum_{xyc}\begin{cases}(1-\hat{Y}_{xyc})^{\alpha}\log(\hat{Y}_{xyc})&\\!Y_{xyc}=1\vspace{2mm}\\\
(1-Y_{xyc})^{\beta}(\hat{Y}_{xyc})^{\alpha}\log(1-\hat{Y}_{xyc})&\\!\text{otherwise}\end{cases},$
where $N$ is the number of objects,
$Y\in[0,1]^{\frac{W}{R}\times\frac{H}{R}\times C}$ is the annotated objects’
ground-truth heatmap and $\alpha$ and $\beta$ are focal loss hyperparameters.
## 4 CenterFusion
In this section we present our approach to radar and camera sensor fusion for
3D object detection. The overall CenterFusion architecture is shown in Fig. 1.
We adopt [34] as our center-based object detection network to detect objects’
center points on the image plane, and regress to other object properties such
as 3D location, orientation and dimensions. We propose a middle-fusion
mechanism that associates radar detections to their corresponding object’s
center point and exploits both radar and image features to improve the
preliminary detections by re-estimating their depth, velocity, rotation and
attributes.
The key in our fusion mechanism is accurate association of radar detections to
objects. The center point object detection network generates a heat map for
every object category in the image. The peaks in the heat map represent
possible center points for objects, and the image features at those locations
are used to estimate other object properties. To exploit the radar information
in this setting, radar-based features need to be mapped to the center of their
corresponding object on the image, which requires an accurate association
between the radar detections and objects in the scene.
### 4.1 Center Point Detection
We adopt the CenterNet [34] detection network for generating preliminary
detections on the image. The image features are first extracted using a fully
convolutional encoder-decoder backbone network. We follow CenterNet [34] and
use a modified version of the Deep Layer Aggregation (DLA) network [32] as the
backbone. The extracted image features are then used to predict object center
points on the image, as well as the object 2D size (width and height), center
offset, 3D dimensions, depth and rotation. These values are predicted by the
primary regression heads as shown in Fig. 1. Each primary regression head
consists of a $3\times 3$ convolution layer with 256 channels and a $1\times
1$ convolutional layer to generate the desired output. This provides an
accurate 2D bounding box as well as a preliminary 3D bounding box for every
detected object in the scene.
### 4.2 Radar Association
The center point detection network only uses the image features at the center
of each object to regress to all other object properties. To fully exploit
radar data in this process, we first need to associate the radar detections to
their corresponding object on the image plane. To accomplish this, a naïve
approach would be mapping each radar detection point to the image plane and
associating it to an object if the point is mapped inside the 2D bounding box
of that object. This is not a very robust solution, as there is not a one-to-
one mapping between radar detections and objects in the image; Many objects in
the scene generate multiple radar detections, and there are also radar
detections that do not correspond to any object. Additionally, because the $z$
dimension of the radar detection is not accurate (or does not exist at all),
the mapped radar detection might end up outside the 2D bounding box of its
corresponding object. Finally, radar detections obtained from occluded objects
would map to the same general area in the image, which makes differentiating
them in the 2D image plane difficult, if possible at all.
Frustum Association Mechanism: We develop a frustum association method that
uses the object’s 2D bounding box as well as its estimated depth and size to
create a 3D Region of Interest (RoI) frustum for the object. Having an
accurate 2D bounding box for an object, we create a frustum for that object as
shown in Fig. 3. This significantly narrows down the radar detections that
need to be checked for association, as any point outside this frustum can be
ignored. We then use the estimated object depth, dimension and rotation to
create a RoI around the object, to further filter out radar detections that
are not associated with this object. If there are multiple radar detections
inside this RoI, we take the closest point as the radar detection
corresponding to this object.
In the training phase, we use the object’s 3D ground truth bounding box to
create a tight RoI frustum and associate radar detections to the object. In
the test phase, the RoI frustum is calculated using the object’s estimated 3D
bounding box as explained before. In this case, we use a parameter $\delta$ to
control the size of the RoI frustum as shown in Fig. 3. This is to account for
inaccuracy in the estimated depth values, as the depth of the object at this
stage is solely determined using the image-based features. Enlarging the
frustum using this parameter increases the chance of including the
corresponding radar detections inside the frustum even if the estimated depth
is slightly off. The value of $\delta$ should be carefully selected, as a
large RoI frustum can include radar detections of nearby objects.
The RoI frustum approach makes associating overlapping objects effortless, as
objects are separated in the 3D space and would have separate RoI frustums. It
also eliminates the multi-detection association problem, as only the closest
radar detection inside the RoI frustum is associated to the object. It does
not, however, help with the inaccurate $z$ dimension problem, as radar
detections might be outside the ROI frustum of their corresponding object due
to their inaccurate height information.
Pillar Expansion: To address the inaccurate height information problem, we
introduce a radar point cloud preprocessing step called pillar expansion,
where each radar point is expanded to a fixed-size pillar, as illustrated in
Fig. 4. Pillars create a better representation for the physical objects
detected by the radar, as these detections are now associated with a dimension
in the 3D space. Having this new representation, we simply consider a radar
detection to be inside a frustum if all or part of its corresponding pillar is
inside the frustum, as shown in Fig. 1.
### 4.3 Radar Feature Extraction
After associating radar detections to their corresponding objects, we use the
depth and velocity of the radar detections to create complementary features
for the image. Particularly, for every radar detection associated to an
object, we generate three heat map channels centered at and inside the
object’s 2D bounding box, as shown in Fig. 4. The width and height of the
heatmaps are proportional to the object’s 2D bounding box, and are controlled
by a parameter $\alpha$. The heatmap values are the normalized object depth
($d$) and also the $x$ and $y$ components of the radial velocity ($v_{x}$ and
$v_{y}$) in the egocentric coordinate system:
$F_{x,y,i}^{j}=\frac{1}{M_{i}}\begin{cases}f_{i}&\\!|x-c_{x}^{j}|\leq\alpha
w^{j}\hskip 8.0pt\text{and}\\\ &|y-c_{y}^{i}|\leq\alpha h^{j}\\\
0&\\!\text{otherwise}\end{cases},$
where $i\in{1,2,3}$ is the feature map channel, $M_{i}$ is a normalizing
factor, $f_{i}$ is the feature value ($d$, $v_{x}$ or $v_{y}$), $c_{x}^{j}$
and $c_{y}^{j}$ are the $x$ and $y$ coordinates of the $j$th object’s center
point on the image and $w^{j}$ and $h^{j}$ are the width and height of the
$j$th object’s 2D bounding box. If two objects have overlapping heatmap areas,
the one with a smaller depth value dominates, as only the closest object is
fully visible in the image.
The generated heat maps are then concatenated to the image features as extra
channels. These features are used as inputs to the secondary regression heads
to recalculate the object’s depth and rotation, as well as velocity and
attributes. The velocity regression head estimates the $x$ and $y$ components
of the object’s actual velocity in the vehicle coordinate system. The
attribute regression head estimates different attributes for different object
classes, such as moving or parked for the Car class and standing or sitting
for the Pedestrian class. The secondary regression heads consist of three
convolutional layers with $3\times 3$ kernels followed by a $1\times 1$
convolutional layer to generate the desired output. The extra convolutional
layers compared to the primary regression heads help with learning higher
level features from the radar feature maps. The last step is decoding the
regression head results into 3D bounding boxes. The box decoder block uses the
estimated depth, velocity, rotation, and attributes from the secondary
regression heads, and takes the other object properties from the primary
heads.
## 5 Implementation Details
We use the pre-trained CenterNet [34] network with the DLA [32] backbone as
our object detection network. DLA uses iterative deep aggregation layers to
increase the resolution of feature maps. CenterNet compares its performance
using different backbone architectures, with the Hourglass network [21]
performing better than others. We choose to use the DLA network as it takes
significantly less time to train while providing a reasonable performance.
We directly use the released CenterNet model that is trained for 140 epochs on
the nuScenes dataset. This model by default does not provide velocity and
attribute predictions. We train the velocity and attribute heads for 30
epochs, and use the resulting model as our baseline. The secondary regression
heads in our network are added on top of the CenterNet backbone network, and
are trained using the image and radar features for an additional 60 epochs
with a batch size of 26 on two Nvidia P5000 GPUs.
During both training and testing, we reduce the image resolution from the
original 1600$\times$900 pixels to 800$\times$450 pixels. Data augmentation is
used during training, with random right-left flipping (with a probability of
0.5) and random shifting (from 0 to 20 percent of image size). The same
augmentations are also applied to the radar point cloud in reference to the
camera coordinate system. We do not apply any scaling augmentation as it
changes the 3D measurements. At testing time, we only use flip test
augmentation where an image and its flipped version are fed into the network
and the average of the network outputs is used for decoding the 3D bounding
boxes. We do not use the multi-scale test augmentation as used by CenterNet.
The pillar size is set to $[0.2,0.2,1.5]$ meters in the $[x,y,z]$ directions
and $\delta$ is set to increase the length of the RoI frustum by 20% in the
radial direction at test time.
We use the L1 loss for most of the regression heads, with the exception of the
center point heat map head which uses the focal loss and the attributes
regression head that uses the Binary Cross Entropy (BCE) loss.
Table 1: Performance comparison for 3D object detection on nuScenes dataset. mATE, mASE, mAOE, mAVE and mAAE stand for average translation, scale, orientation, velocity and attribute errors respectively. $\uparrow$ indicates that higher is better and $\downarrow$ indicates that lower is better. ”C”, ”R” and ”L” specify camera, radar and LIDAR modalities respectively. | | Modality | | | Error $\downarrow$
---|---|---|---|---|---
Method | Dataset | C | R | L | NDS $\uparrow$ | mAP $\uparrow$ | mATE | mASE | mAOE | mAVE | mAAE
InfoFocus [27] | test | | | ✓ | 0.395 | 0.395 | 0.363 | 0.265 | 1.132 | 1.000 | 0.395
OFT [24] | test | ✓ | | | 0.212 | 0.126 | 0.820 | 0.360 | 0.850 | 1.730 | 0.480
MonoDIS [26] | test | ✓ | | | 0.384 | 0.304 | 0.738 | 0.263 | 0.546 | 1.533 | 0.134
CenterNet (HGLS) [34] | test | ✓ | | | 0.400 | 0.338 | 0.658 | 0.255 | 0.629 | 1.629 | 0.142
Ours (DLA) | test | ✓ | ✓ | | 0.449 | 0.326 | 0.631 | 0.261 | 0.516 | 0.614 | 0.115
CenterNet (DLA) [34] | val | ✓ | | | 0.328 | 0.306 | 0.716 | 0.264 | 0.609 | 1.426 | 0.658
Ours (DLA) | val | ✓ | ✓ | | 0.453 | 0.332 | 0.649 | 0.263 | 0.535 | 0.540 | 0.142
Table 2: Per-class performance comparison for 3D object detection on nuScenes dataset. | | Modality | mAP $\uparrow$
---|---|---|---
Method | Dataset | C | R | L | Car | Truck | Bus | Trailer | Const. | Pedest. | Motor. | Bicycle | Traff. | Barrier
InfoFocus [27] | test | | | ✓ | 0.779 | 0.314 | 0.448 | 0.373 | 0.107 | 0.634 | 0.290 | 0.061 | 0.465 | 0.478
MonoDIS [26] | test | ✓ | | | 0.478 | 0.220 | 0.188 | 0.176 | 0.074 | 0.370 | 0.290 | 0.245 | 0.487 | 0.511
CenterNet (HGLS) [34] | test | ✓ | | | 0.536 | 0.270 | 0.248 | 0.251 | 0.086 | 0.375 | 0.291 | 0.207 | 0.583 | 0.533
Ours (DLA) | test | ✓ | ✓ | | 0.509 | 0.258 | 0.234 | 0.235 | 0.077 | 0.370 | 0.314 | 0.201 | 0.575 | 0.484
CenterNet (DLA) [34] | val | ✓ | | | 0.484 | 0.231 | 0.340 | 0.131 | 0.035 | 0.377 | 0.249 | 0.234 | 0.550 | 0.456
Ours (DLA) | val | ✓ | ✓ | | 0.524 | 0.265 | 0.362 | 0.154 | 0.055 | 0.389 | 0.305 | 0.229 | 0.563 | 0.470
## 6 Results
We compare our radar and camera fusion network with the state-of-the-art
camera-based models on the nuScenes benchmark, and also a LIDAR based method.
Table 1 shows the results on both test and validation splits of the nuScenes
dataset. We compare with OFT [24], MonoDIS [26] and CenterNet [34] which are
camera-based 3D object detection networks, as well as InfoFocus [27] which is
a LIDAR-based method. As seen in Table 1, CenterFusion outperforms all other
methods in the nuScenes NDS score, which is a weighted sum of the mAP and the
error metrics. On the test dataset, CenterFusion shows a 12.25% and 16.9%
relative increase in the NDS score compared to CenterNet and MonoDIS
respectively. The LIDAR-based method InfoFocus shows a better performance in
the mAP score compared to other methods, but is significantly outperformed by
CenterFusion in the orientation, velocity and attribute error metrics. While
CenterNet with the Hourglass [21] backbone network results in a better mAP
score compared to CenterFusion (1.2% difference) on the test split, the
results on the validation split show that CenterFusion outperforms CenterNet
by 2.6% when both networks use the same DLA [32] backbone. The validation set
results also show CenterFusion improving CenterNet in all the other metrics.
CenterFusion shows an absolute gain of 38.1% and 62.1% relative increase in
the NDS and velocity error metrics compared to CenterNet, which demonstrates
the effectiveness of using radar features.
Table 2 compares the per-class mAP results for both test and validation
splits. While CenterNet with an Hourglass backbone has a higher mAP than
CenterFusion for most classes in the test set, it is outperformed by
CenterFusion on the validation set where the DLA backbone is used for both
methods. The most improved classes on the validation set are the motorcycle
and car with 5.6% and 4.0% absolute mAP increase respectively.
Fig. 5 demonstrates the 3D object detection results in both camera and BEV. It
shows the detection results from CenterFusion (row 1 & 2) and CenterNet (row 3
& 4) for 4 different scenes. The radar point clouds are also shown in the
CenterFusion BEV results. Compared to CenterNet, the results from CenterFusion
show a better fit for 3D boxes in most cases, especially objects at a larger
distance, such as the far vehicle in the second scene. Additionally, the
velocity vectors estimated by CenterFusion show a significant improvement
compared to the CenterNet results, as seen in the second and third scenes.
Table 3: Overall ablation study on nuScenes validation set. Improvement percentages in each row are relative to the baseline method. (PE: Pillar Expansion, FA: Frustum Association, FT: Flip Test) Method | Cam | Rad | PE | FA | FT | NDS $\uparrow$ | mAP $\uparrow$ | mATE $\downarrow$ | mASE $\downarrow$ | mAOE $\downarrow$ | mAVE $\downarrow$ | mAAE $\downarrow$
---|---|---|---|---|---|---|---|---|---|---|---|---
Baseline | ✓ | - | - | - | - | 0.328 | 0.306 | 0.716 | 0.264 | 0.609 | 1.426 | 0.658
Ours | ✓ | ✓ | ✓ | - | - | +15.4% | +1.0% | -2.0% | +1.1% | -4.4% | -13.1% | -68.6%
Ours | ✓ | ✓ | - | ✓ | - | +25.9% | +2.0% | -2.8% | +1.0% | -7.4% | -48.1% | -75.9%
Ours | ✓ | ✓ | ✓ | ✓ | - | +34.5% | +4.3% | -5.3% | +1.1% | -10.0% | -61.9% | -78.0%
Ours | ✓ | ✓ | ✓ | ✓ | ✓ | +37.8% | +8.4% | -9.4% | -0.5% | -11.6% | -62.0% | -78.3%
Table 4: Class-based ablation study results on nuScenes validation set. Method | Cam | Rad | PE | FA | FT | Car | Truck | Bus | Trailer | Const. | Pedest. | Motor. | Bicycle | Traff. | Barrier
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Baseline | ✓ | - | - | - | - | 48.4 | 23.1 | 34.0 | 13.1 | 3.5 | 37.7 | 24.9 | 23.4 | 55.0 | 45.6
Ours | ✓ | ✓ | ✓ | - | - | +0.6 | +0.7 | -2.1 | +0.9 | +0.6 | +0.9 | +1.9 | -2.5 | +0.1 | +0.8
Ours | ✓ | ✓ | - | ✓ | - | +1.0 | +1.0 | -2.1 | +0.9 | +0.9 | 0.0 | +2.1 | -1.9 | +0.2 | +0.8
Ours | ✓ | ✓ | ✓ | ✓ | - | +2.8 | +2.1 | -1.2 | +1.4 | +1.1 | +0.1 | +3.8 | -1.1 | +0.4 | +0.8
Ours | ✓ | ✓ | ✓ | ✓ | ✓ | +4.1 | +3.4 | +2.7 | +1.8 | +1.8 | +1.2 | +5.5 | -0.7 | +1.3 | +1.5
Figure 5: Qualitative results from CenterFusion (row 1 & 2) and CenterNet (row
3 & 4) in camera view and BEV. In the BEV plots, detection boxes are shown in
cyan and ground truth boxes in red. The radar point cloud is shown in green.
Red and blue arrows on objects show the ground truth and predicted velocity
vectors respectively.
## 7 Ablation Study
We validate the effectiveness of our fusion approach by conducting an ablation
study on the nuScenes validation set. We use the CenterNet model as our
baseline, and study the effectiveness of the pillar expansion, frustum
association and flip testing on the detection results. Table 3 shows the
overall detection results of the ablation study.
In the first experiment, we only apply pillar expansion to the radar point
clouds, and map the 3D pillars to the image plane and obtain their equivalent
2D bounding boxes. These boxes are then filled with the depth and velocity
values of their corresponding radar detections and used as the radar feature
maps, as shown in Fig. 4. According to Table 3, this simple association method
results in a 15.4% relative improvement on the NDS score and 1.0% absolute
improvement on the mAP compared to the baseline.
In the next experiment we only use the frustum association method by directly
applying it on the radar point clouds without converting them to pillars
first. This improves the NDS score by 25.9% relatively and mAP by 2.0%.
Applying both pillar expansion and frustum association results in a relative
35.5% and absolute 4.3% improvement on the NDS and mAP scores respectively.
Flip testing adds another 3.3% improvement on the NDS score and 3.9% on the
mAP, resulting in a total of 37.8% and 8.4% improvement on NDS and mAP
compared to the baseline method.
Table 4 shows the per-class contribution of each step on the mAP. According to
the results, both pillar expansion and frustum association steps have
contributed to the improvement of mAP in most object classes. The only class
that has not improved from the baseline is the bicycle class, in which the
CenterNet mAP score is better than CenterFusion by 0.5%.
## 8 Conclusion
In summary, we proposed a new radar and camera fusion algorithm called
CenterFusion, to exploit radar information for robust 3D object detection.
CenterFusion accurately associates radar detections to objects on the image
using a frustum-based association method, and creates radar-based feature maps
to complement the image features in a middle-fusion approach. Our frustum
association method uses preliminary detection results to generate a RoI
frustum around objects in 3D space, and maps the radar detection to the center
of objects on the image. We also used a pillar expansion method to compensate
for the inaccuracy in radar detections’ height information, by converting
radar points to fixed-size pillars in the 3D space. We evaluated our proposed
method on the challenging nuScenes 3D detection benchmark, where CenterFusion
outperformed the state-of-the-art camera-based object detection methods.
## References
* [1] A. Asvadi, P. Girão, P. Peixoto, and U. Nunes. 3d object tracking using rgb and lidar data. In 2016 IEEE 19th International Conference on Intelligent Transportation Systems (ITSC), pages 1255–1260, 2016.
* [2] Holger Caesar, Varun Bankiti, Alex H. Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. arXiv preprint arXiv:1903.11027, 2019.
* [3] Simon Chadwick, Will Maddern, and Paul Newman. Distant Vehicle Detection Using Radar and Vision. arXiv:1901.10951 [cs], May 2019.
* [4] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. Multi-view 3d object detection network for autonomous driving. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017.
* [5] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. Multi-View 3D Object Detection Network for Autonomous Driving. arXiv:1611.07759 [cs], June 2017.
* [6] David Eigen, Christian Puhrsch, and Rob Fergus. Depth map prediction from a single image using a multi-scale deep network. In Advances in neural information processing systems, pages 2366–2374, 2014.
* [7] Yongkun Fang, Huijing Zhao, Hongbin Zha, Xijun Zhao, and Wen Yao. Camera and lidar fusion for on-road vehicle tracking with reinforcement learning. In 2019 IEEE Intelligent Vehicles Symposium (IV), pages 1723–1730. IEEE, 2019.
* [8] Di Feng, Christian Haase-Schuetz, Lars Rosenbaum, Heinz Hertlein, Claudius Glaeser, Fabian Timm, Werner Wiesbeck, and Klaus Dietmayer. Deep multi-modal object detection and semantic segmentation for autonomous driving: Datasets, methods, and challenges. arXiv:1902.07830 [cs], Feb. 2020.
* [9] Ross Girshick. Fast r-cnn. 2015 IEEE International Conference on Computer Vision (ICCV), Dec 2015.
* [10] Jason Ku, Melissa Mozifian, Jungwook Lee, Ali Harakeh, and Steven L. Waslander. Joint 3d proposal generation and object detection from view aggregation. 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Oct 2018.
* [11] Abhijit Kundu, Yin Li, and James M. Rehg. 3D-RCNN: Instance-Level 3D Object Reconstruction via Render-and-Compare. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3559–3568, Salt Lake City, UT, USA, June 2018\. IEEE.
* [12] Bo Li. 3D Fully Convolutional Network for Vehicle Detection in Point Cloud. arXiv:1611.08069 [cs], Jan. 2017.
* [13] Bo Li, Tianlei Zhang, and Tian Xia. Vehicle Detection from 3D Lidar Using Fully Convolutional Network. arXiv:1608.07916 [cs], Aug. 2016.
* [14] Ming Liang, Bin Yang, Yun Chen, Rui Hu, and Raquel Urtasun. Multi-task multi-sensor fusion for 3d object detection. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
* [15] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár. Focal Loss for Dense Object Detection. arXiv:1708.02002 [cs], Feb. 2018.
* [16] Gregory P. Meyer, Jake Charland, Darshan Hegde, Ankit Laddha, and Carlos Vallespi-Gonzalez. Sensor fusion for joint 3d object detection and semantic segmentation. 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), Jun 2019.
* [17] Arsalan Mousavian, Dragomir Anguelov, John Flynn, and Jana Kosecka. 3D Bounding Box Estimation Using Deep Learning and Geometry. arXiv:1612.00496 [cs], Apr. 2017.
* [18] Arsalan Mousavian, Dragomir Anguelov, John Flynn, and Jana Kosecka. 3d bounding box estimation using deep learning and geometry. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7074–7082, 2017.
* [19] Ramin Nabati and Hairong Qi. RRPN: Radar region proposal network for object detection in autonomous vehicles. 2019 IEEE International Conference on Image Processing (ICIP), Sep 2019.
* [20] Ramin Nabati and Hairong Qi. Radar-Camera Sensor Fusion for Joint Object Detection and Distance Estimation in Autonomous Vehicles. arXiv:2009.08428 [cs], Sept. 2020.
* [21] Alejandro Newell, Kaiyu Yang, and Jia Deng. Stacked Hourglass Networks for Human Pose Estimation. arXiv:1603.06937 [cs], July 2016.
* [22] Felix Nobis, Maximilian Geisslinger, Markus Weber, Johannes Betz, and Markus Lienkamp. A Deep Learning-based Radar and Camera Sensor Fusion Architecture for Object Detection. In 2019 Sensor Data Fusion: Trends, Solutions, Applications (SDF), pages 1–7, Oct. 2019.
* [23] Charles R. Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J. Guibas. Frustum PointNets for 3D Object Detection from RGB-D Data. arXiv:1711.08488 [cs], Apr. 2018.
* [24] Thomas Roddick, Alex Kendall, and Roberto Cipolla. Orthographic feature transform for monocular 3d object detection. arXiv preprint arXiv:1811.08188, 2018.
* [25] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. PointRCNN: 3D Object Proposal Generation and Detection from Point Cloud. arXiv:1812.04244 [cs], May 2019.
* [26] Andrea Simonelli, Samuel Rota Rota Bulò, Lorenzo Porzi, Manuel López-Antequera, and Peter Kontschieder. Disentangling Monocular 3D Object Detection. arXiv:1905.12365 [cs], May 2019.
* [27] Jun Wang, Shiyi Lan, Mingfei Gao, and Larry S Davis. Infofocus: 3d object detection for autonomous driving with dynamic information modeling. arXiv preprint arXiv:2007.08556, 2020.
* [28] Danfei Xu, Dragomir Anguelov, and Ashesh Jain. PointFusion: Deep Sensor Fusion for 3D Bounding Box Estimation. arXiv:1711.10871 [cs], Aug. 2018.
* [29] Yan Yan, Yuxing Mao, and Bo Li. SECOND: Sparsely Embedded Convolutional Detection. Sensors, 18(10):3337, Oct. 2018.
* [30] Bin Yang, Runsheng Guo, Ming Liang, Sergio Casas, and Raquel Urtasun. RadarNet: Exploiting Radar for Robust Perception of Dynamic Objects. arXiv:2007.14366 [cs], July 2020.
* [31] Bin Yang, Wenjie Luo, and Raquel Urtasun. PIXOR: Real-time 3D Object Detection from Point Clouds. arXiv:1902.06326 [cs], Mar. 2019.
* [32] Fisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep Layer Aggregation. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2403–2412, Salt Lake City, UT, June 2018. IEEE.
* [33] R. Zhang, S. A. Candra, K. Vetter, and A. Zakhor. Sensor fusion for semantic segmentation of urban scenes. In 2015 IEEE International Conference on Robotics and Automation (ICRA), pages 1850–1857, 2015.
* [34] Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Objects as points. arXiv preprint arXiv:1904.07850, 2019.
* [35] Yin Zhou and Oncel Tuzel. VoxelNet: End-to-End Learning for Point Cloud Based 3D Object Detection. arXiv:1711.06396 [cs], Nov. 2017.
|
# Evaluating Gradient Inversion Attacks and Defenses in Federated Learning
Yangsibo Huang
Princeton University
Princeton, NJ 08540
<EMAIL_ADDRESS>
&Samyak Gupta
Princeton University
Princeton, NJ 08540
<EMAIL_ADDRESS>
&Zhao Song
Adobe Research
San Jose, CA 95110
<EMAIL_ADDRESS>
&Kai Li
Princeton University
Princeton, NJ 08540
<EMAIL_ADDRESS>
&Sanjeev Arora
Princeton University
Princeton, NJ 08540
<EMAIL_ADDRESS>
###### Abstract
Gradient inversion attack (or input recovery from gradient) is an emerging
threat to the security and privacy preservation of Federated learning, whereby
malicious eavesdroppers or participants in the protocol can recover
(partially) the clients’ private data. This paper evaluates existing attacks
and defenses. We find that some attacks make strong assumptions about the
setup. Relaxing such assumptions can substantially weaken these attacks. We
then evaluate the benefits of three proposed defense mechanisms against
gradient inversion attacks. We show the trade-offs of privacy leakage and data
utility of these defense methods, and find that combining them in an
appropriate manner makes the attack less effective, even under the original
strong assumptions. We also estimate the computation cost of end-to-end
recovery of a single image under each evaluated defense. Our findings suggest
that the state-of-the-art attacks can currently be defended against with minor
data utility loss, as summarized in a list of potential strategies. Our code
is available at: https://github.com/Princeton-SysML/GradAttack.
## 1 Introduction
Federated learning (McMahan et al., 2016; Kairouz et al., 2021) is a framework
that allows multiple clients in a distributed environment to collaboratively
train a neural network model at a central server, without moving their data to
the central server.
At every training step, each client computes a model update —i.e., gradient—
on its local data using the latest copy of the global model, and then sends
the gradient to the central server. The server aggregates these updates
(typically by averaging) to construct a global model, and then sends the new
model parameters to all clients. By allowing clients to participate in
training without directly sharing their data, such protocols align better with
data privacy regulations such as Health Insurance Portability and
Accountability Act (HIPPA) (Act, 1996), California Consumer Privacy Act (CCPA)
(Legislature, 2018), and General Data Protection Regulation (Commission,
2018).
While sharing gradients was thought to leak little information about the
client’s private data, recent papers (Zhu et al., 2019; Zhao et al., 2020;
Geiping et al., 2020; Yin et al., 2021) developed a “gradient inversion
attack” by which an attacker eavesdropping on a client’s communications with
the server can begin to reconstruct the client’s private data. The attacker
can also be a malicious participant in the Federated Learning scheme,
including a honest-but-curious server who wishes to reconstruct private data
of clients, or a honest-but-curious client who wishes to reconstruct private
data of other clients. These attacks have been shown to work with batch sizes
only up to $100$ but even so they have created doubts about the level of
privacy ensured in Federated Learning. The current paper seeks to evaluate the
risks and suggest ways to minimize them.
Several defenses against gradient inversion attacks have been proposed. These
include perturbing gradients (Zhu et al., 2019; Wei et al., 2020) and using
transformation for training data that clients can apply on the fly (Zhang et
al., 2018a; Huang et al., 2020). More traditional cryptographic ideas
including secure aggregation (Bonawitz et al., 2016) or homomorphic encryption
(Phong et al., 2018) for the gradients can also be used and presumably stop
any eavesdropping attacks completely. They will not be studied here due to
their special setups and overhead.
We are not aware of a prior systematic evaluation of the level of risk arising
from current attacks and the level of security provided by various defenses,
as well as the trade-off (if any) between test accuracy, computation overhead,
and privacy risks.
The paper makes two main contributions. First, we draw attention to two strong
assumptions that a current gradient inversion attack (Geiping et al., 2020)
implicitly makes. We show that by nullifying these assumptions, the
performance of the attack drops significantly and can only work for low-
resolution images. The findings are explored in Section 3 and already imply
some more secure configurations in Federated Learning (Section 6).
Second, we summarize various defenses (Section 4) and systematically evaluate
(Section 5) some of their performance of defending against a state-of-the-art
gradient inversion attack, and present their data utility and privacy leakage
trade-offs. We estimate the computation cost of end-to-end recovery of a
single image under each evaluated defense. We also experimentally demonstrate
the feasibility and effectiveness of combined defenses. Our findings are
summarized as strategies to further improve Federated Learning’s security
against gradient inversion attacks (Section 6).
In Appendix B, we provide theoretical insights for mechanism of each evaluated
defense.
## 2 Gradient Inversion Attacks
Previous studies have shown the feasibility of recovering input from gradient
(i.e. gradient inversion) for image classification tasks, by formulating it as
an optimization problem: given a neural network with parameters $\theta$, and
the gradient $\nabla_{\theta}\mathcal{L}_{\theta}(x^{*},y^{*})$ computed with
a private data batch $(x^{*},y^{*})\in\mathbb{R}^{b\times
d}\times\mathbb{R}^{b}$ ($b,d$ being the batch size, image size), the attacker
tries to recover $x\in\mathbb{R}^{b\times d}$, an approximation of $x^{*}$:
$\arg\min_{x}\mathcal{L}_{\rm
grad}(x;\theta,\nabla_{\theta}\mathcal{L}_{\theta}(x^{*},y^{*}))+\alpha\mathcal{R}_{\rm
aux}(x)$ (1)
The optimization goal consists of two parts: $\mathcal{L}_{\rm
grad}(x;\theta,\nabla_{\theta}\mathcal{L}_{\theta}(x^{*},y^{*}))$ enforces
matching of the gradient of recovered batch $x$ with the provided gradients
$\mathcal{L}_{\theta}(x^{*},y^{*})$, and $\mathcal{R}_{\rm aux}(x)$
regularizes the recovered image based on image prior(s).
(Phong et al., 2017) brings theoretical insights on this task by proving that
such reconstruction is possible with a single-layer neural network. (Zhu et
al., 2019) is the first to show that accurate pixel-level reconstruction is
practical for a maximum batch size of 8. Their formulation uses
$\ell_{2}$-distance as $\mathcal{L}_{\rm grad}(\cdot,\cdot)$ but no
regularization term $\mathcal{R}_{\rm aux}(x)$. The approach works for low-
resolution CIFAR datasets (Krizhevsky et al., 2009), with simple neural
networks with sigmoid activations, but cannot scale up to high-resolution
images, or larger models with ReLU activations. A follow-up (Zhao et al.,
2020) proposes a simple approach to extract the ground-truth labels from the
gradient, which improves the attack but still cannot overcome its limitations.
With a careful choice of $\mathcal{L}_{\rm grad}$ and $\mathcal{R}_{\rm
aux}(x)$, (Geiping et al., 2020) substantially improves the attack and
succeeds in recovering a single ImageNet (Deng et al., 2009) image from
gradient: their approach uses cosine distance as $\mathcal{L}_{\rm grad}$, and
the total variation as $\mathcal{R}_{\rm aux}(x)$. Their approach is able to
reconstruct low-resolution images with a maximum batch size of 100 or a single
high-resolution image. Based on (Geiping et al., 2020), (Wei et al., 2020)
analyzes how different configurations in the training may affect attack
effectiveness.
A more recent work (Yin et al., 2021) further improves the attack on high-
resolution images, by introducing to $\mathcal{R}_{\rm aux}(x)$ a new image
prior term based on batch normalization (Ioffe and Szegedy, 2015) statistics,
and a regularization term which enforces consistency across multiple attack
trials.
An orthogonal line of work (Zhu and Blaschko, 2021) proposes to formulate the
gradient inversion attack as a closed-form recursive procedure, instead of an
optimization problem. However, their implementation can recover only low-
resolution images under the setting where batch size = 1.
## 3 Strong Assumptions Made by SOTA Attacks
### 3.1 The state-of-the-art attacks
Two recent attacks (Geiping et al., 2020; Yin et al., 2021) achieve best
recovery results. Our analysis focuses on the former as the implementation of
the latter is not available at the time of writing this paper. We plan to
include the analysis for the latter attack in the final version of this paper
if its implementation becomes available.
(Geiping et al., 2020)’s attack optimizes the following objective function:
$\arg\min_{x}1-\frac{\left\langle\nabla_{\theta}\mathcal{L}_{\theta}(x,y),\nabla_{\theta}\mathcal{L}_{\theta}\left(x^{*},y^{*}\right)\right\rangle}{\left\|\nabla_{\theta}\mathcal{L}_{\theta}(x,y)\left|\left\|\mid\nabla_{\theta}\mathcal{L}_{\theta}\left(x^{*},y^{*}\right)\right\|\right.\right.}+\alpha_{\rm
TV}\mathcal{R}_{\rm TV}(x)$ (2)
where $\langle\cdot,\cdot\rangle$ is the inner-product between vectors, and
$\mathcal{R}_{\rm TV}(\cdot)$ is the total variation of images.
We notice that Geiping et al. has made two strong assumptions (Section 3.2).
Changing setups to invalidate those assumptions will substantially weaken the
attacks (Section 3.3). We also summarize whether other attacks have made
similar assumptions in Table 1.
### 3.2 Strong assumptions
We find that previous gradient inversion attacks have made different
assumptions about whether the attacker knows Batch normalization statistics or
private labels, as shown in Table 3.2. Note that (Geiping et al., 2020)’s
attack makes both strong assumptions.
#### Assumption 1: Knowing BatchNorm statistics.
Batch normalization (BatchNorm) (Ioffe and Szegedy, 2015) is a technique for
training neural networks that normalizes the inputs to a layer for every mini-
batch. It behaves differently during training and evaluation. Assume the model
has $L$ batch normalization layers. Given $x^{*}$, a batch of input images, we
use $x^{*}_{l}$ to denote the input features to the $l$-th BatchNorm layer,
where $l\in[L]$. During training, the $l$-th BatchNorm layer normalizes
$x^{*}_{l}$ based on the batch’s mean ${\rm mean}(x^{*}_{l})$ and variance
${\rm var}(x^{*}_{l})$, and keeps a running estimate of mean and variance of
all training data points, denoted by $\mu_{l}$ and $\sigma^{2}_{l}$. During
inference, $\\{\mu_{l}\\}_{l=1}^{L}$ and $\\{\sigma^{2}_{l}\\}_{l=1}^{L}$ are
used to normalize test images. In the following descriptions, we leave out
$\\{\cdot\\}_{l=1}^{L}$ for simplicity (i.e. use $\mu,\sigma^{2}$ to denote
$\\{\mu_{l}\\}_{l=1}^{L},\\{\sigma^{2}_{l}\\}_{l=1}^{L}$, and ${\rm
mean}(x^{*})$, ${\rm var}(x^{*})$ to denote $\\{{\rm
mean}(x^{*}_{l})\\}_{l=1}^{L}$, $\\{{\rm var}(x^{*}_{l})\\}_{l=1}^{L}$).
We notice that (Geiping et al., 2020)’s implementation111The official
implementation of (Geiping et al., 2020):
https://github.com/JonasGeiping/invertinggradients. assumes that BatchNorm
statistics of the private batch, i.e., ${\rm mean}(x^{*})$, ${\rm
var}(x^{*})$, are jointly provided with the gradient. Knowing BatchNorm
statistics would enable the attacker to apply the same batch normalization
used by the private batch on his recovered batch, to achieve a better
reconstruction. This implicitly increases the power of the attacker, as
sharing private BatchNorm statistics are not necessary in Federated learning
(Andreux et al., 2020; Li et al., 2021).
Note that this assumption may be realistic in some settings: 1) the neural
network is shallow, thus does not require using BatchNorm layers, or 2) the
neural network is deep, but adapts approaches that normalize batch inputs with
a fixed mean and variance (as alternative to BatchNorm), e.g. Fixup
initialization (Zhang et al., 2019).
#### Assumption 2: Knowing or able to infer private labels.
Private labels are not intended to be shared in Federated learning, but
knowing them would improve the attack. (Zhao et al., 2020) finds that label
information of a single private image can be inferred from the gradient (see
Section 3.3 for details). Based on this, (Geiping et al., 2020) assumes the
attacker knows private labels (see remark at the end of Section 4 in their
paper). However, this assumption may not hold true when multiple images in a
batch share the same label, as we will show in the next section.
Assumptions | (Zhu et al., 2019) | (Zhao et al., 2020) | (Geiping et al., 2020) | (Yin et al., 2021)
---|---|---|---|---
Knowing BN statistics | N/A† | N/A† | Yes | Yes∗
Knowing private labels | No | No‡ | Yes | No‡
Table 1: Assumptions of gradient inversion attacks. †: its evaluation uses a
simple model without a BatchNorm layer; ‡: it proposes a method to infer
private labels, which works when images in a batch have unique labels (see
Section 3.3); ∗: although the paper discusses a setting where BatchNorm
statistics are unknown, its main results assume knowing BatchNorm statistics.
### 3.3 Re-evaluation under relaxed assumptions
We re-evaluate the performance of the gradient inversion attack in settings
where two assumptions above are relaxed. For each relaxation, we re-design the
attack (if needed) based on the knowledge that the attacker has.
#### Relaxation 1: Not knowing BatchNorm statistics.
We refer to the previous threat model as $\rm BN_{exact}$, where the attacker
knows exact BatchNorm statistics of the private batch. We consider a more
realistic threat model where these statistics are not exposed, and re-design
the attack based on it.
Threat model. In each training step, the client normalizes its private batch
$x^{*}$ using the batch’s mean ${\rm mean}(x^{*})$ and variance ${\rm
var}(x^{*})$, keeps the running estimate of mean and variance locally as in
(Li et al., 2021), and shares the gradient. The client releases the final
aggregated mean $\mu$, and aggregated variance $\sigma^{2}$ of all training
data points at the end of training. Same as before, the attacker has access to
the model and the gradient during training.
Re-design A: $\rm BN_{proxy}$, attacker naively uses $\mu$ and $\sigma^{2}$. A
simple idea is that the attacker uses $(\mu,\sigma^{2})$ as the proxy for
$({\rm mean}(x^{*}),{\rm var}(x^{*}))$, and uses them to normalize $x$, his
guesses of the private batch. Other operations of the gradient inversion
attack remain the same as before. However, Figure 1.d and 1.h show poor-
quality reconstruction with this re-design.
Re-design B: $\rm BN_{infer}$, attacker infers $({\rm mean}(x^{*}),{\rm
var}(x^{*}))$ based on $(\mu,\sigma^{2})$. A more reasonable attacker will try
to infer $({\rm mean}(x^{*}),{\rm var}(x^{*}))$ while updating $x$, his
guesses of the private batch, and uses $({\rm mean}(x),{\rm var}(x))$ to
normalize the batch. In this case, $(\mu,\sigma^{2})$ could be used as a prior
of BatchNorm statistics to regularize the recovery, as suggested in (Yin et
al., 2021):
$\arg\min_{x}1-\frac{\left\langle\nabla_{\theta}\mathcal{L}_{\theta}(x,y),\nabla_{\theta}\mathcal{L}_{\theta}\left(x^{*},y^{*}\right)\right\rangle}{\left\|\nabla_{\theta}\mathcal{L}_{\theta}(x,y)\left|\left\|\mid\nabla_{\theta}\mathcal{L}_{\theta}\left(x^{*},y^{*}\right)\right\|\right.\right.}+\alpha_{\rm
TV}\mathcal{R}_{\rm TV}(x)+\alpha_{\rm BN}\mathcal{R}_{\rm BN}(x)$ (3)
where $\mathcal{R}_{\mathrm{BN}}(x)=\sum_{l}\|{\rm
mean}(x_{l})-\mu_{l}\|_{2}+\sum_{l}\|{\rm var}(x_{l})-\sigma_{l}^{2}\|_{2}$.
We tune $\alpha_{\rm BN}$ and present the best result in Figure 1.c and 1.g
(see results of different $\alpha_{\rm BN}$’s in Appendix A). As shown, for a
batch of low-resolution images, $\rm BN_{infer}$ gives a much better
reconstruction result than $\rm BN_{proxy}$, but still cannot recover some
details of the private batch when compared with $\rm BN_{exact}$. The result
for a single high-resolution image is worse: the attacker fails to return a
recognizable reconstruction with $\rm BN_{infer}$. This suggests not having
access to BatchNorm statistics of the private batch already weakens the state-
of-the-art gradient inversion attack.
(a) Original
(b) $\rm BN_{exact}$
(c) $\rm BN_{infer}$
(d) $\rm BN_{proxy}$
(e) Original
(f) $\rm BN_{exact}$
(g) $\rm BN_{infer}$
(h) $\rm BN_{proxy}$
Figure 1: Attacking a batch of 16 low-resolution images from CIFAR-10 (a-d)
and a single high-resolution image from ImageNet (e-h) with different
knowledge of BatchNorm statistics. Attack is weakened when BatchNorm
statistics are not available (c, d versus b, and g, h versus f). See Appendix
A for more examples and quantitative results.
#### Relaxation 2: Not knowing private labels.
(Zhao et al., 2020) notes that label information of a single private image can
be computed analytically from the gradients of the layer immediately before
the output layer. (Yin et al., 2021) further extends this method to support
recovery of labels for a batch of images. However, if multiple images in the
private batch belong to a same label, neither approach can tell how many
images belong to that label, let alone which subset of images belong to that
label. Figure 2a demonstrates that with CIFAR-10, for batches of various sizes
it is possible for many of the training samples to be have the same label, and
the distribution of labels is not uniform - and hence, inferring labels
becomes harder and the attack would be weakened. In Figure 2b we evaluate the
worst-case for an attacker in this setting by comparing recoveries where the
batch labels are simultaneously reconstructed alongside the training samples.
(a) Distribution of labels in a batch
(b) Reconstructions with and without private labels
Figure 2: Attack is weakened when private labels are not available. (a) shows
that for CIFAR-10, when the batch size is large, many images in the batch
belong to the same class, which essentially weakens label restoration (Zhao et
al., 2020; Yin et al., 2021). (b) visualizes a reconstructed batch of 16
images with and without private labels known. The quality of the
reconstruction drops without knowledge of private labels.
## 4 Defenses Against the Gradient Inversion Attack
Several defense ideas have been proposed to mitigate the risks of gradient
inversion.
### 4.1 Encrypt gradients
Cryptography-based approaches encrypt gradient to prevent gradient inversion.
(Bonawitz et al., 2016) presents a secure aggregation protocol for Federated
learning by computing sum of gradient vectors based on secret sharing (Shamir,
1979). (Phong et al., 2018) proposes using homomorphic encryption to encrypt
the gradients before sending. These approaches require special setup and can
be costly to implement.
Moreover, with secure aggregation protocol, an honest-but-curious server can
still launch the gradient inversion attack on the summed gradient vector.
Similarly, an honest-but-curious client can launch the gradient inversion
attack on the model returned by the server to reconstruct other clients’
private data, even with homomorphic encryption.
As alternatives, two other types of defensive mechanisms have been proposed to
mitigate the risks of attacks on plain-text gradient.
### 4.2 Perturbing gradients
#### Gradient pruning.
When proposing the first practical gradient inversion attack, (Zhu et al.,
2019) also suggests a defense by setting gradients of small magnitudes to zero
(i.e. gradient pruning). Based on their attack, they demonstrate that pruning
more than $70\%$ of the gradients would make the recovered images no longer
visually recognizable. However, the suggested prune ratio is determined based
on weaker attacks, and may not remain safe against the state-of-the-art
attack.
#### Adding noise to gradient.
Motivated by DPSGD (Abadi et al., 2016) which adds noise to gradients to
achieve differential privacy (Dwork, 2009; Dwork and Roth, 2014), (Zhu et al.,
2019; Wei et al., 2020) also suggests defending by adding Gaussian or
Laplacian noise to gradient. They show that a successful defense requires
adding high noise level such that its accuracy drops by more than $30\%$ with
CIFAR-10 tasks. Recent works (Papernot et al., 2020; Tramèr and Boneh, 2021)
suggest using better pre-training techniques and a large batch size (e.g.
$4,096$) to achieve a better accuracy for DPSGD training.
Since most DPSGD implementations for natural image classification tasks (Abadi
et al., 2016; Papernot et al., 2020; Tramèr and Boneh, 2021) use a pre-
training and fine-tuning pipeline, it is hard to fairly compare with other
defense methods that can directly apply when training the model from scratch.
Thus, we leave the comparison with DPSGD to future work.
### 4.3 Weak encryption of inputs (i.e. encoding inputs)
#### MixUp.
MixUp data augmentation (Zhang et al., 2018a) trains neural networks on
composite images created via linear combination of image pairs. It has been
shown to improve the generalization of the neural network and stabilizes the
training. Recent work also suggests that MixUp increases the model’s
robustness to adversarial examples (Pang et al., 2020; Lamb et al., 2019).
#### InstaHide.
Inspired by MixUp, (Huang et al., 2020) proposes InstaHide as a light-weight
instance-encoding scheme for private distributed learning. To encode an image
$x\in{\mathbb{R}}^{d}$ from a private dataset, InstaHide first picks $k-1$
other images $s_{2},s_{3},\ldots,s_{k}$ from that private dataset, or a large
public dataset, and $k$ random nonnegative coefficients
$\\{\lambda_{i}\\}_{i=1}^{k}$ that sum to $1$, and creates a composite image
$\lambda_{1}x+\sum_{i=2}^{k}\lambda_{i}s_{i}$ ($k$ is typically small, e.g.,
$4$). A composite label is also created using the same set of
coefficients.222Only the labels of examples from the private dataset will get
combined. See (Huang et al., 2020) for details. Then it adds another layer of
security: pick a random sign-flipping pattern $\sigma\in\\{-1,1\\}^{d}$ and
output the encryption
$\tilde{x}=\sigma\circ(\lambda_{1}x+\sum_{i=2}^{k}\lambda_{i}s_{i})$, where
$\circ$ is coordinate-wise multiplication of vectors. The neural network is
then trained on encoded images, which look like random pixel vectors to the
human eye and yet lead to good classification accuracy ($<6\%$ accuracy loss
on CIFAR-10, CIFAR-100, and ImageNet).
Recently, (Carlini et al., 2020) gives an attack to recover private images of
a small dataset, when the InstaHide encodings are revealed to the attacker
(not in a Federated learning setting). Their first step is to train a neural
network on a public dataset for similarity annotation, to infer whether a pair
of InstaHide encodings contain the same private image. With the inferred
similarities of all pairs of encodings, the attacker then runs a combinatorial
algorithm (cubic time in size of private dataset) to cluster all encodings
based on their original private images, and finally uses a regression
algorithm (with the help of composite labels) to recover the private images.
Neither (Huang et al., 2020) or (Carlini et al., 2020) has evaluated their
defense or attack in the Federated learning setting, where the attacker
observes gradients of the encoded images instead of original encoded images.
This necessitates the systematic evaluation in our next section.
## 5 Evaluation of defenses
The main goal of our experiments is to understand the trade-offs between data
utility (accuracy) and securely defending the state-of-the-art gradient
inversion attack even in its strongest setting, without any relaxation of its
implicit assumptions. Specifically, we grant the attacker the knowledge of 1)
BatchNorm statistics of the private batch, and 2) labels of the private batch.
We vary key parameters for each defense, and evaluate their performance in
terms of the test accuracy, computation overhead, and privacy risks (Section
5.2). We then investigate the feasibility of combining defenses (Section 5.3).
We also estimate the computation cost of end-to-end recovery of a single image
under evaluated defenses (Section 5.4).
As the feasibility of the state-of-the-art attack (Geiping et al., 2020) on a
batch of high-resolution images remain elusive when its implicit assumptions
no longer hold (see Figure 1), we focus on the evaluation with low-resolution
in trying to understand whether current attacks can be mitigated.
### 5.1 Experimental setup
#### Key parameters of defenses.
We evaluate following defenses on CIFAR-10 dataset (Krizhevsky et al., 2009)
with ResNet-18 architecture (He et al., 2016).
* •
GradPrune (gradient pruning): gradient pruning set gradients of small
magnitudes to zero. We vary the pruning ratio $p$ in
$\\{0.5,0.7,0.9,0.95,0.99,0.999\\}$.
* •
MixUp: MixUp encodes a private image by linearly combining it with $k-1$ other
images from the training set. Following (Huang et al., 2020), we vary $k$ in
$\\{4,6\\}$, and set the upper bound of a single coefficient to $0.65$
(coefficients sum to $1$).
* •
Intra-InstaHide: InstaHide (Huang et al., 2020) proposes two versions: Inter-
InstaHide and Intra-InstaHide. The only difference is that at the mixup step,
Inter-Instahide mixes up an image with images from a public dataset, whereas
Intra-InstaHide only mixes with private images. Both versions apply a random
sign flipping pattern on each mixed image. We evaluate Intra-InstaHide in our
experiments, which is a weaker version of InstaHide. Similar to the evaluation
of MixUp, we vary $k$ in $\\{4,6\\}$, and set the upper bound of a single
coefficient to $0.65$. Note that InstaHide flips signs of pixels in the image,
which destroys the total variation prior. However, the absolute value of
adjacent pixels should still be close. Therefore, for the InstaHide defense,
we apply the total variation regularizer on $|x|$, i.e. taking absolute value
of each pixel in the reconstruction.
We train the ResNet-18 architecture on CIFAR-10 using different defenses, and
launch the attack. We provide more details of the experiments in Appendix A.
#### The attack.
We use a subset of 50 CIFAR-10 images to evaluate the attack performance. Note
that attacking MixUp and InstaHide involves another step to decode private
images from the encoded images. We apply (Carlini et al., 2020)’s attack here
as the decode step, where the attacker needs to eavesdrop $T$ epochs of
training, instead of a single training step. We set $T=20$ in our evaluation.
We also grant the attacker the strongest power for the decode step to evaluate
the upper bound of privacy leakage. Given a MixUp or Intra-InstaHide image
which encodes $k$ private images, we assume the attacker knows:
1. 1.
The indices of $k$ images in the private dataset. In a realistic scenario, the
attacker of (Carlini et al., 2020) would need to train a neural network to
detect similarity of encodings, and run a combinatorial algorithm to solve an
approximation of this mapping.
2. 2.
The mixing coefficients for each of the $k$ private image. In real Federated
learning, this information is not available.
#### Hyper-parameters of the attack.
The attack minimize the objective function given in Eq.3. We search
$\alpha_{\rm TV}$ in $\\{0,0.001,0.005,0.01,0.05,0.1,0.5\\}$ for all defenses,
and apply the best choice for each defense: $0.05$ for GradPrune, $0.1$ for
MixUp, and $0.01$ for Intra-InstaHide. We apply $\alpha_{\rm BN}=0.001$ for
all defenses after searching it in $\\{0,0.0005,0.001,0.01,0.05,0.01\\}$. We
optimize the attack for $10,000$ iterations using Adam (Kingma and Ba, 2015),
with initial learning rate $0.1$. We decay the learning rate by a factor of
$0.1$ at $3/8,5/8,7/8$ of the optimization.
#### Batch size of the attack.
(Zhu et al., 2019; Geiping et al., 2020) have shown that a small batch size is
important for the success of the attack. We intentionally evaluate the attack
with three small batch sizes to test the upper bound of privacy leakage,
including the minimum (and unrealistic) batch size 1, and two small but
realistic batch sizes, 16 and 32.
#### Metrics for reconstruction quality.
We visualize reconstructions obtained under different defenses. Following (Yin
et al., 2021), we also use the learned perceptual image patch similarity
(LPIPS) score (Zhang et al., 2018b) to measure mismatch between reconstruction
and original images: higher values suggest more mismatch (less privacy
leakage).
Figure 3: Reconstruction results under different defenses with batch size
being 1, 16 and 32. When batch size is 32, combining gradient pruning and
Intra-InstaHide makes the reconstruction almost unrecognizable (the last
column). See Figure 7 in Appendix A for the full version.
| None | GradPrune ($p$) | MixUp ($k$) | Intra-InstaHide ($k$) | GradPrune ($p=0.9$)
---|---|---|---|---|---
| \+ MixUp | \+ Intra-InstaHide
Parameter | - | 0.5 | 0.7 | 0.9 | 0.95 | 0.99 | 0.999 | 4 | 6 | 4 | 6 | $k=4$ | $k=4$
Test Acc. | 93.37 | 93.19 | 93.01 | 90.57 | 89.92 | 88.61 | 83.58 | 92.31 | 90.41 | 90.04 | 88.20 | 91.37 | 86.10
Time (train) | $1\times$ | $1.04\times$ | $1.06\times$ | $1.06\times$ | $1.10\times$
Attack batch size $=1$
Avg. LPIPS $\downarrow$ | 0.19 | 0.19 | 0.22 | 0.35 | 0.42 | 0.52 | 0.52 | 0.34 | 0.46 | 0.58 | 0.61 | 0.41 | 0.60
Best LPIPS $\downarrow$ | 0.02 | 0.02 | 0.05 | 0.14 | 0.22 | 0.32 | 0.36 | 0.12 | 0.25 | 0.41 | 0.42 | 0.21 | 0.43
(LPIPS std.) | 0.16 | 0.17 | 0.16 | 0.13 | 0.11 | 0.08 | 0.06 | 0.08 | 0.07 | 0.06 | 0.09 | 0.07 | 0.09
Attack batch size $=16$
Avg. LPIPS $\downarrow$ | 0.45 | 0.46 | 0.47 | 0.51 | 0.55 | 0.58 | 0.61 | 0.34 | 0.31 | 0.62 | 0.63 | 0.46 | 0.68
Best LPIPS $\downarrow$ | 0.18 | 0.19 | 0.19 | 0.31 | 0.43 | 0.47 | 0.51 | 0.11 | 0.13 | 0.41 | 0.44 | 0.22 | 0.54
(LPIPS std.) | 0.12 | 0.12 | 0.11 | 0.07 | 0.05 | 0.04 | 0.03 | 0.09 | 0.09 | 0.08 | 0.08 | 0.10 | 0.07
Attack batch size $=32$
Avg. LPIPS $\downarrow$ | 0.45 | 0.46 | 0.48 | 0.52 | 0.54 | 0.58 | 0.63 | 0.50 | 0.49 | 0.69 | 0.69 | 0.62 | 0.73
Best LPIPS $\downarrow$ | 0.18 | 0.18 | 0.22 | 0.31 | 0.43 | 0.48 | 0.54 | 0.31 | 0.28 | 0.56 | 0.56 | 0.37 | 0.65
(LPIPS std.) | 0.11 | 0.11 | 0.09 | 0.07 | 0.05 | 0.04 | 0.04 | 0.10 | 0.10 | 0.06 | 0.07 | 0.10 | 0.05
(a) (b)
Table 2: Utility-security trade-off of different defenses. We train the
ResNet-18 model on the whole CIFAR-10 dataset, and report the averaged test
accuracy and running time of 5 independent runs. We evaluate the attack on a
subset of 50 CIFAR-10 images, and report the LPIPS score ($\downarrow$: lower
values suggest more privacy leakage). We mark the least-leakage defense
measured by the metric in green.
### 5.2 Performance of defense methods
We summarize the performance of each defense in Table 2, and visualize
reconstructed images in Figure 3. We report the averaged and the best results
for the metric of reconstruction quality, as a proxy for average-case and
worst-case privacy leakage.
#### No defense.
Without any defense, when batch size is 1, the attack can recover images well
from the gradient. Increasing the batch size makes it difficult to recover
well, but the recovered images are visually similar to the originals (see
Figure 3).
#### Gradient pruning (GradPrune).
Figure 3 shows that as the pruning ratio $p$ increases, there are more
artifacts in the reconstructions. However, the reconstructions are still
recognizable even when the pruning ratio $p=0.9$, thus the previous suggestion
of using $p=0.7$ by (Zhu et al., 2019) is no longer safe against the state-of-
the-art attack. Our results suggest that, for CIFAR-10, defending the
strongest attack with gradient pruning may require the pruning ratio $p\geq
0.999$. As a trade-off, such a high pruning ratio would introduce an accuracy
loss of around $10\%$ (see Table 2).
#### MixUp.
MixUp introduces a small computational overhead to training. MixUp with $k=4$
only has a minor impact (~$2\%$) on test accuracy, but it is not sufficient to
defend the gradient inversion attack (see Figure 3). Increasing $k$ from $4$
to $6$ slightly reduces the leakage, however, the reconstruction is still
highly recognizable. This suggests that MixUp alone may not be a practical
defense against the state-of-the-art gradient inversion attack.
#### Intra-InstaHide.
Intra-InstaHide with $k=4$ incurs an extra ~$2\%$ accuracy loss compared with
MixUp, but it achieves better defense performance: when batch size is 32,
there are obvious artifacts and color shift in the reconstruction (see Figure
3). However, with batch size 32, Intra-InstaHide alone also cannot defend the
state-of-the-art gradient inversion, as structures of private images are still
vaguely identifiable in reconstructions.
Appendix A provides the whole reconstructed dataset under MixUp and Intra-
InstaHide.
### 5.3 Performance of combined defenses
We notice that two types of defenses (i.e perturbing gradient and encoding
inputs) are complementary to each other, which motivates an evaluation of
combining gradient pruning with MixUp or Intra-InstaHide.
As shown in Figure 3, when the batch size is 32, combining Intra-InstaHide
($k=4$) with gradient pruning ($p=0.9$) makes the reconstruction almost
unrecognizable. The combined defense yields a higher LPIPS score than using
gradient pruning with $p=0.999$, but introduces a smaller accuracy loss
(~$7\%$ compared with a no-defense pipeline).
Note that our evaluation uses the strongest attack and relatively small batch
sizes. As shown in Appendix A, invalidating assumptions in Section 3 or
increasing the batch size may hinder the attack with an even weaker defense
(e.g. with a lower $p$, or smaller $k$), which gives a better accuracy.
### 5.4 Time estimate for end-to-end recovery of a single image
Table 3 shows time estimates for the end-to-end recovery of a single image in
a Federated learning setting with GradPrune or InstaHide defense. We do not
estimate for MixUp since it has been shown to be a weak defense (see Section
5.2).
Our time estimates consider three fairly small dataset sizes. The largest size
in our estimate is a small fraction of a dataset of ImageNet scale. We
consider a client holds a dataset of $N$ private images and participates in
Federated learning, which trains a ResNet-18 model with batch size $b=128$.
Assumes that the resolution of the client’s data is $32\times 32\times 3$. If
the attacker uses a single NVIDIA GeForce RTX 2080 Ti GPU as his computation
resource, and runs gradient inversion with 10,000 iterations of optimization,
then $t$, the running time for attacking a single batch is ~$0.25$ GPU hours
(batch size $b$ has little impact on the attack’s running time, but a larger
$b$ makes the attack less effective).
#### Non-defended and gradient pruning.
Recovering a single image in a non-defended pipeline (or a pipeline that
applies gradient pruning alone as the defense) only requires the attacker to
invert gradient of a single step of training, which takes time $t$.
#### InstaHide.
When InstaHide is applied, the current attack (Carlini et al., 2020) suggests
that recovering a single image would involve recovering the whole dataset
first. As discussed in Section 4, Carlini et al.’s attack consists of two
steps: 1) recover InstaHide images from gradient of $T$ epochs. This would
take $(NT/b)\times t$ GPU hours. 2) Run the decode attack (Carlini et al.,
2020) on InstaHide images to recover the private dataset, which involves:
1. 2a
Train a neural network to detect similarity in recovered InstaHide images.
Assume that training the network requires at least $n$ recovered InstaHide
images, then collecting these images by running gradient inversion would take
$(n/b)\times t$ GPU hours. The training takes $10$ GPU hours according to
(Carlini et al., 2020), so training the similarity network would take
$(n/b)\times t+10$ GPU hours in total.
2. 2b
Run the combinotorial algorithm to recover original images. Running time of
this step has been shown to be at least quadratic in $m$, the number of
InstaHide encodings (Chen et al., 2021). This step takes $1/6$ GPU hours with
$m=5\times 10^{3}$. Therefore for $m=NT$, the running time is at least
$1/6\times(\frac{NT}{5\times 10^{3}})^{2}$ GPU hours.
In total, an attack on InstaHide in this real-world setting would take
$(NT/b)\times t+(n/b)\times t+10+1/6\times(\frac{NT}{5\times 10^{3}})^{2}$ GPU
hours. We use $T=50$ (used by (Carlini et al., 2020)), $n=10,000$ and give
estimate in Table 3. As shown, when InstaHide is applied on a small dataset
($N=5,000$), the end-to-end recovery of a single image takes $>3,000\times$
longer than in a no-defense pipeline or GradPrune pipeline; when InstaHide is
applied on a larger dataset ($N=500,000$), the computation cost for end-to-end
recovery is enormous.
Size of client’s dataset ($N$) | No defense | GradPrune | InstaHide
---|---|---|---
5,000 | 0.25 | 0.25 | 934.48
50,000 | 46,579.01 ($\approx$ 5.5 GPU years)
500,000 | 4,215,524.32 ($\approx$ 493.4 GPU years)
Table 3: Time estimates (NVIDIA GeForce RTX 2080 Ti GPU hours) of recovering a
single image from the client’s dataset using the state-of-the-art gradient
inversion attack (Geiping et al., 2020) under different defenses. We assume
image resolution of the client’s data is $32\times 32\times 3$.
## 6 Conclusions
This paper first points out that some state-of-the-art gradient inversion
attacks have made strong assumptions about knowing BatchNorm statistics and
private labels. Relaxing such assumptions can significantly weaken these
attacks.
The paper then reports the performance of a set of proposed defenses against
gradient inversion attacks, and estimates the computation cost of an end-to-
end recovery of a single image in different dataset sizes. Our evaluation
shows that InstaHide without mixing with data from a public dataset combined
with gradient pruning can defend the state-of-the-art attack, and the
estimated time to recover a single image in a medium-size client dataset (e.g.
of 500,000 images) is enormous.
Based on our evaluation of the attack by (Geiping et al., 2020) and multiple
defenses for plain-text gradients, we have the following observations:
* •
Using BatchNorm layers in your deep net but don’t share BatchNorm statistics
of the private batch during Federated learning weakens the attack. We have
demonstrated in Section 3 that exposing BatchNorm statistics to the attacker
significantly improves the quality of gradient inversion. So a more secure
configuration of Federated Learning would be to use BatchNorm layers, but do
not share BatchNorm statistics in training, which has been shown feasible in
(Andreux et al., 2020; Li et al., 2021).
* •
Using a large batch size weakens the attack; a batch size smaller than 32 is
not safe. We have shown that a larger batch size hinders the attack by making
it harder to guess the private labels (Section 3) and to recover the private
images even with correct private labels (Section 5). Our experiments suggest
that even with some weak defenses applied, a batch size smaller than 32 is not
safe against the strongest gradient inversion attack.
* •
Combining multiple defenses may achieve a better utility-privacy trade-off. In
our experiment, for a batch size of 32, combining InstaHide ($k=4$) with
gradient pruning ($p=0.9$) achieves the best utility-privacy trade-off, by
making the reconstruction almost unrecognizable at a cost of ~$7\%$ accuracy
loss (using InstaHide also makes the end-to-end recovery of a single image
more computationally expensive). Best parameters would vary for different deep
learning tasks, but we strongly encourage Federated learning participants to
explore the possibility of combining multiple defensive mechanisms, instead of
only using one of them.
We hope to extend our work by including evaluation of defenses for high-
resolution images, the attack by (Yin et al., 2021) (when its implementation
becomes available), and more defense mechanisms including those rely on adding
noise to gradients.
## Acknowledgments
This project is supported in part by Ma Huateng Foundation, Schmidt
Foundation, NSF, Simons Foundation, ONR and DARPA/SRC. Yangsibo Huang and
Samyak Gupta are supported in part by the Princeton Graduate Fellowship.
We would like to thank Quanzheng Li, Xiaoxiao Li, Hongxu Yin and Aoxiao Zhong
for helpful discussions, and members of Kai Li’s and Sanjeev Arora’s research
groups for comments on early versions of the work.
## References
* Abadi et al. [2016] Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In _Proceedings of the 2016 ACM SIGSAC conference on computer and communications security_ , pages 308–318, 2016.
* Act [1996] Accountability Act. Health insurance portability and accountability act of 1996. _Public law_ , 104:191, 1996.
* Andreux et al. [2020] Mathieu Andreux, Jean Ogier du Terrail, Constance Beguier, and Eric W Tramel. Siloed federated learning for multi-centric histopathology datasets. In _Domain Adaptation and Representation Transfer, and Distributed and Collaborative Learning_ , pages 129–139. Springer, 2020.
* Bonawitz et al. [2016] K. A. Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H. Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth. Practical secure aggregation for federated learning on user-held data. In _NIPS Workshop on Private Multi-Party Machine Learning_ , 2016\.
* Carlini et al. [2020] Nicholas Carlini, Samuel Deng, Sanjam Garg, Somesh Jha, Saeed Mahloujifar, Mohammad Mahmoody, Shuang Song, Abhradeep Thakurta, and Florian Tramer. An attack on instahide: Is private learning possible with instance encoding? In _IEEE Symposium on Security and Privacy_ , 2020.
* Chen et al. [2021] Sitan Chen, Xiaoxiao Li, Zhao Song, and Danyang Zhuo. On instahide, phase retrieval, and sparse matrix factorization. In _ICLR_ , 2021.
* Cohen et al. [2019] Michael B Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In _Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC)_ , 2019.
* Commission [2018] European Commission. 2018 reform of eu data protection rules. https://gdpr-info.eu/, 2018.
* Deng et al. [2009] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In _CVPR_ , 2009.
* Deng [2012] Li Deng. The mnist database of handwritten digit images for machine learning research. _IEEE Signal Processing Magazine_ , 29(6):141–142, 2012.
* Dwork [2009] Cynthia Dwork. The differential privacy frontier. In _Theory of Cryptography Conference (TCC)_ , pages 496–502, 2009\.
* Dwork and Roth [2014] Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. _Foundations and Trends in Theoretical Computer Science_ , 9(3–4):211–407, 2014.
* Geiping et al. [2020] Jonas Geiping, Hartmut Bauermeister, Hannah Dröge, and Michael Moeller. Inverting gradients–how easy is it to break privacy in federated learning? In _NeurIPS_ , 2020.
* He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In _CVPR_ , 2016.
* Huang et al. [2020] Yangsibo Huang, Zhao Song, Kai Li, and Sanjeev Arora. Instahide: Instance-hiding schemes for private distributed learning. In _ICML_ , 2020.
* Ioffe and Szegedy [2015] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In _ICML_ , 2015.
* Kairouz et al. [2021] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. _Foundations and Trends in Machine Learning_ , 14(1–2):1–210, 2021.
* Kingma and Ba [2015] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In _ICLR_ , 2015.
* Krizhevsky et al. [2009] Alex Krizhevsky et al. Learning multiple layers of features from tiny images. 2009\.
* Lamb et al. [2019] Alex Lamb, Vikas Verma, Juho Kannala, and Yoshua Bengio. Interpolated adversarial training: Achieving robust neural networks without sacrificing too much accuracy. In _Proceedings of the 12th ACM Workshop on Artificial Intelligence and Security_ , pages 95–103, 2019.
* Legislature [2018] California State Legislature. California consumer privacy act. https://oag.ca.gov/privacy/ccpa, 2018.
* Li et al. [2021] Xiaoxiao Li, Meirui Jiang, Xiaofei Zhang, Michael Kamp, and Qi Dou. Fedbn: Federated learning on non-iid features via local batch normalization. In _ICLR_ , 2021.
* McMahan et al. [2016] H Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, et al. Communication-efficient learning of deep networks from decentralized data. In _Artificial Intelligence and Statistics (AISTATS)_ , pages 1273–1282, 2016.
* Pang et al. [2020] Tianyu Pang, Kun Xu, and Jun Zhu. Mixup inference: Better exploiting mixup to defend adversarial attacks. In _ICLR_ , 2020.
* Papernot et al. [2020] Nicolas Papernot, Steve Chien, Shuang Song, Abhradeep Thakurta, and Ulfar Erlingsson. Making the shoe fit: Architectures, initializations, and tuning for learning with privacy, 2020. URL https://openreview.net/forum?id=rJg851rYwH.
* Phong et al. [2017] Le Trieu Phong, Yoshinori Aono, Takuya Hayashi, Lihua Wang, and Shiho Moriai. Privacy-preserving deep learning: Revisited and enhanced. In _ICATIS_ , pages 100–110, 2017.
* Phong et al. [2018] Le Trieu Phong, Yoshinori Aono, Takuya Hayashi, Lihua Wang, and Shiho Moriai. Privacy-preserving deep learning via additively homomorphic encryption. _IEEE Transactions on Information Forensics and Security_ , 2018.
* Shamir [1979] Adi Shamir. How to share a secret. _Communications of the ACM_ , 22(11):612–613, 1979.
* Tramèr and Boneh [2021] Florian Tramèr and Dan Boneh. Differentially private learning needs better features (or much more data). In _ICLR_ , 2021.
* Wei et al. [2020] Wenqi Wei, Ling Liu, Margaret Loper, Ka-Ho Chow, Mehmet Emre Gursoy, Stacey Truex, and Yanzhao Wu. A framework for evaluating gradient leakage attacks in federated learning. _arXiv preprint arXiv:2004.10397_ , 2020.
* Yin et al. [2021] Hongxu Yin, Arun Mallya, Arash Vahdat, Jose M Alvarez, Jan Kautz, and Pavlo Molchanov. See through gradients: Image batch recovery via gradinversion. _arXiv preprint arXiv:2104.07586_ , 2021.
* Zhang et al. [2018a] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. Mixup: Beyond empirical risk minimization. In _ICLR_ , 2018a.
* Zhang et al. [2019] Hongyi Zhang, Yann N Dauphin, and Tengyu Ma. Fixup initialization: Residual learning without normalization. In _ICLR_ , 2019.
* Zhang et al. [2018b] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In _CVPR_ , 2018b.
* Zhao et al. [2020] Bo Zhao, Konda Reddy Mopuri, and Hakan Bilen. idlg: Improved deep leakage from gradients. _arXiv preprint arXiv:2001.02610_ , 2020.
* Zhu and Blaschko [2021] Junyi Zhu and Matthew Blaschko. R-gap: Recursive gradient attack on privacy. In _ICLR_ , 2021.
* Zhu et al. [2019] Ligeng Zhu, Zhijian Liu, and Song Han. Deep leakage from gradients. In _NeurIPS_ , 2019.
## Appendix A Experimental details and more results
We run all the experiments on Nvidia RTX 2080 Ti GPUs and V100 GPUs. Table 4
summarizes the set of images used in each figure or table in the main paper.
Figure/Table | Comments
---|---
Figure 1a | We’ve tuned hyperparams for the attack (see Appendix A.1) and carried out evaluations on the whole CIFAR-subset. The first sampled batch of size 16 from CIFAR-subset was used in Figure 1a to demonstrate the quality of recovery for low-resolution images when BatchNorm statistics are not assumed to be known.
Figure 1b | We’ve tuned hyperparams for the attack (see Appendix A.1) and carried out evaluations on the whole ImageNet-subset. The best-reconstructed image in ImageNet-subset was used in Figure 1b to demonstrate the quality of recovery for high-resolution images when BatchNorm statistics are not assumed to be known.
Figure 2a | Percentages of class labels per batch were evaluated over the entire CIFAR10 dataset, for a random seed.
Figure 2b | The first sampled batch of size 16 was used in Figure 2b to demonstrate the quality of recovery when labels are not assumed to be known.
Table 2 and Figure 3 | We’ve tuned hyperparams for the attack and carried out evaluations on the whole CIFAR-subset. Table 2 summarizes the performance of the attack on the whole CIFAR-subset and Figure 3 shows example images.
Table 4: Summary of experimental testbed for each evaluation.
### A.1 Hyper-parameters
#### Training.
For all experiments, we train ResNet-18 for 200 epochs, with a batch size of
128. We use SGD with momentum 0.9 as the optimizer. The initial learning rate
is set to 0.1 by default, except for gradient pruning with $p=0.99$ and
$p=0.999$. where we set the initial learning rate to 0.02. We decay the
learning rate by a factor of 0.1 every 50 epochs.
#### The attack.
We report the performance under different $\alpha_{\rm TV}$’s (Figure 4) and
$\alpha_{\rm BN}$’s (Figure 5).
(a) Original
(b) $\alpha_{\rm TV}$=0
(c) $\alpha_{\rm TV}$=1e-3
(d) $\alpha_{\rm TV}$=5e-3
(e) $\alpha_{\rm TV}$=1e-2
(f) $\alpha_{\rm TV}$=5e-2
(g) $\alpha_{\rm TV}$=1e-1
(h) $\alpha_{\rm TV}$=5e-1
Figure 4: Attacking a single CIFAR-10 images in $\rm BN_{exact}$ setting, with
different coefficients for the total variation regularizer ($\alpha_{\rm
TV}$’s). $\alpha_{\rm TV}$=1e-2 gives the best reconstruction.
(a) Original
(b) $\alpha_{\rm BN}$=0
(c) $\alpha_{\rm BN}$=5e-4
(d) $\alpha_{\rm BN}$=1e-3
(e) $\alpha_{\rm BN}$=5e-3
(f) $\alpha_{\rm BN}$=1e-2
Figure 5: Attacking a batch of 16 CIFAR-10 images in $\rm BN_{infer}$ setting,
with different coefficients for the BatchNorm regularizer ($\alpha_{\rm
BN}$’s). $\alpha_{\rm TV}$=1e-3 gives the best reconstruction.
### A.2 Details and more results for Section 3
#### Attacking a single ImageNet image.
We launched the attack on ImageNet using the objective function in Eq. 3,
where $\alpha_{\rm TV}=0.1$, $\alpha_{\rm BN}=0.001$. We run the attack for
24,000 iterations using Adam optimizer, with initial learning rate 0.1, and
decay the learning rate by a factor of $0.1$ at $3/8,5/8,7/8$ of training. We
rerun the attack 5 times and present the best results measured by LPIPS in
Figure 1.
#### Qualitative and quantitative results for a more realistic attack.
We also present results of a more realistic attack in Table 5 and Figure 6,
where the attacker does not know BatchNorm statistics but knows the private
labels. We assume the private labels to be known in this evaluation, because
for those batches whose distribution of labels is uniform, the restoration of
labels should still be quite accurate [Yin et al., 2021]. As shown, in the
evaluated setting, the attack is no longer effective when the batch size is 32
and Intra-InstaHide with $k=4$ is applied. The accuracy loss to stop the
realistic attack is only around $3\%$ (compared to around $7\%$ to stop the
strongest attack) .
(a) Figure 6: Reconstruction results under different defenses for a more realistic setting (when the attacker knows private labels but does not know BatchNorm statistics). We also present the results for the strongest attack from Figure 3 for comparison. Using Intra-InstaHide with $k=4$ and batch size 32 seems to stop the realistic attack. | None | GradPrune ($p$) | MixUp ($k$) | Intra-InstaHide ($k$) | GradPrune ($p=0.9$)
---|---|---|---|---|---
| \+ MixUp | \+ Intra-InstaHide
Parameter | - | 0.5 | 0.7 | 0.9 | 0.95 | 0.99 | 0.999 | 4 | 6 | 4 | 6 | $k=4$ | $k=4$
Test Acc. | 93.37 | 93.19 | 93.01 | 90.57 | 89.92 | 88.61 | 83.58 | 92.31 | 90.41 | 90.04 | 88.20 | 91.37 | 86.10
Time (train) | $1\times$ | $1.04\times$ | $1.06\times$ | $1.06\times$ | $1.10\times$
Attack batch size $=16$, the strongest attack
Avg. LPIPS $\downarrow$ | 0.41 | 0.41 | 0.42 | 0.46 | 0.48 | 0.50 | 0.55 | 0.50 | 0.49 | 0.69 | 0.69 | 0.62 | 0.73
Best LPIPS $\downarrow$ | 0.21 | 0.22 | 0.27 | 0.29 | 0.30 | 0.29 | 0.48 | 0.31 | 0.28 | 0.56 | 0.56 | 0.37 | 0.65
(LPIPS std.) | 0.09 | 0.08 | 0.07 | 0.06 | 0.06 | 0.06 | 0.04 | 0.10 | 0.10 | 0.06 | 0.07 | 0.10 | 0.05
Attack batch size $=16$, attacker knows private labels but does not know
BatchNorm statistics
Avg. LPIPS $\downarrow$ | 0.49 | 0.51 | 0.48 | 0.51 | 0.52 | 0.56 | 0.60 | 0.71 | 0.71 | 0.75 | 0.75 | 0.74 | 0.74
Best LPIPS $\downarrow$ | 0.30 | 0.33 | 0.31 | 0.33 | 0.34 | 0.39 | 0.44 | 0.48 | 0.53 | 0.65 | 0.63 | 0.61 | 0.63
(LPIPS std.) | 0.08 | 0.09 | 0.08 | 0.08 | 0.07 | 0.07 | 0.05 | 0.08 | 0.07 | 0.04 | 0.05 | 0.08 | 0.05
Attack batch size $=32$, the strongest attack
Avg. LPIPS $\downarrow$ | 0.45 | 0.46 | 0.48 | 0.52 | 0.54 | 0.58 | 0.63 | 0.50 | 0.49 | 0.69 | 0.69 | 0.62 | 0.73
Best LPIPS $\downarrow$ | 0.18 | 0.18 | 0.22 | 0.31 | 0.43 | 0.48 | 0.54 | 0.31 | 0.28 | 0.56 | 0.56 | 0.37 | 0.65
(LPIPS std.) | 0.11 | 0.11 | 0.09 | 0.07 | 0.05 | 0.04 | 0.04 | 0.10 | 0.10 | 0.06 | 0.07 | 0.10 | 0.05
Attack batch size $=32$, attacker knows private labels but does not know
BatchNorm statistics
Avg. LPIPS $\downarrow$ | 0.48 | 0.50 | 0.53 | 0.53 | 0.55 | 0.60 | 0.63 | 0.73 | 0.72 | 0.76 | 0.76 | 0.76 | 0.77
Best LPIPS $\downarrow$ | 0.29 | 0.32 | 0.32 | 0.31 | 0.40 | 0.41 | 0.55 | 0.63 | 0.60 | 0.68 | 0.63 | 0.66 | 0.65
(LPIPS std.) | 0.08 | 0.07 | 0.07 | 0.08 | 0.08 | 0.06 | 0.04 | 0.06 | 0.06 | 0.04 | 0.05 | 0.06 | 0.05
Table 5: Utility-security trade-off of different defenses for a more realistic
setting (when the attacker knows private labels but does not know BatchNorm
statistics). We also present the results for the strongest attack from Table 2
for comparison. We evaluate the attack on 50 CIFAR-10 images and report the
LPIPS score ($\downarrow$: lower values suggest more privacy leakage). We mark
the least-leakage defense measured by the metric in green.
### A.3 More results for the strongest attack
#### Full version of Figure 3.
Figure 7 provides more examples for reconstructed images by the strongest
attack under different defenses and batch sizes.
(a) Batch size $=1$
(b) Batch size $=16$
(c) Batch size $=32$
Figure 7: Reconstruction results under different defenses with batch size 1
(a), 16 (b) and 32 (c). Full version of Figure 3.
#### Results with MNIST dataset.
We’ve repeated our main evaluation of defenses and attacks (Table 2) on MNIST
dataset [Deng, 2012] with a simple 6-layer ConvNet model. Note that the simple
ConvNet does not contain BatchNorm layers. We evaluate the following defenses
on the MNIST dataset with a 6-layer ConvNet architecture against the strongest
attack (private labels known):
* •
GradPrune (gradient pruning): gradient pruning sets gradients of small
magnitudes to zero. We vary the pruning ratio $p$ in {0.5, 0.7, 0.9, 0.95,
0.99, 0.999, 0.9999}.
* •
MixUp: we vary $k$ in {4,6}, and set the upper bound of a single coefficient
to 0.65 (coefficients sum to 1).
* •
Intra-InstaHide: we vary $k$ in {4,6}, and set the upper bound of a single
coefficient to 0.65 (coefficients sum to 1).
* •
A combination of GradPrune and MixUp/Intra-InstaHide.
We run the evaluation against the strongest attack and batch size 1 to
estimate the upper bound of privacy leakage. Specifically, we assume the
attacker knows private labels, as well as the indices of mixed images and
mixing coefficients for MixUp and Intra-InstaHide.
Figure 8: Reconstruction results of MNIST digits under different defenses with
the strongest atttack and batch size 1.
For MNIST with a simple 6-layer ConvNet, defending the strongest attack with
gradient pruning may require the pruning ratio $p\geq 0.9999$. MixUp with
$k=4$ or $k=6$ are not sufficient to defend the gradient inversion attack.
Combining MixUp ($k=4$) with gradient pruning ($p=0.99$) improves the defense,
however, the reconstructed digits are still highly recognizable. Intra-
InstaHide alone ($k=4$ or $k=6$) gives a bit better defending performance than
MixUp and GradPrune. Combining InstaHide ($k=4$) with gradient pruning
($p=0.99$) further improves the defense and makes the reconstruction almost
unrecognizable.
### A.4 More results for encoding-based defenses
We visualize the whole reconstructed dataset under MixUp and Intra-InstaHide
defenses with different batch sizes in Figure 10, 11 and 12. Sample results of
the original and the reconstructed batches are provided in Figure 9.
Figure 9: Original and reconstructed batches of 16 images under MixUp and
Intra-InstaHide defenses. We visualize both the original and the absolute
images for the Intra-InstaHide defense. Intra-InstaHide makes pixel-wise
matching harder.
(a) Original
(b) MixUp, $k$=4
(c) MixUp, $k$=6
(d) MixUp+GradPrune, $k$=4, $p$=0.9
(e) Original
(f) InstaHide, $k$=4
(g) InstaHide, $k$=6
(h) InstaHide+GradPrune, $k$=4, $p$=0.9
Figure 10: Reconstrcuted dataset under MixUp and Intra-InstaHide against the
strongest attack (batch size is 1).
(a) Original
(b) MixUp, $k$=4
(c) MixUp, $k$=6
(d) MixUp+GradPrune, $k$=4, p=0.9
(e) Original
(f) InstaHide, $k$=4
(g) InstaHide, $k$=6
(h) InstaHide+GradPrune, $k$=4, $p$=0.9
Figure 11: Reconstrcuted dataset under MixUp and Intra-InstaHide against the
strongest attack (batch size is 16).
(a) Original
(b) MixUp, $k$=4
(c) MixUp, $k$=6
(d) MixUp+GradPrune, $k$=4, $p$=0.9
(e) Original
(f) InstaHide, $k$=4
(g) InstaHide, $k$=6
(h) InstaHide+GradPrune, $k$=4, $p$=0.9
Figure 12: Reconstrcuted dataset under MixUp and Intra-InstaHide against the
strongest attack (batch size is 32).
## Appendix B Theoretical Insights for Defenses’ Working Mechanism
In this section, we provide some theoretical insights for the mechanism of
each defense.
### B.1 Gradient pruning
Gradient pruning is a non-oblivious case of applying sketching techniques
[Cohen et al., 2019] to compress the gradient vector. Usually, if we only
observe the vector after sketching, it is hard to recover the original vector
unless certain assumptions of the vector itself and the sketching technique
have been made. Therefore, gradient pruning prevents the attacker from seeing
the original gradient and make the inversion harder.
### B.2 MixUp and InstaHide
Intuitively, MixUp and InstaHide’s working mechanism may come from mixing $k$
images in a single encoded image, which appears to be similar to multiplying
the batch size by a factor of $k$, thus makes the attack less effective. In
Section B.3, we provide theoretical analysis for this intuition, by showing
that mixing $k$ images and using a batch size of $k$ are essentially similar,
with any neural network that has a fully connected layer as its first layer.
Another layer of security of InstaHide seems to come from applying random
sign-flipping on the mixed images. As mentioned in Section 5.1, for an
InstaHide-encoded image $x\in\operatorname{{\mathbb{R}}}^{d}$, we apply the
total variation regularizer on $|x|$ instead of $x$, which pushes the gap
between absolute value of adjacent pixels (i.e., $||x_{j}|-|x_{j+1}||$) to be
small. However having $||x_{j}|-|x_{j+1}||=\delta$ for some small
$\delta<10^{-4}$ does not imply that $|x_{j}-x_{j+1}|=\delta$; in fact,
$|x_{j}-x_{j+1}|$ can be as large as $1-\delta$. Therefore, the random sign
flipping operation in InstaHide could potentially make the total variation
image prior less effective in some sense (see Figure 9).
### B.3 Property of gradient in a small batch
The goal of this section is to present the following results,
###### Lemma B.1.
Given a neural network with ReLU activation function, each row of the gradient
of first layer weights is a linear combination of images, i.e.
$\displaystyle(\frac{\partial{\cal L}(W)}{\partial
W_{1}})_{i}=\sum_{j=1}^{b}\alpha_{i,j}x_{i}^{\top}$
where the $b$ is the number of images in a small batch,
$\\{x_{1},\cdots,x_{b}\\}\in\operatorname{{\mathbb{R}}}^{d}$ are images in
that small batch.
In Section B.4 and B.5, we show the above observation holds for one/two-hidden
layer neural network. In Section B.6, we generalize it to multiple layer
neural network.
The standard batched $k$-vector sum can be defined as follows:
###### Definition B.2.
Give a database $X$ list of vectors $x_{1},\cdots,x_{n}$. Given a list of
observations $y_{1},\cdots,y_{m}$ where for each $j\in[m]$, there is a set
$S_{j}$ such that $y_{j}=\sum_{i\in S_{j}}x_{i}$ and $|S_{j}|=b$. We can
observe $y_{1},\cdots y_{m}$ but has no access to database, the goal is to
recover $S_{j}$ and the vectors $x_{i}$ being use, for each $j$.
The above definition is a mathematical abstraction of MixUp recovery/attack.
It can be further generalized to InstaHide, if we only observe the $|y_{j}|$.
We also want to remark that in the above definition, we simplify the
formulation by using coefficients $1$ for all vectors. It can be easily
generalized to settings where random coefficients are assigned to vectors in
the database for MixUp/InstaHide.
Using Lemma B.1, we notice that
###### Lemma B.3.
Under the condition of Lemma B.1, given a list of observation of gradients,
and the problem recovering images is also a batched vector sum problem.
Thus, gradient attack is essentially an variation of MixUp/Instahide attack.
### B.4 One Hidden Layer
We consider a one-hidden layer ReLU activated neural network with $m$ neurons
in the hidden layer:
$\displaystyle f(x)=a^{\top}\phi(Wx)$
where $a\in\operatorname{{\mathbb{R}}}^{m}$ and
$W\in\operatorname{{\mathbb{R}}}^{m\times d}$. We define objective function
$L$ as follows:
$\displaystyle L(W)=\frac{1}{2}\sum_{i=1}^{n}(y_{i}-f(W,x_{i},a))^{2}$
We can compute the gradient of $L$ in terms of $w_{r}$
$\displaystyle\frac{\partial L(W)}{\partial
w_{r}}=\sum_{i=1}^{n}(f(W,x_{i},a)-y_{i})a_{r}x_{i}{\bf 1}_{\langle
w_{r},x_{i}\rangle}$
Let $\widetilde{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$,
$\displaystyle\frac{\partial L(W)}{\partial
w_{r}}=(f(W,\widetilde{x},a)-y)a_{r}\widetilde{x}{\bf 1}_{\langle
w_{r},\widetilde{x}\rangle}$
Another version
$\displaystyle\frac{\partial L(W)}{\partial
w_{r}}=\sum_{i=1}^{n}(f(W,x_{i},a)-y_{i})\cdot\Big{(}a_{r}\widetilde{x}{\bf
1}_{\langle w_{r},\widetilde{x}\rangle}\Big{)}$
### B.5 Two Hidden Layers
Suppose $a\in\operatorname{{\mathbb{R}}}^{m}$,
$V\in\operatorname{{\mathbb{R}}}^{m\times
d},W\in\operatorname{{\mathbb{R}}}^{m\times m}$. The neural network is defined
as $f:\operatorname{{\mathbb{R}}}^{d}\rightarrow\operatorname{{\mathbb{R}}}$,
here we slightly deviate from the standard setting and assume the input
dimension is $m$, in order to capture the general setting.
$\displaystyle f(x)=a^{\top}\phi(W\phi(Vx))$
Consider the mean square loss
$\displaystyle L(W,V,a)=\frac{1}{2}\sum_{i=1}^{n}|f(x_{i})-y_{i}|^{2}$
The gradient with respect to $W$ is
$\displaystyle\frac{\partial L(W,V,a)}{\partial
W}=\sum_{i=1}^{n}(f(x_{i})-y_{i})\underbrace{\mathrm{diag}\\{\phi^{\prime}(W\phi(Vx_{i}))\\}}_{m\times
m}\underbrace{a}_{m\times 1}\underbrace{\phi(Vx_{i})^{\top}}_{1\times m}$
and the gradient with respect to $V$ is
$\displaystyle\frac{\partial L(W,V,a)}{\partial
V}=\sum_{i=1}^{n}(f(x_{i})-y_{i})\underbrace{\mathrm{diag}\\{\phi^{\prime}(Vx_{i})\\}}_{m\times
m}\underbrace{W^{\top}}_{m\times
m}\underbrace{\mathrm{diag}\\{\phi^{\prime}(W\phi(Vx_{i}))\\}}_{m\times
m}\underbrace{a}_{m\times 1}\underbrace{x_{i}^{\top}}_{1\times d}$
### B.6 The multi-layers case
The following multiple layer neural network definition is standard in
literature.
Consider a $L$ layer neural network with one vector
$a\in\operatorname{{\mathbb{R}}}^{m_{L}}$ and $L$ matrices
$W_{L}\in\operatorname{{\mathbb{R}}}^{m_{L}\times m_{L-1}}$, $\cdots$,
$W_{2}\in\operatorname{{\mathbb{R}}}^{m_{2}\times m_{1}}$ and
$W_{1}\in\operatorname{{\mathbb{R}}}^{m_{1}\times m_{0}}$. Let $m_{0}=d$. In
order to write gradient in an elegant way, we define some artificial variables
as follows
$\displaystyle g_{i,1}=$ $\displaystyle~{}W_{1}x_{i},$ $\displaystyle
h_{i,1}=\phi(W_{1}x_{i}),$
$\displaystyle\in\operatorname{{\mathbb{R}}}^{m_{1}}$ $\displaystyle\forall
i\in[n]$ $\displaystyle g_{i,\ell}=$ $\displaystyle~{}W_{\ell}h_{i,\ell-1},$
$\displaystyle h_{i,\ell}=\phi(W_{\ell}h_{i,\ell-1}),$
$\displaystyle\in\operatorname{{\mathbb{R}}}^{m_{\ell}}$ $\displaystyle\forall
i\in[n],\forall\ell\in\\{2,3,\cdots,L\\}$ (4) $\displaystyle D_{i,1}=$
$\displaystyle~{}\text{diag}\big{(}\phi^{\prime}(W_{1}x_{i})\big{)},$
$\displaystyle\in\operatorname{{\mathbb{R}}}^{m_{1}\times m_{1}}$
$\displaystyle\forall i\in[n]$ $\displaystyle D_{i,\ell}=$
$\displaystyle~{}\text{diag}\big{(}\phi^{\prime}(W_{\ell}h_{i,\ell-1})\big{)},$
$\displaystyle\in\operatorname{{\mathbb{R}}}^{m_{\ell}\times m_{\ell}}$
$\displaystyle\forall i\in[n],\forall\ell\in\\{2,3,\cdots,L\\}$
where $\phi(\cdot)$ is the activation function and $\phi^{\prime}(\cdot)$ is
the derivative of activation function.
Let
$f:\operatorname{{\mathbb{R}}}^{m_{0}}\rightarrow\operatorname{{\mathbb{R}}}$
denote neural network function:
$\displaystyle f(W,x)=a^{\top}\phi(W_{L}(\phi(\cdots\phi(W_{1}x))))$
Thus using definition of $f$ and $h$, we have
$\displaystyle
f(W,x_{i})=a^{\top}h_{i,L},~{}~{}~{}\in\operatorname{{\mathbb{R}}},~{}~{}~{}\forall
i\in[n]$
Given $n$ input data points
$(x_{1},y_{1}),(x_{2},y_{2}),\cdots(x_{n},y_{n})\in\operatorname{{\mathbb{R}}}^{d}\times\operatorname{{\mathbb{R}}}$.
We define the objective function $\mathcal{L}$ as follows
$\displaystyle\mathcal{L}(W)=\frac{1}{2}\sum_{i=1}^{n}(y_{i}-f(W,x_{i}))^{2}.$
We can compute the gradient of ${\cal L}$ in terms of
$W_{\ell}\in\operatorname{{\mathbb{R}}}^{m_{\ell}\times m_{\ell-1}}$, for all
$\ell\geq 2$
$\displaystyle\frac{\partial\mathcal{L}(W)}{\partial
W_{\ell}}=\sum_{i=1}^{n}(f(W,x_{i})-y_{i})\underbrace{D_{i,\ell}}_{m_{\ell}\times
m_{\ell}}\left(\prod_{k=\ell+1}^{L}\underbrace{W_{k}^{\top}}_{m_{k-1}\times
m_{k}}\underbrace{D_{i,k}}_{m_{k}\times
m_{k}}\right)\underbrace{a}_{m_{L}\times
1}\underbrace{h_{i,\ell-1}^{\top}}_{1\times m_{\ell-1}}$ (5)
Note that the gradient for $W_{1}\in\operatorname{{\mathbb{R}}}^{m_{1}\times
m_{0}}$ (recall that $m_{0}=d$) is slightly different and can not be written
by general form. Here is the form
$\displaystyle\frac{\partial\mathcal{L}(W)}{\partial
W_{1}}=\sum_{i=1}^{n}(f(W,x_{i})-y_{i})\underbrace{D_{i,1}}_{m_{1}\times
m_{1}}\left(\prod_{k=2}^{L}\underbrace{W_{k}^{\top}}_{m_{k-1}\times
m_{k}}\underbrace{D_{i,k}}_{m_{k}\times
m_{k}}\right)\underbrace{a}_{m_{L}\times 1}\underbrace{x_{i}^{\top}}_{1\times
m_{0}}$ (6)
|
# On the Baire class of n-Dimensional Boundary Functions
Connor Paul Wilson 530 Church Street Ann Arbor, MI 48109<EMAIL_ADDRESS>
###### Abstract.
We show an extention of a theorem of Kaczynski to boundary functions in
n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote
its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into
some metric space $Y.$ If $E$ is any subset of $X,$ we will say that a
function $\varphi:E\rightarrow Y$ is a boundary function for $f$ if and only
if for each $x\in E$ there exists an arc $\gamma$ at $x$ such that
$\lim_{z\rightarrow x\atop z\in\gamma}f(z)=\varphi(x)$
## 1\. Introduction
### 1.1. Preliminaries and Notation
Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the
$x$-axis. If $x\in X,$ then by an arc at $x$ we mean a a simple arc $\gamma$
with one endpoint at $x$ such that $\gamma-\\{x\\}\subseteq H.$ Suppose that
$f$ is a function mapping $H$ into some metric space $Y.$ If $E$ is any subset
of $X,$ we will say that a function $\varphi:E\rightarrow Y$ is a boundary
function for $f$ if and only if for each $x\in E$ there exists an arc $\gamma$
at $x$ such that
$\lim_{z\rightarrow x\atop z\in\gamma}f(z)=\varphi(x)$
We will also define Baire classes as Kaczynski does in [1], such that a
function $f:M\rightarrow Y$ is said to be of Baire class $O(M,Y)$ if and only
if it is continuous; if $\xi$ is an ordinal number greater than or equal to
$1,$ then f is said to be of Baire class $\xi(M,Y)$ if and only if there
exists a sequence of functions $\left\\{f_{n}\right\\}_{n=1}^{\infty}$ mapping
M into $Y,f_{n}$ being of Baire class $\eta_{n}(M,Y)$ for some $\eta_{n}<\xi$,
such that $f_{n}\rightarrow f$ pointwise.
## 2\. Boundary functions for discontinuous functions
###### Theorem 2.1.
Let $Y$ be a separable arc-wise connected metric space, with $f:H\rightarrow
Y$ a function of Baire class $\xi(H,Y)$ where $\xi\geq 1,$ $E$ a subset of
$X$, and $\varphi:E\rightarrow Y$ a boundary function for $f$. Therefore
$\varphi$ is of Baire class $\xi+1(E,Y).$
###### Proof.
Let $U$ be an open subset of $Y$ such that $V=Y-\operatorname{clos}(U)$. Set
$A=\varphi^{-1}(U),\ B=\varphi^{-1}(V),\ C=A\cup B.$ Notice that we clearly
have an empty intersection between $A$ and $B$. $\forall x\in C$, choose an
arc $\gamma_{x}$ at $x$ such that:
$\lim_{z\rightarrow x\atop z\in\gamma_{x}}f(z)=\varphi(x)$
with
$\gamma_{x}\subseteq\\{z:\mid z-x\mid\leq 1\\}$
where
$\begin{cases}\gamma_{x}-\\{x\\}\subseteq f^{-1}(U)&\text{ if }x\in A\\\
\gamma_{x}-\\{x\\}\subseteq f^{-1}(V)&\text{ if }x\in B\end{cases}$
Note once again the empty intersection $\gamma_{x}\cap\gamma_{y}=\emptyset$
for $x\in A\ \wedge\ y\in B$
Let us define the terminology $\gamma_{x}$ meets $\gamma_{y}$ in
$\operatorname{clos}(H_{n})$ provided that the two arcs have respective
subarcs, $\gamma_{x}\prime$ and $\gamma_{y}\prime$ with
$x\in\gamma_{x}\prime\subseteq\operatorname{clos}(H_{n})$ and
$x\in\gamma_{y}\prime\subseteq\operatorname{clos}(H_{n})$, with
$\gamma_{x}\prime\cap\gamma_{y}\prime\neq\varnothing$
Let:
$L_{a}:={x\in A:\forall n\exists y,\text{ such that }y\in C,\ y\neq x,\
\gamma_{y}\text{ meets }\gamma_{x}\text{ in }\operatorname{clos}(H_{n})}$
$L_{b}:={x\in B:\forall n\exists y,\text{ such that }y\in C,\ y\neq x,\
\gamma_{y}\text{ meets }\gamma_{x}\text{ in }\operatorname{clos}(H_{n})}$
$M_{a}:={x\in A:\exists n,\gamma_{x}\text{ meets no }\gamma_{y}\text{ in
}\operatorname{clos}(H_{n})}$ $M_{b}:={x\in B:\exists n,\gamma_{x}\text{ meets
no }\gamma_{y}\text{ in }\operatorname{clos}(H_{n})}$ $L=L_{a}\cup L_{b}$
$M=M_{a}\cup M_{b}$
$L_{a},L_{b},M_{a},$ and $M_{b}$ are notably pairwise disjoint, and
$A=L_{a}\cup M_{a},$ $B=L_{b}\cup M_{b}.$
Let $n(x)\in\mathbb{Z}^{+}$ for each $x\in M$ such that $\gamma_{x}$ meets no
$\gamma_{y}$ in $\operatorname{clos}(H_{n(x)})$ such that $x\neq y,$ where
$n\geq n(x)$ gives the obvious no meeting case in
$\operatorname{clos}(H_{n})$. Moreover, take
$K_{n}:=\\{x\in C:\gamma_{x}\text{ meets }X_{n}\wedge\text{ if }x\in M,n\geq
n(x)\\}$
It is clear that we have for every $n,K_{n}\subseteq K_{n+1}$, as well as
$C=\bigcup_{n=1}^{\infty}K_{n}.$ It follows by the work of Kaczynski[1] and
the following lemma that we have Theorem $2.1$.
###### Lemma 2.2.
Let Y be a separable arc-wise connected metric space, E any metric space, and
$\varphi:E\rightarrow Y$ a function such that for every open set $U\subseteq
Y$ there exists a set $T\in P^{\xi+1}(E)$ where $\varphi^{-1}(U)\subseteq
T\subseteq\varphi^{-1}(\operatorname{clos}(U))$. Thus for $\xi\geq 2,$
$\varphi$ is of Baire class $\xi(E,Y).$
###### Proof.
Let $\mathcal{B}$ be a countable base for $Y,$ and suppose $W$ is some open
subset of $Y.$ Let:
$\mathcal{A}(W)=\\{U\in\mathcal{B}:\operatorname{clos}(U)\subseteq W\\}$
By Kaczynski, we take:
$W=\bigcup_{U\in\mathcal{A}(W)}U=\bigcup_{U\in\mathcal{A}(W)}\operatorname{clos}(U).$
$\forall U\in\mathcal{B},$ let $T(U)\in P^{\xi+1}(E)$ be chosen so that
$\varphi^{-1}(U)\subseteq T(U)\subseteq\varphi^{-1}(\operatorname{clos}(U)).$
Thus we have:
$\displaystyle\varphi^{-1}(W)$
$\displaystyle=\bigcup_{U\in\mathcal{A}(W)}\varphi^{-1}(U)\subseteq\bigcup_{U\in\mathcal{A}(W)}T(U)$
$\displaystyle\subseteq\bigcup_{U\in\mathcal{A}(W)}\varphi^{-1}(\operatorname{clos}(U))=\varphi^{-1}(W).$
Therefore $\varphi^{-1}(W)=\bigcup_{U\in\mathcal{A}(W)}T(U)$, and given
$P^{\xi+1}(E)$ is closed under countable unions, we have $\varphi^{-1}(W)\in
P^{\xi+1}(E),$ and $\varphi$ is of Baire class $\xi(E,Y)$ ∎
And following from above we therefore have:
$\varphi^{-1}(U)=A\subseteq T\cap E\subseteq
E-B=E-\varphi^{-1}(V)=\varphi^{-1}(\operatorname{clos}(U))$
for some $T\in P^{\xi+2}(X)$, and we know $T\cap E\in P^{\xi+2}(E)$ which by
the above lemma gives us $\varphi$ of Baire class $\xi+1(E,Y).$ ∎
## 3\. Sets of curvilinear convergence for $\mathbb{R}^{3}$
Let $f:H\rightarrow Y$ of Baire class $\xi(H,Y)$, and $\varphi:E\rightarrow Y$
a boundary function of $f.$ Let us define some function to analyse the
properties of $M_{a},$ following Kaczynski. Take
$\pi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ such that
$\pi(\langle x,y,z\rangle)=\left\|\langle x,y\rangle\right\|_{2}.$
If $\left\|\langle x,y\rangle\right\|_{2}\in M\cap K_{n},$ then let us define
$p_{n}(\left\|\langle x,y\rangle\right\|_{2})$ as the first point of $X_{n}$
reached along $\gamma_{x}$ starting from $x.$ It is clear thus that by
Kaczynski, the function $\pi(p_{n}(\left\|\langle x,y\rangle\right\|_{2})$ is
strictly increasing on $M\cap K_{n}.$ Thus by the above logic, and Lemma
$2.2$, we can show the following theorem:
###### Theorem 3.1.
Let $Y$ be a separable arc-wise connected metric space in $\mathbb{R}^{n}$,
with $f:H\rightarrow Y$ a function of Baire class $\xi+n-1(H,Y)$ where
$\xi\geq 1,$ $E$ a subset of $X$, and $\varphi:E\rightarrow Y$ a boundary
function for $f$. Therefore $\varphi$ is of Baire class $\xi+n(E,Y).$
Although this does not resolve the fourth open problem of Kaczynski’s work, it
does provide an extension to a theorem shown, and is valuable nonetheless to
the field.
## References
* [1] Kaczynski, Theodore J., Boundary functions, Doctoral Dissertation, University of Michigan (1967).
|
# Active Learning for Video Classification with Frame Level Queries
††thanks: This research was supported in part by the National Science
Foundation under Grant Number: 2143424
Debanjan Goswami Department of Computer Science
Florida State University
Shayok Chakraborty Department of Computer Science
Florida State University
###### Abstract
Deep learning algorithms have pushed the boundaries of computer vision
research and have depicted commendable performance in a variety of
applications. However, training a robust deep neural network necessitates a
large amount of labeled training data, acquiring which involves significant
time and human effort. This problem is even more serious for an application
like video classification, where a human annotator has to watch an entire
video end-to-end to furnish a label. Active learning algorithms automatically
identify the most informative samples from large amounts of unlabeled data;
this tremendously reduces the human annotation effort in inducing a machine
learning model, as only the few samples that are identified by the algorithm,
need to be labeled manually. In this paper, we propose a novel active learning
framework for video classification, with the goal of further reducing the
labeling onus on the human annotators. Our framework identifies a batch of
exemplar videos, together with a set of informative frames for each video; the
human annotator needs to merely review the frames and provide a label for each
video. This involves much less manual work than watching the complete video to
come up with a label. We formulate a criterion based on uncertainty and
diversity to identify the informative videos and exploit representative
sampling techniques to extract a set of exemplar frames from each video. To
the best of our knowledge, this is the first research effort to develop an
active learning framework for video classification, where the annotators need
to inspect only a few frames to produce a label, rather than watching the end-
to-end video. Our extensive empirical analyses corroborate the potential of
our method to substantially reduce human annotation effort in applications
like video classification, where annotating a single data instance can be
extremely tedious.
###### Index Terms:
active learning, video classification, deep learning
## I Introduction
With the widespread deployment of modern sensors and cameras, images and
videos have become ubiquitous. This has encouraged the development of video
classification algorithms to analyze their semantic content for various
applications, such as search, summarization, security and surveillance among
others. Deep neural networks (CNN and LSTM architectures) have depicted
commendable performance in this field [1]. Common methods include obtaining
global video-level descriptors using CNN architectures [2], processing videos
at two spatial resolutions: a low-resolution context stream and a high-
resolution fovea stream [3], fusion technique to integrate data
representations at the frame level and video level [4] among others. However,
for all these models to work reliably, a large amount of labeled training data
is essential, gathering which is an expensive process in terms of time, labor
and human expertise. Thus, an algorithm to reduce the human labeling effort is
of immense importance in video classification applications.
(a) Conventional AL query for video classification
(b) Proposed active query and annotation mechanism
Figure 1: (a) Conventional AL query for video classification, where the human
annotator has to watch a video end-to-end to provide a label. (b) Proposed
active query and annotation mechanism where we actively select a batch of
videos, together with a subset of frames from each video; the human annotator
merely needs to inspect the frames and provide a label for the video.
Active Learning (AL) is a machine learning paradigm, where the goal is to
automatically identify the salient and exemplar samples from large amounts of
redundant data [5]. This tremendously reduces the human annotation effort in
inducing a machine learning model, since the human expert only has to label
the samples queried by the algorithm. Further, since the model gets trained on
the exemplar samples from the data population, it typically depicts better
generalization performance than a model where the training data is selected at
random. This is an extremely relevant paradigm in today’s world, where an
enormous amount of digital data is being generated, but there is a serious
dearth of human labor to annotate the data to induce learning models. AL has
been successfully used in a variety of applications, including computer vision
[6], text analysis [7], computational biology [8] and medical diagnosis [9]
among others. Active learning is particularly relevant in the context of deep
learning, in order to reduce human annotation effort in training the data-
hungry deep neural networks [10].
Designing an AL algorithm for a video classification application entails the
human annotator to meticulously watch each queried video end-to-end in order
to furnish a label 111we use the terms annotators, oracles, labelers and users
synonymously in this paper. This is an extremely time-consuming and laborious
process; the annotators may get bored and fatigued quickly and lose interest
in the task. This necessitates specialized and more user-friendly query and
annotation mechanisms, to utilize the available human labor more efficiently.
In this paper, we propose a novel active learning algorithm to address this
challenging and practical problem. Our algorithm identifies a batch of
informative videos, together with a set of exemplar frames from each; the
human annotator merely has to review the queried frames and furnish a label
for each video. This is illustrated in Figure 1. Providing such feedback is
significantly less time-consuming and burdensome than watching an end-to-end
video. We formulate an optimization problem based on an uncertainty and
diversity based criterion to identify a batch of informative videos, and
exploit representative sampling techniques to select a subset of exemplar
frames from each. To our knowledge, this is the first active learning
framework for video classification which poses label queries based on a set of
exemplar frames, rather than the complete video. We hope this research will
motivate the development of AL algorithms with other novel annotation
mechanisms, with the goal of further reducing the labeling burden on human
oracles in a video classification application.
The rest of the paper is organized as follows: we present a survey of related
research in Section II, our active sampling framework is detailed in Section
III, the results of our empirical studies are presented in Section IV, and we
conclude with discussions in Section V.
## II Related Work
In this section, we present an overview of active learning in general,
followed by a survey of AL for video classification.
Active Learning: AL has received significant research attention in the machine
learning community. Uncertainty sampling is by far the most common strategy
for active learning, where unlabeled samples with highest classification
uncertainties are queried for their labels. The uncertainty of an unlabeled
sample can be computed by its entropy [11], its distance from the separating
hyperplane in the feature space for SVM classifiers [7], the disagreement
among a committee of classifiers regarding the label of the sample [12], the
expected error reduction of the future learner [13] and so on. Submodular
optimization techniques have also been exploited for active data sampling [14,
15].
The growing success and popularity of deep learning has motivated research in
the field of deep active learning (DAL), where the goal is to select
informative unlabeled samples to efficiently train a deep neural network [10].
A task agnostic AL framework was proposed by Yoo and Kweon [6] that
incorporated a loss prediction module in the network architecture, to predict
the loss value of an unlabeled sample and query samples accordingly. A DAL
framework based on core-set selection was proposed by Sener and Savarese [16],
which selected a batch of samples, such that the deep model trained on the
selected samples depicts similar performance to that trained on the whole
dataset. DAL has also been studied in conjunction with neural architecture
search [17], which queries samples for labeling and simultaneously searches
for the best neural architectures on-the-fly. A novel training loss function
for DAL was proposed by Shui et al., where active sample selection and traning
the network parameters were achieved through alternating optimization [18].
Deep active learning techniques based on adversarial training have depicted
particularly impressive performance [19, 20, 21, 22]. Active learning has also
been studied in conjunction with other learning paradigms such as transfer
learning [23], reinforcement learning [24] etc. Moreover, the idea of
identifying an informative set of samples for human inspection has been
extended to other problem domains, such as matrix completion [25], video
summarization [26] and feature selection [27] among others.
Recently, there have been efforts to design AL systems with novel query and
annotation mechanisms, with the goal of further reducing the labeling burden
on human annotators. Joshi et al. [28] proposed a binary query mechanism which
queried an unlabeled sample together with a potential class label and the user
had to provide the binary answer as to whether the queried unlabeled sample
belonged to the selected class or not. Along similar lines, Biswas and Jacobs
proposed an AL algorithm for clustering, which queried a pair of samples and
the oracles needed to specify whether or not the samples in a pair correspond
to the same cluster [29]. Xiong et al. [30] proposed a triplet query framework
to learn approximate distance metrics for a nearest neighbor classifier; the
algorithm queried unlabeled data triplets $(x_{i},x_{j},x_{k})$ and posed the
question whether instance $x_{i}$ was more similar to $x_{j}$ than to $x_{k}$.
Qian et al. [31] proposed an active learning algorithm where the query
strategy was to place an ordering (or partial ordering) on the similarity of
the neighbors of the selected unlabeled sample, rather than querying its
actual label.
Active Learning for Video Classification: While AL has been extensively
studied for image recognition [6, 32, 22, 33], it is much less explored for
video classification. Similar to image classification, uncertainty sampling
(using metrics like entropy, error reduction) is a popular AL query strategy
for video recognition [34, 35]. Yan et al. [36] proposed a multi-class AL
framework for video classification using expected error reduction. Since the
estimation of the posterior probability distribution $P(y|x)$ may be
unreliable due to the lack of sufficient training data, simple heuristics were
also proposed to simplify the sample selection strategies. Another approach
was developed in the context of SVMs, which queried a set of samples which can
produce the maximum expected reduction in the SVM objective [37, 38]. Bandla
and Grauman [39] used AL to train an action detector for videos which selected
the video which was expected to maximally reduce the entropy among all
unlabeled videos. The core idea was to use the current trained detector to
extract relevant portions in the video where the action of interest occurs, so
that the video segment outside the interval does not introduce noise in the
entropy computation. However, this method is specifically designed to actively
learn an action detector from videos. Active contrastive learning has also
been explored for learning audio-visual representations from unlabeled videos
[40].
All these methods require the human annotator to watch an unlabeled video end-
to-end in order to provide a label, which may be extremely time-consuming and
arduous. In contrast, our framework identifies a subset of exemplar frames,
and the human labeler has to label a video by merely reviewing the frames,
which is a much more efficient annotation strategy. Our method is applicable
to any type of videos and does not make any assumptions about the contents of
the video. Other related efforts include AL for video tracking [41], video
description [42], video recommendation [43] and video segmentation [44].
However, these methods attempt to solve a different problem than video
classification, which is the focus of this paper. We now describe our
framework.
## III Proposed Framework
Consider an active learning problem for video classification, where we are
given a labeled training set $L$ and an unlabeled set $U$, with $|L|\ll|U|$.
Each data sample $x$ in $L$ and $U$ is a video. Let $w$ be the deep neural
network trained on $L$ and $C$ be the number of classes in the data. Our
objective is two-fold: $(i)$ select a batch $B$ containing $b$ unlabeled
videos so that the model trained on $L\cup B$ has maximum generalization
capability; $(ii)$ however, we are not allowed to show an entire video to a
human annotator and ask for its label; we are required to select a subset of
$k$ exemplar frames from each queried video, so that only those can be shown
to an annotator for labeling the video.
Both these objectives are critical in improving the generalization capability
of the deep model. The first objective ensures that the salient videos are
selected from the unlabeled set for active query. The second objective ensures
that the most representative frames are selected from each video for query.
This is important, as otherwise, the annotator may not be confident enough to
provide a label or may provide an incorrect label, both of which will result
in a wastage of query budget and degrade the performance of the model. In the
following sections, we discuss our active sampling strategies for sampling
videos and frames.
### III-A Active Video Sampling
We quantified the utility of a batch of $b$ videos and selected a batch
furnishing the maximal utility. The informativeness and diversity metrics were
used to compute the utility of a batch of videos in this research. An active
learning framework driven by these conditions ensures that the video samples
in the batch augment useful knowledge to the underlying deep neural network,
and there is high diversity (minimum redundancy) of information among the
samples in the batch. These conditions have been used in previous active
learning research [45].
Computing informativeness: The informativeness of an unlabeled video sample
$x_{i}$ was computed as the uncertainty of the deep model $w$ in predicting a
label for $x_{i}$. The Shannon’s entropy was used to compute the prediction
uncertainty:
$e(x_{i})=-\sum_{y=1}^{C}P(y|x_{i},w)\log P(y|x_{i},w)$ (1)
Computing diversity: We computed a diversity matrix $R\in\Re^{|U|\times|U|}$
where $R(i,j)$ denotes the diversity between videos $x_{i}$ and $x_{j}$ in the
unlabeled set. We used the kernelized distance on the deep feature
representations to compute the diversity between a pair of videos in this
research:
$R(i,j)=K(x_{i},x_{j})$ (2)
where $K=(.,.)$ denotes the distance in the Reproducing Kernel Hilbert Space
(RKHS) [46].
#### III-A1 Active Video Selection
By definition, all the entries in $e$ and $R$ are non-negative, that is,
$e_{i}\geq 0$ and $R(i,j)\geq 0,\forall i,j$. Given $e$ and $R$, our objective
is to select a batch of videos with high uncertainties (given by the entries
in $e$) and high mutual diversity (given by the entries in $R$). We define a
binary selection vector $z\in\Re^{|U|\times 1}$ where $z_{i}$ denotes whether
the unlabeled video $x_{i}$ will be selected in the batch $(z_{i}=1)$ or not
$(z_{i}=0)$. Our batch selection task (with batch size $b$) can thus be posed
as the following NP-hard integer quadratic programming (IQP) problem:
$\max_{z}e^{T}z+\mu z^{T}Rz$ $\text{s.t.}\hskip
7.22743ptz_{i}\in\\{0,1\\},\forall i\hskip 7.22743pt\text{and}\hskip
7.22743pt\sum_{i=1}^{|U|}z_{i}=b$ (3)
where $\mu$ is a weight parameter governing the relative importance of the two
terms. The binary integer constraints on $z$ allow us to combine $e$ and $R$
into a single matrix $Q\in\Re^{|U|\times|U|}$ and express the optimization
problem as follows:
$\max_{z}z^{T}Qz$ $\text{s.t.}\hskip 7.22743ptz_{i}\in\\{0,1\\},\forall
i\hskip 7.22743pt\text{and}\hskip 7.22743pt\sum_{i=1}^{|U|}z_{i}=b$ (4)
where the matrix $Q$ is constructed as follows:
$Q(i,j)=\begin{cases}\mu R(i,j),&\text{if}\hskip 2.168pti\neq j\\\
e(i),&\text{if}\hskip 2.168pti=j\end{cases}$ (5)
The binary integer constraints on the variable $z$ make the IQP in Equation
(4) NP-hard. We used the Iterative Truncated Power algorithm [47] to solve
this optimization problem.
#### III-A2 The Iterative Truncated Power Algorithm
This algorithm was originally proposed in the context of the sparse eigenvalue
and the densest $k$-subgraph problems. It attempts to solve an optimization
problem similar to that in Equation (4). The algorithm starts with an initial
solution $z_{0}$ and then generates a sequence of solutions
$z_{1},z_{2},\ldots$. The solution $z_{t}$ at iteration $t$ is obtained by
multiplying the solution $z_{t-1}$ at iteration $(t-1)$ by the matrix $Q$ and
then truncating all the entries to $0$, except the $b$ largest entries. The
process is repeated until convergence. The algorithm is guaranteed to converge
monotonically for a positive semi-definite (psd) matrix $Q$. When the matrix
$Q$ is not psd, the algorithm can be run on the shifted quadratic function
(with a positive scalar added to the diagonal elements) to guarantee a
monotonic convergence [47]. The algorithm is computationally efficient and
converges fast. It benefits from a good starting point. In our empirical
studies, the initial solution $z_{0}$ was taken as the indicator vector
corresponding to the $b$ largest column sums of the matrix $Q$, as it produced
competitive results in our preliminary experiments. The pseudo-code for our
active video sampling algorithm is presented in Algorithm 1.
Algorithm 1 The Proposed Active Video Sampling Algorithm
0: Training set $L$, Unlabeled set $U$, batch size $b$ and weight parameter
$\mu$
1: Train a deep neural network $w$ on the training set $L$
2: Compute the entropy vector $e$ (Equation (1)) and the diversity matrix $R$
(Equation (2))
3: Compute the matrix $Q$, as described in Equation (5)
4: Derive the initial solution $z_{0}$ as the indicator vector containing the
$b$ largest column sums of the matrix $Q$
5: t = 1
6: repeat
7: Compute $\widehat{z_{t}}=Q.z_{t-1}$
8: Identify $F_{t}$ as the index set of $\widehat{z_{t}}$ with top $b$ values
9: Set $z_{t}$ to be $1$ on the index set $F_{t}$ and $0$ otherwise
10: t = t + 1
11: until Convergence
12: Select a batch of $b$ unlabeled videos based on the final solution $z_{t}$
#### III-A3 Computational Considerations
Computing the diversity matrix $R$ involves quadratic complexity. We first
note that $R$ needs to be computed only once in our framework, before the
start of the AL iterations. As the unlabeled videos get queried through AL, we
can keep deleting the corresponding rows and columns in $R$ to derive the new
diversity matrix. Moreover, random projection algorithms can be used to speed
up computations. The theory of random projections states that, if we have a
point cloud in a high dimensional space, they may be projected into a suitable
lower-dimensional space such that the distances between the points are
approximately preserved [48]. A data matrix $A\in\Re^{N\times D}$ in the $D$
dimensional space is multiplied by a random projection matrix
$X\in\Re^{D\times d}$ $(d\ll D)$ to obtain a projected matrix
$B\in\Re^{N\times d}$ in the lower dimensional space $d$: $B=AX$ [49]. This
can be used to substantially reduce the computational overhead, as distance
computations are more efficient in the low dimensional space. We will explore
this as part of future research.
### III-B Active Frame Sampling
Once we select $b$ videos from the unlabeled set, our next task is to identify
a subset of $k$ frames from each of these videos; we exploited representative
sampling techniques for this purpose. These techniques identify the exemplar
data points which well-represent a given dataset. In particular, the coreset
algorithm selects a subset of points such that a model trained over the
selected subset is maximally similar to that trained on the whole dataset. For
the sake of completeness, we discuss the main ideas here and request
interested readers to refer to [16] for further details. Coreset poses the
subset selection problem as:
$\min_{s:|s|=k}\Bigg{|}\frac{1}{n}\sum_{i\in[n]}l(x_{i},y_{i},A_{i})-\frac{1}{|s|}\sum_{j\in
s}l(x_{j},y_{j},A_{j})\Bigg{|}$ (6)
where $(x_{i},y_{i})$ denotes a training sample and its label, $A_{i}$ denotes
a learning algorithm which outputs a set of parameters by minimizing a loss
function $l(.,.,.)$ on a given labeled set $i$. Informally, given a budget
$k$, the goal is to select a set of samples $s$, such that the model trained
on $s$ depicts similar performance as the model trained on the whole dataset
with $n$ samples.
This function cannot be directly optimized, as the labels of the samples in
the unlabeled set are unknown. An upper bound of this function was derived and
the problem of active sampling was shown to be equivalent to the $k$-center
problem (also called min-max facility location problem) [50]. The objective of
this problem is to select $k$ center points from $n$ samples, such that the
largest distance between a data point and its nearest center is minimized.
Formally, this can be posed as follows:
$\min_{s:|s|=k}\max_{i}\min_{j\in s}\Delta(x_{i},x_{j})$ (7)
This problem is NP-Hard [51]. However, a greedy algorithm, as detailed in
Algorithm 2, is guaranteed to produce a solution $s$ such that:
$\max_{i}\min_{j\in s}\Delta(x_{i},x_{j})\leq 2\times OPT$, where $OPT$ is the
optimal solution. We used this algorithm to select a subset of $k$ frames from
each of the queried videos. As evident from the formulation, our method does
not make any assumptions about the contents of the video, and is applicable to
any type of video.
Algorithm 2 The Active Frame Sampling Algorithm [16]
0: A video with $n$ frames ($x_{1},x_{2},\ldots x_{n}$) and frame budget $k$
1: Initialize $s=\Phi$
2: repeat
3: $u=\operatorname*{arg\,max}_{i\in[n]\backslash s}\min_{j\in
s}\Delta(x_{i},x_{j})$
4: $s=s\cup\\{u\\}$
5: until $|s|=k$
6: Select $k$ frames from the video based on the set $s$
## IV Experiments and Results
### IV-A Datasets
We used the UCF-101 [52] and the Kinetics datasets [53] to study the
performance of our algorithm. Both these datasets contain videos of humans
performing a variety of actions, captured under unconstrained, real-world
conditions, and are extensively used to study the performance of video
classification algorithms. We used data from $5$ classes at random from each
dataset for our experiments.
### IV-B Oracle Simulation
All the publicly available video datasets contain annotations for the complete
videos; we did not find any datasets which contain annotations based on a
subset of frames. Also, different active sampling algorithms will select
different subsets of frames, and it is challenging to obtain annotations from
a human labeler for every possible subset of frames for a given video, to
conduct experiments. We therefore used a deep neural network to simulate the
human labeling oracle in our empirical studies. The oracle model was trained
on a completely different subset of the data. No information about the oracle
model was used in the design and development of our active learning algorithm.
During AL, when a video sample was selected for query, the selected frames
were passed as an input to the trained oracle model and its prediction entropy
on the sample was computed. If the entropy exceeded a particular threshold
$\tau_{oracle}$, the oracle was assumed to be not confident enough to produce
a label, and no label was returned; otherwise, the oracle returned the
predicted label (which may be correct or incorrect). These were done to
appropriately mimic a real-world data annotation setup with a human annotator.
### IV-C Implementation Details
Base Model: We used a CNN-RNN architecture in our experiments where
InceptionV3 pretrained on the ImageNet-1k dataset was used as the feature
extractor and a GRU network as the decoder
222https://keras.io/examples/vision/video_classification/. The input frames
were scaled and normalized to a fixed input size of $224\times 224$ pixels and
fed into the Convolutional Neural Network (CNN). The features extracted were
fed into a $5$-layer GRU network which consists of $2$ GRU layers and $1$
fully connected layer with one dropout layer. The $2$ GRU layers had $20$ and
$12$ neurons, while the first fully connected layer had $8$ neurons with the
ReLU activation function. We used the adam optimizer with a learning rate of
$0.001$, momentum of $0.99$, batch size of $32$, and the network was trained
for $20$ epochs in each active learning iteration.
Oracle Model: We used a similar CNN-RNN architecture as the oracle model.
However, for the oracle model, the $2$ GRU layers of the GRU network had $40$
and $16$ neurons. We used the adam optimizer with a learning rate of $0.001$
for the UCF dataset and $0.01$ for the Kinetics dataset, momentum of $0.99$,
batch size of $64$, and the network was trained for $30$ epochs. As part of
future research, we plan to study the performance of our framework with other
architectures for the oracle model, and also conduct experiments with real
people as annotators.
### IV-D Experimental Setup
Each dataset was split into $5$ parts: $(i)$ an initial training set $L$;
$(ii)$ unlabeled set $U$; $(iii)$ test set $T$; $(iv)$ training set to train
the oracle model $L_{oracle}$; and $(v)$ test set $T_{oracle}$ to test the
oracle model and compute the entropy threshold $\tau_{oracle}$. The number of
samples (videos) in each of these sets, together with the accuracy of the
oracle model ($A_{oracle}$) for each dataset are depicted in Table I. We note
that a better trained oracle could have potentially improved the performance
of our algorithm; however, we wanted to validate our algorithm in a
challenging real-world setup, where the annotators can abstain from labeling
samples and can also provide incorrect labels. We therefore used an oracle
model with moderate accuracy ($\approx 70-75\%$) in our empirical studies.
Dataset | L | U | T | $L_{oracle}$ | $T_{oracle}$ | $A_{oracle}$
---|---|---|---|---|---|---
UCF | 250 | 320 | 150 | 697 | 185 | $75.61\%$
Kinetics | 500 | 750 | 300 | 584 | 211 | $71.34\%$
TABLE I: Dataset split used in our experiments, together with the accuracy of
the oracle model for the two datasets.
(a) UCF
(b) Kinetics
Figure 2: AL performance comparison. The $x$-axis denotes the iteration number
and the $y$-axis denotes the accuracy on the test set. Best viewed in color.
UCF Dataset: Video budget $b=25$, Frame Budget $k=100$
---
Methods | Total Videos Queried | Correct (%) | Incorrect (%) | Discarded (%)
RR | 250 | $54.4\pm 1.38$ | $42.4\pm 1.6$ | $3.2\pm 0.8$
ER | 250 | $55.2\pm 1.05$ | $40.4\pm 1.05$ | $4.4\pm 0.69$
EK | 250 | $53.06\pm 1.40$ | $30.93\pm 2.20$ | $16\pm 0.8$
Proposed | 250 | $58.66\pm 1.8$ | $31.06\pm 0.61$ | $10.26\pm 2.41$
TABLE II: Performance of the oracle on the UCF dataset. Kinetics Dataset:
Video budget $b=25$, Frame Budget $k=100$
---
Methods | Total Videos Queried | Correct (%) | Incorrect (%) | Discarded (%)
RR | 250 | $67.33\pm 1.40$ | $23.33\pm 1.40$ | $9.33\pm 1.40$
ER | 250 | $64\pm 1.2$ | $23.86\pm 2.83$ | $12.13\pm 4.02$
EK | 250 | $64.66\pm 1.28$ | $25.06\pm 3.02$ | $10.26\pm 4.31$
Proposed | 250 | $66\pm 1.44$ | $22.93\pm 1.51$ | $11.06\pm 1.28$
TABLE III: Performance of the oracle on the Kinetics dataset.
The oracle model was trained on $L_{oracle}$; each sample in $T_{oracle}$ was
then passed as an input to the trained oracle and the prediction entropy was
noted. The $50^{th}$ percentile of the prediction entropy distribution was
taken as the entropy threshold $\tau_{oracle}$; during the AL iterations, if
the entropy of any queried video exceeded this threshold, the oracle was
assumed to abstain from labeling. The base model was first trained on the set
$L$. In each AL iteration, each algorithm queried $b$ videos from the set $U$,
and $k$ frames from each of the $b$ videos. The $k$ frames of each video were
then passed as an input to the oracle model. Based on its prediction entropy
on the sample, the oracle may or may not furnish a label for a given unlabeled
video sample. If the oracle does not furnish a label for a given video, it was
discarded. The other unlabeled videos (which were labeled by the oracle),
together with the returned labels were then appended to the training set, the
base model was updated, and its accuracy was computed on the test set. The
process was repeated for $10$ iterations, which was taken as the stopping
criterion in this work. All the results were averaged over $3$ runs (with
different training, unlabeled and test sets) to rule out the effects of
randomness. The video budget $b$ was taken as $25$ and the frame budget $k$ as
$100$ in each AL iteration. The weight parameter $\mu$ in Equation (3) was
taken as $0.01$ and a Gaussian kernel was used to compute the diversity matrix
in Equation (2).
### IV-E Comparison Baselines
As mentioned in Section II, existing AL techniques for video classification
query the complete videos for annotation and the labels obtained are assumed
to be always correct. In our framework, the labeling oracle may refuse to
provide a label to a queried video and may also provide an incorrect label.
This is a more challenging and real-world scenario. It will thus not be fair
to compare our method against the existing techniques. We used three
comparison baselines in this work: $(i)$ Random-Random (RR), where we selected
a batch of $b$ videos at random and a subset of $k$ frames from each video at
random; $(ii)$ Entropy-Random (ER), where the $b$ videos with the highest
classification entropies were queried and $k$ frames were queried from each at
random; and $(iii)$ Entropy-kmeans (EK), where $b$ videos were first selected
using entropy sampling; $k$-means clustering was then performed and the $k$
frames corresponding to the $k$ cluster centroids were selected for query from
each video.
(a) Frame Budget k = 10
(b) Frame Budget k = 20
(c) Frame Budget k = 50
(d) Frame Budget k = 100
Figure 3: Study of the number of queried frames per video. The result with
$k=100$ (default setting) is the same as in Figure 2(a) and is included here
for the sake of completeness. Best viewed in color.
(a) Video Budget b = 15
(b) Video Budget b = 20
(c) Video Budget b = 25
(d) Video Budget b = 30
Figure 4: Study of the number of queried videos in each AL iteration. The
result with $b=25$ (default setting) is the same as in Figure 2(a) and is
included here for the sake of completeness. Best viewed in color.
### IV-F Active Learning Performance
The AL performance results are depicted in Figure 2. In each figure, the
$x$-axis represents the iteration number, and the $y$-axis denotes the
accuracy on the test set. The proposed method comprehensively outperforms the
RR method on both datasets. The ER method depicts random fluctuations in the
test accuracy over the AL iterations; our method, on the other hand, depicts a
more steady growth in the test accuracy. The EK method depicts the best
performance among the baselines, but is not as good as our method. Our method
outperforms EK in most of the AL iterations across both the datasets. It also
attains the highest accuracy after $10$ AL iterations for both the datasets.
We can conclude the following: $(i)$ our video selection criterion based on
uncertainty and diversity identifies the most informative videos in the
unlabeled set; and $(ii)$ our frame selection criterion based on
representative sampling selects a subset of exemplar frames from each queried
video, so that a large percentage of them can be correctly annotated by the
oracle, which enriches the quality of our training data. As a result, our
method augments maximal useful information to the deep neural network, which
boosts its generalization capability. These results unanimously corroborate
the potential of our framework in substantially reducing the human annotation
effort in real-world video classification applications, where labeling a
single sample involves significant time and human labor.
The performance of the oracle model is reported in Tables II and III for the
UCF and Kinetics datasets respectively. A total of $250$ videos were queried
from these datasets ($25$ videos in each of the $10$ AL iterations). The
tables show the percentage of these videos that were correctly annotated,
incorrectly annotated and discarded by the labeling oracle. For the UCF
dataset, and for the proposed method, the oracle correctly annotated $58.66\%$
of the queried videos (the highest among all the methods). This shows that
representative sampling through coreset is an effective strategy to identify
the exemplar frames from a queried video, which have a high probability of
receiving the correct label from the oracle, and augmenting useful information
to the training set. For the Kinetics dataset, $66\%$ of the videos queried by
our method were correctly annotated by the oracle, where as $67.33\%$ of the
videos queried by Random Sampling were annotated correctly by the oracle.
However, we note that, having a high percentage of unlabeled videos correctly
labeled by the oracle does not necessarily mean that the useful samples are
being queried. For instance, it is easy to select a batch of videos, which do
not have much useful content and are easy to label, and get a high percentage
of them correctly labeled by the oracle. However, these videos, even if
correctly labeled, will not augment much useful information to the training
set, as they are devoid of any useful content. Even though RR depicts a
slightly higher percentage of correctly labeled samples than our method in
Table III, its generalization accuracy is much worse than our method, as
evident from Figure 2(b). The key challenge is to query a set of informative
videos and get a high percentage of them correctly labeled by the oracle; both
of these are crucial in improving the generalization capability of the model
over time. The results in Figure 2 jointly capture both these aspects, and
show that our method outperforms the baselines.
### IV-G Effect of the Number of Queried Frames per Video
In this experiment, we studied the effect of the frame budget $k$ (number of
frames allowed to be selected from a queried video) on the AL performance. The
results on the UCF dataset, with frame budgets $10$, $20$, $50$ and $100$ are
presented in Figure 3. Our method depicts impressive performance across
different frame budgets. For frame budgets $20$, $50$ and $100$, our framework
attains the highest test accuracy after $10$ AL iterations. Note that querying
lesser number of frames from a video lessens the labeling burden on the
oracle, as the oracle has to review an even smaller number of frames to
furnish a label. These results show the promise and potential of our technique
to further reduce human annotation effort in a video classification
application.
### IV-H Effect of the Number of Queried Videos
The goal of this experiment was to study the effect of the video budget $b$
(number of videos queried in each AL iteration) on the AL performance. The
results on the UCF dataset with $b=15,20,25$ and $30$ are shown in Figure 4.
Our framework once again surpasses the baselines across the different budgets.
These results are important from the standpoint of a real-world application,
where the batch size is governed by the time, man-power and other available
resources in a given application, and is different for different applications.
## V Conclusion and Future Work
The goal of this research was to devise an efficient annotation mechanism to
reduce human annotation effort in video classification applications, where
annotating a single data instance is extremely tedious. Our framework
identifies a batch of informative videos, together with a set of exemplar
frames from each; the human annotator has to produce a label for each video
just by reviewing the subset of frames, instead of watching the complete video
end-to-end. To the best of our knowledge, this is the first research effort to
develop an AL technique for video classification, with this kind of a query
and annotation mechanism. Our empirical results validated the promise and
potential of our framework to drastically reduce human annotation effort in
training a deep neural network for video classification. We hope this research
will motivate the development of AL algorithms with other annotation
mechanisms, with the goal of further reducing the human annotation effort in
video classification.
As part of future work, we plan to validate the performance of our algorithm
on other applications where the data has a temporal nature. For instance, the
proposed query mechanism will also be very relevant in a text classification
application to identify informative text snippets, so that a human annotator
can furnish a label by reviewing only the snippets, rather than reading the
document end-to-end. We will also study the performance of our framework with
different size of the data splits, as outlined in Table I.
## References
* [1] V. Sharma, M. Gupta, A. Kumar, and D. Mishra, “Video processing using deep learning techniques: A systematic literature review,” _IEEE Access_ , vol. 9, 2021.
* [2] J. Ng, M. Hausknecht, S. Vijayanarasimhan, O. Vinyals, R. Monga, and G. Toderici, “Beyond short snippets: Deep networks for video classification,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2015.
* [3] A. Karpathy, G. Toderici, S. Shetty, T. Leung, R. Sukthankar, and L. Fei-Fei, “Large-scale video classification with convolutional neural networks,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2014\.
* [4] H. Tian, Y. Tao, S. Pouyanfar, S. Chen, and M. Shyu, “Multimodal deep representation learning for video classification,” _World Wide Web_ , vol. 22, no. 3, pp. 1325 – 1341, 2019.
* [5] B. Settles, “Active learning literature survey,” in _Technical Report: University of Wisconsin-Madison_ , 2010.
* [6] D. Yoo and I. Kweon, “Learning loss for active learning,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2019.
* [7] S. Tong and D. Koller, “Support vector machine active learning with applications to text classification,” _Journal of Machine Learning Research (JMLR)_ , vol. 2, pp. 45–66, 2001.
* [8] H. Osmanbeyoglu, J. Wehner, J. Carbonell, and M. Ganapathiraju, “Active machine learning for transmembrane helix prediction,” _BMC Bioinformatics_ , vol. 11, no. 1, 2010.
* [9] M. Gorriz, A. Carlier, E. Faure, and X. G. i Nieto, “Cost-effective active learning for melanoma segmentation,” in _Neural Information processing Systems (NeurIPS) Workshop_ , 2017.
* [10] P. Ren, Y. Xiao, X. Chang, P. Huang, Z. Li, B. Gupta, X. Chen, and X. Wang, “A survey of deep active learning,” _ACM Computing Surveys_ , vol. 54, no. 9, 2021.
* [11] A. Holub, P. Perona, and M. Burl, “Entropy-based active learning for object recognition,” in _IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPR-W)_ , 2008.
* [12] Y. Freund, S. Seung, E. Shamir, and N. Tishby, “Selective sampling using the query by committee algorithm,” _Machine Learning_ , vol. 28, no. 2-3, pp. 133–168, 1997.
* [13] W. Fu, M. Wang, S. Hao, and X. Wu, “Scalable active learning by approximated error reduction,” in _ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , 2018.
* [14] K. Wei, R. Iyer, and J. Bilmes, “Submodularity in data subset selection and active learning,” in _International Conference on Machine Learning (ICML)_ , 2015.
* [15] K. Fujii and H. Kashima, “Budgeted stream-based active learning via adaptive submodular maximization,” in _Neural Information Processing Systems (NeurIPS)_ , 2016.
* [16] O. Sener and S. Savarese, “Active learning for convolutional neural networks: A core-set approach,” in _International Conference on Learning Representations (ICLR)_ , 2018.
* [17] Y. Geifman and R. El-Yaniv, “Deep active learning with a neural architecture search,” in _Neural Information Processing Systems (NeurIPS)_ , 2019.
* [18] C. Shui, F. Zhou, C. Gagne, and B. Wang, “Deep active learning: Unified and principled method for query and training,” in _International Conference on Artificial Intelligence and Statistics (AISTATS)_ , 2020.
* [19] M. Ducoffe and F. Precioso, “Adversarial active learning for deep networks: a margin based approach,” in _International Conference on Machine Learning (ICML)_ , 2018.
* [20] C. Mayer and R. Timofte, “Adversarial sampling for active learning,” in _IEEE Winter Conference on Applications of Computer Vision (WACV)_ , 2020\.
* [21] B. Zhang, L. Li, S. Yang, S. Wang, Z. Zha, and Q. Huang, “State-relabeling adversarial active learning,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2020.
* [22] S. Sinha, S. Ebrahimi, and T. Darrell, “Variational adversarial active learning,” in _IEEE International Conference on Computer Vision (ICCV)_ , 2019.
* [23] R. Chattopadhyay, W. Fan, I. Davidson, S. Panchanathan, and J. Ye, “Joint transfer and batch-mode active learning,” in _International Conference on Machine Learning (ICML)_ , 2013.
* [24] D. Krueger, J. Leike, O. Evans, and J. Salvatier, “Active reinforcement learning: Observing rewards at a cost,” in _Neural Information Processing Systems (NeurIPS) Workshop_ , 2016.
* [25] N. Ruchansky, M. Crovella, and E. Terzi, “Matrix completion with queries,” in _ACM Conference on Knowledge Discovery and Data Mining (KDD)_ , 2015.
* [26] A. Molino, X. Boix, J. Lim, and A. Tan, “Active video summarization: Customized summaries via on-line interaction with the user,” in _Association for the Advancement of Artificial Intelligence (AAAI)_ , 2017\.
* [27] H. Shim, S. Hwang, and E. Yang, “Joint active feature acquisition and classification with variable-size set encoding,” in _Neural Information Processing Systems (NeurIPS)_ , 2018.
* [28] A. Joshi, F. Porikli, and N. Papanikolopoulos, “Breaking the interactive bottleneck in multi-class classification with active selection and binary feedback,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2010.
* [29] A. Biswas and D. Jacobs, “Active image clustering: Seeking constraints from humans to complement algorithms,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2012.
* [30] S. Xiong, Y. Pei, R. Rosales, and X. Fern, “Active learning from relative comparisons,” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 27, no. 12, 2015.
* [31] B. Qian, X. Wang, F. Wang, H. Li, J. Ye, and I. Davidson, “Active learning from relative queries,” in _International Joint Conference on Artificial Intelligence (IJCAI)_ , 2013.
* [32] A. Bhattacharya and S. Chakraborty, “Active learning with n-ary queries for image recognition,” in _IEEE Winter Conference on Applications of Computer Vision (WACV)_ , 2019.
* [33] A. Joshi, F. Porikli, and N. Papanikolopoulos, “Scalable active learning for multiclass image classification,” _IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_ , vol. 34, no. 11, pp. 2259 – 2273, 2012.
* [34] T. Sabata, P. Pulc, and M. Holena, “Semi-supervised and active learning in video scene classification from statistical features,” in _Workshop at the European Conference on Machine Learning (ECML)_ , 2018.
* [35] S. Sivaraman and M. Trivedi, “A general active-learning framework for on-road vehicle recognition and tracking,” _IEEE Transactions on Intelligent Transportation Systems (TITS)_ , vol. 11, no. 2, pp. 267 – 276, 2010.
* [36] R. Yan, J. Yang, and A. Hauptmann, “Automatically labeling video data using multi-class active learning,” in _IEEE International Conference on Computer Vision (ICCV)_ , 2003.
* [37] S. Vijayanarasimhan, P. Jain, and K. Grauman, “Far-sighted active learning on a budget for image and video recognition,” in _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2010.
* [38] L. Zhao, G. Sukthankar, and R. Sukthankar, “Robust active learning using crowdsourced annotations for activity recognition,” in _Workshop at the AAAI Conference on Artificial Intelligence_ , 2011.
* [39] S. Bandla and K. Grauman, “Active learning of an action detector from untrimmed videos,” in _IEEE International Conference on Computer Vision (ICCV)_ , 2013.
* [40] S. Ma, Z. Zeng, D. McDuff, and Y. Song, “Active contrastive learning of audio-visual video representations,” in _International Conference on Learning Representations (ICLR)_ , 2021.
* [41] S. Behpour, “Active learning in video tracking,” in _arXiv:1912.12557_ , 2020\.
* [42] D. Chan, S. Vijayanarasimhan, D. Ross, and J. Canny, “Active learning for video description with cluster-regularized ensemble ranking,” in _Asian Conference on Computer Vision (ACCV)_ , 2020.
* [43] J. Cai, J. Tang, Q. Chen, Y. Hu, X. Wang, and S. Huang, “Multi-view active learning for video recommendation,” in _International Joint Conference on Artificial Intelligence (IJCAI)_ , 2019.
* [44] A. Fathi, M. Balcan, X. Ren, and J. Rehg, “Combining self training and active learning for video segmentation,” in _British Machine Vision Conference (BMVC)_ , 2011.
* [45] D. Shen, J. Zhang, J. Su, G. Zhou, and C. Tan, “Multi-criteria based active learning for named entity recognition,” in _Association for Computational Linguistics (ACL)_ , 2004.
* [46] B. Sriperumbudur, K. Fukumizu, and G. Lanckriet, “Universality, characteristic kernels and rkhs embedding of measures,” _Journal of Machine Learning Research (JMLR)_ , vol. 12, 2011.
* [47] X. Yuan and T. Zhang, “Truncated power method for sparse eigenvalue problems,” _Journal of Machine Learning Research (JMLR)_ , vol. 14, pp. 899 – 925, 2013.
* [48] W. Johnson and J. Lindenstrauss, “Extensions of lipschitz mappings into a hilbert space,” in _Conference in Modern Analysis and Probability_ , 1984\.
* [49] S. Vempala, “The random projection method,” in _Americal Mathematical Society_ , 2004.
* [50] R. Farahani and M. Hekmatfar, _Facility Location: Concepts, Models, Algorithms and Case Studies_. Physica-Verlag HD, 2009.
* [51] W. Cook, W. Cunningham, W. Pulleyblank, and A. Schrijver, _Combinatorial Optimization_. Springer, 1998.
* [52] K. Soomro, A. Zamir, and M. Shah, “Ucf 101: A dataset of 101 human action classes from videos in the wild,” in _Techical Report, UCF_ , 2012.
* [53] W. Kay, J. Carreira, K. Simonyan, B. Zhang, C. Hillier, S. Vijayanarasimhan, F. Viola, T. Green, T. Back, P. Natsev, M. Suleyman, and A. Zisserman, “The kinetics human action video dataset,” in _arXiv:1705.06950_ , 2017.
|
# Pressure Tuning of Berry Curvature in CrGeTe3
G. Scharf Raymond and Beverly Sackler School of Physics and Astronomy, Tel-
Aviv University, Tel Aviv, 69978, Israel B. Hen Raymond and Beverly Sackler
School of Physics and Astronomy, Tel-Aviv University, Tel Aviv, 69978, Israel
P. M. Sarte Materials Department, University of California, Santa Barbara,
California, 93106, USA B. R. Ortiz Materials Department, University of
California, Santa Barbara, California, 93106, USA G. Kh. Rozenberg Raymond
and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel
Aviv, 69978, Israel S. D. Wilson Materials Department, University of
California, Santa Barbara, California, 93106, USA A. Ron Raymond and Beverly
Sackler School of Physics and Astronomy, Tel-Aviv University, Tel Aviv, 69978,
Israel
###### Abstract
The integrated Berry curvature is a geometric property that has dramatic
implications on material properties. This study investigates the integrated
Berry curvature and other scattering mechanisms through their contribution to
the anomalous Hall effect in CrGeTe3. The Anomalous Hall effect is absent in
the insulating phase of CrGeTe3 and evolves with pressure in a dome-like
fashion as pressure is applied. The dome’s edges are characterized by Fermi
surface deformations, manifested as mixed electron and hole transport. We
discuss the possibility that in CrGeTe3 the integrated Berry curvature is
tuned by the application of hydrostatic pressure due to its relation to the
Fermi surface deformations.
For electrons in solids, Berry phase is a geometric property of the band
structure that has dramatic implications on materials’ properties [1]. It is
acquired when a system is subject to a cyclic adiabatic transformation in its
parameter space, and it is determined by the integrated Berry curvature. As a
band-structure property, one may conjecture that dramatic changes to the Fermi
surface will result in considerable changes to the Berry curvature and thus
may result in variation of its integrated value. An extreme case of such a
change would be the metal-insulator transition [2], where at the insulating
state, there is no Fermi surface, and at the metallic state, a Fermi surface
forms, which may host a nonzero integrated Berry curvature. Such effects were
recently observed and understood theoretically in graphene moiré superlattices
where Berry curvature features were tuned by varying the electric displacement
field and carrier density [3, 4].
A common manifestation of the Berry phase in electronic transport properties
is the anomalous Hall effect (AHE). The AHE is an additional contribution to
the transverse resistivity ($\rho_{xy}$) on top of the ordinary Hall effect.
It occurs in materials where time-reversal symmetry is broken in the presence
of spin-orbit interaction [5]. As such, it can be probed through measurements
of $\rho_{xy}$ as a function of the magnetic field and serve as a superb probe
for investigating the integrated Berry curvature of electrons in solids. We
chose CrGeTe3, a ferromagnetic insulator undergoing a metal-insulator
transition with the application of hydrostatic pressure, as a laboratory for
investigating the evolution of the integrated Berry curvature when the Fermi
surface is strongly deformed.
CrGeTe3 is a layered ferromagnetic insulator with a Curie temperature (TCurie)
of $\mathrm{\sim 67\ K}$ [6] which has recently attracted a lot of attention.
Inelastic neutron scattering suggests that CrGeTe3 is a topological magnonic
insulator [7] that is predicted to sustain ferromagnetism to the 2D limit [8,
9, 10]. Additionally, short-range fluctuations seem to play an important role
above the transition temperature [11, 12], in great similarity to the closely
related compound CrSiTe3 [13, 14, 15].
Application of hydrostatic pressure changes TCurie of CrGeTe3. Up to
$\mathrm{4.5\ GPa}$, the Curie temperature decreases as pressure is applied
[16, 2]. Above $\mathrm{4.5\ GPa}$, TCurie increases dramatically, rising from
$\mathrm{\sim 54\ K}$ to $\mathrm{\sim 250\ K}$ at $\mathrm{9.1\ GPa}$ [2].
Within this pressure range, CrGeTe3 undergoes a metal-insulator transition at
$\mathrm{\sim 6\ GPa}$ [2]. The coexistence of time-reversal symmetry breaking
with the metal-insulator transition in a material with a high Z element (Te)
makes CrGeTe3 an ideal candidate to search for an evolution of the Berry
curvature as the system is tuned through the metal-insulator transition.
In this letter, we demonstrate that the various contributions to the AHE in
CrGeTe3 can be tuned by hydrostatic pressure which at certain pressures is
present also above $\mathrm{300\ K}$ suggesting possible enhancement of
TCurie. Measurements of the AHE at a wide range of hydrostatic pressures at
T=2K reveal a dome-like behavior that onsets at the metal-insulator transition
and is quenched towards higher pressures. The AHE dome coincides with the
pressure range where Fermi surface deformations are observed through the
ordinary Hall effect. We discuss a scenario where the AHE dome emerges due to
the appearance of a nonzero integrated Berry curvature due to the observed
Fermi surface deformation.
## I Methods
To create the CrGeTe3 crystals, Cr powder (99.95%, alfa), Ge powder
(99.9999%), and Te lump (99.999+%) were sealed in a fused silica ampule at an
approximate ratio of 1:1:8 Cr:Ge:Te. Fluxes were heated to 900°C at a rate of
200°C/hr, soaked at 900°C for 24h, and then slowly cooled down to 550°C at a
rate of 2°C/h. The resulting fluxes were centrifuged at 550°C to remove molten
Te from the crystals, after which thin platelets of dimensions 1mm x 2mm x
0.1mm were mechanically isolated.
The pressure was exerted on the samples using miniature diamond anvil cells
(DACs) [17], with diamond anvil culets of $\mathrm{300\ \mu m}$. A rhenium
gasket was drilled, then filled and covered with a powder layer of 75% Al2O3
and 25% NaCl for electrical insulation. Two pressure cells were loaded with
$\mathrm{\sim 5\ \mu m}$ thick CrGeTe3 flakes and placed on top of the
insulating layer, which functions as a pressure-transmitting medium. A
$\mathrm{\sim 5\ \mu m}$ thick Pt foil was cut into triangular pieces and
placed in contact with the CrGeTe3 flakes allowing electrical transport
measurements at elevated pressures in the Van der Pauw geometry. As such, the
$\rho_{xx}$ and $\rho_{xy}$ are inferred from the measured resistance up to a
factor of order unity due to uncertainties in the sample geometry and
thickness, which are inevitable inside a DAC. In addition, Ruby fragments were
placed between the Pt leads for pressure determination [18]. The samples were
compressed in steps of $\mathrm{2-4\ GPa}$ and then cooled down from ambient
temperature to $\mathrm{2\ K}$. Measurements from the two cells are shown in
this manuscript.
## II Results and Discussion
Figure 1(a) shows that gradual application of pressure results in a
significant drop to the sample resistance, which begins to saturate at
pressures of $\mathrm{\sim 6\ GPa}$ where a metal-insulator transition occurs
in agreement with Ref. [19, 2] (see sections S1 and S4 in the supplementary
material for resistivity versus temperature plots). TCurie as a function of
pressure from our AHE at high temperatures, is also shown in Figure 1(a) and
will be discussed later in this section. We note that the pressure in which we
observe the metal-insulator transition, and the dependence of Curie
temperatures on pressure, are consistent with previous reports, indicating the
similarity in sample quality and pressure conditions.
The inset of Figure 1(a) shows the Hall coefficient as a function of pressure
at $\mathrm{T=2\ K}$, which exhibits a dome-like behavior as a function of
pressure. In the metallic state (at $\mathrm{6<P<14.5\ GPa}$), the negative
sign of the Hall coefficient indicates that transport is electron-dominated at
all temperatures, as it is demonstrated for $\mathrm{10.6\ GPa}$ in Figure
1(b). The full data set is available in section S5. Figure 1(c) shows that at
the edges of the dome, both electrons and holes contribute to transport, as
can be seen by the flattening and sign change of the Hall slope as a function
of temperature. We note that a similar behavior also occurs at other pressures
in the vicinity of $\mathrm{14\ GPa}$, as shown in supplementary sections S5
and S6. These most likely originate from hole-like Te 5p and electron-like Cr
3d bands. It should be mentioned that measurements of the Hall effect were
infeasible at pressures below $\mathrm{3.2\ GPa}$ due to the large
longitudinal resistivity relative to the magnitude of the Hall effect
(supplementary material section S6).
Figure 1: The left axis of the panel (a) shows a significant decrease in the
samples’ resistance as pressure is applied due to the metal-insulator
transition. The Blue and red points represent measurements from cells 1 and 2,
respectively. The values are scaled by a single geometric factor of order
unity, which is used throughout the manuscript for any longitudinal
resistivity measurement. On the right axis, we show the Curie temperature as a
function of applied pressure. The cyan dots represent measurements from this
work based on the AHE, which extend the pressure range covered in Ref. [2],
shown in green. The uncertainties in the value of TCurie are estimated as the
interval between our sampling points, and the pressure uncertainties are
estimated to be about $\mathrm{0.5\ GPa}$. The arrows signify that the value
of $\mathrm{300\ K}$ is a lower bound for the Curie point, as it is the
highest temperature in which data was taken. The inset shows the Hall
coefficient as a function of applied pressure at $\mathrm{2\ K}$ extracted
from a linear fit in the field range between $\mathrm{4\ kOe}$ and
$\mathrm{10\ kOe}$. The Blue and the red points are measurements of
$\rho_{xy}$ from the first and second cells, respectively. A single geometric
factor of order unity was used to scale the values of $\rho_{xy}$ here and
throughout the manuscript. Panels (b) and (c) show the antisymmetrized Hall
measurements at pressures of $\mathrm{10.6\ GPa}$ and $\mathrm{3.2\ GPa}$ at
different temperatures as a function of the magnetic field at the range which
was used to calculate the Hall coefficient.
Above $\mathrm{5.6\ GPa}$, when CrGeTe3 enters the metallic state, a
significant AHE signal is observed. Figure 2(a) shows a characteristic
behavior of the AHE in the intermediate pressure regime. Here, the AHE is the
strongest at low temperatures and monotonically weakens as the temperature
increases. From this data, it is clear that the AHE persists to much higher
temperatures than the Curie temperature (TCurie) under ambient pressure
($\mathrm{\sim 67\ K}$), which we interpret as an increase of TCurie. We note
that the AHE can also occur in paramagnetic materials [20, 21], and therefore
its presence at high temperatures does not necessarily signify enhancement of
TCurie. However, we find this scenario less likely since the trend observed by
our measurements is a smooth continuation of the trend observed by
magnetometry measurements [2] shown in Figure 1(a). Figure 2(b) shows
measurements of the AHE at $\mathrm{13.5\ GPa}$, characteristic of the high-
pressure regime between $\mathrm{13.5}$ and $\mathrm{17.6\ GPa}$. At low
temperatures (blue curves), the AHE is completely absent from measurements of
$\rho_{xy}$, as can be seen by the absence of the steep low field AHE slope.
As the temperature increases, the AHE gradually appears and is enhanced at
elevated temperatures. We note that the AHE does not decay even at room
temperature, continuing the trend observed in Ref. [2], thus possibly
indicating that TCurie in CrGeTe3 surpasses room temperature at this pressure
range.
Figure 2: Measurements of the Hall effect in sample 1 at different
temperatures at pressures of $\mathrm{10.6\ GPa}$ (panel (a)) and
$\mathrm{13.5\ GPa}$ (panel (b)). The steep slopes at low fields are due to
the AHE.
The qualitative differences in the evolution of the AHE with temperature
suggest that different mechanisms are at play in different pressure regimes.
To disentangle the contributions to the AHE, we separate the anomalous Hall
resistivity $\rho_{AHE}$ to intrinsic and extrinsic sources and follow Hou
$et\ al.$ [22] who further distinguish between extrinsic scattering
originating from static (temperature independent) and dynamic (temperature
dependent) scattering mechanisms. At low temperatures, quasiparticles are
frozen, and there are no dynamic scattering events. Therefore,
$\rho_{AHE}$$(T=0)$ is a static effect, either intrinsic with geometric
origins or extrinsic emanating from disorder. At higher temperatures,
quasiparticles are thermally activated and dynamic extrinsic scattering
mechanisms may contribute to the AHE. We turn to interpret our results in
light of these distinctions in the three pressure regimes.
$\rho_{AHE}$ is extracted using a procedure similar to Ref. [23] detailed in
the supplementary section S3. In the insulating state, the values of the
longitudinal resistance ($\mathrm{R_{xx}}$) intermixed in the measurements of
$\mathrm{R_{xy}}$ are dominant for fields smaller than $\mathrm{0.2\ T}$ (see
section S6 in the supplementary material). Therefore, our measurements are
insensitive to small anomalous Hall signals at this pressure range. Figure 3
shows $\rho_{AHE}$ as a function of temperature for various pressures above
$\mathrm{5.6\ GPa}$. At the intermediate pressure regime, at low temperatures,
$\rho_{AHE}$ $\neq 0$. This suggests that the pressure tuning into the
metallic state activates either an intrinsic contribution to the AHE or a
static extrinsic scattering mechanism. The signal decays as the temperature
increases, and its disappearance marks the Curie temperature. In the high-
pressure regime ($\mathrm{P>13.5\ GPa}$), $\rho_{AHE}$ $=0$ at low
temperatures and smoothly increases as the temperature increases, persisting
up to room temperature above which we could not heat our DACs. The absence of
AHE indicates either a perfect cancellation of contributions from various
scattering mechanisms, meaning that the sum of all the various contributions
to the AHE is zero, or the complete nullification of all of them. Perfect
cancellation typically occurs when two mechanisms contribute to $\rho_{AHE}$
with opposite signs, which typically occurs at a specific temperature as seen,
for example, in Ref. [24, 25, 26, 27]. In contrast, in CrGeTe3 at
$\mathrm{P>13\ GPa}$, $\rho_{AHE}$ $=0$ at a wide temperature range (over
$\mathrm{150\ K}$ at $\mathrm{17.6\ GPa}$) rather than crossing zero at a
particular temperature. Perfect accidental cancellation of various scattering
mechanisms at such a wide temperature range is unlikely. Therefore we deduce
that for $\mathrm{P>13\ GPa}$, all scattering mechanisms are negligible at low
temperatures, and the behavior shown in Figure 3 is dominated by scattering
off of thermally activated quasi-particles.
Figure 3: $\rho_{AHE}$ as a function of temperature for various pressures,
measured in sample 2. A similar plot for sample 1 is shown in S2 in the
supplementary material. At the intermediate pressure regime, between the
metal-insulator transition and $\mathrm{13\ GPa}$, $\rho_{AHE}$ $\neq 0$ at
low temperatures and decays smoothly as the temperature increases. In
contrast, in the high-pressure regime, at low temperatures $\rho_{AHE}$ $=0$,
and increases as the temperature increases.
In Figure 4, we plot $\rho_{AHE}$ at $\mathrm{2\ K}$ as a function of the
pressure, which exhibits a dome-like shape starting at the metal-insulator
transition and finishing around $\mathrm{13\ GPa}$. To the best of our
knowledge, this behavior has not been observed in the past. Typically,
ferromagnets exhibit a monotonic behavior of the AHE as a function of
pressure, as seen, for example, in $\mathrm{CeAlSi}$ [28] and
$\mathrm{Co_{3}Sn_{2}S_{2}}$ [29], where in the former, the AHE is generated
by skew scattering and in the latter by the intrinsic Berry phase. To
understand this behavior, we look at the relation between $\rho_{AHE}$ and
$\rho_{xx}$, which at low temperatures, in the absence of dynamic scattering,
simplifies to [22]:
$\rho_{AHE}=\alpha\rho_{xx}+\beta_{0}\rho_{xx}^{2}$ (1)
where $\alpha$ represents contributions from skew scattering, and $\beta_{0}$
is a mixture of intrinsic and static side jump mechanisms. In our experiment,
we tune $\rho_{xx}$ by changing the hydrostatic pressure P. The inset to
Figure 4 shows $\rho_{AHE}$ as a function of $\rho_{xx}$$(P)$ at a constant
temperature $\mathrm{T=2\ K}$, where a clear non-parabolic hysteretic behavior
is observed. This means that the application of pressure changes not only
$\rho_{xx}$ but also $\alpha$ and $\beta_{0}$. Transport measurements alone
could not disentangle the contributions of the intrinsic and the static
extrinsic mechanisms to the AHE. However, in CrGeTe3, the AHE dome onsets and
ends in regimes where mixed transport of electrons and holes is observed
(Figure 1(a) inset), indicative of Fermi surface deformations. We suggest that
the Fermi surface deformations result in the appearance of nonzero integrated
Berry curvature, which is in some similarity to what has been observed in
graphene moiré superlattices [3, 4]. Generally, the intrinsic contribution to
the AHE depends on the integrated Berry curvature over all occupied states
[1]. Therefore, it can be tuned by either changing the Fermi level or by
causing changes to the band structure that manifest changes to the Berry
curvature. A toy model calculation, based on Ref [30], where the integrated
Berry curvature is tuned by changing the Fermi energy can be found in Ref [1].
Similar phenomena were observed experimentally and understood theoretically in
graphene moiré superlattices, where both the band structure and the Fermi
level were changed[3, 4]. We suggest that in CrGeTe3, at the metal-insulator
transition, the effects of nonzero integrated Berry’s curvature appear,
contributing to $\rho_{AHE}$ at low temperatures. As pressure increases, those
effects get stronger due to the hybridization of the Te and the Cr bands,
which mark the rising part of the AHE dome. As the pressure further increases,
the AHE weakens and goes to zero at the point where mixed electron/hole
transport is observed again. We note that at the end of the AHE dome,
structural anomalies were observed by x-ray diffraction as nonmonotonic
behavior of the $\angle$Te-Cr-Te angle [19]. In light of the nonmonotonic
behavior of atomic positions, one may not be surprised by the nonmonotonic
behavior we report in the electronic properties of CrGeTe3. Additionally, Dong
and coauthors [31] observed a kink in the axial ratio at $\mathrm{14\ GPa}$ in
CrGeTe3 and claim it is indicative of an isostructural phase transition, which
is possibly related to the mixed electron-hole transport we observe. It is
worth noting that the behavior depicted in Figure 4 can also be explained
through changes in the skew-scattering and static side jumps contributions to
the AHE as a function of the pressure. However, since the AHE dome onsets at
the metal-insulator transition and ends where another Fermi surface
deformation was observed strengthens our belief that the observed dome-like
behavior of the AHE at low temperatures is due to changes in the integrated
Berry’s curvature tuned by the application of hydrostatic pressure.
Figure 4: $\rho_{AHE}$ measured at $\mathrm{2\ K}$ as a function of applied
pressure. The black line is a guide to the eye. The inset shows $\rho_{AHE}$,
measured at $\mathrm{2\ K}$, as a function of the longitudinal resistivity
$\rho_{xx}$, showing a hysteretic behavior that deviates from the parabolic
relation in equation 1. The Blue and the red points are from the first and
second cells, respectively. Their resistivity values are scaled by a single
geometric factor of order unity, as was previously mentioned in Figure 1.
In summary, we have measured the AHE in CrGeTe3 as a function of applied
hydrostatic pressure and temperature. We suggest that the application of
hydrostatic pressure to CrGeTe3 results in the tuning of the intrinsic
contribution to the AHE, which originates from nonzero integrated Berry’s
curvature in proximity to the metal-insulator transition and a Fermi surface
deformation. We also found that at elevated pressures, the AHE appears at
temperatures above $\mathrm{300\ K}$ suggestive of possible enhancement of
TCurie, continuing and agreeing with the trend observed by Bhoi $et\ al.$ [2].
###### Acknowledgements.
A. R. acknowledges support from the Zuckerman Foundation, and the Israel
Science Foundation (Grant No. 1017/20). S.D.W., B.R.O., and P.S. gratefully
acknowledge support via the UC Santa Barbara NSF Quantum Foundry funded via
the Q-AMASE-i program under award DMR-1906325. G. Kh. R. acknowledges the
Israel Science Foundation (Grants No. 1748/20). G. S. thanks Shay Sandik, Itai
Silber, and Gal Tuvia for the help with the cryogenic equipment.
## References
* Xiao _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
* Bhoi _et al._ [2021] D. Bhoi, J. Gouchi, N. Hiraoka, Y. Zhang, N. Ogita, T. Hasegawa, K. Kitagawa, H. Takagi, K. H. Kim, and Y. Uwatoko, Physical Review Letters 127, 217203 (2021).
* Sinha _et al._ [2022] S. Sinha, P. C. Adak, A. Chakraborty, K. Das, K. Debnath, L. V. Sangani, K. Watanabe, T. Taniguchi, U. V. Waghmare, A. Agarwal, _et al._ , Nature Physics 18, 765 (2022).
* Kuiri _et al._ [2022] M. Kuiri, C. Coleman, Z. Gao, A. Vishnuradhan, K. Watanabe, T. Taniguchi, J. Zhu, A. H. MacDonald, and J. Folk, Nature Communications 13, 6468 (2022).
* Nagaosa _et al._ [2010] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Reviews of modern physics 82, 1539 (2010).
* Carteaux _et al._ [1995] V. Carteaux, D. Brunet, G. Ouvrard, and G. Andre, Journal of Physics: Condensed Matter 7, 69 (1995).
* Zhu _et al._ [2021] F. Zhu, L. Zhang, X. Wang, F. J. Dos Santos, J. Song, T. Mueller, K. Schmalzl, W. F. Schmidt, A. Ivanov, J. T. Park, _et al._ , Science advances 7, eabi7532 (2021).
* Li and Yang [2014] X. Li and J. Yang, Journal of Materials Chemistry C 2, 7071 (2014).
* Xu _et al._ [2018] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Computational Materials 4, 1 (2018).
* Zhuang _et al._ [2015] H. L. Zhuang, Y. Xie, P. Kent, and P. Ganesh, Physical Review B 92, 035407 (2015).
* Chen _et al._ [2022] L. Chen, C. Mao, J.-H. Chung, M. B. Stone, A. I. Kolesnikov, X. Wang, N. Murai, B. Gao, O. Delaire, and P. Dai, Nature communications 13, 1 (2022).
* Lin _et al._ [2017] G. Lin, H. Zhuang, X. Luo, B. Liu, F. Chen, J. Yan, Y. Sun, J. Zhou, W. Lu, P. Tong, _et al._ , Physical Review B 95, 245212 (2017).
* Ron _et al._ [2019] A. Ron, E. Zoghlin, L. Balents, S. Wilson, and D. Hsieh, Nature communications 10, 1 (2019).
* Ron _et al._ [2020] A. Ron, S. Chaudhary, G. Zhang, H. Ning, E. Zoghlin, S. Wilson, R. Averitt, G. Refael, and D. Hsieh, Physical Review Letters 125, 197203 (2020).
* Williams _et al._ [2015] T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler, M. B. Stone, J.-Q. Yan, and D. Mandrus, Physical Review B 92, 144404 (2015).
* Sakurai _et al._ [2021] T. Sakurai, B. Rubrecht, L. Corredor, R. Takehara, M. Yasutani, J. Zeisner, A. Alfonsov, S. Selter, S. Aswartham, A. Wolter, _et al._ , Physical Review B 103, 024404 (2021).
* Sterer _et al._ [1990] E. Sterer, M. Pasternak, and R. Taylor, Review of scientific instruments 61, 1117 (1990).
* Dewaele _et al._ [2008] A. Dewaele, M. Torrent, P. Loubeyre, and M. Mezouar, Phys. Rev. B 78, 104102 (2008).
* Yu _et al._ [2019] Z. Yu, W. Xia, K. Xu, M. Xu, H. Wang, X. Wang, N. Yu, Z. Zou, J. Zhao, L. Wang, _et al._ , The Journal of Physical Chemistry C 123, 13885 (2019).
* Maryenko _et al._ [2017] D. Maryenko, A. Mishchenko, M. Bahramy, A. Ernst, J. Falson, Y. Kozuka, A. Tsukazaki, N. Nagaosa, and M. Kawasaki, Nature communications 8, 1 (2017).
* Culcer _et al._ [2003] D. Culcer, A. MacDonald, and Q. Niu, Phys. Rev. B 68, 045327 (2003).
* Hou _et al._ [2015] D. Hou, G. Su, Y. Tian, X. Jin, S. A. Yang, and Q. Niu, Physical review letters 114, 217203 (2015).
* Liu _et al._ [2018] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, _et al._ , Nature physics 14, 1125 (2018).
* Fang _et al._ [2003] Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R. Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K. Terakura, Science 302, 92 (2003).
* Pureur _et al._ [2004] P. Pureur, F. W. Fabris, J. Schaf, and I. A. Campbell, Europhysics Letters 67, 123 (2004).
* Zeng _et al._ [2006] C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Phys. Rev. Lett. 96, 037204 (2006).
* Haham _et al._ [2011] N. Haham, Y. Shperber, M. Schultz, N. Naftalis, E. Shimshoni, J. W. Reiner, and L. Klein, Physical Review B 84, 174439 (2011).
* Piva _et al._ [2023] M. M. Piva, J. C. Souza, V. Brousseau-Couture, S. Sorn, K. R. Pakuszewski, J. K. John, C. Adriano, M. Côté, P. G. Pagliuso, A. Paramekanti, and M. Nicklas, Phys. Rev. Res. 5, 013068 (2023).
* Liu _et al._ [2020] Z. Liu, T. Zhang, S. Xu, P. Yang, Q. Wang, H. Lei, Y. Sui, Y. Uwatoko, B. Wang, H. Weng, _et al._ , Physical Review Materials 4, 044203 (2020).
* Onoda _et al._ [2006] S. Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. Lett. 97, 126602 (2006).
* Dong _et al._ [2020] E. Dong, B. Liu, Q. Dong, X. Shi, X. Ma, R. Liu, X. Zhu, X. Luo, X. Li, Y. Li, _et al._ , Physica B: Condensed Matter 595, 412344 (2020).
## III Supplemental material
## Appendix S2 - S1 - The MIT - R(T) at different pressures
The longitudinal resistance as a function of temperature, normalized at
$\mathrm{9.5\ K}$, measured at different pressures in the first cell. The
change in the behavior of the graph from decreasing to increasing as a
function of the temperature in the low-temperature regime by application of
pressure indicates a metal-insulator transition driven by the application of
pressure on the CrGeTe3.
Figure S1: The longitudinal resistance as a function of temperature,
normalized at $\mathrm{9.5\ K}$, measured at different pressures in the first
cell. The change in the behavior of the graph from decreasing to increasing as
a function of the temperature by application of pressure indicates a MIT
driven by the application of pressure on the CrGeTe3.
## Appendix S3 - S2 - THE AHE measured in the first cell
Here we present our measurements of $\rho_{AHE}$ as a function of temperature
for the various pressures measured in the first sample. As was also observed
in the second sample ( in the main text), At pressures below $\mathrm{13\
GPa}$, $\rho_{AHE}\neq 0$ at low temperatures and decays smoothly as the
temperature increases. In contrast, for $\mathrm{P>13\ GPa}$, at low
temperatures $\rho_{AHE}=0$, and increases as the temperature increases.
Figure S2: $\rho_{AHE}$ as a function of temperature for the various pressures
measured for sample 1.
## Appendix S4 - S3 - The extraction of $\rho_{AHE}$ from the measurements
$\rho_{AHE}$ in a specific temperature and pressure, is extracted from the
measurements by measuring $\mathrm{R_{xy}(H)}$ and antisymmetrize it. This
results in graphs as shown in section S6. Then we do a linear fit to the high-
field regime ($\mathrm{4\ kOe<H}$), and the intersection of the fit with the
y-axis is the AHE resistance ($\mathrm{R_{AHE}}$) (see Fig.S3). Finally, by
multiplying $\mathrm{R_{AHE}}$ with the width of the sample, we get
$\rho_{AHE}$.
Figure S3: Here we show an example of how we extracted the AHE resistance
($\mathrm{R_{AHE}}$) for each pressure at different temperatures. The figure
displays the antisymmetrization of the raw data of $\mathrm{R_{xy}}$ as a
function of the applied field H, measured at several different temperatures
for the first sample in the metallic state at a pressure of $\mathrm{13.5\
GPa}$. The solid lines are the linear fit for the high-fields regime
($\mathrm{4\ kOe<H}$) at each temperature, and the big dots represent the
intersection of each fit with the y-axis. The intersection of each fit is
$\mathrm{R_{AHE}}$ measured at each temperature.
## Appendix S5 - S4 - $\rho_{xx}$ at different pressures and temperatures
In Figure S5, we present the longitudinal resistivity as a function of
temperature in the metallic state, measured on both samples. As can be seen,
they show very similar behavior as all of them are monotonic - increasing with
the temperature and showing similar values. As such, going back to our
measurements of the AHE as a function of the temperature (see Figures 2 and
S2), the observed behaviors cannot be explained just by the scaling of
$\rho_{AHE}$ with the $\rho_{xx}$. First, the scaling of $\rho_{xx}$ cannot
explain the change in the behavior of the AHE between the intermediate
pressure regime ($\mathrm{5.6\ GPa<P<13\ GPa}$) and the high-pressure regime
($\mathrm{13\ Gpa<P}$). Second, it cannot explain why at $\mathrm{13.5\ GPa}$,
the AHE is stronger than at higher pressures in the first sample and thus
probably not also in the second cell. Finally, going back to the low-
temperature behavior of the AHE as shown in Figure 4, the values of
$\rho_{xx}$ at low temperatures (shown in log-scale in Figure S6) cannot
explain the dome-like behavior of the AHE.
Figure S4: The longitudinal resistivity as a function of temperature at
different pressures presented in log-scale, on the left in the first sample
and on the right in the second sample. Figure S5: The longitudinal resistivity
as a function of temperature at different pressures in the metallic state, on
the left in the first sample and on the right in the second sample.
Figure S6: The longitudinal resistivity at low temperature (2K) as a function
of the pressure in log-scale. The Blue and the red points are from the first
and second cells, respectively. Their resistivity values are scaled by a
single geometric factor of order unity which was used throughout the
manuscript for each longitudinal measurement.
## Appendix S6 - S5 - The Hall slopes measured at different pressures and
temperatures
Here, we present the Hall slopes as a function of temperature, measured at
different pressures. In most measurements, the Hall slope is negative, meaning
that although there is a mix of electrons and holes in all pressures, in most
of the pressures, we can treat the transport as of electron-like charge
carriers. However, at $\mathrm{3.2\ GPa}$ and at $\mathrm{14.5\ GPa}$, there
is a change in the sign of the Hall slope, indicating that these pressures,
both holes and electrons contribute to the transport where their contributions
are temperature dependent, which can be seen in Figure 1(b) in the text as
well. At these pressures, we cannot treat the transport as dominated by a
single charge carrier. The fact that the Hall slope changes sign as a function
of the temperature at those pressures but not before might indicate changes in
the band structure of the CrGeTe3, which may result in a change in the
integrated Berry curvature.
Figure S7: The Hall slopes as a function of temperature, measured at
$\mathrm{0.87\ GPa}$ and at $\mathrm{3.2\ GPa}$. Figure S8: The Hall slopes as
a function of temperature, measured at $\mathrm{0.87\ GPa}$, $\mathrm{3.2\
GPa}$, and $\mathrm{5.6\ GPa}$. Figure S9: The Hall slopes as a function of
temperature, measured at different pressures. The dots represent data from the
first cell, and the Xs denote measurements from the second cell.
## Appendix S7 - S6 - Raw Data measurements of $\mathrm{R_{xy}}$ and the
resulted anti-symmetric plots for all pressures and cells
Here we present measurements of $\mathrm{R_{xy}}$ as a function of the applied
field H at various pressures and temperatures, both in their raw form and
after undergoing antisymmetrization. When there is significant mixing of
$\mathrm{R_{xx}}$ and $\mathrm{R_{xy}}$ in the measurements, it is reflected
in the raw data, which appears neither symmetric nor antisymmetric. This
effect has been observed multiple times, particularly in the low-pressure
regime before the metal-insulator transition.
Figure S10: The left panel displays the raw data of $\mathrm{R_{xy}}$ as a
function of the applied field H for the first sample in the metallic state at
a pressure of $\mathrm{7.5\ GPa}$. The right panel shows the same data after
antisymmetrization. Figure S11: The left panel displays the raw data of
$\mathrm{R_{xy}}$ as a function of the applied field H for the first sample in
the metallic state at a pressure of $\mathrm{9.5\ GPa}$. The right panel shows
the same data after antisymmetrization. Figure S12: The left panel displays
the raw data of $\mathrm{R_{xy}}$ as a function of the applied field H for the
first sample in the metallic state at a pressure of $\mathrm{10.6\ GPa}$. The
right panel shows the same data after antisymmetrization. Figure S13: The left
panel displays the raw data of $\mathrm{R_{xy}}$ as a function of the applied
field H for the first sample in the metallic state at a pressure of
$\mathrm{11.7\ GPa}$. The right panel shows the same data after
antisymmetrization. Figure S14: The left panel displays the raw data of
$\mathrm{R_{xy}}$ as a function of the applied field H for the first sample in
the metallic state at a pressure of $\mathrm{13.5\ GPa}$. The right panel
shows the same data after antisymmetrization. Figure S15: The left panel
displays the raw data of $\mathrm{R_{xy}}$ as a function of the applied field
H for the first sample in the metallic state at a pressure of $\mathrm{14.5\
GPa}$. The right panel shows the same data after antisymmetrization. Figure
S16: The left panel displays raw data of $\mathrm{R_{xy}}$ as a function of
the applied field H for the second sample in the insulating state at a
pressure of $\mathrm{0.87\ GPa}$. The right panel presents the same data after
antisymmetrization. The presence of significant intermixing between
$\mathrm{R_{xx}}$ and $\mathrm{R_{xy}}$ can be easily identified by the
absence of symmetry or antisymmetry in the raw data. Figure S17: The left
panel displays raw data of $\mathrm{R_{xy}}$ as a function of the applied
field H for the second sample in the insulating state at a pressure of
$\mathrm{3.2\ GPa}$. The right panel presents the same data after
antisymmetrization. The presence of significant intermixing between
$\mathrm{R_{xx}}$ and $\mathrm{R_{xy}}$ can be easily identified by the
absence of symmetry or antisymmetry in the raw data. Figure S18: The left
panel displays the raw data of $\mathrm{R_{xy}}$ as a function of the applied
field H for the second sample in the metallic state at a pressure of
$\mathrm{5.6\ GPa}$. The right panel shows the same data after
antisymmetrization. Figure S19: The left panel displays the raw data of
$\mathrm{R_{xy}}$ as a function of the applied field H for the second sample
in the metallic state at a pressure of $\mathrm{7.5\ GPa}$. The right panel
shows the same data after antisymmetrization. Figure S20: The left panel
displays the raw data of $\mathrm{R_{xy}}$ as a function of the applied field
H for the second sample in the metallic state at a pressure of $\mathrm{8.9\
GPa}$. The right panel shows the same data after antisymmetrization. Figure
S21: The left panel displays the raw data of $\mathrm{R_{xy}}$ as a function
of the applied field H for the second sample in the metallic state at a
pressure of $\mathrm{10.8\ GPa}$. The right panel shows the same data after
antisymmetrization. Figure S22: The left panel displays the raw data of
$\mathrm{R_{xy}}$ as a function of the applied field H for the second sample
in the metallic state at a pressure of $\mathrm{13.5\ GPa}$. The right panel
shows the same data after antisymmetrization. Figure S23: The left panel
displays the raw data of $\mathrm{R_{xy}}$ as a function of the applied field
H for the second sample in the metallic state at a pressure of $\mathrm{17.6\
GPa}$. The right panel shows the same data after antisymmetrization.
|
# Newsalyze: Enabling News Consumers to Understand Media Bias
Felix Hamborg1, Anastasia Zhukova2, Karsten Donnay3,4, Bela Gipp2 1Dept. of
Computer and Information Science, University of Konstanz, Germany,
<EMAIL_ADDRESS>2Data & Knowledge Engineering Group, University
of Wuppertal, Wuppertal, Germany<EMAIL_ADDRESS>3Dept. of Political
Science, University of Zurich, Switzerland<EMAIL_ADDRESS>4Dept. of
Politics and Public Administration, University of Konstanz, Germany,
<EMAIL_ADDRESS>
(2020)
###### Abstract.
News is a central source of information for individuals to inform themselves
on current topics. Knowing a news article’s slant and authenticity is of
crucial importance in times of “fake news,” news bots, and centralization of
media ownership. We introduce _Newsalyze_ , a bias-aware news reader focusing
on a subtle, yet powerful form of media bias, named bias by word choice and
labeling (WCL). WCL bias can alter the assessment of entities reported in the
news, e.g., “freedom fighters” vs. “terrorists.” At the core of the analysis
is a neural model that uses a news-adapted BERT language model to determine
target-dependent sentiment, a high-level effect of WCL bias. While the
analysis currently focuses on only this form of bias, the visualizations
already reveal patterns of bias when contrasting articles (overview) and in-
text instances of bias (article view).
††journalyear: 2020††copyright: rightsretained††conference: Proceedings of the
ACM/IEEE Joint Conference on Digital Libraries in 2020; August 1–5, 2020;
Virtual Event, China††booktitle: Proceedings of the ACM/IEEE Joint Conference
on Digital Libraries in 2020 (JCDL ’20), August 1–5, 2020, Virtual Event,
China††doi: 10.1145/3383583.3398561††isbn: 978-1-4503-7585-6/20/06††copyright:
none
(0cm,1cm) This is a preprint. The accepted manuscript can be found at:
https://doi.org/10.1145/3383583.3398561
## 1\. Introduction and Related Work
People rely on the news to inform themselves on current topics and events.
Especially news articles, which the public commonly deems most trustworthy
(Urban, 1999), are a central part of individual and societal opinion formation
and decision making. Media bias, e.g., slanted or biased news coverage, thus
can have severe effects on democratic processes (Meyrowitz, 1986). A subtle,
yet powerful form of media bias is bias by word choice and labeling (WCL),
which occurs when news authors sway readers’ perception of persons, actions,
or other semantic concepts by using different terms or phrases to refer to the
concepts, e.g., ”undocumented immigrant” vs. ”illegal alien.” Previous works
have struggled to automatically identify WCL bias (Balahur et al., 2013;
Godbole et al., 2007; Hamborg et al., 2019a), mainly due its implicitness,
subjectivity, and high context dependence (Card et al., 2015), requiring
actual understanding of the text at hand. However, the advent of deep learning
and language models, such as BERT, has led to a significant leap towards
natural language understanding (NLU), thereby strongly improving the
performance in many tasks deemed traditionally as difficult (Wang et al.,
2018).
To our knowledge, there is no news reader that enables bias comparison of
articles reporting on the same topic and exploration of bias instances within
an article. More importantly, no bias-related approaches leverage most recent
advancements in NLU, which could help to significantly improve the detection
performance of biases that could not be addressed well before. We propose
_Newsalyze_ , a news reader that analyzes and visualizes WCL bias in news
articles. The prototype currently focuses on visualizing a high-level effect
of WCL bias and determines whether a target, i.e., a semantic concept, is
portrayed positively or negatively within a sentence.
## 2\. System and User’s Workflow
The system performs a five task workflow (cf. (Hamborg et al., 2019a, b)):
article gathering, preprocessing, target concept analysis, frame
identification, and visualization. For article gathering, we crawl and extract
news articles, currently for given a set of user-defined URLs (Hamborg et al.,
2017) for each topic. We then perform state-of-the-art NLP preprocessing using
Stanford CoreNLP. Target concept analysis finds and resolves semantic
concepts, such as persons or countries, across each topic’s articles, going
beyond regular coreference resolution by finding also broadly or abstractly
defined as well as contrarily mentioned concepts, such as ”freedom fighters”
vs. ”terrorists” (Hamborg et al., 2019a). Frame identification determines how
concepts are portrayed in their mentions, e.g., ranging from sentiment
polarity (positive or negative) to fine-grained framing effects, e.g., whether
a person is portrayed as being ”competent”, ”weak” or ”aggressive” (Hamborg et
al., 2019a). Identifying frames is a challenging task, for human coders (Card
et al., 2015) as well as for previous automated approaches, which either yield
mixed results if aiming to find universally valid frames (Hamborg et al.,
2019a) or are specialized to only one or a few topics (Greussing and
Boomgaarden, 2017). Thus, we currently focus on targeted sentiment, which is a
high-level effect of WCL bias but also a universal perception dimension. To
achieve state of-the-art performance in target-dependent sentiment
classification (TSC) on news articles, we use NewsTSC, a BERT-based neural
model (Hamborg, 2020).
Lastly, the system visualizes the identified instances of WCL bias using two
visualizations, which follow the overview first, details on demand mantra
(Shneiderman, 1996). First, an overview, similar to the overview offered by
news aggregators such as Google News, shows current topics and for each topic
a selection of articles reporting on it. Newsalyze’s overview enables users to
efficiently compare how articles portray the topic’s most important concepts:
besides each article snippet, the visualization shows a histogram representing
the article’s normalized sentiment of the topic’s most frequent concepts.
Figure 2 shows histograms of two articles reporting on the Iran deal topic
published by HuffPost (left-slanted outlet) and Breitbart (right) in April
2018. Second, an article-view helps users to understand WCL bias while reading
an article, e.g., by visually highlighting concept mentions as to the bias
categories identified for them on sentence-level. Figure 1 shows an excerpt of
the left-slanted article.
Figure 1. Newsalyze’s article view highlights mentions of semantic concepts,
such as persons, according to their target-dependent sentiment, a high-level
effect of bias by word choice and labeling (green: positive, red: negative).
Using the overview, users can quickly understand current topics. In contrast
to common news aggregators, the overview is bias-aware: its framing histogram
shown besides each article snippet enables users to quickly compare how
important actors are portrayed across the topic. For example, Figure 2 shows
aggregated polarities of Trump and other most frequent NEs of the Iran deal
topic. The visual comparison immediately reveals that Trump is portrayed
rather negatively in the left outlet but strongly positively in the right
outlet. In common news aggregators, users would have to read whole articles to
come to this conclusion. Lastly, the article-view aids user to understand bias
simply while reading the article, because, for example, phrases of WCL bias
are visually highlighted.
Figure 2. Framing histograms of a topic’s most frequent semantic concepts,
shown for a left-slanted (L) and a right-slanted (R) article. Each bar’s
height represents the frequency of its concept, the color aggregated positive
(green) or negative (red) sentiment of the concept.
Framing histogram of a topic’s most frequent semantic concepts.
## 3\. Conclusion and Future Work
Newsalyze is the first bias-aware news reader that supports the full news
consumption process, from getting an overview of current topics as well as
reading articles. By contrasting how a topic’s actors are portrayed by each
article, users can efficiently get an overview not only of the topic but also
of the slant of each article. Afterward, when reading an article of interest,
users are aided to see bias with the help of in-text bias markers. The system
currently analyzes and visualizes a high-level effect of bias by word choice
and labeling (WCL), i.e., target-dependent sentiment. In the future, we plan
to devise and train a neural model to additionally classify more fine-grained
perception dimensions, e.g., framing effects such as whether a person is
portrayed as competent or incompetent. We also plan to classify causes of the
identified WCL instances, e.g., the use of emotional language (see Figure 1).
We hope that in the future systems such as Newsalyze will help people to
become aware of bias conveniently during their daily news consumption. The
recently increased interest in this topic, not only in research communities
but also in society, emphasizes the issue’s importance.
###### Acknowledgements.
The work described in this paper is partially funded by the WIN program of the
Heidelberg Academy of Sciences and Humanities, financed by the Ministry of
Science, Research and the Arts of the State of Baden-Wurttemberg, Germany.
## References
* (1)
* Balahur et al. (2013) Alexandra Balahur, Ralf Steinberger, Mijail Kabadjov, Vanni Zavarella, Erik Van Der Goot, Matina Halkia, Bruno Pouliquen, and Jenya Belyaeva. 2013. Sentiment analysis in the news. _arXiv preprint arXiv:1309.6202_ (2013).
* Card et al. (2015) Dallas Card, Amber E. Boydstun, Justin H. Gross, Philip Resnik, and Noah A. Smith. 2015\. The Media Frames Corpus: Annotations of Frames Across Issues. In _Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)_. Association for Computational Linguistics, Stroudsburg, PA, USA, 438–444. https://doi.org/10.3115/v1/P15-2072
* Godbole et al. (2007) Namrata Godbole, Manja Srinivasaiah, and Steven Skiena. 2007\. Large-Scale Sentiment Analysis for News and Blogs. _ICWSM_ 7, 21 (2007), 219–222.
* Greussing and Boomgaarden (2017) Esther Greussing and Hajo G. Boomgaarden. 2017. Shifting the refugee narrative? An automated frame analysis of Europe’s 2015 refugee crisis. _Journal of Ethnic and Migration Studies_ 43, 11 (8 2017), 1749–1774. https://doi.org/10.1080/1369183X.2017.1282813
* Hamborg (2020) Felix Hamborg. 2020\. Media Bias, the Social Sciences, and NLP: Automating Frame Analyses to Identify Bias by Word Choice and Labeling. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics (ACL): Student Research Workshop_. Association for Computational Linguistics, 1–9.
* Hamborg et al. (2017) Felix Hamborg, Norman Meuschke, Corinna Breitinger, and Bela Gipp. 2017. news-please: A Generic News Crawler and Extractor. In _Proceedings of the 15th International Symposium of Information Science_. Verlag Werner Hülsbusch, 218–223.
* Hamborg et al. (2019a) Felix Hamborg, Anastasia Zhukova, and Bela Gipp. 2019a. Automated Identification of Media Bias by Word Choice and Labeling in News Articles. In _2019 ACM/IEEE Joint Conference on Digital Libraries (JCDL)_. IEEE, Urbana-Champaign, IL, USA, 196–205. https://doi.org/10.1109/JCDL.2019.00036
* Hamborg et al. (2019b) Felix Hamborg, Anastasia Zhukova, and Bela Gipp. 2019b. Illegal Aliens or Undocumented Immigrants? Towards the Automated Identification of Bias by Word Choice and Labeling. In _Proceedings of the iConference 2019_. Springer, Cham, Washington, DC, USA, 179–187. https://doi.org/10.1007/978-3-030-15742-5{_}17
* Meyrowitz (1986) Joshua Meyrowitz. 1986\. _No sense of place: The impact of electronic media on social behavior_. Oxford University Press.
* Shneiderman (1996) B. Shneiderman. 1996\. The eyes have it: a task by data type taxonomy for information visualizations. _Proceedings 1996 IEEE Symposium on Visual Languages_ (1996), 336–343. https://doi.org/10.1109/VL.1996.545307
* Urban (1999) Christine D Urban. 1999\. _Examining Our Credibility: Perspectives of the Public and the Press_. Asne Foundation. 1–108 pages.
* Wang et al. (2018) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel Bowman. 2018\. GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding. In _Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP_. Association for Computational Linguistics, Stroudsburg, PA, USA, 353–355. https://doi.org/10.18653/v1/W18-5446
|
# Random Fully Connected Neural Networks as Perturbatively Solvable
Hierarchies
Boris Hanin111BH gratefully acknowledges support by the NSF through
DMS-2143754, DMS-1855684, and DMS-2133806 as well support from an ONR MURI on
Foundations of Deep Learning.
Department of Operations Research
and Financial Engineering
Princeton University
<EMAIL_ADDRESS>
###### Abstract
This article considers fully connected neural networks with Gaussian random
weights and biases as well as $L$ hidden layers, each of width proportional to
a large parameter $n$. For polynomially bounded non-linearities we give sharp
estimates in powers of $1/n$ for the joint cumulants of the network output and
its derivatives. Moreover, we show that network cumulants form a
perturbatively solvable hierarchy in powers of $1/n$ in that $k$-th order
cumulants in one layer have recursions that depend to leading order in $1/n$
only on $j$-th order cumulants at the previous layer with $j\leq k$. By
solving a variety of such recursions, however, we find that the depth-to-width
ratio $L/n$ plays the role of an effective network depth, controlling both the
scale of fluctuations at individual neurons and the size of inter-neuron
correlations. Thus, while the cumulant recursions we derive form a hierarchy
in powers of $1/n$, contributions of order $1/n^{k}$ often grow like $L^{k}$
and are hence non-negligible at positive $L/n$. We use this to study a
somewhat simplified version of the exploding and vanishing gradient problem,
proving that this particular variant occurs if and only if $L/n$ is large.
Several key ideas in this article were first developed at a physics level of
rigor in a recent monograph of Daniel A. Roberts, Sho Yaida, and the author.
This article not only makes these ideas mathematically precise but also
significantly extends them, opening the way to obtaining corrections to all
orders in $1/n$.
## 1 Introduction
We live in an era of big data and cheap computation. This has led to
remarkable progress in domains ranging from self-driving cars [35] to
automatic drug discovery [33] and machine translation [8]. Underpinning many
of these exciting practical developments is a class of computational models
called neural networks. While they were originally developed in the 1940’s and
1950’s [26, 54], the complexity of state-of-the-art neural nets is
unprecedented. And yet, despite their empirical utility, a theoretical
understanding of how they work and how to make them better is nascent. In
fact, it is sometimes said that neural networks are simply too complicated to
allow for a rigorous understanding of their key features.
This article adds to the growing body of literature to the contrary. Namely,
in the simplest important setting of fully connected networks, we develop a
flexible set of probabilistic tools for studying correlation functions of
random fully connected neural networks at finite width.
A random fully connected neural network, defined precisely in §2.1, is a
random field whose law is determined by a fixed and typically non-linear
function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ as well as two structural
parameters: a depth $L\in\mathbb{N}$ and a width $n\in\mathbb{N}$. For a fixed
depth $L$, in the limit of infinite width $n\rightarrow\infty$ random neural
networks converge in distribution to Gaussian processes (see Theorem 2.2). To
study finite width effects we therefore consider the higher cumulants for the
distribution of the outputs. Our approach is recursive in the network depth
$L$ and perturbatively in the reciprocal $1/n$ of the network width. Our main
results are:
* •
We give in Theorem 3.1 sharp estimates in powers of $1/n$ for the the size of
all joint cumulants of networks outputs and their derivatives. These can be
viewed as quantitative results refinements of the central limit theorem at
fixed $L$ and large $n$ (cf Theorems 2.2 and Theorem 3.2).
* •
We derive exact recursions with respect to depth that describe cumulants at
one layer in terms of cumulants at the previous layer (see e.g. Corollary 3.4
and Proposition 11.2). These recursions take the form of perturbatively
solvable hierarchies in the sense that the recursion for the $k$-th cumulant
at layer $\ell+1$ involves, at leading order in $1/n$ only the $j$-th
cumulants at layer $\ell$ with $j\leq k$ (see Corollaries 3.3 and 3.4). This
is distinct from but similar in spirit to the remarkable article [29], which
derived a dynamical perturbative hierarchy for the so-called neural tangent
kernel. As in our work, the perturbative parameter is $1/n$. However, the
emphasis in [29] is primarily on training dynamics while ours is on
understanding the effect of depth.
* •
Solving some special cases of the cumulant recursions to leader order in $1/n$
reveals that it is the depth-to-width ratio $L/n$, which we term the effective
network depth, rather than the apparent depth $L$, that is a more informative
measure of neural network depth and complexity. Mathematically, this suggests
a non-trivial double scaling limit for random neural networks in which
$n,L\rightarrow\infty\qquad\text{and}\qquad L/n\rightarrow\xi\in[0,\infty).$
For non-linear networks this scaling limit has only started to be considered
[17, 21, 53, 40]. Even in the very special case of product of $L$ iid random
$n\times n$ matrices (sometimes called deep linear networks) the simultaneous
large $n,L$ regime has revealed a range of interesting and not fully
understood properties (see e.g. references in [1, 2] as well as [16, 21, 22,
42]). In contrast to the $\xi=0$ regime typically considered in previous work
on neural networks (c.f. e.g. [12, 13, 32, 41]), we show that when $\xi>0$ our
double scaling limit is capable of exhibiting non-Gaussian and non-linear
effects. We find the following effects to leading order in $1/n$:
* –
We prove in Corollary 4.1 that for any non-linearity $\sigma$ from the
$K_{*}=0$ universality class (defined in §4.2.2) and for $k=2,3,4$, the
$2k^{th}$ cumulants of the output of a random neural network with non-
linearity $\sigma$ grow like $(L/n)^{k/2-1}$. This implies, for instance, that
both the correlations between neurons and the fluctuations of a single neuron
grow like the effective depth $L/n$ at large $L,n$ (see Remark 4.2 and just
after Theorem 4.4). Since the components of the output of a depth $0$ neural
network are simply iid Gaussians, we see that the effective depth $L/n$ can be
thought of as a measure of how close a random fully connected network is to a
Gaussian process. Related questions were considered in [4, 15].
* –
We show in Theorems 4.4 and 4.5 that, for random neural networks initialized
as in practice (see §4.1), the variance of the gradient of the network output
with respect to either its input or a trainable parameter in its first layer
grows like $L/n$ at large $L$. As explained in §4.4, this gives the first
mathematical characterization for fairly general fully connected networks of
the so-called exploding and vanishing gradient problem (EVGP). Previously,
this problem was solved rigorously in the important special case of random
ReLU networks [17, 21] and solved at a physics level of rigor by a somewhat
different but related set of ideas in the monograph [53] of Roberts, Yaida,
and the author.
The remainder of the introduction is structured as follows. We begin by giving
in §2.1 the precise definition of random fully connected neural networks. We
then formulate and motivate the main question taken up in this article in
§2.2.
## 2 Background and Motivation
### 2.1 What is a (Random) Fully Connected Neural Network?
Neural networks are parameterized families of functions. The simplest kind of
networks are called fully connected. Each such network is specified by its
architecture, which consists of an input dimension $n_{0}\in\mathbb{Z}_{+}$, a
network depth $L\in\mathbb{Z}_{+}$, hidden layer widths
$n_{1},\ldots,n_{L}\in\mathbb{Z}_{+}$, an output dimension
$n_{L+1}\in\mathbb{Z}_{+}$, and a non-linearity
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$. The functions computed by a fully
connected network with a given architecture are all maps from
$\mathbb{R}^{n_{0}}$ to $\mathbb{R}^{n_{L+1}}$ that associate to each network
input $x_{\alpha}\in\mathbb{R}^{n_{0}}$ an output
$z_{\alpha}^{{}_{(L+1)}}\in\mathbb{R}^{n_{L+1}}$ through a sequence of
intermediate representations
$z_{\alpha}^{{}_{(\ell)}}\in\mathbb{R}^{n_{\ell}}$ as follows:
$z_{\alpha}^{(\ell+1)}:=\begin{cases}b^{(\ell+1)}+W^{(\ell+1)}\sigma(z_{\alpha}^{(\ell)}),&\quad\ell\geq
1\\\ b^{(1)}+W^{(1)}x_{\alpha},&\quad\ell=0\end{cases},\qquad
W^{(\ell+1)}\in\mathbb{R}^{n_{\ell+1}\times
n_{\ell}},\,b^{(\ell+1)}\in\mathbb{R}^{n_{\ell+1}}.$ (2.1)
In this recursion, the univariate function $\sigma$ applied to a vector
$z_{\alpha}^{{}_{(\ell)}}\in\mathbb{R}^{n_{\ell}}$ is short-hand for applying
it separately to each component. The entries of the matrices $W^{(\ell)}$ and
the components of the vectors $b^{(\ell)}$ are called the weights and biases
in layer $\ell$, respectively. One typically refers to
$z_{\alpha}^{(\ell)}=\left(z_{1;\alpha}^{(\ell)},\ldots,z_{n_{\ell};\alpha}^{(\ell)}\right)\in\mathbb{R}^{n_{\ell}}$
as the vector of pre-activations at layer $\ell$ corresponding to the input
$x_{\alpha}$. The most popular choices of $\sigma$ in practice include
$\mathrm{ReLU}(t):=\max\left\\{0,t\right\\}$ as well the hyperbolic tangent
and their variations. We will analyze these cases in detail later (see §4.2),
but for our general results make only the following mild assumption
###### Assumption 2.1.
There exists $r\geq 1$ so that the $r$-th derivative of $\sigma$ exists almost
everywhere and grows at most polynomially:
$\exists k\geq 1\text{ s.t.
}\mathrm{sup}_{x\in\mathbb{R}}\left|(1+\left|x\right|)^{-k}\frac{d^{r}}{dx^{r}}\sigma(x)\right|<\infty.$
The primary objects of study in this article are random fully connected neural
networks, obtained by choosing network weights and biases to be independent
centered Gaussians:
$W_{ij}^{(\ell)}\sim\mathcal{N}(0,C_{W}/n_{\ell-1}),\qquad
b_{i}^{(\ell)}\sim\mathcal{N}(0,C_{b})\qquad\text{independent}.$ (2.2)
Here $C_{b}\geq 0,C_{W}>0$ are fixed constants. The $1/n_{\ell-1}$ scaling in
the weight variance ensures that the moments of the outputs
$z_{\alpha}^{{}_{(L+1)}}$ remain uniformly bounded as
$n_{1},\ldots,n_{L},\rightarrow\infty$ (see e.g. Theorem 2.2 and (3.6)). While
we carry out a substantial portion of our analysis for any choice of
$C_{b},C_{W}$, as we will see in §4.1, there often exist distinguished
$\sigma$-dependent settings of $C_{b},C_{W}$ for which random neural networks
at infinite width $n_{1},\ldots,n_{L}\rightarrow\infty$ are well-behaved at
large depth $L$.
### 2.2 Statement and Motivation for Questions Addressed in this Article
The main problem we take up in the present article is to characterize the
finite dimensional distributions of a random neural network $x_{\alpha}\mapsto
z_{\alpha}^{{}_{(L+1)}}$ (and its derivatives with respect to $x_{\alpha}$) in
the regime where the input dimension $n_{0}$ is arbitrary with the inputs
$x_{\alpha}$ satisfying
$\frac{1}{n_{0}}\left|\left|x_{\alpha}\right|\right|_{2}^{2}<\infty,$
the output dimension $n_{L+1}$ is fixed, and the hidden layer widths
$n_{\ell}$ are large but finite:
$\exists c,C>0\text{ s.t. }\qquad cn\leq n_{1},\ldots,n_{L}\leq
Cn,\qquad\qquad n\gg 1.$ (2.3)
Our approach will be to describe the random field $x_{\alpha}\mapsto
z_{\alpha}^{{}_{(L+1)}}$ perturbatively in $1/n$ and recursively in $L$.
Before proceeding to the technical statements of our results, we pause to
address below the following motivational questions:
* •
Why study random neural networks?
* •
Why treat $1/n$ as a perturbative parameter?
* •
Why study networks at finite width?
* •
What role does $\sigma$ play?
The primary use of a neural network with a fixed architecture is to find a
setting of its parameters $W^{{(\ell)}},b^{{(\ell)}}$ giving rise to a mapping
$x_{\alpha}\mapsto z_{\alpha}^{{}_{(L+1)}}$ as in (2.1) that matches a dataset
$\left\\{(x_{i},f(x_{i}))\right\\}$ of values for some otherwise unknown
function $f:\mathbb{R}^{n_{0}}\rightarrow\mathbb{R}^{n_{L+1}}$. The
optimization procedure for finding these weights and biases is almost always
some variant of gradient descent starting with $W^{{(\ell)}},b^{{(\ell)}}$
drawn at random from the distribution (2.2). Thus, studying the properties of
random neural networks gives direct insights into the starting conditions for
neural network optimization. For instance, understanding the behavior of
neural networks at the start of training gives a principled way to set
optimization hyperparameters (e.g. the variances $C_{b},C_{W}$ and the step
size used for gradient descent). We refer the interested reader our discussion
of criticality in §4 as well as to §3.2 in [53] the articles articles [25, 43,
52, 67] for more on this point.
Next, to address why $1/n$ can reasonably be treated perturbatively, we recall
that networks used in practice often have a very large number of parameters.
This is reflected in the fact that both the layer widths $n_{\ell}$ and the
network depth $L$ are often big. Thus, it is sensible to first understand
various limits of neural networks in which the number of parameters tends to
infinity. The most well-studied regime of this type (though not the only
option cf eg [45, 55, 57, 58, 64]) is the infinite width limit, also known as
the NTK regime. By definition, this regime is accessed by fixing the depth
$L$, the input and output dimensions $n_{0},n_{L+1}$, and the non-linearity
$\sigma$, the initialization scheme (2.2) and considering the limit when
$n_{1},\ldots,n_{L}\rightarrow\infty$. From the random matrix theory point of
view this is the free probability regime. In view of the relation (2.3) the
NTK regime is obtained by taking $n\rightarrow\infty$ at fixed $L$ and has two
salient features:
* •
At the start of training, neural networks converge to a Gaussian process (see
Theorem 2.2 below) [36, 44, 48, 49, 62, 63, 65].
* •
For the purposes of optimization of a squared loss, the network can be
replaced by its linearization at the start of training (see [9, 12, 32, 37,
41]).
Taken together, these two points show that at least at infinite width and
finite depth, it is the structure of the network at initialization that
determines not only the start of training but really the entire training
trajectory. However, the infinite width limit is too rigid to capture the
ability of real work networks to learn data-dependent features (see e.g. [20,
29, 53, 64]). Only finite width networks can capture these effects! Since the
starting point for our analysis of random neural networks at finite width is
their infinite width behavior, we take this opportunity to record the
following result about random neural networks in the NTK regime.
###### Theorem 2.2 (Random Networks at fixed $L$ and Infinite Width are
Gaussian Processes).
Fix a non-negative integer $r\geq 0$, and suppose
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is $r$-times differentiable and that
its $r$-th derivative is polynomially bounded:
$\exists k\geq 1\text{ s.t.
}\sup_{x\in\mathbb{R}}\left|(1+\left|x\right|)^{-k}\frac{d^{r}}{dx^{r}}\sigma(x)\right|<\infty.$
Then the finite-dimensional distributions of the stochastic process
$x_{\alpha}\mapsto z_{\alpha}^{(L+1)}$ and its derivatives of order up to $r$
converge to those of a centered Gaussian process with $n_{L+1}$ iid
components. The limiting variance of each component
$K_{\alpha\beta}^{(L+1)}:=\lim_{n_{1},\ldots,n_{L}\rightarrow\infty}\mathrm{Cov}\left(z_{i;\alpha}^{(L+1)},z_{i;\beta}^{(L+1)}\right),\qquad
x_{\alpha},x_{\beta}\in\mathbb{R}^{n_{0}},$
satisfies the recursion
$\displaystyle
K_{\alpha\beta}^{(\ell+1)}=\begin{cases}C_{b}+C_{W}\left\langle\sigma(z_{\alpha})\sigma(z_{\beta})\right\rangle_{K^{(\ell)}},&\quad\ell\geq
1\\\
C_{b}+\frac{C_{W}}{n_{0}}\sum_{j=1}^{n_{0}}x_{j;\alpha}x_{j;\beta},&\quad\ell=0\end{cases}.$
(2.4)
In the statement of Theorem 2.2 and henceforth we reserve the symbol
$\left\langle f(z_{\alpha},z_{\beta})\right\rangle_{\kappa}$ to denote the
expectation of $f(z_{\alpha},z_{\beta})$ with respect to the Gaussian
distribution
$\left(z_{\alpha},z_{\beta}\right)\sim\mathcal{N}\left(0,\left(\begin{array}[]{cc}\kappa_{\alpha\alpha}&\kappa_{\alpha\beta}\\\
\kappa_{\alpha\beta}&\kappa_{\beta\beta}\end{array}\right)\right),$
where $\kappa_{\alpha\beta}=\kappa(x_{\alpha},x_{\beta})$ is a given
covariance function. The conclusion in Theorem 2.2 is not new, having been
obtained many times and under a variety of different assumptions (including
for more general architectures) [19, 36, 44, 51, 63]. We refer the interested
reader to [19] for a discussion of prior work and note only that convergence
of the derivatives of the field $z_{\alpha}^{{}_{(L+1)}}$ to its Gaussian
limit does not seem to have been previously considered. We give a short proof
that includes convergence of derivatives along the lines of the arguments in
[19, 36] in Appendix §A.
Let us remark that the relation between $K^{{(\ell+1)}}$ to $K^{(\ell)}$
supplied by the recursion (2.4) is in general non-linear due to the presence
of $\sigma$ and depends crucially on the weight and bias variances
$C_{W},C_{b}$. Given $\sigma$, finding a choice of $C_{b},C_{W}$ so that this
recursion is well-behaved (e.g. not exponentially growing or decaying) at
large $\ell$ is an important practical matter [30, 36, 50]. We explain this in
§4.1, where we also point out that the different possible large $\ell$ limits
suggest the existence of universality classes of random neural networks.
Finally, as we alluded to above, the tremendous simplification of random
neural networks in the NTK regime comes at a steep cost in terms of the
descriptive power of the resulting model in the sense that such models cannot
learn data-dependent features. We again refer to reader to articles such as
[10, 45, 55, 57, 58, 64] in which a different initialization scheme leads to
infinite width limits capable of feature learning. Since we do not treat
feature learning in this article, we will not elaborate further on this point,
referring the interested reader to [9, 20, 29, 53, 60, 64] instead.
Finally, before formulating the new results in this article, we point the
interested to a final tranche of physics-style references such as [3, 11, 14,
34, 38, 46, 47, 56], which analyze a variety of questions related to either
very wide or infinitely wide networks.
## 3 Results
In this section we give precise formulations of our results on random neural
networks. We start in §3.1 by stating a structural result, Theorem 3.1, on the
size of the joint cumulants of a random neural networks and its derivatives in
powers of $1/n$. We then extend this order of magnitude estimate to obtain in
Theorem 3.2 a general prescription for obtaining series expansions in powers
of $1/n$ for expectation values of functions of a random depth $\ell+1$ neural
network in terms of those of a random depth $\ell$ network. An application of
Theorem 3.2 yields explicit recursions for the $4^{th},6^{th},8^{th}$ joint
cumulants for components $z_{i;\alpha}^{{}_{(\ell)}}$ of the vector of pre-
activations at layer $\ell$ corresponding to a fixed network input
$x_{\alpha}.$ These recursions are recorded in Corollary 3.4. See §5 for an
overview of the proof of Theorems 3.1 and 3.2 as well as Corollary 3.4.
After stating these results, we describe in §4 the important notion of
criticality in random neural networks, which corresponds to choosing values of
the weight and bias variance parameters $C_{b},C_{W}$ in (2.2) depending on
$\sigma$ in such a way that the infinite width covariance
$K_{\alpha\beta}^{{}_{(\ell)}}$ (and in fact the higher cumulants at finite
width as well) is well-behaved at large $\ell$. The variety of resulting large
$\ell$ behaviors determines universality classes of random neural networks, as
we explain in §4.2.
We then turn in §4.2.1 and §4.2.2 to solving to the $2k$-th cumulant
recursions for $k=1,2,3,4$ from Corollary 3.4 in random networks tuned to
criticality with non-linearities either from the universality class of ReLU or
of tanh. We will see that, at large network depth $L$ and leading order in
$1/n$, these cumulants depend only on the effective network depth $L/n$. We
formalize our belief that this is a universal phenomenon in Conjecture 4.3. We
then solve in §4.4 a simplified version of the so-called exploding and
vanishing gradient problem (EVGP), which concerns the empirical variance of
parameter gradients in a random neural network. More precisely, for non-
linearities in the $K_{*}=0$ universality class in prove (see Theorem 4.5)
that to leading order in $1/n$ the empirical variance over weights in the
first layer of network gradients grows linearity in the effective network
depth $L/n$. These results depend on Theorem 4.4, which gives for non-
linearities in the $K_{*}=0$ universality class asymptotics at large $\ell$
for joint fourth cumulants between the values of partial derivatives of the
output of a random neural network.
### 3.1 Cumulants of Random Neural Networks at Finite Width
Since in the infinite $n$ limit, the field $z_{\alpha}^{{}_{(L+1)}}$ is
Gaussian (see Theorem 2.2), it is natural to measure perturbations around this
regime by considering the behavior of the cumulants of
$z_{\alpha}^{{}_{(L+1)}}$ and its derivatives. Let us therefore agree that,
given random variables $X_{1},\ldots,X_{k}$ with finite moments defined on the
same probability space, we will denote their mixed cumulant by
$\kappa\left(X_{1},\ldots,X_{k}\right):=i^{k}\frac{\partial^{k}}{\partial
t_{1}\cdots\partial
t_{k}}\bigg{|}_{t=0}\log\mathbb{E}\left[\exp\left[-i(t_{1}X_{1}+\cdots+t_{k}X_{k})\right]\right].$
(3.1)
Thus, for example, $\kappa(X_{1})=\mathbb{E}\left[X_{1}\right]$ and
$\kappa(X_{1},X_{2})=\mathrm{Cov}(X_{1},X_{2}).$ We refer the reader to §6.1
for background on cumulants. Our first result, Theorem 3.1, gives estimates on
the order in $1/n$ of the cumulants of $z_{\alpha}^{{}_{(L+1)}}$ and its
derivatives. To state it, we fix a finite collection
$\left\\{x_{\alpha},\,\alpha\in\mathcal{A}\right\\}\subseteq\mathbb{R}^{n_{0}},$
of $\left|\mathcal{A}\right|$ distinct network inputs. Moreover, we fix a
collection of $p$ directional derivatives:
$D=\left(d_{1},\ldots,d_{p}\right),\qquad
d_{j}:=\nabla_{v_{j}}=\sum_{i=1}^{n_{0}}v_{ij}\partial_{x_{i}}.$ (3.2)
and for any multi-index $J=\left(j_{1},\ldots,j_{p}\right)\in\mathbb{N}^{p}$
set
$D_{\alpha}^{J}:=d_{1}^{j_{1}}\cdots d_{m}^{j_{p}}\bigg{|}_{x=x_{\alpha}}$
for the corresponding differential operator of order
$\left|J\right|:=j_{1}+\cdots+j_{p}.$
###### Theorem 3.1 (order of magnitude for cumulants of random neural
networks).
Fix $r,L\geq 1$ and suppose that $\sigma:\mathbb{R}\rightarrow\mathbb{R}$
satisfies Assumption (2.1) with this value of $r$. Suppose further that one of
the following two conditions holds:
* •
$\sigma$ is smooth
* •
the limiting covariance matrix
$\left(\lim_{n\rightarrow\infty}\mathrm{Cov}\left(D_{\alpha_{1}}^{J_{1}}z_{1;\alpha_{1}}^{(\ell)},D_{\alpha_{2}}^{J_{2}}z_{1;\alpha_{2}}^{(\ell)}\right)\right)_{\begin{subarray}{c}\left|J_{1}\right|,\left|J_{2}\right|\leq
r\\\ \alpha_{1},\alpha_{2}\in\mathcal{A}\end{subarray}}$ (3.3)
of derivatives of order at most $r$ in the directional derivatives
$d_{1},\ldots,d_{p}$ of the scalar field $z_{1;\alpha}^{(\ell)}$ is strictly
positive definite in the infinite width limit for all $\ell\leq L$.
Then, for each $k,\ell\geq 1$, as $n\rightarrow\infty$
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell)},\ldots,D_{\alpha_{k}}^{J_{k}}z_{i_{k};\alpha_{k}}^{(\ell)}\right)=\begin{cases}0,&\quad
k\text{ odd}\\\ O(n^{-\frac{k}{2}+1}),&\quad k\text{ even}\end{cases},$ (3.4)
where the implicit constant in the error term depends on $k$, the inputs
$x_{\alpha_{1}},\ldots,x_{\alpha_{k}},$ the multi-indices
$J_{1},\ldots,J_{k}$, the weight and bias variances $C_{b},C_{W},$ the non-
linearity $\sigma$, and the layer index $\ell$.
We prove Theorem 3.1 in §7. At a physics level of rigor, Theorem 3.1 with
$k=4$ and no derivatives was already derived in the breakthrough work of Yaida
[61]. In fact, Yaida’s original article went much further: it obtained a
recursive formula with respect to $\ell$ for the fourth cumulant
$\kappa(z_{i_{1};\alpha_{1}}^{{}_{(\ell)}},\ldots,z_{i_{4};\alpha_{4}}^{{}_{(\ell)}})$
at layer $\ell$ in terms of the second and fourth cumulants at layer $\ell-1$.
This is analogous to the recursion (2.4) for the infinite width covariance
$K_{\alpha_{1}\alpha_{2}}^{{}_{(\ell)}}$. This theme was then picked up and
significantly expanded upon in the physics monograph [53], which computes,
among other things, at order $1/n$ the leading corrections to the field
$z_{\alpha}^{{}_{(\ell)}}$ and its derivatives with respect to both
$x_{\alpha}$ and model parameters. We will reproduce some of these recursions
and obtain new ones of a similar flavor below.
Compared to this prior work the main novelty of Theorem 3.1 is two-fold.
First, it gives sharp estimates in powers of $1/n$ for cumulants of all orders
(the sharpness can already be seen when $\ell=2$). Second, it treats in a
uniform way the cumulants for not only the values but also all derivatives of
$z_{\alpha}^{{}_{(\ell)}}$.
In order to put Theorem 3.1 into context, we take this opportunity to make an
important remark. Namely, it is no accident that the order of magnitude
$O(n^{-k+1})$ for the $2k$-th cumulant in Theorem 3.1 is the same as that of
the $k$-th cumulant of an average of $n$ iid random variables. To see why, let
us denote by $\mathcal{F}^{(\ell)}$ the sigma algebra generated by the weights
and biases in layers up to and including $\ell$. Since we’ve assumed weights
and biases to be Gaussian and independent for different neurons, we find that
conditional on $\mathcal{F}^{(\ell)}$ the vectors
$z_{i;\mathcal{A}}^{(\ell+1)}:=\left(z_{i;\alpha}^{(\ell+1)},\,\alpha\in\mathcal{A}\right)$
are independent centered Gaussians with the covariance
$\Sigma_{\alpha\alpha^{\prime}}^{(\ell)}:=\mathrm{Cov}\left(z_{i;\alpha}^{(\ell+1)},z_{i;\alpha^{\prime}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right)=C_{b}+\frac{C_{W}}{n_{\ell}}\sum_{j=1}^{n_{\ell}}\sigma\left(z_{j;\alpha}^{(\ell)}\right)\sigma\left(z_{j;\alpha^{\prime}}^{(\ell)}\right).$
(3.5)
Put another way, we have the following equality in distribution:
$\left(z_{i;\alpha}^{(\ell+1)}\right)_{i=1}^{n_{\ell+1}}~{}\stackrel{{\scriptstyle
d}}{{=}}~{}\left(\left(\Sigma^{(\ell)}\right)^{1/2}Z_{i}\right)_{i=1}^{n_{\ell+1}},$
(3.6)
where $Z_{i}$ are i.i.d. standard Gaussians and are independent of the random
covariance matrix
$\Sigma^{(\ell)}:=\left(\Sigma_{\alpha\alpha^{\prime}}^{(\ell)}\right)_{\begin{subarray}{c}\alpha,\alpha^{\prime}\in\mathcal{A}\end{subarray}}.$
Inspecting (3.5) shows this conditional covariance has the structure of an
average of $n_{\ell}$ identically distributed random matrices, each depending
only on the pre-activations $z_{i;\mathcal{A}}^{{}_{(\ell)}}$ of a single
neuron at layer $\ell$. We will refer to such random variables as collective
observables. The key point is that while the $z_{i;\mathcal{A}}^{{}_{(\ell)}}$
are not independent for different $i$ at finite width, we will show that they
are sufficiently weakly correlated that the cumulants of
$\Sigma_{\alpha\alpha^{\prime}}^{{}_{(\ell)}}$ have the same order in $n$ as
if they were exactly independent (see Theorem 7.3 and Lemma 7.5). This
reasoning applies equally well to derivatives of
$z_{i;\alpha}^{{}_{(\ell+1)}}$ and explains the order of magnitude of the
estimates in Theorem 3.1.
Collective observables such as $\Sigma_{\alpha\alpha^{\prime}}^{{}_{(\ell)}}$
are a convenient book-keeping device for studying the fully distribution of
$D^{\leq
r}z_{i;\mathcal{A}}^{(\ell+1)}:=\left(D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)},\quad\alpha\in\mathcal{A},\,J\in\mathbb{N}^{p},\,\left|J\right|\leq
r\right).$
Indeed, the conditional Gaussian structure (3.6) means the cumulants of
$D^{\leq r}z_{i;\mathcal{A}}^{{}_{(\ell+1)}}$ are easily expressible in terms
of those of
$D_{\alpha}^{J}D_{\alpha^{\prime}}^{J^{\prime}}\Sigma_{\alpha\alpha^{\prime}}^{{}_{(\ell)}}$.
For example, using Wick’s Theorem and the multi-linearity of cumulants reveals
that the fourth cumulant
$\displaystyle\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},D_{\alpha_{2}}^{J_{2}}z_{i_{2};\alpha_{2}}^{(\ell+1)},D_{\alpha_{3}}^{J_{3}}z_{i_{3};\alpha_{3}}^{(\ell+1)},D_{\alpha_{4}}^{J_{4}}z_{i_{4};\alpha_{4}}^{(\ell+1)}\right)$
equals
$\delta_{i_{1}i_{2}}\delta_{i_{3}i_{4}}\kappa_{(\alpha_{1}\alpha_{2})(\alpha_{3}\alpha_{4})}^{(J_{1}J_{2})(J_{3}J_{4}),(\ell)}+\delta_{i_{1}i_{3}}\delta_{i_{2}i_{4}}\kappa_{(\alpha_{1}\alpha_{3})(\alpha_{2}\alpha_{4})}^{(J_{1}J_{3})(J_{2}J_{4}),(\ell)}+\delta_{i_{1}i_{4}}\delta_{i_{2}i_{3}}\kappa_{(\alpha_{1}\alpha_{4})(\alpha_{2}\alpha_{3})}^{(J_{1}J_{4})(J_{2}J_{3}),(\ell)},$
where we’ve abbreviated
$\kappa_{(\alpha_{1}\alpha_{1}^{\prime}),\ldots,(\alpha_{k}\alpha_{k}^{\prime})}^{{(J_{1}J_{1}^{\prime}),\ldots,(J_{k}J_{k}^{\prime}),(\ell)}}:=\kappa\left(D_{\alpha_{1}}^{J_{1}}D_{\alpha_{1}^{\prime}}^{J_{1}^{\prime}}\Sigma_{\alpha_{1}\alpha_{1}^{\prime}}^{(\ell)},\ldots,D_{\alpha_{k}}^{J_{k}}D_{\alpha_{k}^{\prime}}^{J_{k}^{\prime}}\Sigma_{\alpha_{k}\alpha_{k}^{\prime}}^{(\ell)}\right).$
(3.7)
The general pattern is that odd mixed cumulants in the $z_{\alpha}^{(\ell)}$
and its derivatives vanish by symmetry and $2k$-th cumulants are finite sums
of cumulants
$\kappa_{(\alpha_{1}\alpha_{1}^{\prime}),\ldots,(\alpha_{k}\alpha_{k}^{\prime})}^{{}_{(J_{1}J_{1}^{\prime}),\ldots,(J_{k}J_{k}^{\prime}),(\ell)}}$.
In Corollary 3.4 and Proposition 10.2 below we will obtain a range of
recursions for these cumulants and thereby implicitly for the cumulants of the
original field $z_{\alpha}^{{}_{(\ell)}}$ as well.
To obtain such recursions note that, at first glance, the estimate (3.4) only
gives the order of magnitude in powers of $1/n$ of the cumulants of
$\kappa(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell)},\ldots,D_{\alpha_{k}}^{J_{k}}z_{i_{k};\alpha_{k}}^{{}_{(\ell)}})$
but does not seem to provide information about their structural dependence on
the remaining model parameters $C_{b},C_{W},\sigma,\ell$. However, such
information can be obtained by combining (3.4) with an additional argument. To
state the result, let us agree to write
$\kappa_{\alpha_{1}\alpha_{2}}^{(\ell)}:=\mathrm{Cov}\left(z_{i;\alpha_{1}}^{(\ell)},z_{i;\alpha_{2}}^{(\ell)}\right),\qquad\qquad\kappa_{\alpha_{1}\alpha_{2}}^{J_{1}J_{2},(\ell)}:=D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\kappa_{\alpha_{1}\alpha_{2}}^{(\ell)}.$
(3.8)
Physicists might refer to $\kappa_{\alpha_{1}\alpha_{2}}^{{}_{(\ell)}}$ as a
dressed two point function. Moreover, let us denote by
$\left\langle\cdot\right\rangle_{\kappa^{(\ell)}}$ the expectation with
respect to a collection of centered jointly Gaussian random vectors
$D_{\mathcal{A}}^{\leq
r}z_{i}=\left(D_{\alpha}^{J}z_{i;\alpha},\,\alpha\in\mathcal{A},\,\left|J\right|\leq
r\right)$
with the same covariance
$\mathrm{Cov}\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}},\,D_{\alpha_{2}}^{J_{2}}z_{i_{2};\alpha_{2}}\right):=\mathrm{Cov}\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell)},\,D_{\alpha_{2}}^{J_{2}}z_{i_{2};\alpha_{2}}^{(\ell)}\right)=\delta_{i_{1}i_{2}}\kappa_{\alpha_{1}\alpha_{2}}^{J_{1}J_{2},(\ell)}$
as the true vectors of derivatives $D_{\mathcal{A}}^{\leq
r}z_{i;\mathcal{A}}^{{}_{(\ell)}}$ in each component separately but zero
covariance for different $i$.
###### Theorem 3.2 (perturbative expansions of expectations of observables at
finite width).
Fix $r\geq 0$ and suppose that $f$ is both a continuous function and a
tempered distribution. Then for any $q_{*}\geq 0$ we have
$\displaystyle\mathbb{E}\left[f\left(D^{\leq
r}z_{1,\mathcal{A}}^{(\ell+1)},\ldots,D^{\leq
r}z_{m,\mathcal{A}}^{(\ell+1)}\right)\right]$
$\displaystyle\quad=\sum_{q=0}^{2q_{*}}\frac{(-1)^{q}}{2^{q}q!}\bigg{\langle}\mathbb{E}\bigg{[}\bigg{(}\sum_{\begin{subarray}{c}\left|J\right|,\left|J^{\prime}\right|\leq
r\\\
\alpha,\alpha^{\prime}\in\mathcal{A}\end{subarray}}\Delta_{\alpha\alpha^{\prime}}^{JJ^{\prime},(\ell)}\sum_{j=1}^{m}\partial_{D_{\alpha}^{J}z_{j;\alpha}}\partial_{D_{\alpha^{\prime}}^{J^{\prime}}z_{j;\alpha^{\prime}}}\bigg{)}^{q}\bigg{]}f\left(D_{\mathcal{A}}^{\leq
r}z_{1},\ldots,D_{\mathcal{A}}^{\leq
r}z_{m}\right)\bigg{\rangle}_{\kappa^{(\ell+1)}}$
$\displaystyle\quad+O(n^{-q_{*}-1}),$ (3.9)
where the sum is over multi-indices $J,J^{\prime}\in\mathbb{N}^{p}$ of order
at most $r$, we’ve set
$\Delta_{\alpha\alpha^{\prime}}^{JJ^{\prime},(\ell)}:=D_{\alpha}^{J}D_{\alpha^{\prime}}^{J^{\prime}}\Sigma_{\alpha\alpha^{\prime}}^{(\ell)}-\mathbb{E}\left[D_{\alpha}^{J}D_{\alpha^{\prime}}^{J^{\prime}}\Sigma_{\alpha\alpha^{\prime}}^{(\ell)}\right],$
(3.10)
and the derivatives $\partial_{D_{\alpha}^{J}z_{j;\alpha}}$ are interpreted in
the weak sense if $f$ is not differentiable.
We prove Theorem 3.2 in §9 and give the main idea of the proof in §5. By
substituting various polynomials in $z_{\alpha}^{{}_{(\ell+1)}}$ and its
derivatives for $f$ into the perturbative expansion (3.9), it is now
straightforward in principle to deduce recursions for the cumulants
$\kappa_{(\alpha_{1}\alpha_{1}^{\prime}),\ldots,(\alpha_{k}\alpha_{k}^{\prime})}^{{}_{(J_{1}J_{1}^{\prime}),\ldots,(J_{k}J_{k}^{\prime}),(\ell+1)}}$
at layer $\ell+1$ in terms of objects of the same type at layer $\ell$. In
particular, we have the following
###### Corollary 3.3 (cumulants in random neural networks form a hierarchy to
leading order in $1/n$).
With the assumptions of Theorem 3.1,the mixed cumulant
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},\ldots,D_{\alpha_{2k}}^{J_{2k}}z_{i_{2k};\alpha_{2k}}^{(\ell+1)}\right)$
equals
$\sum_{\begin{subarray}{c}j\leq k\\\ J_{i}^{\prime},\,i=1,\ldots,2j\\\
\left|J_{1}^{\prime}\right|+\cdots+\left|J_{2j}^{\prime}\right|\\\
\quad\leq\left|J_{1}\right|+\cdots+\left|J_{2k}\right|\end{subarray}}C(J_{i}^{\prime},K_{\alpha_{i}\alpha_{i^{\prime}}}^{(\ell)}i,i^{\prime}=1,\ldots,2j)~{}\kappa\left(D_{\alpha_{1}}^{J_{1}^{\prime}}z_{1;\alpha_{1}}^{(\ell)},\ldots,D_{\alpha_{2j}}^{J_{2j}^{\prime}}z_{2j;\alpha_{2j}}^{(\ell)}\right)+O\left(n^{-k}\right),$
(3.11)
where the sum if over multi-indices $J_{i}^{\prime}$, the constants
$C(J_{i}^{\prime},K_{\alpha_{i}\alpha_{j}}^{(\ell)}i,j=1,\ldots,2k)$ depend
only on the multi-indices $J_{i}^{\prime}$ and the infinite width covariance
$K^{(\ell)}$, while the implicit constant in the error term depends on $k$,
the inputs $x_{\alpha_{1}},\ldots,x_{\alpha_{k}},$ the multi-indices
$J_{1},\ldots,J_{k}$, the weight and bias variances $C_{b},C_{W},$ the non-
linearity $\sigma$, and the layer index $\ell$.
We do not know how to efficiently compute the coefficients $C$ in the
recursion (3.11) for arbitrary $k$. Instead, we will compute them by hand for
small values of $k$ in the two settings:
* •
We fix a single input $x_{\alpha}\in\mathbb{R}^{n_{0}}$ and obtain a recursion
for the cumulants:
$\kappa_{2k;\alpha}^{(\ell)}:=\kappa_{(\alpha\alpha)\cdots(\alpha\alpha)}^{(00)\cdots(00),(\ell)}=\frac{1}{(2k-1)!!}\kappa\bigg{(}\underbrace{z_{i;\alpha}^{(\ell+1)},\ldots,z_{i;\alpha}^{(\ell+1)}}_{2k\text{
times}}\bigg{)}$ (3.12)
when $k=2,3,4$ (the pairs $(\alpha\alpha)$ and $(00)$ both appear $k$ times).
See Corollary 3.4.
* •
We again fix a single input $x_{\alpha}\in\mathbb{R}^{n_{0}}$ and consider any
two partial derivatives
$d_{j}:=\sum_{i=1}^{n_{0}}c_{i,j}\partial_{x_{j}}\bigg{|}_{x=x_{\alpha}},\qquad
j=1,2.$
We obtain recursions for
$\displaystyle\kappa_{(i_{1}i_{1}^{\prime})(i_{2}i_{2}^{\prime});\alpha}^{(\ell)}:=\kappa\left(d_{i_{1}}d_{i_{1}^{\prime}}\Delta_{\alpha\alpha}^{(\ell)},d_{i_{2}}d_{i_{2}^{\prime}}\Delta_{\alpha\alpha}^{(\ell)}\right).$
(3.13)
These cumulants are the building blocks for understanding the fourth mixed
cumulants of
$z_{i;\alpha}^{{}_{(\ell+1)}},d_{1}z_{i;\alpha}^{{}_{(\ell+1)}},d_{2}z_{i;\alpha}^{{}_{(\ell+1)}}$.
For example,
$\kappa\left(d_{1}z_{i;\alpha}^{(\ell+1)},d_{1}z_{i;\alpha}^{(\ell+1)},d_{2}z_{i;\alpha}^{(\ell+1)},d_{2}z_{i;\alpha}^{(\ell+1)}\right)=\kappa_{(11)(22);\alpha}^{(\ell)}+2\kappa_{(12)(12);\alpha}^{(\ell)}.$
This will be done in the course of proving Theorem 4.5. We refer the
interested reader to Proposition 10.2 for the precise recursions and to
Proposition 11.2 their solutions for non-linearities from the $K_{*}=0$
universality class (including $\tanh$).
In order to facilitate a compact form for the recursions described in first
bullet point, let us write
$T_{i,j;\alpha}^{(\ell)}:=C_{W}^{j}\left\langle\partial_{z}^{i}\left\\{\left(\sigma^{2}(z)-\left\langle\sigma^{2}(z)\right\rangle_{K_{\alpha\alpha}^{(\ell)}}\right)^{j}\right\\}\right\rangle_{K_{\alpha\alpha}^{(\ell)}},$
(3.14)
with the derivatives interpreted in the weak sense if $\sigma$ is not
sufficiently smooth and where we remind the reader of our standing notation
$\left\langle
f(z)\right\rangle_{K}=\int_{-\infty}^{\infty}f(zK^{1/2})e^{-\frac{z^{2}}{2}}\frac{dz}{\sqrt{2\pi}},\qquad
K\geq 0.$
###### Corollary 3.4.
Fix $r\geq 1$ and suppose that $\sigma:\mathbb{R}\rightarrow\mathbb{R}$
satisfies Assumption (2.1) with this value of $r$. Consider a depth $L$ random
neural network with input dimension $n_{0},$ hidden layer widths
$n_{1},\ldots,n_{L}$, output dimension $n_{L+1}$ and non-linearity $\sigma$.
Fix $x_{\alpha}\in\mathbb{R}^{n_{0}}$ and define
$\chi_{||;\alpha}^{(\ell)}:=\frac{1}{2}T_{2,1;\alpha}^{(\ell)}=\frac{C_{W}}{2}\left\langle\partial_{z}^{2}\sigma(z)^{2}\right\rangle_{K_{\alpha\alpha}^{(\ell)}},$
where the second derivative is interpreted in the weak sense if $\sigma$ is
not twice differentiable. For each $\ell=1,\ldots,L$, in the notation of
(3.12), the fourth cumulant satisfies
$\displaystyle\kappa_{4;\alpha}^{(\ell+1)}$
$\displaystyle=\frac{T_{0,2;\alpha}^{(\ell)}}{n_{\ell}}+\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}\kappa_{4,\alpha}^{(\ell)}+O(n^{-2}).$
(3.15)
Further, the $6$-th cumulant satisfies
$\displaystyle\kappa_{6;\alpha}^{(\ell+1)}$
$\displaystyle=\frac{T_{0,3;\alpha}}{n_{\ell}^{2}}+\frac{3T_{2,2;\alpha}^{(\ell)}}{2n_{\ell}}\chi_{||;\alpha}^{(\ell)}\kappa_{4;\alpha}^{(\ell)}-\frac{3T_{4,1;\alpha}^{(\ell)}}{8}\left(\chi_{||;\alpha}^{(\ell)}\kappa_{4;\alpha}^{(\ell)}\right)^{2}+\left(\chi_{||;\alpha}^{(\ell)}\right)^{3}\kappa_{6;\alpha}^{(\ell)}+O(n^{-3}).$
(3.16)
Finally, the $8$-th cumulant satisfies:
$\displaystyle\kappa_{8;\alpha}^{(\ell+1)}$
$\displaystyle=\frac{1}{n_{\ell}^{3}}\left(T_{0,4;\alpha}^{(\ell)}-3\left(T_{0,2;\alpha}^{(\ell)}\right)^{2}\right)$
$\displaystyle+\frac{1}{n_{\ell}^{2}}\left[2T_{2,3;\alpha}^{(\ell)}\chi_{||;\alpha}^{(\ell)}-12T_{0,2;\alpha}^{(\ell)}\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}+\frac{3}{2}\left(T_{2,2;\alpha}^{(\ell)}\right)^{2}-\frac{3}{2}T_{4,1;\alpha}^{(\ell)}T_{0,2;\alpha}^{(\ell)}\right]\kappa_{4;\alpha}^{(\ell)}$
$\displaystyle-\frac{1}{n_{\ell}}\left[2T_{2,2;\alpha}^{(\ell)}T_{4,1;\alpha}^{(\ell)}\chi_{||;\alpha}^{(\ell)}-\frac{1}{2}T_{4,2;\alpha}^{(\ell)}\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}+\left(\chi_{||;\alpha}^{(\ell)}\right)^{4}\right]\left(\kappa_{4;\alpha}^{(\ell)}\right)^{2}$
$\displaystyle+\frac{1}{n_{\ell}}\left[5T_{0,2;\alpha}^{(\ell)}T_{4,1;\alpha}^{(\ell)}\chi_{||;\alpha}^{(\ell)}+12T_{2,2;\alpha}^{(\ell)}\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}\right]\kappa_{6;\alpha}^{(\ell)}$
$\displaystyle+\frac{3}{32}\left(T_{4,1;\alpha}^{(\ell)}\right)^{2}\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}\left(\kappa_{4;\alpha}^{(\ell)}\right)^{3}-\frac{1}{2}\left(\chi_{||;\alpha}^{(\ell)}\right)^{3}T_{4,1;\alpha}^{(\ell)}\kappa_{4;\alpha}^{(\ell)}\kappa_{6;\alpha}^{(\ell)}$
$\displaystyle+\left(\chi_{||;\alpha}^{(\ell)}\right)^{4}\kappa_{8;\alpha}^{(\ell)}+O(n^{-4}).$
(3.17)
The initial condition for the recursions (3.15)-(3.17) is that
$\kappa_{2k;\alpha}^{{}_{(1)}}=0$ for all $k\geq 2$.
###### Remark 3.5.
Note that for $k=2,3,4$, we therefore see that to leading order in $1/n$ the
recursions for $\kappa_{2k;\alpha}^{{}_{(\ell+1)}}$ depends only on
$\kappa_{2j;\alpha}^{{}_{(\ell)}}$ for $j\leq k$. This allows us to interpret
(3.15) - (3.17) as a forming the start of hierarchy in powers of $1/n$
describing the cumulants of the output of a random neural network.
Let us briefly compare Corollary 3.4 to results in prior work:
* •
In the special case when $\sigma$ is $1-$homogeneous (i.e. is linear, ReLU,
leaky ReLU, etc, see (4.5)), the full distribution of a neuron pre-activation
$z_{i;\alpha}^{{}_{(\ell)}}$ can be worked out in closed form. Namely, as we
explain in §4.2.1 and Appendix C, is has the same distribution as a Gaussian
with an independent random variance given by a product of independent weighted
chi-squared random variables. This was first pointed out in [17, 21] and
described in the language of special functions (namely Meijer G functions) in
[66]. For such non-linearities obtaining the recursions (3.15)-(3.17) is not
new.
* •
The breakthrough work of Yaida [61] was the first to obtain, at a physics
level of rigor, the recursion (3.15) and probe its solutions at large $\ell$.
* •
The ideas of Yaida in [61] then seeded the development in the monograph [53]
of Roberts, Yaida, and the author a much richer analysis, producing at a
physics level of rigor many recursions similar in flavor to (3.15)-(3.17) that
describe the the behavior of objects such as network derivatives at the start
of training, the NTK at the start of training, and even the change in the NTK
and the resulting output of a trained network. Many of these results go far
beyond what we are currently capable of doing mathematically.
* •
The analysis in [53] never required studying cumulants
$\kappa_{2k;\alpha}^{{}_{(\ell)}}$ for $k\geq 3$, and while the techniques
developed there can certainly be used to obtain the recursions (3.16) and
(3.17) we take a rather different approach in this article that produces such
recursions more directly.
The functional $\chi_{||;\alpha}^{{}_{(\ell)}}$ plays a fundamental role in
the recursive description of random neural networks supplied by Corollary 3.4,
whose proof is in §9. In the following section we explain a principled
procedure, called tuning to criticality, that reveals its origin (as well as
that of a similar object we denote $\chi_{\perp;\alpha}^{{}_{(\ell)}}$) and
explains how to choose $C_{b},C_{W}$ so that these functionals are
approximately equal to $1$ at large $\ell$. As we will see, such a choice will
ensure that the recursions in Corollary 3.4 and their infinite width
counterpart (2.4) have well-behaved solutions at large $\ell$. We will then
return in §4.2.1 and §4.2.2 to solving the recursions from Corollary 3.4 in
random networks tuned to criticality (see (4.7) and Corollary 4.1).
## 4 Criticality and Universality in Random Neural Networks at Large Depth
In the previous section we presented two kinds of results about the structure
of random neural networks at large but finite width. The first, Theorem 3.1,
concerned the order of magnitude for cumulants of the output of such a random
network and its derivatives. The second, Theorem 3.2 and Corollary 3.4,
spelled out recursions with respect to the layer index $\ell$ that describe,
to leading order in $1/n$, network cumulants at layer $\ell+1$ in terms of
those at layer $\ell$. Our purpose in forthcoming sections is to analyze these
recursions at large $\ell$ and to apply this analysis to obtain results about
the structure of gradients in deep fully connected networks. Before doing
this, we must take a step back and ask: for which $\sigma,C_{b},C_{W}$ are the
recursions (2.4) describing the infinite width covariance $K^{(\ell)}$ well-
behaved at large $\ell$?
In section §4.1, we recall a more or less canonical answer to this question
whose roots are in the early articles [51, 52] and that was recently spelled
out in the generality presented here in [53]. This procedure, called tuning to
criticality, prescribes combinations of $\sigma,C_{b},C_{W}$ for which
$K^{(\ell)}$ is indeed well-behaved at large $\ell$. As we shall see below,
the term criticality is meant to be evocative of it’s use in the analysis of
2d systems in statistical mechanics in that tuning to criticality consists of
choosing $C_{b},C_{W}$ so that the infinite width covariance function
$K^{{(\ell)}}$ is as close to constant as a function $\ell$ as possible.
At a high level, there are two reasons to ask that $K^{(\ell)}$ be slowly
varying as a function of $\ell$. First, it arguably only makes sense to study
perturbative corrections in $1/n$ recursively in $\ell$ if the limiting
$n\rightarrow\infty$ covariance structure does not change too rapidly between
consecutive layers. Second, and perhaps more importantly, as explained and
thoroughly validated in [50, 52], deep fully connected networks (without
residual connections [24], batch normalization [31], etc) are numerically
stable enough for gradient-based optimization to succeed only if they are
tuned to criticality.
We discuss in §4.2 how considerations underlying criticality naturally give
rise to a notion of universality classes for random neural networks. Even the
correct definition of universality is still not fully understood. Unlike in
random matrix theory, universality for random neural networks depends not on
the statistics of the individual weights and biases (though this is also an
interesting direction to consider e.g. [19]) but rather on the effect of the
non-linearity $\sigma$ on the behavior of the infinite width covariance
$K^{(\ell)}$ at large values of the depth $\ell$.
Before giving the details, we take this opportunity to emphasize, as we have
elsewhere, that the definitions of criticality and universality, the approach
to solving the recursions for $\kappa_{2k;\alpha}^{{}_{(\ell)}}$ from
Corollary 3.4, and the resulting lessons learned about the role of the
effective network depth $L/n$ closely follow the ideas developed in the
monograph [53]. Though we pursue them in a somewhat different way, the author
would nonetheless like to acknowledge that his co-authors Dan Roberts and Sho
Yaida in the book deserve significant credit.
### 4.1 Tuning to Criticality
As originally explained in [51, 52] and recently spelled out in a definitive
way in [53], tuning a neural network to criticality means seeking choices of
$(C_{b},C_{W})$ that lead to critical fixed points of the form
$(K_{*},K_{*},K_{*})$ for the recursion (2.4), viewed as a dynamical system
describing
$(K_{\alpha\alpha}^{{}_{(\ell)}},K_{\beta\beta}^{{}_{(\ell)}},K_{\alpha\beta}^{{}_{(\ell)}})$
with time parameter $\ell$. Specifically, criticality requires
$\displaystyle\exists K_{*}\geq 0\qquad\text{s.t.}\qquad$ $\displaystyle
K_{*}=C_{b}+C_{W}\left\langle\sigma^{2}(z)\right\rangle_{K_{*}}$ ($*$)
$\displaystyle\forall\ell\geq 1\qquad$ $\displaystyle\frac{\partial
K_{\alpha\alpha}^{(\ell)}}{\partial
K_{\alpha\alpha}^{(1)}}\bigg{|}_{K_{\alpha\alpha}^{(1)}=K_{*}}=1$ ($||$)
$\displaystyle\forall\ell\geq 1\qquad$ $\displaystyle\frac{\partial
K_{\alpha\beta}^{(\ell)}}{\partial
K_{\alpha\beta}^{(1)}}\bigg{|}_{K_{\alpha\alpha}^{(1)}=K_{\alpha\alpha}^{(1)}=K_{\alpha\beta}^{(1)}=K_{*}}=1,$
($\perp$)
where
$K_{\alpha\alpha}^{(1)}=C_{b}+C_{W}K_{\alpha\beta}^{(0)},\qquad
K_{\alpha\beta}^{(0)}:=\frac{1}{n_{0}}\sum_{j=1}^{n_{0}}x_{j;\alpha}x_{j;\beta},\qquad
x_{\alpha},x_{\beta}\in\mathbb{R}^{n_{0}}.$
The intuitive meaning of these conditions is as follows. Due to Theorem 2.2,
the first guarantees the existence of a fixed point $K_{*}$ for of the
recursion
$K_{\alpha\alpha}^{(\ell+1)}=C_{b}+C_{W}\left\langle\sigma^{2}(z)\right\rangle_{K_{\alpha\alpha}^{(\ell)}}$
(4.1)
of the infinite width variance. In particular, ($*$ ‣ 4.1) implies
$K_{\alpha\alpha}^{(1)}=C_{b}+\frac{C_{W}}{n_{0}}\left|\left|x_{\alpha}\right|\right|^{2}=K_{*}\qquad\Longrightarrow\qquad
K_{\alpha\alpha}^{(\ell)}=\lim_{n\rightarrow\infty}\mathrm{Var}\left[z_{i;\alpha}^{(\ell)}\right]=K_{*}\quad\forall\ell\geq
1.$
Thus, if a network is tuned to criticality, there is a critical radius
$K_{\text{crit}}^{2}:=n_{0}C_{W}^{-1}(K_{*}-C_{b})$
such that for inputs $x_{\alpha}$ on the sphere of radius $K_{\text{crit}}$
the variance of $z_{i;\alpha}^{{}_{(\ell)}}$ is independent of $\ell$ in the
infinite width limit. In non-critical networks, we expect this variance to
either grow or decay exponentially in $\ell$, leading to numerical
instabilities. The second condition ($||$ ‣ 4.1) considers the infinite width
limit of the variance of $z_{i;\alpha}^{{}_{(\ell)}}$ for an input
$x_{\alpha}$ for which $K_{\alpha\alpha}^{{}_{(1)}}$ is close to $K_{*}$.
Specifically, condition ($||$ ‣ 4.1) requires for all $\ell\geq 1$ that
$K_{\alpha\alpha}^{(1)}=\mathrm{Var}[z_{i;\alpha}^{(1)}]=K_{*}+\delta
K\qquad\Longrightarrow\qquad
K_{\alpha\alpha}^{(\ell)}=\lim_{n\rightarrow\infty}\mathrm{Var}\left[z_{i;\alpha}^{(\ell)}\right]=K_{*}+\delta
K+o(\delta K).$
This guarantees that the fixed point $K_{*}$ of the recursion (4.1) is
critical and hence that for inputs near the sphere of radius $K_{\text{crit}}$
the variance of the resulting pre-activations $z_{i;\alpha}^{{}_{(\ell)}}$ is
approximately constant in $\ell$ at large $n$. The final condition ($\perp$ ‣
4.1) considers instead the covariance between two inputs on the sphere of
radius $K_{\text{crit}}$. Namely, given two nearby network inputs
$x_{\alpha},x_{\beta}\in\mathbb{R}^{n_{0}}$ with
$K_{\alpha\alpha}^{(1)}=K_{\beta\beta}^{(1)}=K_{*},\qquad
K_{\alpha\beta}^{(1)}=C_{b}+\frac{C_{W}}{n_{0}}\sum_{j=1}^{n_{0}}x_{j;\alpha}x_{j;\beta}=K_{*}-\delta
K,$
the third condition asks that
$K_{\alpha\beta}^{(\ell)}=\lim_{n\rightarrow\infty}\mathrm{Cov}\left(z_{i;\alpha}^{(\ell)},z_{i;\beta}^{(\ell)}\right)=K_{*}-\delta
K+o(\delta K),\quad\forall\ell.$
This ensures that the covariance between pre-activations
$z_{i;\alpha}^{{}_{(\ell)}}$ and $z_{i;\beta}^{{}_{(\ell)}}$ corresponding to
two nearby inputs on the $K_{\text{crit}}$-sphere are approximately
independent of $\ell$ at large $n$. A simple computation directly from the
recursion (2.4) shows that
$\displaystyle\chi_{||}(K)$ $\displaystyle:=\frac{\partial
K_{\alpha\alpha}^{(\ell+1)}}{\partial
K_{\alpha\alpha}^{(\ell)}}\bigg{|}_{K_{\alpha\alpha}^{(\ell)}=K}=\frac{C_{W}}{2}\left\langle\partial_{z}^{2}(\sigma^{2}(z))\right\rangle_{K}$
(4.2) $\displaystyle\chi_{\perp}(K)$ $\displaystyle:=\frac{\partial
K_{\alpha\beta}^{(\ell+1)}}{\partial
K_{\alpha\beta}^{(\ell)}}\bigg{|}_{K_{\alpha\alpha}^{(\ell)}=K_{\beta\beta}^{(\ell)}=K_{\alpha\beta}^{(\ell)}=K}=C_{W}\left\langle(\partial_{z}\sigma(z))^{2}\right\rangle_{K}.$
(4.3)
Hence, all together, tuning to criticality requires
$\boxed{K_{*}\geq 0\text{ s.t.
}K_{*}=C_{b}+C_{W}\left\langle\sigma^{2}(z)\right\rangle_{K_{*}}\qquad\text{and}\qquad\chi_{||}(K_{*})=\chi_{\perp}(K_{*})=1.}$
(4.4)
### 4.2 Universality Classes of Random Neural Networks: Two Examples In
Search of a General Definition
We now turn to discussing the notion of universality classes for random neural
networks. To start, recall from Theorem 2.2 that the behavior at large depth
$\ell$ of random fully connected neural networks at infinite width is
completely specified by the asymptotics of the limiting covariance function
$K^{{(\ell)}}$. Observe, moreover, that the coefficients in the recursions for
$k=2,3,4$ of the cumulants $\kappa_{2k;\alpha}^{{}_{(\ell+1)}}$ from Corollary
3.4, which by Theorem 3.1 determine the behavior of random neural networks at
finite width to the first four orders in $1/n$, are completely determined by
$\sigma$, the infinite width covariance $K^{(\ell)}$, and cumulants
$\kappa_{2j;\alpha}^{{}_{(\ell)}},\,j\leq k$. It is therefore in terms of the
large $\ell$ behavior of $K^{{(\ell)}}$ that we should hope to define
universality classes of random neural networks at large depth. At present it
is not clear what the correct general definition of such a universality class
should be. We content ourselves instead with studying two important classes of
examples.
#### 4.2.1 The Universality Class of ReLU
The most popular non-linearities used in practice are positively homogeneous
of degree $1$, i.e. have the form
$\sigma(t)=(a_{-}{\bf 1}_{\left\\{t<0\right\\}}+a_{+}{\bf
1}_{\left\\{t>0\right\\}})t,\qquad a_{-},a_{+}\in\mathbb{R},\quad a_{-}\neq
a_{+},\quad a_{-}^{2}+a_{+}^{2}\neq 0.$ (4.5)
Such non-linearities include the ReLU ($a_{-}=0,a_{+}=1$) and the leaky ReLU
($a_{-}\in(0,1),a_{+}=1$). A direct computation, left to the reader, shows
that criticality is achieved if and only if
$K_{*}\geq 0\text{ is arbitrary}\quad\text{and}\quad
C_{b}=0,\,C_{W}=\frac{2}{a_{+}^{2}+a_{-}^{2}}.$
Thus, the first property of the ReLU universality class is that setting
$(C_{b},C_{W})=(0,2/(a_{+}^{2}+a_{-}^{2}))$ allows all non-negative $K_{*}$ to
satisfy ($*$). In fact, at criticality, a simple symmetrization argument shows
that the variance of neuron pre-activations is preserved exactly even at
finite width
$\mathrm{Var}\left[z_{i;\alpha}^{(\ell)}\right]=\mathrm{Var}\left[z_{i;\alpha}^{(1)}\right]=\frac{C_{W}}{n_{0}}\left|\left|x_{\alpha}\right|\right|^{2}\qquad\forall\ell,n_{0},\ldots,n_{\ell}\geq
1,\,x_{\alpha}\in\mathbb{R}^{n_{0}}$ (4.6)
and, relatedly, that we have
$\chi_{||;\alpha}^{(\ell)}:=\chi_{||}(K_{\alpha\alpha}^{(\ell)})=1=\chi_{\perp;}(K_{\alpha\alpha}^{(\ell)})=:\chi_{\perp;\alpha}^{(\ell)},\qquad\forall\ell\geq
1,\,x_{\alpha}\in\mathbb{R}^{n_{0}}.$
The remarkable property (4.6) is much stronger than the criticality condition
$(*)$, which requires only that this condition holds for some value of
$n_{0}^{-1}\left|\left|x_{\alpha}\right|\right|^{2}$ and only in the limit
when $n\rightarrow\infty$. It implies that the cumulant recursions from
Corollary 3.4 for $1-$homogeneous non-linearities have constant coefficients
and are therefore particularly simple to solve. For instance, we find at
leading order in $1/n$
$\displaystyle\kappa_{4;\alpha}^{(\ell+1)}$
$\displaystyle=\frac{C_{W}^{2}}{n_{\ell}}\left[\left\langle\sigma(z)^{4}\right\rangle_{K_{\alpha\alpha}^{(\ell)}}-\left\langle\sigma(z)^{2}\right\rangle_{K_{\alpha\alpha}^{(\ell)}}^{2}\right]+\left(\chi_{||;\alpha}^{(\ell)}\right)^{2}\kappa_{4;\alpha}^{(\ell)}$
$\displaystyle=\left(\frac{2}{(a_{+}^{2}+a_{-}^{2})n_{0}}\left|\left|x_{\alpha}\right|\right|^{2}\right)^{2}\left(6\frac{a_{+}^{4}+a_{-}^{4}}{(a_{+}^{2}+a_{-}^{2})^{2}}-1\right)\sum_{\ell^{\prime}=1}^{\ell}\frac{1}{n_{\ell^{\prime}}},$
which shows that while $\kappa_{4,\alpha}^{{}_{(\ell)}}$ is suppressed by one
power of $1/n$ relative to the infinite width variance
$K_{\alpha\alpha}^{{}_{(\ell)}}$, it also grows one order faster in $\ell$.
This illustrates an important and general theme: depth amplifies finite width
effects. It is the effective depth $\ell/n$ of neurons at layer $\ell$ that
measures the distance to the infinite width Gaussian regime.
Moreover, in the special setting of $1$-homogeneous non-linearities there is a
simple method for obtaining the full distribution of the pre-activation vector
$z_{\alpha}^{{}_{(\ell)}}$ at a single input at any finite values of
$n_{0},\ldots,n_{\ell}$. This was first pointed out in [17, 21, 66] and is
briefly reviewed in Appendix C. A key takeaway is that if we take the hidden
layer widths $n_{1}=\cdots=n_{L}=n$, then we have following convergence in
distribution to product of independent normal and log-normal random variables:
$\lim_{\begin{subarray}{c}n,L\rightarrow\infty\\\
L/n\rightarrow\xi\in[0,\infty)\end{subarray}}z_{i;\alpha}^{(L)}\quad\stackrel{{\scriptstyle
d}}{{=}}\quad\left(\frac{2\left|\left|x_{\alpha}\right|\right|^{2}}{(a_{+}^{2}+a_{-}^{2})n_{0}}\right)^{1/2}Z_{1}\exp\left[-\mu(\xi,a_{+},a_{-})+\sigma(\xi,a_{+},a_{-})Z_{2}\right],$
(4.7)
where
$\mu(\xi,a_{+},a_{-})=\sigma^{2}(\xi,a_{+},a_{-}):=\frac{\xi}{4}\left(6\frac{a_{+}^{4}+a_{-}^{4}}{(a_{+}^{2}+a_{+}^{2})^{2}}-1\right),\quad
Z_{1},Z_{2}\sim\mathcal{N}(0,1)\text{ iid}.$
The convergence (4.7) reveals that for a fixed input the distribution of the
output of a random with $1-$homogeneous non-linearities at large depth and
width depends in a simple way on the limiting effective network depth $\xi$.
This bolsters the claim that they are all part of the same universality class.
It also means that increasing the network depth $L$ drives it away from the
infinite width Gaussian behavior observed at $\xi=0$ and that the outputs of
deep and wide networks are not well-approximated by a Gaussian at all, unless
$\xi$ is infinitesimal, in which case the log-normal term
$\exp\left[-\mu(\xi,a_{+},a_{-})+\sigma(\xi,a_{+},a_{-})Z_{2}\right]$ is
negligible.
Prior work [20, 21, 23] of the author shows that when $\sigma=\mathrm{ReLU}$
(or any other $1$-homogeneous non-linearity), the distribution at large $n,L$
of not only the network output $z_{i;\alpha}^{{}_{(L+1)}}$ but also is
derivatives with respect to inputs $x_{\alpha}$ and model parameters (e.g.
weights and biases) depends only on the effective depth $L/n.$ We will return
to this theme in our discussion of exploding and vanishing gradients (Theorem
4.5). Finally, we note that it has also been observed that log-normal random
variables describe the structure of gradients in residual networks, even
during/after training [39].
To complete our discussion of the ReLU universality class, we make two final
remarks. First, a direct computation (reviewed briefly in Proposition C.1 of
Appendix C) shows that at criticality for any non-zero inputs
$x_{\alpha_{1}},x_{\alpha_{2}}\in\mathbb{R}^{n_{0}}$ with the same norm we
have
$\lim_{n\rightarrow\infty}\mathrm{Corr}\left(z_{i;\alpha_{1}}^{(\ell)},z_{i;\alpha_{2}}^{(\ell)}\right)=1-\frac{2(a_{+}-a_{-})^{2}}{3\pi(a_{+}^{2}+a_{-}^{2})}\ell^{-2}(1+o(1)).$
(4.8)
The power law exponent $2$ that appears in this estimate is common to all
$1-$homogeneous non-linearities and is another reason to believe they fall
into the same universality class. In contrast, this exponent equals one for
non-linearities in the $K_{*}=0$ universality class presented below. The
estimate (4.8) suggests that in order to define a double scaling limit
$n,L\rightarrow\infty$ and $L/n\rightarrow\xi$ in which the entire field
$x_{\alpha}\mapsto z_{\alpha}^{{}_{(L+1)}}$ is non-degenerate (rather than
just its value at a single input) one must rescale distances in the input
space to prevent the collapse of correlations otherwise guaranteed in (4.8).
We leave this as an interesting direction for future work.
#### 4.2.2 The Universality Class of Hyperbolic Tangent
The second class of non-linearities we study is what [53] termed the $K_{*}=0$
universality class, which we take to mean non-linearities $\sigma$ such that
* •
$\sigma$ is a smooth, odd function satisfying Assumption 2.1.
* •
$\sigma$ satisfies
$\sigma_{1}\sigma_{3}<0,\qquad\sigma_{j}:=\frac{1}{j!}\frac{d^{j}}{dt^{j}}\bigg{|}_{t=0}\sigma(t).$
(4.9)
* •
$K_{*}=0$ is the unique fixed point of equation ($*$ ‣ 4.1).
* •
At criticality, for every non-zero network input
$x_{\alpha}\in\mathbb{R}^{n_{0}}$ and each $\delta\in(0,1)$ we have as
$L\rightarrow\infty$ that
$K_{\alpha\alpha}^{(L)}=\frac{1}{aL}\left(1+O(L^{-1+\delta})\right),$ (4.10)
where the implicit constant depends on $\delta$ and $x_{\alpha}$ and we’ve set
$a:=-6\frac{\sigma_{3}}{\sigma_{1}}.$
This specific value of $a$, which is positive by (4.9), is the only possible
candidate for decay of the form (4.10) that is consistent with the recursion
(2.4).
Some remarks are in order. First, if $K_{*}=0$ is the unique fixed point for
($*$ ‣ 4.1), then a simple computation shows that criticality is achieved if
and only if
$K_{*}=0,\qquad C_{b}=0,\qquad C_{W}=\sigma_{1}^{-2}.$ (4.11)
Next, our definition of the $K_{*}=0$ universality class does not make
apparent whether it is empty. As we will see in Proposition B.1, however, the
$K_{*}=0$ universality class is in fact quite large and contains for example
the hyperbolic tangent and more generally any non-linearity that is
$\tanh$-like in the sense that is smooth with $\sigma_{1}\neq 0$, has the
opposite sign as its second derivative
$\text{for almost every
}z,~{}\mathrm{sgn}\left(\sigma(z)\sigma^{\prime\prime}(z)\right)=-1,$
is sub-linear
$\exists C>0\,\text{ s.t. }\forall
z\in\mathbb{R}\,\left|\sigma(z)\right|\leq\left|\sigma_{1}z\right|,$
and is controlled by its first few non-zero Taylor series coefficients at $0$:
$\exists C\geq 0\text{ s.t. }\forall z\geq
0,\quad\sigma_{1}z+\sigma_{3}z^{3}\leq\sigma(z)\leq\sigma_{1}z+\sigma_{3}z^{3}+Cz^{4}.$
Further, by definition, for the $K_{*}=0$ universality class, the infinite
width variance $K_{\alpha\alpha}^{{}_{(\ell)}}$ of neuron pre-activations
$z_{i;\alpha}^{(\ell)}$ is qualitatively different from that of
$1-$homogeneous non-linearities. Indeed, $K_{\alpha\alpha}^{{}_{(L)}}$ depends
on $L$, decaying polynomially to $0$. Moreover, at large $L$, the value of
$K_{\alpha\alpha}^{{}_{(L)}}$ is independent of the initial condition
$K_{\alpha\alpha}^{{}_{(0)}}$ to leading order in $L$. As a final remark let
us point out that searching for non-linearities $\sigma$ so that $K_{*}=0$ at
criticality is quite natural. Indeed, for any $\sigma$ that is twice
differentiable, we have
$\chi_{||}(K)=\chi_{\perp}(K)+C_{W}\left\langle\sigma(z)\sigma^{\prime\prime}(z)\right\rangle_{K}$
Hence, if $K>0$, then
$\chi_{||}(K)=1,\,\chi_{\perp}(K)=1\qquad\Longrightarrow\qquad\left\langle\sigma(z)\sigma^{\prime\prime}(z)\right\rangle_{K}=0.$
But if $\sigma$ is a sigmoidal function such as $\tanh$, then
$\sigma(z)\sigma^{\prime\prime}(z)<0$ for all $z\neq 0$. Hence,
$\left\langle\sigma(z)\sigma^{\prime\prime}(z)\right\rangle_{K}=0$ can only
occur when $K=0$.
As in the monograph [53], let us now probe the role of network depth by
studying the large $L$ behavior of the cumulants
$\kappa_{2k;\alpha}^{{}_{(L)}},\,k=2,3,4,$ in networks with non-linearities
from the $K_{*}=0$ universality class tuned to criticality. Note that in
(4.10) the limiting behavior of the variance $K_{\alpha\alpha}^{{}_{(L)}}$
depends (mildly) on the non-linearity $\sigma$ in terms of is first few Taylor
coefficients at $0$. As we are about to see, however, the behavior of the
higher cumulants $\kappa_{2k;\alpha}^{{}_{(L)}},\,k=2,3,4,$ when normalized by
the appropriate power of $K_{\alpha\alpha}^{{}_{(L)}}$, is independent of
$\sigma$ at leading order in $n$ and $L$ and depends only on universal
constants and the effective network depth $L/n$.
###### Corollary 4.1.
Suppose $\sigma$ is a non-linearity from the $K_{*}=0$ universality class, and
consider a depth $L$ random neural network with input dimension $n_{0}$,
output dimension $n_{L+1}$, hidden layer widths satisfying
$n_{1},\ldots,n_{L}=n\gg 1,$
and non-linearity $\sigma$ that has been tuned to criticality as in (4.11).
Write $\xi=L/n$ and define the normalized cumulants
$\widehat{\kappa}_{2k;\alpha}^{(L)}:=\frac{\kappa_{2k;\alpha}^{(\ell)}}{(K_{2;\alpha}^{(\ell)})^{k}}.$
We have for $k=2,3,4$ that
$\displaystyle\widehat{\kappa}_{2k;\alpha}^{(L)}$
$\displaystyle=C_{2k}\xi^{k-1}\left(1+O(L^{-1})\right)+O(n^{-k}),$ (4.12)
where
$C_{4}=\frac{2}{3},\qquad C_{6}=\frac{28}{15},\qquad C_{8}=\frac{8756}{315}$
are some positive universal constants. The implicit constants in error terms
$O(L^{-1})$ depend on $\sigma,C_{b},C_{W}$ and constants in
$O(n^{-j}),\,j=2,3,4$ may depend in addition on $L$.
###### Remark 4.2.
Combining estimate (4.12) for $k=2$ with the definition (4.10) of the
$K_{*}=0$ universality class and (3.12) yields that in the setting of
Corollary 3.4 we have to first order in $1/n$ and to leading order in $1/L$
that
$\displaystyle\widehat{\mathrm{Var}}\left[\left(z_{1;\alpha}^{(\ell+1)}\right)^{2}\right]$
$\displaystyle=2(1+\xi),\qquad\mathrm{Corr}\left(\left(z_{1;\alpha}^{(\ell+1)}\right)^{2},\,\left(z_{2;\alpha}^{(\ell+1)}\right)^{2}\right)=\frac{2}{3}\xi,\qquad\xi:=\frac{L}{n},$
where
$\widehat{\mathrm{Var}}[X]=\mathrm{Var}[X]/\mathbb{E}\left[X^{2}\right]$.
Thus, both the fluctuations of a single neuron pre-activation and the
correlation between different neurons is controlled to first order in
$1/n,1/L$ by the effective network depth $\xi$.
We prove Corollary 3.4 in §11.2. The formula (4.12) derived in [61] for $k=2$
at a physics level of rigor. Moreover, Corollary 3.4 represents a partial
analog of the convergence result (4.7), which lends credence to the following
###### Conjecture 4.3 (Existence of Double Scaling Limit for Random Neural
Networks).
Consider a random depth $L$ neural network with input dimension $n_{0}$,
hidden layer widths
$n_{1},\ldots,n_{L}=n\gg 1,$
output dimension $n_{L+1}$ and non-linearity $\sigma$. Suppose further that
this network is tuned to criticality in the sense that (4.4) is satisfied. Fix
a non-zero network input $x_{\alpha}\in\mathbb{R}^{n_{0}}$ and write
$\xi=L/n$. For each $k\geq 1$ there exists $C_{2k}>0$ depending on the
universality class of $\sigma$ so that
$\frac{\kappa_{2k;\alpha}^{(L)}}{\left(K_{2;\alpha}^{(L)}\right)^{k}}=C_{2k}\xi^{k-1}+O\left(\xi^{k}\right).$
Moreover, for each $\xi\in[0,\infty)$ there exists a probability distribution
$\mathbb{P}_{\xi,\sigma}$ on $\mathbb{R}$, depending only on $\xi$ and
$\sigma$, such that in the double scaling limit
$n,L\rightarrow\infty,\qquad\frac{L}{n}\rightarrow\xi,$
the random variable $z_{i;\alpha}^{(\ell)}$ converges in distribution to a
random variable with law $\mathbb{P}_{\xi,\sigma}$.
### 4.3 Gradients in Random Neural Networks
We presented in §3 and §4 a range of results about random fully connected
neural networks. The high-level takeaway was that such networks are succinctly
described when $n,L$ are large by the effective network depth $\xi=L/n$. We
saw that when $\xi=0$ the network outputs are a Gaussian process (Theorem
2.2). When $\xi>0$, however, a different picture emerges. Namely, the higher
order cumulants of the network output grow like powers of $\xi$ (see Corollary
3.4 and Conjecture 4.3). In this section we continue to consider a random
neural network $x_{\alpha}\mapsto z_{\alpha}^{{}_{(L+1)}}$ and study for any
$\ell$ the partial derivatives
$\partial_{x_{j;\alpha}}z_{i;\alpha}^{(\ell)},\qquad j=0,\ldots,n_{0},$
where by definition when $j=0$ we set
$\partial_{x_{0;\alpha}}z_{i;\alpha}^{(\ell)}:=z_{i;\alpha}^{(\ell)}.$
In §10.2 \- §10.4 we obtain recursions with respect to $\ell$ for both the
infinite width covariances
$K_{(ij)}^{(\ell)}:=\lim_{n\rightarrow\infty}\mathrm{Cov}\left(\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{i;\alpha}},\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{j;\alpha}}\right),\quad i,j=1,2.$
and the fourth cumulants
$\displaystyle\kappa_{(ii)(jj)}^{(\ell)}$
$\displaystyle:=(1-\frac{2}{3}\delta_{ij})\kappa\left(\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{i;\alpha}},\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{i;\alpha}},\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{j;\alpha}},\frac{\partial
z_{i;\alpha}^{(\ell+1)}}{\partial x_{j;\alpha}}\right),\qquad i,j=0,1,2.$
$\displaystyle=\kappa\left(\frac{\partial z_{1;\alpha}^{(\ell+1)}}{\partial
x_{i;\alpha}},\frac{\partial z_{1;\alpha}^{(\ell+1)}}{\partial
x_{i;\alpha}},\frac{\partial z_{2;\alpha}^{(\ell+1)}}{\partial
x_{j;\alpha}},\frac{\partial z_{2;\alpha}^{(\ell+1)}}{\partial
x_{j;\alpha}}\right).$
These recursions are valid for any non-linearity and any choice of
$C_{b},C_{W}$. Then, in §11.2 we solve these recursions in the context of a
random neural network with a non-linearity from the $K_{*}=0$ universality
class tuned to criticality. We record several of these solutions in the
following result.
###### Theorem 4.4 (Second and Fourth Joint Cumulants for Values and
Derivatives of Random Neural Networks).
Fix $r\geq 1$ and suppose that $\sigma:\mathbb{R}\rightarrow\mathbb{R}$
satisfies Assumption 2.1 with this value of $r$. Fix also an $x_{\alpha}\neq
0\in\mathbb{R}^{n_{0}}$. Consider a random neural network
$z_{\alpha}^{{}_{(L+1)}}$ with input dimension $n_{0}$, output dimension
$n_{L+1}$, hidden layer widths $n_{1},\ldots,n_{L}=n$, and non-linearity
$\sigma$ belonging to the $K_{*}=0$ universality class that has been tuned to
criticality as in (4.11). We have for any $\delta\in(0,1)$
$\displaystyle K_{(00)}^{(\ell+1)}$
$\displaystyle=\frac{1}{a\ell}+O_{\delta}(\ell^{-2+\delta})$ (4.13)
$\displaystyle K_{(10)}^{(\ell+1)}$
$\displaystyle=\frac{C_{W}e^{-2\gamma}x_{1;\alpha}}{n_{0}\ell^{2}}\left(1+O(\ell^{-1})\right)$
(4.14) $\displaystyle K_{(11)}^{(\ell+1)}$
$\displaystyle=\frac{C_{W}e^{-\gamma}}{n_{0}\ell}\left(1+O(\ell^{-1})\right)+O\left(\frac{(x_{1;\alpha})^{2}}{n_{0}^{2}\ell^{3}}\right)$
(4.15) $\displaystyle K_{(12)}^{(\ell+1)}$
$\displaystyle=O\left(\frac{x_{1;\alpha}x_{2;\alpha}}{n_{0}^{2}\ell^{3}}\right),$
(4.16)
where $\gamma$ is the Euler-Mascheroni constant and the implicit constant in
the $O_{\delta}(\ell^{-2+\delta})$ error depends on $\delta$. Moreover,
denoting by $\xi=L/n$ the effective network depth, we have
$\displaystyle\frac{\kappa_{(11)(00)}^{(\ell+1)}}{K_{(11)}^{(\ell)}K_{(00)}^{(\ell)}}$
$\displaystyle=-\frac{1}{3}\xi(1+O(\ell^{-1}))+O(n^{-2})$ (4.17)
$\displaystyle\frac{\kappa_{(11)(11)}^{(\ell+1)}}{(K_{(11)}^{(\ell)})^{2}}$
$\displaystyle=\frac{8}{3}\xi(1+O(\ell^{-1}))+O(n^{-2})$ (4.18)
$\displaystyle\frac{\kappa_{(11)(22)}^{(\ell+1)}}{K_{(11)}^{(\ell)}K_{(22)}^{(\ell)}}$
$\displaystyle=\frac{2}{3}\xi(1+O(\ell^{-1}))+O(n^{-2}),$ (4.19)
where the implicit constants in the $O(\ell^{-1})$ error terms depend only on
$x_{\alpha},\sigma$ and the implicit constants in the $O(n^{-2})$ error terms
may depend in addition on $\ell$.
Theorem 4.4 reveals several interesting phenomena. First, we find from (4.13)
and (4.14) that at infinite width values and derivatives become uncorrelated
at large $L$:
$\lim_{L\rightarrow\infty}\lim_{n\rightarrow\infty}\mathrm{Corr}\left(z_{1;\alpha}^{(L+1)},\,\partial_{x_{1;\alpha}}z_{1;\alpha}^{(L+1)}\right)=0.$
Moreover, at finite width, the effective network depth $\xi$ controls the
distribution of gradients both in terms of their fluctuations at a single
neuron and their correlations between neurons. For instance, to leading order
in $n,L$:
$\displaystyle\frac{\mathrm{Var}\left[\left(\partial_{x_{1;\alpha}}z_{1;\alpha}^{(L+1)}\right)^{2}\right]}{\mathbb{E}\left[\left(\partial_{x_{1;\alpha}}z_{1;\alpha}^{(L+1)}\right)^{2}\right]}$
$\displaystyle=\frac{8}{3}\xi,\qquad\mathrm{Corr}\left(\left(\frac{\partial
z_{1;\alpha}^{(L+1)}}{\partial x_{1;\alpha}}\right)^{2},\left(\frac{\partial
z_{1;\alpha}^{(L+1)}}{\partial
x_{2;\alpha}}\right)^{2}\right)=\frac{2}{3}\xi.$
In the next section, we will apply such estimates to understand a simple
version of the so-called exploding and vanishing gradient problem.
### 4.4 Exploding and Vanishing Gradient Problem for First Layer Weights
On its face, Theorem 4.4 concerns only derivatives of
$z_{i;\alpha}^{{}_{(L+1)}}$ with respect to the network input $x_{\alpha}$.
However, it is the derivatives of $z_{i;\alpha}^{{}_{(L+1)}}$ with respect to
the model parameters (weights and biases) that are arguably more important. To
explain this more precisely, consider a fully connected neural network
$x_{\alpha}\mapsto z_{\alpha}^{{}_{(L+1)}}$ with input dimension $n_{0}$,
output dimension $n_{L+1}$, and hidden layer widths $n_{1},\ldots,n_{L}$. For
any $m\in\mathbb{N}$ let us set
$[m]:=\left\\{1,\ldots,m\right\\}$
and denote by
$\theta=\left(\theta_{\mu},\,\mu\in[\\#\text{params}]\right)=\left(W_{ij}^{(\ell)},b_{i}^{(\ell)},\,\ell\in[L+1],\,i\in[n_{\ell}],\,j\in[n_{\ell-1}]\right)$
the vector of its trainable parameters, where we’ve set
$\\#\text{params}:=\\#\text{weights and
biases}=\sum_{\ell=1}^{L+1}n_{\ell}\left(1+n_{\ell-1}\right).$
As we briefly discussed in §2.2 the typical use case for a network is to
optimize its parameters $\theta$ by some variant of gradient descent
$\theta_{\mu}(t+1)=\theta_{\mu}(t)-\eta\partial_{\mu}\mathcal{L}(\theta(t)),\qquad\mu\in[\\#\text{params}]$
(4.20)
starting from a random initialization $\theta(0)$ drawn from the distribution
(2.2). The goal of this procedure is to minimize a loss
$\mathcal{L}(\theta)=\mathcal{L}(z_{\mathcal{A}_{\text{train}}}^{(L+1)}(\theta)),\qquad
z_{\mathcal{A}_{\text{train}}}^{(L+1)}:=\left(z_{\alpha}^{(L+1)},\,\alpha\in\mathcal{A}_{\text{train}}\right),$
which often depends only on the network output and measures for each $\theta$
how well the resulting neural net function $x_{\alpha}\mapsto
z_{\alpha}^{(L+1)}(\theta)$ matches given training data
$\left\\{(x_{\alpha},f(x_{\alpha})),\,\alpha\in\mathcal{A}_{\text{train}}\right\\}$
for a function $f:\mathbb{R}^{n_{0}}\rightarrow\mathbb{R}^{n_{L+1}}$. For
example, we might have
$\mathcal{L}(\theta)=\frac{1}{\left|\mathcal{A}_{\text{train}}\right|}\sum_{\alpha\in\mathcal{A}_{\text{train}}}\left|\left|f(x_{\alpha})-z_{\alpha}^{(L+1)}\right|\right|_{2}^{2},$
though many different losses are used in practice. The parameter $\eta$ is
called the learning rate, or step size. An important and well-known [5, 17,
27] numerical stability issue that comes up in the course of using a first
order method such as (4.20) to optimize the parameters of a deep neural
network is called the exploding and vanishing gradient problem (EVGP).
Informally, the EVGP occurs when the update (4.20) is numerically unstable:
$\mathrm{EVGP}\qquad\longleftrightarrow\qquad\text{for many parameters
}\theta_{\mu}\quad\frac{\left|\eta\partial_{\mu}\mathcal{L}(\theta)\right|}{\left|\theta_{\mu}\right|}\approx
0\text{ or }\infty.$
The presence of the EVGP causes optimization to break down since if
$\left|\partial_{\mu}\mathcal{L}(\theta)\right|/\left|\theta_{\mu}\right|\approx
0\text{ or }\infty$, then the relative change in the parameter $\theta_{\mu}$
is either too small to be useful or so large that it amounts to a large random
step in the space of parameters and therefore is unlikely to decrease the
loss.
It has long been known that sufficiently deep fully connected neural networks
are prone to suffer from the EVGP [5, 27, 28]. Indeed, many important
practical innovations such as residual connections [24] were invented in part
to address such numerical issues, though by now they are seen as useful for
many other reasons. To see why deep enough neural nets tend to have
numerically unstable gradients note that since the loss only depends on the
network outputs on a finite training set
$z_{\mathcal{A}_{\text{train}}}^{{}_{(L+1)}}$, we may use the chain rule to
write
$\partial_{\mu}\mathcal{L}(\theta)=\sum_{\alpha\in\mathcal{A}_{\text{train}}}J_{\theta_{\mu}}z_{\alpha}^{(L+1)}J_{z_{\alpha^{(L+1)}}}\mathcal{L}(z_{\mathcal{A}_{\text{train}}}^{(L+1)}),$
where for any function $f$ we denote by $J_{x}f(x)$ its Jacobian with respect
to $x.$ Moreover, suppose that $\theta_{\mu}$ is a weight or bias in layer
$\ell_{0}$. Then, again by the chain rule, the Jacobian of the network output
with respect to $\mu$ equals
$J_{\theta_{\mu}}z_{\alpha}^{(L+1)}=J_{\theta_{\mu}}z_{\alpha}^{(\ell_{0})}J_{z_{\alpha}^{(\ell_{0})}}z_{\alpha}^{(L+1)}.$
Both of these terms may have large fluctuations. Perhaps most importantly, the
second term can be rewritten as a product
$J_{z_{\alpha}^{(\ell_{0})}}z_{\alpha}^{(L+1)}=\prod_{\ell=\ell_{0}}^{L}J_{z_{\alpha}^{(\ell)}}z_{\alpha}^{(\ell+1)},$
of $L-\ell_{0}+1$ layer-to-layer Jacobian matrices (which are random at the
start of training), with each term having the following explicit
representation:
$J_{z_{\alpha}^{(\ell)}}z_{\alpha}^{(\ell+1)}=D^{(\ell+1)}W^{{(\ell+1)}},\qquad
D^{(\ell+1)}:=\mathrm{Diag}\left(\sigma^{\prime}\left(z_{i;\alpha}^{(\ell+1)}\right),\,i=1,\ldots,n_{\ell+1}\right).$
Except in the simple case when $\sigma$ is the identity and we require
$W^{{(\ell+1)}}$ to be orthogonal, the top singular value of each layer-to-
layer Jacobian will tend to deviate from $1$, causing the norms of their
products to either grow or decay exponentially with $L$. This poor
conditioning for large matrix products is a key culprit behind the EVGP.
Often, though not always, the gradients $\partial_{\mu}\mathcal{L}$ have the
largest fluctuations at the start of training. Thus, the EVGP is primarily a
property of random neural networks, and the techniques in this article can be
used to shed light on when it occurs. At the start of training the entries of
the Jacobian $J_{z_{\alpha}^{(\ell_{0})}}z_{\alpha}^{{}_{(L+1)}}$ are
precisely the derivatives of a random neural network with depth $L-\ell_{0}+1$
with respect to its input. One can therefore hope to use Theorems 3.1 and 3.2
to address the following more precise formulation of the EVGP:
$\displaystyle\text{EVGP}\quad\longleftrightarrow\quad$ for a typical random
draw of weights and biases the size of the relative Jacobians of the network
output
$\displaystyle\qquad\qquad\qquad\qquad\qquad|J_{\theta_{\mu}}z_{j;\alpha}^{(L+1)}|/\left|\theta_{\mu}\right|$
$\displaystyle\text{has a large variance over $\mu$ (i.e. over network
parameters)}.$
This characterization of the EVGP says in essence that if we can choose only a
single learning rate $\eta$ for a group of otherwise indistinguishable
parameters, such as the weights in a single layer, then it make sense to set
$\eta=\left(\text{average relative gradient
}|J_{\theta_{\mu}}z_{j;\alpha}^{(L+1)}|/\left|\theta_{\mu}\right|\right)^{-1}.$
The EVGP will then occur when the size of relative gradients have a large
variance across parameters in a typical random draw of weights and biases, so
that for some parameters our setting of $\eta$ is too small whereas for others
it is too large.
We will leave the study of the EVGP in this general form as pertaining to
future work and consider here a special case. Specifically, we compute in
Theorem 4.5 below the average of the empirical variance of the squared
gradients $|J_{\theta_{\mu}}z_{j;\alpha}^{{}_{(L+1)}}|^{2}$ when
$\theta_{\mu}=W_{ij}^{{}_{(1)}}$ varies over all weights in the first layer.
By construction, the typical size of $\left|\theta_{\mu}\right|$ is the same
for all such $\mu$. The EVGP will thus occur for weights in the first layer if
and only if the empirical fluctuations over $\mu$ of the raw Jacobians
$|J_{{W_{ij}^{(1)}}}z_{j;\alpha}^{{}_{(L+1)}}|$ is large. To state the precise
result we fix $x_{\alpha}\neq 0\in\mathbb{R}^{n_{0}}$ and $q\in[n_{L+1}]$. We
then write
$\mathrm{Grad~{}Mean}^{(1)}:=\frac{1}{n_{0}n_{1}}\sum_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ 1\leq j\leq
n_{0}\end{subarray}}\left(\partial_{W_{ij}^{(1)}}z_{q;\alpha}^{(L+1)}\right)^{2}$
for the empirical mean of the squared gradients of $z_{q;\alpha}^{(L+1)}$ with
respect to the weights $W_{ij}^{{}_{(1)}}$ in the first layer and analogously
$\mathrm{Grad~{}Var}^{(1)}:=\frac{1}{n_{0}n_{1}}\sum_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ 1\leq j\leq
n_{0}\end{subarray}}\left(\partial_{W_{ij}^{(1)}}z_{q;\alpha}^{(L+1)}\right)^{4}-\left(\mathrm{Grad~{}Mean}^{(1)}\right)^{2}$
for the empirical variance of the squared gradients.
###### Theorem 4.5 (EVGP for First Layer Weights in Fully Connected
Networks).
Let $x_{\alpha}\mapsto z_{\alpha}^{(L+1)}$ be a random neural network with
input dimension $n_{0}$, output dimension $n_{L+1}$, hidden layer widths
$n_{1},\ldots,n_{L}=n\gg 1,$
and non-linearity $\sigma$. Suppose also that
* •
$\sigma$ satisfies Assumption. 2.1
* •
$\sigma$ belongs to the $K_{*}=0$ universality class in the sense described in
§4.1 as well as §4.2.2.
* •
$C_{b},C_{W}$ are tuned to criticality in the sense described as in (4.11).
Then, for any non-zero $x_{\alpha}\in\mathbb{R}^{n_{0}}$ we have
$\frac{\mathbb{E}\left[\mathrm{Grad~{}Var}^{(1)}\right]}{\mathbb{E}\left[\mathrm{Grad~{}Mean}^{(1)}\right]^{2}}=C_{\sigma,\alpha}\bigg{(}1+\frac{8}{3}\xi+O(L^{-1})+O(n^{-1})+O_{L}(n^{-2})\bigg{)},$
where $\xi=\frac{L}{n}$ is the effective network depth and
$C_{\sigma,\alpha}:=3\frac{\left\langle(\sigma^{\prime})^{4}\right\rangle_{K_{\alpha\alpha}^{(1)}}}{\left\langle(\sigma^{\prime})^{2}\right\rangle_{K_{\alpha\alpha}^{(1)}}^{2}}\frac{n_{0}^{-1}\left|\left|x_{\alpha}\right|\right|_{4}^{4}}{(n_{0}^{-1}\left|\left|x_{\alpha}\right|\right|_{2}^{2})^{2}}-1$
is a strictly positive constant depending only on $x_{\alpha},\sigma$, the
implicit constants in the error terms $O(n^{-1}),O(L^{-1})$ depend only on
$x_{\alpha},\sigma$ whereas the implicit constant in the error term
$O_{L}(n^{-2})$ depends on $x_{\alpha},\sigma,L$.
###### Remark 4.6.
Theorem 4.5 shows that, at least to leading order in $1/n$, the exploding and
vanishing gradient problem occurs for first layer weights in a critically
tuned random fully connected with non-linearity from the $K_{*}=0$
universality class if and only if the effective network depth $L/n$ is large.
We prove Theorem 4.5 in §11. To put this result into the context, let us make
several remarks. First, the analog of Theorem 4.5 for $1-$homogeneous non-
linearities was derived in prior work [17, 21]. As we’ve alluded to before,
the behavior of a random fully connected neural network with such non-
linearities can be studied by more specialized methods that reveal not only
the leading order dependence in $1/n$ but actually the full dependence on
$L/n$ plus errors of size $L/n^{2}$ (see Appendix C). The conclusion, just as
for non-linearities from the $K_{*}=0$ universality class, is that the EVGP
occurs if and only if $L/n$ is large.
Second, Theorem 4.5 is the first time the EVGP has been characterized
mathematically in random fully connected networks for a broad class of non-
linearities beyond those that are $1-$homogeneous. However, as throughout, the
author would like to acknowledge the prior work [53] of Roberts, Yaida, and
the author which also studies the EVGP and shows, at a physics level of rigor,
that the variance over random initialization of a single squared gradient
$(\partial_{\mu}z_{q;\alpha}^{{}_{(L+1)}})^{2}$ also scales like $L/n$ at
large $L$. This is conceptually different from showing that in a typical
initialization the empirical variance of
$(\partial_{\mu}z_{q;\alpha}^{{}_{(L+1)}})^{2}$ over $\mu$ in the first layer
is large, which is what Theorem 4.5 establishes. Nonetheless, the conclusions
are the essentially the same and the basic techniques used to derive the
results are similar.
## 5 Overview of Proofs
In this section, we present the essential idea for how we analyze a random
fully connected neural network $x_{\alpha}\mapsto z_{\alpha}^{(L+1)}$ at
finite width. Our approach is based on the following structural properties:
* •
The sequence of fields $z_{\alpha}^{(\ell)}$ is a Markov Chain with respect to
$\ell$.
* •
Conditional on the sigma algebra $\mathcal{F}^{(\ell)}$ defined by
$z_{\alpha}^{(\ell)}$ is a Gaussian field with independent components
$z_{i;\alpha}^{(\ell+1)}$. See Lemma 7.1.
* •
The variance of each component $z_{i;\alpha}^{(\ell+1)}$ depends on
$z_{\alpha}^{(\ell)}$ only through random variables of the form
$\mathcal{O}_{f}^{(\ell)}:=\frac{1}{n_{\ell}}\sum_{j=1}^{n_{\ell}}f(z_{i;\alpha}^{(\ell)}),$
which we refer to as collective observables. See (3.6).
* •
Collective observables are self-averaging to a similar extent as if the random
variables $f(z_{i;\alpha}^{(\ell)})$ were independent in the sense that for
any $q\geq 0$, we have
$\mathbb{E}\left[\left(\mathcal{O}_{f}^{(\ell)}-\mathbb{E}\left[\mathcal{O}_{f}^{(\ell)}\right]\right)^{q}\right]=O_{q}\left(n^{-\lceil\frac{q}{2}\rceil}\right).$
(5.1)
See Theorem 7.3 and Lemma 7.5.
Let us briefly explain, mostly dispensing with rigor, how these four ideas
come together to obtain a recursive description of the distribution of the
field $z_{\alpha}^{(\ell+1)}$ in terms of that of $z_{\alpha}^{(\ell)}$. To
keep the notation to a minimum, we fix a network input $x_{\alpha}$ and focus
on describing the joint distribution of
$z_{i;\alpha}^{(\ell+1)},\,i=1,\ldots,m$. Extensions to multiple inputs and
derivatives proceed along very similar lines. Denoting by
$\xi=\left(\xi_{1},\ldots,\xi_{m}\right)$ dual variables, consider the
characteristic function
$p^{(\ell+1)}(\xi):=\mathbb{E}\left[\exp\left[-i\sum_{i=1}^{m}\xi_{i}z_{i;\alpha}^{(\ell+1)}\right]\right].$
Conditioning on $z_{\alpha}^{(\ell)}$ and using (3.6) allows us to write
$p^{(\ell+1)}(\xi):=\mathbb{E}\left[\exp\left[-\frac{1}{2}\left|\left|\xi\right|\right|^{2}\Sigma_{\alpha\alpha}^{(\ell)}\right]\right],$
where we remind the reader that
$\Sigma_{\alpha\alpha}^{(\ell)}=\mathrm{Var}\left[z_{i;\alpha}^{(\ell+1)}~{}\big{|}~{}\mathcal{F}^{(\ell)}\right]=C_{b}+\frac{C_{W}}{n_{\ell}}\sum_{j=1}^{n_{\ell}}\sigma(z_{j;\alpha}^{(\ell)})^{2}$
is a collective observable at the previous layer. Writing
$\kappa_{\alpha\alpha}^{(\ell)}:=\mathbb{E}\left[\Sigma_{\alpha\alpha}^{(\ell)}\right],\qquad\Delta_{\alpha\alpha}^{(\ell)}:=\Sigma_{\alpha\alpha}^{(\ell)}-\mathbb{E}\left[\Sigma_{\alpha\alpha}^{(\ell)}\right],$
we find
$p^{(\ell+1)}(\xi):=\mathbb{E}\left[\exp\left[-\frac{1}{2}\left|\left|\xi\right|\right|^{2}\Delta_{\alpha\alpha}^{(\ell)}\right]\right]\exp\left[-\frac{1}{2}\left|\left|\xi\right|\right|^{2}\kappa_{\alpha\alpha}^{(\ell)}\right].$
The second term is precisely the characteristic function of a centered
$m$-dimensional Gaussian with iid components of variance
$\kappa_{\alpha\alpha}^{(\ell)}$. Moreover, at least heuristically, the first
term can be written
$\mathbb{E}\left[\exp\left[-\frac{1}{2}\left|\left|\xi\right|\right|^{2}\Delta_{\alpha\alpha}^{(\ell)}\right]\right]=\sum_{q\geq
0}\mathbb{E}\left[\left(\Delta_{\alpha\alpha}^{(\ell)}\right)^{q}\right]\frac{(-1)^{q}}{2^{q}q!}\left|\left|\xi\right|\right|^{2q}.$
The concentration estimates (5.1) ensure that this series converges. Moreover,
since the Fourier transform turns polynomials into derivatives we have
$-\left|\left|\xi\right|\right|^{2}=\text{ Laplacian in the variables
}z_{i;\alpha}^{(\ell+1)}.$
Hence, we obtain for any reasonable test function $f$ that
$\mathbb{E}\left[f(z_{i;\alpha}^{(\ell+1)},\,i=1,\ldots,m)\right]=\sum_{q=0}^{\infty}\frac{1}{2^{q}q!}\mathbb{E}\left[\left(\Delta_{\alpha\alpha}^{(\ell)}\right)^{q}\right]\left\langle\left(\sum_{i=1}^{m}\partial_{z_{i;\alpha}}^{2}\right)^{q}f(z_{i;\alpha},\,i=1,\ldots,m)\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}},$
where $(z_{i;\alpha},\,i=1,\ldots,m)$ is a vector of iid centered Gaussians
with variance $\kappa_{\alpha\alpha}^{(\ell)}$. The concentration estimates
(5.1) ensure that this expression is a power series in $1/n$. In particular,
$\displaystyle\mathbb{E}\left[f(z_{i;\alpha}^{(\ell+1)},\,i=1,\ldots,m)\right]$
$\displaystyle=\left\langle
f(z_{i;\alpha},\,i=1,\ldots,m)\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}$
(5.2)
$\displaystyle+\frac{\mathbb{E}\left[(\Delta_{\alpha\alpha}^{(\ell)})^{2}\right]}{8}\left\langle\left(\sum_{i=1}^{m}\partial_{z_{i;\alpha}}^{2}\right)^{2}f(z_{i;\alpha},\,i=1,\ldots,m)\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}+O(n^{-2}).$
This is the essence of Theorem 3.2. To derive usable recursions for cumulants
of $z_{i;\alpha}^{(\ell+1)}$, note for instance that, in the notation of
Corollary 3.4,
$\kappa_{4;\alpha}^{(\ell)}:=\frac{1}{3}\kappa\left(z_{i;\alpha}^{(\ell+1)},z_{i;\alpha}^{(\ell+1)},z_{i;\alpha}^{(\ell+1)},z_{i;\alpha}^{(\ell+1)}\right)=\mathbb{E}\left[(\Delta_{\alpha\alpha}^{(\ell)})^{2}\right].$
Writing
$X_{j}:=\sigma(z_{j;\alpha}^{(\ell+1)})^{2}-\mathbb{E}\left[\sigma(z_{j;\alpha}^{(\ell+1)})^{2}\right]$
we thus have
$\kappa_{\alpha\alpha}^{(\ell+1)}=\mathbb{E}\left[\left(\Delta_{\alpha\alpha}^{(\ell)}\right)^{2}\right]=\frac{C_{W}^{2}}{n_{\ell}}\mathbb{E}\left[X_{1}^{2}\right]+C_{W}^{2}\left(1-n_{\ell}^{-1}\right)\mathbb{E}\left[X_{1}X_{2}\right].$
Applying the expansion (5.2) to both these terms and a bit of algebra already
yields
$\displaystyle\kappa_{\alpha\alpha}^{(\ell+1)}$
$\displaystyle=\mathbb{E}\left[\left(\Delta_{\alpha\alpha}^{(\ell)}\right)^{2}\right]$
$\displaystyle=\frac{C_{W}^{2}}{n_{\ell}}\left(\left\langle\sigma^{4}\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}-\left\langle\sigma^{2}\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}^{2}\right)$
$\displaystyle+C_{W}^{2}\left(1-n_{\ell}^{-1}\right)\left(\left(\left\langle\sigma^{2}\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}-\mathbb{E}\left[\sigma(z_{1;\alpha}^{(\ell)})^{2}\right]\right)^{2}+\frac{1}{4}\left\langle\partial^{2}\sigma^{2}\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}^{2}\kappa_{4;\alpha}^{(\ell)}\right)+O(n^{-2}).$
A short argument supplied in §9 shows that
$\left\langle\sigma^{2}\right\rangle_{\kappa_{\alpha\alpha}^{(\ell)}}=\mathbb{E}\left[\sigma(z_{1;\alpha}^{(\ell)})^{2}\right]+O(n^{-1})$
and that we may replace $\kappa_{\alpha\alpha}^{(\ell)}$ by its infinite width
limit $K_{\alpha\alpha}^{(\ell)}$ in all remaining expectations at the cost of
an $O(n^{-1})$ error. This already yields the recursion (3.15) of Corollary
3.4.
## 6 Background
### 6.1 Properties of Cumulants
Recall that, given random variables $X_{1},\ldots,X_{k}$ on the same
probability space, we denote their mixed cumulant by
$\kappa\left(X_{1},\ldots,X_{k}\right):=i^{k}\frac{\partial^{k}}{\partial
t_{1}\cdots\partial
t_{k}}\bigg{|}_{t=0}\log\mathbb{E}\left[\exp\left[-i(t_{1}X_{1}+\cdots+t_{k}X_{k})\right]\right].$
In the following result, we recall the key properties of these mixed cumulants
that we will need.
###### Proposition 6.1 (See Theorem 2.3.1 in [7]).
Mixed cumulants satisfy the following properties.
1. 1.
Suppose $X=\left(X_{1},\ldots,X_{k}\right)$ is a random vector with finite
moments of all orders. Then
$\displaystyle\kappa\left(X_{1},\ldots,X_{k}\right)=\sum_{\pi=\left(\pi_{1},\ldots\pi_{b}\right)}\kappa\left(\kappa\left(X_{\pi_{1}}~{}|~{}\mathcal{F}\right),\ldots,\kappa\left(X_{\pi_{b}}~{}|~{}\mathcal{F}\right)\right),$
(6.1)
where the sum is over all partitions $\pi$ of $[k]$ and for each
$a=1,\ldots,b$
$X_{\pi_{a}}:=\left(X_{i},\,i\in\pi_{a}\right).$
This is known as the law of total cumulance. See [6].
2. 2.
Suppose $X=\left(X_{1},\ldots,X_{k}\right)$ is a random vector with finite
moments of all orders. When $X$ can be partitioned into two independent
subsets, the mixed cumulant $\kappa(X)$ vanishes. More precisely, suppose
$I\subseteq[k]$ and that $I,I^{c}\neq\emptyset.$ Then
$X_{I}:=\left(X_{i},\,i\in I\right)\perp X_{I^{c}}=\left(X_{i},\,i\not\in
I\right)\qquad\Longrightarrow\qquad\kappa\left(X_{1},\ldots,X_{k}\right)=0.$
(6.2)
3. 3.
Mixed cumulants are multi-linear. More precisely if
$\left\\{X_{i,j},\,1\leq j\leq k,\,i\leq T_{j}\right\\}$
are random variables with finite moments defined on the same probability
space, then
$\kappa\left(\sum_{i_{1}=1}^{T_{1}}a_{i_{1},1}X_{i_{1},1},\ldots,\sum_{i_{k}=1}^{T_{k}}a_{i_{k},k}X_{i_{k},k}\right)=\sum_{i_{1}=1}^{T_{1}}\cdots\sum_{i_{k}=1}^{T_{k}}a_{i_{1},1}\cdots
a_{i_{k},k}\kappa\left(X_{i_{1},1},\ldots,X_{i_{k},k}\right)$ (6.3)
for any $a_{i,j}\in\mathbb{R}$.
4. 4.
Suppose $X=\left(X_{1},\ldots,X_{j}\right)\sim\mathcal{N}(0,\Sigma)$ is a
centered Gaussian with covariance $\Sigma$. Then for any
$i_{1},\ldots,i_{k}\in[j]$
$\kappa\left(X_{i_{1}},\ldots,X_{i_{k}}\right)=\begin{cases}\Sigma_{i_{1}i_{2}},&\quad
k=2\\\ 0,&\quad\text{otherwise}\end{cases}.$ (6.4)
5. 5.
Moments are polynomials in cumulants. Specifically, suppose
$X=\left(X_{1},\ldots,X_{k}\right)$ is a random vector with finite moments of
all orders. Then,
$\mathbb{E}\left[X_{1}\cdots
X_{k}\right]=\sum_{\pi=\left(\pi_{1},\ldots,\pi_{b}\right)}\prod_{a=1}^{b}\kappa\left(X_{\pi_{a}}\right),$
(6.5)
where the sum is over all partitions of $[k]$ and for each $a\in[b]$ we’ve
written
$X_{\pi_{a}}:=\left(X_{i},\,i\in\pi_{a}\right).$
6. 6.
Cumulants are polynomials in moments. Specifically,
$\kappa\left(X_{1},\ldots,X_{k}\right)=\sum_{\pi=\left(\pi_{1},\ldots,\pi_{b}\right)}(-1)^{b-1}(b-1)!\prod_{a=1}^{b}\mathbb{E}\left[\prod_{i\in\pi_{a}}X_{i}\right],$
(6.6)
where the sum is over all partitions of $[k]$ and for each $a\in[b]$ we’ve
written
$X_{\pi_{a}}:=\left(X_{i},\,i\in\pi_{a}\right).$
### 6.2 Lemmata
In this section, we collect two simple auxiliary results that we will need.
The first is a Lemma for Solving Certain Recursions.
###### Lemma 6.2.
Fix $C_{1},C_{2},\psi>0$ satisfying
$C_{2}\geq 1,\quad\psi\neq C_{2}+1$
as well as $*\in\left\\{\leq,\,\geq\right\\}$. Suppose also that for each
$\ell\geq 0$ we have
$a_{\ell+1}~{}*~{}\xi_{\ell}+(1-\zeta_{\ell})a_{\ell},\qquad\zeta_{\ell}\in[0,1]$
(6.7)
with $a_{0}\in\mathbb{R}$ given and that there exist
$C_{1}^{\prime},C_{2}^{\prime}>0$ so that
$\left|\xi_{\ell}-C_{1}\ell^{-\psi}\right|\leq
C_{1}^{\prime}\ell^{-1-\psi},\qquad\left|\zeta_{\ell}-C_{2}\ell^{-1}\right|\leq
C_{2}^{\prime}\ell^{-2}.$
Then
$a_{\ell+1}*\frac{\ell^{1-\psi}}{1-\psi+C_{2}}\left(1+O(\ell^{-1})\right)+e^{-C_{2}\gamma}\ell^{-C_{2}}a_{0}\left(1+O(\ell^{-1})\right)$
(6.8)
where $\gamma$ is the Euler-Mascheroni constant and the implied constants
depend only $C_{1},C_{2},C_{1}^{\prime},C_{2}^{\prime}.$
###### Proof.
By unfolding the recursion (6.7) we find
$\displaystyle
a_{\ell+1}~{}*~{}\sum_{\ell^{\prime}=1}^{\ell}\xi_{\ell^{\prime}}\prod_{\ell^{\prime\prime}=\ell^{\prime}+1}^{\ell}\left(1-\zeta_{\ell^{\prime\prime}}\right)+a_{0}\prod_{\ell^{\prime\prime}=0}^{\ell}(1-\zeta_{\ell^{\prime\prime}}).$
We have
$\displaystyle\prod_{\ell^{\prime\prime}=1}^{\ell}(1-\zeta_{\ell^{\prime\prime}})$
$\displaystyle=\exp\left[\sum_{\ell^{\prime\prime}=1}^{\ell}\log\left(1-C_{2}(\ell^{\prime\prime})^{-1}+O(\ell^{-2})\right)\right]$
$\displaystyle=\exp\left[O(\ell^{-1})+\sum_{\ell^{\prime\prime}=1}^{\ell}-C_{2}(\ell^{\prime\prime})^{-1}\right]$
$\displaystyle=\exp\left[O(\ell^{-1})-C_{2}\log(\ell)-C_{2}\gamma\right]$
$\displaystyle=e^{-C_{2}\gamma}\ell^{-C_{2}}\left(1+O(\ell^{-1})\right).$
This gives the second term in (6.8). For the first term, we write
$\displaystyle\sum_{\ell^{\prime}=1}^{\ell}\xi_{\ell^{\prime}}\prod_{\ell^{\prime\prime}=\ell^{\prime}+1}^{\ell}\left(1-\zeta_{\ell}\right)$
$\displaystyle=\sum_{\ell^{\prime}=1}^{\ell}\xi_{\ell^{\prime}}\exp\left[\sum_{\ell^{\prime\prime}=\ell^{\prime}+1}^{\ell}\log\left(1-\zeta_{\ell}\right)\right]$
$\displaystyle=\sum_{\ell^{\prime}=1}^{\ell}\xi_{\ell^{\prime}}\exp\left[\sum_{\ell^{\prime\prime}=\ell^{\prime}+1}^{\ell}-C_{2}(\ell^{\prime\prime})^{-1}+O((\ell^{\prime\prime})^{-2})\right]$
$\displaystyle=\sum_{\ell^{\prime}=1}^{\ell}C_{1}(\ell^{\prime})^{-\psi}(1+O(\ell^{\prime})^{-1})\exp\left[-C_{2}\log\left(\frac{\ell}{\ell^{\prime}}\right)+O((\ell^{\prime})^{-1})\right]$
$\displaystyle=\ell^{-C_{2}}\sum_{\ell^{\prime}=1}^{\ell}C_{1}(\ell^{\prime})^{-\psi+C_{2}}(1+O(\ell^{\prime})^{-1})$
$\displaystyle=\frac{C_{1}}{1+C_{2}-\psi}\ell^{1-\psi}\left(1+O(\ell^{-1})\right).$
This completes the proof of (6.8). ∎
The second result we will need is the following simple Lemma about Gaussian
Integrals.
###### Lemma 6.3.
Fix $r\geq 1$, a $r\times r$ matrix $\Sigma$ and measurable function
$g:\mathbb{R}^{r}\rightarrow\mathbb{R}$ that is polynomially bounded:
$\exists k\geq 1\text{ s.t.
}\sup_{x\in\mathbb{R}^{k}}\left|\left(1+\left|\left|x\right|\right|\right)^{-k}g(x)\right|<\infty.$
If $X$ is a standard Gaussian random vector in $\mathbb{R}^{r}$, then the
function
$\Sigma\mapsto\mathbb{E}\left[g\left(\Sigma^{1/2}X\right)\right]$ (6.9)
is smooth on the open set of strictly positive definite $k\times k$ matrices.
Further, if $g$ is a smooth function and each of its derivatives is
polynomially bounded, then the map (6.9) is extends to a smooth function on
the closed set of positive semi-definite matrices and, in particular,
$\frac{\partial}{\partial\Sigma_{ij}}\mathbb{E}\left[g\left(\Sigma^{1/2}X\right)\right]=\mathbb{E}\left[(\partial_{i}\partial_{j}g)(\Sigma^{1/2}X)\right].$
(6.10)
###### Proof.
On the open set of strictly positive definite matrices, the Gaussian density
$\Sigma\mapsto\exp\left[-\frac{1}{2}x^{T}\Sigma^{-1}x-\frac{1}{2}\log\det(2\pi\Sigma)\right]$
is a smooth function of $\Sigma$ with derivatives that are polynomials in $x$
and the entries of $\Sigma,\Sigma^{-1}$. The assumption that $f$ is
polynomially bounded shows that we may differentiate under the integral sign
and see that that
$\mathbb{E}\left[g(\Sigma^{1/2}X)\right]=\int_{\mathbb{R}^{r}}g(x)\exp\left[-\frac{1}{2}x^{T}\Sigma^{-1}x-\frac{1}{2}\log\det(2\pi\Sigma)\right]dx$
is indeed a smooth function of $\Sigma$. Suppose instead that $g$ is a smooth
function and that it’s derivatives are all polynomially bounded. Suppose first
that $g$ is in fact a Schwartz function. Then, writing $\widehat{g}$ for its
Fourier transform we have
$\mathbb{E}\left[g(\Sigma^{1/2}X)\right]=\int_{\mathbb{R}^{r}}\widehat{g}(\xi)\exp\left[-\frac{1}{2}\xi^{T}\Sigma\xi\right]d\xi.$
Since $\widehat{g}$ is also Schwartz, we may differentiate under the integral
sign to obtain
$\frac{\partial}{\partial\Sigma_{ij}}\mathbb{E}\left[g(\Sigma^{1/2}X)\right]=-\int_{\mathbb{R}^{r}}\xi_{i}\xi_{j}\widehat{g}(\xi)\exp\left[-\frac{1}{2}\xi^{T}\Sigma\xi\right]d\xi=\mathbb{E}\left[\partial_{x_{i}}\partial_{x_{j}}\bigg{|}_{x=\Sigma^{1/2}X}g(x)\right].$
(6.11)
Finally, if $g$ is not Schwartz but is smooth with all derivatives being
polynomially bounded, we consider the convolution
$g_{\epsilon}(x):=(g*\psi_{\epsilon})(x),\qquad\psi_{\epsilon}(y)=\exp\left[-\frac{\left|\left|y\right|\right|^{2}}{2\epsilon}-\frac{1}{2}\log(2\pi\epsilon)\right].$
Then, $g_{\epsilon}$ is Schwartz for all $\epsilon>0$. Moreover, note that
$g_{\epsilon}(\Sigma^{1/2}x)$ is also polynomially bounded for any PSD matrix
$\Sigma.$ Specifically, for any fixed PSD matrix $\Sigma_{0}$ we have for any
$k\geq 1$
$\displaystyle\sup_{\epsilon\in[0,1]}\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\\\ \Sigma\text{
PSD}\end{subarray}}\sup_{x\in\mathbb{R}^{r}}\left|(1+\left|\left|x\right|\right|)^{-k}g_{\epsilon}(\Sigma^{1/2}x)\right|$
$\displaystyle\qquad=\sup_{\epsilon\in[0,1]}\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\\\ \Sigma\text{
PSD}\end{subarray}}\sup_{x\in\mathbb{R}^{r}}\left|(1+\left|\left|x\right|\right|)^{-k}\int_{\mathbb{R}^{r}}g(\Sigma^{1/2}(x-y))\psi_{\epsilon}(y)dy\right|$
$\displaystyle\qquad\leq\sup_{\epsilon\in[0,1]}\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\\\ \Sigma\text{
PSD}\end{subarray}}\sup_{x\in\mathbb{R}^{r}}\left\\{(1+\left|\left|x\right|\right|)^{-k}\int_{\mathbb{R}^{r}}\left(1+\left|\left|\Sigma^{1/2}(x-y)\right|\right|^{k}\right)\psi_{\epsilon}(y)dy\right\\}$
$\displaystyle\qquad<\infty,$ (6.12)
Note that there exists $K>0$ depending only $k,r,\Sigma_{0}$ so that
$\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\end{subarray}}\left|\left|\Sigma^{1/2}(x-y)\right|\right|^{k}\leq
K\left(1+\left|\left|\Sigma_{0}^{1/2}\right|\right|\right)^{k}(\left|\left|x\right|\right|^{k}+\left|\left|y\right|\right|^{k}).$
Hence, since
$\sup_{\epsilon\in[0,1]}\int_{\mathbb{R}^{r}}\left|\left|y\right|\right|^{k}\psi_{\epsilon}(y)dy<\infty$
we find that
$\displaystyle\sup_{\epsilon\in[0,1]}\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\\\ \Sigma\text{
PSD}\end{subarray}}\sup_{x\in\mathbb{R}^{r}}\left|(1+\left|\left|x\right|\right|)^{-k}g_{\epsilon}(\Sigma^{1/2}x)\right|<\infty.$
(6.13)
The estimate above allows us to use dominate convergence to see that for any
PSD $\Sigma$
$\mathbb{E}\left[g(\Sigma^{1/2}X)\right]=\lim_{\epsilon\rightarrow
0}\mathbb{E}\left[g_{\epsilon}(\Sigma^{1/2}X)\right].$ (6.14)
To complete the proof we note that $g_{\epsilon}$ and
$\partial_{i}\partial_{j}g_{\epsilon}$ are both Schwartz for any positive
$\epsilon$. Moreover,
$\partial_{i}\partial_{j}\partial_{k}\partial_{m}g_{\epsilon}$ satisfies
(6.13). Hence, we conclude by applying (6.11) that for any PSD matrix
$\Sigma_{0}$ there exists $C>0$ so that
$\displaystyle\sup_{\begin{subarray}{c}\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1\\\ \Sigma\text{
PSD}\end{subarray}}\sup_{\epsilon\in[0,1]}\frac{\left|\mathbb{E}\left[g_{\epsilon}(\Sigma^{1/2}X)\right]-\mathbb{E}\left[g_{\epsilon}(\Sigma_{0}^{1/2}X)\right]-\sum_{i,j=1}^{r}\mathbb{E}\left[(\partial_{i}\partial_{j}g_{\epsilon})(\Sigma_{0}^{1/2}X)\right]\left(\Sigma-\Sigma_{0}\right)_{ij}\right|}{\left|\left|\Sigma-\Sigma_{0}\right|\right|^{2}}$
$\displaystyle\qquad\leq\sup_{\epsilon\in[0,1]}\sup_{\left|\left|\Sigma-\Sigma_{0}\right|\right|\leq
1}\sum_{i,j,k,m=1,\ldots,r}\left|\mathbb{E}\left[(\partial_{i}\partial_{j}\partial_{k}\partial_{m})g_{\epsilon}(\Sigma^{1/2}X)\right]\right|$
$\displaystyle\qquad\leq C.$
Thus, if
$\Sigma-\Sigma_{0}/\left|\left|\Sigma-\Sigma_{0}\right|\right|\rightarrow\Sigma_{1}$,
we find by applying (6.14) to $\partial_{i}\partial_{j}g$ that
$\displaystyle\lim_{\Sigma\rightarrow\Sigma_{0}}\frac{\mathbb{E}\left[g(\Sigma^{1/2}X)\right]-\mathbb{E}\left[g(\Sigma_{0}^{1/2}X)\right]}{\left|\left|\Sigma-\Sigma_{0}\right|\right|}$
$\displaystyle=\lim_{\Sigma\rightarrow\Sigma_{0}}\lim_{\epsilon\rightarrow
0}\frac{\mathbb{E}\left[g_{\epsilon}(\Sigma^{1/2}X)\right]-\mathbb{E}\left[g_{\epsilon}(\Sigma_{0}^{1/2}X)\right]}{\left|\left|\Sigma-\Sigma_{0}\right|\right|}$
$\displaystyle=\lim_{\Sigma\rightarrow\Sigma_{0}}\lim_{\epsilon\rightarrow
0}\left\\{\sum_{i,j=1}^{r}\mathbb{E}\left[(\partial_{i}\partial_{j}g_{\epsilon}(\Sigma_{0}^{1/2}X))\right]\frac{(\Sigma-\Sigma_{0})_{ij}}{\left|\left|\Sigma-\Sigma_{0}\right|\right|}\right\\}$
$\displaystyle=\sum_{i,j=1}^{r}\mathbb{E}\left[\partial_{i}\partial_{j}g(\Sigma_{0}^{1/2}X)\right](\Sigma_{1})_{ij}.$
This shows that (6.10) holds for any $g$ that is smooth, completing the proof
of Lemma 6.3. ∎
## 7 Proof of Theorem 3.1
Let us first recall the notation. We fix $r\geq 1$ and assume that
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ satisfies assumption 2.1 with this
value of $r$. We also fix a finite collection
$x_{\mathcal{A}}=\left\\{x_{\alpha},\alpha\in\mathcal{A}\right\\}\subseteq\mathbb{R}^{n_{0}}$
of distinct network inputs and $p$ directional derivatives
$d_{1},\ldots,d_{p}$ as in (3.2). We denote by
$N(p,r)=\\#\left\\{J=\left(j_{1},\ldots,j_{p}\right)\in\mathbb{N}^{p}~{}|~{}j_{1}+\cdots+j_{p}\leq
r\right\\},$
which computes the number of possible derivatives of order at most $r$ in the
$p$ directional derivatives $d_{j}$. We also denote by $\mathcal{F}^{(\ell)}$
the sigma algebra generated by the weights and biases in layers up to and
including $\ell$. The starting point for proving Theorem 3.1 is the following
simple but fundamental observation.
###### Lemma 7.1.
For each $\ell\geq 0$, conditional on $\mathcal{F}^{(\ell)}$,
$\left\\{\left(D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)},\,\alpha\in\mathcal{A},\,\left|J\right|\leq
r\right)\right\\}_{i=1}^{n_{\ell+1}}$
is a collection of $n_{\ell+1}$ iid centered Gaussians of dimension $N(p,r)$.
###### Proof.
The defining recursion (2.1) of a fully connected network yields for each
$\alpha,J$
$\displaystyle D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)}$
$\displaystyle=D_{\alpha}^{J}\left\\{b_{i}^{(\ell+1)}+\sum_{j=1}^{n_{\ell}}W_{ij}^{(\ell+1)}\sigma\left(z_{j;\alpha}^{(\ell)}\right)\right\\}=\delta_{\left|J\right|=0}b_{i}^{(\ell+1)}+\sum_{j=1}^{n_{\ell}}W_{ij}^{(\ell+1)}D_{\alpha}^{J}\sigma\left(z_{j;\alpha}^{(\ell)}\right).$
(7.1)
Note that $D_{\alpha}^{J}\sigma(z_{j;\alpha}^{{}_{(\ell)}})$ are measurable
with respect to $\mathcal{F}^{(\ell)}$. The conclusion now follows since the
weights $W_{ij}^{{}_{(\ell+1)}},\,j=1\ldots,n_{\ell}$ and bias
$b_{i}^{{}_{(\ell+1)}}$ are centered Gaussians and are independent for
different $i$. ∎
Thus, the structure of $z_{\alpha}^{{}_{(\ell+1)}}$ and its derivatives is
always that of a Gaussian field after conditioning on
$\mathcal{F}^{{}_{(\ell)}}$. To ease the notation in what comes given
$f:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R}$, let us abbreviate
$f\left(z_{j;\mathcal{A}}^{(\ell)}\right):=f\left(D_{\alpha}^{J}z_{j;\alpha}^{(\ell)},\,\alpha\in\mathcal{A},\,\left|J\right|\leq
r\right),\qquad j\in[n_{\ell}].$
Next, we remind the reader that given
$f:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R}$, which is measurable and polynomially
bounded, the corresponding collective observable $\mathcal{O}_{f}^{(\ell)}$ at
layer $\ell$ is
$\mathcal{O}_{f}^{(\ell)}=\frac{1}{n_{\ell}}\sum_{j=1}^{n_{\ell}}f\left(z_{j;\mathcal{A}}^{(\ell)}\right)$
and that the statement (A.3) in Proposition A.1 ensures
$\sup_{n\geq
1}\mathbb{E}\left[\left|\mathcal{O}_{f}^{(\ell)}\right|\right]<\infty.$ (7.2)
Recall also our notation for the conditional covariance
$\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}:=\mathrm{Cov}\left(z_{i;\alpha_{1}}^{(\ell+1)},z_{i;\alpha_{2}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right)=C_{b}+\frac{C_{W}}{n_{\ell}}\sum_{j=1}^{n_{\ell}}\sigma\left(z_{j;\alpha_{1}}^{(\ell)}\right)\sigma\left(z_{j;\alpha_{2}}^{(\ell)}\right)$
and note that both it and its derivatives
$\displaystyle
D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}$
$\displaystyle=\mathrm{Cov}\left(D_{\alpha_{1}}^{J_{1}}z_{i;\alpha_{1}}^{(\ell+1)},D_{\alpha_{2}}^{J_{2}}z_{i;\alpha_{2}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right)$
$\displaystyle=D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\left(C_{b}+\frac{C_{W}}{n_{\ell}}\sum_{j=1}^{n_{\ell}}\sigma\left(z_{j;\alpha_{1}}^{(\ell)}\right)\sigma\left(z_{j;\alpha_{2}}^{(\ell)}\right)\right)$
are collective observables at layer $\ell$. Our first application of Lemma 7.1
is the following reduction of the study of cumulants of
$D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)}$ to the cumulants of certain collective
observables.
###### Proposition 7.2.
Fix $k,\ell\geq 1$ and $p$-dimensional multi-indices $J_{1},\ldots,J_{k}$ with
$\left|J_{i}\right|\leq r.$ If $k$ is odd, then
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},\ldots,D_{\alpha_{k}}^{J_{k}}z_{i_{k};\alpha_{k}}^{(\ell+1)}\right)=0$
In contrast, if $k$ is even
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},\ldots,D_{\alpha_{k}}^{J_{k}}z_{i_{k};\alpha_{k}}^{(\ell+1)}\right)~{}=~{}\text{
finite sums of
}\kappa\left(\mathcal{O}_{f_{1}}^{(\ell)},\ldots,\mathcal{O}_{f_{k/2}}^{(\ell)}\right),$
where $\mathcal{O}_{f_{j}}^{(\ell)}$ are collective observables of the form
$D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)},\qquad\left|J_{1}\right|,\left|J_{2}\right|\leq
r.$ (7.3)
###### Proof.
Using (6.1) and recalling that $\mathcal{F}^{(\ell)}$ is the sigma algebra
generated by weights and biases in layers up to and including $\ell$, we have
that
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},\ldots,D_{\alpha_{k}}^{J_{k}}z_{i_{k};\alpha_{k}}^{(\ell+1)}\right)$
equals
$\displaystyle\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\end{subarray}}\kappa\left(\kappa\left(\left(D^{J}z^{(\ell+1)}\right)_{\pi_{1}}\big{|}\mathcal{F}^{(\ell)}\right),\ldots,\kappa\left(\left(D^{J}z^{(\ell+1)}\right)_{\pi_{B}}\big{|}\mathcal{F}^{(\ell)}\right)\right),$
(7.4)
where the sum is over partitions $\pi$ of $[k]$ and for $b=1,\ldots,B$ we’ve
abbreviated
$\left(D^{J}z^{(\ell+1)}\right)_{\pi_{b}}:=\left(D_{\alpha_{t}}^{J_{t}}z_{i_{t};\alpha_{t}}^{(\ell+1)},\quad
t\in\pi_{b}\right).$
By Lemma 7.1,
$\\{(D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)},\,\alpha\in\mathcal{A},\,\left|J\right|\leq
d),\,i=1,\ldots,n_{\ell+1}\\}$ are iid centered Gaussians conditional on
$\mathcal{F}^{(\ell)}$. Hence, by the properties (6.2) and (6.3) and (6.4)
from Proposition 6.1, in the sum over partitions above, a term is non-zero
only if
$\forall b\in[B],\qquad\left|\pi_{b}\right|=2\quad\text{and}\quad
i_{\pi_{b}(1)}=i_{\pi_{b}(2)}$
This proves that
$\kappa\left(D^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},\ldots,D^{J_{k}}z_{i_{k};\alpha_{k}}^{(\ell+1)}\right)$
vanishes if $k$ is odd. To treat the case when $k$ is even observe that by
(7.1)
$\displaystyle\kappa\left(D^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},D^{J_{2}}z_{i_{2};\alpha_{2}}^{(\ell+1)}\big{|}\mathcal{F}^{(\ell)}\right)$
$\displaystyle=\delta_{i_{1}i_{2}}D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}.$
Substituting this into (7.4) completes the proof. ∎
When $k=2$, Proposition 7.2 and our assumption (2.1) shows that for each
$\ell\geq 1$, any $i_{1},i_{2}\in[n_{\ell+1}],\,\alpha\in\mathcal{A},$ and
multi-indices $J_{1},J_{2}$ of order at most $d$, there exists a polynomially
bounded function $f:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},d)}\rightarrow\mathbb{R}$ for which
$\kappa\left(D_{\alpha_{1}}^{J_{1}}z_{i_{1};\alpha_{1}}^{(\ell+1)},D_{\alpha_{2}}^{J_{2}}z_{i_{2};\alpha_{2}}^{(\ell+1)}\right)=\mathbb{E}\left[\mathcal{O}_{f}^{(\ell)}\right]$
In light of (7.2) this proves Theorem 3.1 when $k=2$. Further, since the
cumulant of $2$ or more random variables is shift-invariant, we may assume for
$k\geq 3$ that the collective observables
$D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}$
in Proposition 7.2 are replaced by their zero mean versions:
$\Delta_{\alpha_{1}\alpha_{2}}^{J_{1},J_{2},(\ell)}:=D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}-\mathbb{E}\left[D_{\alpha_{1}}^{J_{1}}D_{\alpha_{2}}^{J_{2}}\Sigma_{\alpha_{1}\alpha_{2}}^{(\ell)}\right].$
(7.5)
Hence, Theorem 3.1 is a special case of the following result.
###### Theorem 7.3.
Fix $k,m\geq 1$. Consider any $m$-tuple $F=\left(f_{1},\ldots,f_{m}\right)$
consisting of measurable, functions
$f_{i}:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R},\quad i=1,\ldots,m\qquad$
that are polynomially bounded and satisfy
$\mathbb{E}\left[\mathcal{O}_{f_{i}}^{(\ell)}\right]=\mathbb{E}\left[f_{i}\left(z_{1;\mathcal{A}}^{(\ell)}\right)\right]=0,\qquad
i=1,\ldots,m.$
Define the $m-$tuple of collective observables
$\overrightarrow{\mathcal{O}}_{F}^{(\ell)}:=\left(\mathcal{O}_{f_{i}}^{(\ell)},\,i=1,\ldots,m\right).$
Consider further any measurable polynomially bounded functions
$g_{j}:\mathbb{R}^{m}\rightarrow\mathbb{R},\quad j=1,\ldots,k.$
which are smooth in a neighborhood of $0$. If $f_{i}$ and $\sigma$ are in fact
smooth, then, for every $\ell\geq 1$
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(g_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell)}\right),\ldots,g_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell)}\right)\right)\right|<\infty$
(7.6)
Moreover, (7.6) holds without the assumption that $f_{i},\sigma$ are smooth
provided that for each $\ell$ the vector of iterated directional derivatives
$(D_{\alpha}^{J}z_{i;\alpha}^{(\ell)},\,\left|J\right|\leq
r,\alpha\in\mathcal{A})$ of order at most $r$ is non-degenerate in the sense
of (3.3).
###### Proof.
Our starting point is a reduction of Theorem 7.3 to the case when $g_{j}$ are
polynomials. This is related to a technique called the delta method in some
parts of the mathematical statistics literature [59].
###### Proposition 7.4 (Polynomials are Enough for Theorem 7.3).
Fix $m\geq 1$ and suppose that for each $n\geq 1$ we have an $m-$tuple
$X_{n}=\left(X_{n,1},\ldots,X_{n,m}\right)$ of mean $0$ random variables that
possess bounded moments of all orders:
$\sup_{n\geq 1}\left|\mathbb{E}\left[X_{n,1}^{q_{1}}\cdots
X_{n,m}^{q_{m}}\right]\right|<\infty,\qquad\forall\,q_{1},\ldots,q_{m}\geq 0.$
(7.7)
Suppose for any given polynomials $p_{1},\ldots,p_{k}$ in $m$ variables we
have
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(p_{1}(X_{n}),\ldots,p_{k}(X_{n})\right)\right|<\infty.$
(7.8)
Then, for any measurable, polynomially bounded functions
$g_{j}:\mathbb{R}^{m}\rightarrow\mathbb{R},\,j=1,\ldots,k,$ which are smooth
in some fixed neighborhood of $0$
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(g_{1}\left(X_{n}\right),\ldots,g_{k}\left(X_{n}\right)\right)\right|<\infty.$
(7.9)
###### Proof.
We begin with the following simple Lemma, which allows us to translate between
the cumulants bounds (7.8) and high probability bounds.
###### Lemma 7.5.
For any $q\geq 1$
$\sup_{n\geq 1}\sup_{1\leq i\leq
m}\left|n^{\lceil\frac{q}{2}\rceil}\mathbb{E}\left[X_{i,n}^{q}\right]\right|<\infty.$
###### Proof.
We have by property (6.5) from Proposition 6.1 that
$\displaystyle\mathbb{E}\left[X_{i;n}^{q}\right]=\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\\\
\pi\in
P(m)\end{subarray}}\prod_{b=1}^{B}\kappa\big{(}{\underbrace{X_{i;n},\ldots,X_{i;n}}_{\left|\pi_{b}\right|\text{
times}}}\big{)}.$
Since by assumption $X_{i;n}$ has mean $0$, we have
$\kappa(X_{i;n})=\mathbb{E}\left[X_{i;n}\right]=0.$
Thus, the only partitions $\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\in S(m)$
that give rise to non-zero terms in the expression above must have
$B\leq\lfloor\frac{q}{2}\rfloor$. Moreover, for any such partition, we have
$\left\lceil\frac{q}{2}\right\rceil=q-\left\lfloor\frac{q}{2}\right\rfloor=-\left\lfloor\frac{q}{2}\right\rfloor+\sum_{b=1}^{B}\left|\pi_{b}\right|\leq\sum_{b=1}^{B}\left(\left|\pi_{b}\right|-1\right).$
Hence, we find
$\displaystyle\sup_{n\geq
1}\left|n^{\lceil\frac{q}{2}\rceil}\mathbb{E}\left[X_{i;n}^{q}\right]\right|$
$\displaystyle\leq\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\\\
\pi\in P(m),\,\left|\pi_{b}\right|\geq
2\end{subarray}}\prod_{b=1}^{B}\sup_{n\geq
1}\left|n^{\left|\pi_{b}\right|-1}\kappa\big{(}{\underbrace{X_{i;n},\ldots,X_{i;n}}_{\left|\pi_{b}\right|\text{
times}}}\big{)}\right|<\infty,$
where the final inequality follows from the assumption (7.8). ∎
Applying Markov’s inequality and Lemma 7.5 shows that for any $q\geq 1$ we
have
$\sup_{n\geq 1}n^{q}\mathbb{P}\left(S_{n}^{c}\right)<\infty,\qquad
S_{n}:=\left\\{\left|X_{i;n}\right|\leq n^{-1/4},\qquad
i=1,\ldots,m\right\\}.$ (7.10)
This localization estimate allows us to replace each $g_{i}$ by its Taylor
expansion around $0$. Indeed, note that
$\kappa\left(g_{1}(X_{n}),\ldots,g_{k}(X_{n})\right)=P\left(\mathbb{E}\left[g_{1}(X_{n})^{q_{1}}\cdots
g_{k}(X_{n})^{q_{k}}\right],\quad q_{1}+\cdots+q_{k}\leq k\right)$
for some universal polynomial $P$ evaluated at the mixed moments of $X_{n}$
(the formula for this polynomial is given in (6.6) but is not important).
Moreover, using the growth assumption (7.7) on $X$ and the fact that $g_{i}$
are polynomially bounded we find that
$\sup_{n\geq 1}\mathbb{E}\left[g(X_{n})^{q_{1}}\cdots
g_{k}(X_{n})^{q_{k}}\right]<\infty,\qquad\forall\,q_{1},\ldots,q_{k}\geq 1.$
(7.11)
This, in combination with the localization estimate (7.10) applied with
$q=k-1$ yields
$\kappa\left(g_{1}(X_{n}),\ldots,g_{k}(X_{n})\right)=P\left(\mathbb{E}\left[{\bf
1}_{S_{n}}g_{1}(X_{n})^{q_{1}}\cdots g_{k}(X_{n})^{q_{m}}\right],\quad
q_{1}+\cdots+q_{m}\leq k\right)+O(n^{-k+1}).$
Note that for $n$ sufficiently large, on the event $S_{n}$, the argument
$X_{n}$ is any fixed neighborhood of $0$. Hence, we may write
$g_{j}(X_{n})=p_{j}(X_{n})+O(n^{-k+1}),$
where $p_{j}$ represents the $q-$th order Taylor expansion of $g_{j}$ around
$0$ with $q$ sufficiently large and the constant in the error term is
uniformly bounded. This yields
$\kappa\left(g_{1}(X_{n}),\ldots,g_{k}(X_{n})\right)=P\left(\mathbb{E}\left[{\bf
1}_{S_{n}}p_{1}(X_{n})^{q_{1}}\cdots p_{k}(X_{n})^{q_{m}}\right],\quad
q_{1}+\cdots+q_{m}\leq k\right)+O(n^{-k+1}).$
Finally, using the mixed moment estimates (7.7) and the localization estimate
(7.10), we conclude
$\displaystyle\kappa\left(g_{1}(X_{n}),\ldots,g_{k}(X_{n})\right)$
$\displaystyle=P\left(\mathbb{E}\left[p_{1}(X_{n})^{q_{1}}\cdots
p_{k}(X_{n})^{q_{m}}\right],\quad q_{1}+\cdots+q_{m}\leq k\right)+O(n^{-k+1})$
$\displaystyle=\kappa\left(p_{1}(X_{n}),\ldots,p_{k}(X_{n})\right)+O(n^{-k+1}).$
Recalling (7.8) completes the proof. ∎
Proposition 7.4 shows that, in establishing the conclusion (7.6) of Theorem
7.3, it is sufficient to assume that $g_{j}$ are polynomials. The remainder of
the proof of Theorem 7.3 is by induction on $\ell$, starting with $\ell=1$. In
view of Proposition 7.4, the following result establishes the base case.
###### Proposition 7.6 (Base Case: Theorem 7.3 holds for polynomials at
$\ell=1$).
Fix $k,m\geq 1$ and suppose $f_{i},\,i=1,\ldots,m$ are as in the statement of
Theorem 7.3. Then, if $p_{1},\ldots,p_{k}$ are any polynomials in $m$
variables, we have
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)\right|<\infty.$
###### Proof.
Since cumulants are multi-linear, we may and shall assume that $p_{a}$ are
monomials:
$p_{a}(x)=x^{Q^{(a)}}:=x_{1}^{q_{1}^{(a)}}\cdots x_{m}^{q_{m}^{(a)}},\qquad
x=\left(x_{1},\ldots,x_{m}\right),\quad
Q^{(a)}=\left(q_{1}^{(a)},\ldots,q_{m}^{(a)}\right).$ (7.12)
Recall that
$\mathcal{O}_{f_{i}}^{(1)}:=n_{1}^{-1}\sum_{j=1}^{n_{1}}f_{i}\left(z_{j;\mathcal{A}}^{(1)}\right).$
Therefore, writing $q^{(a)}:=q_{1}^{(a)}+\cdots+q_{m}^{(a)}$ we find
$\displaystyle
p_{a}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)=n_{1}^{-q^{(a)}}\sum_{J^{(a)}}f_{J^{(a)}},\qquad
f_{J^{(a)}}:=\prod_{i=1}^{m}\prod_{q=1}^{q_{i}^{(a)}}f_{i}\big{(}z_{j_{q;i}^{(a)};\mathcal{A}}^{(1)}\big{)},$
where the sum is over tuples of multi-indices
$J^{(a)}=\left(J_{1}^{(a)},\ldots,J_{m}^{(a)}\right),\qquad
J_{i}^{(a)}=\left(j_{q;i}^{(a)}\in[n_{1}],\,i\in[m],\,q\in[q_{i}^{(a)}]\right).$
(7.13)
Hence, using that cumulants are multi-linear (see (6.3)), we obtain
$\displaystyle\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)=n_{1}^{-(q^{(1)}+\cdots+q^{(k)})}\sum_{\begin{subarray}{c}J^{(1)},\ldots,J^{(k)}\end{subarray}}\kappa\left(f_{J^{(1)}},\ldots,f_{J^{(k)}}\right),$
where the sum extends over ordered collections $\left(J^{(a)},\,1\leq a\leq
k\right)$ of multi-indices as in (7.13). The expression on the right hand side
can be interpreted as an average. Namely, we can think of the indices
$j_{q;i}^{(a)}\in[n_{1}]$ are chosen uniformly from $[n_{1}]$ and
independently for all $i,q,a$. Writing $\mathcal{E}$ for the average with
respect to this distribution, we obtain
$\displaystyle\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)=\mathcal{E}\left[\kappa\left(f_{J^{(1)}},\ldots,f_{J^{(k)}}\right)\right].$
Our goal is to show that this average is small. To quantify this, let us
associate to each collection $\left(J^{(a)},\,a\in[k]\right)$ a graph
$\mathcal{G}\left(J^{(a)},\,a\in[k]\right)=\left([k],\,E\left(J^{(a)},\,a\in[k]\right)\right),$
(7.14)
with vertex set $[k]$ and edge set defined by
$(a,a^{\prime})\in\mathcal{E}\left(J^{(a)},\,a\in[k]\right)\quad\Longleftrightarrow\quad\exists
i,i^{\prime}\in[m],\,q\in[q_{i}^{(a)}],\,q^{\prime}\in[q_{i^{\prime}}^{(a^{\prime})}]\text{
s.t. }j_{q;i}^{(a)}=j_{q^{\prime};i^{\prime}}^{(a^{\prime})}.$
The key point is that in light of the vanishing property (6.2) of cumulants
and the fact that neurons at layer $1$ are independent
$\mathcal{G}\left(J^{(a)},\,a\in[k]\right)\text{ disconnected
}\quad\Longrightarrow\quad\kappa\left(f_{J^{(1)}}^{(1)},\ldots,f_{J^{(k)}}^{(1)}\right)=0.$
Hence,
$\displaystyle\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)=\mathcal{E}\left[{\bf
1}_{\left\\{\mathcal{G}\left(J^{(a)},\,a\in[k]\right)\text{
connected}\right\\}}\kappa\left(f_{J^{(1)}}^{(1)},\ldots,f_{J^{(k)}}^{(1)}\right)\right].$
Since $f_{i}$ are assumed to be polynomially bounded and the distribution of
the neuron pre-activations $z_{i;\alpha}^{{}_{(1)}}$ is that of centered
Gaussians with mean $0$ and covariance
$\mathrm{Cov}\left(z_{i_{1};\alpha_{1}}^{{}_{(1)}},z_{i_{2};\alpha_{2}}^{{}_{(1)}}\right)=\delta_{i_{1}i_{2}}\left(C_{b}+\frac{C_{W}}{n_{0}}\sum_{j=1^{n_{0}}}x_{j;\alpha_{1}}x_{j;\alpha_{2}}\right),$
we have for any fixed $k$ that
$\sup_{n\geq
1}\sup_{J^{(1)},\ldots,J^{(a)}}\left|\kappa\left(f_{J^{(1)}}^{(1)},\ldots,f_{J^{(k)}}^{(1)}\right)\right|<\infty.$
Hence,
$\displaystyle\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)=O\left(\mathcal{P}\left(\mathcal{G}\left(J^{(a)},\,a\in[k]\right)\text{
connected}\right)\right),$
where $\mathcal{P}$ is the probability measure associated to our random draw
of $J^{(1)},\ldots,J^{(k)}$. To complete the proof, note that since
$m,q_{i}^{(a)}$ are fixed, by a simple union bound, we obtain
$\displaystyle\mathcal{P}\left(\mathcal{G}\left(J^{(a)},\,a\in[k^{\prime}]\right)\text{
connected}~{}\bigg{|}~{}\mathcal{G}\left(J^{(a)},\,a\in[k^{\prime}-1]\right)\text{
connected}\right)=O(n^{-1}).$
Hence,
$\displaystyle\mathcal{P}\left(\mathcal{G}\left(J^{(a)},\,a\in[k]\right)\text{
connected}\right)$
$\displaystyle\qquad=\prod_{k^{\prime}=2}^{k}\mathcal{P}\left(\mathcal{G}\left(J^{(a)},\,a\in[k^{\prime}]\right)\text{
connected}~{}\bigg{|}~{}\mathcal{G}\left(J^{(a)},\,a\in[k^{\prime}-1]\right)\text{
connected}\right)$ $\displaystyle\qquad=O(n^{-k+1}).$ (7.15)
Thus,
$\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(1)}\right)\right)=O(n^{-k+1}),$
as desired. ∎
Propositions 7.4 and 7.6 together show that the conclusion (7.6) of Theorem
7.3 holds at layer $1$. In conjunction with Proposition 7.4, the following
result establishes that if the conclusion (7.6) of Theorem 7.3 holds at some
layer $\ell\geq 1$ then it also holds at layer $\ell+1$. This will complete
the proof by inductive of Theorem 7.3.
###### Proposition 7.7 (Inductive Step: Reducing polynomial cumulants in
layer $\ell+1$ to smooth cumulants in layer $\ell$).
Fix $\ell\geq 1$.
Case 1: Suppose that $\sigma$ is smooth. Assume that for any collection
$F^{\prime}=\left(f_{i}^{\prime}:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R},\,i=1,\ldots,m\right)$
of smooth and polynomially bounded functions and any $g_{j}$ as in the
statement of Theorem 7.3 the conclusion (7.6) of Theorem 7.3 holds at layer
$\ell$:
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(g_{1}\left(\overrightarrow{\mathcal{O}}_{F^{\prime}}^{(\ell)}\right),\ldots,g_{k}\left(\overrightarrow{\mathcal{O}}_{F^{\prime}}^{(\ell)}\right)\right)\right|<\infty.$
Then, if $p_{1},\ldots,p_{k}$ are any polynomials in $m$ variables, and
$F=\left(f_{i},\,i=1,\ldots,m\right)$ is an arbitrary collection of smooth and
polynomially bounded functions
$f_{i}:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R}$, then
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right)\right)\right|<\infty.$
Case 2: Suppose $\sigma$ is not smooth but satisfies Assumption 2.1 and that
$(D_{\alpha}^{J}z_{i;\alpha}^{(\ell)},\,\alpha\in\mathcal{A},\,\left|J\right|\leq
r)$ is non-degenerate in the infinite width in the sense of (3.3). Assume that
for any collection
$F^{\prime}=\left(f_{i}^{\prime}:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},r)}\rightarrow\mathbb{R},\,i=1,\ldots,m\right)$
of measurable and polynomially bounded functions and any $g_{j}$ as in the
statement of Theorem 7.3 the conclusion (7.6) of Theorem 7.3 holds at layer
$\ell$:
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(g_{1}\left(\overrightarrow{\mathcal{O}}_{F^{\prime}}^{(\ell)}\right),\ldots,g_{k}\left(\overrightarrow{\mathcal{O}}_{F^{\prime}}^{(\ell)}\right)\right)\right|<\infty.$
Then, if $p_{1},\ldots,p_{k}$ are any polynomials in $m$ variables, and
$F=\left(f_{i},\,i=1,\ldots,m\right)$ is an arbitrary collection of measurable
and polynomially bounded functions
$f_{i}:\mathbb{R}^{\left|\mathcal{A}\right|\times
N(n_{0},d)}\rightarrow\mathbb{R}$, then
$\sup_{n\geq
1}\left|n^{k-1}\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right)\right)\right|<\infty.$
###### Proof.
The proof of Proposition 7.7 is similar but somewhat more involved than that
of Proposition 7.6. Moreover, the two cases are proved in essentially the same
way, except that we will employ the different cases in Lemma 6.3. We give the
details in the case when $\sigma$ is smooth and indicate where the proof is
modified slightly to handle the non-smooth case.
To start, as in the proof of Proposition 7.6, note that since cumulants are
multi-linear (see (6.3)), it is enough to assume that $p_{j}$ are monomials.
Thus, borrowing the notation from the proof of Proposition 7.6 (see starting
(7.12)), we find
$\displaystyle\kappa\left(p_{1}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right),\ldots,p_{k}\left(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}\right)\right)=n_{\ell+1}^{-(q^{(1)}+\cdots+q^{(a)})}\sum_{\begin{subarray}{c}J^{(1)},\ldots,J^{(k)}\end{subarray}}\kappa\left(f_{J^{(1)}}^{(\ell+1)},\ldots,f_{J^{(k)}}^{(\ell+1)}\right),$
where
$f_{J^{(a)}}^{(\ell+1)}:=\prod_{i=1}^{m}\prod_{q=1}^{q_{i}^{(a)}}f_{j_{\alpha}^{(a)}}^{(\ell+1)},\qquad
f_{j}^{(\ell+1)}:=f\left(z_{j;\mathcal{A}}^{(\ell+1)}\right).$
Note that, as in Proposition A.1, the polynomially bounded assumption on
$f_{j}$ and the non-linearity $\sigma$ together with the Gaussianity of
weights and biases show that
$\sup_{n\geq
1}\left|\kappa\left(f_{J^{(1)}}^{(\ell+1)},\ldots,f_{J^{(k)}}^{(\ell+1)}\right)\right|<\infty.$
(7.16)
Using the law of total cumulance (6.1), we find that
$\kappa\left(p_{1}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}),\ldots,p_{k}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)})\right)$
equals
$\displaystyle\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\end{subarray}}n_{\ell+1}^{-(q^{(1)}+\cdots+q^{(a)})}\sum_{\begin{subarray}{c}J^{(1)},\ldots,J^{(k)}\end{subarray}}\kappa\left(\kappa\left(f_{J^{(\pi_{1})}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right),\ldots,\kappa\left(f_{J^{(\pi_{B})}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right)\right),$
where $\pi$ is any partition of $[k]$ and
$f_{J^{(\pi_{b})}}:=\left(f_{J^{(a)}},\,a\in\pi_{b}\right).$
Just as in the proof of Proposition 7.6, we may interpret the sum over
$J^{(1)},\ldots,J^{(k)}$ as an average over the distribution in which
$j_{q;i}^{(a)}$ are drawn iid uniformly on $[n_{\ell+1}]$. Writing
$\mathcal{E}$ for averages with respect to this distribution yields
$\kappa\left(p_{1}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}),\ldots,p_{k}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)})\right)=\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{b}\right)\end{subarray}}\mathcal{E}\left[\kappa\left(\kappa\left(f_{J^{(\pi_{1})}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right),\ldots,\kappa\left(f_{J^{(\pi_{b})}}^{(\ell+1)}~{}|~{}\mathcal{F}^{(\ell)}\right)\right)\right]$
As in (7.14), we may associate to each collection $J^{(\pi_{t})}$ the graph
$\mathcal{G}\left(J^{(\pi_{t})}\right)$. Recall that by Lemma 7.1, the neurons
pre-activations $D_{\alpha}^{J}z_{i;\alpha}^{(\ell+1)}$ in layer $\ell+1$ are
independent for different $i$ conditional on $\mathcal{F}^{(\ell)}$. Hence, in
view of the vanishing property (6.2) of cumulants, we obtain that
$\kappa\left(p_{1}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)}),\ldots,p_{k}(\overrightarrow{\mathcal{O}}_{F}^{(\ell+1)})\right)$
equals
$\displaystyle\sum_{\begin{subarray}{c}\pi=\left(\pi_{1},\ldots,\pi_{B}\right)\end{subarray}}\mathcal{E}\left[{\bf
1}_{\left\\{\mathcal{G}\left(J^{(\pi_{b})}\right)\text{ connected }\forall
|
# Mid-infrared Single-photon Detection Using Superconducting NbTiN Nanowires
with Sub-15 ps Time Resolution in a Gifford-McMahon Cryocooler
Jin Chang Optics Research Group, ImPhys Department, Faculty of Applied
Sciences, Delft University of Technology, Delft 2628 CJ, The Netherlands.
Single Quantum B.V., Delft 2628 CJ, The Netherlands. _Corresponding
<EMAIL_ADDRESS>Johannes W. N. Los Single Quantum B.V., Delft
2628 CJ, The Netherlands. Ronan Gourgues Single Quantum B.V., Delft 2628 CJ,
The Netherlands. Stephan Steinhauer KTH Royal Institute of Technology,
Department of Applied Physics, Albanova University Centre, Roslagstullsbacken
21, 106 91 Stockholm, Sweden S. N. Dorenbos Single Quantum B.V., Delft 2628
CJ, The Netherlands. Silvania F. Pereira Optics Research Group, ImPhys
Department, Faculty of Applied Sciences, Delft University of Technology, Delft
2628 CJ, The Netherlands. H. Paul Urbach Optics Research Group, ImPhys
Department, Faculty of Applied Sciences, Delft University of Technology, Delft
2628 CJ, The Netherlands. Val Zwiller KTH Royal Institute of Technology,
Department of Applied Physics, Albanova University Centre, Roslagstullsbacken
21, 106 91 Stockholm, Sweden Iman Esmaeil Zadeh Optics Research Group,
ImPhys Department, Faculty of Applied Sciences, Delft University of
Technology, Delft 2628 CJ, The Netherlands.
###### Abstract
Shortly after their inception [1], superconducting nanowire single-photon
detectors (SNSPDs) became the leading quantum light detection technology [2].
With the capability of detecting single-photons with near-unity efficiency [3,
4, 5], high time resolution [6, 7], low dark count rate [8], and fast recovery
time [9], SNSPDs outperform conventional single-photon detection techniques.
However, detecting lower energy single-photons (<0.8 eV) with high efficiency
and low timing jitter has remained a challenge. To achieve unity internal
efficiency at mid-infrared wavelengths, previous works [10, 11] used amorphous
superconducting materials with low energy gaps at the expense of reduced time
resolution (close to a nanosecond [12]), and by operating them in complex mK
dilution refrigerators. In this work, we provide an alternative approach with
SNSPDs fabricated from 5-9.5 nm thick NbTiN superconducting films and devices
operated in conventional Gifford-McMahon (GM) cryocoolers. By optimizing the
superconducting film deposition process, film thickness and nanowire design,
our fiber-coupled devices achieved > 70% system detection efficiency (SDE) at
2 µm and sub-15 ps timing jitter. Furthermore, detectors from the same batch
demonstrated unity internal detection efficiency at 3 µm and 80% internal
efficiency at 4 µm, paving the road for an efficient mid-infrared single-
photon detection technology with unparalleled time resolution and without mK
cooling requirements. We also systematically studied the dark count rates
(DCRs) of our detectors coupled to different types of mid-infrared optical
fibers and black-body radiation filters. This offers insight into the trade-
off between bandwidth and dark count rates for mid-infrared SNSPDs. To
conclude, this paper significantly extends the working wavelength range for
SNSPDs made from polycrystalline NbTiN to 1.5-4 µm, and we expect quantum
optics experiments and applications in the mid-infrared range to benefit from
this far-reaching technology.
## Introduction
Detecting light at the single photon level has enabled novel scientific and
industrial applications in recent decades [2]. Specifically, near- and mid-
infrared detection are crucial for areas such as infrared fluorescence and
spectroscopy [13, 14, 15], semiconductor and industrial production monitoring
[16, 17], planetary soil studies [18], remote light detection and ranging [19]
as well as two-photon entanglement and interference [20] experiments. However,
since photon energy is inversely proportional to wavelength, detecting long
wavelength photons is intrinsically more challenging than detecting shorter
wavelength photons. Generally, Si-based detectors can be used for infrared
detection but suffer from a low cut-off wavelength, typically around 1.1 µm
[21], making them inefficient for long wavelength photon detection. Si:Sb
based impurity band conduction detectors show mid-infrared light detection
capability but not at the single-photon level [22]. Similarly, narrow-bandgap
photoconductive semiconductors, like HgCdTe, InAs and InGaAs detectors suffer
from low efficiency, large dark counts and poor time resolution [23]. In
contrast, SNSPDs have high detection efficiency [3, 4, 5], high detection
rates [24], low dark count rates (DCR) [8], unprecedented temporal resolution
[7, 6] and thus outperform traditional infrared single-photon detectors.
In 2001, NbN-based SNSPDs were first demonstrated by detecting 810 nm single-
photons [1]. Subsequently, SNSPDs fabricated on different platforms were
explored and developed [2]. Although high system detection efficiencies have
been realized and reported for the UV [25], visible [26] and near-
infrared/telecom [27, 3, 4, 5], detecting single-photons beyond 1550 nm with
high efficiency and time resolution has remained a challenge [28]. Early works
showed amorphous WSi based SNSPDs could be used for mid-infrared detection.
However, these studies employed 4-6 nm thin superconducting films, resulting
in low critical currents which is detrimental to the timing jitter. The
reported temporal resolution was close to the nanosecond scale [12]. Also,
these devices must be operated at sub-Kelvin temperatures, requiring complex
dilution refrigerators. NbN-based SNSPDs with ultra-narrow line widths showed
sensitivity up to 5 µm (saturated internal efficiency until 2.7 µm) [29]. A
consequence of squeezing the nanowire width to around 30 nm makes fabrication
challenging and degrades the detectors’ time resolution with the reduced
critical current.
Alternatively, our previous work [30] showed that by optimizing the
stoichiometry of polycrystalline NbTiN film during reactive magnetron co-
sputtering deposition, it is possible to make SNSPDs with strongly saturated
efficiency plateaus in the near-infrared region at 2.8 K operating
temperature, and also high performance at visible wavelengths up to 7 K [31].
Also, relatively thick NbTiN superconducting films were used [6, 32] to
improve our detectors’ optical absorption and critical current, therefore
enhancing efficiency and time resolution. Building on our previous results, in
this work we made SNSPDs from 5, 6.5, 7.5 and 9.5 nm thick NbTiN films with
different nanowire designs. First, by characterizing our SNSPDs using flood
illumination, we optimized the meander design in terms of internal detection
efficiency. Encouraged by our initial characterization results, we fabricated
fiber-coupled SNSPDs and achieved a system detection efficiency of >70% at 2
µm in Gifford-McMahon (GM) cryo-coolers (2.4-2.8 K). Broadband detectors were
also demonstrated with >50% SDE over the entire 1550-2000 nm range with sub-15
ps timing jitter. Furthermore, devices made from 7.5 and 6.5 nm films showed
unity internal detection efficiency at 3 µm and 80% internal efficiency at 4
µm. We also systematically studied the DCRs from the detector itself
(intrinsic DCRs) and from black-body radiation delivered by different types of
fibers as well as coated fibers as a technique to reduce the DCR. These
results offer a comprehensive understanding of the origin of dark counts in
mid-infrared SNSPD systems.
## 1 SNSPD Fabrication and Measurement Setup
Similar to [30], we deposited superconducting NbTiN films by a reactive
magnetron co-sputtering deposition process. The stoichiometry of the films was
controlled by adjusting the sputtering powers on the Ti and Nb targets. Film
thickness was determined by a calibrated crystal microbalance and SNSPDs were
fabricated as described in [6]. Our fabricated detectors were either tested
under flood illumination (figure 1 (a)), or etched into a key-hole die shape
and packaged using a standard ferrule and mating sleeves approach [33] (figure
1 (b)). This coupling method guarantees automatic alignment between detector
and optical fibers for accurate system efficiency measurements. Both the flood
illumination and fiber-coupled setup are shown in figure 1 (c).
Figure 1: Illustration of (a) detector flood illumination, (b) a fiber-coupled
detector, and (c) schematic of the efficiency measurement set-up.
As shown in figure 1 (c), we employ a near-infrared tunable laser (JGR-TLS5,
1260-1650 nm) and mid-infrared CW lasers with different wavelengths (2000 and
2700 nm laser from Thorlabs, 3001 and 4013 nm laser from Nanoplus) as input
photon sources. Single mode optical fibers are used to couple the light to the
first fiber-to-fiber coupler (containing neutral density filters and a
polarizer). A beam splitter is then used to create a reference arm with the
majority of the power coupled to a calibrated power meter $P_{1}$. The signal
beam with the lowest power is sent to the second fiber-to-fiber coupler, also
containing a polarizer and neutral density filters. A polarization controller
is used to tune the polarization state of the light after the second fiber-to-
fiber coupler. After recording the light power intensity emerging from the
polarization controller with power meter $P_{2}$, the light is guided to the
system to carry out either flood illumination or fiber-coupled measurements.
For flood illumination measurements, the input light is heavily attenuated to
the single-photon regime. For fiber-coupled device measurements we use the
following procedure: we first set the ratio $P_{1}/P_{2}$ to 50 dB by placing
ND (neutral density) filters in both fiber-to-fiber couplers, and then add
additional ND filters to the first coupler to reach $P_{1}$ = 10 nW. In this
way, the input photon flux can be back calculated. For example, 10 nW with 50
dB attenuation at 2000 nm corresponds to an input photon flux of $1.006\times
10^{6}$ photons per second. More measurement details can be found in our
previous work [4].
## 2 Characterization of SNSPDs with Flood Illumination
In this work, SNSPDs with different nanowire widths (40/60/80 nm) and
diameters (8/9/10 µm) were fabricated from 5-9.5 nm thick NbTiN films. For
example, 60-120-r4 refers to a meandering nanowire design with 60 nm wide
lines, a pitch of 120 nm, and 4 µm radius (see insert in figure 3 (b)).
Figure 2: Internal efficiency measurements of SNSPDs fabricated from (a) 9.5
nm and (b) 7.5 nm thick NbTiN films.
As shown in figure 2 (a), at 1550 and 1625 nm, the 40 nm width device
(yellow), 60 nm width device (green ) and 80 nm width device ( red ) all
showed saturated internal efficiencies. When the laser wavelength was
increased to 2000 nm, the 40 and 60 nm wide nanowires (yellow and green)
devices maintained saturated internal efficiencies, while the 80 nm wide
nanowire device (red curve) does not reach unity internal efficiency. To
obtain greater saturated internal efficiency, one possible solution is to make
narrower lines. However, with narrower line width (<40 nm) the nanofabrication
patterning, development, and etching, become more critical. This will, in
general, affect the fabrication yield. Alternatively, we sputtered 7.5 nm
thick NbTiN films and made detectors with the same designs and nanofabrication
process. As shown in figure 2(b), by using 7.5 nm thick NbTiN film, all
devices with line widths ranging from 40-80 nm showed unity internal
efficiency at 2000 nm. Detailed performance of devices based on 9.5 and 7.5 nm
films is summarized in table 1. A thinner film leads to lower critical
currents (for the same meander design), timing jitter is thus higher because
the output pulse has a lower signal to noise ratio [34]. The rise time (time
interval for signal to go from 20%-80% of the pulse amplitude) of the devices
on 7.5 nm NbTiN was longer than for the 9.5 nm devices and dead-time (width of
pulse at level of 1/e of the amplitude) was also slightly longer, this can be
explained by the fact that devices made with thinner films have higher kinetic
inductance.
Meander Structure | $I_{c}$ (µA) 9.5/7.5 nm | Rise-time (ps) 9.5/7.5 nm | Dead-time (ns) 9.5/7.5 nm | Jitter (ps) 9.5/7.5 nm
---|---|---|---|---
80-160-r4.5 | 25.0/18.4 | 350/375 | 9.3/10.6 | 30/44
60-120-r4 | 18.2/14 | 325/335 | 11.6/12.6 | 40/45
40-80-r5 | 8.40/6 | 400/425 | 38.4/49.2 | 93/97
| | | |
Table 1: Flood illumination measurement results of 9.5 and 7.5 nm NbTiN based
SNSPDs. Figure 3: (a) Photon counting rate (PCR) curves at 2700 nm of
60-120-r4 detectors from 7.5 nm (green) and 9.5 nm (blue) film, (b) PCR curves
at 3001 nm of 60-120-r4 detector/purple and 1.1$\times$9 µm detector/red from
7.5 nm film, (c) PCR curves at 4013 nm of a 40-120-r5 detector from 6.5 nm
film, and (d) statistics of device yield made from films with different
thicknesses.
The above results show that saturated internal efficiency until 2000 nm can be
obtained with detectors made from 9.5/7.5 nm thick films. In order to explore
the internal saturation limit, we carried out longer wavelength flood
illumination measurements at 2700 and 3001 nm for a number of selected
detectors. Figure 3 (a) shows detectors with 60-120-r4 meander design from
both 9.5/7.5 nm films at 2700 nm. Both detectors reach unity internal
efficiency and detectors from 7.5 nm film (dark green curve) show stronger
saturated internal efficiency than detectors from 9.5 nm film (light blue
curve). This is because by reducing the thickness of the superconducting film,
the superconducting energy gap is reduced with the same input photon power, it
is easier to break the superconducting state and form a resistive region [35].
Previous measurements at 2700 nm indicate that detectors made from 7.5 nm
films are still promising for detecting single photons beyond 2700 nm, we
fabricated two types of SNSPDs from a 7.5 nm NbTiN film and measured their
detection performances at 3001 nm. As shown in figure 3 (b), a ’large’
meandering nanowire detector design of 60-120-r4 (purple), and a ’small’
detector design with line-width 60 nm, filling factor 50%, $1.1\times 9$ µm
(red) were employed. Both detectors show saturated internal efficiency at 3001
nm and the smaller detector shows superior saturated internal efficiency over
the larger one. According to a previous study [36], the performance of SNSPDs
is influenced by the inhomogeneity of the superconducting film. Since the
total length of the small detector ($\sim 90$ µm) is more than 3 times shorter
than the large ($\sim 418$ µm), less inhomogeneity can be expected and better
detection performance is observed. This shows that by reducing SNSPD’s total
length, better detection performance can be potentially achieved. In previous
works [11], the best performing device was a single 10-µm-long line. As a
consequence the active area is smaller, which can be increased by using
different detector architectures, for example multi-pixel [37] or interleaved
nanowire designs [38].
Finally, we evaluated detectors made from even thinner films (5 and 6.5 nm).
In figure 3 (c), we demonstrate that a detector (40-120-r5) from 6.5 nm film
achieves 80% internal efficiency at 4013 nm (determined using a sigmoid curve
fitting). This represents the state-of-the-art mid-infrared polycrystalline
material based SNSPDs. To get a better understanding of the film thickness on
the detector performance, we created an overview in figure 3 (d). We present
the statistics of 32 fabricated SNSPDs from 4 different films. It is clear
that 5 and 6.5 nm films suffer from low yield. The detectors made from the 5
nm films do not work well, because of their low critical current (1-2 µA). The
non-working detectors from the 6.5 nm film did not show unity internal
efficiency at 1550 nm possibly caused by lower film homogeneity of the thin
film [6]. In contrast, 7.5 and 9.5 nm films show higher yield but detectors
from the 9.5 nm film start to show decreased internal efficiency in the mid-
infrared compared to detectors from the 7.5 nm film. Here, we suggest two
practical solutions to solve the trade-off between film thickness and
performance for future mid-infrared SNSPDs study: Bias-assisted sputtering can
be applied to improve the critical current of SNSPDs [39] and post-processing
treatment (for example, helium ion irradiation [40]) can enhance the internal
efficiency of SNSPDs made from thicker films.
## 3 Measurements of Fiber-coupled SNSPDs
For most quantum optics experiments and applications, a fiber-coupled
detector/system is preferred because of mature fiber optics technology and
instruments.
Figure 4: Fiber-coupled SNSPDs measurements of (a) SDE of detector #1 at 1310,
1625, 1550 and 2000 nm, (b) timing jitter of detector #1, (c) SDE of detector
#2 at 2001 nm, and (d) DCRs of detector #2 under different conditions: no
fiber (green), low-pass coated fiber (turquoise), UHNA fiber (red), and mid-IR
fiber (purple) plugged in.
The previous section provides evidence that both 7.5 and 9.5 nm NbTiN
superconducting films are suitable for making mid-infrared SNSPDs in terms of
good yield and internal efficiency while a reduced thickness (5-6.5 nm) leads
to fewer working devices. Thus, we fabricated fiber-coupled SNSPDs from 7.5 nm
thick NbTiN film. Similar to [4], the NbTiN films were initially deposited on
SiO2 grown by thermal oxidation process and nanowire meanders were eventually
located on top of a Au/SiO2 membrane acting as optical cavity. After packaging
and wire bonding, detectors were mounted in a closed-cycle cryocooler with a
base temperature of 2.4-2.8 K and coupled to single mode fibers. Afterwards,
lasers with different wavelengths were used for system detection efficiency
(SDE) measurements as described in the previous section.
Figure 4 (a) shows the performance of detector #1 (60 nm line width) made from
a 7.5 nm NbTiN film. The SDEs for 1300, 1550, 1625 and 2000 nm are 50%, 60%,
61%, and 63% respectively. The inset in figure 4 (a) shows detector #1’s
photon counting rate (PCR) curves at several wavelengths. Besides high SDE,
high timing resolution is also highly desirable for many applications, for
example, LiDAR [19], fluorescence microscopy and spectroscopy [14, 13]. The
Instrument Response Function (IRF) of detector #1 was characterized with a ps-
pulsed laser (4.2 ps pulse-width at 1064 nm wavelength) and a fast
oscilloscope (4 GHz bandwidth, 40 GHz sampling rate) as described in [4]. As
shown in figure 4 (b), using a low-noise cryogenic amplifier operated at 40 K,
the IRF of this device shows a Gaussian shaped histogram. After fitting, we
obtain 14.3$\pm$0.1 ps timing jitter (full width at half maximum, FWHM).
Compared to previous reported values for mid-infrared SNSPDs [12], we improved
time resolution by nearly two orders of magnitude. The inset picture in figure
4 (b) shows the pulse trace of detector #1. It shows a dead-time of 11.6 ns,
indicating good performance at high count rates [41]. Similarly, in figure 4
(c), we show the 2000 nm SDE measurement of detector #2, which is made from
another 7.5 nm NbTiN film but has a slightly higher meander filling factor
(approximately 10% higher). The benefit of a higher filling factor is an
increased optical absorption, which means if the internal efficiency is
saturated, a higher SDE can be achieved compared to a similar device with a
lower filling factor. As can be seen, detector #2 shows well saturated
internal efficiency (when Ib $\geqslant$ 0.8Ic). After sigmoid fitting (to
improve the efficiency estimation accuracy), we obtained a peak SDE over 70%
at 2000 nm.
Besides achieving high system detection efficiency, high dark count rates of
mid-infrared SNSPDs are a major challenge. Previous work [19] showed that the
DCRs of mid-infrared SNSPDs is typically in the order of $10^{4}$ Hz without
using additional filters. In figure 4 (d), we systematically studied the DCRs
of detector #2 in four different schemes: DCR without any fiber connected to
the detector (green curve), DCR with end-face coated SM2000 fiber (fiber
operating wavelength 1.7-2.3 µm, cyan curve), DCR with ultra-high NA fiber
without coating (fiber operating wavelength 1.5-2 µm, red curve) and DCR with
mid-infrared ZrF4 fiber without coating (fiber operating wavelength 2.3-4.1
µm, purple curve). As a result, at Ib=0.8Ic, the DCRs of detector #2 for the
above mentioned four schemes are around the order of 101 Hz, 102 Hz, 105 Hz
and 106 Hz, respectively. It is clear that the DCR of detector #2 when coupled
to an end-face coated fiber is 3 orders of magnitude lower than coupled to
ultra-high NA fiber without coating. By using this fiber end-facet coating
(low-pass filter), the DCR of detector #2 is below 240 Hz when it reaches
unity internal efficiency at 2000 nm. In contrast, when detector #2 is
connected to mid-infrared ZrF4 fiber for SDE measurements at 3-4 µm, the
detector showed over 2.78 MHz DCR at 0.8 Ic bias and starts latching [42].
This prevents further SDE measurements of detector #2 at 3-4 µm. To solve this
issue, low-pass filters should be employed before the detectors, either by
using fiber end-face coating, or adding cold filtering stages inside the
cryostat [43].
## 4 Discussion and Conclusion
In the past, amorphous materials were mainly used for mid-infrared single
photon detection motivated by the intuition that their superconducting energy
gap (0.59-0.61 meV for WSi [44]) is lower than polycrystalline material (2.46
meV for NbN [45]). This work pinpoints that NbTiN (polycrystalline) based
SNSPDs can also achieve high mid-infrared single photon detection efficiency
while maintaining unprecedented time resolution. Furthermore, given that the
energy of a single photon even at 10 µm wavelength (123.9 meV) is still
significantly larger than both materials’ superconducting energy gap, other
physical properties of the superconducting materials need to be investigated
to enhance SNSPDs’ mid-infrared detection response. Besides improving internal
detection efficiency, reducing the dark count rates is another outstanding
challenge for mid-infrared SNSPD systems. As shown in this work, only room
temperature black body radiation delivered to the detector by ZrF4 fiber has
led to >106 Hz DCR. To overcome this issue, either extra cryogenic filters
need to be added before the detectors or the entire experiment has to be
performed at cryogenic temperatures.
In conclusion, we demonstrated SNSPDs made from magnetron co-sputtered NbTiN
superconducting films (5-9.5 nm) with unity internal efficiency at 3 µm and
80% internal efficiency at 4013 nm when operated in closed-cycle Gifford-
McMahon coolers (2.4-2.8 K). Our fiber coupled device achieves over 70% system
detection efficiency at 2 µm and > 50% system detection efficiency from 1300
to 2000 nm with sub-15 ps time resolution. By employing an end-facet coated
fiber, the dark count rate of mid-infrared SNSPDs was reduced by 3 orders of
magnitude compared to uncoated single mode fibers. The DCR when coupled to
mid-infrared ZrF4 fiber is also studied, which offers valuable information for
building mid-infrared fiber-coupled SNSPDs systems in the future. To the best
of our knowledge, the detectors presented in this work have the best system
detection efficiency and temporal resolution among the mid-infrared SNSPDs
reported so far, and NbTiN is a solid choice for making mid-infrared SNSPDs
without mK dilution refrigerators.
## Acknowledgements
J.C. acknowledges China Scholarships Council (CSC, No.201603170247) . I.E.Z.,
V.Z., and Single Quantum B.V. acknowledge the supports from the ATTRACT
project funded by the EC under Grant Agreement 777222. R.B.M.G. acknowledges
support by the European Commission via the Marie-Sklodowska Curie action
Phonsi (H2020-MSCA-ITN-642656). S.N.D., S.S., V.Z. and Single Quantum B.V.
acknowledge EU FET-Open project funding (No. 899580). V.Z. acknowledges
funding from the Knut and Alice Wallenberg Foundation Grant “Quantum Sensors”,
and support from the Swedish Research Council (VR) through the VR Grant for
International Recruitment of Leading Researchers (Ref 2013-7152) and Research
Environment Grant (Ref 2016-06122).
## References
* [1] Gol’Tsman, G. _et al._ Picosecond superconducting single-photon optical detector. _Applied Physics Letters_ 79, 705–707 (2001).
* [2] Esmaeil Zadeh, I. _et al._ Superconducting nanowire single-photon detectors: A perspective on evolution, state-of-the-art, future developments, and applications. _Applied Physics Letters_ 118, 190502 (2021).
* [3] Hu, P. _et al._ Detecting single infrared photons toward optimal system detection efficiency. _Optics Express_ 28, 36884–36891 (2020).
* [4] Chang, J. _et al._ Detecting telecom single photons with 99.5- 2.07+ 0.5% system detection efficiency and high time resolution. _APL Photonics_ 6, 036114 (2021).
* [5] Reddy, D. V., Nerem, R. R., Nam, S. W., Mirin, R. P. & Verma, V. B. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550 nm. _Optica_ 7, 1649–1653 (2020).
* [6] Esmaeil Zadeh, I. _et al._ Efficient single-photon detection with 7.7 ps time resolution for photon-correlation measurements. _ACS Photonics_ 7, 1780–1787 (2020).
* [7] Korzh, B. _et al._ Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector. _Nature Photonics_ 14, 250–255 (2020).
* [8] Shibata, H., Shimizu, K., Takesue, H. & Tokura, Y. Ultimate low system dark-count rate for superconducting nanowire single-photon detector. _Optics Letters_ 40, 3428–3431 (2015).
* [9] Münzberg, J. _et al._ Superconducting nanowire single-photon detector implemented in a 2D photonic crystal cavity. _Optica_ 5, 658–665 (2018).
* [10] Chen, Q. _et al._ Mid-infrared single photon detector with superconductor Mo80Si20 nanowire. _arXiv preprint arXiv:2011.06699_ (2020).
* [11] Verma, V. _et al._ Single-photon detection in the mid-infrared up to 10 µm wavelength using tungsten silicide superconducting nanowire detectors. _APL Photonics_ 6, 056101 (2021).
* [12] Chen, L. _et al._ Ultra-sensitive mid-infrared emission spectrometer with sub-ns temporal resolution. _Optics Express_ 26, 14859–14868 (2018).
* [13] Verma, V. _et al._ Towards single-photon spectroscopy in the mid-infrared using superconducting nanowire single-photon detectors. In _Advanced Photon Counting Techniques XIII_ , vol. 10978, 109780N (International Society for Optics and Photonics, 2019).
* [14] Chen, L. _et al._ Mid-infrared laser-induced fluorescence with nanosecond time resolution using a superconducting nanowire single-photon detector: New technology for molecular science. _Accounts of Chemical Research_ 50, 1400–1409 (2017).
* [15] Elsinger, L. _et al._ Integration of colloidal pbs/cds quantum dots with plasmonic antennas and superconducting detectors on a silicon nitride photonic platform. _Nano Letters_ 19, 5452–5458 (2019).
* [16] Mc Manus, M. _et al._ Pica: Backside failure analysis of cmos circuits using picosecond imaging circuit analysis. _Microelectronics Reliability_ 40, 1353–1358 (2000).
* [17] Willer, U., Saraji, M., Khorsandi, A., Geiser, P. & Schade, W. Near-and mid-infrared laser monitoring of industrial processes, environment and security applications. _Optics and Lasers in Engineering_ 44, 699–710 (2006).
* [18] Sprague, A. _et al._ Mercury: Mid-infrared (3–13.5 $\mu$m) observations show heterogeneous composition, presence of intermediate and basic soil types, and pyroxene. _Meteoritics & Planetary Science_ 37, 1255–1268 (2002).
* [19] Taylor, G. G. _et al._ Photon counting lidar at 2.3 µm wavelength with superconducting nanowires. _Optics Express_ 27, 38147–38158 (2019).
* [20] Prabhakar, S. _et al._ Two-photon quantum interference and entanglement at 2.1 µm. _Science Advances_ 6, eaay5195 (2020).
* [21] Tan, C. L. & Mohseni, H. Emerging technologies for high performance infrared detectors. _Nanophotonics_ 7, 169–197 (2018).
* [22] Huffman, J. E., Crouse, A., Halleck, B., Downes, T. & Herter, T. L. Si: Sb blocked impurity band detectors for infrared astronomy. _Journal of Applied Physics_ 72, 273–275 (1992).
* [23] Rogalski, A. Next decade in infrared detectors. In _Electro-Optical and Infrared Systems: Technology and Applications XIV_ , vol. 10433, 104330L (International Society for Optics and Photonics, 2017).
* [24] Zhang, W. _et al._ A 16-pixel interleaved superconducting nanowire single-photon detector array with a maximum count rate exceeding 1.5 ghz. _IEEE Transactions on Applied Superconductivity_ 29, 1–4 (2019).
* [25] Wollman, E. E. _et al._ Uv superconducting nanowire single-photon detectors with high efficiency, low noise, and 4 k operating temperature. _Optics Express_ 25, 26792–26801 (2017).
* [26] Chang, J. _et al._ Multimode-fiber-coupled superconducting nanowire single-photon detectors with high detection efficiency and time resolution. _Applied Optics_ 58, 9803–9807 (2019).
* [27] Le Jeannic, H. _et al._ High-efficiency wsi superconducting nanowire single-photon detectors for quantum state engineering in the near infrared. _Optics Letters_ 41, 5341–5344 (2016).
* [28] Taylor, G. G., Morozov, D. & Hadfield, R. H. Mid-infrared photon counting with superconducting nanowires. In _Quantum Optics and Photon Counting 2021_ , vol. 11771, 1177106 (International Society for Optics and Photonics, 2021).
* [29] Marsili, F. _et al._ Efficient single photon detection from 500 nm to 5 µm wavelength. _Nano Letters_ 12, 4799–4804 (2012).
* [30] Zichi, J. _et al._ Optimizing the stoichiometry of ultrathin nbtin films for high-performance superconducting nanowire single-photon detectors. _Optics Express_ 27, 26579–26587 (2019).
* [31] Gourgues, R. _et al._ Superconducting nanowire single photon detectors operating at temperature from 4 to 7 k. _Optics Express_ 27, 24601–24609 (2019).
* [32] Chang, J., Zadeh, I. E., Los, J. W., Zichi, J. & Zwiller, V. Superconducting nanowire single photon detector with high efficiency and time resolution for multimode fiber coupling. In _CLEO: QELS_Fundamental Science_ , FF1A–2 (Optical Society of America, 2019).
* [33] Miller, A. J. _et al._ Compact cryogenic self-aligning fiber-to-detector coupling with losses below one percent. _Optics express_ 19, 9102–9110 (2011).
* [34] You, L. _et al._ Jitter analysis of a superconducting nanowire single photon detector. _AIP Advances_ 3, 072135 (2013).
* [35] Hofherr, M. _et al._ Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses. _Journal of Applied Physics_ 108, 014507 (2010).
* [36] Cheng, Y., Gu, C. & Hu, X. Inhomogeneity-induced timing jitter of superconducting nanowire single-photon detectors. _Applied Physics Letters_ 111, 062604 (2017).
* [37] Wollman, E. E. _et al._ Kilopixel array of superconducting nanowire single-photon detectors. _Optics Express_ 27, 35279–35289 (2019).
* [38] Huang, J. _et al._ High speed superconducting nanowire single-photon detector with nine interleaved nanowires. _Superconductor Science and Technology_ 31, 074001 (2018).
* [39] Dane, A. E. _et al._ Bias sputtered nbn and superconducting nanowire devices. _Applied Physics Letters_ 111, 122601 (2017).
* [40] Zhang, W. _et al._ Saturating intrinsic detection efficiency of superconducting nanowire single-photon detectors via defect engineering. _Physical Review Applied_ 12, 044040 (2019).
* [41] Esmaeil Zadeh, I. _et al._ Single-photon detectors combining high efficiency, high detection rates, and ultra-high timing resolution. _APL Photonics_ 2, 111301 (2017).
* [42] Annunziata, A. J. _et al._ Reset dynamics and latching in niobium superconducting nanowire single-photon detectors. _Journal of Applied Physics_ 108, 084507 (2010).
* [43] Shibata, H., Fukao, K., Kirigane, N., Karimoto, S. & Yamamoto, H. Snspd with ultimate low system dark count rate using various cold filters. _IEEE Transactions on Applied Superconductivity_ 27, 1–4 (2016).
* [44] Zhang, X. _et al._ Characteristics of superconducting tungsten silicide WxSi1-x for single photon detection. _Physical Review B_ 94, 174509 (2016).
* [45] Antonova, E., Dzhuraev, D., Motulevich, G. & Sukhov, V. Superconducting energy gap in niobium nitride. _Zhurnal Ehksperimental’noj i Teoreticheskoj Fiziki_ 80, 2426–2429 (1981).
|
# The Future of Hybrid Meetings
Marios Constantinides Nokia Bell LabsCambridgeUnited Kingdom
<EMAIL_ADDRESS>and Daniele Quercia Nokia Bell
LabsCambridgeUnited Kingdom<EMAIL_ADDRESS>
(2022)
###### Abstract.
Meetings are typically considered to be the fuel of an organization’s
productivity—a place where employees discuss ideas and make collective
decisions. However, it is no secret that meetings are also often perceived as
wasteful vacuums, depleting employee morale and productivity, likely due to
the fact that current technologies fall short in fully supporting physical or
virtual meeting experience. In this position paper, we discuss the three key
elements that make a meeting successful (i.e., execution, psychological
safety, and physical comfort), and present new tools for hybrid meetings that
incorporate those elements. As past research has focused on supporting meeting
execution (the first element), we set the roadmap for future research on the
two other elements: on psychological safety by articulating how new
technologies could make meeting useful for all participants, ensure all
participants give and receive appropriate levels of attention, and enable all
participants to feel and make others feel comfortable; and on physical comfort
by dwelling on how new technologies could make the meeting experience
comfortable by integrating all human senses. We also discuss the potential
danger of these technologies inadvertently becoming surveillance tools.
workplace, meetings, psychological safety, physical comfort
††journalyear: 2022††copyright: rightsretained††conference: 2022 Symposium on
Human-Computer Interaction for Work; June 8–9, 2022; Durham, NH,
USA††booktitle: 2022 Symposium on Human-Computer Interaction for Work
(CHIWORK), June 8–9, 2022, Durham, NH, USA††price: 15.00††doi:
10.1145/3533406.3533415††isbn: 978-1-4503-9655-4/22/06††ccs: Human-centered
computing Collaborative and social computing systems and tools
## 1\. Introduction
While meeting tools, to a great extent, have increasingly simplified and
augmented the ways meetings are conducted, they still fall short in supporting
the nuanced experience of offline meetings. To see why, consider, a virtual
all-hands meeting, where one would expect a one-to-many form of communication.
One could imagine that not everyone would feel comfortable sharing their video
streams in such a setting and, as such, the meeting host might miss the
opportunity to “read the room” (i.e., interpret non-verbal cues that are
generally expressed in offline settings). The interplay between offline and
online worlds has slowly but surely produced a new breed of meetings: _the
hybrid meeting_. This type of meeting typically involves a mixture of in-
person and remote attendees: remote attendees join the meeting via a virtual
meeting platform (e.g., Zoom), and in-person attendees sit together in the
typical meeting room. Meeting experience is therefore shaped not only by the
participants’ interactions but also by their diverse physical environments. In
this position paper, we:
* •
Discuss the three key elements that make a meeting successful (Section 2).
* •
Present recent research that incorporates those three elements (Section 3).
* •
Set the roadmap for future research, focusing on the two elements out of the
three that are often neglected in the literature—psychological safety and
physical comfort (Section 4).
* •
Discuss the potential danger of future meeting technologies inadvertently
becoming surveillance tools (Section 5).
## 2\. What makes meetings successful
Figure 1. Three key elements that make a meeting successful along with key
research questions for future research: (a) _Execution_ : whether the meeting
had a clear purpose, structure, and resulted in actionable items; (b)
_Psychological safety_ : whether participants felt listened to and were
motivated to contribute during the meeting; and (c) _Physical comfort_ :
whether participants felt physically comfortable by, say, being in an
environment with good air quality and sufficient natural light. Most
innovations in meeting technologies have focused on execution, and future work
should focus on the two other aspects - psychological safety and physical
comfort. A sample of key research questions are reported in this infographic.
A meeting’s perceived experience has been evaluated through standardized
questionnaires, or ad-hoc post meeting surveys. For example, the Event
Performance Indices (EPI) (Kerns, 2015) is a six-item questionnaire that
focuses on a meeting’s overall performance results, and, in turn, measures
meeting effectiveness. The questionnaire prompts participants’ satisfaction
levels including, for example, whether the meeting was worth the time
investment, and whether participants were personally motivated. In the fields
of Management and Organizational Science, meeting productivity has been
directly linked with a meeting’s agenda, structure, and purpose (Lent, 2015;
Cohen, 2016; Schwarz, 2016). Inclusiveness (Axtell, 2016, 2017), dominance
(Romano and Nunamaker, 2001), peripheral activities (Niemantsverdriet and
Erickson, 2017), and psychological safety (Grenny, 2017; Axtell, 2018) are
aspects that have also been found to influence meeting experience and
productivity. The physical comfort of the workplace environment is an
additional factor that makes up the list (Alavi et al., 2017). For example, a
recent survey found that light and outdoor views were among the most popular
perks employees craved for (Meister, 2018), whereas stuffy and stale air
offices tended to reduce productivity (Allen et al., 2016; Allen, 2017; Tom Y.
Chang and Neidell, 2016).
More recently, to capture the factors that generally make a meeting
successful, Constantinides et al. (Constantinides et al., 2020) designed a
28-question survey and administered it to 363 individuals whose answers were
then statistically analyzed. The survey covered an array of themes, previously
identified in the Organization and Management Science literature, including a
meeting’s psychological experience (e.g., contribution, balance, turn-taking,
attention), its structure, and its content (e.g., agenda, physical
environment, use of technology). The survey responses were then statistically
analyzed using a Principal Component Analysis, with 11 items of the
28-question survey explaining 62% of the total variance. These 11 items loaded
on three factors - the three factors that make a meeting successful:
:
Execution. Whether participants ultimately “got things done”—more
specifically, whether the meeting had a clear purpose, structure, and resulted
in actionable items.
:
Psychological Safety. Whether participants felt listened to and were motivated
to contribute during the meeting.
:
Physical Comfort. Whether participants felt physically comfortable by, for
example, be in a meeting room with good air quality and sufficient natural
light.
## 3\. Current Support For The Three Factors Determining Success
Next, we discuss recent research aiming at supporting these three key elements
in either physical or virtual meetings.
### 3.1. Execution
Execution refers to whether the meeting had a clear purpose, structure, and
resulted in actionable items. Besides the mainstream communication tools of
the likes of MS Teams, WebEx, Skype, Zoom, just to name a few, a large body of
scientific work has been focused on supporting meeting execution through
contextual information, text, audio, and video support. For example, one work
focused on detecting key decisions in dialogues (Kim and Rudin, 2014), while
another on generating an “action items” list from these dialogues (McGregor
and Tang, 2017). Using an agenda planning technique, Garcia et al. (Garcia et
al., 2004) developed a tool that allows meeting participants to vote for
agendas items. As the balance of conversational turn-taking is important for
group performance (Woolley et al., 2010), technologies were developed to
create awareness. This was done by highlighting salient moments and
visualizing participants’ contributions (Kim et al., 2008). Content recording
was also a focus for technologies supporting execution. NoteLook (Chiu et al.,
1999) exploited video streams to support note taking. Catchup (Tucker et al.,
2010) automatically identified the gist of what was missed in a meeting,
allowing people to join the meeting late and still participate effectively.
Video Threads (Barksdale et al., 2012) supported asynchronous video sharing
for geographically distributed teams. Finally, Banerjee et al.’s playback
system allowed for revisiting a recorded meeting (Banerjee et al., 2005).
### 3.2. Psychological Safety
Psychological safety refers to whether participants felt listened to and were
motivated to contribute during the meeting. As Edmondson described it,
psychological safety refers to “the absence of interpersonal fear that allows
people to speak up with work-relevant content” (Edmondson, 1999). In face-to-
face interactions (e.g., during a social encounter or a meeting), we primarily
read faces through visual cues and read bodies through a multi-sensory
integration. All these cues, to a great extent, allow us to understand the
dynamics in a conversation (e.g., whether one feels listened to and receives
the appropriate attention from his/her peers). In virtual interactions,
however, these natural cues are often lost, flattening the psychological and
social experience. Furthermore, current meeting technologies need to deliver
this experience in an attention-enhancing way as opposed to the current
attention-depleting way: currently, in a virtual meeting, attendees need to
pay attention not only to what is being said, but also to peripheral modes of
communication (e.g., text messages, raising of virtual hands).
To bring the offline meeting experience into the virtual world, recent
research developed an app called MeetCues. This captures three types of cues,
aiming at augmenting psychological safety in hybrid meetings. First, the
MeetCues app allows meeting attendees to provide real-time feedback (Aseniero
et al., 2020). The app enabled attendees to react on what was being discussed
by tagging key points and action items. These virtual crowdsourced tags were
translated into an emoji cloud visualization, which allowed attendees to infer
the overall atmosphere. In this way, these crowdsourced cues served as
indicators of moments during which, for example, participants agreed with what
was being said or whether further clarification was needed, cultivating a safe
environment for contributions.
However, these crowdsourced feedbacks did not fully capture the multi-sensory
integration that we are attuned to in physical meetings. A case in point is
the almost universally acceptable non-verbal cue of nodding (Peleckis and
Peleckienė, 2015). In face-to-face interactions, it is natural to observe
bodily expressions to understand, for example, whether one agrees on what is
being said (nodding), or a clarification is needed. However, the picture is
different in virtual meetings. Participants of virtual meetings who often turn
off their cameras, leave the speaker staring at a sea of black squares,
feeling psychologically disoriented.
That is why the second type of cues MeetCues captured was body movements. It
did so by integrating wearable devices with the MeetCues app, which captured
participants’ heart rates, head and hand movements, and changes in postures
(Choi et al., 2021; Park et al., 2020). These body cues were predictive of the
meeting’s vibrancy and multi-tasking activities, and ultimately of the
meeting’s success; these cues were even more predictive than the meeting’s
emotional content derived from the meeting’s transcript. For example, head
movements such as nodding served as a proxy for (dis)agreement, while changes
in postures served as a proxy for (dis)comfort. In particular, these two body
cues metrics proved useful in helping attendees infer the levels of
psychological safety “in the room”.
The third type of cues MeetCues captured was types of conversations (Choi et
al., 2020) (e.g., a heated discussion resulting in a conflict eventually being
resolved, a supportive conversation, an exchange of knowledge). By analyzing
more than four thousand minutes of conversations from eighty-five real-world
meetings, Zhou et al. (Zhou et al., 2021b) found that conversation types were
more predictive of meeting success than traditional voice and text analytics.
By monitoring these conversations during a meeting (or after it), one could
potentially measure specific aspects of organizational productivity, and
proactively take actions for improvement. To see how, consider conflict
resolution. Having new real-time ways of marking indicators of, for example,
conflicts, not only would increase attendees’ awareness but would also help
cultivate an approach of conflict resolution.
### 3.3. Physical Comfort
Physical comfort refers to whether participants felt physically comfortable
by, for example, be in a meeting room with good air quality and sufficient
natural light. To see what we mean by the word ‘comfort’, let us draw a
parallel with architecture. In that area, comfort mainly describes four main
qualities that a good building must possess: visually pleasurable, thermally
livable, acoustically tolerant, and respiratory humane (in terms of air
quality) (Alavi et al., 2017).
To paraphrase comfort in the meeting context, recent research has initially
focused on the specific aspects of good air quality and natural light. A team
of researchers, in particular, developed miniaturized devices, called Geckos,
which are fitted with cheap-to-produce sensors capturing light, temperature,
and the presence of a broad range of gases such as volatile organic compounds
(VOCs). They integrated Geckos with the MeetCues app (Aseniero et al., 2020),
and created a new indoor environmental sensing infrastructure ComFeel
(Constantinides et al., 2020). At the end of each meeting, the MeetCues app
asks each participant: _(a)_ whether the meeting had a clear purpose and
structure, and resulted in a list of actionable points, and _(b)_ whether each
participant felt listened to and was motivated to contribute.
To explore the extent to which air quality alone determined whether a meeting
was perceived to be productive or not, they deployed ComFeel in a corporate
office and gathered data from 29 meetings in different rooms. As one expects,
productive meetings were those in which participants felt safe to contribute:
the probability of a meeting being productive increased by 35% for each
standard deviation increase in psychological safety. Surprisingly, the results
for air quality were dramatic too. The productivity probability increased by
as much as 25% for each standard deviation increase in room pleasantness.
Furthermore, among all the sensors, the air quality one was the most
important: indeed, room pleasantness was achieved through improved temperature
(with a relative contribution of 25%), lighting (30%), and air quality (45%).
These results suggest that, if a meeting takes place in a stuffy conference
room, even if it is run well, people will still struggle to pay attention. To
fix that, one just needs to do a handful of things, from manually or
automatically adjusting ventilation, lighting, temperature, which could
increase a meeting’s productivity by a considerable extent.
## 4\. The three factors in Future Hybrid Meetings
Most innovations in meeting technologies have focused on supporting physical
or virtual meetings independently, and they did so by extensively focusing on
how to support meeting execution. That is why future work should focus on the
two other overlooked aspects (Figure 1): on psychological safety (Section
§4.1) by making meetings useful for all participants, ensuring all
participants give and receive appropriate levels of attention, and enabling
all participants to feel and make others feel comfortable; and on physical
comfort (Section §4.2) by integrating all human senses, and, as such, making
the meeting sensory experience pleasurable. At the same time though, future
work should also consider how these new technologies could inadvertently
become surveillance tools (Section §5).
### 4.1. Rethinking Psychological Safety
The design space of meeting technologies could be well enriched with features
that fully support all three facets of psychological safety (Constantinides et
al., 2020): _(a)_ making the meeting useful and satisfying for all
participants, _(b)_ ensuring all participants give and receive appropriate
levels of attention, and _(c)_ enabling all participants to feel and make
others feel comfortable.
First, to make a meeting useful and satisfying, existing tools already provide
analytics in the form of summaries or real-time feedback (Aseniero et al.,
2020). However, these analytics could be enriched with aspects that are hard
to quantify such as the types of conversations happening in a meeting (Zhou et
al., 2021b), nuanced emotional states (Zhou et al., 2022), or other language
markers that are hidden in conversations (e.g., presence of stress, engagement
in emphatic conversations) (Scepanovic et al., 2020; Robertson et al., 2019;
Zhou et al., 2021a).
Second, as turn-taking and balance in conversation helps to cultivate a safe
environment for contribution (Woolley et al., 2010), future meeting tools need
to ensure that all attendees give and receive the appropriate levels of
attention. For example, some meeting technologies already do provide
indicators about “speaking up” time, and often do so through attention-
depleting notifications. However, future technologies must ensure attention-
enhancing ways of notifying users. As Weiser and Brown argued in their seminar
work _The Coming Age of Calm Technology_ , “the information display should
move easily from the periphery of users’ attention to the centre, and back”
(Weiser and Brown, 1997). Future design elements need to smoothly capture the
user’s attention only when necessary, while calmly remain in the user’s
periphery most of the time. For example, while current meeting tools allow
various modes of communication (e.g., voice, text) and notifications (e.g.,
emojis) in a virtual meeting, personalized or timely delivery of those
notifications could remove unnecessary information overload. Imagine a
scenario during which five attendees need to focus on the speaker, yet one of
them types a question and another raises his/her virtual hand. If these
notifications were to be delivered straight away, they may well overload the
speaker and distract the audience. More work needs to go into identifying the
right moment for delivering a notification, or that for delivering a
notification to the best person in a group, allowing for personalized
delivery.
Third, it is also important for meeting technologies to facilitate an
inclusive environment where all attendees feel comfortable, and can make
others feel comfortable. One way of achieving it is to create social
interactions that scaffold learning, and then use computational methods to
weave these interactions into the fabric of meeting tools. For example, as
demonstrated in massive online classes (MOOCs) (Quintana et al., 2018), future
meeting technologies could coach people skills that foster inclusion and
diversity, and collaboratively help them reflect on their behavior. Another
way of achieving inclusivity is through broadening the design space. Design
elements such as emojis allow, to a great extent, attendees to non-verbally
communicate and express their emotional state; it can be seen as a way of
developing trust or empathy for others. Future meeting technologies could also
borrow concepts from biophilic design (Wilson, 1984) and embrace new types of
visual cues (Qin et al., 2020) (e.g., the use of different symbols, imagery,
and artificial artifacts), thus making additional layers of non-verbal
communication available to users.
### 4.2. Rethinking Physical Comfort
Aligned with the Human-Building Interaction vision (Alavi et al., 2019) (an
emerging area aiming at unifying HCI research in the built environment),
physical comfort goes well beyond air quality, integrating all the human
senses. To begin with, let’s take sight. Could we make the physical
environment visually pleasurable? Recent studies showed that adjusting light
conditions in a room could serve as a stress reducing intervention mechanism,
or even as a biofeedback relaxation training (Ren et al., 2019; Yu et al.,
2018). Moving to hearing. Could we design ubiquitous technologies that
facilitate meetings anywhere and everywhere? Consider a knowledge worker who
attends meetings from a variety of places: from the office to home to a
cafeteria to even an autonomous car (Kun et al., 2020). How could future
technologies support smooth transition between different places, and provide a
pleasurable sound experience? Smart earable devices can be one of the answers
wherein an array of sensors could be embedded in them to control not only what
sound the end user is hearing, but also how that sound is delivered in a
specific physical setting (Kawsar et al., 2018). While addressing sight and
hearing is to some extent easier, smell and touch require future research
efforts. Which odour is more relaxing and energizing during a meeting, and, to
what extent, certain odour could be adapted as a meeting unfolds to facilitate
better execution? How could we stimulate the sense of touch to increase
physical comfort during a meeting? For example, as most of white collar jobs
require employees to spend long hours sitting, previous studies explored the
design of ergonomic chairs that would increase physical comfort levels. While
these studies typically employ various objective methods to capture physical
comfort such as pressure sensors measurement (Zemp et al., 2015) or an array
of sensors including, for example, temperature, gas, illuminance, VOC, CO2
(Zhong et al., 2020), new sensing modalities such as Electromyography (EMG)
could prove useful to fully incorporate the sense of touch in the design space
of these chairs.
## 5\. The dangers of meeting technologies
While these technologies hold the promise of enabling employees to be
productive, report after report has highlighted the outcries of workplace AI-
based solutions being biased and unfair, and lacking transparency and
accountability. During the COVID-19 pandemic, systems were being used to
analyze footage from security cameras in workplace to detect when employees
are not complying with social distancing
rules111https://www.ft.com/content/58bdc9cd-18cc-44f6-bc9b-8ca4ac598fc8; while
there is a handful of good intentions behind such a technology, the very same
technology could be used for tracking employees’ movements, or time away from
desk. As we move towards a future likely ruled by big data and powerful AI
algorithms, important questions arise relating to the psychological impacts of
surveillance, data governance, leadership and organizational culture, and
compliance with ethical and moral concerns. The historian and philosopher
Yuval Noah Harari argued that digital platforms such as those that allow us to
work from remote need to follow three basic
rules222https://www.ft.com/content/f1b30f2c-84aa-4595-84f2-7816796d6841 to
protect us from “digital dictatorships”. First, any data collection on people
should be used to help people rather than to manipulate, control, or harm
them. In the meetings context, this translates into providing analytics that
help employees reflect on their experience rather than causing them to receive
low performance review in the event of a meeting not being executed well.
Second, surveillance must always go both ways. That is, whenever an
organization increases surveillance of individuals, at the same time,
organizational accountability needs to increase. If organizations could
establish processes to monitor their workforce, they may well establish
processes to audit their own actions as well. Third, data should not be
concentrated in a single entity, not least because data monopolies are the
recipe for dictatorship. In the workplace context, this translates into newly
created divisions in a company that oversee data collection and, ideally,
ensure that data about their workers is in the hands of the workers
themselves.
In a global modernized workplace, new meeting tools are likely to be
developed, facilitating and augmenting the experience of hybrid meetings. As a
result of the COVID-19 pandemic, fully remote or hybrid meetings removed any
physical barriers and often contributed to improved work-life balance
(Rudnicka et al., 2020). On the downside, tools for supporting hybrid meetings
may inadvertently becoming surveillance tools, compromising employees’ privacy
(Constantinides and Quercia, 2022). To unpack the AI ethics of workplace
technologies, Constantinides and Quercia (Constantinides and Quercia, 2022)
conducted a crowdsourcing study to understand how employees judge such
technologies and determine which ones are desirable, and why. They considered
16 workplace technologies that track productivity based on diverse inputs
(e.g., tracking audio conversation during virtual meetings, tracking text
messages in collaboration tools), and asked crowd-workers to judge these
scenarios along five moral dimensions. They found that workplace technologies
were judged harshly depending on three aspects (heuristics) participants used
to assess the scenarios. In increasing importance, these aspects reflected
whether a scenario: 1) was not currently supported by existing technologies
(_hard to adopt_); 2) interfered with current ways of working (_intrusive_);
and, more importantly, 3) was not fit for tracking productivity or infringed
on individual rights (_harmful_). Tracking eye movements in virtual meetings
and the visited websites in remote work, despite being possible, were
considered to be “on the way” of getting the job done (they were easy to adopt
but intrusive). By contrast, tracking text messages in collaboration tools
such as Slack was considered to not interfere with work (unobtrusive).
Finally, tracking audio conversations in virtual meetings was considered to be
possible (easy to adopt), and not interfere with work (unobtrusive), yet it
was considered to be harmful, as it entailed tracking not only whether a
meeting took place but also its content, causing a loss of control. The above
heuristics offer a guide on how workplace technologies are likely to be
morally judged. Having a technology that is easy to implement and does not
interfere with work is not necessarily a technology that should be deployed.
_Tracking facial expressions_ (even beyond the nefarious uses - of dubious
effectiveness - of inferring political orientation or sexual preferences (Wang
and Kosinski, 2018)) is possible and could be done in seamless ways (e.g.,
with existing off-the-shelf cameras), yet it would be still considered harmful
and unethical. _Tracking eye movements_ , _task completion_ , or _typing
behavior_ was considered a proxy for focus (harmless) yet intrusive as it
would “get in the way”. _Tracking social media use in remote work_ was
considered not only intrusive but also harmful, as it infringes on privacy
rights.
On a very pragmatic level, there is a handful of reasons as to why
organizations opt for employee surveillance (e.g., maintaining productivity,
monitoring resources used, protecting the organization from legal
liabilities). Critics, however, rightly argue that there is a fine line
between what organizations could be monitoring and what they should be
monitoring. If this line is crossed, it will have consequences on employees,
affecting their well-being, work culture, and productivity (Ball, 2010). If
future meeting tools incorporate any kind of employee monitoring, they need to
preserve individual rights, including that of privacy. New meetings tools need
to also ensure non-discrimination, while promoting inclusivity, and, at the
same time, provide explainable and understandable outputs (e.g., the decision
upon which the system decided that a meeting was successful or not).
Hybrid meetings also create asymmetries of interactions stemming from the
social and cultural contexts. As Saatci et al.(Saatçi et al., 2019) found, the
most challenging asymmetry is the diverse experience between co-located and
remote meeting participants. Remote participants often feel isolated, while
co-located participants dominate the interaction. Differences in language and
accent, cultural behaviors, digital literacy, physical location, and the
boundaries between work and personal life (Rudnicka et al., 2020) also
contribute to interaction asymmetries. To overcome such challenges, new
meetings tools should focus on making meetings more inclusive for everyone by
maximizing psychological safety and optimizing physical comfort.
###### Acknowledgements.
We thank those who actively supported this research at Nokia Bell Labs; in
particular, Sagar Joglekar, Bon Adriel Aseniero, and Jun-Ho Choi for their
active role in the development of the meeting companion app MeetCues; and Mark
Clougherty, Sean Kennedy, Michael Eggleston, and Marcus Weldon for their
guidance during the development. We also thank Nigel Oseland for sharing his
research at the intersection of environmental psychology and workplace
strategy.
## References
* (1)
* Alavi et al. (2019) Hamed S Alavi, Elizabeth F Churchill, Mikael Wiberg, Denis Lalanne, Peter Dalsgaard, Ava Fatah gen Schieck, and Yvonne Rogers. 2019. Introduction to Human-Building Interaction (HBI) Interfacing HCI with Architecture and Urban Design. _ACM Transactions on Computer-Human Interaction (TOCHI)_ 26, 2 (2019). https://doi.org/10.1145/3309714
* Alavi et al. (2017) Hamed S Alavi, Himanshu Verma, Michael Papinutto, and Denis Lalanne. 2017. Comfort: A Coordinate of User Experience in Interactive Built Environments. In _IFIP Conference on Human-Computer Interaction_. Springer, 247–257. https://doi.org/10.1007/978-3-319-67687-6_16
* Allen (2017) Joseph G. Allen. 2017\. Stale Office Air Is Making You Less Productive. _Harvard Business Review_ (2017). https://hbr.org/2017/03/research-stale-office-air-is-making-you-less-productive
* Allen et al. (2016) Joseph G Allen, Piers MacNaughton, Usha Satish, Suresh Santanam, Jose Vallarino, and John D Spengler. 2016. Associations of Cognitive Function Scores with Carbon Dioxide, Ventilation, and Volatile Organic Compound Exposures in Office Workers: A Controlled Exposure Study of Green and Conventional Office Environments. _Environmental Health Perspectives_ 124, 6 (2016), 805–812. https://doi.org/10.1289/ehp.1510037
* Aseniero et al. (2020) Bon Adriel Aseniero, Marios Constantinides, Sagar Joglekar, Ke Zhou, and Daniele Quercia. 2020\. MeetCues: Supporting Online Meetings Experience. In _Proceedings of the IEEE Visualization Conference (VIS)_. IEEE, 236–240. https://doi.org/10.1109/VIS47514.2020.00054
* Axtell (2016) Paul Axtell. 2016\. 6 Reasons to Get Better at Leading Meetings. _Harvard Business Review_ (2016). https://hbr.org/2016/12/6-reasons-to-get-better-at-leading-meetings
* Axtell (2017) Paul Axtell. 2017\. How to Design Meetings Your Team Will Want to Attend. _Harvard Business Review_ (2017). https://hbr.org/2017/04/how-to-design-meetings-your-team-will-want-to-attend
* Axtell (2018) Paul Axtell. 2018\. How to Respond When You’re Put on the Spot in a Meeting. _Harvard Business Review_ (2018). https://hbr.org/2017/12/how-to-save-a-meeting-thats-gotten-tense
* Ball (2010) Kirstie Ball. 2010\. Workplace Surveillance: An Overview. _Labor History_ 51, 1 (2010), 87–106. https://doi.org/10.1080/00236561003654776
* Banerjee et al. (2005) Satanjeev Banerjee, Carolyn Rose, and Alexander I Rudnicky. 2005. The Necessity of a Meeting Recording and Playback System, and the Benefit of Topic–level Annotations to Meeting Browsing. In _IFIP Conference on Human-Computer Interaction_. Springer, 643–656. https://doi.org/10.1007/11555261_52
* Barksdale et al. (2012) Jeremy Barksdale, Kori Inkpen, Mary Czerwinski, Aaron Hoff, Paul Johns, Asta Roseway, and Gina Venolia. 2012. Video Threads: Asynchronous Video Sharing for Temporally Distributed Teams. In _Proceedings of the ACM Conference on Computer Supported Cooperative Work (CSCW)_. ACM, 1101–1104. https://doi.org/10.1145/2145204.2145367
* Chiu et al. (1999) Patrick Chiu, Ashutosh Kapuskar, Sarah Reitmeier, and Lynn Wilcox. 1999. NoteLook: Taking Notes in Meetings with Digital Video and Ink. In _Proceedings of ACM International Conference on Multimedia_. ACM, 149–158. https://doi.org/10.1145/319463.319483
* Choi et al. (2021) Jun-Ho Choi, Marios Constantinides, Sagar Joglekar, and Daniele Quercia. 2021. KAIROS: Talking Heads and Moving Bodies for Successful Meetings. In _Proceedings of the International Workshop on Mobile Computing Systems and Applications (HotMobile)_. 1–7. https://doi.org/10.1145/3446382.3448361
* Choi et al. (2020) Minje Choi, Luca Maria Aiello, Krisztián Zsolt Varga, and Daniele Quercia. 2020. Ten Social Dimensions of Conversations and Relationships. In _Proceedings of The Web Conference (WWW)_. 1514–1525. https://doi.org/10.1145/3366423.3380224
* Cohen (2016) Jordan Cohen. 2016\. Use Subtle Cues to Encourage Better Meetings. _Harvard Business Review_ (2016). https://hbr.org/2016/09/use-subtle-cues-to-encourage-better-meetings
* Constantinides and Quercia (2022) Marios Constantinides and Daniele Quercia. 2022. Good Intentions, Bad Inventions: How Employees Judge Pervasive Technologies in the Workplace. _arXiv_ (2022).
* Constantinides et al. (2020) Marios Constantinides, Sanja Šćepanović, Daniele Quercia, Hongwei Li, Ugo Sassi, and Michael Eggleston. 2020. ComFeel: Productivity is a Matter of the Senses Too. _Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies (IMWUT)_ 4, 4 (2020), 1–21. https://doi.org/10.1145/3432234
* Edmondson (1999) Amy Edmondson. 1999\. Psychological Safety and Learning Behavior in Work Teams. _Administrative Science Quarterly_ 44, 2 (1999), 350–383. https://doi.org/10.2307/2666999
* Garcia et al. (2004) AC Bicharra Garcia, John Kunz, and Martin Fischer. 2004\. Cutting to the Chase: Improving Meeting Effectiveness by Focusing on the Agenda. In _Proceedings of the ACM Conference on Computer Supported Cooperative Work (CSCW)_ , Vol. 6. ACM, 346–349. https://doi.org/10.1145/1031607.1031664
* Grenny (2017) Joseph Grenny. 2017\. How to Save a Meeting That’s Gotten Tense. _Harvard Business Review_ (2017).
* Kawsar et al. (2018) Fahim Kawsar, Chulhong Min, Akhil Mathur, and Allesandro Montanari. 2018. Earables for Personal-scale Behavior Analytics. _IEEE Pervasive Computing_ 17, 3 (2018), 83–89. https://doi.org/10.1109/MPRV.2018.03367740
* Kerns (2015) Ira Kerns. 2015\. _The 6 Post-Event Survey Questions That Will Reveal Your Meeting’s Effectiveness_. https://www.meetingsnet.com/corporate-meetings/6-post-event-survey-questions-will-reveal-your-meeting-s-effectiveness
* Kim and Rudin (2014) Been Kim and Cynthia Rudin. 2014. Learning About Meetings. _Data Mining and Knowledge Discovery_ 28, 5-6 (2014), 1134–1157. https://doi.org/10.1007/s10618-014-0348-z
* Kim et al. (2008) Taemie Kim, Agnes Chang, Lindsey Holland, and Alex Sandy Pentland. 2008. Meeting Mediator: Enhancing Group Collaboration Using Sociometric Feedback. In _Proceedings of the ACM Conference on Computer Supported Cooperative Work and Social Computing (CSCW)_. ACM, 457–466. https://doi.org/10.1145/1460563.1460636
* Kun et al. (2020) Andrew L Kun, Orit Shaer, Raffaella Sadun, Linda Ng Boyle, and John D Lee. 2020. The Future of Work and Play: From Automated Vehicles to Working from Home. _New Future of Work_ (2020). https://par.nsf.gov/biblio/10186623
* Lent (2015) Richard M Lent. 2015\. _Leading Great Meetings: How to Structure Yours for Success_. Meeting for Results.
* McGregor and Tang (2017) Moira McGregor and John C Tang. 2017. More to Meetings: Challenges in Using Speech-based Technology to Support Meetings. In _Proceedings of the ACM Conference on Computer Supported Cooperative Work and Social Computing (CSCW)_. ACM, 2208–2220. https://doi.org/10.1145/2998181.2998335
* Meister (2018) Jeanne C. Meister. 2018\. The #1 Office Perk? Natural Light. _Harvard Business Review_ (2018).
* Niemantsverdriet and Erickson (2017) Karin Niemantsverdriet and Thomas Erickson. 2017. Recurring Meetings: An Experiential Account of Repeating Meetings in a Large Organization. _Proceedings of the ACM on Human-Computer Interaction_ 1, CSCW (2017), 84. https://doi.org/10.1145/3134719
* Park et al. (2020) Sungkyu Park, Marios Constantinides, Luca Maria Aiello, Daniele Quercia, and Paul Van Gent. 2020\. WellBeat: A Framework for Tracking Daily Well-Being Using Smartwatches. _IEEE Internet Computing_ 24, 5 (2020), 10–17. https://doi.org/10.1109/MIC.2020.3017867
* Peleckis and Peleckienė (2015) Kestutis Peleckis and Valentina Peleckienė. 2015. Nonverbal Communication in Business Negotiations and Business Meetings. _International Letters of Social and Humanistic Sciences_ 62 (2015), 62–72. https://www.learntechlib.org/p/176672/
* Qin et al. (2020) Chao Ying Qin, Marios Constantinides, Luca Maria Aiello, and Daniele Quercia. 2020. HeartBees: Visualizing Crowd Affects. In _Proceedings of the IEEE VIS Arts Program (VISAP)_. IEEE, 1–8. https://doi.org/10.1109/VISAP51628.2020.00007
* Quintana et al. (2018) Rebecca M Quintana, Christopher Brooks, Cinzia Villanucci Smothers, Yuanru Tan, Zheng Yao, and Chinmay Kulkarni. 2018. Mentor Academy: Engaging Global Learners in the Creation of Data Science Problems for MOOCs. International Society of the Learning Sciences. https://doi.org/10.22318/cscl2018.1415
* Ren et al. (2019) Xipei Ren, Bin Yu, Yuan Lu, Biyong Zhang, Jun Hu, and Aarnout Brombacher. 2019\. LightSit: An Unobtrusive Health-promoting System for Relaxation and Fitness Microbreaks at Work. _Sensors_ 19, 9 (2019), 2162. https://doi.org/10.3390/s19092162
* Robertson et al. (2019) Alexander Robertson, Luca Maria Aiello, and Daniele Quercia. 2019. The Language of Dialogue is Complex. In _Proceedings of the International AAAI Conference on Web and Social Media (ICWSM)_ , Vol. 13. 428–439. https://ojs.aaai.org/index.php/ICWSM/article/view/3241
* Romano and Nunamaker (2001) Nicholas C Romano and Jay F Nunamaker. 2001. Meeting Analysis: Findings from Research and Practice. In _Proceedings of the Annual Hawaii International Conference on System Sciences_. IEEE, 13–pp. https://doi.org/10.1109/HICSS.2001.926253
* Rudnicka et al. (2020) Anna Rudnicka, Joseph W Newbold, Dave Cook, Marta E Cecchinato, Sandy Gould, and Anna L Cox. 2020\. Eworklife: Developing Effective Strategies for Remote Working During the COVID-19 Pandemic. In _Eworklife: developing effective strategies for remote working during the COVID-19 pandemic_. The New Future of Work Online Symposium.
* Saatçi et al. (2019) Banu Saatçi, Roman Rädle, Sean Rintel, Kenton O’Hara, and Clemens Nylandsted Klokmose. 2019\. Hybrid meetings in the modern workplace: stories of success and failure. In _International Conference on Collaboration and Technology_. Springer, 45–61.
* Scepanovic et al. (2020) Sanja Scepanovic, Enrique Martin-Lopez, Daniele Quercia, and Khan Baykaner. 2020. Extracting Medical Entities From Social Media. In _Proceedings of the ACM Conference on Health, Inference, and Learning (CHIL)_. 170–181. https://doi.org/10.1145/3368555.3384467
* Schwarz (2016) Roger Schwarz. 2016\. 8 Ground Rules for Great Meetings. _Harvard Business Review_ (2016). https://hbr.org/2016/06/8-ground-rules-for-great-meetings
* Tom Y. Chang and Neidell (2016) Tal Gross Tom Y. Chang, Joshua Graff Zivin and Matthew Neidell. 2016. Air Pollution Is Making Office Workers Less Productive. _Harvard Business Review_ (2016). https://hbr.org/2016/09/air-pollution-is-making-office-workers-less-productive
* Tucker et al. (2010) Simon Tucker, Ofer Bergman, Anand Ramamoorthy, and Steve Whittaker. 2010. Catchup: A Useful Application of Time-travel in Meetings. In _Proceedings of ACM Conference on Computer Supported Cooperative Work (CSCW)_. ACM, 99–102. https://doi.org/10.1145/1718918.1718937
* Wang and Kosinski (2018) Yilun Wang and Michal Kosinski. 2018. Deep neural networks are more accurate than humans at detecting sexual orientation from facial images. _Journal of Personality and Social Psychology_ 114, 2 (2018), 246. https://doi.org/10.1037/pspa0000098
* Weiser and Brown (1997) Mark Weiser and John Seely Brown. 1997. The Coming Age of Calm Technology. In _Beyond Calculation_. Springer, 75–85. https://doi.org/10.1007/978-1-4612-0685-9_6
* Wilson (1984) Edward O Wilson. 1984\. _Biophilia_. Harvard University Press. https://doi.org/10.4159/9780674045231
* Woolley et al. (2010) Anita Williams Woolley, Christopher F Chabris, Alex Pentland, Nada Hashmi, and Thomas W Malone. 2010\. Evidence for a Collective Intelligence Factor in the Performance of Human Groups. _Science_ 330, 6004 (2010), 686–688. https://doi.org/10.1126/science.1193147
* Yu et al. (2018) Bin Yu, Jun Hu, Mathias Funk, and Loe Feijs. 2018\. DeLight: Biofeedback Through Ambient Light for Stress Intervention and Relaxation Assistance. _Personal and Ubiquitous Computing_ 22, 4 (2018), 787–805. https://doi.org/10.1007/s00779-018-1141-6
* Zemp et al. (2015) Roland Zemp, William R Taylor, and Silvio Lorenzetti. 2015\. Are Pressure Measurements Effective in the Assessment of Office Chair Comfort/Discomfort? A Review. _Applied Ergonomics_ 48 (2015), 273–282. https://doi.org/10.1016/j.apergo.2014.12.010
* Zhong et al. (2020) Sailin Zhong, Hamed S Alavi, and Denis Lalanne. 2020\. Hilo-wear: Exploring Wearable Interaction with Indoor Air Quality Forecast. In _Proceedings of the ACM CHI Conference Extended Abstracts on Human Factors in Computing Systems (CHI)_. 1–8. https://doi.org/10.1145/3334480.3382813
* Zhou et al. (2021a) Ke Zhou, Luca Maria Aiello, Sanja Scepanovic, Daniele Quercia, and Sara Konrath. 2021a. The Language of Situational Empathy. _Proceedings of the ACM on Human-Computer Interaction_ 5, CSCW1 (2021), 1–19. https://doi.org/10.1145/3449087
* Zhou et al. (2021b) Ke Zhou, Marios Constantinides, Luca Maria Aiello, Sagar Joglekar, and Daniele Quercia. 2021b. The Role of Different Types of Conversations for Meeting Success. _IEEE Pervasive Computing_ 20, 4 (2021), 35–42. https://doi.org/10.1109/MPRV.2021.3115879
* Zhou et al. (2022) Ke Zhou, Marios Constantinides, Sagar Joglekar, and Daniele Quercia. 2022. Predicting Meeting Success With Nuanced Emotions. _IEEE Pervasive Computing_ (2022). https://doi.org/10.1109/MPRV.2022.3145047
|
# End-to-End Anti-Backdoor Learning on Images and Time Series
Yujing Jiang1, Xingjun Ma2, Sarah Monazam Erfani1, Yige Li3, James Bailey1
1Faculty of Engineering and Information Technology, The University of
Melbourne
<EMAIL_ADDRESS>sarah.erfani<EMAIL_ADDRESS>2School of Computer
Science, Fudan University
<EMAIL_ADDRESS>3School of Computer Science and Technology, Xidian
University
<EMAIL_ADDRESS>
###### Abstract
Backdoor attacks present a substantial security concern for deep learning
models, especially those utilized in applications critical to safety and
security. These attacks manipulate model behavior by embedding a hidden
trigger during the training phase, allowing unauthorized control over the
model’s output during inference time. Although numerous defenses exist for
image classification models, there is a conspicuous absence of defenses
tailored for time series data, as well as an end-to-end solution capable of
training clean models on poisoned data. To address this gap, this paper builds
upon Anti-Backdoor Learning (ABL) and introduces an innovative method, End-to-
End Anti-Backdoor Learning (E2ABL), for robust training against backdoor
attacks. Unlike the original ABL, which employs a two-stage training
procedure, E2ABL accomplishes end-to-end training through an additional
classification head linked to the shallow layers of a Deep Neural Network
(DNN). This secondary head actively identifies potential backdoor triggers,
allowing the model to dynamically cleanse these samples and their
corresponding labels during training. Our experiments reveal that E2ABL
significantly improves on existing defenses and is effective against a broad
range of backdoor attacks in both image and time series domains.
## Introduction
Deep learning has achieved remarkable performance in computer vision tasks
such as object detection [1], motion tracking [2], and autonomous driving [3],
as well as time series analysis in fields like finance [4], smart
manufacturing [5], and healthcare [6]. With the increasing deployment of Deep
Neural Networks (DNNs) in real-world applications, their vulnerability to
backdoor attacks has become a significant concern. Backdoor attacks occur
either by poisoning a few training samples with a trigger pattern [7] or by
manipulating the training procedure [8], thereby implanting a backdoor in the
DNN model. The compromised model learns a strong correlation between the
trigger pattern and a chosen backdoor label. At inference time, it predicts
correct labels on clean inputs while exhibiting a systematic bias towards the
backdoor label in the presence of the trigger. This issue is especially
serious as these technologies are being deployed in safety-critical
applications. Therefore, developing defenses against such backdoor attacks is
becoming a critical necessity.
Backdoor attacks generally have two primary objectives: 1) high effectiveness,
i.e., high attack success rate (ASR), and 2) high stealthiness, i.e.,
maintaining a high clean accuracy (CA) while visually undetectable. High
effectiveness allows the attacker to manipulate the model’s prediction in a
more precise manner, while high stealthiness ensures that the attacks cannot
be easily detected by rudimentary filtering or manual inspection. Stealthiness
also involves designing subtle trigger patterns that do not impact the model’s
clean performance (performance on clean data), making detection even more
challenging. Several works [7, 8, 9, 10, 11, 12] have studied backdoor attacks
on both image and time series data.
As more and more DNNs are being trained and employed in different types of
real-world applications, defending against malicious and stealthy backdoor
attacks on different tasks and data modalities has become an imperative task.
This work takes the first attempt to design one single defense method that
could work for two data modalities, i.e., images and time series. The reason
why we chose images and time series is that both modalities are continuous
(unlike discrete texts), their classification tasks are well-studied, and
there exist multiple backdoor attacks for both types of data. However, current
defense methods are mostly tailored for image data and have not been well
studied for time series data. It is thus unclear whether defenses developed
against image backdoor attacks are suitable for time series backdoor attacks.
Anti-Backdoor Learning (ABL) [13] is a robust training method that was
initially introduced to train clean models on poisoned data. It has
demonstrated promising results against a diverse set of backdoor attacks on
image datasets. Nonetheless, ABL has some limitations. One notable limitation
is its two-stage training process. In the first stage, the model undergoes a
training phase for a specified number of epochs, following which a subset of
suspected backdoor samples is isolated. The model then enters a secondary
training phase aimed at “unlearning” these potentially harmful patterns. Each
of these stages demands distinct training objectives, thereby complicating the
process and potentially reducing its efficiency.
In this paper, we propose an End-to-End Anti-Backdoor Learning (E2ABL)
training method that is capable of training a clean model on a poisoned
dataset in an end-to-end manner. Our approach eliminates the need to change
the training objectives or extend training epochs. Specifically, we introduce
a secondary classification head attached to the shallow layers of the DNN
model. This second head traps potential backdoor samples and corrects their
labels. The second head is specifically designed to be sensitive to backdoor
correlations and samples. This ensures backdoor samples are captured in the
shallow layers, safeguarding the primary head and steering the model training
toward a more secure and trustworthy direction.
To summarize, our main contributions are:
* •
We introduce End-to-End Anti-Backdoor Learning (E2ABL), a novel end-to-end
robust training method that can train backdoor-free models on backdoored
datasets. E2ABL works effectively for both image and time series data, and to
the best of our knowledge, is the first backdoor defense method for time
series models.
* •
E2ABL proposes a novel strategy of using a second head to safeguard the
learning of the main network against backdoor attacks. The second head is
designed to learn, capture, and rectify backdoored samples in real-time, and
is trained concurrently with the main head to neutralize the impact of
backdoor attacks.
* •
Through extensive empirical evaluations, we demonstrate that E2ABL can serve
as an effective backdoor defense method against a broad spectrum of backdoor
attacks on both image and time series data. Further, models trained using our
E2ABL consistently outperform those trained by other defense methods,
exhibiting higher clean accuracy (CA) and lower attack success rate (ASR).
## Related Work
In this section, we provide a brief overview of the existing literature
focusing on both backdoor attacks and defenses.
### Backdoor Attack
#### Image Attacks
Backdoor attacks optimize two primary objectives including attack
effectiveness, and stealthiness. These objectives are fulfilled by optimizing
metrics such as the attack success rate and clean accuracy, while also
focusing on the design of increasingly subtle and inconspicuous trigger
patterns. Additionally, efforts are made to minimize the rate at which
training samples are poisoned. The seminal work in this domain, BadNets [7],
laid the foundation for backdoor attacks on Deep Neural Networks (DNNs) by
introducing a simplistic checkerboard pattern affixed to the lower-right
corner of a clean image. In the wake of this pioneering study, subsequent
research has ventured into more sophisticated techniques, such as the
integration of blended backgrounds [14], the inclusion of natural reflections
[8], and the utilization of imperceptible noise [8, 9, 10, 15]. There are also
works utilizing adversarial patterns [16] and sample-wise patterns [17, 18] as
backdoor attack methods. Remarkably, some attacks have even demonstrated the
ability to reverse-engineer training data without requiring access to the
original dataset [19]. Moreover, clean-label attacks, which insert triggers
without altering the actual class labels, have gained attention in recent
research [16, 20, 21, 22, 23]. Many of these methods achieve considerable
attack success rates while contaminating less than 10% of the training
dataset, some being effective at a surprisingly low poisoning rate of 0.1%.
These trends underscore the stealthy and evasive nature of backdoor attacks,
thereby accentuating the urgent need for robust anti-backdoor learning
mechanisms. It is worth noting that the majority of these attacks and
subsequent studies have been primarily focused on image data and image
classification models.
#### Time Series Attacks
The field of backdoor attacks on time series data is still in its nascent
stage. One of the pioneering works in this domain is by [24], which
transformed time series data into 2D images. This transformation allowed them
to apply existing backdoor attack techniques originally designed for image
data. Building on this initial exploration, [11] introduced TimeTrojan, a
specialized backdoor attack tailored for DNN-based time series classifiers.
TimeTrojan utilizes a multi-objective optimization framework, enabling it to
establish strong and stealthy links between the hidden trigger and the target
label, thereby making the attack more effective and less detectable. More
recently, the research landscape has seen the advent of a Generative
Adversarial Network (GAN)-based approach presented by [12]. This method
overlays a unique, sample-specific trigger on each compromised time series
data point. The GAN-based approach not only elevates the level of stealthiness
but also enhances the natural appearance of the poisoned data, further
complicating detection efforts.
In addition to data poisoning-based backdoor attacks, it is worth mentioning
another category of attacks that directly target the model’s parameters [25,
26]. These parameter-based attacks can be executed independently or in
conjunction with data poisoning-based strategies, thereby presenting a multi-
faceted threat landscape. However, the primary focus of this paper remains on
countermeasures against data poisoning-based backdoor attacks. The exploration
of defenses against model parameter manipulation-based backdoor attacks
constitutes an avenue for our future work, given its distinct set of
challenges and implications.
### Backdoor Defenses (On Image Data)
While a large number of defense methods have been proposed to combat backdoor
attacks on image data, there is a noticeable absence of techniques
specifically tailored for time series data. Prominent existing defenses, such
as Mode Connectivity Repair (MCR) [27], Neural Attention Distillation (NAD)
[28], Adversarial Neuron Pruning (ANP) [29], and Reconstructive Neural Pruning
(RNP) [30] are principally engineered to counter the harmful influence of
backdoor triggers in image-based neural networks. These methods largely
neglect the unique characteristics and vulnerabilities inherent to time series
models. Moreover, earlier defense strategies like Fine-Pruning [31] have been
found to be less effective in the face of evolving, more sophisticated
backdoor attacks [8, 32].
In response to these challenges, [13] introduced the concept of Anti-Backdoor
Learning (ABL) where the goal is to train clean models directly on poisoned
data. Their proposed ABL method consists of two distinctive stages. In the
first stage, the target model undergoes initial training for several epochs,
following which a limited number of samples with the lowest loss values are
isolated as backdoor samples. The second stage involves fine-tuning the model
in conjunction with maximizing the model’s loss on the isolated backdoor
samples. By employing different training objectives in the two stages, ABL
diverges from standard training procedures which are mostly end-to-end
training with one single loss.
Despite these advancements in the image domain, the time series domain is
notably under-researched. Particularly, there are no established defense
methods specifically designed to counter backdoor attacks on time series
models, highlighting an urgent gap in the current literature. In this paper,
we advocate the concept of ABL and propose a novel End-to-End Anti-Backdoor
Learning (E2ABL) method that works for both images and time series. The second
head attached to the main network in E2ABL serves as an innovative mechanism
that is capable of real-time identification, capture, and rectification of
backdoor samples, thereby protecting the integrity of the learning process
conducted by the main network.
## End-to-End Anti-Backdoor Learning
This section presents our innovative E2ABL method, which features two key
advancements: a dual-head model architecture and a true class recovery
mechanism. We also outline the threat model, formulation, and motivation for
E2ABL.
### Threat Model
In this study, we focus on classification tasks involving both image and time
series data. The methodology could potentially be extended to other types of
data and applications, such as natural language processing or anomaly
detection. We adopt a classic data poisoning-based threat model where the
adversary can poison the training data by injecting backdoor trigger patterns
into a few clean samples. The backdoor-poisoned dataset is then used by the
defender to train a target DNN model. The defender has full control over the
training process but has no prior knowledge of the poisoning statistics,
including the existence of an attack, the number of poisoned samples, the
trigger pattern(s), or the backdoor target label. The defender’s objective is
to train a clean, backdoor-free model from a potentially poisoned dataset,
aiming to attain a clean performance equivalent to models trained on clean
data. This scenario embodies a robust training environment where backdoor
mitigation or elimination strategies can still be applied effectively, even
for those devised in different settings.
### Problem Formulation
Consider a benign training set comprised of $N$ independently and identically
distributed samples, denoted as
${\mathcal{D}}=\\{({\bm{x}}_{i},y_{i})\\}_{i=1}^{N}$. In this dataset, each
${\bm{x}}_{i}$ represents a single training sample, and $y_{i}$ is its
corresponding ground truth label. A classification model represented as
$f_{\theta}$, learns the function
$f_{\theta}:{\mathcal{X}}\rightarrow{\mathcal{Y}}$, which maps the input space
to the label space. This learning process is usually achieved by minimizing
the empirical error, as shown below:
$\mathcal{L}=\mathbb{E}_{({\bm{x}},y)\sim{\mathcal{D}}}[\ell(f_{\theta}({\bm{x}}),y)],$
(1)
where $\ell(\cdot)$ denotes the loss function, such as the widely used cross-
entropy loss. A backdoor adversary manipulates a portion of the benign dataset
${\mathcal{D}}$, creating a backdoor-poisoned subset ${\mathcal{D}}_{p}$,
while leaving the remaining dataset ${\mathcal{D}}_{c}$ intact. This forms a
compromised dataset denoted as
${\mathcal{D}}^{\prime}={\mathcal{D}}_{p}\cup{\mathcal{D}}_{c}$. The poisoned
samples are created through the operation
${\bm{x}}_{i}^{\prime}={\bm{x}}_{i}\odot{\bm{p}}_{i}$, where ${\bm{p}}_{i}$
denotes a backdoor trigger pattern, which can either be specific to each
sample or consistent across samples (in this case,
${\bm{p}}_{1}={\bm{p}}_{2}=\cdots={\bm{p}}_{|{\mathcal{D}}_{p}|}$). The
operator $\odot$ denotes an element-wise modification process, such as
addition, subtraction, or replacement. Typically, all compromised samples
within ${\mathcal{D}}_{p}$ share a common backdoor target label $y^{\prime}$.
Training a model on the (potentially) manipulated dataset
${\mathcal{D}}^{\prime}$ can be considered as a dual-task learning problem
involving: 1) a “clean task” focusing on the clean subset of data
${\mathcal{D}}_{c}$, and 2) a “backdoor task” concentrating on the poisoned
subset of data ${\mathcal{D}}_{p}$. In standard (unsecured) training, the
model is trained on both the clean and poisoned data by minimizing the
following empirical error:
${\mathcal{L}}=\underbrace{{\mathbb{E}}_{({\bm{x}},y)\sim{\mathcal{D}}_{c}}[\ell(f_{\theta}({\bm{x}}),y)]}_{\textup{clean
task}}+\underbrace{{\mathbb{E}}_{({\bm{x}}^{\prime},y^{\prime})\sim{\mathcal{D}}_{p}}[\ell(f_{\theta}({\bm{x}}^{\prime}),y^{\prime})]}_{\textup{backdoor
task}}.$ (2)
The outcome of the above optimization is a backdoored model, denoted as
$f_{\theta}^{\prime}$, which consistently outputs the backdoor target label
whenever the trigger pattern appears, i.e.,
$f_{\theta}^{\prime}({\bm{x}}\odot{\bm{p}})=y^{\prime}$.
To inhibit the learning of backdoor samples, ABL employs its first stage to
segregate clean samples into $\hat{{\mathcal{D}}}_{c}$ and possibly poisoned
samples into $\hat{{\mathcal{D}}}_{p}$ . It then seeks to minimize the
model’s error on $\hat{{\mathcal{D}}}_{c}$ while maximizing its error on
$\hat{{\mathcal{D}}}_{p}$ in its second stage by minimizing the following
objective:
${\mathcal{L}}=\underbrace{{\mathbb{E}}_{({\bm{x}},y)\sim{\hat{\mathcal{D}}}_{c}}[\ell(f_{\theta}({\bm{x}}),y)]}_{\textup{clean
training}}-\underbrace{{\mathbb{E}}_{({\bm{x}}^{\prime},y^{\prime})\sim{\hat{\mathcal{D}}}_{p}}[\ell(f_{\theta}({\bm{x}}^{\prime}),y^{\prime})]}_{\textup{backdoor
unlearning}}.$ (3)
Notably, the opposite optimization direction on $\hat{{\mathcal{D}}}_{p}$ can
help unlearn the backdoor trigger from the model. Following ABL, we next
introduce our E2ABL method, which leverages an enhanced optimization objective
with the integration of a second classification head.
### Proposed E2ABL Method
##### Motivation
As discovered in ABL [13], a strong correlation exists between the trigger
pattern and the backdoor label. The more potent the attack, the stronger the
correlation becomes, which in turn allows the backdoor samples to be learned
more quickly by the model. This indicates that the backdoor task, as defined
in Equation (2), is generally less complex than the clean task. This suggests
that a simpler task could be efficiently learned by either a smaller network
or the shallow layers of a deeper network. Such insight motivates us to add a
second classification head to the shallow layers of the target model,
specifically designed to learn and trap the backdoor samples. The placement of
the second head in a ResNet-34 model [33] is illustrated in Figure 1.
Leveraging the second head, our E2ABL methodology trains a model by minimizing
the following empirical error:
$\displaystyle{\mathcal{L}}$
$\displaystyle=\underbrace{{\mathbb{E}}_{({\bm{x}},y)\sim\hat{{\mathcal{D}}}_{c}}[\ell(h_{1}({\bm{x}}),y)]}_{\textup{main
head: clean
leaning}}+\underbrace{{\mathbb{E}}_{({\bm{x}}^{\prime},y^{\prime})\sim\hat{{\mathcal{D}}}_{p}}[\ell(h_{2}({\bm{x}}^{\prime}),y^{\prime})]}_{\textup{second
head: backdoor learning}}$
$\displaystyle+\underbrace{{\mathbb{E}}_{({\bm{x}}^{\prime},y_{\textup{rectified}})\sim{{\mathcal{D}}}^{*}}[\ell(h_{1}({\bm{x}}^{\prime}),y_{\textup{rectified}})]}_{\textup{main
head: backdoor recovery}},$ (4)
where $h_{1}(\cdot)$ and $h_{2}(\cdot)$ denote the primary and second heads
of the network $f_{\theta}$ respectively. The detected clean and poisoned
samples are denoted as $\hat{{\mathcal{D}}}_{c}$ and
$\hat{{\mathcal{D}}}_{p}$ (maintaining
$\hat{{\mathcal{D}}}_{c}\cap\hat{{\mathcal{D}}}_{p}=\emptyset$ and
${\mathcal{D}}^{\prime}=\hat{{\mathcal{D}}}_{c}\cup\hat{{\mathcal{D}}}_{p}$ ).
And, we use ${\mathcal{D}}^{*}$ to denote samples from the detected poison
set $\hat{{\mathcal{D}}}_{p}$ with corrected class labels.
In our design, the second head plays a crucial role in enhancing the
robustness of the training process. But it will be removed when training is
completed, i.e., only the network and the main head will be used as the final
model ($f=h_{1}$). A noteworthy distinction from the ABL model, as outlined in
Equation (3), is that the second head’s objective is to minimize the error on
$\hat{{\mathcal{D}}}_{p}$ (as opposed to maximizing it). This is to
effectively detect and trap backdoor samples to the second head. Moreover,
E2ABL utlizes a specialized detection and recovery strategy for both
$\hat{{\mathcal{D}}}_{c}$ and $\hat{{\mathcal{D}}}_{p}$ . This involves a
method premised on the drop rate of training loss for detecting backdoor
samples and the subsequent recovery of their true classes. The specifics of
this approach will be described in subsequent sections.
Figure 1: The second classification head attached to ResNet-34.
#### Backdoor Sample Detection
Through empirical observation, we note that several convolutional channels
within the shallow layers are closely tied to the backdoor trigger. These
channels consistently produce specific features for nearly every backdoored
input, which is so even for dynamic backdoor attacks that utilize sample-wise
trigger patterns. The deep layers of the model will amplify these features,
prompting the model to predict the backdoor class. More significantly, due to
their short-cut nature [34], these trigger-related features can be rapidly and
adequately learned even with very shallow networks [35, 36]. In light of this,
E2ABL utilizes an additional classification head to capture the backdoor
features at the shallow layers and trap those features to safeguard the main
head during the training procedure.
Specifically, the second head is designed to learn backdoor features at
several shallow layers, constituting two new convolutional layers and one
fully connected (FC) layer, as depicted in Figure 1. The final output of this
second head aligns with that of the main head, producing class probabilities.
The pathway from the input to the output of the second head operates as a
shallow model, adept at learning the backdoor features. Prior to training the
main network or executing backdoor sample detection, the second head first
undergoes a self-training phase (lasting for only a few epochs) on the entire
dataset ${\mathcal{D}}^{\prime}$ , as a warm-up. Subsequently, it partitions
all samples into two subsets: $\hat{{\mathcal{D}}}_{c}$ and
$\hat{{\mathcal{D}}}_{p}$ , based on the loss reduction during the warm-up
phase. Samples that exhibit the most precipitous decline in loss are allocated
to the poison subset, $\hat{{\mathcal{D}}}_{p}$ . The metric used to
bifurcate the training data is defined as follows:
$\Delta\ell=\sum_{i=2}^{n}\
\frac{\mathcal{L}_{i}-\mathcal{L}_{i-1}}{(i-1)^{2}},$ (5)
where $\mathcal{L}_{i}$ is the historical loss value in the $i^{th}$ epoch.
Following the poisoning rate (less than 20%) assumption made in ABL, we
segregate the top 20% of training samples (more discussions on this percentage
are given in the ablation studies section), exhibiting the most significant
loss drops $\Delta\ell$, into $\hat{{\mathcal{D}}}_{p}$ , while the remainder
is retained in $\hat{{\mathcal{D}}}_{c}$ . These two subsets form the basis
for training the main and second heads as detailed in Equation (Motivation).
This detection process is performed at the conclusion of each training epoch,
subsequent to the initial warm-up phase.
Our backdoor sample detection strategy as described above is notably simpler
than the loss-restricted training and filtering strategy employed in ABL.
Unlike ABL which strives to accurately identify the backdoor samples at this
stage, our method partitions the dataset into two broad subsets. While it is
likely that the detected poison subset $\hat{{\mathcal{D}}}_{p}$ will
encompass the majority of the backdoor samples (as some clean “easy-to-learn”
samples could also have exceptionally high loss drop, as demonstrated in
[13]), we cannot fully separate the backdoor samples from the rest at this
stage. Following this, we will introduce the second key technique of E2ABL
that makes it work more effectively than ABL.
#### True Class Recovery
This operation purifies the labels of certain samples in the detected poison
subset $\hat{{\mathcal{D}}}_{p}$ and incorporates these corrected samples
into an additional subset, ${\mathcal{D}}^{*}$ . This additional subset is
then used to train the main head in conjunction with
$\hat{{\mathcal{D}}}_{c}$ . Backdoor attacks are commonly tied to a specific
label deliberately chosen by the adversary. This particular label is referred
to as the backdoor label (or class). Intuitively, if we manage to successfully
identify and modify this backdoor label, we could effectively break the
correlation between the trigger pattern and the backdoor label, thereby
mitigating the attack’s impact. Furthermore, through empirical experiments, we
discovered that the true label of a backdoor sample can be recovered using the
output of the main head. As the main head remains unaffected by the attack
during training, it allows us to recover the authentic identity of the sample,
ensuring the model learns the correct information.
Intuitively, samples in $\hat{{\mathcal{D}}}_{p}$ with the largest loss drop,
such as the top 1% of all training samples, are most likely to be backdoor
samples. We take their predominant label prediction (the one with the highest
softmax value) from the second head as the backdoor label. This detection
strategy is built upon two assumptions: 1) backdoor samples have the most
significant loss drop, and 2) the second head is designed to be highly skilled
in learning the backdoor. Following this phase, samples with the most notable
loss drop are relabeled based on the main head’s prediction and incorporated
into the ${\mathcal{D}}^{*}$ .
The entire training procedure of E2ABL is described in Algorithm 1. Throughout
the training process, the second head functions as a backdoor supervisor,
segregating the input samples into clean and backdoor subsets, identifying the
most suspicious samples within the backdoor subset and correcting their
labels, and discovering samples from the backdoor subset that are not
particularly suspicious (those bearing a non-backdoor label). These operations
are all contingent on the predictions made by the second head. In summary,
both heads are concurrently trained on the corresponding subsets defined in
Equation (Motivation). These subsets are dynamically updated at the end of
each epoch, after the warm-up of the second head.
Algorithm 1 Training Procedure of E2ABL
Input: ${\mathcal{D}}^{\prime}$ : backdoor-poisoned dataset; $h_{1}(\cdot)$ ,
$h_{2}(\cdot)$ : the main and second head; $\ell_{\text{CE}}(\cdot)$ : cross-
entropy loss; $\hat{{\mathcal{D}}}_{c}$ : detected clean subset;
$\hat{{\mathcal{D}}}_{p}$ : detected poison subset; ${\mathcal{D}}^{*}$ :
samples with corrected labels.
Output: $h_{1}(\cdot)$
Hyper-parameters: $T_{\text{warm\\_start}}$ : warm start epochs for the second
head; $T_{\text{training}}$: total training epochs; $\gamma$: clean percentage
of samples for detection; $y_{\textup{rectified}}$: the non-backdoor class
with the maximum probability.
1: for $i$ in [1, …,
$T_{\text{warm\\_start}}$
] do
2: # Warm up the second head
3:
$h_{2}\leftarrow\operatorname*{arg\,min}_{h_{2}}\mathbb{E}_{({\bm{x}},y)\sim{\mathcal{D}}^{\prime}}[\ell_{\text{CE}}(h_{2}({\bm{x}}),y)]$
4: end for
5: for $i$ in [1, …, $T_{\text{training}}$] do
6: # Detect backdoor samples and fix their labels
7:
$\hat{{\mathcal{D}}}_{c}$
$\leftarrow$ for $\gamma_{c}\%$ low loss drop samples in
${\mathcal{D}}^{\prime}$
w.r.t.
$h_{2}(\cdot)$
8:
$\hat{{\mathcal{D}}}_{p}$
$\leftarrow$ for $\gamma_{p}\%$ high loss drop samples in
${\mathcal{D}}^{\prime}$
w.r.t.
$h_{2}(\cdot)$
9:
${\mathcal{D}}^{*}$
$\leftarrow(\scalebox{0.85}{${\mathcal{D}}^{\prime}\setminus\hat{{\mathcal{D}}}_{c}$})$
with corrected labels $y_{\textup{rectified}}$
10: # Update the two heads
11:
$h_{1}\leftarrow\operatorname*{arg\,min}_{h_{1}}\mathbb{E}_{({\bm{x}},y)\sim\hat{{\mathcal{D}}}_{c}}[\ell_{\text{CE}}(h_{1}({\bm{x}}),y)]$
12:
$h_{1}\leftarrow\operatorname*{arg\,min}_{h_{1}}\mathbb{E}_{({\bm{x}}^{\prime},y_{\textup{rectified}})\sim{\mathcal{D}}^{*}}[\ell_{\text{CE}}(h_{1}({\bm{x}}^{\prime}),y_{\textup{rectified}})]$
13:
$h_{2}\leftarrow\operatorname*{arg\,min}_{h_{2}}\mathbb{E}_{({\bm{x}}^{\prime},y^{\prime})\sim\hat{{\mathcal{D}}}_{p}}[\ell_{\text{CE}}(h_{2}({\bm{x}}^{\prime}),y^{\prime}]$
14: end for
15: return $h_{1}$
## Experiments
TABLE I: The attack success rate (ASR) (lower is better) and the clean
accuracy (CA) (higher is better) of 4 backdoor defense methods against state-
of-the-art backdoor attacks on both image and time series datasets. ‘None’
means no attack.
Dataset | Attack | No Defense | FP | NAD | ABL | E2ABL (Ours)
---|---|---|---|---|---|---
ASR | CA | ASR | CA | ASR | CA | ASR | CA | ASR | CA
CIFAR-10 | None | N/A | 89.32% | N/A | 86.07% | N/A | 87.43% | N/A | 88.04% | N/A | 89.39%
BadNets | 100.0% | 87.51% | 99.87% | 82.90% | 3.48% | 84.11% | 3.18% | 86.44% | 0.17% | 87.96%
Blend | 100.0% | 85.64% | 86.40% | 82.16% | 4.97% | 83.11% | 16.85% | 84.93% | 8.95% | 85.21%
Trojan | 100.0% | 88.77% | 65.17% | 82.46% | 16.43% | 76.59% | 3.45% | 87.38% | 1.87% | 88.26%
Dynamic | 100.0% | 86.40% | 87.63% | 82.48% | 31.59% | 73.14% | 18.83% | 85.93% | 12.15% | 86.22%
CL | 99.81% | 84.11% | 51.94% | 82.16% | 14.95% | 81.14% | 0.00% | 89.05% | 0.18% | 89.11%
SIG | 99.45% | 84.58% | 74.81% | 83.04% | 2.37% | 82.18% | 0.08% | 88.44% | 0.25% | 88.92%
LBA | 99.02% | 82.89% | 56.72% | 81.19% | 10.07% | 78.28% | 0.12% | 81.26% | 0.03% | 82.34%
CBA | 89.14% | 85.71% | 75.94% | 81.32% | 34.94% | 81.12% | 29.28% | 84.75% | 25.64% | 85.27%
DFST | 99.55% | 84.92% | 78.47% | 81.67% | 35.01% | 79.39% | 5.47% | 81.14% | 3.21% | 81.95%
Average | 98.55% | 85.61% | 75.22% | 82.15% | 17.09% | 79.90% | 8.58% | 85.48% | 5.83% | 86.14%
GTSRB | None | N/A | 97.91% | N/A | 90.48% | N/A | 95.64% | N/A | 96.78% | N/A | 97.95%
BadNets | 100.0% | 97.50% | 99.40% | 88.12% | 0.22% | 89.62% | 0.05% | 96.42% | 0.09% | 96.89%
Blend | 100.0% | 96.12% | 99.18% | 87.34% | 7.54% | 93.16% | 25.81% | 93.27% | 12.18% | 93.95%
Trojan | 99.84% | 96.74% | 93.41% | 85.72% | 0.46% | 90.55% | 0.43% | 95.24% | 0.27% | 95.68%
Dynamic | 100.0% | 97.13% | 99.82% | 85.16% | 69.64% | 79.15% | 6.48% | 95.87% | 5.69% | 96.23%
SIG | 96.58% | 97.02% | 81.04% | 86.43% | 4.97% | 90.42% | 5.45% | 96.41% | 4.78% | 96.79%
Average | 99.28% | 96.90% | 94.57% | 86.55% | 16.57% | 88.58% | 7.64% | 95.44% | 4.60% | 95.91%
ArabicDigits | None | N/A | 86.24% | N/A | 82.15% | N/A | 83.44% | N/A | 84.95% | N/A | 86.10%
TT-FGSM | 83.43% | 72.27% | 21.14% | 62.78% | 7.63% | 69.48% | 0.04% | 83.56% | 0.13% | 84.32%
TT-DE | 96.06% | 69.12% | 42.83% | 60.46% | 26.82% | 68.17% | 4.16% | 81.83% | 2.54% | 82.85%
TSBA-B | 97.70% | 83.49% | 63.22% | 62.15% | 57.63% | 71.27% | 24.11% | 79.19% | 16.89% | 81.52%
Average | 92.40% | 74.96% | 42.40% | 61.80% | 30.69% | 69.64% | 9.44% | 81.53% | 6.52% | 82.90%
ECG5000 | None | N/A | 99.60% | N/A | 95.50% | N/A | 96.90% | N/A | 97.40% | N/A | 99.40%
TT-FGSM | 76.10% | 88.20% | 20.00% | 64.00% | 8.10% | 72.60% | 0.00% | 92.90% | 0.00% | 96.40%
TT-DE | 98.20% | 86.40% | 29.40% | 63.10% | 18.20% | 70.40% | 1.90% | 92.60% | 0.60% | 96.00%
TSBA-B | 98.70% | 98.10% | 58.70% | 63.50% | 45.20% | 74.60% | 10.80% | 91.70% | 8.60% | 94.80%
Average | 91.00% | 90.90% | 36.03% | 63.53% | 23.83% | 72.53% | 4.23% | 92.40% | 3.07% | 95.73%
UWave | None | N/A | 92.47% | N/A | 86.43% | N/A | 88.96% | N/A | 90.17% | N/A | 92.54%
TT-FGSM | 87.15% | 81.10% | 14.49% | 72.37% | 5.62% | 76.14% | 0.37% | 88.72% | 0.69% | 91.55%
TT-DE | 96.64% | 78.12% | 28.41% | 71.68% | 11.27% | 74.40% | 3.32% | 86.62% | 1.73% | 91.15%
TSBA-B | 94.13% | 89.67% | 54.73% | 73.69% | 48.49% | 77.13% | 15.34% | 84.76% | 14.64% | 89.27%
Average | 92.64% | 82.96% | 32.54% | 72.58% | 21.79% | 75.89% | 6.34% | 86.70% | 5.69% | 90.66%
### Attack Configurations
Our analysis encompasses 9 backdoor attacks on image datasets, including
CIFAR-10 [37] and GTSRB [38] datasets. The evaluated attacks consist of 4
classic backdoor attacks, including BadNets [7], Blend [14], Trojan [19], and
Dynamic [17]; 2 clean-label backdoor attacks, including Clean-Label attack
(CL) [21] and Sinusoidal signal attack (SIG) [39]; and 3 feature-space
backdoor attacks, including Latent Backdoor Attack (LBA) [32], Composite
Backdoor Attack (CBA) [40], and DFST [10]. Furthermore, we tested our E2ABL
against 3 time series backdoor attacks, including TimeTrojan-FGSM [11],
TimeTrojan-DE [11], and TSBA-B [12], using 3 multivariate signal datasets.
These datasets include ArabicDigits, ECG5000, and UWave, all sourced from the
MTS Archive [41]. All the tested image and time series-based attacks adopt a
consistent poison rate of 10%, with all other training parameters being set as
per their original configurations. It is worth noting that several attacks,
including CL, LBA, CBA, and DFST, could not be reproduced on certain image
datasets such as GTSRB. Consequently, those experiments have been excluded
from our reported results.
### Defense and Training Details
We assess our E2ABL in comparison with 3 representative defense methods: Fine-
pruning (FP) [31], Neural Attention Distillation (NAD) [28], and Anti-backdoor
Learning (ABL) [13]. Given that ResNet has been demonstrated to be an
effective baseline method for time series classification, as supported by [42,
12], we utilize ResNet-34 as the backbone model for all poisoned datasets in
each attack scenario, on both image and time series data.
For FP, we prune the final convolutional layer of each model until the CA
falls below the minimum CA under the no-defense condition. In terms of NAD, we
follow the standard distillation procedure, which necessitates fine-tuning the
backdoored student network for 10 epochs with a 5% clean data subset.
Regarding the original ABL defense, we train the model for 20 epochs, applying
a learning rate of 0.1 on CIFAR-10 and 0.01 on GTSRB prior to the turning
epoch. We followed the ABL work and set the backdoor isolation and unlearning
rate ($\gamma_{p}$) as 1%. After successfully segregating 1% of the potential
backdoor samples, we proceed to fine-tune the model for 60 additional epochs
with all training samples to restore the model’s clean accuracy. We then carry
out backdoor unlearning with the 1% isolated backdoor samples, applying a
learning rate of 0.0001 for 20 epochs.
As for our E2ABL defense, we follow the training procedure outlined in
Algorithm 1. We initially trained the second head on the full training dataset
for 2 epochs with a learning rate of 0.1 and monitored the loss reduction for
each training sample. The subsets defined in Equation (Motivation) are
dynamically updated based on the weighted sum of the total loss reductions
specified in Equation (5), with the loss drop threshold $\gamma_{c}$ set as 80
and $\gamma_{p}$ set as 1. The E2ABL model undergoes a total of 60 epochs of
training, inclusive of the 2 warm-up epochs, with a learning rate of 0.01 for
the main head and 0.005 for the second head.
### Effectiveness of E2ABL Defense
#### Comparison to Existing Defenses
Table I shows that our E2ABL achieves the best clean accuracy among all
backdoor defense techniques while maintaining an exceptionally low Attack
Success Rate (ASR) against state-of-the-art backdoor attacks. In the case of
clean-label attacks, E2ABL fully recovers clean accuracy while maintaining an
almost negligible ASR. On average, our E2ABL outperforms the original ABL by a
margin of 2.76% in terms of lower ASR and 0.66% in higher Clean Accuracy (CA)
across all 9 experiments on the image datasets. For time series, compared to
the original ABL, our E2ABL achieves a lower ASR by 1.58% and a higher CA by
2.89% on average, spanning all time series experiments.
Previous research has shown that implementing backdoor defense methods on
clean training datasets can negatively impact the clean accuracy of the final
model [13]. However, compared to existing defenses, our E2ABL achieved an even
higher CA when the training data is completely clean, as shown in the ‘None’
rows for each dataset in Table I. Interestingly, on certain datasets such as
CIFAR-10 and UWave, the models trained by our E2ABL exhibit higher CAs than
those trained using standard training procedures. For instance, when the
training data is clean, a standard model achieves a CA of 89.32%, but a model
trained with our E2ABL reaches a CA of 89.39%. This phenomenon may be
attributed to the exclusion of those “easy-to-learn” samples, which could have
a negative impact on the model’s overall performance. This underlines a unique
advantage of our defense method in real-world scenarios where the presence of
a backdoor attack remains uncertain. Note that our defense requires no prior
knowledge of the attack, whereas it uses an additional backdoor detection head
(i.e., the second head) to determine whether there are any backdoor samples in
the training set and recover the potential backdoor label.
#### Effectiveness with Different Subset Sizes
We also study the correlation between the isolation size (the ratio between
the isolated clean subset and the full set, $\gamma_{c}$ ) and the
performance of our E2ABL. We test E2ABL with the clean subset size varying
from 50% to 90% against all the nine state-of-the-art backdoor attacks on the
CIFAR-10 dataset and show their ASR and CA results in Figure 2. It shows that,
with a higher clean subset size, more training samples will be used to train
the main head, resulting in moderately higher CA. However, achieving a perfect
separation between backdoor and clean samples is not feasible, increasing the
clean subset size will reduce the precision of clean sample detection so that
poisoned samples have more chance to be mixed into the clean subset, causing a
significant rise in ASR (indicates worse performance). We also find that even
with a few ($<5$) poisoned samples mixed into the clean subset, a noticeable
increase in ASR will be observed in the final model. This also indicates the
importance of our proposed strategy that first performs a less accurate but
more secure clean-vs-poison isolation and then gradually refines the samples
in the poison subset $\hat{{\mathcal{D}}}_{p}$ to improve the clean
performance.
Figure 2: Performance of E2ABL regarding ASR and CA with different isolation
rate (
$\gamma_{c}$
) on CIFAR-10.
### Detection and Recovery Performances
During the E2ABL procedure, it can distinguish between clean and backdoor
samples with high precision. Samples present in $\hat{{\mathcal{D}}}_{c}$
were classified as clean samples. However, not all samples in
$\hat{{\mathcal{D}}}_{p}$ should be considered as backdoor samples.
Conceptually, we could designate the 1% of samples in
$\hat{{\mathcal{D}}}_{p}$ that have the largest loss drop rate $\Delta\ell$ as
the “most probable” backdoor samples. In this subsection, we delve into the
precision of these two subsets to offer further insights into their detection
performance.
TABLE II: Performance of E2ABL in differentiating between clean and backdoor
samples, and restoring true labels. The second column shows the precision of
identifying backdoor-infected samples within the subset characterized by the
top 1% of loss reductions. The results are computed at the 10th epoch,
following the warm start of the second head.
Attack | | Precision
---
Clean $\hat{{\mathcal{D}}}_{c}$
| Precision
---
Backdoor${}_{\text{top}1\%}$
| Recall
---
Backdoor $\hat{{\mathcal{D}}}_{p}$
| Precision
---
True Label
(1) BadNets | 100% | 99.8% | 100% | 83.8%
(2) Blend | 99.0% | 98.4% | 99.6% | 76.2%
(3) Trojan | 100% | 99.6% | 100% | 72.0%
(4) Dynamic | 98.4% | 95.2% | 94.4% | 88.6%
(5) CL | 100% | 100% | 100% | 0%
(6) SIG | 100% | 99.6% | 100% | 0%
(7) LBA | 100% | 99.8% | 100% | 68.4%
(8) CBA | 97.2% | 96.4% | 91.6% | 76.8%
(9) DFST | 98.8% | 98.6% | 99.8% | 63.0%
#### Precision of the Detected Clean and Backdoor Samples
As demonstrated in Table II, our E2ABL exhibits high precision in separating
clean and backdoor samples based on the loss drops captured by the second
head. However, it is worth noting that stronger backdoor attacks, such as
Dynamic and CBA, result in lower detection precision. This means that a small
fraction of backdoor samples may not fall within the 1% of samples in
$\hat{{\mathcal{D}}}_{p}$ that have the most substantial loss drops.
Furthermore, some are not even captured in the $\hat{{\mathcal{D}}}_{p}$ set
(which contains top 20% samples based on loss drops), as illustrated by the
third column in Table II. This phenomenon explains the higher (worse) ASR
observed in some of the experiments and serves as an area where E2ABL could be
further refined. Generally speaking, our method highlights an effective
technique for segregating backdoor samples, thereby allowing E2ABL to train
clean models on potentially compromised training data.
#### Precision of the Recovered True Class Labels
Our E2ABL method introduces a dynamic recovery of the true class labels of
poisoned training samples during the training process to recover certain
poisoned samples in $\hat{{\mathcal{D}}}_{p}$ to enhance the clean accuracy
of the main head. As shown in the last column of Table II, these recovered
labels present high precision for dirty-label and feature-based attacks. Note
that as long as the sample is not a backdoor sample, its loss value with
respect to the backdoor label at the second head will be notably high, as the
second head does not have the capacity to sufficiently learn the clean task.
In the case of clean-label attacks (such as CL and SIG), the backdoor-poisoned
samples at the second head will point to the adversary-chosen target.
Accordingly, the poisoned samples will be corrected to different (although
might be incorrect) class labels other than the backdoor target. This disrupts
the correlation between the trigger pattern and the backdoor label, making it
more challenging for the main head to learn and recognize the backdoor.
## Ablation Studies
TABLE III: Ablation studies of E2ABL on CIFAR-10. The full names of the
attacks are in Table II. The $\Delta$CA and $\Delta$ASR are calculated based
on the E2ABL results in Table I.
| Unlearn Top 1% | With No Control | Use Two Models
---|---|---|---
| $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA
(1) | +1.01% | -3.54% | +7.55% | -0.62% | +0.93% | +0.13%
(2) | +1.64% | -5.41% | +14.64% | -4.13% | +2.16% | +0.64%
(3) | +0.83% | -2.55% | +9.62% | -1.59% | +1.14% | +0.20%
(4) | +0.96% | -2.94% | +12.25% | -1.41% | +0.85% | +0.35%
(5) | +0.56% | -0.94% | +2.36% | -2.15% | +1.02% | -0.06%
(6) | +1.12% | -1.15% | +4.17% | +0.05% | +0.81% | -0.81%
(7) | +0.98% | -5.42% | +7.42% | +0.11% | +1.23% | -0.67%
(8) | +0.51% | -4.73% | +6.39% | -1.04% | +0.79% | +0.13%
(9) | +1.26% | -5.10% | +8.59% | -2.98% | +1.34% | +0.29%
avg. | +0.99% | -3.53% | +8.11% | -1.53% | +1.14% | +0.02%
### E2ABL Without Label Correction
To achieve higher CA and lower ASR, our E2ABL attempts to rectify the labels
of the detected backdoor samples, specifically targeting the 1% of samples in
$\hat{\mathcal{D}}_{p}$ with the largest loss drops. These corrected samples
are subsequently included in the subset $\mathcal{D}^{*}$, enabling the main
head to learn the clean task from them. To explore alternative approaches to
managing the detected backdoor samples, we conducted two experiments without
using the corrected samples. First, instead of training the main head with
$\mathcal{D}^{*}$, we applied the unlearning operation of the top 1% high
loss-drop samples utilizing the original ABL method (using negative cross-
entropy loss defined with respect to the backdoor label). Additionally, we
conducted an experiment that entails training the main head without the
corrected samples in $\mathcal{D}^{*}$. As presented in the first two columns
of Table III, training the main head with the “rectified” samples in
$\mathcal{D}^{*}$ leads to improvements in both ASR and CA. In contrast, when
no backdoor control (namely backdoor unlearning and true class recovery) is
applied in the main head’s training, the ASR increases dramatically, and CA
declines against the majority of attacks. This indicates the significance of
the true class recovery step in our E2ABL method, emphasizing its role in
enhancing accuracy and robustness. In most cases, the proposed backdoor
recovery method not only significantly reduces ASR but also boosts CA,
resulting from the recovered true labels.
### A Second Head or a Second Model?
Our E2ABL methodology incorporates a secondary head that is attached to the
shallow layers of the DNN, aimed at detecting and rectifying backdoor samples.
This approach is based on the assumption that the backdoor task is
substantially more straightforward than the clean task. Such an implementation
will naturally lead to the question: “Can a second model achieve the same
result?” The most significant difference between employing a second model as
opposed to a second head lies in whether they share the same shallow layer
weights with the main network. In essence, without this weight sharing,
attaching a second head to the DNN model is equivalent to using two separate
DNN models.
To further explore this concept, we conducted an experiment utilizing an
alternative two-model setting. In this setting, one smaller model is
exclusively trained to differentiate between clean and backdoor subsets,
mirroring the function of the second head in E2ABL. Concurrently, a full
ResNet-34 model is trained following the same procedure as $h_{1}$, as
outlined in Algorithm 1 of the manuscript. The results, presented in the third
column of Table III, reveal that the two-model design can only marginally
enhance the CA by 0.02% with an average 1.14% decline in ASR.
These findings illuminate that utilizing separate weights might compromise the
secondary model’s proficiency in detecting and isolating backdoor samples. The
underlying cause of this limitation is that the two models are not learned
synchronously, and thus their learning pace may vary. The shared layer design
in E2ABL ensures a coordinated learning process, maximizing both detection
efficiency and correction effectiveness. This illustrates the advantages of
employing a second head in comparison to a separate two-model approach.
### Different Isolation and Recovery Rates
TABLE IV: Performance of E2ABL under different isolation and recovery rates
($\gamma_{rec}$, $\gamma_{iso}$): $\gamma_{iso}$ is the isolation rate,
$\gamma_{rec}$ is the recovery rate. The $\Delta$CA and $\Delta$ASR are
calculated w.r.t. the result of our default experiment setting with
($\gamma_{rec}$, $\gamma_{iso}$) = (1%, 80%) shown in Table I.
$\gamma$ | $(1\%,70\%)$ | $(2\%,80\%)$ | $(5\%,80\%)$
---|---|---|---
| $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA
(1) | +1.05% | +0.74% | -0.02% | -0.21% | -0.06% | -0.36%
(2) | +3.68% | +0.92% | -2.94% | -0.42% | -3.17% | -0.96%
(3) | +1.22% | +0.47% | -0.32% | -0.09% | -0.79% | -0.17%
(4) | +4.11% | +0.39% | -3.25% | -0.47% | -7.12% | -0.41%
(5) | +0.84% | +0.28% | -0.02% | -0.10% | -0.08% | -0.05%
(6) | +0.59% | +0.37% | -0.10% | -0.15% | -0.17% | -0.23%
(7) | +0.63% | +1.01% | +0.01% | +0.08% | -0.01% | +0.00%
(8) | +3.89% | +0.87% | -4.58% | -0.27% | -8.12% | -0.95%
(9) | +1.20% | +0.59% | -0.89% | -0.39% | -1.27% | -0.62%
avg. | +1.91% | +0.63% | -1.35% | -0.22% | -2.31% | -0.42%
We perform experiments employing three distinct sets of hyperparameters:
isolation rate ($\gamma_{p}$) and recovery rate ($\gamma_{c}$). The findings,
as detailed in Table IV, indicate that our proposed method demonstrates
robustness in different hyperparameter settings. The following conclusions can
be derived:
1) A higher recovery rate (increasing from $1\%$ to $5\%$) can further
diminish ASR, while the CA is primarily preserved. This suggests that
calibration of the recovery rate will have a limited adverse effect on the
system’s overall accuracy.
2) Conversely, a higher isolation rate (increasing from $10\%$ to $20\%$) can
lead to an improvement in CA, though it causes a marginal increase in ASR by
less than $2\%$. Importantly, the overall ASR still remains minimal,
indicating that the method’s ability to defend against attacks is maintained,
even when the isolation rate is reduced.
These observations underscore the robustness of E2ABL, confirming that
adjustments to these particular hyper-parameters have a controlled impact on
the system’s performance for both image and time series, thereby offering
flexibility in tuning according to specific requirements.
### Model behavior between two heads
Figure 3: Behavior of dual-head learning over training epochs. The experiments
are performed using the CIFAR-10 dataset, incorporating Trojan attacks.
To clearly depict the training behavior of the dual-head model, we carried out
supplementary experiments without a warm start (lasting for 2 epochs). The
comparative results are presented in the first row of Figure 3. We also
plotted the precision of clean data and backdoor isolation in the secondary
head. The following conclusion can be derived:
1) The cold start approach significantly enhances the second head’s ability to
effectively segregate backdoor samples. As our E2ABL model is designed to
train clean models on poisoned data, only the clean data is channeled into the
main head for training purposes, primarily after the removal of the majority
of backdoor samples. Simultaneously, these backdoor samples are utilized in
the unlearning process.
2) Without the cold start, the final ASR witnesses a 6% reduction. This
outcome is attributed to the exposure of backdoor samples in the early stages
of the main head training, which proves challenging to eliminate in the later
stages of backdoor unlearning. Conversely, the cold start method first
empowers the second head to distinguish between clean and backdoor samples,
thereby effectively supporting the subsequent clean training process of the
main head.
3) As demonstrated in the bottom row of the figure, the precision of backdoor
isolation converges more rapidly compared to clean isolation, reaching nearly
98% in just 2 epochs. However, the convergence of clean isolation occurs over
a longer period, taking nearly 10 epochs. This observation also indicates that
the backdoor task is easier to learn compared to the clean task, primarily
attributable to the higher loss values incorporated during training.
### Where To Attach the Second Head?
In this work, we introduce a secondary classification head, integrated into
the shallow layers of the DNN. Functioning as a trap for backdoor samples,
this secondary head plays a dual role: it 1) detects these deceptive samples
and 2) concurrently corrects their labels. Specifically designed to be
sensitive to the presence of backdoors, this innovative secondary head
performs essential detection tasks, identifying backdoor samples and striving
to recover their true labels. By confining the backdoor samples within the
shallow layers, this approach protects the primary head, guiding the model
training towards a more secure and trustworthy trajectory. In our experiment,
utilizing ResNet-34 as the backbone model, the secondary head is constructed
of two convolutional layers, strategically attached to the termination of the
second convolutional stage, as illustrated in Figure 1. It is worth noting
that the specific attachment point of the second head and its size (including
the number of convolutional layers and the number of neurons in fully
connected layers) require further investigation. This assessment can lead to a
deeper understanding of the second head, contributing to a more resilient
model.
Figure 4: A generalized view of attaching an additional classification head to
any DNN models.
As depicted in Figure 4, the entire dual-head model can be understood from the
following perspective. First, the early convolutional layers are responsible
for extracting high-level feature representations from the input sample. In
the case of a backdoor-poisoned sample, these feature representations embody
both clean features (which infer the true class of the given input) and
backdoor features (which lead the model to misclassify the sample into the
target class). Subsequently, these mixed feature representations serve as
inputs for both the main head and the secondary head. Intriguingly, the two
heads are trained with opposing objectives: the main head aims for backdoor-
free classification, while the secondary head is sensitized to detect backdoor
features. The variations in the attachment point of the secondary head control
the depth of the feature representation generated for the latter tasks.
To investigate the corresponding consequences of varying the attachment point
of the secondary head, we conducted controlled experiments with the ResNet-34
model. The location of attachment was systematically altered, ranging from the
initial convolutional group (after Block 1) to a position immediately prior to
the FC layer (after Block 4). This experimental design allowed us to explore
the effects of the secondary head’s placement on the model’s overall
performance and behavior, contributing to our understanding of its optimal
integration.
TABLE V: Ablation studies of E2ABL on the attachment point of the secondary
head. The full names of the attacks can be found in Table 2 of the manuscript.
The $\Delta$CA and $\Delta$ASR are calculated based on the CIFAR-10 results in
Table 1 of the manuscript (attachment point is after Block 2).
| After Block 1 | After Block 3 | After Block 4
---|---|---|---
| $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA | $\Delta$ASR | $\Delta$CA
(1) | +1.23% | +0.61% | -0.09% | -1.63% | +0.98% | -6.14%
(2) | +1.54% | +0.69% | -0.04% | -2.57% | +1.12% | -7.18%
(3) | +2.46% | -0.92% | +0.17% | -3.68% | -0.14% | -5.12%
(4) | +3.17% | +0.80% | -0.20% | -6.40% | +0.68% | -9.19%
(5) | +0.64% | -0.56% | -0.34% | -6.77% | +1.63% | -13.10%
(6) | +0.96% | +0.75% | -0.16% | -4.69% | +1.32% | -12.57%
(7) | +1.15% | -0.64% | +0.31% | -5.93% | +0.65% | -8.16%
(8) | +0.99% | -0.61% | +0.07% | -7.11% | +0.84% | -11.24%
(9) | +1.10% | +0.73% | -0.14% | -5.74% | +0.79% | -10.64%
avg. | +1.47% | +0.54% | -0.05% | -4.95% | +0.87% | -9.26%
The results, as presented in Table V, lead to the following conclusion:
attaching the secondary head to the output of deeper convolutional groups
yields a lower ASR (which is favorable from a defense perspective), but also
results in a lower CA (indicating worse performance in prediction). However,
when the secondary head is affixed to deeper convolutional groups (such as
after Block 4), the dual-head model exhibits a noticeable decline in
performance across both metrics, leading to unintended negative consequences.
The possible cause of this decline in performance may be attributed to the
convolutional layers picking up an excessive number of backdoor features,
while more benign features are overshadowed or ignored. As a result, the
secondary head’s capacity to provide protection to the main head in terms of
backdoor robustness becomes limited. The current configuration of the dual
head model contains $1.2\times$ of parameters with $1.25\times$ run-time
compared with the ResNet-34 model.
In our supplementary experiments with alternative backbone models like
ResNet-18 and ResNet-50, we noticed consistent trends related to the placement
of the secondary head. Generally speaking, our dual-head model tends to
achieve an optimal balance between ASR and CA when the secondary head is
attached around the midpoint of the convolutional layers.
## Conclusion
In this paper, we proposed the End-to-End Anti-Backdoor Learning (E2ABL)
methodology, a simple but innovative technique specifically engineered to
train models that remain clean even when exposed to potentially backdoor-
poisoned training data. The E2ABL approach, by connecting a second head to the
shallow layers of a Deep Neural Network (DNN), serves as a backdoor supervisor
that learns, detects, and segregates backdoor samples. E2ABL also incorporates
a partitioning mechanism to distinguish clean samples from potentially
poisoned ones, thus creating a subset of backdoor samples. It then employs a
novel, dynamic true class recovery process to rectify the labels of a certain
proportion of samples within the poisoned subset. Through extensive
experiments on both image and time series data, we have proven E2ABL’s
effectiveness in defending against 9 backdoor attacks. It can train clean and
reliable models even when confronted with sophisticated backdoor attacks. This
work presents an actionable solution for safety-critical industries seeking to
train models devoid of backdoor vulnerabilities using real-world datasets.
While there are still many open problems, this work has made a first attempt
toward a single unified defense for multiple tasks and data modalities. This
work can thus serve as a useful baseline for future research.
## References
* [1] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You only look once: Unified, real-time object detection,” in CVPR, 2016.
* [2] A. Balasundaram and C. Chellappan, “Vision based motion tracking in real time videos,” in ICCIC, 2017.
* [3] M. Cordts, M. Omran, S. Ramos, T. Rehfeld, M. Enzweiler, R. Benenson, U. Franke, S. Roth, and B. Schiele, “The cityscapes dataset for semantic urban scene understanding,” in CVPR, 2016.
* [4] O. Peia and K. Roszbach, “Finance and growth: time series evidence on causality,” Journal of Financial Stability, 2015.
* [5] A. Essien and C. Giannetti, “A deep learning model for smart manufacturing using convolutional lstm neural network autoencoders,” IEEE Transactions on Industrial Informatics, 2020.
* [6] R. B. Penfold and F. Zhang, “Use of interrupted time series analysis in evaluating health care quality improvements,” Academic pediatrics, 2013.
* [7] T. Gu, B. Dolan-Gavitt, and S. Garg, “Badnets: Identifying vulnerabilities in the machine learning model supply chain,” arXiv preprint arXiv:1708.06733, 2017.
* [8] Y. Liu, X. Ma, J. Bailey, and F. Lu, “Reflection backdoor: A natural backdoor attack on deep neural networks,” in ECCV, 2020.
* [9] A. Turner, D. Tsipras, and A. Madry, “Label-consistent backdoor attacks,” arXiv preprint arXiv:1912.02771, 2019.
* [10] S. Cheng, Y. Liu, S. Ma, and X. Zhang, “Deep feature space trojan attack of neural networks by controlled detoxification,” in AAAI, 2021.
* [11] D. Ding, M. Zhang, Y. Huang, X. Pan, F. Feng, E. Jiang, and M. Yang, “Towards backdoor attack on deep learning based time series classification,” in ICDE, 2022.
* [12] Y. Jiang, X. Ma, S. M. Erfani, and J. Bailey, “Backdoor attacks on time series: A generative approach,” arXiv preprint arXiv:2211.07915, 2022.
* [13] Y. Li, X. Lyu, N. Koren, L. Lyu, B. Li, and X. Ma, “Anti-backdoor learning: Training clean models on poisoned data,” NeurIPS, 2021.
* [14] X. Chen, C. Liu, B. Li, K. Lu, and D. Song, “Targeted backdoor attacks on deep learning systems using data poisoning,” arXiv preprint arXiv:1712.05526, 2017.
* [15] E. Bagdasaryan and V. Shmatikov, “Blind backdoors in deep learning models,” in USENIX Security, 2021.
* [16] S. Zhao, X. Ma, X. Zheng, J. Bailey, J. Chen, and Y.-G. Jiang, “Clean-label backdoor attacks on video recognition models,” in CVPR, 2020.
* [17] A. Nguyen and A. Tran, “Input-aware dynamic backdoor attack,” NeurIPS, 2020.
* [18] Y. Li, Y. Li, B. Wu, L. Li, R. He, and S. Lyu, “Invisible backdoor attack with sample-specific triggers,” in CVPR, 2021.
* [19] Y. Liu, S. Ma, Y. Aafer, W.-C. Lee, J. Zhai, W. Wang, and X. Zhang, “Trojaning attack on neural networks,” in NDSS, 2018.
* [20] A. Shafahi, W. R. Huang, M. Najibi, O. Suciu, C. Studer, T. Dumitras, and T. Goldstein, “Poison frogs! targeted clean-label poisoning attacks on neural networks,” NeurIPS, 2018.
* [21] A. Turner, D. Tsipras, and A. Madry, “Clean-label backdoor attacks,” https://people.csail.mit.edu/madry/lab/, 2019.
* [22] C. Zhu, W. R. Huang, H. Li, G. Taylor, C. Studer, and T. Goldstein, “Transferable clean-label poisoning attacks on deep neural nets,” in International Conference on Machine Learning, pp. 7614–7623, PMLR, 2019.
* [23] A. Saha, A. Subramanya, and H. Pirsiavash, “Hidden trigger backdoor attacks,” in AAAI, 2020.
* [24] S. Wang, S. Nepal, C. Rudolph, M. Grobler, S. Chen, and T. Chen, “Backdoor attacks against transfer learning with pre-trained deep learning models,” IEEE Transactions on Services Computing, 2020.
* [25] J. Dumford and W. Scheirer, “Backdooring convolutional neural networks via targeted weight perturbations,” in IJCB, 2020.
* [26] S. Garg, A. Kumar, V. Goel, and Y. Liang, “Can adversarial weight perturbations inject neural backdoors,” in CIKM, 2020.
* [27] P. Zhao, P.-Y. Chen, P. Das, K. N. Ramamurthy, and X. Lin, “Bridging mode connectivity in loss landscapes and adversarial robustness,” arXiv preprint arXiv:2005.00060, 2020.
* [28] Y. Li, X. Lyu, N. Koren, L. Lyu, B. Li, and X. Ma, “Neural attention distillation: Erasing backdoor triggers from deep neural networks,” ICLR, 2021.
* [29] D. Wu and Y. Wang, “Adversarial neuron pruning purifies backdoored deep models,” NeurIPS, 2021.
* [30] Y. Li, X. Lyu, X. Ma, N. Koren, L. Lyu, B. Li, and Y.-G. Jiang, “Reconstructive neuron pruning for backdoor defense,” ICML, 2023.
* [31] K. Liu, B. Dolan-Gavitt, and S. Garg, “Fine-pruning: Defending against backdooring attacks on deep neural networks,” in RAID, 2018.
* [32] Y. Yao, H. Li, H. Zheng, and B. Y. Zhao, “Latent backdoor attacks on deep neural networks,” in CCS, 2019.
* [33] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in CVPR, 2016.
* [34] R. Geirhos, J.-H. Jacobsen, C. Michaelis, R. Zemel, W. Brendel, M. Bethge, and F. A. Wichmann, “Shortcut learning in deep neural networks,” Nature Machine Intelligence, vol. 2, no. 11, pp. 665–673, 2020.
* [35] S. Li, S. Ma, M. Xue, and B. Z. H. Zhao, “Deep learning backdoors,” in Security and Artificial Intelligence, pp. 313–334, Springer, 2022.
* [36] S. Yang, Y. Li, Y. Jiang, and S.-T. Xia, “Backdoor defense via suppressing model shortcuts,” arXiv preprint arXiv:2211.05631, 2022.
* [37] A. Krizhevsky, G. Hinton, et al., “Learning multiple layers of features from tiny images,” 2009.
* [38] J. Stallkamp, M. Schlipsing, J. Salmen, and C. Igel, “The german traffic sign recognition benchmark: a multi-class classification competition,” in IJCNN, 2011.
* [39] M. Barni, K. Kallas, and B. Tondi, “A new backdoor attack in cnns by training set corruption without label poisoning,” in ICIP, 2019.
* [40] J. Lin, L. Xu, Y. Liu, and X. Zhang, “Composite backdoor attack for deep neural network by mixing existing benign features,” in SIGSAC Conference on Computer and Communications Security, 2020.
* [41] Y. Chen, E. Keogh, B. Hu, N. Begum, A. Bagnall, A. Mueen, and G. Batista, “The ucr time series classification archive,” July 2015. www.cs.ucr.edu/~eamonn/time_series_data/.
* [42] Z. Wang, W. Yan, and T. Oates, “Time series classification from scratch with deep neural networks: A strong baseline,” in IJCNN, 2017.
|
# ZeroGen: Zero-shot Multimodal Controllable Text Generation with Multiple
Oracles
Haoqin Tu, Bowen Yang, Xianfeng Zhao
State Key Laboratory of Information Security, Institute of Information
Engineering,
School of Cyber Security, University of Chinese Academy of Sciences
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Automatically generating textual content with desired attributes is an
ambitious task that people have pursued long. Existing works have made a
series of progress in incorporating unimodal controls into language models
(LMs), whereas how to generate controllable sentences with multimodal signals
and high efficiency remains an open question. To tackle the puzzle, we propose
a new paradigm of zero-shot controllable text generation with multimodal
signals (ZeroGen). Specifically, ZeroGen leverages controls of text and image
successively from token-level to sentence-level and maps them into a unified
probability space at decoding, which customizes the LM outputs by weighted
addition without extra training. To achieve better inter-modal trade-offs, we
further introduce an effective dynamic weighting mechanism to regulate all
control weights. Moreover, we conduct substantial experiments to probe the
relationship of being in-depth or in-width between signals from distinct
modalities. Encouraging empirical results on three downstream tasks show that
ZeroGen not only outperforms its counterparts on captioning tasks by a large
margin but also shows great potential in multimodal news generation with a
higher degree of control. Our code will be released at
https://github.com/ImKeTT/ZeroGen.
Figure 1: Traditional CTG only has unimodal guidance (up), while our ZeroGen
follows Multimodal CTG (down) that incorporates multimodal controls to
generate relevant texts. We mark words/sentences that are relevant to the
textual control or visual control.
## 1 Introduction
Large-scale pre-trained models (PTMs) have recently achieved great success and
become a milestone in the field of AI. Owing to their sophisticated pre-
training objectives and huge model parameters, PTMs can benefit a variety of
downstream tasks just like Oracles. In the domain of language, pre-trained
language models (PLMs) have become a cornerstone of versatile generation tasks
including controllable text generation (CTG). By controlling the presence of
certain linguistic attributes, these PLMs can be trained to generate texts
with desired aspects such as length, and topic Kikuchi et al. (2016); Ficler
and Goldberg (2017). Conventional approaches usually construct a conditional
LM with supervision (e.g., by fine-tuning), which is unscalable due to the
combinatorially numerous conceivable compositions and the lack of annotated
data Keskar et al. (2019); Liu et al. (2022). Most recent studies have begun
to look into “plug-and-play” (PnP) solutions. Those techniques plug in
arbitrary restrictions to guide the generation of desired sentences with PLMs
and little training expenses. And the control signals of this paradigm are
typically limited to unimodal domains, such as provided keywords or topics
Dathathri et al. (2019); Pascual et al. (2021); Yang and Klein (2021); Tu et
al. (2022). Rapidly, the PnP fashion has been adopted to bridge multimodal
knowledge, recent works have introduced pre-trained multimodal models like
CLIP Radford et al. (2021) into cross-modal tasks with vision-only controls
such as captioning. These approaches obtained exceptional performances with
minimal or no task-oriented training Su et al. (2022a); Tewel et al. (2022);
Nukrai et al. (2022).
On the one hand, meaningful interactions between human speakers often
necessitate real-world experiences Bisk et al. (2020), and the text-only
instruction alone may not be sufficient to fulfill such communication purpose
Harnad (1990). As a result, using unimodal controls for CTG may conflict
between how to reliably regulate current PLMs and real-world scenarios (e.g.,
multimodal controlled news generation in Figure 1). On the other hand, unlike
some keyword-guided PnP works Pascual et al. (2021); Gu et al. (2022), models
that incorporate visual guidance into language generation insert constant
controls at LM decoding instead of considering the dynamic nature of such
process, which may lead to task under-performance Su et al. (2022a); Tewel et
al. (2022).
To overcome those shortcomings, we take a step further to extend the current
unimodal PnP paradigm into a multimodal setting and propose ZeroGen. To
accomplish multimodal CTG task, we are aware that inputs from different
domains affect different granularities of presences in texts. As shown in
Figure 1, while textual control steers generated news to the science topic by
presenting related keywords, visual control provides more abundant ambient
information by producing sentence descriptions. In order to plug in multimodal
signals, we propose to unify controls into the LM output probability using
token- or sentence-level similarity with several Oracles. Specifically, we
first regard the textual guidance as the token-level similarity between
keywords and the LM vocabulary from a textual Oracle before decoding, then we
incorporate such guidance to LM outputs by weighted addition at generation.
For visual guidance, we use a multimodal score Su et al. (2022a) based on
sentence-level probability determined by a multimodal Oracle. Finally, we
employ beam search to find the token with the highest score at each step. To
adapt to the dynamic nature of LM decoding and further promote model
performance, we provide a dynamic weighting mechanism on the word-level that
can not only enhance visual information expression but also maintain output
fluency.
We conduct three tasks (image captioning, stylized captioning, and
controllable news generation) with ZeroGen. We explore the relationship
between textual and visual control being either vertical or lateral.
Specifically, in two captioning tasks, textual objects of the image extend the
visual signal as a complement (vertical extension). For news generation, a
collection of positive or negative words are used to embody generated news a
specific sentiment (lateral extension). The effectiveness of our approach in
providing better captions and easily controlled news is demonstrated by
results on both automatic metrics and human evaluations.
Contributions. (1) We explore the task of multimodal controllable text
generation under zero-shot setting and propose ZeroGen that utilizes token-
and sentence-level multimodal guidance to fulfill this task. (2) We present a
dynamic weighting scheme on the word-level that can be applied to different
modalities and boost the fluency and controllability of generated texts. (3)
Extensive experiments on two captioning tasks and the controllable news
generation task not only justify the effectiveness of ZeroGen but also
investigate the relationship between different types of modal controls.
Figure 2: Workflow of ZeroGen at decoding step $t$. Through multiple LM output
changing stages, ZeroGen is essentially a decoding scheme that finds a word
related to both textual ($\textbf{C}_{T}$) and visual control
($\textbf{C}_{V}$) at each step. It then feeds the word back to the base LM
for the future conditional generation.
## 2 Related Work
#### Efficient Image Captioning.
The prerequisite of supervised captioning for a large amount of paired image-
text data is unrealistic in real-life scenarios. Various attempts have been
made to reduce the dependence on large paired image-text data. For example,
some works Anderson et al. (2018); Laina et al. (2019); Chen et al. (2020);
Honda et al. (2021) have sought to incorporate objects from given images into
model training. Despite their efficiency in comparing supervised methods, they
still need to be trained with partial cross-modal guidance as supervision.
CLIP Radford et al. (2021) as a milestone for vision-language alignment has
shown impressive zero-shot capabilities on various multimodal generation
tasks. For example, Tewel et al. (2022) proposed the first zero-shot
captioning model with CLIP and a base LM (i.e., GPT-2). It constantly updates
the model’s transformer cache under the direction of CLIP guidance decoding.
Nevertheless, it still demands gradient computation and optimization during
generation, introducing additional generation overhead. Su et al. (2022a)
proposed MAGIC that utilizes a token decoding score based on CLIP to produce
plausible captions without task-specified training. Most recently, Nukrai et
al. (2022) employs text-only training with gaussian noises parameterized by a
few images to connect CLIP and the base LM textual embedding. Still, Nukrai et
al. (2022) requires a small amount of external visual knowledge during
training. As for our model, ZeroGen expands MAGIC with additional capabilities
to facilitate multimodal guided generation with dynamic weighting, supporting
several downstream applications while keeping its ability to transfer to
different base LMs. Most recently, Zeng et al. (2023) proposed to employ
sample-based sequential polishing during language decoding to produce
plausible and fluent captions.
#### PnP Controllable Text Generation.
To avoid excessive training costs from fine-tuning PLMs into CTG tasks,
researchers have turned their attention to specialized training-free methods
such as the “plug-and-play” (PnP) framework by Dathathri et al. (2019). This
framework can be used along an existing generative LM (the base LM) with
minimum or no training procedure between PnP components and the base LM. In
comparison to conventional methods, these PnP approaches typically follow two
aspects. In-model guidance approaches including “prompt tuning” Lester et al.
(2021) that either aim at optimizing the input prompts and additional
parameters that are fed into the base LM Houlsby et al. (2019); Li and Liang
(2021); Lester et al. (2021) or seek to alter certain hidden representations
that are not model input or output layers, by plugging a trainable model into
the middle of the base LM Dathathri et al. (2019); Duan et al. (2020); Mai et
al. (2020); Tu et al. (2022). Out-model guidance techniques, on the contrary,
focus on building controllable language models that only modify the output
probabilities from the base LMs at inference time Krause et al. (2021);
Pascual et al. (2021); Yang and Klein (2021); Liu et al. (2021a); Krause et
al. (2021). And our ZeroGen belongs to the last category that only imposes
control signals at LM decoding.
## 3 ZeroGen Methodology
For the multimodal CTG task, we formally define it as: given the visual
control $\textbf{C}_{V}$ (i.e., an image) and $N$ representative words from a
topic or an image as the textual control
$\textbf{C}_{T}=\\{C_{T_{1}},...,C_{T_{N}}\\}$, we aim at getting the textual
output $\textbf{X}=\\{x_{1},x_{2},...\\}$ to meet the two control aspects
simultaneously.
ZeroGen focuses on the output probability space of the base LM. As shown in
Figure 2, at decoding step $t$, it first adjusts the original LM output
probability $p_{\text{LM}_{t}}$ to $p^{\prime}_{\text{LM}_{t}}$ follows the
token-level textual guidance from keywords-vocabulary similarities, then it
completes word searching on $p^{\prime}_{\text{LM}_{t}}$ using a sentence-
level multimodal scoring function and beam search. Note that, instead of
calculating the token similarity constantly Pascual et al. (2021), we only
compute it once before decoding and turning it into the overall textual
control with options. Finally, being processed on word-level, the dynamic
weighting scheme is applied to regulate both control weights for every
generation step.
### 3.1 Token-level Textual Guidance
Since the appearance of keywords from a certain topic can drive sentences in
such direction, we consider token-level similarity between LM tokens and
keywords in $\textbf{C}_{T}$ as the textual guidance. To avoid the additional
computational costs, we unify the textual control into probability space by a
set of cosine similarities between word $C_{T_{n}}\in\textbf{C}_{T}$ and the
full base LM vocabulary $\textbf{V}\in\mathbb{R}^{V}$ before decoding. These
word similarities are obtained using the textual Oracle $\phi_{T}$ (e.g., pre-
trained word embedding):
$\displaystyle
p(\textbf{V},\textbf{C}_{T})=\left\\{\cos(\phi_{T}(\textbf{V}),\phi_{T}(C_{T_{n}}))\right\\}_{n=1}^{N},$
where $p(\textbf{V},\textbf{C}_{T})\in\mathbb{R}^{N\times V}$, $V$ is the
vocabulary size. To fully utilize all the given keywords, we explore three
selection methods at time $t$ when $N>1$ to get the overall textual control
$p_{t}(\textbf{C}_{T})\in\mathbb{R}^{V}$:
#### Step-wise Random (SR):
we first provide changing controls through the generation. At different steps,
we sample one keyword-vocabulary similarity uniformly from
$p(\textbf{V},\textbf{C}_{T})$ as textual guidance.
#### Mean Pooling (MP):
an intuitive way to consider all textual information is to average their
guiding similarities w.r.t. V across distinct keywords.
#### Word-wise Max (WM):
for every token $w$ from V, we choose the most similar keyword in
$\textbf{C}_{T}$ with $w$ (with the highest cosine similarity score) to
compute its guiding probability and compose all the highest similarities
together as $p_{t}(\textbf{C}_{T})$.
The overall textual control $p_{t}(\textbf{C}_{T})\in\mathbb{R}^{V}$ is
available after this selection, we introduce it to $p_{\text{LM}_{t}}$ as a
control bias by simple addition operation with weighting:
$p^{\prime}_{\text{LM}_{t}}=p_{\text{LM}_{t}}+\alpha\times
p_{t}(\textbf{C}_{T})$.
### 3.2 Sentence-level Visual Guidance
Image information carries more general and higher-level global information
than a single word. As discussed in Sec. 1, we thus consider sentence-level
similarity between generated texts and visual control $\textbf{C}_{V}$ as the
visual guidance.
We employ scoring function $S_{t}$ for word $w\in\textbf{V}$ at $t$-th step
with weighted visual guidance as in Su et al. (2022a), and use beam search for
generation:
$\displaystyle S_{t}\left(w,\textbf{C}_{V}\mid x_{<t},W_{t}^{(k)}\right)=$
$\displaystyle\left\\{\begin{aligned} &\begin{aligned}
p^{\prime}_{\text{LM}_{t}}&(w\mid x_{<t})+\\\
&\beta\times\frac{e^{p_{\phi_{M}}\left(\left[x_{<t};w\right],\textbf{C}_{V}\right)}}{\sum_{z\in
W_{t}^{(k)}}e^{p_{\phi_{M}}\left(\left[x_{<t};z\right],\textbf{C}_{V}\right)}}\end{aligned},\text{if
}w\in W_{t}^{(k)}\\\
&-\inf,\qquad\qquad\qquad\qquad\qquad\quad\quad\text{otherwise}.\end{aligned}\right.$
Here $[x_{<t};w]$ means appending $w$ to the generated texts before step $t$,
$W_{t}^{(k)}$ is the searching beam consists of words with the $k$ highest
probabilities in $p^{\prime}_{\text{LM}_{t}}$. In detail, we bridge texts and
$\textbf{C}_{V}$ using multimodal Oracle $\phi_{M}$ (e.g., CLIP) and compute
their similarity:
$p_{\phi_{M}}([x_{<t};w],\textbf{C}_{V})=\cos(\phi_{M}([x_{<t};w]),\phi_{M}(\textbf{C}_{V}))$.
Our final goal is to find
$x_{t}=\arg\max_{w\in\textbf{V}}S_{t}(w,\textbf{C}_{V}\mid
x_{<t},W_{t}^{(k)})$.
### 3.3 Multimodal Dynamic Weighting
To further boost the model performance and make the model get attuned to
different generation steps, a novel dynamic weighting mechanism is proposed to
achieve step-wise multimodal weights adjustment. Concretely, we replace
$\alpha,\beta$ with dynamic $\alpha_{t},\beta_{t}$ severally. The design
should consider such principles: (1) It is necessary to seek a certain balance
between textual control (i.e., shifts the LM output probability) and the
original LM modeling to avoid inconsistent outputs. (2) During generation,
visual-relevant words ought to be encouraged, while those irrelevant are
punished. Since the smallest comprehensible output of an LM is one word, we
then apply this framework on the word-level.
#### Dynamic $\boldsymbol{\alpha_{t}}$.
To maintain the original language modeling ability, and also make the most out
of provided textual guidance. We re-scale the textual control using a step-
wise weighting calibration that incorporates the original LM output confidence
$p_{\text{LM}_{t}}$. Specifically, we compute the average probability of
$\hat{N}\in[1,N]$ keywords in $\textbf{C}_{T}$ from the unshifted LM output as
the $t$-th textual control weight:
$\displaystyle
D_{T}=\sum_{n=1}^{\hat{N}}\frac{p_{\text{LM}_{t}}\left(C_{T_{n}}\mid
x_{<t}\right)}{\hat{N}},$
$\displaystyle\alpha_{t}=\min\left(\frac{D_{T}}{\lambda},\hat{\alpha}\right).$
If $D_{T}$ is high, keywords from $\textbf{C}_{T}$ are encouraged to be
spoken. Since this is the exact time when unchanged base LM has high
confidence to produce these words, we can avoid jeopardizing output fluency
while generating controlled texts.
#### Dynamic $\boldsymbol{\beta_{t}}$.
To reward the generation stages where all words in $W_{t}^{(k)}$ are highly
associated with the knowledge in $\textbf{C}_{V}$ and penalize those do not,
we employ average word-level similarity between current candidate words and
the visual control:
$\displaystyle D_{V}=\sum_{w\in
W_{t}^{(k)}}\frac{p\left(w,\textbf{C}_{V}\right)}{k},$
$\displaystyle\beta_{t}=\min\left(\frac{D_{V}}{\lambda},\hat{\beta}\right).$
If $D_{V}$ is high, words in $W_{t}^{(k)}$ are relevant to $\textbf{C}_{V}$
and should be expressed with a higher chance.
Inspired by Gu et al. (2022), $\lambda$ in this framework serves as a
threshold that amplifies the control signal if $D_{V}$ or $D_{T}$ is larger
than it and vice versa. Meanwhile, $\hat{\alpha},\hat{\beta}$ are weighting
upper bounds.
Model | MS-COCO | Flickr30k | Speed
---|---|---|---
B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$ | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$
Weakly Supervised Approaches
IC-SME Laina et al. (2019) | - | 6.5 | 12.9 | 35.1 | 22.7 | - | - | 7.9 | 13.0 | 32.8 | 9.9 | - | -
S2S-GCC Honda et al. (2021) | 50.4 | 7.6 | 13.5 | 37.3 | 31.8 | 8.4 | - | - | - | - | - | - | -
CapDec Nukrai et al. (2022) | 69.2 | 26.4 | 25.1 | 51.9 | 91.8 | - | 55.5 | 17.7 | 20.0 | 43.9 | 39.1 | - | -
Unsupervised Approaches
CLIPRe | 39.5 | 4.9 | 11.4 | 29.0 | 13.6 | 5.3 | 38.5 | 5.2 | 11.6 | 27.6 | 10.0 | 5.7 | -
ZeroCap Tewel et al. (2022) | 49.8 | 7.0 | 15.4 | 31.8 | 34.5 | 9.2 | 44.7 | 5.4 | 11.8 | 27.3 | 16.8 | 6.2 | 1.0 $\times$
MAGIC Su et al. (2022a) | 56.5 | 12.4 | 17.3 | 39.6 | 48.3 | 11.2 | 43.3 | 6.8 | 12.3 | 30.8 | 20.5 | 6.8 | 26.6 $\times$
ConZIC Zeng et al. (2023) | - | 1.3 | 11.5 | - | 12.8 | 5.2 | - | - | - | - | - | - | -
DeCap Li et al. (2023) | - | 8.9 | 17.5 | - | 50.6 | 13.1 | - | - | - | - | - | - | -
ZeroGen | 59.4 | 15.5 | 18.7 | 42.3 | 55.4 | 12.1 | 54.9 | 13.1 | 15.2 | 37.4 | 26.4 | 8.3 | 16.4 $\times$
-TDW | 58.9 | 15.2 | 18.4 | 41.8 | 54.4 | 11.9 | 54.1 | 12.8 | 14.7 | 36.8 | 24.5 | 7.7 | 16.5 $\times$
-T | 58.6 | 14.7 | 17.4 | 41.3 | 51.7 | 11.8 | 53.3 | 11.9 | 14.3 | 36.2 | 24.1 | 7.5 | 18.6 $\times$
-VDW | 57.0 | 12.6 | 17.6 | 39.7 | 49.7 | 11.6 | 49.2 | 6.4 | 14.1 | 32.4 | 22.9 | 7.7 | 22.5 $\times$
-DW | 57.0 | 12.6 | 17.6 | 39.7 | 49.7 | 11.6 | 47.7 | 7.1 | 13.8 | 32.3 | 21.9 | 7.6 | 21.6 $\times$
Table 1: Captioning results of ZeroGen with only 1 object as $\textbf{C}_{T}$ (i.e., $N=1$) on MS-COCO and Flickr30k. ZeroGen outperforms most baselines with tolerable efficiency sacrifice. T, TDW/VDW, DW represent textual control, textual/visual dynamic weighting and two dynamic weighting schemes combined respectively. Model | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$
---|---|---|---|---|---
ZeroGen ($N=1$) | 59.4 | 15.5 | 18.7 | 42.3 | 55.4
ZeroGen ($N=2$) | 60.1 | 15.6 | 18.5 | 42.3 | 55.9
ZeroGen ($N=3$) | 60.4 | 15.6 | 18.6 | 42.3 | 56.5
ZeroGen ($N=4$) | 60.5 | 15.7 | 18.7 | 42.4 | 57.0
ZeroGen ($N=5$) | 60.6 | 15.8 | 18.7 | 42.4 | 57.1
Table 2: Captioning results of ZeroGen on MS-COCO with varied size $N$ for textual control $\textbf{C}_{T}$. Model | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$
---|---|---|---|---|---
ZeroGen SR | 59.6 | 15.3 | 18.4 | 42.1 | 55.5
ZeroGen MP | 59.9 | 15.2 | 18.3 | 42.0 | 55.2
ZeroGen WM | 60.6 | 15.8 | 18.7 | 42.4 | 57.1
Table 3: Captioning results of ZeroGen with $N=5$ for $\textbf{C}_{T}$ on MS-
COCO with three $p_{t}(\textbf{C}_{T})$ options.
## 4 General Implementations and Baselines
#### General Implementations.
We take SimCTG Su et al. (2022b) as our base LM and first fine-tune it on
every dataset with text-only data like previous works. Since ZeroGen follows
the zero-shot paradigm, it can leverage any off-the-shelf LM and empower it a
pair of eyes. For Oracles, we employ GloVe Pennington et al. (2014) as the
textual Oracle $\phi_{T}$ and CLIP Radford et al. (2021) as the multimodal
Oracle $\phi_{M}$. The $\hat{N}$ for $\alpha_{t}$ is $N$ itself on two
captioning tasks, while $\hat{N}=2$ on controllable news generation task
through ablation study. The amplifying factor $\lambda$ is $0.2$ throughout
the paper. See Appendix A for full model details.
#### Baseline Models.
For the image captioning task, we select both weakly supervised and
unsupervised methods as our baselines, (1) IC-SME Laina et al. (2019), S2S-GCC
Honda et al. (2021), and CapDec Nukrai et al. (2022) are three weakly
supervised approaches, the former two adapt neural network modules to align
visual features with pseudo captions, CapDec introduces CLIP guidance and few
images in training. (2) CLIPRe, ZeroCap Tewel et al. (2022), and MAGIC Su et
al. (2022a) are three zero-shot methods, which follow the retrieval manner,
CLIP-guided gradient update, and decoding scheme respectively. For fair
comparisons, we use the same base LM as ours for ZeroCap and MAGIC. In
stylized captioning, MemCap Zhao et al. (2020) is additionally considered.
Model | FlickrStyle10k Romantic | FlickrStyle10k Humorous
---|---|---
B@1 $\uparrow$ | B@3 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$ | B@1 $\uparrow$ | B@3 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$
MemCap Zhao et al. (2020) | 21.2 | 4.8 | 8.4 | - | 22.4 | - | 19.9 | 4.3 | 7.4 | - | 19.4 | -
ZeroCap Tewel et al. (2022) | 19.3 | 2.7 | 7.6 | 16.5 | 14.9 | 7.0 | 18.4 | 2.7 | 7.7 | 16.5 | 15.6 | 7.7
MAGIC Su et al. (2022a) | 23.3 | 4.9 | 8.6 | 21.7 | 24.4 | 8.6 | 23.7 | 5.2 | 9.0 | 21.2 | 27.8 | 10.1
CapDec∗ Nukrai et al. (2022) | 21.4 | 5.0 | 9.6 | - | 26.9 | - | 24.9 | 4.3 | 10.2 | - | 34.1 | -
ConZIC Zeng et al. (2023) | - | 1.2 | 6.1 | - | - | - | - | 1.2 | 6.1 | - | - | -
ZeroGen | 24.4 | 5.5 | 9.2 | 22.3 | 27.3 | 9.8 | 24.2 | 5.6 | 9.6 | 22.0 | 30.5 | 11.2
-TDW | 23.5 | 5.4 | 8.7 | 21.9 | 26.1 | 9.0 | 24.2 | 5.6 | 9.6 | 21.9 | 30.5 | 11.2
-T | 23.3 | 4.9 | 8.6 | 21.8 | 24.7 | 8.6 | 23.7 | 5.2 | 9.1 | 21.2 | 28.3 | 10.2
-VDW | 24.0 | 5.5 | 9.0 | 22.1 | 26.9 | 9.4 | 24.1 | 5.6 | 9.5 | 21.9 | 30.2 | 11.1
-DW | 23.4 | 5.1 | 8.7 | 21.8 | 25.0 | 9.0 | 23.8 | 5.3 | 9.1 | 21.4 | 29.1 | 10.3
Table 4: Stylized captioning results on two subsets of FlickrStyle10k with
$N=1$. * meas CapDec is a weakly supervised method that requires additional
visual knowledge from several images during training.
For controllable news generation task, we take MAGIC and MAGIC+PPLM as two
baseline models. Specifically, MAGIC+PPLM is the combination of two existing
PnP works that take image and keywords as input respectively. PPLM Dathathri
et al. (2019) is the first controllable PnP LM that requires gradient descents
of model hidden states at decoding time. More details and code links of
baselines are available in Appendix A.5.
## 5 Experiments and Analysis
### 5.1 Image Captioning
#### Dataset and Metrics.
We conduct experiments on MS-COCO and Flickr30k using Karpathy split Karpathy
and Fei-Fei (2015). For the visual control, we take images for captioning task
as $\textbf{C}_{V}$. For the textual control, we take textual objects of the
corresponding image as $\textbf{C}_{T}$.111Textual objects are extracted from
each picture ahead of the generation using a pre-trained DETR Carion et al.
(2020). We use five relevance-based metrics for evaluation: BLEU-1 (B@1),
BLEU-4 (B@4) Papineni et al. (2002), METEOR (M) Denkowski and Lavie (2014),
ROUGE-L (R-L) Lin and Och (2004), CIDEr Vedantam et al. (2015), and SPICE
Anderson et al. (2016). Besides, we also compare the decoding speed of ZeroGen
against baselines.222We measure the model’s decoding speed on the same machine
with one NVIDIA GeForce 3090 GPU sequentially.
#### Main Results.
Since both modal controls aim to enhance the model’s ability to understand
image content and to generate better captions, we consider $\textbf{C}_{T}$ to
be a vertical augmentation (or a complement) of $\textbf{C}_{V}$ in this task.
From results in Table 1, we can draw the following conclusions, (1) the
proposed ZeroGen model consistently outperforms unsupervised baselines and
most of the weakly supervised methods (except CapDec) by a great margin,
demonstrating the superiority of the proposed method. (2) Textual guidance as
a vertical augmentation of the visual guidance, provides extra information
about an image, thus promoting the model performance by more than 2 absolute
points in CIDEr on both datasets (comparing model -T). (3) Both dynamic
weighting techniques help strengthen the model’s capacity, especially VDW, we
ascribe this situation to its direct optimization of certain token appearances
that are recognized in the image. (4) However, ZeroGen falls short in
efficiency comparing MAGIC, but still largely outperforms ZeroCap. This is
because additional computations are required for multimodal controls and
dynamic weightings, but there is no need for gradient calculation in our model
like ZeroCap.
We also make a series of cross-domain evaluations in Appendix B.3, which
further verifies the robustness of ZeroGen across various domains.
#### Number of Objects in $\boldsymbol{\textbf{C}_{T}}$.
Is the more objects the better? To answer this question, we conduct an
experiment over MS-COCO with varied numbers of objects from the image (size
$N$ in $\textbf{C}_{T}$) using word-wise max (WM) for $p(\textbf{C}_{T})$
selection. In Table 2, we can observe that, as the increase of object number,
our model generally performs better on most metrics, which verifies that more
textual object guidance brings more information for captioning task. Similar
results also prove the answer on Flickr30k as shown in Appendix B.1.
#### $\boldsymbol{p(\textbf{C}_{T})}$ Selection Method.
In Table 3, the most effective method for $p(\textbf{C}_{T})$ selection is
word-wise max (WM). It is attributed to WM’s highlight of textual objects with
all their relevant tokens in the vocabulary. While mean pooling (MP) also
takes all given keywords into consideration, by presenting them equally, it
may introduce biases in token similarity calculation and output controlling.
Hence, we use WM for the rest experiments.
Model | Positive | Negative | Speed $\uparrow$
---|---|---|---
D-2 $\uparrow$ | D-4 $\uparrow$ | C-S $\uparrow$ | $\Delta\text{Acc}$ $\uparrow$ | PPL $\downarrow$ | D-2 $\uparrow$ | D-4 $\uparrow$ | C-S $\uparrow$ | $\Delta\text{Acc}$ $\uparrow$ | PPL $\downarrow$
Human∗ | 96.25 | 96.98 | 23.36 | 0.00 | 14.59 | 96.25 | 96.98 | 23.36 | 0.00 | 14.59 | -
$\textsc{MAGIC}^{*}$ Su et al. (2022a) | 95.62 | 95.92 | 20.07 | - | 10.01 | 95.62 | 95.92 | 20.07 | - | 10.01 | 25.0 $\times$
+PPLM Dathathri et al. (2019) | 74.22 | 81.44 | 20.44 | 11.00 | 29.07 | 74.47 | 83.66 | 20.79 | 18.76 | 27.32 | 1.0 $\times$
ZeroGen | 72.04 | 79.32 | 18.11 | 22.12 | 12.22 | 76.42 | 83.01 | 19.11 | 31.75 | 13.04 | 9.8 $\times$
-TDW | 71.87 | 78.90 | 18.08 | 21.87 | 11.75 | 76.29 | 82.52 | 19.14 | 29.88 | 12.53 | 10.4 $\times$
-VDW | 75.44 | 82.06 | 17.56 | 20.50 | 11.62 | 77.80 | 83.84 | 18.20 | 29.63 | 12.62 | 11.7 $\times$
-DW | 81.70 | 86.38 | 17.22 | 19.00 | 12.62 | 77.73 | 83.60 | 18.19 | 29.13 | 12.13 | 12.4 $\times$
-T∗ | 95.27 | 95.80 | 21.19 | - | 10.84 | 95.27 | 95.80 | 21.19 | - | 10.84 | 17.9 $\times$
ZeroGen w/ obj | 81.56 | 87.93 | 19.42 | 16.37 | 12.93 | 82.23 | 87.93 | 19.66 | 29.76 | 13.30 | 9.8 $\times$
Table 5: Results of controllable news generation on VisNews. With $\textbf{C}_{T}$ controlling the sentiment, we regard the textual and the visual control as vertical elements in this task. Methods with * cannot be controlled w.r.t. sentiment. Model | Positive | Negative
---|---|---
Flue.$\uparrow$ | Relv.$\uparrow$ | Sent.$\uparrow$/$\downarrow$ | Flue.$\uparrow$ | Relv.$\uparrow$ | Sent.$\uparrow$/$\downarrow$
MAGIC | 3.37 | 2.77 | 28.7/22.0 | 3.85 | 3.13 | 46.0/14.7
+PPLM | 2.24 | 2.85 | 34.0/7.3 | 3.12 | 3.11 | 52.0/10.7
ZeroGen | 3.38 | 2.94 | 80.0/10.7 | 3.80 | 2.85 | 84.7/6.0
Table 6: Human evaluation results. Sent. scores are percentages of news that
obeys/disobeys given sentiment.
### 5.2 Stylized Captioning
To explore the sufficiency of our model to adapt to different styles, such as
“romantic” or “humorous”. We follow Nukrai et al. (2022) to conduct stylized-
text fine-tuning in base LM for stylized captioning.
#### Dataset and Metrics.
In this task we still take textual objects from images as $\textbf{C}_{T}$ and
we follow the exact experimental setting in previous works Zhao et al. (2020);
Nukrai et al. (2022) on FlickrStyle10k dataset Gan et al. (2017). As for
metrics, we take the same ones in Sec. 5.1. Refer to Appendix A.3 for more
detailed settings.
#### Main Results.
Table 4 shows quantitative results of the task, (1) ZeroGen outperforms most
baselines on two stylized data, including weakly supervised CapDec on Romantic
and MemCap with task-oriented training. (2) While it under-performs CapDec on
some metrics over Humorous data, our method produces more fluent and plausible
captions with consistently higher B@3 scores. (3) From two stylized sets,
textual guidance takes a large credit to boost the model performance
(comparing model -T), verifying the effectiveness of the proposed multimodal
guidance in ZeroGen.
### 5.3 Controllable News Generation
Textual guidance can not only serve as a complement to visual knowledge but
can also be a lateral extension. In this task, we assign textual control for
news sentiment guidance and visual control for image-relevant news generation.
#### Dataset and Metrics.
We conduct experiments on VisNews Liu et al. (2021b). We fine-tune the base LM
(i.e., SimCTG) on news data with the news title as an input prompt. We follow
Dathathri et al. (2019) to obtain word lists for two types of sentiment
guidance respectively. We take four aspects for evaluation, diversity:
Distinct-2 (D-2) and Distinct-4 (D-4) Li et al. (2016). Image-text relevance:
CLIP score (C-S) is the image-text similarity calculated by a CLIP. Control
degree: $\Delta\text{Acc}$ (%) evaluates the accuracy gain between generated
sentences and human written news.333Human written news in the test set
consists of $62.88\%$ positive content and $37.12\%$ negative content.
Fluency: perplexity (PPL) measures the model output confidence.
For human evaluation, we take Fluency (Flue.) for content fluency evaluation,
Relevance (Relv.) for image/title-news relevance evaluation, and Sentiment
(Sent.) to measure sentiment control. We strictly obey a double-blind
procedure, where three annotators know nothing about the models. We sample 100
instances across every model.444Details of automatic metrics and human
evaluation settings are in Appendix A.4 and D respectively.
#### Main Results.
From results in Table 5, we can draw the following conclusions: (1) ZeroGen
has the highest classification accuracy gain and competitive CLIP scores among
all presented statistics, proving that the proposed method can successfully
produce controllable outputs under both modal supervisions. But our model
generally degenerates the diversity, which we consider a trade-off. (2)
Introducing dynamic weighting enhances the overall model performance. While
VDW augments connections between the given image and generated news content
with higher CLIP scores (C-S), TDW is able to make the output more
recognizable w.r.t. the sentiment control without sacrificing content
diversity. These findings validate the vastness and efficacy of the dynamic
weighting mechanism even when their functional domains (i.e.,
$\textbf{C}_{T},\textbf{C}_{V}$) are not complementary. (3) External controls
jeopardize our model’s output confidence with slightly higher PPL than MAGIC,
yet ZeroGen still largely outperforms MAGIC+PPLM, the only controllable
counterpart on PPL. Also, ZeroGen without parts of the dynamic weighting
(e.g., -VDW) can advantageously outgain MAGIC+PPLM on both controllability and
diversity metrics. (4) ZeroGen requires no task-oriented training, thus
registering its superiority in decoding efficiency over MAGIC+PPLM by nearly
10 times faster.
We also present human evaluation results in Table 6,555The average Fleiss’s
Kappa Fleiss and Cohen (1973) is 0.28, indicating three annotators reached a
fair agreement. which can further verify our findings above.
$\hat{\alpha}$ | 5.0 | 8.0 | 10.0
---|---|---|---
Pos. | D-2 $\uparrow$ | 91.60 | 72.04 | 52.37
D-4 $\uparrow$ | 95.00 | 79.32 | 60.00
C-S $\uparrow$ | 19.96 | 18.11 | 17.19
$\Delta\text{Acc}\uparrow$ | 10.75 | 22.12 | 26.62
PPL $\downarrow$ | 11.76 | 12.22 | 10.28
Neg. | D-2 $\uparrow$ | 92.95 | 76.42 | 52.09
D-4 $\uparrow$ | 95.87 | 83.01 | 60.07
C-S $\uparrow$ | 21.59 | 19.11 | 18.81
$\Delta\text{Acc}\uparrow$ | 11.26 | 31.75 | 51.26
PPL $\downarrow$ | 11.71 | 13.04 | 11.52
Table 7: Effect of different $\hat{\alpha}$ on news generation task.
#### Effect of $\boldsymbol{\alpha}$ Upper Bound.
$\textbf{C}_{T}$ is the only source for sentiment manipulation in this task,
the upper bound of which can decide how distinguishable an output sentence is
w.r.t. sentiment. In Table 7, with the increase of $\hat{\alpha}$, both
accuracy and fluency will gain significant benefits. However, the diversity of
texts and image relevance indicators will fall precipitously. This phenomenon
is explained as more guiding information from sentiment may make the model
inclined to only express the desired sentiment, trading for wider imagination
associated with image or diversity. At the user end, we can twitch
$\hat{\alpha}$ according to tasks to fit in different situations.
#### $\boldsymbol{\textbf{C}_{T}}$ Plays Two Roles.
Now we have examined the function of $\textbf{C}_{T}$ as a complement
(captioning) or additional control element (news generation). We wonder can
$\textbf{C}_{T}$ play both roles well at the same time? We conduct experiments
using our ZeroGen with both objects from the image and sentiment words as
textual guidance. Our method with objects as a part of $\textbf{C}_{T}$ is
marked with “w/ obj”. Though ZeroGen w/ obj reaches a higher CLIP score, the
accuracy and PPL generally decline comparing methods without textual objects
guidance, which can also be accomplished by switching $\hat{\alpha}$ as
mentioned earlier. That is to say, $\textbf{C}_{T}$ may be confused to play
both roles as a complement and a lateral extension of images at the same time.
Figure 3: An example of news from varied models. We highlight Positive and
Negative words respectively.
#### Case Analysis.
We exhibit an example of generated controllable news in Figure 3. As the image
shows Culture secretary Jeremy Hunt is giving a talk. All methods are able to
produce image/title-relevant sentences, but MAGIC+PPLM generates some false
evidence such as recognizing Jeremy Hunt as the “leader of Conservative and
Nationalist Labour groups”. Besides, our ZeroGen can produce more diverse and
controllable words like “good”, “benefits” for positive and “deadbeat”, “lose”
for negative, while MAGIC+PPLM is not competent to fulfill the controllable
aspect. More cases are exhibited in Appendix C.
## 6 Conclusion
In this paper, we present ZeroGen, a paradigm of zero-shot controllable text
generation with multimodal signals. We explicitly separate visual control and
textual control into sentence-level and token-level guidance. And we use two
Oracles to unify the control signals to LM output probability space. A dynamic
weighting mechanism is applied to adapt to all multimodal controls, which
further boosts the model generation ability. Three tasks from captioning to
controllable news generation justify the effectiveness of ZeroGen and help us
explore the relationship between distinct signals. By providing multimodal
knowledge, we demonstrate LMs without task-specified training can
substantially achieve astonishing performance in multimodal tasks across
different setups and domains.
## 7 Limitations
Although ZeroGen successfully achieves zero-shot controllable text generation,
our technique is still subject to a few limitations to be addressed in follow-
up work. (1) There is still a large gap between weakly and fully supervised
methods. We believe the rich semantic information contained in these large-
scale pre-trained Oracles we employed can further narrow it. (2) The diversity
in our controllable news generation task is insufficient. Since this is a
widespread problem in the field of zero-shot research, we plan to alleviate
the issue by incorporating more diverse language decoding schemes Xu et al.
(2022) or partial training parameters in the model such as adapters Houlsby et
al. (2019). (3) The existence of spurious correlations Tu et al. (2020); Chai
et al. (2022) in bad cases (as shown in Appendix C) is nonnegligible, one of
our future work directions is to handle it by introducing causal inference
Pearl (2009).
## 8 Ethics Statement
We are well aware that text generation technologies may be abused to create
deceptive, harmful, or objectionable content. For our ZeroGen, we can conduct
experiments on detoxification datasets Gehman et al. (2020) to make it a
useful tool for combating hate speech and eliminating harmful information in
PLMs. As we are considering components to make our method more robust and
effective in multimodal controllable tasks, we believe it is meaningful and
beneficial to progress research on controllable text generation.
## References
* Anderson et al. (2016) Peter Anderson, Basura Fernando, Mark Johnson, and Stephen Gould. 2016. Spice: Semantic propositional image caption evaluation. In _European conference on computer vision_ , pages 382–398. Springer.
* Anderson et al. (2018) Peter Anderson, Stephen Gould, and Mark Johnson. 2018. Partially-supervised image captioning. _Advances in Neural Information Processing Systems_ , 31.
* Bisk et al. (2020) Yonatan Bisk, Ari Holtzman, Jesse Thomason, Jacob Andreas, Yoshua Bengio, Joyce Chai, Mirella Lapata, Angeliki Lazaridou, Jonathan May, Aleksandr Nisnevich, et al. 2020. Experience grounds language. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 8718–8735.
* Carion et al. (2020) Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. 2020. End-to-end object detection with transformers. In _European conference on computer vision_ , pages 213–229. Springer.
* Chai et al. (2022) Junyi Chai, Reid Pryzant, Victor Ye Dong, Konstantin Golobokov, Chenguang Zhu, and Yi Liu. 2022. Fast: Improving controllability for text generation with feedback aware self-training. _arXiv preprint arXiv:2210.03167_.
* Chen et al. (2020) Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. 2020. A simple framework for contrastive learning of visual representations. In _International conference on machine learning_ , pages 1597–1607. PMLR.
* Dathathri et al. (2019) Sumanth Dathathri, Andrea Madotto, Janice Lan, Jane Hung, Eric Frank, Piero Molino, Jason Yosinski, and Rosanne Liu. 2019. Plug and play language models: A simple approach to controlled text generation. _arXiv preprint arXiv:1912.02164_.
* Denkowski and Lavie (2014) Michael Denkowski and Alon Lavie. 2014. Meteor universal: Language specific translation evaluation for any target language. In _Proceedings of the ninth workshop on statistical machine translation_ , pages 376–380.
* Duan et al. (2020) Yuguang Duan, Canwen Xu, Jiaxin Pei, Jialong Han, and Chenliang Li. 2020. Pre-train and plug-in: Flexible conditional text generation with variational auto-encoders. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 253–262.
* Ficler and Goldberg (2017) Jessica Ficler and Yoav Goldberg. 2017. Controlling linguistic style aspects in neural language generation. _EMNLP 2017_ , page 94.
* Fleiss and Cohen (1973) Joseph L Fleiss and Jacob Cohen. 1973. The equivalence of weighted kappa and the intraclass correlation coefficient as measures of reliability. _Educational and psychological measurement_ , 33(3):613–619.
* Gan et al. (2017) Chuang Gan, Zhe Gan, Xiaodong He, Jianfeng Gao, and Li Deng. 2017. Stylenet: Generating attractive visual captions with styles. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 3137–3146.
* Gehman et al. (2020) Samuel Gehman, Suchin Gururangan, Maarten Sap, Yejin Choi, and Noah A Smith. 2020\. Realtoxicityprompts: Evaluating neural toxic degeneration in language models. In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 3356–3369.
* Gu et al. (2022) Yuxuan Gu, Xiaocheng Feng, Sicheng Ma, Jiaming Wu, Heng Gong, and Bing Qin. 2022\. Improving controllable text generation with position-aware weighted decoding. In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 3449–3467.
* Harnad (1990) Stevan Harnad. 1990. The symbol grounding problem. _Physica D: Nonlinear Phenomena_ , 42(1-3):335–346.
* Hartmann et al. (2022) Jochen Hartmann, Mark Heitmann, Christian Siebert, and Christina Schamp. 2022. More than a feeling: Accuracy and application of sentiment analysis. _International Journal of Research in Marketing_.
* Honda et al. (2021) Ukyo Honda, Yoshitaka Ushiku, Atsushi Hashimoto, Taro Watanabe, and Yuji Matsumoto. 2021. Removing word-level spurious alignment between images and pseudo-captions in unsupervised image captioning. In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 3692–3702.
* Houlsby et al. (2019) Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin De Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. 2019. Parameter-efficient transfer learning for nlp. In _International Conference on Machine Learning_ , pages 2790–2799. PMLR.
* Karpathy and Fei-Fei (2015) Andrej Karpathy and Li Fei-Fei. 2015. Deep visual-semantic alignments for generating image descriptions. In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 3128–3137.
* Keskar et al. (2019) Nitish Shirish Keskar, Bryan McCann, Lav R Varshney, Caiming Xiong, and Richard Socher. 2019. Ctrl: A conditional transformer language model for controllable generation. _arXiv preprint arXiv:1909.05858_.
* Kikuchi et al. (2016) Yuta Kikuchi, Graham Neubig, Ryohei Sasano, Hiroya Takamura, and Manabu Okumura. 2016. Controlling output length in neural encoder-decoders. In _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pages 1328–1338.
* Krause et al. (2021) Ben Krause, Akhilesh Deepak Gotmare, Bryan McCann, Nitish Shirish Keskar, Shafiq Joty, Richard Socher, and Nazneen Fatema Rajani. 2021. Gedi: Generative discriminator guided sequence generation. In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 4929–4952.
* Laina et al. (2019) Iro Laina, Christian Rupprecht, and Nassir Navab. 2019. Towards unsupervised image captioning with shared multimodal embeddings. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 7414–7424.
* Lester et al. (2021) Brian Lester, Rami Al-Rfou, and Noah Constant. 2021. The power of scale for parameter-efficient prompt tuning. _arXiv preprint arXiv:2104.08691_.
* Li et al. (2016) Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and William B Dolan. 2016\. A diversity-promoting objective function for neural conversation models. In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 110–119.
* Li et al. (2023) Wei Li, Linchao Zhu, Longyin Wen, and Yi Yang. 2023. Decap: Decoding clip latents for zero-shot captioning via text-only training. _arXiv preprint arXiv:2303.03032_.
* Li and Liang (2021) Xiang Lisa Li and Percy Liang. 2021. Prefix-tuning: Optimizing continuous prompts for generation. In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 4582–4597.
* Lin and Och (2004) Chin-Yew Lin and Franz Josef Och. 2004. Automatic evaluation of machine translation quality using longest common subsequence and skip-bigram statistics. In _Proceedings of the 42nd Annual Meeting of the Association for Computational Linguistics (ACL-04)_ , pages 605–612.
* Liu et al. (2021a) Alisa Liu, Maarten Sap, Ximing Lu, Swabha Swayamdipta, Chandra Bhagavatula, Noah A Smith, and Yejin Choi. 2021a. Dexperts: Decoding-time controlled text generation with experts and anti-experts. In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 6691–6706.
* Liu et al. (2021b) Fuxiao Liu, Yinghan Wang, Tianlu Wang, and Vicente Ordonez. 2021b. Visual news: Benchmark and challenges in news image captioning. In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, EMNLP 2021, Virtual Event / Punta Cana, Dominican Republic, 7-11 November, 2021_ , pages 6761–6771. Association for Computational Linguistics.
* Liu et al. (2022) Guangyi Liu, Zeyu Feng, Yuan Gao, Zichao Yang, Xiaodan Liang, Junwei Bao, Xiaodong He, Shuguang Cui, Zhen Li, and Zhiting Hu. 2022. Composable text controls in latent space with odes. _arXiv preprint arXiv:2208.00638_.
* Mai et al. (2020) Florian Mai, Nikolaos Pappas, Ivan Montero, Noah A Smith, and James Henderson. 2020\. Plug and play autoencoders for conditional text generation. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 6076–6092.
* Nukrai et al. (2022) David Nukrai, Ron Mokady, and Amir Globerson. 2022. Text-only training for image captioning using noise-injected clip. _arXiv preprint arXiv:2211.00575_.
* Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002. Bleu: a method for automatic evaluation of machine translation. In _Proceedings of the 40th annual meeting of the Association for Computational Linguistics_ , pages 311–318.
* Pascual et al. (2021) Damian Pascual, Beni Egressy, Clara Meister, Ryan Cotterell, and Roger Wattenhofer. 2021. A plug-and-play method for controlled text generation. In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 3973–3997.
* Pearl (2009) Judea Pearl. 2009. Causal inference in statistics: An overview. _Statistics surveys_ , 3:96–146.
* Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher D Manning. 2014. Glove: Global vectors for word representation. In _Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP)_ , pages 1532–1543.
* Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. 2021. Learning transferable visual models from natural language supervision. In _International Conference on Machine Learning_ , pages 8748–8763. PMLR.
* Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. 2019. Language models are unsupervised multitask learners. _OpenAI blog_ , 1(8):9.
* Socher et al. (2013) Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew Y Ng, and Christopher Potts. 2013. Recursive deep models for semantic compositionality over a sentiment treebank. In _Proceedings of the 2013 conference on empirical methods in natural language processing_ , pages 1631–1642.
* Su et al. (2022a) Yixuan Su, Tian Lan, Yahui Liu, Fangyu Liu, Dani Yogatama, Yan Wang, Lingpeng Kong, and Nigel Collier. 2022a. Language models can see: Plugging visual controls in text generation. _arXiv preprint arXiv:2205.02655_.
* Su et al. (2022b) Yixuan Su, Tian Lan, Yan Wang, Dani Yogatama, Lingpeng Kong, and Nigel Collier. 2022b. A contrastive framework for neural text generation. _arXiv preprint arXiv:2202.06417_.
* Tewel et al. (2022) Yoad Tewel, Yoav Shalev, Idan Schwartz, and Lior Wolf. 2022. Zerocap: Zero-shot image-to-text generation for visual-semantic arithmetic. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 17918–17928.
* Tu et al. (2022) Haoqin Tu, Zhongliang Yang, Jinshuai Yang, Siyu Zhang, and Yongfeng Huang. 2022\. Pcae: A framework of plug-in conditional auto-encoder for controllable text generation. _Knowledge-Based Systems_ , 256:109766.
* Tu et al. (2020) Lifu Tu, Garima Lalwani, Spandana Gella, and He He. 2020. An empirical study on robustness to spurious correlations using pre-trained language models. _Transactions of the Association for Computational Linguistics_ , 8:621–633.
* Vedantam et al. (2015) Ramakrishna Vedantam, C Lawrence Zitnick, and Devi Parikh. 2015. Cider: Consensus-based image description evaluation. In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 4566–4575.
* Xu et al. (2022) Jin Xu, Xiaojiang Liu, Jianhao Yan, Deng Cai, Huayang Li, and Jian Li. 2022. Learning to break the loop: Analyzing and mitigating repetitions for neural text generation. _arXiv preprint arXiv:2206.02369_.
* Yang and Klein (2021) Kevin Yang and Dan Klein. 2021. Fudge: Controlled text generation with future discriminators. In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 3511–3535.
* Zeng et al. (2023) Zequn Zeng, Hao Zhang, Zhengjue Wang, Ruiying Lu, Dongsheng Wang, and Bo Chen. 2023\. Conzic: Controllable zero-shot image captioning by sampling-based polishing. _arXiv preprint arXiv:2303.02437_.
* Zhao et al. (2020) Wentian Zhao, Xinxiao Wu, and Xiaoxun Zhang. 2020. Memcap: Memorizing style knowledge for image captioning. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 12984–12992.
## Appendix A Implementation Details
### A.1 General Model Details
For the base LM, we use SimCTG Su et al. (2022b), which is extended from a
pre-trained GPT-2 Radford et al. (2019) model. SimCTG essentially consists of
contrastive training and contrastive searching. As for training period, it
introduces a $\mathcal{L}_{\mathrm{CL}}$ term to learn discriminative and
isotropic token representations:
$\displaystyle\mathcal{L}_{\mathrm{CL}}$
$\displaystyle=\frac{1}{V\times(V-1)}\sum_{i=1}^{V}\sum_{j=1,j\neq
i}^{V}\max\\{0,$ (1)
$\displaystyle\rho-s\left(h_{x_{i}},h_{x_{i}}\right)+s\left(h_{x_{i}},h_{x_{j}}\right)\\},$
where $V$ is the vocabulary size, function $s(\cdot,\cdot)$ is the similarity
function, $h_{x_{i}}$ is the LM hidden state of token $x_{i}$ and $\rho$ is a
pre-defined margin. The final training objective of a LM turns into:
$\mathcal{L}_{\text{SimCTG}}=\mathcal{L}_{\mathrm{MLE}}+\mathcal{L}_{\mathrm{CL}},$
(2)
with $\mathcal{L}_{\mathrm{MLE}}$ to be the vanilla MLE objective of LM.
As for contrastive decoding, at decoding time $t$, the token to be generated
is formalized as:
$\displaystyle
S_{\text{SimCTG}}(x_{<t})=(1-\eta)\times\underbrace{p_{\theta}\left(w\mid
x_{<t}\right)}_{\text{model confidence}}-$
$\displaystyle\eta\times\underbrace{\left(\max\left\\{s\left(h_{w},h_{x_{j}}\right):1\leq
j\leq t-1\right\\}\right)}_{\text{degeneration penalty}}$ $\displaystyle
x_{t}=\underset{w\in W_{t}^{(k)}}{\arg\max}S_{\text{SimCTG}}(x_{<t})$
where $\eta$ is a parameter to balance generation diversity and consistency.
Then for our ZeroGen, we have the following decoding objective based on the
shifted LM output $p^{\prime}_{\text{LM}_{t}}$:
$\displaystyle x_{t}$ $\displaystyle=\underset{w\in
W_{t}^{(k)}}{\arg\max}\\{S_{\text{SimCTG}}(x_{<t})$
$\displaystyle+\beta_{t}\times S_{t}(w,\textbf{C}_{V}\mid x_{<t})\\},$
here $W_{t}^{(k)}$ is grouped from $p^{\prime}_{\text{LM}_{t}}$ and $S_{t}$ is
the MAGIC score we introduced in Sec. 3.2.
When we apply ZeroGen, there are several parameters should be decided in
advance: $k$ in $W_{t}^{(k)}$, $\eta$ for contrastive decoding,
$\beta,\alpha,\hat{\beta},\hat{\alpha}$ for dynamic weighting mechanism. We
present detailed parameters in Table 9 to aid reproducibility. We also present
the workflow of our system in Algorithm 1.
We implement all the experiments on the same machine with one NVIDIA GeForce
RTX 3090 GPU with 24G memory and one Intel 3.70GHz i9-10900K CPU. We will
release the code of all methods (including baselines) and datasets processing
once the paper is accepted.
Algorithm 1 ZeroGen
Input: Visual control: $\textbf{C}_{V}$, textual control: $\textbf{C}_{T}$
Output: Generated content $\textbf{X}=[x_{1},x_{2},...]$
1:initialize V; //LM vocabulary
2:initialize $\phi_{M},\phi_{T}$; //oracles
3:initialize $\hat{\beta},\hat{\alpha},k,\lambda$; //hyper params
4:compute $p(\textbf{V},\textbf{C}_{T})$ using $\phi_{T}$;
5:$x_{0}\longleftarrow\texttt{[BOS]},\textbf{X}\longleftarrow\text{[}x_{0}\text{]},t\longleftarrow
0$;
6:while $x_{t}\not=\texttt{[EOS]}$ do
7: $t\leftarrow t+1$;
8: compute $p_{\text{LM}_{t}}$ using base LM and $x_{<t}$;
9: compute $p_{t}(\textbf{C}_{T})$ using $p(\textbf{V},\textbf{C}_{T})$;
10: $D_{T}\leftarrow\sum_{n}p_{\text{LM}_{t}}(C_{T_{n}}\mid x_{<t})/N$;
11: $\alpha_{t}\leftarrow\min(D_{T}/\lambda,\hat{\alpha})$;
12: $p^{\prime}_{\text{LM}_{t}}\leftarrow p_{\text{LM}_{t}}+\alpha_{t}\times
p_{t}(\textbf{C}_{T})$;
13: $D_{V}\leftarrow\sum_{w\in
W_{t}^{(k)}}p_{\phi_{M}}(w_{t},\textbf{C}_{V})/k$;
14: $\beta_{t}\leftarrow\min(D_{V}/\lambda,\hat{\beta})$;
15: compute $S_{t}(w,\textbf{C}_{V}\mid x_{<t})$ using $\phi_{M}$;
16: $x_{t}\leftarrow\arg\max_{w}S_{t}(w,\textbf{C}_{V}\mid x_{<t})$;
17: add $x_{t}$ to content X;
18:return generated content X;
Dataset | Train | Val | Test | # Voc | # Len
---|---|---|---|---|---
F10k Humor | 6,000 | 1,000 | 1,000 | 7,186 | 14.07
F10k Romantic | 6,000 | 1,000 | 1,000 | 6,434 | 14.55
VisNews | 13,098 | 200 | 800 | 23,274 | 136.20
Table 8: Detailed statistics of data employed in our tasks. # Voc and # Len represent the vocabulary size and average sentence length of current dataset. F10k represents the Flickr10k dataset. # Params / Data | MS-COCO | Flickr30k | F10k Romantic | F10k Humor | VisNews
---|---|---|---|---|---
$k$ (int) | 45 | 25 | 45 | 45 | 5
$\eta$ (float) | 0.10 | 0.10 | 0.10 | 0.10 | 0.65
$\alpha$ (float) | 1.0 | 2.0 | 1.0 | 1.0 | 8.0
$\beta$ (float) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
$\hat{\alpha}$ (float) | 2.5 | 2.0 | 3.0 | 2.5 | 8.0
$\hat{\beta}$ (float) | 1.0 | 0.5 | 0.5 | 0.5 | 0.5
$N$ (int) | 1$\sim$5 | 1$\sim$5 | 1 | 1 | 40
Table 9: Detailed parameters of ZeroGen for different tasks.
### A.2 Image Captioning Details
For both MS-COCO and Flickr30k datasets, we take the Karpathy split Karpathy
and Fei-Fei (2015) and use the train, valid sets for base LM trainin and test
set for the task evaluation. In detail, MS-COCO dataset is under a Creative
Commons Attribution 4.0 License, and is publicly available at
https://cocodataset.org, while Flickr30k is under a Creative Commons Public
Domain Dedication License, and is publicly available at
https://www.kaggle.com/datasets/hsankesara/flickr-image-dataset. For the base
LM, we load the publicly available pre-trained language model
weights666https://huggingface.co/cambridgeltl/magic_flickr30k and
https://huggingface.co/cambridgeltl/magic_mscoco and set the maximum decoding
length to be 16 for this task.
### A.3 Stylized Captioning Details
For stylized captioning and controllable news generation task, the Flickr10k
Stylized dataset is introduced by Gan et al. (2017) and under an unknown
license, it can be publicly downloaded from
https://zhegan27.github.io/Papers/FlickrStyle_v0.9.zip. On model side, we
follow Zhao et al. (2020) to randomly sample 6,000 instances from the original
corpus as training data and 1,000 as test data. Their detailed statistics are
shown in Table 8. Following Nukrai et al. (2022), we fine-tune two base LMs on
two training sets respectively to achieve stylized outputs. As for the base LM
fine-tuning, we use SimCTG with 0.5 to be the margin $\rho$ and 1e-5 as the
learning rate to train the model until it has no further loss decrease on the
valid set. We set the maximum sentence length to 128 for base LM training and
25 for decoding.
### A.4 Controllable News Generation Details
For dataset and metrics, we conduct experiments based on VisNews Liu et al.
(2021b). This dataset is under an unknown license, and can be acquired by
asking the author directly.777https://github.com/FuxiaoLiu/VisualNews-
Repository Specifically, for image-text data, we sampled 13000, 200, and 800
image-news pairs from the original VisNews dataset as train, valid, and test
set. We use train and valid set for base LM (i.e., SimCTG) fine-tuning with
the news title as an input prompt. The test set is employed for the final
evaluation. Details are shown in Table 8. The max training news length is 200,
and the max decoding length is set to 130.
Figure 4: Classification accuracy (Prob) and CLIP-score (C-S) with varied
topic word number $N$.
For word bags of two sentiments, we follow Dathathri et al. (2019) to obtain
word lists of “happiness” and “negative words” for positive and negative
sentiment guidance respectively.888Word lists are downloaded from
www.enchantedlearning.com/wordlist. As for evaluation, diversity, we use the
CLIP model that is different from the one guides multimodal generation in
ZeroGen to compute the CLIP score (C-S). For $\Delta\text{Acc}$ (%), we take
the pre-trained SiEBRT model Hartmann et al. (2022) that has SOTA performance
on SST-2 dataset Socher et al. (2013) as the classifier.
From Table 9, we can notice that we select the topic number $N$ to be 40 in
this task, meaning at every decoding time, we input 40 sentiment words as
textual control $\textbf{C}_{T}$. We conduct ablation experiments w.r.t. topic
number $N$ and the classification accuracy as well as CLIP score in Figure 4.
From the picture, we can figure out that, as the increase of $N$, the control
degree gradually gets higher with better accuracy while the image-news
relevance declines with lower C-S. This is because more words from one
sentiment as $\textbf{C}_{T}$ make the model more focused on sentiment control
but sacrifice some image-related text generative ability.
Model | MS-COCO | Flickr30k
---|---|---
B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$ | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$
ZeroGen ($N=1$) | 59.4 | 15.5 | 18.7 | 42.3 | 55.4 | 12.1 | 54.9 | 13.1 | 15.2 | 37.4 | 26.4 | 8.3
ZeroGen ($N=2$) | 60.1 | 15.6 | 18.5 | 42.3 | 55.9 | 11.9 | 55.3 | 13.3 | 15.4 | 37.7 | 27.5 | 8.3
ZeroGen ($N=3$) | 60.4 | 15.6 | 18.6 | 42.3 | 56.5 | 12.1 | 55.0 | 13.1 | 15.2 | 37.4 | 27.1 | 8.2
ZeroGen ($N=4$) | 60.5 | 15.7 | 18.7 | 42.4 | 57.0 | 12.1 | 54.5 | 13.1 | 15.2 | 37.5 | 27.2 | 8.3
ZeroGen ($N=5$) | 60.6 | 15.8 | 18.7 | 42.4 | 57.1 | 12.1 | 55.0 | 13.0 | 15.2 | 37.5 | 27.3 | 8.2
Table 10: Caption results of ZeroGen with only varied number of objects as textual control for each generation. Model | MS-COCO | Flickr30k
---|---|---
B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$ | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$
ZeroGen SR | 59.6 | 15.3 | 18.4 | 42.1 | 55.5 | 11.8 | 54.8 | 13.1 | 15.0 | 37.2 | 26.7 | 8.1
ZeroGen MP | 59.9 | 15.2 | 18.3 | 42.0 | 55.2 | 11.8 | 54.8 | 13.2 | 15.1 | 37.5 | 26.7 | 8.1
ZeroGen WM | 60.6 | 15.8 | 18.7 | 42.4 | 57.1 | 12.1 | 55.3 | 13.3 | 15.4 | 37.7 | 27.5 | 8.3
Table 11: Caption results of ZeroGen with only three different $p_{t}(\textbf{C}_{T})$ selection methods. WM, MP, SR represent Word-wise Max, Mean Pooling and Step-wise Random in Sec. 3.1 respectively. Model | MS-COCO $\implies$ Flickr30k | Flickr30k $\implies$ MS-COCO
---|---|---
B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$ | B@1 $\uparrow$ | B@4 $\uparrow$ | M $\uparrow$ | R-L $\uparrow$ | CIDEr $\uparrow$ | SPICE $\uparrow$
ZeroCap | 49.2 | 6.2 | 11.9 | 29.3 | 18.3 | - | 46.3 | 6.0 | 13.7 | 30.1 | 27.3 | -
MAGIC | 46.4 | 6.2 | 12.2 | 31.3 | 17.5 | 5.9 | 41.4 | 5.2 | 12.5 | 30.7 | 18.3 | 5.7
ZeroGen | 50.5 | 8.1 | 13.1 | 34.5 | 17.3 | 6.0 | 46.9 | 7.6 | 14.0 | 34.4 | 26.1 | 6.8
-TDW | 50.1 | 8.0 | 12.7 | 34.0 | 17.0 | 5.8 | 46.2 | 7.1 | 13.5 | 33.7 | 23.9 | 6.2
-T | 49.3 | 8.1 | 12.5 | 33.7 | 16.7 | 5.8 | 45.1 | 6.7 | 13.1 | 33.3 | 22.4 | 5.9
-VDW | 49.3 | 7.2 | 13.0 | 33.5 | 18.5 | 6.2 | 43.8 | 6.2 | 13.5 | 32.6 | 24.5 | 6.3
-DW | 48.2 | 7.1 | 12.5 | 32.8 | 17.4 | 5.9 | 43.7 | 6.2 | 13.5 | 32.6 | 24.4 | 6.3
Table 12: Cross-domain results on two image-caption datasets MS-COCO and
Flickr30k.
### A.5 Baseline Model Details
For IC-SME, S2S-GCC and CapDec results, we directly take them from their
respect paper. For ZeroCap, we take their official implementation from
https://github.com/YoadTew/zero-shot-image-to-text and use its default
parameter setting for captioning tasks. For MAGIC, we take the official code
from https://github.com/yxuansu/MAGIC to reproduce the results.
For MAGIC+PPLM implementation, we take two codebases into consideration,
including the official PPLM code from https://github.com/uber-research/PPLM
and a simpler version of PPLM from https://github.com/hit-scma/CAT-PAW. And we
add MAGIC process at the decoding stage of PPLM as MAGIC+PPLM, we provide a
minimum reproducible code of it in this anonymous repository:
https://anonymous.4open.science/r/Pplm_Magic-3E15. For positive sentiment
control, we set the iteration for each LM hidden state update to be 5 with
step size to be 0.03, while 15 iterations for negative control. And we choose
(15+5)/2=10 iterations for decoding time measure in Sec. 5.3. We use the same
SimCTG model in ZeroGen on VisNews for MAGIC+PPLM generation. We set the max
decoding length to be 130, $k=5$ for MAGIC searching in MAGIC+PPLM like our
method. Other hyper-parameters are the default values in the official code
repositories.
## Appendix B Additional Experimental Results
### B.1 Number of Objects in $\boldsymbol{\textbf{C}_{T}}$ for Captioning
We display the full results of varying the number of objects in
$\textbf{C}_{T}$ (number $N$) for captioning task in Table 10. For images with
less than $N$ objects extracted, we just use their all existing objects as the
textual control $\textbf{C}_{T}$. We can observe that, for both datasets, more
textual guidance can bring performance gain. Nevertheless, increasing objects
in this process do not necessarily gain better metrics. For instance, on
Flickr30k, the model with $N=2$ performs the best among other $N$ settings.
This may be because too much textual guidance cause confusion for ZeroGen on
relatively easy tasks (i.e., shorter captions, smaller vocabulary size).
### B.2 $\boldsymbol{p(\textbf{C}_{T})}$ Selection Method in Captioning
We also present full results of $p(\textbf{C}_{T})$ selection methods in Table
11. On two datasets, the results are consistent and indicate that WM is the
best way for calculating $p_{t}(\textbf{C}_{T})$ at $t$-th step.
### B.3 Image Captioning in Cross Domain
For cross-domain captioning evaluation, we follow the setting in Su et al.
(2022b) and fine-tune the base LM on one dataset (e.g., MS-COCO), while
evaluating the model’s captioning capacity on another dataset (e.g.,
$\texttt{MS-COCO}\implies$ Flickr30k). Results in Table 12 can verify findings
from in-domain evaluations in Sec. 5.1.
Figure 5: Good case of news generated by our models comparing baselines.
Figure 6: Good case of news generated by our models comparing baselines.
## Appendix C More Cases of ZeroGen
In the presented cases, we highlight Positive words and Negative words
respectively.
#### Good Cases.
As shown in Figure 5 and 6, the proposed method is capable to generate image-
related and sentiment-controllable texts even with very few prompt words
(Figure 5). And our ZeroGen generally produce more diverse sentiment-specified
words like “beautiful”, “unique”, “great-looking” and “good” for positive
sentiment and “terrible”, “damaged”, “dirty” and “evil” for negative
sentiment. The compared baselines are unable to generate controllable news.
For instance, MAGIC+PPLM generates very short texts given negative sentiment
for Hair today image in Figure 5 and negative contents given positive
sentiment in Figure 6.
#### Bad Cases.
We present bad cases of ZeroGen in Figure 7 and 8. For both cases, we can
observe that there may exist generating biases caused by spurious correlation
in dataset Tu et al. (2020); Chai et al. (2022). In Figure 7, the image
describes a smiling woman with a flowered sweater, which means the visual
control may be confounded with textual control when $\textbf{C}_{T}$ includes
negative words (under the case where no task-oriented training is conducted).
Our method struggles to generate negative-only content given the image and
negative keywords. In Figure 8, the title Morrisons faces gloomy week gives
away its sentimental preference (i.e., negative) for the image. Similarly, for
both ZeroGen and MAGIC+PPLM, they fail to generate news with only positive
sentiment given positive textual control. We plan to take causal-related
solutions such as self-training and causal intervention Pearl (2009); Chai et
al. (2022) towards this issue in the future.
Figure 7: Bad case of news generated by our models comparing baselines. Figure
8: Bad case of news generated by our models comparing baselines.
## Appendix D Human Evaluation
For annotators, we hire three graduate students from America or China with
fluent English reading skills. Each annotator is assigned $100$
(instances)$\times 3$ (models)$\times 3$ (aspects) $=900$ rating tasks,
resulting in $900$ (tasks)$\times 3$ (annotators) $=2,700$ human ratings in
total. We use a three-scale scheme (i.e., integrate score 1, 2 to the poor
class, 3 to the moderate class and 4, 5 to the good class) for Fluency and
Relevance to compute the Fleiss’s kappa Fleiss and Cohen (1973). The
annotators have acknowledged the use of annotated data sets and are paid an
average annotation salary. All annotators were aware of the potential risks or
ethical concerns of machine-generated texts.
#### Annotation Instruction
Here we present the human evaluation standard:
Fluency(Flue.): Whether the generated news is fluent and easy to understand.
1. 1.
The system’s result does not make sense and it is unreadable.
2. 2.
Choose this score when you are hesitant between score 1 and score 3.
3. 3.
The system’s result contains minor errors but they do not affect your
understanding.
4. 4.
Choose this score when you are hesitant between score 3 and score 5.
5. 5.
The system’s result is human-like, grammatically correct, and very easy to
understand.
Relevance (Relv.): Whether the generated news is related to the given image
and the corresponding title.
1. 1.
The system’s result is completely irrelevant to the given image.
2. 2.
Choose this score when you are hesitant between score 1 and score 3.
3. 3.
The system’s result is partially related to the image and some of its content
can be found in the image.
4. 4.
Choose this score when you are hesitant between score 3 and score 5.
5. 5.
The system’s result is very related to the given image and contains a diverse
set of concepts in the image.
Sentiment (Sent.): Does the generated news have positive or negative
sentiment.
* •
Positive: The system’s result has a positive sentiment.
* •
Negative: The system’s result has a negative sentiment.
* •
Can’t Tell: The system’s result is neither negative nor positive.
|
[Theory of computation]: Formal languages and automata theory — Automata over
infinite objects — Quantitative automata Primary: 68Q45, 68Q70; Secondary:
68Q42
# Weighted $\omega$-Restricted One-Counter Automata
Manfred Droste Universität Leipzig, Institut für Informatik
<EMAIL_ADDRESS>and Werner Kuich Technische Universität
Wien, Institut für Diskrete Mathematik und Geometrie<EMAIL_ADDRESS>
###### Abstract.
Let $S$ be a complete star-omega semiring and $\Sigma$ be an alphabet. For a
weighted $\omega$-restricted one-counter automaton $\mathcal{C}$ with set of
states $\\{1,\dots,n\\}$, $n\geq 1$, we show that there exists a mixed
algebraic system over a complete semiring-semimodule pair
${((S\ll\Sigma^{*}\gg)^{n\times n},(S\ll\Sigma^{\omega}\gg)^{n})}$ such that
the behavior $\|\mathcal{C}\|$ of $\mathcal{C}$ is a component of a solution
of this system. In case the basic semiring is $\mathbb{B}$ or
$\mathbb{N}^{\infty}$ we show that there exists a mixed context-free grammar
that generates $\|\mathcal{C}\|$. The construction of the mixed context-free
grammar from $\mathcal{C}$ is a generalization of the well-known triple
construction in case of restricted one-counter automata and is called now
triple-pair construction for $\omega$-restricted one-counter automata.
###### Key words and phrases:
weighted pushdown automata, algebraic series, weighted contextfree grammar,
formal power series, complete semiring
This work was partially supported by DFG Graduiertenkolleg 1763 (QuantLA)
The second author was partially supported by Austrian Science Fund (FWF):
grant no. I1661 – N25.
## 1\. Introduction
Restricted one-counter pushdown automata and languages were introduced by
Greibach [13] and considered in Berstel [1], Chapter vii@ 4. These restricted
one-counter pushdown automata are pushdown automata having just one pushdown
symbol accepting by empty tape, and the family of restricted one-counter
languages is the family of languages accepted by them.
Let L be the Lukasiewicz language, i.e., the formal language over the alphabet
$\Sigma=\\{a,b\\}$ generated by the context-free grammar with productions
$S\to aSS,S\to b$. Then the family of restricted one-counter languages is the
principal cone generated by L, while the family of one-counter languages is
the full AFL generated by L.
All these results can be transferred to formal power series and restricted
one-counter automata over them (see Kuich, Salomaa [16], Example 11.5).
Restricted one-counter automata can also be used to accept infinite words and
it is this aspect we generalize in our paper.
We consider weighted $\omega$-restricted one-counter automata and their
relation to algebraic systems over the complete semiring-semimodule pair
$(S^{n\times n},V^{n})$, where $S$ is a complete star-omega semiring. It turns
out that the well-known triple construction for pushdown automata in case of
unweighted restricted one-counter automata can be generalized to a triple-pair
construction for weighted $\omega$-restricted one-counter automata. In the
classical theory, the triple construction yields for a given pushdown
automaton an equivalent context-free grammar. (See Harrison [14], Theorem
5.4.3; Bucher, Maurer [3], Sätze 2.3.10, 2.3.30; Kuich, Salomaa [16], pages
178, 306; Kuich [15], page 642; Ésik, Kuich [11], pages 77, 78.)
The paper consists of this and three more sections. In Section 2, we review
the necessary preliminaries. In Section 3, restricted one-counter matrices are
introduced and their properties are studied. The main result is that, for such
a matrix $M$, the $p$-block of the infinite column vector $M^{\omega,k}$ is a
solution of the linear equation
$z=(M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p}+M_{p,p^{2}})z$. In Section 4,
weighted $\omega$-restricted one-counter automata are introduced as a special
case of weighted $\omega$-pushdown automata. We show that for a weighted
$\omega$-restricted one-counter automaton $\mathcal{C}$ there exists a mixed
algebraic system such that the behavior $\|\mathcal{C}\|$ of $\mathcal{C}$ is
a component of a solution of this system. In Section 5 we consider the case
that the complete star-omega semiring $S$ is equal to $\mathbb{B}$ or
$\mathbb{N}^{\infty}$. Then for a given weighted $\omega$-restricted one-
counter automaton $\mathcal{C}$ a mixed context-free grammar is constructed
that generates $\|\mathcal{C}\|$. This construction is a generalization of the
well-known triple construction in case of restricted one-counter automata and
is called _triple-pair construction_ for $\omega$-restricted one-counter
automata.
## 2\. Preliminaries
For the convenience of the reader, we quote definitions and results of Ésik,
Kuich [7, 8, 10] from Ésik, Kuich [11]. The reader should be familiar with
Sections 5.1-5.6 of Ésik, Kuich [11].
A semiring $S$ is called _complete_ if it is possible to define sums for all
families $(a_{i}\mid i\in I)$ of elements of $S$, where $I$ is an arbitrary
index set, such that the following conditions are satisfied (see Conway [4],
Eilenberg [6], Kuich [15]):
(i)
$\displaystyle\sum\limits_{i\in\emptyset}a_{i}=0,\qquad\sum\limits_{i\in\\{j\\}}a_{i}=a_{j},\qquad\sum\limits_{i\in\\{j,k\\}}a_{i}=a_{j}+a_{k}\text{
for }j\neq k\,,$ (ii) $\displaystyle\sum\limits_{j\in J}\big{(}\sum_{i\in
I_{j}}a_{i}\big{)}=\sum_{i\in I}a_{i}\,,\text{ if }\ \bigcup_{j\in
J}\\!I_{j}=I\ \text{ and }\ I_{j}\cap I_{j^{\prime}}=\emptyset\ \text{ for }\
j\neq j^{\prime}\,,$ (iii) $\displaystyle\sum_{i\in I}(c\cdot
a_{i})=c\cdot\big{(}\sum_{i\in I}a_{i}\big{)},\qquad\sum_{i\in I}(a_{i}\cdot
c)=\big{(}\sum_{i\in I}a_{i}\big{)}\cdot c\,.$
This means that a semiring $S$ is complete if it is possible to define
“infinite sums” (i) that are an extension of the finite sums, (ii) that are
associative and commutative and (iii) that satisfy the distribution laws. If
$S$ is a monoid and conditions (i) and (ii) are satisfied then $S$ is called a
_complete monoid_.
A semiring S equipped with an additional unary star operation ${}^{*}:S\to S$
is called a starsemiring. In complete semirings for each element $a$, the
_star_ $a^{*}$ of $a$ is defined by
$a^{*}=\sum_{j\geq 0}a^{j}\,.$
Hence, each complete semiring is a starsemiring, called a _complete
starsemiring_.
Suppose that $S$ is a semiring and $V$ is a commutative monoid written
additively. We call $V$ a (left) $S$-semimodule if $V$ is equipped with a
(left) action
$\displaystyle S\times V$ $\displaystyle\ \to\ V$ $\displaystyle(s,v)$
$\displaystyle\ \mapsto\ sv$
subject to the following rules:
$\displaystyle s(s^{\prime}v)$ $\displaystyle=(ss^{\prime})v$ $\displaystyle
1v$ $\displaystyle=v$ $\displaystyle(s+s^{\prime})v$
$\displaystyle=sv+s^{\prime}v$ $\displaystyle\hskip 56.9055pt0v$
$\displaystyle=0$ $\displaystyle s(v+v^{\prime})$
$\displaystyle=sv+sv^{\prime}$ $\displaystyle s0$ $\displaystyle=0,$
for all $s,s^{\prime}\in S$ and $v,v^{\prime}\in V$. When V is an
$S$-semimodule, we call $(S,V)$ a _semiring-semimodule pair_.
Suppose that $(S,V)$ is a semiring-semimodule pair such that $S$ is a
starsemiring and $S$ and $V$ are equipped with an omega operation
${}^{\omega}:S\to V$. Then we call $(S,V)$ a _starsemiring-omegasemimodule
pair_.
Ésik, Kuich [9] define a _complete semiring-semimodule pair_ to be a semiring-
semimodule pair $(S,V)$ such that $S$ is a complete semiring and V is a
complete monoid with
$\displaystyle s\big{(}\sum_{i\in I}v_{i}\big{)}$ $\displaystyle=\sum_{i\in
I}sv_{i}$ $\displaystyle\big{(}\sum_{i\in I}s_{i}\big{)}v$
$\displaystyle=\sum_{i\in I}s_{i}v\,,$
for all $s\in S$, $v\in V$, and for all families $(s_{i})_{i\in I}$ over $S$
and $(v_{i})_{i\in I}$ over $V$; moreover, it is required that an _infinite
product operation_
$(s_{1},s_{2},\ldots)\ \mapsto\ \prod_{j\geq 1}s_{j}$
is given mapping infinite sequences over $S$ to $V$ subject to the following
three conditions:
$\displaystyle\prod_{i\geq 1}s_{i}$ $\displaystyle\ =\ \prod_{i\geq
1}(s_{n_{i-1}+1}\cdot\dots\cdot s_{n_{i}})$ $\displaystyle
s_{1}\cdot\prod_{i\geq 1}s_{i+1}$ $\displaystyle\ =\ \prod_{i\geq 1}s_{i}$
$\displaystyle\prod_{j\geq 1}\sum_{i_{j}\in I_{j}}s_{i_{j}}$ $\displaystyle\
=\ \sum_{(i_{1},i_{2},\dots)\in I_{1}\times I_{2}\times\dots}\prod_{j\geq
1}s_{i_{j}}\,,$
where in the first equation $0=n_{0}\leq n_{1}\leq n_{2}\leq\dots$ and
$I_{1},I_{2},\dots$ are arbitrary index sets. Suppose that $(S,V)$ is
complete. Then we define
$\displaystyle s^{*}$ $\displaystyle\ =\ \sum_{i\geq 0}s^{i}$ $\displaystyle
s^{\omega}$ $\displaystyle\ =\ \prod_{i\geq 1}s\,,$
for all $s\in S$. This turns $(S,V)$ into a starsemiring-omegasemimodule pair.
Observe that, if $(S,V)$ is a complete semiring-semimodule pair, then
$0^{\omega}=0$.
For a starsemiring $S$, we denote by $S^{n\times n}$ the semiring of $n\times
n$-matrices over $S$. If $(S,V)$ is a complete semiring-semimodule pair then,
by Ésik, Kuich [12], $(S^{n\times n},V^{n})$ is again a complete semiring-
semimodule pair.
A _star-omega semiring_ is a semiring $S$ equipped with unary operations ∗ and
${}^{\omega}:S\to S$. A star-omega semiring $S$ is called _complete_ if
$(S,S)$ is a complete semiring semimodule pair, i.e., if $S$ is complete and
is equipped with an infinite product operation that satisfies the three
conditions stated above. For the theory of infinite words and finite automata
accepting infinite words by the Büchi condition consult Perrin, Pin [17].
## 3\. Restricted one-counter matrices
In this section we introduce restricted one-counter (roc) matrices. Restricted
one-counter matrices are a special case of pushdown matrices introduced by
Kuich, Salomaa [16]. A matrix $M\in(S^{n\times
n})^{\Gamma^{*}\times\Gamma^{*}}$ is termed a _pushdown transition matrix_
(with _pushdown alphabet_ $\Gamma$ and _set of states_ $\\{1,\dots,n\\}$) if
1. (i)
for each $p\in\Gamma$ there exist only finitely many blocks $M_{p,\pi}$,
$\pi\in\Gamma^{*}$, that are non-zero;
2. (ii)
for all $\pi_{1},\pi_{2}\in\Gamma^{*}$,
$M_{\pi_{1},\pi_{2}}=\left\\{\begin{array}[]{ll}M_{p,\pi}&\hskip
5.69046pt\text{if there exist }p\in\Gamma,\pi,\pi^{\prime}\in\Gamma^{*}\text{
with }\pi_{1}=p\pi^{\prime}\text{ and }\pi_{2}=\pi\pi^{\prime},\\\ 0&\hskip
5.69046pt\text{otherwise.}\end{array}\right.$
Theorem 10.5 of Kuich, Salomaa [16] states that for pushdown matrices over
power series semirings with particular properties,
$(M^{*})_{\pi_{1}\pi_{2},\varepsilon}=(M^{*})_{\pi_{1},\varepsilon}(M^{*})_{\pi_{2},\varepsilon}$
holds for all $\pi_{1},\pi_{2}\in\Gamma^{*}$. This result is generalized in
the case of roc-matrices to arbitrary roc-matrices over complete starsemirings
in Corollary 2. Then we prove some important equalities for roc-matrices. In
Theorem 1 and Corollary 2, $S$ denotes a complete starsemiring; afterwards in
this section, $(S,V)$ denotes a complete semiring-semimodule pair.
A _restricted one-counter_ (abbreviated _roc_) _matrix (with counter symbol
$p$)_ is a matrix $M$ in $(S^{n\times n})^{p^{*}\times p^{*}}$, for some
$n\geq 1$, subject to the following condition: There exist matrices $A,B,C\in
S^{n\times n}$ such that, for all $k\geq 1$,
$M_{p^{k},p^{k+1}}=A\,,\quad M_{p^{k},p^{k}}=C\,\quad M_{p^{k},p^{k-1}}=B\,,$
and these blocks of $M$ are the only ones which may be non-zero. (Here,
$p^{*}=\\{p^{n}\mid n\geq 0\\}$. A block of $M$ is an element of the matrix
$M$ which is itself a matrix in $S^{n\times n}$.)
Observe that, for $k\geq 1$,
$\displaystyle M_{p^{k},p^{k+1}}$ $\displaystyle\ =\ M_{p,p^{2}}$
$\displaystyle\ =\ A\,,$ $\displaystyle M_{p^{k},p^{k}}$ $\displaystyle\ =\
M_{p,p}$ $\displaystyle\ =\ C\,,$ $\displaystyle M_{p^{k},p^{k-1}}$
$\displaystyle\ =\ M_{p,\varepsilon}$ $\displaystyle\ =\ B\,,$ $\displaystyle
M_{\varepsilon,p^{k}}$ $\displaystyle\ =\ M_{\varepsilon,\varepsilon}$
$\displaystyle\ =\ 0\,.$
Also note that the matrix $A$ (resp $B,C$) in $S^{n\times n}$ describes the
weight of transitions when pushing (resp., popping, not changing) an
additional symbol $p$ to (resp., from) the pushdown counter.
###### Theorem 1.
Let S be a complete starsemiring and $M$ be a roc-matrix. Then, for all $i\geq
0$,
$(M^{*})_{p^{i+1},\varepsilon}\ =\
(M^{*})_{p,\varepsilon}(M^{*})_{p^{i},\varepsilon}\,.$
###### Proof 3.1.
First observe that
$(M^{*})_{p^{i+1}\\!,\varepsilon}=\\!\\!\sum_{m\geq
0}(M^{m+1})_{p^{i+1}\\!,\varepsilon}=\\!\\!\sum_{m\geq 0}\sum_{\hskip
8.5359pti_{1},\dots,i_{m}\geq
1}\\!\\!\\!\\!\\!M_{p^{i\text{+}1}\\!,p^{i_{1}}}M_{p^{i_{1}}\\!,p^{i_{2}}}\dots
M_{p^{i_{m\text{-}1}}\\!,p^{i_{m}}}M_{p^{i_{m}}\\!,\varepsilon}\,,$
where, for $m=0$, the product equals $M_{p^{i+1},\varepsilon}$. Now we obtain
$\displaystyle(M^{*})_{p^{i+1},\varepsilon}=\ $ $\displaystyle\sum_{m\geq
0}\sum_{i_{1},\dots,i_{m}\geq 1}M_{p^{i+1},p^{i_{1}}}\dots
M_{p^{i_{m-1}},p^{i_{m}}}M_{p^{i_{m}},\varepsilon}$ $\displaystyle=\ $
$\displaystyle\sum_{m_{1}\geq 0}\big{(}\sum_{j_{1},\dots,j_{m_{1}}\geq
1}M_{p^{i+1},p^{i+j_{1}}}\dots
M_{p^{i+j_{m_{1}-1}},p^{i+j_{m_{1}}}}M_{p^{i+j_{m_{1}}},p^{i}}\big{)}\cdot$
$\displaystyle\sum_{m_{2}\geq 0}\big{(}\sum_{i_{1},\dots,i_{m_{2}}\geq
1}M_{p^{i},p^{i_{1}}}\dots
M_{p^{i_{m_{2}-1}},p^{i_{m_{2}}}}M_{p^{i_{m_{2}}},\varepsilon}\big{)}$
$\displaystyle=\ $ $\displaystyle\sum_{m_{1}\geq
0}\big{(}\sum_{j_{1},\dots,j_{m_{1}}\geq 1}M_{p,p^{j_{1}}}\dots
M_{p^{j_{m_{1}-1}},p^{j_{m_{1}}}}M_{p^{j_{m_{1}}},\varepsilon}\big{)}(M^{*})_{p^{i},\varepsilon}$
$\displaystyle=\ $
$\displaystyle(M^{*})_{p,\varepsilon}(M^{*})_{p^{i},\varepsilon}\,.$
Clearly, in each sequence leading from $p^{i+1}$ to $\varepsilon$, there is a
first time at which the top $p$ is reduced to $\varepsilon$ and at which
$p^{i}$ is seen. This moment is reached at the end of the second line. Hence,
in the second line the pushdown contents $p^{i+j_{1}},\dots,p^{i+j_{m_{1}}}$,
$m_{1}\geq 0$ are always nonempty.
###### Corollary 2.
For all $i\geq 0$, $\ (M^{*})_{p^{i},\varepsilon}\ =\
((M^{*})_{p,\varepsilon})^{i}$.
###### Lemma 3.
Let $(S,V)$ be a complete semiring-semimodule pair. Let $M\in(S^{n\times
n})^{p^{*}\times p^{*}}$ be a roc-matrix. Then
$(M^{\omega})_{p^{2}}\ =\
(M^{\omega})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega})_{p}\,.$
###### Proof 3.2.
Subsequently in the first equation we split the summation so that in the first
summand there is no factor $M_{p^{2},p}$, while in the second summand there is
at least one factor $M_{p^{2},p}$; since $k_{1},\dots,k_{m}\geq 2$,
$M_{p^{k_{m}},p}$ is the first such factor. In the second equality we use the
property of $M$ being a roc-matrix: $M_{p^{i},p^{j}}=M_{p^{i-1},p^{j-1}}$ for
$i\geq 2$, $j\geq 1$. We compute:
$\displaystyle(M^{\omega})_{p^{2}}=\ $
$\displaystyle\sum_{i_{1},i_{2},\dots\geq
2}M_{p^{2},p^{i_{1}}}M_{p^{i_{1}},p^{i_{2}}}\dots+$ $\displaystyle\sum_{m\geq
0}\sum_{k_{1},k_{2},\dots,k_{m}\geq
2}\\!\\!\\!M_{p^{2},p^{k_{1}}}M_{p^{k_{1}},p^{k_{2}}}\dots
M_{p^{k_{m}},p}\sum_{j_{1},j_{2},\dots\geq
1}\\!\\!M_{p,p^{j_{1}}}M_{p^{j_{1}},p^{j_{2}}}\dots$ $\displaystyle=\ $
$\displaystyle\sum_{i_{1},i_{2},\dots\geq
2}M_{p,p^{i_{1}-1}}M_{p^{i_{1}-1},p^{i_{2}-1}}\dots+$
$\displaystyle\sum_{m\geq 0}\sum_{k_{1},k_{2},\dots,k_{m}\geq
2}\\!\\!\\!M_{p,p^{k_{1}-1}}M_{p^{k_{1}-1},p^{k_{2}-1}}\dots
M_{p^{k_{m}-1},\varepsilon}(M^{\omega})_{p}$ $\displaystyle\
=(M^{\omega})_{p}+\sum_{m\geq 0}(M^{m+1})_{p,\varepsilon}(M^{\omega})_{p}$
$\displaystyle\ =(M^{\omega})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega})_{p}\,.$
Intuitively, our next theorem states that infinite computations starting with
$p$ on the pushdown tape yield the same matrix $(M^{\omega})_{p}$ as the sum
of the following three matrix products:
* •
$M_{p,p^{2}}(M^{\omega})_{p}$ (i.e., changing the contents of the pushdown
tape from $p$ to $pp$ and starting the infinite computations with the leftmost
$p$; the second $p$ is never read),
* •
$M_{p,p^{2}}(M^{*})_{p,\varepsilon}(M^{\omega})_{p}$ (i.e., changing the
contents of the pushdown tape from $p$ to $pp$, emptying the leftmost $p$ by
finite computations and starting the infinite computations with the rightmost
$p$),
* •
$M_{p,p}(M^{\omega})_{p}$ (i.e., changing the contents of the pushdown tape
from $p$ to $p$ and starting the infinite computations with this $p$).
The forthcoming Theorem 7 has an analogous intuitive interpretation.
###### Theorem 4.
Let $(S,V)$ be a complete semiring-semimodule pair and let $M\in(S^{n\times
n})^{p^{*}\times p^{*}}$ be a roc-matrix. Then
$(M^{\omega})_{p}\ =\
(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})(M^{\omega})_{p}\,.$
###### Proof 3.3.
We obtain, by Lemma 3
$\displaystyle(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})(M^{\omega})_{p}$
$\displaystyle=\quad$ $\displaystyle
M_{p,p^{2}}((M^{\omega})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega})_{p})+M_{p,p}(M^{\omega})_{p}$
$\displaystyle=\quad$ $\displaystyle
M_{p,p^{2}}(M^{\omega})_{p^{2}}+M_{p,p}(M^{\omega})_{p}\ =\ (MM^{\omega})_{p}\
=\ (M^{\omega})_{p}\,.$
###### Corollary 5.
Let $M\in(S^{n\times n})^{p^{*}\times p^{*}}$ be a roc-matrix. Then
$(M^{\omega})_{p}$ is a solution of
$z=(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})z\,.$
When we say “$G$ is the graph with matrix $M\in(S^{n\times n})^{p^{*}\times
p^{*}}$” then it means that $G$ is the graph with adjacency matrix
$M^{\prime}\in S^{(p^{*}\times n)\times(p^{*}\times n)}$, where $M^{\prime}$
corresponds to $M$ with respect to the canonical isomorphism between
$(S^{n\times n})^{p^{*}\times p^{*}}$ and $S^{(p^{*}\times
n)\times(p^{*}\times n)}$.
Let now $M$ be a roc-matrix and $0\leq k\leq n$. Then $M^{\omega,k}$ is the
column vector in $(V^{n})^{p^{*}}$ defined as follows: For $i\geq 1$ and
$1\leq j\leq n$, let $((M^{\omega,k})_{p^{i}})_{j}$ be the sum of all weights
of paths in the graph with matrix $M$ that have initial vertex $(p^{i},j)$ and
visit vertices $(p^{i^{\prime}},j^{\prime})$, $i^{\prime}\in\mathbb{N}$,
$j^{\prime}\in\\{1,\ldots,k\\}$, infinitely often. Observe that
$M^{\omega,0}=0$ and $M^{\omega,n}=M^{\omega}$. Later on it will be seen that
this formalizes the Büchi acceptance condition with repeated states
$\\{1,\ldots,k\\}$.
Let $P_{k}=\\{(j_{1},j_{2},\dots)\in\\{1,\dots,n\\}^{\omega}\mid j_{t}\leq
k\text{ for infinitely many }t\geq 1\\}$.
Then for $1\leq j\leq n$, we obtain
$((M^{\omega,k})_{p})_{j}=\sum_{i_{1},i_{2},\dots\geq
1}\sum_{(j_{1},j_{2},\dots)\in
P_{k}}(M_{p,p^{i_{1}}})_{j,j_{1}}(M_{p^{i_{1}},p^{i_{2}}})_{j_{1},j_{2}}(M_{p^{i_{2}},p^{i_{3}}})_{j_{2},j_{3}}\dots\,.$
By Theorem 5.4.1 of Ésik, Kuich [11], we obtain for a finite matrix $A\in
S^{n\times n}$ and for $0\leq k\leq n$, $1\leq j\leq n$,
$(A^{\omega,k})_{j}=\sum_{(j_{1},j_{2},\dots)\in
P_{k}}A_{j,j_{1}}A_{j_{1},j_{2}}A_{j_{2},j_{3}}\dots\,.$
Observe that again $A^{\omega,0}=0$ and $A^{\omega,n}=A^{\omega}$.
In the next lemma, we use the following summation identity: Assume that
$A_{1},A_{2},\dots$ are matrices in $S^{n\times n}$. Then for $0\leq k\leq n$,
$1\leq j\leq n$, and $m\geq 1$,
$\displaystyle\sum_{(j_{1},j_{2},\dots)\in
P_{k}}(A_{1})_{j,j_{1}}(A_{2})_{j_{1},j_{2}\dots}=$ $\displaystyle\sum_{1\leq
j_{1},\dots,j_{m}\leq
n}(A_{1})_{j,j_{1}}\dots(A_{m})_{j_{m-1},j_{m}}\sum_{(j_{m+1},j_{m+2},\dots)\in
P_{k}}(A_{m+1})_{j_{m},j_{m+1}}\dots\,.$
###### Lemma 6.
Let $(S,V)$ be a complete semiring-semimodule pair. Let $M\in(S^{n\times
n})^{\Gamma^{*}\times\Gamma^{*}}$ be a roc-matrix and $0\leq k\leq n$. Then
$(M^{\omega,k})_{p^{2}}\ =\
(M^{\omega,k})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega,k})_{p}\,.$
###### Proof 3.4.
We use the proof of Lemma 3, i.e., the proof for the case
$M^{\omega,n}=M^{\omega}$. For $1\leq j\leq n$, we obtain
$((M^{\omega,k})_{p^{2}})_{j}=$
$\displaystyle\sum_{i_{1},i_{2},\dots\geq 2}\sum_{(j_{1},j_{2},\dots)\in
P_{k}}(M_{p,p^{i_{1}-1}})_{j,j_{1}}(M_{p^{i_{1}-1},p^{i_{2}-1}})_{j_{1},j_{2}}\dots+$
$\displaystyle\Big{(}\sum_{1\leq j^{\prime}\leq n}\sum_{m\geq
0}\sum_{k_{1},k_{2},\dots,k_{m}\geq 2}\sum_{1\leq j_{1},\dots,j_{m}\leq
n}\\!\\!(M_{p,p^{k_{1}-1}})_{j,j_{1}}\dots(M_{p^{k_{m}-1},\varepsilon})_{j_{m},j^{\prime}}\Big{)}\
\cdot$ $\displaystyle\Big{(}\sum_{k_{m+2},k_{m+3},\dots\geq
1}\sum_{(j_{m+2},j_{m+3},\dots)\in
P_{k}}\\!\\!(M_{p,p^{k_{m+2}}})_{j^{\prime},j_{m+2}}(M_{p^{k_{m+2}},p^{k_{m+3}}})_{j_{m+2},j_{m+3}}\dots\Big{)}=$
$\displaystyle((M^{\omega,k})_{p})_{j}+\sum_{1\leq j^{\prime}\leq
n}((M^{*})_{p,\varepsilon})_{j,j^{\prime}}((M^{\omega,k})_{p})_{j^{\prime}}=$
$\displaystyle((M^{\omega,k})_{p})_{j}+((M^{*})_{p,\varepsilon}(M^{\omega,k})_{p})_{j}=$
$\displaystyle((M^{\omega,k})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega,k})_{p})_{j}\,.$
###### Theorem 7.
Let $(S,V)$ be a complete semiring-semimodule pair and let $M\in(S^{n\times
n})^{p^{*}\times p^{*}}$ be a roc-matrix. Then
$(M^{\omega,k})_{p}\ =\
(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})(M^{\omega,k})_{p}\,,$
for all $0\leq k\leq n$.
###### Proof 3.5.
We obtain, by Lemma 6, for all $0\leq k\leq n$,
$\displaystyle(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})(M^{\omega,k})_{p}$
$\displaystyle=\quad$ $\displaystyle
M_{p,p^{2}}((M^{\omega,k})_{p}+(M^{*})_{p,\varepsilon}(M^{\omega,k})_{p})+M_{p,p}(M^{\omega,k})_{p}$
$\displaystyle=\quad$ $\displaystyle
M_{p,p^{2}}(M^{\omega,k})_{p^{2}}+M_{p,p}(M^{\omega,k})_{p}\ =\
(MM^{\omega,k})_{p}\ =\ (M^{\omega,k})_{p}\,.$
###### Corollary 8.
Let $M\in(S^{n\times n})^{p^{*}\times p^{*}}$ be a roc-matrix. Then, for all
$0\leq k\leq n$, $(M^{\omega,k})_{p}$ is a solution of
$z=(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})z\,.$
## 4\. $\omega$-restricted one-counter automata
In this section, we define $\omega$-roc automata as a special case of
$\omega$-pushdown automata. We show that for an $\omega$-roc automaton
$\mathcal{C}$ there exists an algebraic system over a complete semiring-
semimodule pair such that the behavior $\|\mathcal{C}\|$ of $\mathcal{C}$ is a
component of a solution of this system.
In the sequel, $(S,V)$ is a complete semiring-semimodule pair and $S^{\prime}$
is a subset of $S$ containing $0$ an $1$. An _$S^{\prime}$ -$\omega$-pushdown
automaton_
$\mathcal{P}=(n,\Gamma,I,M,P,p_{0},k)$
is given by
1. (i)
a finite set of _states_ $\\{1,\dots,n\\}$, $n\geq 1$,
2. (ii)
an alphabet $\Gamma$ of _pushdown symbols_ ,
3. (iii)
a _pushdown transition matrix_ $M\in({S^{\prime}}^{n\times
n})^{\Gamma^{*}\times\Gamma^{*}}$,
4. (iv)
an _initial state vector_ $I\in{S^{\prime}}^{1\times n}$,
5. (v)
a _final state vector_ $P\in{S^{\prime}}^{n\times 1}$,
6. (vi)
an _initial pushdown symbol_ $p_{0}\in\Gamma$,
7. (vii)
a set of _repeated states_ $\\{1,\dots,k\\}$, $0\leq k\leq n$.
The definition of a pushdown transition matrix is given at the beginning of
Section 3. (See also Kuich, Salomaa [16], Kuich [15] and Ésik, Kuich [11].)
Clearly, any roc-matrix is a pushdown transition matrix.
The _behavior of $\mathcal{P}$_ is an element of $S\times V$ and is defined by
$\|\mathcal{P}\|=(I(M^{*})_{p_{0},\varepsilon}P,I(M^{\omega,k})_{p_{0}})\,.$
Here $I(M^{*})_{p_{0},\varepsilon}P$ is the behavior of the
$S^{\prime}$-$\omega$-pushdown automaton
$\mathcal{P}_{1}=(n,\Gamma,I,M,P,p_{0},0)$ and $I(M^{\omega,k})_{p_{0}}$ is
the behavior of the $S^{\prime}$-$\omega$-pushdown automaton
$\mathcal{P}_{2}=(n,\Gamma,I,M,0,p_{0},k)$. Observe that $\mathcal{P}_{2}$ is
an automaton with the Büchi acceptance condition: if $G$ is the graph with
adjacency matrix $M$, then only paths that visit the repeated states
${1,\dots,k}$ infinitely often contribute to $\|\mathcal{P}_{2}\|$.
Furthermore, $\mathcal{P}_{1}$ contains no repeated states and behaves like an
ordinary $S^{\prime}$-pushdown automaton.
An $S^{\prime}$-$\omega$-roc automaton is an $S^{\prime}$-$\omega$-pushdown
automaton with just one pushdown symbol such that its pushdown matrix is a
roc-matrix.
In the sequel, an $S^{\prime}$-$\omega$-roc automaton
$\mathcal{P}=(n,\\{p\\},I,M,P,p,k)$ is denoted by $\mathcal{C}=(n,I,M,P,k)$
with behavior
$\|\mathcal{C}\|=(I(M^{*})_{p,\varepsilon}P,I(M^{\omega,k})_{p})\,.$
###### Remark 9.
Consider an $S^{\prime}$-$\omega$-pushdown automaton $\mathcal{P}$ with just
one pushdown symbol. By the construction in the proof of Theorem 13.28 of
Kuich, Salomaa [16], an $S^{\prime}$-$\omega$-roc automaton $\mathcal{C}$ can
be constructed such that $\|\mathcal{C}\|=\|\mathcal{P}\|$.
The next definitions and results are taken from Ésik, Kuich [11, Section 5.6]
For the definition of an $S^{\prime}$-algebraic system over a quemiring
$S\times V$ we refer the reader to [11], page 136, and for the definition of
quemirings to [11], page 110. Here we note that a quemiring $T$ is isomorphic
to a quemiring $S\times V$ determined by the semiring-semimodule pair $(S,V)$,
cf. [11], page 110.
Observe that the forthcoming system (1) is a system over the quemiring
$S^{n\times n}\times V^{n}$. Compare the forthcoming algebraic system (2) with
the algebraic systems occurring in the proofs of Theorem 14.15 of Kuich,
Salomaa [16] and Theorem 6.4 of Kuich [15], both in the case of a roc-matrix.
Let $M$ be a roc-matrix. Consider the ${S^{\prime}}^{n\times n}$-algebraic
system over the complete semiring-semimodule pair $(S^{n\times n},V^{n})$
$y\ =\ M_{p,p^{2}}yy+M_{p,p}y+M_{p,\varepsilon}\,.$ (1)
Then by Theorem 5.6.1 of Ésik, Kuich [11] $(A,U)\in(S^{n\times n},V^{n})$ is a
solution of (1) iff $A$ is a solution of the ${S^{\prime}}^{n\times
n}$-algebraic system over $S^{n\times n}$
$x=M_{p,p^{2}}xx+M_{p,p}x+M_{p,\varepsilon}$ (2)
and $U$ is a solution of the $S^{n\times n}$-linear system over $V^{n}$
$z=M_{p,p^{2}}z+M_{p,p^{2}}Az+M_{p,p}z\,.$ (3)
###### Theorem 10.
Let $S$ be a complete starsemiring and $M$ be a roc-matrix. Then
$(M^{*})_{p,\varepsilon}$ is a solution of the ${S^{\prime}}^{n\times
n}$-algebraic system (2). If $S$ is a continuous starsemiring, then
$(M^{*})_{p,\varepsilon}$ is the least solution of (2).
###### Proof 4.1.
We obtain, by Theorem 1
$\displaystyle
M_{p,p^{2}}(M^{*})_{p,\varepsilon}(M^{*})_{p,\varepsilon}+M_{p,p}(M^{*})_{p,\varepsilon}+M_{p,\varepsilon}$
$\displaystyle=\ $ $\displaystyle
M_{p,p^{2}}(M^{*})_{p^{2},\varepsilon}+M_{p,p}(M^{*})_{p,\varepsilon}+M_{p,\varepsilon}$
$\displaystyle=\ $
$\displaystyle(MM^{*})_{p,\varepsilon}=(M^{+})_{p,\varepsilon}=(M^{*})_{p,\varepsilon}\,.$
This proves the first sentence of our theorem. The second sentence of Theorem
10 is proved by Theorem 6.4 of Kuich [15].
###### Theorem 11.
Let $(S,V)$ be a complete semiring-semimodule pair and $M$ be a roc-matrix.
Then
$((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p})\,,$
is a solution of the ${S^{\prime}}^{n\times n}$-algebraic system (1), for each
$0\leq k\leq n$.
###### Proof 4.2.
Let $0\leq k\leq n$, and consider the $S^{n\times n}$-linear system
$z=(M_{p,p^{2}}+M_{p,p^{2}}(M^{*})_{p,\varepsilon}+M_{p,p})z\,.$
By Corollary 8, $(M^{\omega,k})_{p}$ is a solution of this system. Hence, by
Theorem 5.6.1 of Ésik, Kuich [11] (see the remark above) and Theorem 10,
$((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p})$ is a solution of the system
(1).
Observe that, if $S$ is a continuous semiring, then the $S^{n\times n}$-linear
system in the proof of Theorem 11 is in fact an
$\mathfrak{Alg}({S^{\prime}})^{n\times n}$-linear system (see Kuich [15, p.
623]).
###### Theorem 12.
Let $(S,V)$ be a complete semiring-semimodule pair and let
$\mathcal{C}=(n,I,M,P,k)$ be an $S^{\prime}$-$\omega$-roc-automaton. Then
$(\|\mathcal{C}\|,((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p}))$ is a solution
of the ${S^{\prime}}^{n\times n}$-algebraic system
$y_{0}=IyP\ ,\quad y=M_{p,p^{2}}yy+M_{p,p}y+M_{p,\varepsilon}$ (4)
over the complete semiring-semimodule pair $(S^{n\times n},V^{n})$.
###### Proof 4.3.
By Theorem 11, $((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p})$ is a solution of
the second equation. Since
$I((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p})P=(I(M^{*})_{p,\varepsilon}P,I(M^{\omega,k})_{p})=\|\mathcal{C}\|\,,$
$(\|\mathcal{C}\|,((M^{*})_{p,\varepsilon},(M^{\omega,k})_{p}))$ is a solution
of the given ${S^{\prime}}^{n\times n}$-algebraic system.
Let now $S$ be a complete star-omega semiring and $\Sigma$ be an alphabet.
Then by Theorem 5.5.5 of Ésik, Kuich [11],
$(S\ll\\!\Sigma^{*}\\!\gg,S\ll\\!\Sigma^{\omega}\\!\gg)$ is a complete
semiring-semimodule pair.
Let $\mathcal{C}\\!=\\!(n,I,M,P,k)$ be an
$S\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton. Consider
the algebraic system (4) over the complete semiring-semimodule pair
${((S\ll\\!\Sigma^{*}\\!\gg)^{n\times
n},}\allowbreak{(S\ll\\!\Sigma^{\omega}\\!\gg)^{n})}$ and the mixed algebraic
system (5) over ${((S\ll\Sigma^{*}\gg)^{n\times n},}$
${(S\ll\Sigma^{\omega}\gg)^{n})}$ induced by (4)
$\displaystyle x_{0}$ $\displaystyle=IxP,\qquad$ $\displaystyle x$
$\displaystyle=M_{p,p^{2}}xx+M_{p,p}x+M_{p,\varepsilon}\,,$ (5) $\displaystyle
z_{0}$ $\displaystyle=Iz,$ $\displaystyle z$
$\displaystyle=M_{p,p^{2}}z+M_{p,p^{2}}xz+M_{p,p}z\,.$
Then, by Theorem 12,
$(I(M^{*})_{p,\varepsilon}P,(M^{*})_{p,\varepsilon},I(M^{\omega,k})_{p},(M^{\omega,k})_{p}),\
0\leq k\leq n,$
is a solution of (5). It is called _solution of order $k$_. Hence, we have
proved the next theorem.
###### Theorem 13.
Let $S$ be a complete star-omega semiring and $\mathcal{C}=(n,I,M,P,k)$ be an
$S\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton. Then
$(I(M^{*})_{p,\varepsilon}P,(M^{*})_{p,\varepsilon},I(M^{\omega,k})_{p},(M^{\omega,k})_{p}),\
0\leq k\leq n$, is a solution of the mixed algebraic system (5). ∎
Let now in (5)
$x=([i,p,j])_{1\leq i,j\leq n}$
be an $n\times n$-matrix of variables and
$z=([i,p])_{1\leq i\leq n}$
be an $n$-dimensional column vector of variables. If we write the mixed
algebraic system (5) component-wise, we obtain a mixed algebraic system over
$({(S\ll\Sigma^{*}\gg),}{(S\ll\Sigma^{\omega}\gg)})$ with variables $[i,p,j]$
over $S\ll\Sigma^{*}\gg$, where $1\leq i,j\leq n$, and variables $[i,p]$ over
$S\ll\Sigma^{\omega}\gg$, where $1\leq i\leq n$. Observe that we do not really
need $p$ in the notation of the variables. But we want to save the form of the
triple construction in connection with pushdown automata.
Let
$M_{p,p^{2}}=(a_{ij})_{1\leq i,j,\leq n},M_{p,p}=(c_{ij})_{1\leq i,j,\leq
n},M_{p,\varepsilon}=(b_{ij})_{1\leq i,j,\leq n}$
and write (5) with the matrices $x$ and $z$ of variables component-wise then
we obtain:
$\displaystyle x_{0}$ $\displaystyle=$ $\displaystyle\sum_{1\leq
m_{1},m_{2}\leq n}I_{m_{1}}[m_{1},p,m_{2}]P_{m_{2}}$ (6)
$\displaystyle[i,p,j]$ $\displaystyle=$ $\displaystyle\sum_{1\leq
m_{1},m_{2}\leq n}a_{im_{1}}[m_{1},p,m_{2}][m_{2},p,j]+$
$\displaystyle\sum_{1\leq m\leq n}c_{im}[m,p,j]+b_{ij}$ $\displaystyle z_{0}$
$\displaystyle=$ $\displaystyle\sum_{1\leq m\leq n}I_{m}[m,p]$
$\displaystyle[i,p]$ $\displaystyle=$ $\displaystyle\sum_{1\leq m\leq
n}a_{im}[m,p]+\sum_{1\leq m_{1},m_{2}\leq
n}a_{im_{1}}[m_{1},p,m_{2}][m_{2},p]+$ $\displaystyle\sum_{1\leq m\leq
n}c_{im}[m,p]$
for all $1\leq i,j\leq n$.
###### Theorem 14.
Let $S$ be a complete star-omega semiring and $\mathcal{C}=(n,I,M,P,k)$ be an
$S\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton. Then
$(\sigma_{0},((M^{*})_{p,\varepsilon})_{ij},\tau_{0},(M^{\omega,k})_{p})$
is a solution of the system (6) with $\|\mathcal{C}\|=(\sigma_{0},\tau_{0})$ .
###### Proof 4.4.
By Theorem 13.
## 5\. Mixed algebraic systems and mixed context-free grammars
In this section we associate a mixed context-free grammar with finite and
infinite derivations to the algebraic system (6). The language generated by
this mixed context-free grammar is then the behavior $\|\mathcal{C}\|$ of the
$\omega$-roc automaton $\mathcal{C}$. The construction of the mixed context-
free grammar from the $\omega$-roc automaton $\mathcal{C}$ is a generalization
of the well-known triple construction in case of roc automata and is called
now _triple-pair construction for $\omega$-roc automata_. We will consider the
commutative complete star-omega semirings
$\mathbb{B}=(\\{0,1\\},\vee,\land,*,0,1)$ with $0^{*}=1^{*}=1$ and
$\mathbb{N}^{\infty}=(\mathbb{N}\cup\\{\infty\\},+,\cdot,^{*},0,1)$ with
$0^{*}=1$ and $a^{*}=\infty$ for $a\neq\infty$.
If $S=\mathbb{B}$ or $S=\mathbb{N}^{\infty}$ and $1\leq k\leq n$, then we
associate to the mixed algebraic system (6) over
$((S\ll\Sigma^{*}\gg),(S\ll\Sigma^{\omega}\gg))$ the _mixed context-free
grammar_
$G_{k}\ =\ (X,Z,\Sigma,P_{X},P_{Z},x_{0},z_{0},k)\,.$
(See also Ésik, Kuich [11, page 139].) Here
1. (i)
$X=\\{x_{0}\\}\cup\\{[i,p,j]\mid 1\leq i,j\leq n\\}$ is a set of _variables
for finite derivations_ ;
2. (ii)
$Z=\\{z_{0}\\}\cup\\{[i,p]\mid 1\leq i\leq n\\}$ is a set of _variables for
infinite derivations_ ;
3. (iii)
$\Sigma$ is an alphabet of _terminal symbols_ ;
4. (iv)
$P_{X}$ is a finite set of _productions for finite derivations_ given below;
5. (v)
$P_{Z}$ is a finite set of _productions for infinite derivations_ given below;
6. (vi)
$x_{0}$ is the _start variable for finite derivations_ ;
7. (vii)
$z_{0}$ is the _start variable for infinite derivations_ ;
8. (viii)
$\\{[i,p]\mid 1\leq i\leq k\\}$ is the set of _repeated variables for infinite
derivations_.
In the definition of $G_{k}$ the sets $P_{X}$ and $P_{Z}$ are as follows:
$\displaystyle P_{X}=\ $ $\displaystyle\\{x_{0}\to a[m_{1},p,m_{2}]b\mid$
$\displaystyle\ \ (I_{m_{1}},a)\cdot(P_{m_{2}},b)\neq
0,a,b\in\Sigma\cup\\{\varepsilon\\},1\leq m_{1},m_{2}\leq n\\}\ \cup$
$\displaystyle\\{[i,p,j]\to a[m_{1},p,m_{2}][m_{2},p,j]\mid$ $\displaystyle\ \
(a_{im_{1}},a)\neq 0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,j,m_{1},m_{2}\leq
n\\}\ \cup$ $\displaystyle\\{[i,p,j]\to a[m,p,j]\mid(c_{im},a)\neq
0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,j,m\leq n\\}\ \cup$
$\displaystyle\\{[i,p,j]\to a\mid(b_{ij},a)\neq
0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,j\leq n\\}\,,$ $\displaystyle
P_{Z}=\ $ $\displaystyle\\{z_{0}\to a[m,p]\mid(I_{m},a)\neq
0,a\in\Sigma\cup\\{\varepsilon\\},1\leq m\leq n\\}\ \cup$
$\displaystyle\\{[i,p]\to a[m,p]\mid(a_{im},a)\neq
0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,m\leq n\\}\ \cup$
$\displaystyle\\{[i,p]\to a[m_{1},p,m_{2}][m_{2},p]\mid$ $\displaystyle\ \
(a_{im_{1}},a)\neq 0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,m_{1},m_{2}\leq
n\\}\ \cup$ $\displaystyle\\{[i,p]\to a[m,p]\mid(c_{im},a)\neq
0,a\in\Sigma\cup\\{\varepsilon\\},1\leq i,m\leq n\\}\,.$
A _finite leftmost derivation_ $\alpha_{1}\Rightarrow_{\\!L}^{*}\alpha_{2}$,
where $\alpha_{1},\alpha_{2}\in(X\cup\Sigma)^{*}$, by productions in $P_{X}$
is defined as usual. An _infinite (leftmost) derivation_
$\pi:z_{0}\Rightarrow_{\\!L}^{\omega}w$, for $z_{0}\in Z,w\in\Sigma^{\omega}$,
is defined as follows:
$\displaystyle\pi:\ $ $\displaystyle
z_{0}\Rightarrow_{\\!L}\alpha_{0}[{i_{0}},p]\Rightarrow_{\\!L}^{*}w_{0}[i_{0},p]\Rightarrow_{\\!L}w_{0}\alpha_{1}[i_{1},p]\Rightarrow_{\\!L}^{*}w_{0}w_{1}[{i_{1}},p]\Rightarrow_{\\!L}\dots$
$\displaystyle\Rightarrow_{\\!L}^{*}w_{0}w_{1}\dots
w_{m}[{i_{m}},p]\Rightarrow_{\\!L}w_{0}w_{1}\dots
w_{m}\alpha_{m+1}[{i_{m+1}},p]\Rightarrow_{\\!L}^{*}\dots\,,$
where
$z_{0}\to\alpha_{0}[{i_{0}},p],[{i_{0}},p]\to\alpha_{1}[{i_{1}},p],\dots,[{i_{m}},p]\to\alpha_{m+1}[{i_{m+1}},p],\dots$
are productions in $P_{Z}$ and $w=w_{0}w_{1}\dots w_{m}\dots$.
We now define an infinite derivation
$\pi_{k}:z_{0}\Rightarrow_{\\!L}^{\omega,k}w$ for $0\leq k\leq n$, $z_{0}\in
Z$, $w\in\Sigma^{\omega}$: We take the above definition
$\pi:z_{0}\Rightarrow^{\omega}w$ and consider the sequence of the first
elements of the variables of $X$ that are rewritten in the finite leftmost
derivation $\alpha_{m}\Rightarrow_{L}^{*}w_{m}$, $m\geq 0$. Assume this
sequence is $i_{m}^{1},i_{m}^{2},\dots,i_{m}^{t_{m}}$ for some $t_{m}$, $m\geq
1$. Then, to obtain $\pi_{k}$ from $\pi$, the condition
$i_{0},i_{1}^{1},\dots,i_{1}^{t_{1}},i_{1},i_{2}^{1},\dots,i_{2}^{t_{2}},i_{2},\dots,i_{m},i_{m+1}^{1},\dots,i_{m+1}^{t_{m+1}},i_{m+1},\dots\in
P_{k}$ has to be satisfied.
Then
$L(G_{k})=\\{w\in\Sigma^{*}\mid x_{0}\Rightarrow_{\\!L}^{*}w\\}\ \cup\
\\{w\in\Sigma^{\omega}\mid\pi:z_{0}\Rightarrow_{\\!L}^{\omega,k}w\\}\,.$
Observe that the construction of $G_{k}$ from $\mathcal{C}$ is nothing else
than a generalization of the triple construction in the case of a roc-
automaton, if $\mathcal{C}$ is viewed as a pushdown automaton, since the
construction of the context-free grammar $G=(X,\Sigma,P_{X},x_{0})$ is the
triple construction. (See Harrison [14], Theorem 5.4.3; Bucher, Maurer [3],
Sätze 2.3.10, 2.3.30; Kuich, Salomaa [16], pages 178, 306; Kuich [15], page
642; Ésik, Kuich [11], pages 77, 78.)
We call the construction of the mixed context-free grammar $G_{k}$, for $0\leq
k\leq n$, from $\mathcal{C}$ the _triple-pair construction for $\omega$-roc
automata_. This is justified by the definition of the sets of variables
$\\{[i,p,j]\mid 1\leq i,j,\leq n\\}$ and $\\{[i,p]\mid 1\leq i\leq n\\}$ of
$G_{k}$ and by the forthcoming Corollary 16.
In the next theorem we use the isomorphism between
${\mathbb{B}\ll\Sigma^{*}\gg}\times{\mathbb{B}\ll\Sigma^{\omega}\gg}$ and
$2^{\Sigma^{*}}\times 2^{\Sigma^{\omega}}$.
###### Theorem 15.
Assume that $(\sigma,\tau)$ is the solution of order $k$ of the mixed
algebraic system (6) over
$(\mathbb{B}\ll\Sigma^{*}\gg,\mathbb{B}\ll\Sigma^{\omega}\gg)$ for
$k\in\\{0,\dots,n\\}$. Then
$L(G_{k})\ =\ \sigma_{x_{0}}\cup\tau_{z_{0}}\,.$
###### Proof 5.1.
By Theorem<EMAIL_ADDRESS>of Salomaa, Soittola [18] and by Theorem 14, we obtain
$\sigma_{x_{0}}=\\{w\in\Sigma^{*}\mid x_{0}\Rightarrow_{\\!L}^{*}w\\}$. We now
show that $\tau_{z_{0}}$ is generated by the infinite derivations
$\Rightarrow_{\\!L}^{\omega,k}$ from $z_{0}$. First observe that the rewriting
by the typical $[i,p,j]$\- and $[i,p]$\- production corresponds to the
situation that in the graph of the $\omega$-restricted one counter automaton
$\mathcal{C}$ the edge from $(p\rho,i)$ to $(pp\rho,j),(p\rho,j)$ or
$(\rho,j)$, $\rho=p^{t}$ for some $t\geq 0$ is passed after the state $i$ is
visited. The first step of the infinite derivation $\pi_{k}$ is given by
$z_{0}\Rightarrow_{\\!L}\alpha_{0}[i_{0},p]$ and indicates that the path in
the graph of $\mathcal{C}$ corresponding to $\pi_{k}$ starts in state $i_{0}$.
Furthermore, the sequence of the first elements of variables that are
rewritten in $\pi_{k}$, i.e.,
$i_{0},i_{1}^{1},\dots,i_{1}^{t_{1}},i_{1},i_{2}^{2},\dots,i_{2}^{t_{2}},i_{2},\dots,i_{m},i_{m+1}^{1},\dots,i_{m+1}^{t_{m+1}},i_{m+1},\dots$
indicates that the path in the graph of $\mathcal{C}$ corresponding to
$\pi_{k}$ visits these states. Since this sequence is in $P_{k}$ the
corresponding path contributes to $\|\mathcal{C}\|$. Hence, by Theorem 14 we
obtain
$\tau_{z_{0}}=\\{w\in\Sigma^{\omega}\mid\pi:z_{0}\Rightarrow_{\\!L}^{*}w\\}\,.$
###### Corollary 16.
Assume that, for some $k\in\\{0,\dots,n\\}$, the mixed context free grammar
$G_{k}$ associated to the mixed algebraic system (6) is constructed from the
$\mathbb{B}\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton
$\mathcal{C}$. Then
$L(G_{k})=\|\mathcal{C}\|\,.$
###### Proof 5.2.
By Theorems 14 and 15.
For the remainder of this section our basic semiring is $\mathbb{N}^{\infty}$,
which allows us to draw some stronger conclusions.
###### Theorem 17.
Assume that $(\sigma,\tau)$ is the solution of order $k$ of the mixed
algebraic system (6) over
$(\mathbb{N}^{\infty}\ll\Sigma^{*}\gg,\mathbb{N}^{\infty}\ll\Sigma^{\omega}\gg)$,
$k\in\\{0,\dots,n\\}$, where
$I_{m_{1}},P_{m_{1}},a_{m_{1}m_{2}},b_{m_{1}m_{2}},c_{m_{1}m_{2}}$, $1\leq
m_{1},m_{2}\leq n$ are in
$\\{0,1\\}\langle\Sigma\cup\\{\varepsilon\\}\rangle$. Denote by $d(w)$, for
$w\in\Sigma^{*}$, the number (possibly $\infty$) of distinct finite leftmost
derivations of $w$ from $x_{0}$ with respect to $G_{k}$; and by $c(w)$, for
$w\in\Sigma^{\omega}$, the number (possibly $\infty$) of distinct infinite
leftmost derivations $\pi$ of $w$ from $z_{0}$ with respect to $G_{k}$. Then
$\sigma_{x_{0}}=\sum_{w\in\Sigma^{*}}d(w)w\qquad\text{\ and \
}\qquad\tau_{z_{0}}=\sum_{w\in\Sigma^{\omega}}c(w)w\,.$
###### Proof 5.3.
By Theorem<EMAIL_ADDRESS>of Salomaa, Soittola [18], Theorems 5.5.9 and 5.6.3 of
Ésik, Kuich [11] and Theorem 14.
In the forthcoming Corollary 18 we consider, for a given
$\\{0,1\\}\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton
$\mathcal{C}=(n,I,M,P,k)$ the number of distinct computations from an initial
instantaneous description $(i,w,p)$ for $w\in\Sigma^{*}$, $I_{i}\neq 0$, to an
accepting instantaneous description $(j,\varepsilon,\varepsilon)$, with
$P_{j}\neq 0$, $i,j\in\\{0,\dots,n\\}$.
Here $(i,w,p)$ means that $\mathcal{C}$ starts in the initial state $i$ with
$w$ on its input tape and $p$ on its pushdown tape; and
$(j,\varepsilon,\varepsilon)$ means that $\mathcal{C}$ has entered the final
state $j$ with empty input tape and empty pushdown tape.
Furthermore, we consider the number of distinct infinite computations starting
in an initial instantaneous description $(i,w,p)$ for $w\in\Sigma^{\infty}$,
$I_{i}\neq 0$.
###### Corollary 18.
Assume that, for some $k\in\\{0,\dots,n\\}$, the mixed context-free grammar
$G_{k}$ associated to the mixed algebraic system (6) is constructed from the
$\\{0,1\\}\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton
$\mathcal{C}$. Then the number (possibly $\infty$) of distinct finite leftmost
derivations of $w\in\Sigma^{*}$ from $x_{0}$ equals the number of distinct
finite computations from an initial instantaneous description for $w$ to an
accepting instantaneous description; moreover, the number (possibly $\infty$)
of distinct infinite (leftmost) derivations of $w\in\Sigma^{\omega}$ from
$z_{0}$ equals the number of distinct infinite computations starting in an
initial instantaneous description for $w$.
###### Proof 5.4.
By Corollary 3.4.12 of Ésik, Kuich [11, Theorem 4.3] and the definition of
infinite derivations with respect to $G_{k}$.
The context-free grammar $G_{k}$ associated to (6) is called _unambiguous_ if
each $w\in L(G)$, $w\in\Sigma^{*}$ has a unique finite leftmost derivation and
each $w\in L(G)$, $w\in\Sigma^{\omega}$, has a unique infinite (leftmost)
derivation.
An $\mathbb{N}^{\infty}\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc
automaton $\mathcal{C}$ is called _unambiguous_ if
$(\|\mathcal{C}\|,w)\in\\{0,1\\}$ for each
$w\in\Sigma^{*}\cup\Sigma^{\omega}$.
###### Corollary 19.
Assume that, for some $k\in\\{0,\dots,n\\}$, the mixed context-free grammar
$G_{k}$ associated to the mixed algebraic system (6) is constructed from the
$\\{0,1\\}\langle\Sigma\cup\\{\varepsilon\\}\rangle$-$\omega$-roc automaton
$\mathcal{C}$. Then $G_{k}$ is unambiguous iff $\|\mathcal{C}\|$ is
unambiguous.
In the forthcoming paper Droste, Ésik, Kuich [5] we extend the results of this
paper to weighted $\omega$-pushdown automata and obtain the triple-pair
construction for them. In the classical theory this triple-pair constructions
extends the well-known triple construction that, given an $\omega$-pushdown
automaton, yields an equivalent context-free grammar.
## Acknowledgment
The ideas of and personal discussions with Zoltán Ésik were of great influence
in preparing this paper. Thanks are due to two unknown referees for their
helpful remarks.
## References
* [1] Berstel, J.: Transductions and Context-Free Languages. Teubner, 1979.
* [2] Bloom, S. L., Ésik, Z.: Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer, 1993.
* [3] Bucher, W., Maurer, H.: Theoretische Grundlagen der Programmiersprachen. B. I. Wissenschaftsverlag, 1984.
* [4] Conway, J. H.: Regular Algebra and Finite Machines. Chapman & Hall, 1971\.
* [5] Droste, M., Ésik, Z., Kuich, W.: The triple-pair construction for weighted $\omega$-pushdown automata. In: Automata and Formal Languages (AFL 2017), EPTCS (2017) 101-113.
* [6] Eilenberg, S.: Automata, Languages and Machines. Vol. A. Academic Press, 1974.
* [7] Ésik, Z., Kuich, W.: A semiring-semimodule generalization of $\omega$-context-free languages. In: Theory is Forever (Eds.: J. Karhumäki, H. Maurer, G. Paun, G. Rozenberg), LNCS 3113, Springer, 2004, 68–80.
* [8] Ésik, Z., Kuich, W.: A semiring-semimodule generalization of $\omega$-regular languages II. Journal of Automata, Languages and Combinatorics 10 (2005) 243–264.
* [9] Ésik, Z., Kuich, W.: On iteration semiring-semimodule pairs. Semigroup Forum 75 (2007), 129–159.
* [10] Ésik, Z., Kuich, W.: A semiring-semimodule generalization of transducers and abstract $\omega$-families of power series. Journal of Automata, Languages and Combinatorics, 12 (2007), 435–454.
* [11] Ésik, Z., Kuich, W.: Modern Automata Theory. http://www.dmg.tuwien.ac.at/kuich
* [12] Ésik, Z., Kuich, W.: Continuous semiring-semimodule pairs and mixed algebraic systems. Acta Cybernetica 252 (2017) 43-59.
* [13] Greibach S. A.: An infinite hierarchy of context-free languages. Journal of the ACM 16 (1969) 91–106.
* [14] Harrison, M. A.: Introduction to Formal Language Theory. Addison-Wesley, 1978.
* [15] Kuich, W.: Semirings and formal power series: Their relevance to formal languages and automata theory. In: Handbook of Formal Languages (Eds.: G. Rozenberg and A. Salomaa), Springer, 1997, Vol. 1, Chapter 9, 609–677.
* [16] Kuich, W., Salomaa, A.: Semirings, Automata, Languages. EATCS Monographs on Theoretical Computer Science, Vol. 5. Springer, 1986.
* [17] Perrin, D., Pin, J. - E.: Infinite Words – Automata, Semigroups, Logic and Games, Elsevier, 2004.
* [18] Salomaa, A., Soittola, M.: Automata - Theoretic Aspects of Formal Power Series, Springer, 1978.
|
work in progress.5[gray].98
# Universal cutoff for Dyson Ornstein Uhlenbeck process
Jeanne Boursier (JB) Université Paris-Dauphine, PSL University, UMR 7534,
CNRS, CEREMADE, 75016 Paris, France<EMAIL_ADDRESS>, Djalil
Chafaï (DC) École Normale Supérieure, UMR 8553, CNRS, DMA, 75005 Paris,
France & Université Paris-Dauphine, PSL University, UMR 7534, CNRS, CEREMADE,
75016 Paris, France<EMAIL_ADDRESS>https://djalil.chafai.net/ and Cyril
Labbé (CL) Université de Paris, Laboratoire de Probabilités, Statistiques et
Modélisation, UMR 8001, F-75205 Paris, France<EMAIL_ADDRESS>
(Date: Summer 2021, revised Winter 2022, revised Summer 2022, compiled )
###### Abstract.
We study the Dyson–Ornstein–Uhlenbeck diffusion process, an evolving gas of
interacting particles. Its invariant law is the beta Hermite ensemble of
random matrix theory, a non-product log-concave distribution. We explore the
convergence to equilibrium of this process for various distances or
divergences, including total variation, relative entropy, and transportation
cost. When the number of particles is sent to infinity, we show that a cutoff
phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical
time. A remarkable feature is that this critical time is independent of the
parameter beta that controls the strength of the interaction, in particular
the result is identical in the non-interacting case, which is nothing but the
Ornstein–Uhlenbeck process. We also provide a complete analysis of the non-
interacting case that reveals some new phenomena. Our work relies among other
ingredients on convexity and functional inequalities, exact solvability, exact
Gaussian formulas, coupling arguments, stochastic calculus, variational
formulas and contraction properties. This work leads, beyond the specific
process that we study, to questions on the high-dimensional analysis of heat
kernels of curved diffusions.
###### Key words and phrases:
Dyson process; Ornstein–Uhlenbeck process; Coulomb gas; Random Matrix Theory;
High dimensional phenomenon; Cutoff phenomenon; High-dimensional probability;
Functional inequalities; Spectral analysis; Stochastic Calculus; Gaussian
analysis; Markov process; Diffusion process; Interacting Particle System
###### 2000 Mathematics Subject Classification:
60J60 (Diffusion processes); 82C22 (Interacting particle systems)
###### Contents
1. 1 Introduction and main results
2. 2 Additional comments and open problems
3. 3 Cutoff phenomenon for the OU
4. 4 General exactly solvable aspects
5. 5 The random matrix cases
6. 6 Cutoff phenomenon for the DOU in TV and Hellinger
7. 7 Cutoff phenomenon for the DOU in Wasserstein
8. A Distances and divergences
9. B Convexity and its dynamical consequences
## 1\. Introduction and main results
Let us consider a Markov process $X={(X_{t})}_{t\geq 0}$ with state space $S$
and invariant law $\mu$ for which
$\lim_{t\to\infty}\mathrm{dist}(\mathrm{Law}(X_{t})\mid\mu)=0$
where $\mathrm{dist}(\cdot\mid\cdot)$ is a distance or divergence on the
probability measures on $S$. Suppose now that $X=X^{n}$ depends on a
dimension, size, or complexity parameter $n$, and let us set $S=S^{n}$,
$\mu=\mu^{n}$, and $X_{0}=x^{n}_{0}\in S^{n}$. For example $X^{n}$ can be a
random walk on the symmetric group of permutations of $\\{1,\ldots,n\\}$,
Brownian motion on the group of $n\times n$ unitary matrices, Brownian motion
on the $n$-dimensional sphere, etc. In many of such examples, it has been
proved that when $n$ is large enough, the supremum over some set of initial
conditions $x^{n}_{0}$ of the quantity
$\mathrm{dist}(\mathrm{Law}(X_{t}^{n})\mid\mu^{n})$ collapses abruptly to $0$
when $t$ passes a critical value $c=c_{n}$ which may depend on $n$. This is
often referred to as a _cutoff phenomenon_. More precisely, if $\mathrm{dist}$
ranges from $0$ to $\max$, then, for some subset $S^{n}_{0}\subset S^{n}$ of
initial conditions, some critical value $c=c_{n}$ and for all
$\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\sup_{x^{n}_{0}\in
S^{n}_{0}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid\mu^{n})=\begin{cases}\max&\text{if
$t_{n}=(1-\varepsilon)c_{n}$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)c_{n}$}\end{cases}.$
It is standard to introduce, for an arbitrary small threshold $\eta>0$, the
quantity $\inf\\{t\geq 0:\sup_{x_{0}\in
S_{0}^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t})\mid\mu^{n})\leq\eta\\}$ known
as the _mixing time_ in the literature. Of course such a definition fully
makes sense as soon as $t\mapsto{\sup_{x_{0}\in
S_{0}^{n}}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t})\mid\mu^{n})$ is non-
increasing.
When $S^{n}$ is finite, it is customary to take $S^{n}_{0}=S^{n}$. When
$S^{n}$ is infinite, it may happen that the supremum over the whole set
$S^{n}$ of the distance to equilibrium remains equal to $\max$ at all times,
in which case one has to consider strict subspaces of initial conditions. For
some processes, it is possible to restrict $S^{n}_{0}$ to a single state in
which case one obtains a very precise description of the convergence to
equilibrium starting from this initial condition. Note that the constraint
over the initial condition can be made compatible with a limiting dynamics,
for instance a mean-field limit when the process describes an exchangeable
interacting particle system.
The _cutoff phenomenon_ was put forward by Aldous and Diaconis at the origin
for random walks on finite sets, see for instance [1, 28, 26, 52] and
references therein. The analysis of the cutoff phenomenon is the subject of an
important activity, still seeking for a complete theory: let us mention that,
for the total variation distance, Peres proposed the so-called product
condition (the mixing time must be much larger than the inverse of the
spectral gap) as a necessary and sufficient condition for a cutoff phenomenon
to hold, but counter-examples were exhibited [52, Sec. 18.3] and the product
condition is only necessary.
The study of the cutoff phenomenon for Markov diffusion processes goes back at
least to the works of Saloff-Coste [62, 64] in relation notably with
Nash–Sobolev type functional inequalities, heat kernel analysis, and
Diaconis–Wilson probabilistic techniques. We also refer to the more recent
work [55] for the case of diffusion processes on compact groups and symmetric
spaces, in relation with group invariance and representation theory, a point
of view inspired by the early works of Diaconis on Markov chains and of
Saloff-Coste on diffusion processes. Even if most of the available results in
the literature on the cutoff phenomenon are related to compact state spaces,
there are some notable works devoted to non-compact spaces such as [47, 19,
10, 8, 9, 11].
Our contribution is an exploration of the cutoff phenomenon for the
Dyson–Ornstein–Uhlenbeck diffusion process, for which the state space is
$\mathbb{R}^{n}$. This process is an interacting particle system. When the
interaction is turned off, we recover the Ornstein–Uhlenbeck process, a
special case that has been considered previously in the literature but for
which we also provide new results.
### 1.1. Distances
As for $\mathrm{dist}$ we use several standard distances or divergences
between probability measures: Wasserstein, total variation (TV), Hellinger,
Entropy, $\chi^{2}$ and Fisher, surveyed in Appendix A. We take the following
convention for probability measures $\mu$ and $\nu$ on the same space:
$\mathrm{dist}(\mu\mid\nu)=\begin{cases}\mathrm{Wasserstein}(\mu,\nu)&\text{when
$\mathrm{dist}=\mathrm{Wasserstein}$}\\\
\left\|\mu-\nu\right\|_{\mathrm{TV}}&\text{when
$\mathrm{dist}=\mathrm{TV}$}\\\ \mathrm{Hellinger}(\mu,\nu)&\text{when
$\mathrm{dist}=\mathrm{Hellinger}$}\\\
\mathrm{Kullback}(\mu\mid\nu)&\text{when $\mathrm{dist}=\mathrm{Kullback}$}\\\
\chi^{2}(\mu\mid\nu)&\text{when $\mathrm{dist}=\chi^{2}$}\\\
\mathrm{Fisher}(\mu\mid\nu)&\text{when
$\mathrm{dist}=\mathrm{Fisher}$}\end{cases},$ (1.1)
see Appendix A for precise definitions. The maximal value $\max$ taken by
$\mathrm{dist}$ is given by
$\max=\begin{cases}1&\text{ if
}\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\},\\\ +\infty&\text{ if
}\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}.\end{cases}$
(1.2)
### 1.2. The Dyson–Ornstein–Uhlenbeck (DOU) process and preview of main
results
The DOU process is the solution $X^{n}={(X^{n}_{t})}_{t\geq 0}$ on
$\mathbb{R}^{n}$ of the stochastic differential equation
$X^{n}_{0}=x^{n}_{0}\in\mathbb{R}^{n},\quad\mathrm{d}X_{t}^{n,i}=\sqrt{\frac{2}{n}}\mathrm{d}B^{i}_{t}-V^{\prime}(X_{t}^{n,i})\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}},\quad 1\leq i\leq n,$ (1.3)
where ${(B_{t})}_{t\geq 0}$ is a standard $n$-dimensional Brownian motion
(BM), and where
* •
$V(x)=\frac{x^{2}}{2}$ is a “confinement potential” acting through the drift
$-V^{\prime}(x)=-x$
* •
$\beta\geq 0$ is a parameter tuning the interaction strength.
The notation $X^{n,i}_{t}$ stands for the $i$-th coordinate of the vector
$X^{n}_{t}$. The process $X^{n}$ can be thought of as an interacting particle
system of $n$ one-dimensional Brownian particles $X^{n,1},\ldots,X^{n,n}$,
subject to confinement and singular pairwise repulsion when $\beta>0$
(respectively first and second term in the drift). We take an inverse
temperature of order $n$ in (1.3) in order to obtain a mean-field limit
without time-changing the process, see Section 2.5. The spectral gap is $1$
for all $n\geq 1$, see Section 2.6. We refer to Section 2.9 for other
parametrizations or choices of inverse temperature.
In the special cases $\beta\in\\{0,1,2\\}$, the cutoff phenomenon for the DOU
process can be established by using Gaussian analysis and stochastic calculus,
see Sections 1.4 and 1.5. For $\beta=0$, the process reduces to the
Ornstein–Uhlenbeck process (OU) and its behavior serves as a benchmark for the
interaction case $\beta\neq 0$, while when $\beta\in\\{1,2\\}$, the approach
involves a lift to unitary invariant ensembles of random matrix theory. For a
general $\beta\geq 1$, our main results regarding the cutoff phenomenon for
the DOU process are given in Sections 1.6 and 1.7. We are able, in particular,
to prove the following: for all
$\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{TV},\mathrm{Hellinger}\\}$,
$a>0$, $\varepsilon\in(0,1)$, we have
$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a,a]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases},$
where $P_{n}^{\beta}$ is the invariant law of the process, and where
$c_{n}:=\begin{cases}\log(\sqrt{n}a)&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\\\ \log(na)&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\end{cases}.$
This result is stated in a slightly more general form in Corollary 1.7. Our
proof relies crucially on an exceptional exact solvability of the dynamics,
notably the fact that we know explicitly the optimal long time behavior in
entropy and coupling distance, as well as the eigenfunction associated to the
spectral gap which turns out to be linear and optimal. This comes from the
special choice of $V$ as well as the special properties of the Coulomb
interaction. We stress that such an exact solvability is no longer available
for a general strongly convex $V$, even for instance in the simple example
$V(x)=\frac{x^{2}}{2}+x^{4}$ or for general linear forces. Nevertheless, and
as usual, two other special classical choices of $V$ could be explored,
related to Laguerre and Jacobi weights, see Section 2.8.
### 1.3. Analysis of the Dyson–Ornstein–Uhlenbeck process
The process $X^{n}$ was essentially discovered by Dyson in [33], in the case
$\beta\in\\{1,2,4\\}$, because it describes the dynamics of the eigenvalues of
$n\times n$ symmetric/Hermitian/symplectic random matrices with independent
Ornstein–Uhlenbeck entries, see Lemma 5.1 and Lemma 5.2 below for the cases
$\beta=1$ and $\beta=2$ respectively.
* •
Case $\beta=0$ (interaction turned off). The particles become $n$ independent
one-dimensional Ornstein–Uhlenbeck processes, and the DOU process $X^{n}$
becomes exactly the $n$-dimensional Ornstein–Uhlenbeck process $Z^{n}$ solving
(1.8). The process lives in $\mathbb{R}^{n}$. The particles collide but since
they do not interact, this does not raise any issue.
* •
Case $0<\beta<1$. Then with positive probability the particles collide
producing a blow up of the drift, see for instance [22, 25] for a discussion.
Nevertheless, it is possible to define the process for all times, for instance
by adding a local time term to the stochastic differential equation, see [25]
and references therein. It is natural to expect that the cutoff universality
works as for $\beta\not\in(0,1)$, but for simplicity we do not consider this
case here.
* •
Case $\beta\geq 1$. If we order the coordinates by defining the convex domain
$D_{n}=\\{x\in\mathbb{R}^{n}:x_{1}<\cdots<x_{n}\\},$
and if $x^{n}_{0}\in D_{n}$ then the equation (1.3) admits a unique strong
solution that never exits $D_{n}$, in other words the particles never collide
and the order of the initial particles is preserved at all times, see [60].
Moreover if
$\overline{D}_{n}=\\{x\in\mathbb{R}^{n}:x_{1}\leq\cdots\leq x_{n}\\}$
then it is possible to start the process from the boundary
$\overline{D}_{n}\setminus D_{n}$, in particular from $x^{n}_{0}$ such that
$x^{n,1}_{0}=\cdots=x^{n,n}_{0}$, and despite the singularity of the drift, it
can be shown that with probability one, $X^{n}_{t}\in D_{n}$ for all $t>0$. We
refer to [2, Th. 4.3.2] for a proof in the Dyson Brownian Motion case that can
be adapted _mutatis mutandis_.
In the sequel, we will only consider the cases $\beta=0$ with
$x_{0}^{n}\in\mathbb{R}^{n}$ and $\beta\geq 1$ with
$x^{n}_{0}\in\overline{D}_{n}$.
The drift in (1.3) is the gradient of a function, and (1.3) rewrites
$X_{0}^{n}=x_{0}^{n}\in
D_{n},\quad\mathrm{d}X_{t}^{n}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-\frac{1}{n}\nabla
E(X_{t}^{n})\mathrm{d}t,$ (1.4)
where
$E(x_{1},\ldots,x_{n})=n\sum_{i=1}^{n}V(x_{i})+{\beta}\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}$
(1.5)
can be interpreted as the energy of the configuration of particles
$x_{1},\ldots,x_{n}$.
* •
If $\beta=0$, then the Markov process $X^{n}$ is an Ornstein–Uhlenbeck
process, irreducible with unique invariant law
$P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$ which is reversible.
* •
If $\beta\geq 1$, then the Markov process $X^{n}$ is not irreducible, but
$D_{n}$ is a recurrent class carrying a unique invariant law $P_{n}^{\beta}$,
which is reversible and given by
$P_{n}^{\beta}=\frac{\mathrm{e}^{-E(x_{1},\ldots,x_{n})}}{C_{n}^{\beta}}\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n},$
(1.6)
where $C_{n}^{\beta}$ is the normalizing factor given by
$C_{n}^{\beta}=\int_{\overline{D}_{n}}\mathrm{e}^{-E(x_{1},\ldots,x_{n})}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$
(1.7)
In terms of geometry, it is crucial to observe that since $-\log$ is convex on
$(0,+\infty)$, the map
$(x_{1},\ldots,x_{n})\in
D_{n}\mapsto\mathrm{Interaction}(x_{1},\ldots,x_{n})={\beta}\sum_{i>j}\log\frac{1}{x_{i}-x_{j}},$
is convex. Thus, since $V$ is convex on $\mathbb{R}$, it follows that $E$ is
convex on $D_{n}$. For all $\beta\geq 0$, the law $P_{n}^{\beta}$ is log-
concave with respect to the Lebesgue measure as well as with respect to
$\mathcal{N}(0,\frac{1}{n}I_{n})$.
### 1.4. Non-interacting case and Ornstein–Uhlenbeck benchmark
When we turn off the interaction by taking $\beta=0$ in (1.3), the DOU process
becomes an Ornstein–Uhlenbeck process (OU) $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ on
$\mathbb{R}^{n}$ solving the stochastic differential equation
$Z^{n}_{0}=z^{n}_{0}\in\mathbb{R}^{n},\quad\mathrm{d}Z^{n}_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B^{n}_{t}-Z^{n}_{t}\mathrm{d}t,$
(1.8)
where $B^{n}$ is a standard $n$-dimensional BM. The invariant law of $Z^{n}$
is the product Gaussian law
$P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})=\mathcal{N}(0,\frac{1}{n})^{\otimes
n}$. The explicit Gaussian nature of
$Z_{t}^{n}\sim\mathcal{N}(z_{0}^{n}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n})$,
valid for all $t\geq 0$, allows for a fine analysis of convergence to
equilibrium, as in the following theorem.
###### Theorem 1.1 (Cutoff for OU: mean-field regime).
Let $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ be the OU process (1.8) and let
$P_{n}^{0}$ be its invariant law. Suppose that
$\varliminf_{n\to\infty}\frac{|z^{n}_{0}|^{2}}{n}>0\quad\text{and}\quad\varlimsup_{n\to\infty}\frac{|z^{n}_{0}|^{2}}{n}<\infty$
where $|z|=\sqrt{z_{1}^{2}+\cdots+z_{n}^{2}}$ is the Euclidean norm. Then for
all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z^{n}_{t_{n}})\mid
P_{n}^{0})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$
where
$c_{n}=\begin{cases}\frac{1}{2}\log(n)&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$},\\\ \log(n)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$},\\\
\frac{3}{2}\log(n)&\text{if $\mathrm{dist}=\mathrm{Fisher}$}.\end{cases}$
Theorem 1.1 is proved in Section 3. See Figure 1 and Figure 2 for a numerical
experiment.
Theorem 1.1 constitutes a very natural benchmark for the cutoff phenomenon for
the DOU process. Theorem 1.1 is not a surprise, and actually the TV and
Hellinger cases are already considered in [47], see also [7]. Let us mention
that in [9], a cutoff phenomenon for TV, entropy and Wasserstein is proven for
the OU process of _fixed_ dimension $d$ and vanishing noise. This is to be
compared with our setting where the dimension is sent to infinity: the results
(and their proofs) are essentially the same in these two situations, however
we will see below that if one considers more general initial conditions, there
are some substantial differences according to whether the dimension is fixed
or sent to infinity.
The restriction over the initial condition in Theorem 1.1 is spelled out in
terms of the second moment of the empirical distribution, a natural choice
suggested by the mean-field limit discussed in Section 2.5. It yields a mixing
time of order $\log(n)$, just like for Brownian motion on compact Lie groups,
see [64, 55]. For the OU process and more generally for overdamped Langevin
processes, the non-compactness of the space is replaced by the confinement or
tightness due to the drift.
Actually, Theorem 1.1 is a particular instance of the following, much more
general result that reveals that, except for the Wasserstein distance, a
cutoff phenomenon _always_ occurs.
###### Theorem 1.2 (General cutoff for OU).
Let $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ be the OU process (1.8) and let
$P_{n}^{0}$ be its invariant law. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}$.
Then, for all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z^{n}_{t_{n}})\mid
P_{n}^{0})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$,}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$
where
$c_{n}=\begin{cases}\log(\sqrt{n}|z_{0}^{n}|)\vee\frac{1}{4}\log(n)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$},\\\
\log(n|z_{0}^{n}|)\vee\frac{1}{2}\log(n)&\text{if
$\mathrm{dist}=\mathrm{Fisher}$}.\end{cases}$
Regarding the Wasserstein distance, the following dichotomy occurs:
* •
if $\lim_{n\to\infty}|z_{0}^{n}|=+\infty$, then for all $\varepsilon\in(0,1)$,
with $c_{n}=\log|z_{0}^{n}|$,
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(Z_{t_{n}}),P_{n}^{0})=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)c_{n}$,}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)c_{n}$},\end{cases}$
* •
if $\lim_{n\to\infty}|z_{0}^{n}|=\alpha\in[0,\infty)$ then there is _no cutoff
phenomenon_ namely for any $t>0$
$\lim_{n\to\infty}\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{0})=\alpha^{2}\mathrm{e}^{-2t}+2\Bigr{(}1-\sqrt{1-\mathrm{e}^{-2t}}-\tfrac{1}{2}\mathrm{e}^{-2t}\Bigr{)}.$
Theorem 1.2 is proved in Section 3.
The observation that for every distance or divergence, except for the
Wasserstein distance, a cutoff phenomenon occurs _generically_ seems to be
new.
Let us make a few comments. First, in terms of convergence to equilibrium the
relevant observable in Theorem 1.2 appears to be the Euclidean norm
$|z_{0}^{n}|$ of the initial condition. This quantity differs from the
eigenfunction associated to the spectral gap of the generator, which is given
by $z_{1}+\cdots+z_{n}$ as we will recall later on. Note by the way that (3.4)
and (2.3) are equal! Second, cutoff occurs at a time that is _independent_ of
the initial condition provided that its Euclidean norm is small enough: this
cutoff time appears as the time required to regularize the initial condition
(a Dirac mass) into a sufficiently spread out absolutely continuous
probability measure; in particular this cutoff phenomenon would not hold
generically if we allowed for spread out (non-Dirac) initial conditions. Note
that, for the OU process of _fixed_ dimension and vanishing noise, we would
not observe a cutoff phenomenon when starting from initial conditions with
small enough Euclidean norm: this is a high dimensional phenomenon. In this
respect, the Wasserstein distance is peculiar since it is much less stringent
on the local behavior of the measures at stake: for instance
$\lim_{n\to\infty}\mathrm{Wasserstein}(\delta_{0},\delta_{1/n})=0$ while for
all other distances or divergences considered here, the corresponding quantity
would remain equal to $\max$. This explains the absence of _generic_ cutoff
phenomenon for Wasserstein. Third, the explicit expressions provided in our
proof allow to extract the cutoff profile in each case, but we prefer not to
provide them in our statement and refer the interested reader to the end of
Section 3.
### 1.5. Exactly solvable intermezzo
When $\beta\neq 0$, the law of the DOU process is no longer Gaussian nor
explicit. However several exactly solvable aspects are available. Let us
recall that a Cox–Ingersoll–Ross process (CIR) of parameters $a,b,\sigma$ is
the solution $R=(R_{t})_{t\geq 0}$ on $\mathbb{R}_{+}$ of
$R_{0}=r_{0}\in\mathbb{R}_{+},\quad\mathrm{d}R_{t}=\sigma\sqrt{R_{t}}\mathrm{d}W_{t}+(a-bR_{t})\mathrm{d}t,$
(1.9)
where $W$ is a standard BM. Its invariant law is
$\mathrm{Gamma}(2a/\sigma^{2},2b/\sigma^{2})$ with density proportional to
$r\geq 0\mapsto r^{2a/\sigma^{2}-1}\mathrm{e}^{-2br/\sigma^{2}}$, with mean
$a/b$, and variance $a\sigma^{2}/(2b^{2})$. It was proved by William Feller in
[38] that the density of $R_{t}$ at an arbitrary $t$ can be expressed in terms
of special functions.
If ${(Z_{t})}_{t\geq 0}$ is a $d$-dimensional OU process of parameters
$\theta\geq 0$ and $\rho\in\mathbb{R}$, weak solution of
$\mathrm{d}Z_{t}=\theta\mathrm{d}W_{t}-\rho Z_{t}\mathrm{d}t$ (1.10)
where $W$ is a $d$-dimensional BM, then $R={(R_{t})}_{t\geq 0}$,
$R_{t}:=|Z_{t}|^{2}$, is a CIR process with parameters $a=\theta^{2}d$,
$b=2\rho$, $\sigma=2\theta$. When $\rho=0$ then $Z$ is a BM while $R=|Z|^{2}$
is a squared Bessel process.
The following theorem gathers some exactly solvable aspects of the DOU process
for general $\beta\geq 1$, which are largely already in the statistical
physics folklore, see [58]. It is based on our knowledge of eigenfunctions
associated to the first spectral values of the dynamics, see (2.6), and their
remarkable properties. As in (2.6), we set $\pi(x):=x_{1}+\cdots+x_{n}$ when
$x\in\mathbb{R}^{n}$.
###### Theorem 1.3 (From DOU to OU and CIR).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with $\beta=0$ or
$\beta\geq 1$, and let $P_{n}^{\beta}$ be its invariant law. Then:
* •
${(\pi(X^{n}_{t}))}_{t\geq 0}$ is a one-dimensional OU process weak solution
of (1.8) with $\theta=\sqrt{2}$, $\rho=1$. Its invariant law is
$\mathcal{N}(0,1)$. It does not depend on $\beta$, and
$\pi(X^{n}_{t})\sim\mathcal{N}(\pi(x^{n}_{0})\mathrm{e}^{-t},1-\mathrm{e}^{-2t})$,
$t\geq 0$. Furthermore $\pi(X^{n}_{t})^{2}$ is a CIR process of parameters
$a=2$, $b=2$, $\sigma=2\sqrt{2}$.
* •
${(|X^{n}_{t}|^{2})}_{t\geq 0}$ is a CIR process, weak solution of (1.9) with
$a=2+\beta(n-1)$, $b=2$, $\sigma=\sqrt{8/n}$. Its invariant law is
$\mathrm{Gamma}(\frac{1}{2}(n+\beta\frac{n(n-1)}{2}),\frac{n}{2})$ of mean
$1+\frac{\beta}{2}(n-1)$ and variance $\beta+\frac{2-\beta}{n}$. Furthermore,
if $d=n+\beta\frac{n(n-1)}{2}$ is a positive integer, then
${(|X^{n}_{t}|^{2})}_{t\geq 0}$ has the law of ${(|Z_{t}|^{2})}_{t\geq 0}$
where ${(Z_{t})}_{t\geq 0}$ is a $d$-dimensional OU process, weak solution of
(1.8) with $\theta=\sqrt{2/n}$, $\rho=1$, and $Z_{0}=z^{n}_{0}$ for an
arbitrary $z^{n}_{0}\in\mathbb{R}^{d}$ such that $|z^{n}_{0}|=|x^{n}_{0}|$.
At this step it is worth noting that Theorem 1.3 gives in particular, denoting
$\beta_{n}:=1+\frac{\beta}{2}(n-1)$,
$\mathbb{E}[\pi(X^{n}_{t})]=\pi(x^{n}_{0})\mathrm{e}^{-t}\underset{t\to\infty}{\longrightarrow}0\quad\text{and}\quad\mathbb{E}[|X^{n}_{t}|^{2}]=\beta_{n}+(|x^{n}_{0}|^{2}-\beta_{n})\mathrm{e}^{-2t}\underset{t\to\infty}{\longrightarrow}\beta_{n}.$
(1.11)
Following [25, Sec. 2.2], the limits can also be deduced from the
Dumitriu–Edelman tridiagonal random matrix model [32] isospectral to
$\beta$-Hermite. These formulas for the “transient” first two moments
$\mathbb{E}[\pi(X_{t}^{n})]$ and $\mathbb{E}[|X_{t}^{n}|^{2}]$ reveal an
abrupt convergence to their equilibrium values :
* •
If $\lim_{n\to\infty}\frac{\pi(x^{n}_{0})}{n}=\alpha\neq 0$ then for all
$\varepsilon\in(0,1)$,
$\lim_{n\to\infty}|\mathbb{E}[\pi(X^{n}_{t_{n}})]|=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)\log(n)$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)\log(n)$}\end{cases}.$ (1.12)
* •
If $\lim_{n\to\infty}\frac{|x^{n}_{0}|^{2}}{n}=\alpha\neq\frac{\beta}{2}$ then
for all $\varepsilon\in(0,1)$, denoting $\beta_{n}:=1+\frac{\beta}{2}(n-1)$,
$\lim_{n\to\infty}\left|\mathbb{E}[|X^{n}_{t_{n}}|^{2}]-\beta_{n}\right|=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)\frac{1}{2}\log(n)$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)\frac{1}{2}\log(n)$}\end{cases}.$ (1.13)
These critical times are universal with respect to $\beta$. The first two
transient moments are related to the eigenfunctions (2.6) associated to the
first two non-zero eigenvalues of the dynamics. Higher order transient moments
are related to eigenfunctions associated to higher order eigenvalues. Note
that $\mathbb{E}[\pi(X^{n}_{t})]$ and $\mathbb{E}[|X^{n}_{t}|^{2}]$ are the
first two moments of the non-normalized mean empirical measure
$\mathbb{E}[\sum_{i=1}^{n}\delta_{X^{n,i}_{t}}]$, and this lack of
normalization is responsible of the critical times of order $\log(n)$. In
contrast, the first two moments of the normalized mean empirical measure
$\mathbb{E}[\frac{1}{n}\sum_{i=1}^{n}\delta_{X^{n,i}_{t}}]$, given by
$\frac{1}{n}\mathbb{E}[\pi(X^{n}_{t})]$ and
$\frac{1}{n}\mathbb{E}[|X^{n}_{t}|^{2}]$ respectively, do not exhibit a
critical phenomenon. This is related to the exponential decay of the first two
moments in the mean-field limit (2.12), as well as the lack of cutoff for
Wasserstein already revealed for OU by Theorem 1.2. This also reminds the high
dimension behavior of norms in the field of the asymptotic geometric analysis
of convex bodies. In another direction, this elementary observation on the
moments also illustrates that the cutoff phenomenon for a given quantity is
not stable under rather simple transformations of this quantity.
From the first part of Theorem 1.3 and contraction properties available for
_some_ distances or divergences, see Lemma A.2, we obtain the following lower
bound on the mixing time for the DOU, which is independent of $\beta$:
###### Corollary 1.4 (Lower bound on the mixing time).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Set
$c_{n}:=\begin{cases}\log(|\pi(x_{0}^{n})|)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log\Big{(}\frac{|\pi(x_{0}^{n})|}{\sqrt{n}}\Big{)}&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases},$
and assume that $\lim_{n\to\infty}c_{n}=\infty$. Then, for all
$\varepsilon\in(0,1)$, we have
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=\max.$
Theorem 1.3 and Corollary 1.4 are proved in Section 4.
The derivation of an upper bound on the mixing time is much more delicate:
once again recall that the case $\beta=0$ covered by Theorem 1.2 is specific
as it relies on exact Gaussian computations which are no longer available for
$\beta\geq 1$. In the next subsection, we will obtain results for general
values of $\beta\geq 1$ via more elaborate arguments.
In the specific cases $\beta\in\\{1,2\\}$, there are some exactly solvable
aspects that one can exploit to derive, in particular, precise upper bounds on
the mixing times. Indeed, for these values of $\beta$, the DOU process is the
process of eigenvalues of the matrix-valued OU process:
$M_{0}=m_{0},\quad\mathrm{d}M_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-M_{t}\mathrm{d}t,$
where $B$ is a BM on the symmetric $n\times n$ matrices if $\beta=1$ and on
Hermitian $n\times n$ matrices if $\beta=2$, see (5.4) and (5.16) for more
details. Based on this observation, we can deduce an upper bound on the mixing
times by contraction (for _most_ distances or divergences).
###### Theorem 1.5 (Upper bound on mixing time in matrix case).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with
$\beta\in\\{0,1,2\\}$, and invariant law $P_{n}^{\beta}$, and
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Set
$c_{n}:=\begin{cases}\log(\sqrt{n}|x_{0}^{n}|)\vee\log(\sqrt{n})&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log(|x_{0}^{n}|)&\text{if $\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases},$
and assume that $\lim_{n\to\infty}c_{n}=\infty$ if
$\mathrm{dist}=\mathrm{Wasserstein}$. Then, for all $\varepsilon\in(0,1)$, we
have
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$
Combining this upper bound with the lower bound already obtained above, we
derive a cutoff phenomenon in this particular matrix case.
###### Corollary 1.6 (Cutoff for DOU in the matrix case).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with
$\beta\in\\{0,1,2\\}$, and invariant law $P_{n}^{\beta}$. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Let ${(a_{n})}_{n}$ be a real sequence satisfying $\inf_{n}\sqrt{n}a_{n}>0$,
and assume further that $\lim_{n\to\infty}\sqrt{n}a_{n}=\infty$ if
$\mathrm{dist}=\mathrm{Wasserstein}$. Then, for all $\varepsilon\in(0,1)$, we
have
$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$
where
$c_{n}:=\begin{cases}\log(na_{n})&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log(\sqrt{n}a_{n})&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$
Theorem 1.5 and Corollary 1.6 are proved in Section 5.
It is worth noting that $d=n+\beta\frac{n(n-1)}{2}$ in Theorem 1.3 is indeed
an integer in the “random matrix” cases $\beta\in\\{1,2\\}$, and corresponds
then exactly to the degree of freedom of the Gaussian random matrix models GOE
and GUE respectively. More precisely, if we let $X^{n}_{\infty}\sim
P_{n}^{\beta}$ then:
* •
If $\beta=1$ then $P_{n}^{\beta}$ is the law of the eigenvalues of
$S\sim\mathrm{GOE}_{n}$, and $|X^{n}_{\infty}|^{2}=\sum_{j,k=1}^{n}S_{jk}^{2}$
which is the sum of $n$ squared Gaussians of variance $v=1/n$ (diagonal) plus
twice the sum of $\frac{n^{2}-n}{2}$ squared Gaussians of variance
$\frac{v}{2}$ (off-diagonal) all being independent. The duplication has the
effect of renormalizing the variance from $\frac{v}{2}$ to $v$. All in all we
have the sum of $d=\frac{n^{2}+n}{2}$ independent squared Gaussians of same
variance $v$. See Section 5.
* •
If $\beta=2$ then $P_{n}^{\beta}$ is the law of the eigenvalues of
$H\sim\mathrm{GUE}_{n}$, and
$|X^{n}_{\infty}|^{2}=\sum_{j,k=1}^{n}|H_{jk}|^{2}$ is the sum of $n$ squared
Gaussians of variance $v=1/n$ (diagonal) plus twice the sum of $n^{2}-n$
squared Gaussians of variance $\frac{v}{2}$ (off-diagonal) all being
independent. All in all we have the sum of $d=n^{2}$ independent squared
Gaussians of same variance $v$. See Section 5.
Another manifestation of exact solvability lies at the level of functional
inequalities. Indeed, and following [25], the optimal Poincaré constant of
$P_{n}^{\beta}$ is given by $1/n$ and does not depend on $\beta$, and the
extremal functions are tranlations/dilations of
$x\mapsto\pi(x)=x_{1}+\cdots+x_{n}$. This corresponds to a spectral gap of the
dynamics equal to $1$ and its associated eigenfunction. Moreover, the optimal
logarithmic Sobolev inequality of $P_{n}^{\beta}$ (Lemma B.1) is given by
$2/n$ and does not depend on $\beta$, and the extremal functions are of the
form $x\mapsto\mathrm{e}^{c(x_{1}+\cdots+x_{n})}$, $c\in\mathbb{R}$. This
knowledge of the optimal constants and extremal functions and their
independence with respect to $\beta$ is truly remarkable. It plays a crucial
role in the results presented in this article. More precisely, the optimal
Poincaré inequality is used for the lower bound via the first eigenfunctions
while the optimal logarithmic Sobolev inequality is used for the upper bound
via exponential decay of the entropy.
### 1.6. Cutoff in the general interacting case
Our main contribution consists in deriving an upper bound on the mixing times
in the general case $\beta\geq 1$: the proof relies on the logarithmic Sobolev
inequality, some coupling arguments and a regularization procedure.
###### Theorem 1.7 (Upper bound on the mixing time: the general case).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with $\beta=0$ or
$\beta\geq 1$ and invariant law $P_{n}^{\beta}$. Take
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Wasserstein}\\}$.
Set
$c_{n}:=\begin{cases}\log(\sqrt{n}|x_{0}^{n}|)\vee\log({n})&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\\\
\log(|x_{0}^{n}|)\vee\log(\sqrt{n})&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$
Then, for all $\varepsilon\in(0,1)$, we have
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$
Combining this upper bound with the general lower bound that we obtained in
Corollary 1.4, we deduce the following cutoff phenomenon. Observe that it
holds both for $\beta=0$ and $\beta\geq 1$, and that the expression of the
mixing time does not depend on $\beta$.
###### Corollary 1.8 (Cutoff for DOU in the general case).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$ and invariant law $P_{n}^{\beta}$. Take
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Wasserstein}\\}$.
Let ${(a_{n})}_{n}$ be a real sequence satisfying $\inf_{n}a_{n}>0$. Then, for
all $\varepsilon\in(0,1)$, we have
$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$
where
$c_{n}:=\begin{cases}\log(na_{n})&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\\\
\log(\sqrt{n}a_{n})&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$
The proofs of Theorem 1.7 and Corollary 1.8 for the TV and Hellinger distances
are presented in Section 6. The Wasserstein distance is treated in Section 7.
Let us make a comment on the assumptions made on $a_{n}$ in Corollaries 1.6
and 1.8. They are dictated by the upper bounds established in Theorems 1.5 and
1.7, which take the form of maxima of two terms: one that depends on the
initial condition, and another one which is a power of a logarithm of $n$. The
logarithmic term is an upper bound on the time required to regularize a
pointwise initial condition, its precise expression varies according to the
method of proof we rely on: in the matrix case, it is the time required to
regularize a larger object, the matrix-valued OU process; in the general case,
it is related to the time it takes to make the entropy of a pointwise initial
condition small. These bounds are not optimal for $\beta=0$ (compare with
Theorem 1.2), and probably neither for $\beta\geq 1$.
A natural, but probably quite difficult, goal would be to establish a cutoff
phenomenon in the situation where the set of initial conditions is reduced to
_any_ given singleton, as in Theorem 1.2 for the case $\beta=0$. Recall that
in that case, the asymptotic of the mixing time is dictated by the Euclidean
norm of the initial condition. In the case $\beta\geq 1$, this _cannot_ be the
right observable since the Euclidean norm does not measure the distance to
equilibrium. Instead one should probably consider the Euclidean norm
$|x_{0}^{n}-\rho_{n}|$, where $\rho_{n}$ is the vector of the quantiles of
order $1/n$ of the semi-circle law that arises in the mean-field limit
equilibrium (see Subsection 2.5). More precisely
$\rho_{n,i}=\inf\left\\{t\in\mathbb{R}:\int_{-\infty}^{t}\frac{\sqrt{2\beta-x^{2}}}{\beta\pi}\mathbf{1}_{x\in[-\sqrt{2\beta},\sqrt{2\beta}]}\mathrm{d}x\geq\frac{i}{n}\right\\},\quad
i\in\\{1,\ldots,n\\}.$ (1.14)
Note that $\rho_{n}=0$ when $\beta=0$.
A first step in this direction is given by the following result:
###### Theorem 1.9 (DOU in the general case and pointwise initial condition).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. There hold
* •
If $\lim_{n\to\infty}|x^{n}_{0}-\rho_{n}|=+\infty$, then, denoting
$t_{n}=\log(|x_{0}^{n}-\rho_{n}|)$, for all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X_{(1+\varepsilon)t_{n}}),P_{n}^{\beta})=0.$
* •
If $\lim_{n\to\infty}|x^{n}_{0}-\rho_{n}|=\alpha\in[0,\infty)$, then, for all
$t>0$,
$\varlimsup_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X_{t}),P_{n}^{\beta})^{2}\leq\alpha^{2}\mathrm{e}^{-2t}.$
Theorem 1.9 is proved in Section 7.
### 1.7. Non-pointwise initial conditions
It is natural to ask about the cutoff phenomenon when the initial conditions
$X^{n}_{0}$ is not pointwise. Even if we turn off the interaction by taking
$\beta=0$, the law of the process at time $t$ is then no longer Gaussian in
general, which breaks the method of proof used for Theorem 1.1 and Theorem
1.2. Nevertheless, Theorem 1.10 below provides a universal answer, that is
both for $\beta=0$ and $\beta\geq 1$, at the price however of introducing
several objects and notations. More precisely, for any probability measure
$\mu$ on $\mathbb{R}^{n}$, we introduce
$S(\mu)=\begin{cases}\displaystyle\int\frac{\mathrm{d}\mu}{\mathrm{d}x}\log\frac{\mathrm{d}\mu}{\mathrm{d}x}\mathrm{d}x=\text{``}\mathrm{Kullback}(\mu\mid\mathrm{d}x)\text{''}&\text{if
$\displaystyle\frac{\mathrm{d}\mu}{\mathrm{d}x}\log\frac{\mathrm{d}\mu}{\mathrm{d}x}\in
L^{1}(\mathrm{d}x)$}\\\ +\infty&\text{otherwise}\end{cases}.$ (1.15)
Note that $S$ takes its values in the whole $(-\infty,+\infty]$, and when
$S(\mu)<+\infty$ then $-S(\mu)$ is the _Boltzmann–Shannon entropy_ of the law
$\mu$. For all $x\in\mathbb{R}^{n}$ with $x_{i}\neq x_{j}$ for all $i\neq j$,
we have
$E(x_{1},\ldots,x_{n})=n^{2}\iint\Phi(x,y)\mathbf{1}_{\\{x\neq
y\\}}L_{n}(\mathrm{d}x)L_{n}(\mathrm{d}y)$ (1.16)
where $\displaystyle L_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ and where
$\displaystyle\Phi(x,y):=\frac{n}{n-1}\frac{V(x)+V(y)}{2}+\frac{\beta}{2}\log\frac{1}{|x-y|}$.
Let us define the map $\Psi:\mathbb{R}^{n}\mapsto\overline{D}_{n}$ by
$\Psi(x_{1},\ldots,x_{n}):=(x_{\sigma(1)},\ldots,x_{\sigma(n)}).$ (1.17)
where $\sigma$ is any permutation of $\\{1,\ldots,n\\}$ that reorders the
particles non-decreasingly.
###### Theorem 1.10 (Cutoff for DOU with product smooth initial conditions).
Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. Let $S$, $\Phi$, and $\Psi$
be as in (1.15), (1.16), and (1.17). Let us assume that
$\mathrm{Law}(X_{0}^{n})$ is the image law or push forward of a product law
$\mu_{1}\otimes\cdots\otimes\mu_{n}$ by $\Psi$ where $\mu_{1},\ldots,\mu_{n}$
are laws on $\mathbb{R}$. Then:
1. (1)
If $\displaystyle\varliminf_{n\to\infty}\Bigr{|}\frac{1}{n}\sum_{i=1}^{n}\int
x\mu_{i}(\mathrm{d}x)\Bigr{|}\neq 0$ then, for all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1-\varepsilon)\log(n)})\mid
P_{n}^{\beta})=+\infty.$
2. (2)
If
$\displaystyle\varlimsup_{n\to\infty}\frac{1}{n^{2}}\sum_{i=1}^{n}S(\mu_{i})<\infty$
and $\displaystyle\varlimsup_{n\to\infty}\frac{1}{n^{2}}\sum_{i\neq
j}\iint\Phi\,\mathrm{d}\mu_{i}\otimes\mathrm{d}\mu_{j}<\infty$, then, for all
$\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1+\varepsilon)\log(n)})\mid
P_{n}^{\beta})=0.$
Theorem 1.10 is proved in Section 6.3.
It is likely that Theorem 1.10 can be extended to the case
$\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{Hellinger},\mathrm{Fisher}\\}$.
### 1.8. Structure of the paper
* •
Section 2 provides additional comments and open problems.
* •
Section 3 focuses on the OU process ($\beta=0$) and gives the proofs of
Theorems 1.1 and 1.2.
* •
Section 4 concerns the exact solvability of the DOU process for all $\beta$,
and provides the proofs of Theorem 1.3 and Corollary 1.4.
* •
Section 5 is about random matrices and gives the proofs of Theorem 1.5 and
Corollary 1.6.
* •
Section 6 deals with the DOU process for all $\beta$ with the TV and Hellinger
distances, and provides the proofs of Theorem 1.7 and Corollary 1.8.
* •
Section 7 gives the Wasserstein counterpart of Section 6 and the proof of
Theorem 1.9.
* •
Appendix A provides a survey on distances and divergences, with new results.
* •
Appendix B gathers useful dynamical consequences of convexity.
## 2\. Additional comments and open problems
### 2.1. About the results and proofs
The proofs of our results rely among other ingredients on convexity and
optimal functional inequalities, exact solvability, exact Gaussian formulas,
coupling arguments, stochastic calculus, variational formulas, contraction
properties and regularization.
The proofs of Theorems 1.1 and 1.2 are based on the explicit Gaussian nature
of the OU process, which allows to use Gaussian formulas for all the distances
and divergences that we consider (the Gaussian formula for $\mathrm{Fisher}$
seems to be new). Our analysis of the convergence to equilibrium of the OU
process seems to go beyond what is already known, see for instance [47] and
[10, 8, 9, 11].
Theorem 1.3 is a one-dimensional analogue of [15, Th. 1.2]. The proof exploits
the explicit knowledge of eigenfunctions of the dynamics (2.6), associated
with the first two non-zero spectral values, and their remarkable properties.
The first one is associated to the spectral gap and the optimal Poincaré
inequality. It implies Corollary 1.4, which is the provider of all our lower
bounds on the mixing time for the cutoff.
The proof of Theorem 1.5 is based on a contraction property and the upper
bound for matrix OU processes. It is not available beyond the matrix cases.
All the other upper bounds that we establish are related to an optimal
exponential decay which comes from convexity and involves sometimes coupling,
the simplest instance being Theorem 1.7 about the Wasserstein distance. The
usage of the Wasserstein metrics for Dyson dynamics is quite natural, see for
instance [13].
The proof of Theorem 1.7 for the $\mathrm{TV}$ and $\mathrm{Hellinger}$ relies
on the knowledge of the optimal exponential decay of the entropy (with respect
to equilibrium) related to the optimal logarithmic Sobolev inequality. Since
pointwise initial conditions have infinite entropy, the proof proceeds in
three steps: first we regularize the initial condition to make its entropy
finite, second we use the optimal exponential decay of the entropy of the
process starting from this regularized initial condition, third we control the
distance between the processes starting from the initial condition and its
regularized version. This last part is inspired by a work of Lacoin [48] for
the simple exclusion process on the segment, subsequently adapted to
continuous state-spaces [18, 19], where one controls an _area_ between two
versions of the process.
The (optimal) exponential decay of the entropy (Lemma B.2) is equivalent to
the (optimal) logarithmic Sobolev inequality (Lemma B.1). For the DOU process,
the optimal logarithmic Sobolev inequality provided by Lemma B.1 achieves also
the universal bound with respect to the spectral gap, just like for Gaussians.
This sharpness between the best logarithmic Sobolev constant and the spectral
gap also holds for instance for the random walk on the hypercube, a discrete
process for which a cutoff phenomenon can be established with the optimal
logarithmic Sobolev inequality, and which can be related to the OU process,
see for instance [30, 29] and references therein. If we generalize the DOU
process by adding an arbitrary convex function to $V$, then we will still have
a logarithmic Sobolev inequality – see [25] for several proofs including the
proof via the Bakry–Émery criterion – however the optimal logarithmic Sobolev
constant will no longer be explicit nor sharp with respect to the spectral
gap, and the spectral gap will no longer be explicit.
The proof of Theorem 1.10 relies crucially on the tensorization property of
$\mathrm{Kullback}$ and on the asymptotics on the normalizing constant
$C_{n}^{\beta}$ at equilibrium.
### 2.2. Analysis and geometry of the equilibrium
The full space $\mathbb{R}^{n}$ is, up to a bunch of hyperplanes, covered with
$n!$ disjoint isometric copies of the convex domain $D_{n}$ obtained by
permuting the coordinates (simplices or Weyl chambers). Following [25], for
all $\beta\geq 0$ let us define the law $P_{*n}^{\beta}$ on $\mathbb{R}^{n}$
with density proportional to $\mathrm{e}^{-E}$, just like for $P_{n}^{\beta}$
in (1.6) but without the
$\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}$.
If $\beta=0$ then $P_{*n}^{0}=P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$
according to our definition of $P_{n}^{0}$.
If $\beta>0$ then $P_{*n}^{\beta}$ has density
$(C_{*n}^{\beta})^{-1}\mathrm{e}^{-E}$ with $C_{*n}^{\beta}=n!C_{n}^{\beta}$
where $C_{n}^{\beta}$ is the normalization of $P_{n}^{\beta}$. Moreover
$P_{*n}^{\beta}$ is a mixture of $n!$ isometric copies of $P_{n}^{\beta}$,
while $P_{n}^{\beta}$ is the image law or push forward of $P_{*n}^{\beta}$ by
the map $\Psi_{n}:\mathbb{R}^{n}\to\overline{D}_{n}$ defined in (1.17).
Furthermore for all bounded measurable $f:\mathbb{R}^{n}\to\mathbb{R}$,
denoting $\Sigma_{n}$ the symmetric group of permutations of
$\\{1,\ldots,n\\}$,
$\int f\mathrm{d}P_{*n}^{\beta}=\int
f_{\mathrm{sym}}\mathrm{d}P_{n}^{\beta}\quad\text{with}\quad
f_{\mathrm{sym}}(x_{1},\ldots,x_{n}):=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).$
Regarding log-concavity, it is important to realize that if $\beta=0$ then $E$
is convex on $\mathbb{R}^{n}$, while if $\beta>0$ then $E$ is convex on
$D_{n}$ but is not convex on $\mathbb{R}^{n}$ and has $n!$ isometric local
minima.
* •
The law $P_{*n}^{\beta}$ is centered but is not log-concave when $\beta>0$
since $E$ is not convex on $\mathbb{R}^{n}$.
As $\beta\to 0^{+}$ the law $P_{*n}^{\beta}$ tends to
$P_{*n}^{0}=P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$ which is log-concave.
* •
The law $P_{n}^{\beta}$ is not centered but is log-concave for all $\beta\geq
0$.
Its density vanishes at the boundary of $D_{n}$ if $\beta>0$.
As $\beta\to 0^{+}$ the law $P_{n}^{\beta}$ tends to the law of the order
statistics of $n$ i.i.d. $\mathcal{N}(0,\frac{1}{n})$.
### 2.3. Spectral analysis of the generator: the non-interacting case
This subsection and the next deal with analytical aspects of our dynamics. We
start with the OU process ($\beta=0$) for which everything is explicit; the
next subsection deals with the DOU process ($\beta\geq 1$).
The infinitesimal generator of the OU process is given by
$\operatorname{G}f=\frac{1}{n}\Bigr{(}\Delta-\nabla
E\cdot\nabla\Bigr{)}=\frac{1}{n}\sum_{i=1}^{n}\partial_{i}^{2}-\sum_{i=1}^{n}V^{\prime}(x_{i})\partial_{i}.$
(2.1)
It is a self-adjoint operator on $L^{2}(\mathbb{R}^{n},P_{n}^{0})$ that leaves
globally invariant the set of polynomials. Its spectrum is the set of all non-
positive integers, that is,
$\lambda_{0}=0>\lambda_{1}=-1>\lambda_{2}=-2>\cdots$. The corresponding
eigenspaces $F_{0},F_{1},F_{2},\cdots$ are finite dimensional: $F_{m}$ is
spanned by the multivariate Hermite polynomials of degree $m$, in other words
tensor products of univariate Hermite polynomials. In particular, $F_{0}$ is
the vector space of constant functions while $F_{1}$ is the $n$-dimensional
vector space of all linear functions.
Let us point out that $\operatorname{G}$ can be restricted to the set of
$P_{n}^{0}$ square integrable _symmetric_ functions: it leaves globally
invariant the set of _symmetric_ polynomials, its spectrum is unchanged but
the associated eigenspaces $E_{m}$ are the restrictions of the vector spaces
$F_{m}$ to the set of symmetric functions, in other words, $E_{m}$ is spanned
by the multivariate _symmetrized_ Hermite polynomials of degree $m$. Note that
$E_{1}$ is the one-dimensional space generated by $\pi(x)=x_{1}+\cdots+x_{n}$.
The Markov semigroup ${(\mathrm{e}^{t\operatorname{G}})}_{t\geq 0}$ generated
by $\operatorname{G}$ admits $P_{n}^{0}$ as a reversible invariant law since
$\operatorname{G}$ is self-adjoint in $L^{2}(P_{n}^{0})$. Following [62], let
us introduce the _heat kernel_ $p_{t}(x,y)$ which is the density of
$\mathrm{Law}(X^{n}_{t}\mid X^{n}_{0}=x)$ with respect to the invariant law
$P_{n}^{0}$. The long-time behavior reads $\lim_{t\to\infty}p_{t}(x,\cdot)=1$
for all $x\in\mathbb{R}^{n}$. Let $\left\|\cdot\right\|_{p}$ be the norm of
$L^{p}=L^{p}(P_{n}^{0})$. For all $1\leq p\leq q$, $t\geq 0$,
$x\in\mathbb{R}^{n}$, we have
$2\|\mathrm{Law}(X^{n}_{t}\mid
X^{n}_{0}=x)-P_{n}^{0}\|_{\mathrm{TV}}=\|p_{t}(x,\cdot)-1\|_{1}\leq\|p_{t}(x,\cdot)-1\|_{p}\leq\|p_{t}(x,\cdot)-1\|_{q}.$
(2.2)
In the particular case $p=2$ we can write
$\|p_{t}(x,\cdot)-1\|_{2}^{2}=\sum_{m=1}^{\infty}\mathrm{e}^{-2mt}\sum_{\psi\in
B_{m}}|\psi(x)|^{2}.$ (2.3)
where $B_{m}$ is an orthonormal basis of $F_{m}\subset L^{2}(P_{n}^{0})$,
hence
$\|p_{t}(x,\cdot)-1\|_{2}^{2}\geq\mathrm{e}^{-2t}\sum_{\psi\in
B_{1}}|\psi(x)|^{2},$ (2.4)
which leads to a lower bound on the $\chi^{2}$ (in other words $L^{2}$)
cutoff, provided one can estimate $\sum_{\psi\in B_{1}}|\psi(x)|^{2}$ which is
the square of the norm of the projection of $\delta_{x}$ on $B_{1}$.
Following [62, Th. 6.2], an upper bound would follow from a Bakry–Émery
curvature–dimension criterion $\mathrm{CD}(\rho,d)$ with a finite dimension
$d$, in relation with Nash–Sobolev inequalities and dimensional pointwise
estimates on the heat kernel $p_{t}(x,\cdot)$ or ultracontractivity of the
Markov semigroup, see for instance [63, Sec. 4.1]. The OU process satisfies to
$\mathrm{CD}(\rho,\infty)$ but never to $\mathrm{CD}(\rho,d)$ with $d$ finite
and is not ultracontractive. Actually the OU process is a critical case, see
[3, Ex. 2.7.3].
### 2.4. Spectral analysis of the generator: the interacting case
We now assume that $\beta\geq 1$. The infinitesimal generator of the DOU
process is the operator
$\operatorname{G}f=\frac{1}{n}\Bigr{(}\Delta-\nabla
E\cdot\nabla\Bigr{)}=\frac{1}{n}\sum_{i=1}^{n}\partial_{i}^{2}-\sum_{i=1}^{n}V^{\prime}(x_{i})\partial_{i}+\frac{\beta}{2n}\sum_{j\neq
i}\frac{\partial_{i}-\partial_{j}}{x_{i}-x_{j}}.$ (2.5)
Despite the interaction term, the operator leaves globally invariant the set
of _symmetric_ polynomials. Following Lassalle in [49, 4], see also [25], the
operator $\operatorname{G}$ is a self-adjoint operator on the space of
$P_{*n}^{\beta}$ square integrable _symmetric_ functions of $n$ variables, its
spectrum does not depend on $\beta$ and matches the spectrum of the OU process
case $\beta=0$. In particular the spectral gap is $1$. The eigenspaces $E_{m}$
are spanned by the generalized symmetrized Hermite polynomials of degree $m$.
For instance, $E_{1}$ is the one-dimensional space generated by
$\pi(x)=x_{1}+\cdots+x_{n}$ while $E_{2}$ is the two-dimensional space spanned
by
$(x_{1}+\cdots+x_{n})^{2}-1\quad\text{and}\quad
x_{1}^{2}+\cdots+x_{n}^{2}-1-\frac{\beta}{2}(n-1).$ (2.6)
From the isometry between $L^{2}(\overline{D}_{n},P_{n}^{\beta})$ and
$L^{2}_{\mathrm{sym}}(\mathbb{R}^{n},P_{*n}^{\beta})$, the above explicit
spectral decomposition applies to the semigroup of the DOU on
$\overline{D}_{n}$. Formally, the discussion presented at the end of the
previous subsection still applies. However, in the present interacting case
the integrability properties of the heat kernel are not known: in particular,
we do not know whether $p_{t}(x,\cdot)$ lies in $L^{p}(P_{n}^{\beta})$ for
$t>0$, $x\in\overline{D}_{n}$ and $p>1$. This leads to the question, of
independent interest, of pointwise upper and lower Gaussian bounds for heat
kernels similar to the OU process, with explicit dependence of the constants
over the dimension. We refer for example to [65, 36, 41] for some results in
this direction.
### 2.5. Mean-field limit
The measure $P_{n}^{\beta}$ is log-concave since $E$ is convex, and its
density writes
$x\in\mathbb{R}^{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}|x|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(x_{i}-x_{j})^{\beta}\mathbf{1}_{x_{1}\leq\cdots\leq
x_{n}}.$ (2.7)
See [25, Sec. 2.2] for a high-dimensional analysis. The Boltzmann–Gibbs
measure $P_{n}^{\beta}$ is known as the $\beta$-Hermite ensemble or H$\beta$E.
When $\beta=2$, it is better known as the Gaussian Unitary Ensemble (GUE). If
$X^{n}\sim P_{n}^{\beta}$ then the Wigner theorem states that the empirical
measure with atoms distributed according to $P_{n}^{\beta}$ converges in
distribution to a semi-circle law, namely
$\frac{1}{n}\sum_{i=1}^{n}\delta_{X^{n,i}}\underset{n\to\infty}{\overset{\text{weak}}{\longrightarrow}}\frac{\sqrt{2\beta-x^{2}}}{\beta\pi}\mathbf{1}_{x\in[-\sqrt{2\beta},\sqrt{2\beta}]}\mathrm{d}x,$
(2.8)
and this can be deduced in this Coulomb gas context from a large deviation
principle as in [12].
Let ${(X^{n})}_{t\geq 0}$ be the process solving (1.3) with $\beta\geq 0$ or
$\beta\geq 1$, and let
$\mu^{n}_{t}=\frac{1}{n}\sum_{k=1}^{n}\delta_{X^{n,i}_{t}}$ (2.9)
be the empirical measure of the particles at time $t$. Following notably [60,
14, 21, 20, 53, 31], if the sequence of initial conditions
${(\mu_{0}^{n})}_{n\geq 1}$ converges weakly as $n\to\infty$ to a probability
measure $\mu_{0}$, then the sequence of measure valued processes
${({(\mu_{t}^{n})}_{t\geq 0})}_{n\geq 1}$ converges weakly to the unique
probability measure valued deterministic process ${(\mu_{t})}_{t\geq 0}$
satisfying the evolution equation
$\langle\mu_{t},f\rangle=\langle\mu_{0},f\rangle-\int_{0}^{t}\int
V^{\prime}(x)f^{\prime}(x)\mu_{s}(\mathrm{d}x)\mathrm{d}s+\frac{\beta}{2}\int_{0}^{t}\int_{\mathbb{R}^{2}}\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{s}(\mathrm{d}x)\mu_{s}(\mathrm{d}y)\mathrm{d}s\quad$
(2.10)
for all $t\geq 0$ and $f\in\mathcal{C}^{3}_{b}(\mathbb{R},\mathbb{R})$. The
equation (2.10) is a weak formulation of a McKean–Vlasov equation or free
Fokker–Planck equation associated to a free OU process. Moreover, if $\mu_{0}$
has all its moments finite, then for all $t\geq 0$, we have the free Mehler
formula
$\mu_{t}=\mathrm{dil}_{\mathrm{e}^{-2t}}\mu_{0}\boxplus\mathrm{dil}_{\sqrt{1-\mathrm{e}^{-2t}}}\mu_{\infty},$
(2.11)
where $\mathrm{dil}_{\sigma}\mu$ is the law of $\sigma X$ when $X\sim\mu$,
where “$\boxplus$” stands for the free convolution of probability measures of
Voiculescu free probability theory, and where $\mu_{\infty}$ is the semi-
circle law of variance $\frac{\beta}{2}$. In particular, if $\mu_{0}$ is a
semi-circle law then $\mu_{t}$ is a semi-circle law for all $t\geq 0$.
Let us introduce the $k$-th moment $m_{k}(t):=\displaystyle\int
x^{k}\mu_{t}(\mathrm{d}x)$ of $\mu_{t}$. The first and second moments satisfy
the differential equations $m_{1}^{\prime}=-m_{1}$ and
$m_{2}^{\prime}=-2m_{2}+\beta$ respectively, which give
$m_{1}(t)=\mathrm{e}^{-t}m_{1}(0)\underset{t\to\infty}{\longrightarrow}0\quad\text{and}\quad
m_{2}(t)=m_{2}(0)\mathrm{e}^{-2t}+\frac{\beta}{2}(1-\mathrm{e}^{-2t})\underset{t\to\infty}{\longrightarrow}\frac{\beta}{2}.$
(2.12)
More generally, beyond the first two moments, the Cauchy–Stieltjes transform
$z\in\mathbb{C}_{+}=\\{z\in\mathbb{C}:\Im z>0\\}\mapsto
s_{t}(z)=\int_{\mathbb{R}}\frac{\mu_{t}(\mathrm{d}x)}{x-z}$ (2.13)
of $\mu_{t}$ is the solution of the following complex Burgers equation
$\partial_{t}s_{t}(z)=s_{t}(z)+z\partial_{z}s_{t}(z)+\beta
s_{t}(z)\partial_{z}s_{t}(z),\quad t\geq 0,z\in\mathbb{C}_{+}.$ (2.14)
The semi-circle law on $[-c,c]$ has density $\frac{2\sqrt{c^{2}-x^{2}}}{\pi
c^{2}}\mathbf{1}_{x\in[-c,c]}$, mean $0$, second moment or variance
$\frac{c^{2}}{4}$, and Cauchy–Stieltjes transform
$s_{t}(z)=\frac{\sqrt{4z^{2}-4c^{2}}-2z}{c^{2}}$, $t\geq
0,z\in\mathbb{C}_{+}$.
The cutoff phenomenon is in a sense a diagonal $(t,n)$ estimate, melting long
time behavior and high dimension. When $|z_{0}^{n}|$ is of order $n$, cutoff
occurs at a time of order $\approx\log(n)$: this informally corresponds to
taking $t\to\infty$ in $(\mu_{t})_{t\geq 0}$.
When $\mu_{0}$ is centered with same second moment $\frac{\beta}{2}$ as
$\mu_{\infty}$, then there is a Boltzmann H-theorem interpretation of the
limiting dynamics as $n\to\infty$: the steady-state is the Wigner semi-circle
law $\mu_{\infty}$, the second moment is conserved by the dynamics, the
Voiculescu entropy is monotonic along the dynamics, grows exponentially, and
is maximized by the steady-state.
### 2.6. $L^{p}$ cutoff
Following [26], we can deduce an $L^{p}$ cutoff started from $x$ from an
$L^{1}$ cutoff by showing that the heat kernel $p_{t}(x,\cdot)$ is in
$L^{p}(P_{n}^{\beta})$ for some $t>0$. Thanks to the Mehler formula, it can be
checked that this holds for the OU case, despite the lack of
ultracontractivity. The heat kernel of the DOU process is less accessible.
In another exactly solvable direction, the $L^{p}$ cutoff phenomenon has been
studied for instance in [62, 64] for Brownian motion on compact simple Lie
groups, and in [64, 55] for Brownian motion on symmetric spaces, in relation
with representation theory, an idea which goes back at the origin to the works
of Diaconis on random walks on groups.
### 2.7. Cutoff window and profile
Once a cutoff phenomenon is established, one can ask for a finer description
of the pattern of convergence to equilibrium. The _cutoff window_ is the order
of magnitude of the transition time from the value $\max$ to the value $0$:
more precisely, if cutoff occurs at time $c_{n}$ then we say that the cutoff
window is $w_{n}$ if
$\displaystyle\varlimsup_{b\to+\infty}\varlimsup_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})$ $\displaystyle=0,$
$\displaystyle\varliminf_{b\to-\infty}\varliminf_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})$ $\displaystyle=\max,$
and for any $b\in\mathbb{R}$
$0<\varliminf_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})\leq\varlimsup_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})<\max.$
Note that necessarily $w_{n}=o(c_{n})$ by definition of the cutoff phenomenon.
Note also that $w_{n}$ is unique in the following sense: $w^{\prime}_{n}$ is a
cutoff window if and only if $w_{n}/w^{\prime}_{n}$ remains bounded from above
and below as $n\to\infty$.
We say that the _cutoff profile_ is given by $\varphi:\mathbb{R}\to[0,1]$ if
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})=\varphi(b).$
The analysis of the OU process carried out in Theorems 1.1 and 1.2 can be
pushed further to establish the so-called cutoff profiles, we refer to the end
of Section 3 for details.
Regarding the DOU process, such a detailed description of the convergence to
equilibrium does not seem easily accessible. However it is straightforward to
deduce from our proofs that the cutoff window is of order $1$, in other words
the inverse of the spectral gap, in the setting of Corollary 1.6. This is also
the case in the setting of Corollary 1.8 for the Wasserstein distance.
We believe that this remains true in the setting of Corollary 1.8 for the TV
and Hellinger distances: actually, a lower bound of the required order can be
derived from the calculations in the proof of Corollary 1.4; on the other
hand, our proof of the upper bound on the mixing time does not allow to give a
precise enough upper bound on the window.
### 2.8. Other potentials
It is natural to ask about the cutoff phenomenon for the process solving (1.3)
when $V$ is a more general $\mathcal{C}^{2}$ function. The invariant law
$P_{n}^{\beta}$ of this Markov diffusion writes
$\frac{\mathrm{e}^{-n\sum_{i=1}^{n}V(x_{i})}}{C_{n}^{\beta}}\prod_{i>j}(x_{i}-x_{j})^{\beta}\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$
(2.15)
The case where $V-\frac{\rho}{2}\left|\cdot\right|^{2}$ is convex for some
constant $\rho\geq 0$ generalizes the DOU case and has exponential convergence
to equilibrium, see [25]. Three exactly solvable cases are known:
* •
$\mathrm{e}^{-V(x)}=\mathrm{e}^{-\frac{x^{2}}{2}}$: the DOU process associated
to the Gaussian law weight and the $\beta$-Hermite ensemble including
HOE/HUE/HSE when $\beta\in\\{1,2,4\\}$,
* •
$\mathrm{e}^{-V(x)}=x^{a-1}\mathrm{e}^{-x}\mathbf{1}_{x\in[0,\infty)}$: the
Dyson–Laguerre process associated to the Gamma law weight and the
$\beta$-Laguerre ensembles including LOE/LUE/LSE when $\beta\in\\{1,2,4\\}$,
* •
$\mathrm{e}^{-V(x)}=x^{a-1}(1-x)^{b-1}\mathbf{1}_{x\in[0,1]}$: the
Dyson–Jacobi process associated to the Beta law weight and the $\beta$-Jacobi
ensembles including JOE/JUE/JSE when $\beta\in\\{1,2,4\\}$,
up to a scaling. Following Lassalle [49, 51, 50, 4] and Bakry [5], in these
three cases, the multivariate orthogonal polynomials of the invariant law
$P_{n}^{\beta}$ are the eigenfunctions of the dynamics of the process. We
refer to [35, 32, 54] for more information on (H/L/J)$\beta$E random matrix
models.
The contraction property or spectral projection used to pass from a matrix
process to the Dyson process can be used to pass from BM on the unitary group
to the Dyson circular process for which the invariant law is the Circular
Unitary Ensemble (CUE). This provides an upper bound for the cutoff
phenomenon. The cutoff for BM on the unitary group is known and holds at
critical time or order $\log(n)$, see for instance [64, 62, 55].
More generally, we could ask about the cutoff phenomenon for a McKean–Vlasov
type interacting particle system ${(X^{n}_{t})}_{t\geq 0}$ in
$(\mathbb{R}^{d})^{n}$ solution of the stochastic differential equation of the
form
$\mathrm{d}X^{n,i}_{t}=\sigma_{n,t}(X^{n})\mathrm{d}B^{n}_{t}-\sum_{i=1}^{n}\nabla
V_{n,t}(X^{n,i}_{t})\mathrm{d}t-\sum_{j\neq i}\nabla
W_{n,t}(X^{n,i}_{t}-X^{n,j}_{t})\mathrm{d}t,\quad 1\leq i\leq n,$ (2.16)
for various types of confinement $V$ and interaction $W$ (convex, repulsive,
attractive, repulsive-attractive, etc), and discuss the relation with the
propagation of chaos. The case where $V$ and $W$ are both convex and constant
in time is already very well studied from the point of view of long-time
behavior and mean-field limit in relation with convexity, see for instance
[21, 20, 53].
Regarding universality, it is worth noting that if $V=\left|\cdot\right|^{2}$
and if $W$ is convex then the proof by factorization of the optimal Poincaré
and logarithmic Sobolev inequalities and their extremal functions given in
[25] remains valid, paving the way to the generalization of many of our
results in this spirit. On the other hand, the convexity of the limiting
energy functional in the mean-field limit is of Bochner type and suggests to
take for $W$ a power, in other words a Riesz type interaction.
### 2.9. Alternative parametrization
If ${(X^{n}_{t})}_{t\geq 0}$ is the process solution of the stochastic
differential equation (1.3), then for all real parameters $\alpha>0$ and
$\sigma>0$, the space scaled and time changed stochastic process
${(Y^{n}_{t})}_{t\geq 0}={(\sigma X^{n}_{\alpha t})}_{t\geq 0}$ solves the
stochastic differential equation
$Y^{n}_{0}=\sigma
x^{n}_{0},\quad\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2\alpha\sigma^{2}}{n}}\mathrm{d}B^{i}_{t}-\alpha
Y_{t}^{n,i}\mathrm{d}t+\frac{\alpha\beta\sigma^{2}}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n,$ (2.17)
where ${(B_{t})}_{t\geq 0}$ is a standard $n$-dimensional BM. The invariant
law of ${(Y^{n}_{t})}_{t\geq 0}$ is
$\frac{\mathrm{e}^{-\frac{n}{2\sigma^{2}}|y|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(y_{i}-y_{j})^{\beta}\mathbf{1}_{(y_{1},\ldots,y_{n})\in\overline{D}_{n}}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}$
(2.18)
where $C_{n}^{\beta}$ is the normalizing constant. This law and its
normalization $C_{n}^{\beta}$ depend on the “shape parameter” $\beta$, the
“scale parameter” $\sigma$, and does not depend on the “speed parameter”
$\alpha$. When $\beta>0$, taking $\sigma^{2}=\beta^{-1}$, the stochastic
differential equation (2.17) boils down to
$Y^{n}_{0}=\frac{x^{n}_{0}}{\sqrt{\beta}},\quad\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2\alpha}{n\beta}}\mathrm{d}B^{i}_{t}-\alpha
Y_{t}^{n,i}\mathrm{d}t+\frac{\alpha}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n$ (2.19)
while the invariant law becomes
$\frac{\mathrm{e}^{-\frac{n\beta}{2}|y|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(y_{i}-y_{j})^{\beta}\mathbf{1}_{(y_{1},\cdots,y_{n})\in\overline{D}_{n}}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}.$
(2.20)
The equation (2.19) is the one considered in [37, Eq. (12.4)] and in [46, Eq.
(1.1)]. The advantage of (2.19) is that $\beta$ can be now truly interpreted
as an inverse temperature and the right-hand side in the analogue of (2.8)
does not depend on $\beta$, while the drawback is that we cannot turn off the
interaction by setting $\beta=0$ and recover the OU process as in (1.3). It is
worthwhile mentioning that for instance Theorem 1.7 remains the same for the
process solving (2.19) in particular the cutoff threshold is at critical time
$\frac{c_{n}}{\alpha}$ and does not depend on $\beta$.
### 2.10. Discrete models
There are several discrete space Markov processes admitting the OU process as
a scaling limit, such as for instance the random walk on the discrete
hypercube, related to the Ehrenfest model, for which the cutoff has been
studied in [30, 29], and the M/M/$\infty$ queuing process, for which a
discrete Mehler formula is available [24]. Certain discrete space Markov
processes incorporate a singular repulsion mechanism, such as for instance the
exclusion process on the segment, for which the study of the cutoff in [48]
shares similarities with our proof of Theorem 1.7. It is worthwhile noting
that there are discrete Coulomb gases, related to orthogonal polynomials for
discrete measures, suggesting to study discrete Dyson processes. More
generally, it could be natural to study the cutoff phenomenon for Markov
processes on infinite discrete state spaces, under curvature condition, even
if the subject is notoriously disappointing in terms of high-dimensional
analysis. We refer to the recent work [61] for the finite state space case.
## 3\. Cutoff phenomenon for the OU
In this section, we prove Theorems 1.1 and 1.2: actually we only prove the
latter since it implies the former. We start by recalling a well-known fact.
###### Lemma 3.1 (Mehler formula).
If ${(Y_{t})}_{t\geq 0}$ is an OU process in $\mathbb{R}^{d}$ solution of the
stochastic differential equation $Y_{0}=y_{0}\in\mathbb{R}^{d}$ and
$\mathrm{d}Y_{t}=\sigma\mathrm{d}B_{t}-\mu Y_{t}\mathrm{d}t$ for parameters
$\sigma>0$ and $\mu>0$ where $B$ is a standard $d$-dimensional Brownian motion
then
${(Y_{t})}_{t\geq 0}={\Bigr{(}y_{0}\mathrm{e}^{-\mu
t}+\sigma\int_{0}^{t}\mathrm{e}^{\mu(s-t)}\mathrm{d}B_{s}\Bigr{)}}_{t\geq
0}\text{ hence }Y_{t}\sim\mathcal{N}\Bigr{(}y_{0}\mathrm{e}^{-\mu
t},\frac{\sigma^{2}}{2}\frac{1-\mathrm{e}^{-2\mu
t}}{\mu}\mathrm{I}_{d}\Bigr{)}\text{ for all $t\geq 0$.}$
Moreover its coordinates are independent one-dimensional OU processes with
initial condition $y_{0}^{i}$ and invariant law
$\mathcal{N}(0,\frac{\sigma^{2}}{2\mu})$, $1\leq i\leq d$.
###### Proof of Theorem 1.1 and Theorem 1.2.
By using Lemma 3.1, for all $n\geq 1$ and $t\geq 0$,
$Z^{n}_{t}\sim\mathcal{N}\Bigr{(}z^{n}_{0}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n}\Bigr{)}=\otimes_{i=1}^{n}\mathcal{N}\Bigr{(}z^{n,i}_{0}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}\Bigr{)},\
P_{n}^{0}=\mathcal{N}\Bigr{(}0,\frac{I_{n}}{n}\Bigr{)}=\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)}^{\otimes
n}.$ (3.1)
_Hellinger, Kullback, $\chi^{2}$, Fisher, and Wasserstein cutoffs._ A direct
computation from (3.1) or Lemma A.5 either from multivariate Gaussian formulas
or univariate via tensorization gives
$\displaystyle\mathrm{Hellinger}^{2}(\mathrm{Law}(Z^{n}_{t}),P_{n}^{0})$
$\displaystyle=1-\exp\Bigr{(}-\frac{n}{4}\frac{|z^{n}_{0}|^{2}\mathrm{e}^{-2t}}{2-\mathrm{e}^{-2t}}+\frac{n}{4}\log\Bigr{(}4\frac{1-\mathrm{e}^{-2t}}{(2-\mathrm{e}^{-2t})^{2}}\Bigr{)}\Bigr{)},$
(3.2) $\displaystyle 2\mathrm{Kullback}(\mathrm{Law}(Z^{n}_{t})\mid
P_{n}^{0})$
$\displaystyle=n|z^{n}_{0}|^{2}\mathrm{e}^{-2t}-n\mathrm{e}^{-2t}-n\log(1-\mathrm{e}^{-2t}),$
(3.3) $\displaystyle\chi^{2}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})$
$\displaystyle=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{n/2}}\exp\Bigr{(}n|z_{0}^{n}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\Bigr{)},$
(3.4) $\displaystyle\mathrm{Fisher}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})$
$\displaystyle=n^{2}|z^{n}_{0}|^{2}\mathrm{e}^{-2t}+n^{2}\frac{\mathrm{e}^{-4t}}{1-\mathrm{e}^{-2t}},$
(3.5)
$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z^{n}_{t}),P_{n}^{0})$
$\displaystyle=|z^{n}_{0}|^{2}\mathrm{e}^{-2t}+2(1-\sqrt{1-\mathrm{e}^{-2t}}-\frac{1}{2}\mathrm{e}^{-2t}),$
(3.6)
which gives the desired lower and upper bounds as before by using the
hypothesis on $z^{n}_{0}$.
_Total variation cutoff._ By using the comparison between total variation and
Hellinger distances (Lemma A.1) we deduce from (3.2) the cutoff in total
variation distance at the same critical time. The upper bound for the total
variation distance can alternatively be obtained by using the
$\mathrm{Kullback}$ estimate (3.3) and the Pinsker–Csiszár–Kullback inequality
(Lemma A.1). Since both distributions are tensor products, we could use
alternatively the tensorization property of the total variation distance
(Lemma A.4) together with the one-dimensional version of the Gaussian formula
for $\mathrm{Kullback}$ (Lemma A.1) to obtain the result for the total
variation. ∎
###### Remark 3.2 (Competition between bias and variance mixing).
From the computations of the proof of Theorem 1.2, we can show that for
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\chi^{2}\\}$
$A_{t}:=\mathrm{dist}(\mathrm{Law}(Z^{n}_{t})\mid\mathrm{Law}(Z^{n}_{t}-z^{n}_{0}\mathrm{e}^{-t}))$
has a cutoff at time $c_{n}^{A}=\log(\sqrt{n}|z^{n}_{0}|)$, while
$B_{t}:=\mathrm{dist}(\mathrm{Law}(Z^{n}_{t}-z^{n}_{0}\mathrm{e}^{-t})\mid
P_{n}^{0})$
admits a cutoff at time $c_{n}^{B}=\frac{1}{4}\log(n)$. The triangle
inequality for $\mathrm{dist}$ yields
$|A_{t}-B_{t}|\leq\mathrm{dist}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})\leq
A_{t}+B_{t}.$
Therefore the critical time of Theorem 1.2 is dictated by either $A_{t}$ or
$B_{t}$, according to whether $c_{n}^{A}\gg c_{n}^{B}$ or $c_{n}^{A}\ll
c_{n}^{B}$. This can be seen as a competition between bias and variance
mixing.
###### Remark 3.3 (Total variation discriminating event for small initial
conditions).
Let us introduce the random variable $Z^{n}_{\infty}\sim
P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})=\mathcal{N}(0,\frac{1}{n})^{\otimes
n}$, in accordance with (3.1). There holds
$S_{t}^{n}:=\sum_{i=1}^{n}(Z_{t}^{n,i}-z_{0}^{n,i}\mathrm{e}^{-t})^{2}\sim\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2(1-\mathrm{e}^{-2t})}\Bigr{)}\quad\text{and}\quad|Z^{n}_{\infty}|^{2}\sim\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2}\Bigr{)}.$
We can check, using an explicit computation of Hellinger and Kullback between
Gamma distributions and the comparison between total variation and Hellinger
distances (Lemma A.1), that
$C_{t}:=\mathrm{dist}(\mathrm{Law}(S^{n}_{t})\mid\mathrm{Law}(|Z^{n}_{\infty}|^{2}))$
admits a cutoff at time $c_{n}^{C}=c_{n}^{B}=\frac{1}{4}\log(n)$. Moreover,
one can exhibit a discriminating event for the TV distance. Namely, we can
observe that
$\Bigr{\|}\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2(1-\mathrm{e}^{-2t})}\Bigr{)}-\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2}\Bigr{)}\Bigr{\|}_{\mathrm{TV}}=\mathbb{P}(|Z^{n}_{\infty}|^{2}\geq\alpha_{t})-\mathbb{P}(S_{t}^{n}\geq\alpha_{t})$
with $\alpha_{t}$ the unique point where the two densities meet, which happens
to be
$\alpha_{t}=-\mathrm{e}^{2t}\log(1-\mathrm{e}^{-2t})(1-\mathrm{e}^{-2t}).$
From the explicit expressions (3.2), (3.3), (3.4), (3.5), (3.6), it is
immediate to extract the cutoff profile associated to the convergence of
$\mathrm{Law}(Z_{t}^{n})$ to $P_{n}^{0}$ in Hellinger, Kullback, $\chi^{2}$,
Fisher and Wasserstein. For Wasserstein we already know by Theorem 1.2 that a
cutoff occurs if and only if $|z_{0}^{n}|\underset{n\to\infty}{\to}\infty$. In
this case, regarding the profile, we have
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(Z_{t}^{n}),P_{n}^{0})=\phi(b),$
(3.7)
where for all $b\in\mathbb{R}$,
$t_{n,b}=\log|z_{0}^{n}|+b\quad\text{and}\quad\phi(b)=\mathrm{e}^{-b}.$ (3.8)
For the other distances and divergences, let us assume that the following
limit exists
$a:=\lim_{n\to\infty}\sqrt{n}|z_{0}^{n}|^{2}\in[0,+\infty].$ (3.9)
This quantity can be related with
$c_{n}^{A}:=\log(|z_{0}^{n}|\sqrt{n})\quad\text{and}\quad
c_{n}^{B}:=\frac{\log n}{4}$ (3.10)
which were already introduced in Remark 3.2. Indeed
$a=0\Longleftrightarrow c_{n}^{A}\ll c_{n}^{B},\quad
a=+\infty\Longleftrightarrow c_{n}^{A}\gg c_{n}^{B},$
while $a\in(0,\infty)$ is equivalent to $c_{n}^{A}\asymp c_{n}^{B}$.
Then, for
$\mathrm{dist}\in\\{\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}$,
we have, for all $b\in\mathbb{R}$,
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z_{t_{n,b}})\mid
P_{n}^{0})=\phi(b),$ (3.11)
where $t_{n,b}$ and $\phi(b)$ are as in Table 1. The cutoff window is always
of size $1$.
| $a=+\infty$ | $a=0$ | $a\in(0,+\infty)$
---|---|---|---
$t_{n,b}$ | | |
Hellinger | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
Kullback | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
$\chi^{2}$ | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
Fisher | $\log(|z_{0}^{n}|n)+b$ | $\frac{\log n}{2}+b$ | $\frac{\log n}{2}+b$
$\phi(b)$ | | |
Hellinger | $\sqrt{1-\mathrm{e}^{-\frac{1}{8}\mathrm{e}^{-2b}}}$ | $\sqrt{1-\mathrm{e}^{-\frac{1}{16}\mathrm{e}^{-4b}}}$ | $\sqrt{1-\mathrm{e}^{-\frac{1}{8}{a\mathrm{e}^{-2b}}-\frac{1}{16}\mathrm{e}^{-4b}}}$
Kullback | $\frac{1}{2}\mathrm{e}^{-2b}$ | $\frac{1}{4}\mathrm{e}^{-4b}$ | $\frac{1}{2}a\mathrm{e}^{-2b}+\frac{1}{4}\mathrm{e}^{-4b}$
$\chi^{2}$ | $\mathrm{e}^{\mathrm{e}^{-2b}}-1$ | $\mathrm{e}^{\frac{1}{2}\mathrm{e}^{-4b}}-1$ | $\mathrm{e}^{a\mathrm{e}^{-2b}+\frac{1}{2}\mathrm{e}^{-4b}}-1$
Fisher | $\mathrm{e}^{-2b}$ | $\mathrm{e}^{-4b}$ | $a\mathrm{e}^{-2b}+\mathrm{e}^{-4b}$
Table 1. Values of $t_{n,b}$ and $\phi(b)$ for the cutoff profile of the OU
process in (3.11).
Since the total variation distance is not expressed in a simple explicit
manner, further computations are needed to extract the precise cutoff profile,
which is given in the following lemma:
###### Lemma 3.4 (Cutoff profile in $\mathrm{TV}$ for OU).
Let $Z^{n}=(Z_{t}^{n})_{t\geq 0}$ be the OU process (1.8), started from
$z_{0}^{n}\in\mathbb{R}^{n}$, and let $P_{n}^{0}$ be its invariant law. Assume
as in (3.9) that $a:=\lim_{n\to\infty}|z_{0}^{n}|^{2}\sqrt{n}\in[0,+\infty]$,
and let $t_{n,b}$ be as in Table (1) for Hellinger. Then, for all
$b\in\mathbb{R}$, we have
$\lim_{n\to\infty}\|\mathrm{Law}({Z}_{t_{n,b}}^{n})-P_{n}^{0}\|_{\mathrm{TV}}=\phi(b),$
where
$\phi(b):=\begin{cases}\displaystyle\mathrm{erf}\Bigr{(}\frac{\mathrm{e}^{-b}}{2\sqrt{2}}\Bigr{)}&\text{if
$a=+\infty$}\\\\[10.00002pt]
\displaystyle\mathrm{erf}\Bigr{(}\frac{\mathrm{e}^{-2b}}{4}\Bigr{)}&\text{if
$a=0$}\\\\[10.00002pt]
\displaystyle\mathrm{erf}\Bigr{(}\frac{\sqrt{2a\mathrm{e}^{-2b}+\mathrm{e}^{-4b}}}{4}\Bigr{)}&\text{if
$a\in(0,+\infty)$}\end{cases},$
where $\displaystyle\mathrm{erf}(u):=\frac{1}{\sqrt{\pi}}\int_{|t|\leq
u}\mathrm{e}^{-t^{2}}\mathrm{d}t=\mathbb{P}(|X|\leq\sqrt{2}u)$ with
$X\sim\mathcal{N}(0,1)$ is the _error function_.
###### Proof of Lemma 3.4.
The idea is to exploit the fact that we consider Gaussian product measures
(the covariance matrices are multiple of the identity), which allows a finer
analysis than for instance in [27, Le. 3.1]. We begin with a rather general
step. Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^{n}$ with
densities $f$ and $g$ with respect to the Lebesgue measure $\mathrm{d}x$. We
have then
$\|\mu-\nu\|_{\mathrm{TV}}=\frac{1}{2}\int|f-g|\mathrm{d}x=\frac{1}{2}\Bigr{(}\int(f-g)\mathbf{1}_{g\leq
f}\mathrm{d}x-\int(f-g)\mathbf{1}_{f\leq g}\mathrm{d}x\Bigr{)},$ (3.12)
and since
$-\int(f-g)\mathbf{1}_{f\leq
g}\mathrm{d}x=-\int(f-g)(1-\mathbf{1}_{g<f})\mathrm{d}x=\int(f-g)\mathbf{1}_{g<f}\mathrm{d}x=\int(f-g)\mathbf{1}_{g\leq
f}\mathrm{d}x,$
we obtain
$\|\mu-\nu\|_{\mathrm{TV}}=\int(f-g)\mathbf{1}_{g\leq f}\mathrm{d}x=\mu(g\leq
f)-\nu(g\leq f).$ (3.13)
In particular, when $\mu=\mathcal{N}(m_{1},\sigma_{1}^{2}I_{n})$ and
$\nu=\mathcal{N}(m_{2},\sigma_{2}^{2}I_{n})$ then $g(x)\leq f(x)$ is
equivalent to
$\psi(x):=\frac{|x-m_{1}|^{2}}{\sigma_{1}^{2}}-\frac{|x-m_{2}|^{2}}{\sigma_{2}^{2}}\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}},$ (3.14)
for all $x\in\mathbb{R}^{n}$, and therefore, with $Z_{1}\sim\mu$ and
$Z_{2}\sim\nu$, we get
$\|\mu-\nu\|_{\mathrm{TV}}=\mathbb{P}\Bigr{(}\psi(Z_{1})\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\Bigr{)}-\mathbb{P}\Bigr{(}\psi(Z_{2})\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\Bigr{)}.$ (3.15)
Let us assume from now on that $m_{2}=0$ and $\sigma_{1}\neq\sigma_{2}$. We
can then gather the quadratic terms as
$\psi(x)=\Bigr{(}1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\Bigr{)}\frac{|x-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}+\Bigr{(}\frac{1}{\sigma_{1}^{2}}-\frac{1}{(1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}})\sigma_{1}^{2}}\Bigr{)}|m_{1}|^{2}\quad\text{where}\quad\tilde{m}_{1}:=\frac{1}{1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}}m_{1}.$
(3.16)
We observe at this step that the random variable
$\frac{|Z_{1}-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}$ follows a noncentral chi-
squared distribution, which depends only on $n$ and on the noncentrality
parameter
$\lambda_{1}:=\frac{|m_{1}-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}=\frac{\sigma_{1}^{2}}{(\sigma_{2}^{2}-\sigma_{1}^{2})^{2}}|m_{1}|^{2}.$
(3.17)
Similarly, the random variable
$\frac{|Z_{2}-\tilde{m}_{1}|^{2}}{\sigma_{2}^{2}}$ follows a noncentral chi-
squared distribution, which depends only on $n$ and on the noncentrality
parameter
$\lambda_{2}:=\frac{|\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}=\frac{\sigma_{2}^{4}}{\sigma_{1}^{2}(\sigma_{2}^{2}-\sigma_{1}^{2})^{2}}|m_{1}|^{2}.$
(3.18)
It follows that the law of $\psi(Z_{1})$ and the law of $\psi(Z_{2})$ depend
over $m_{1}$ only via $|m_{1}|$. Hence
$\psi(Z_{1})\overset{\mathrm{d}}{=}X_{1}+\cdots+X_{n}\quad\text{and}\quad\psi(Z_{2})\overset{\mathrm{d}}{=}Y_{1}+\cdots+Y_{n}$
(3.19)
where $X_{1},\ldots,X_{n}$ and $Y_{1},\ldots,Y_{n}$ are two sequences of
i.i.d. random variables for which the mean and variance depends only (and
explicitly) on $|m_{1}|$, $\sigma_{1}$, $\sigma_{2}$. Note in particular that
these means and variances are given by $\frac{1}{n}$ the ones of $\psi(Z_{1})$
and $\psi(Z_{2})$. Now we specialize to the case where
$\mu=\mathrm{Law}(Z^{n}_{t})=\mathcal{N}(z_{0}^{n}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n})$
and
$\nu=\mathrm{Law}(Z^{n}_{\infty})=\mathcal{N}(0,\frac{1}{n}I_{n})=P_{n}^{0}$,
and we find
$\mathbb{E}[\psi(Z_{1})]=n\Bigr{(}1-\frac{\sigma_{t}^{2}}{\sigma_{\infty}^{2}}\Bigr{)}-\frac{|z_{0}^{n}|^{2}\mathrm{e}^{-2t}}{\sigma_{\infty}^{2}},\quad\mathbb{E}[\psi(Z_{2})]=n\Bigr{(}\frac{\sigma_{\infty}^{2}}{\sigma_{t}^{2}}-1\Bigr{)}+\frac{|z_{0}^{n}|^{2}\mathrm{e}^{-2t}}{\sigma_{t}^{2}}$
while
$\mathrm{Var}[\psi(Z_{1})]=2n\Bigr{(}\frac{1}{\sigma_{t}^{2}}-\frac{1}{\sigma_{\infty}^{2}}\Bigr{)}^{2}\sigma_{t}^{4}+4\frac{\sigma_{t}^{2}}{\sigma_{\infty}^{4}}|z_{0}^{n}|^{2}\mathrm{e}^{-2t},\quad\mathrm{Var}[\psi(Z_{2})]=2n\Bigr{(}\frac{1}{\sigma_{t}^{2}}-\frac{1}{\sigma_{\infty}^{2}}\Bigr{)}^{2}\sigma_{\infty}^{4}+4\frac{\sigma_{\infty}^{2}}{\sigma_{t}^{4}}|z_{0}^{n}|^{2}\mathrm{e}^{-2t}.$
Let $t=t_{n,b}$ be as in Table 1 for Hellinger. Using (3.15) and the central
limit theorem for the i.i.d. random variables $X_{1},\ldots,X_{n}$ and
$Y_{1},\ldots,Y_{n}$, we get, with $Z\sim\mathcal{N}(0,1)$,
$\bigr{\|}\mathrm{Law}({Z}_{t}^{n})-P_{n}^{0}\bigr{\|}_{\mathrm{TV}}=\mathbb{P}(Z\leq\gamma_{n,t})-\mathbb{P}(Z\leq\tilde{\gamma}_{n,t})+o_{n}(1),$
where
$\gamma_{n,t}:=\frac{-n\log(1-\mathrm{e}^{-2t})-\mathbb{E}[\psi(Z^{n}_{t})]}{\sqrt{\mathrm{Var}[\psi(Z^{n}_{t})]}},\quad\tilde{\gamma}_{n,t}:=\frac{-n\log(1-\mathrm{e}^{-2t})-\mathbb{E}[\psi(Z^{n}_{\infty})]}{\sqrt{\mathrm{Var}[\psi(Z^{n}_{\infty})]}}.$
Expanding $\gamma_{n,t_{n,b}}$ gives the cutoff profile. Let us detail the
computations in the most involved case
$\lim_{n\to\infty}|z_{0}^{n}|^{2}\sqrt{n}=a\in(0,+\infty)$. For all
$b\in\mathbb{R}$, recall $t_{n,b}=\frac{\log n}{4}+b$. One may check that
$-n\log(1-\mathrm{e}^{-2t_{n,b}})-\mathbb{E}[\psi(Z^{n}_{t_{n,b}})]=\frac{1}{2}\mathrm{e}^{-4b}+a\mathrm{e}^{-2b}+o_{n}(1),$
$-n\log(1-\mathrm{e}^{-2t_{n,b}})-\mathbb{E}[\psi(Z^{n}_{\infty})]=-\frac{1}{2}\mathrm{e}^{-4b}-a\mathrm{e}^{-2b}+o_{n}(1),$
$\mathrm{Var}[\psi(Z^{n}_{t_{n,b}})]=2\mathrm{e}^{-4b}+4a\mathrm{e}^{-2b}+o_{n}(1),\quad\mathrm{Var}[\psi(Z^{n}_{\infty})]=2\mathrm{e}^{-4b}+4a\mathrm{e}^{-2b}+o_{n}(1).$
It follows that
$\lim_{n\to\infty}\bigr{\|}\mathrm{Law}({Z}_{t_{n,b}}^{n})-P_{n}^{0}\bigr{\|}_{\mathrm{TV}}=\mathbb{P}\Bigr{(}|Z|\leq\frac{1}{2\sqrt{2}}\sqrt{\mathrm{e}^{-4b}+2a\mathrm{e}^{-2b}}\Bigr{)}=\mathrm{erf}\Bigr{(}\frac{1}{4}\sqrt{\mathrm{e}^{-4b}+2a\mathrm{e}^{-2b}}\Bigr{)}.$
The other cases are similar. ∎
## 4\. General exactly solvable aspects
In this section, we prove Theorem 1.3 and Corollary 1.4.
The proof of Theorem 1.3 is based on the fact that the polynomial functions
$\pi(x)=x_{1}+\cdots+x_{n}$ and $|x|^{2}=x_{1}^{2}+\cdots+x_{n}^{2}$ are, up
to an additive constant for the second, eigenfunctions of the dynamics
associated to the spectral values $-1$ and $-2$ respectively, and that their
“carré du champ” is affine. In the matrix cases $\beta\in\\{1,2\\}$, these
functions correspond to the dynamics of the trace, the dynamics of the squared
Hilbert–Schmidt trace norm, and the dynamics of the squared trace. It is
remarkable that this phenomenon survives beyond these matrix cases, yet
another manifestation of the Gaussian “ghosts” concept due to Edelman, see for
instance [34].
###### Proof of Theorem 1.3.
The process $Y_{t}:=\pi(X_{t}^{n})$ solves
$\mathrm{d}Y_{t}=\sum_{i=1}^{n}\mathrm{d}X_{t}^{n,i}=\sqrt{\frac{2}{n}}\sum_{i=1}^{n}\mathrm{d}B^{i}_{t}-\sum_{i=1}^{n}X_{t}^{n,i}\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}}.$
By symmetry, the double sum vanishes. Note that the process
$W_{t}:=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}B^{i}_{t}$ is a standard one
dimensional BM, so that
$\mathrm{d}Y_{t}=\sqrt{2}\mathrm{d}W_{t}-Y_{t}\mathrm{d}t.$ This proves the
first part of the statement.
We turn to the second part. Recall that $X_{t}\in D_{n}$ for all $t>0$. By
Itô’s formula
$\mathrm{d}(X_{t}^{n,i})^{2}=\sqrt{\frac{8}{n}}X_{t}^{n,i}\mathrm{d}B^{i}_{t}-2(X_{t}^{n,i})^{2}\mathrm{d}t+2\frac{\beta}{n}X_{t}^{n,i}\sum_{j:j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}}+\frac{2}{n}\mathrm{d}t.$
Set
$W_{t}:=\sum_{i=1}^{n}\int_{0}^{t}\frac{X_{s}^{n,i}}{|X^{n}_{s}|}\mathrm{d}B^{i}_{s}$.
The process ${(W_{t})}_{t\geq 0}$ is a BM by the Lévy characterization since
$\langle
W\rangle_{t}=\int_{0}^{t}\frac{\sum_{i=1}^{n}(X^{n,i}_{s})^{2}}{|X^{n}_{s}|^{2}}\mathrm{d}s=t.$
Furthermore, a simple computation shows that
$\sum_{i=1}^{n}X_{t}^{n,i}\sum_{j:j\neq
i}\frac{1}{X_{t}^{n,i}-X_{t}^{n,j}}=\frac{n(n-1)}{2}.$
Consequently the process $R_{t}:=|X_{t}^{n}|^{2}$ solves
$\mathrm{d}R_{t}=\sqrt{\frac{8}{n}R_{t}}\mathrm{d}W_{t}+\Big{(}2+\beta(n-1)-2R_{t}\Big{)}\mathrm{d}t,$
and is therefore a CIR process of parameters $a=2+\beta(n-1)$, $b=2$, and
$\sigma=\sqrt{8/n}$.
When $d=\frac{\beta}{2}n^{2}+(1-\frac{\beta}{2})n$ is a positive integer, the
last property of the statement follows from the connection between OU and CIR
recalled right before the statement of the theorem. ∎
The last proof actually relies on the following general observation. Let $X$
be an $n$-dimensional continuous semi-martingale solution of
$\mathrm{d}X_{t}=\sigma(X_{t})\mathrm{d}B_{t}+b(X_{t})\mathrm{d}t$
where $B$ is a $n$-dimensional standard BM, and where
$x\in\mathbb{R}^{n}\mapsto\sigma(x)\in\mathcal{M}_{n,n}(\mathbb{R})\quad\text{and}\quad
x\in\mathbb{R}^{n}\mapsto b(x)\in\mathbb{R}^{n}$
are Lipschitz. The infinitesimal generator of the Markov semigroup is given by
$\operatorname{G}(f)(x)=\frac{1}{2}\sum_{i,j=1}^{n}a_{i,j}(x)\partial_{i,j}f(x)+\sum_{i=1}^{n}b_{i}(x)\partial_{i}f(x),\quad\text{where}\quad
a(x)=\sigma(x)(\sigma(x))^{\top},$
for all $f\in\mathcal{C}^{2}(\mathbb{R}^{n},\mathbb{R})$ and
$x\in\mathbb{R}^{n}$. Then, by Itô’s formula, the process
$M^{f}={(M^{f}_{t})}_{t\geq 0}$ given by
$M^{f}_{t}=f(X_{t})-f(X_{0})-\int_{0}^{t}(\operatorname{G}f)(X_{s})\mathrm{d}s=\sum_{i,k=1}^{n}\int_{0}^{t}\partial_{i}f(X_{s})\sigma_{i,k}(X_{s})\mathrm{d}B_{s}^{k}$
is a local martingale, and moreover, for all $t\geq 0$,
$\langle
M^{f}\rangle_{t}=\int_{0}^{t}\Gamma(f)(X_{s})\mathrm{d}s\quad\text{where}\quad\Gamma(f)(x)=|\sigma(x)^{\top}\nabla
f(x)|^{2}=a(x)\nabla f\cdot\nabla f.$
The functional quadratic form $\Gamma$ is known as the “carré du champ”
operator.
If $f$ is an eigenfunction of $\operatorname{G}$ associated to the spectral
value $\lambda$ in the sense that $\operatorname{G}f=\lambda f$ (note by the
way that $\lambda\leq 0$ since $\operatorname{G}$ generates a Markov process),
then we get
$f(X_{t})=f(X_{0})+\lambda\int_{0}^{t}f(X_{s})\mathrm{d}s+M_{t}^{f},\quad\text{in
other words}\quad\mathrm{d}f(X_{t})=\mathrm{d}M^{f}_{t}+\lambda
f(X_{t})\mathrm{d}t.$
Now if $\Gamma(f)=c$ (as in the first part of the theorem), then by the Lévy
characterization of Brownian motion, the continuous local martingale
$W:=\frac{1}{\sqrt{c}}M^{f}$ starting from the origin is a standard BM and we
recover the result of the first part of the theorem. On the other hand, if
$\Gamma(f)=cf$ (as in the second part of the theorem), then by the Lévy
characterization of BM the local martingale
$W:=\int_{0}^{t}\frac{1}{\sqrt{cf(X_{s})}}\mathrm{d}M_{s}^{f}$
is a standard BM and we recover the result of the second part.
At this point, we observe that the infinitesimal generator of the CIR process
$R$ is the Laguerre partial differential operator
$L(f)(x)=\frac{4}{n}xf^{\prime\prime}(x)+(2+\beta(n-1)-2x)f^{\prime}(x).$
(4.1)
This operator leaves invariant the set of polynomials of degree less than or
equal to $k$, for all integer $k\geq 0$, a property inherited from (2.5). We
will use this property in the following proof.
### 4.1. Proof of Corollary 1.4
By Theorem 1.3, $Z=\pi(X^{n})$ is an OU process in $\mathbb{R}$ solution of
the stochastic differential equation
$Z_{0}=\pi(X_{0}^{n}),\quad\mathrm{d}Z_{t}=\sqrt{2}\mathrm{d}B_{t}-Z_{t}\mathrm{d}t,$
where $B$ is a standard one-dimensional BM. By Lemma 3.1,
$Z_{t}\sim\mathcal{N}(Z_{0}\mathrm{e}^{-t},1-\mathrm{e}^{-2t})$ for all $t\geq
0$ and the equilibrium distribution is
$P_{n}^{\beta}\circ\pi^{-1}=\mathcal{N}(0,1)$. Using the contraction property
stated in Lemma A.2, the comparison between Hellinger and TV of Lemma A.1 and
the explicit expressions for Gaussian distributions of Lemma A.5, we find
$\displaystyle\|\mathrm{Law}(X^{n}_{t})-P_{n}^{\beta}\|_{\mathrm{TV}}$
$\displaystyle\geq\|\mathrm{Law}(Z_{t})-P_{n}^{\beta}\circ\pi^{-1}\|_{\mathrm{TV}}$
$\displaystyle\geq\mathrm{Hellinger}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=1-\frac{(1-\mathrm{e}^{-2t})^{1/4}}{(1-\frac{1}{2}\mathrm{e}^{-2t})^{1/2}}\exp\Bigr{(}-\frac{\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}}{4(2-\mathrm{e}^{-2t})}\Bigr{)}.$
Setting $c_{n}:=\log(|\pi(X^{n}_{0})|)$ and assuming that
$\lim_{n\to\infty}c_{n}=\infty$, we deduce that for all $\varepsilon\in(0,1)$
$\lim_{n\to\infty}\|\mathrm{Law}(X^{n}_{c_{n}(1-\varepsilon)})-P_{n}^{\beta}\|_{\mathrm{TV}}=1.$
The comparison between $\mathrm{Hellinger}$ and $\mathrm{TV}$ of Lemma A.1
allows to deduce that this remains true for the Hellinger distance.
We turn to Kullback. The contraction property stated in Lemma A.2 and the
explicit expressions for Gaussian distributions of Lemma A.5 yield
$\displaystyle 2\mathrm{Kullback}(\mathrm{Law}(X^{n}_{t})\mid P_{n}^{\beta})$
$\displaystyle\geq 2\mathrm{Kullback}(\mathrm{Law}(Z_{t})\mid
P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}-\mathrm{e}^{-2t}-\log(1-\mathrm{e}^{-2t}).$
This is enough to deduce that
$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=+\infty.$
The situation is similar for $\chi^{2}$: the contraction property stated in
Lemma A.2 and the explicit expressions for Gaussian distributions of Lemma A.5
yield
$\displaystyle\chi^{2}(\mathrm{Law}(X^{n}_{t})\mid P_{n}^{\beta})$
$\displaystyle\geq\chi^{2}(\mathrm{Law}(Z_{t})\mid
P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=-1+\frac{1}{\sqrt{1-\mathrm{e}^{-4t}}}\exp\left(\frac{1}{1+\mathrm{e}^{-2t}}(1-\pi(X^{n}_{0})\mathrm{e}^{-t})^{2}\right),$
so that
$\lim_{n\to\infty}\chi^{2}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=+\infty.$
Regarding the Wasserstein distance, we have
$\left\|\pi\right\|_{\mathrm{Lip}}:=\sup_{x\neq
y}\frac{|\pi(x)-\pi(y)|}{|x-y|}\leq\sqrt{n}$ from the Cauchy–Schwarz
inequality, and by Lemma A.2, for all probability measures $\mu$ and $\nu$ on
$\mathbb{R}^{n}$,
$\mathrm{Wasserstein}(\mu\circ\pi^{-1},\nu\circ\pi^{-1})\leq\sqrt{n}\mathrm{Wasserstein}(\mu,\nu).$
(4.2)
Using the explicit expressions for Gaussian distributions of Lemma A.5, we
thus find
$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})$
$\displaystyle\geq\frac{1}{n}\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=\frac{1}{n}\Bigr{(}\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}\Bigr{)}.$
Setting $c_{n}:=\log\Big{(}\frac{|\pi(x_{0}^{n})|}{\sqrt{n}}\Big{)}$ and
assuming $c_{n}\to\infty$ as $n\to\infty$, we thus deduce that for all
$\varepsilon\in(0,1)$
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}}),P_{n}^{\beta})=+\infty.$
## 5\. The random matrix cases
In this section, we prove Theorem 1.5 and Corollary 1.6 that cover the matrix
cases $\beta\in\\{1,2\\}$. For these values of $\beta$, the DOU process is the
image by the spectral map of a matrix OU process, connected to the random
matrix models $\mathrm{GOE}$ and $\mathrm{GUE}$. We could consider the case
$\beta=4$ related to $\mathrm{GSE}$. Beyond these three algebraic cases, it
could be possible for an arbitrary $\beta\geq 1$ to use random tridiagonal
matrices dynamics associated to $\beta$ Dyson processes, see for instance
[44].
The next two subsections are devoted to the proof of Theorem 1.5 in the
$\beta=2$ and $\beta=1$ cases respectively. The third section provides the
proof of Corollary 1.6.
### 5.1. Hermitian case ($\beta=2$)
Let $\mathrm{Herm}_{n}$ be the set of $n\times n$ complex Hermitian matrices,
namely the set of $h\in\mathcal{M}_{n,n}(\mathbb{C})$ with
$h_{i,j}=\overline{h_{j,i}}$ for all $1\leq i,j\leq n$. An element
$h\in\mathrm{Herm}_{n}$ is parametrized by the $n^{2}$ real variables
$(h_{i,i})_{1\leq i\leq n}$, $(\Re h_{i,j})_{1\leq i<j\leq n}$, $(\Im
h_{i,j})_{1\leq i<j\leq n}$. We define, for $h\in\mathrm{Herm}_{n}$ and $1\leq
i,j\leq n$,
$\pi_{i,j}(h)=\begin{cases}h_{i,i}&\text{if $i=j$}\\\ \sqrt{2}\,\Re
h_{i,j}&\text{if $i<j$}\\\ \sqrt{2}\,\Im h_{j,i}&\text{if $i>j$}\end{cases}.$
(5.1)
Note that
$\mathrm{Tr}(h^{2})=\sum_{i,j=1}^{n}|h_{i,j}|^{2}=\sum_{i=1}^{n}h_{i,i}^{2}+2\sum_{i<j}(\Re
h_{i,j})^{2}+2\sum_{i<j}(\Im h_{i,j})^{2}=\sum_{i,j}\pi_{i,j}(h)^{2}.$
We thus identify $\mathrm{Herm}_{n}$ with
$\mathbb{R}^{n}\times\mathbb{R}^{2\frac{n^{2}-n}{2}}=\mathbb{R}^{n^{2}}$, this
identification is isometrical provided $\mathrm{Herm}_{n}$ is endowed with the
norm $\sqrt{\mathrm{Tr}(h^{2})}$ and $\mathbb{R}^{n^{2}}$ with the Euclidean
norm.
The Gaussian Unitary Ensemble $\mathrm{GUE}_{n}$ is the Gaussian law on
$\mathrm{Herm}_{n}$ with density
$h\in\mathrm{Herm}_{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(h^{2})}}{C_{n}}\quad\text{where}\quad
C_{n}:=\int_{\mathbb{R}^{n^{2}}}\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(h^{2})}\prod_{i=1}^{n}\mathrm{d}h_{i,i}\prod_{i<j}\mathrm{d}\Re
h_{i,j}\prod_{i<j}\mathrm{d}\Im h_{i,j}.$ (5.2)
If $H$ is a random $n\times n$ Hermitian matrix then $H\sim\mathrm{GUE}_{n}$
if and only if the $n^{2}$ real random variables $\pi_{i,j}(H)$, $1\leq
i,j\leq n$, are independent Gaussian random variables with
$\pi_{i,j}(H)\sim\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)},\quad 1\leq i,j\leq
n.$ (5.3)
The law $\mathrm{GUE}_{n}$ is the unique invariant law of the Hermitian matrix
OU process ${(H_{t})}_{t\geq 0}$ on $\mathrm{Herm}_{n}$ solution of the
stochastic differential equation
$H_{0}=h_{0}\in\mathrm{Herm}_{n},\quad\mathrm{d}H_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-H_{t}\mathrm{d}t,$
(5.4)
where $B={(B_{t})}_{t\geq 0}$ is a Brownian motion on $\mathrm{Herm}_{n}$, in
the sense that the stochastic processes $(\pi_{i,j}(B_{t}))_{t\geq 0}$, $1\leq
i\neq j\leq n$, are independent standard one-dimensional BM. The coordinates
stochastic processes ${(\pi_{i,j}(H_{t}))}_{t\geq 0}$, $1\leq i,j\leq n$, are
independent real OU processes.
For any $h$ in $\mathrm{Herm}_{n}$, we denote by $\Lambda(h)$ the vector of
the eigenvalues of $h$ ordered in non-decreasing order. Lemma 5.1 below is an
observation which dates back to the seminal work of Dyson [33], hence the name
DOU for $X^{n}$. We refer to [37, Ch. 12] and [2, Sec. 4.3] for a mathematical
approach using modern stochastic calculus.
###### Lemma 5.1 (From matrix OU to DOU).
The image of $\mathrm{GUE}_{n}$ by the map $\Lambda$ is the Coulomb gas
$P_{n}^{\beta}$ given by (1.6) with $\beta=2$. Moreover the stochastic process
$X^{n}={(X^{n}_{t})}_{t\geq 0}={(\Lambda(H_{t}))}_{t\geq 0}$ is well-defined
and solves the stochastic differential equation (1.3) with $\beta=2$ and
$x_{0}^{n}=\Lambda(h_{0})$.
Let $\beta=2$. Let us assume from now on that the initial value
$h_{0}\in\mathrm{Herm}_{n}$ of ${(H_{t})}_{t\geq 0}$ has eigenvalues
$x_{0}^{n}$ where $x^{n}_{0}$ is as in Theorem 1.5. We start by proving the
upper bound on the $\chi^{2}$ distance stated in Theorem 1.5: it will be an
adaptation of the proof of the upper bound of Theorem 1.1 applied to the
Hermitian matrix OU process ${(H_{t})}_{t\geq 0}$ combined with the
contraction property of the $\chi^{2}$ distance. Indeed, by Lemma 5.1 and the
contraction property of Lemma A.2
$\chi^{2}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GUE}_{n}).$ (5.5)
We claim now that the right-hand side tends to $0$ as $n\to\infty$ when
$t=t_{n}$ is well chosen. Indeed, using the identification between
$\mathrm{Herm}_{n}$ and $\mathbb{R}^{n^{2}}$ mentioned earlier, we have
$\mathrm{GUE}_{n}=\mathcal{N}(m_{2},\Sigma_{2})$ where $m_{2}=0$ and where
$\Sigma_{2}$ is an $n^{2}\times n^{2}$ diagonal matrix with
$(\Sigma_{2})_{(i,j),(i,j)}=\frac{1}{n}.$ (5.6)
On the other hand, the Mehler formula (Lemma 3.1) gives
$\mathrm{Law}(H_{t})=\mathcal{N}(m_{1},\Sigma_{1})$ where
$m_{1}=\mathrm{e}^{-t}h_{0}$ and where $\Sigma_{1}$ is an $n^{2}\times n^{2}$
diagonal matrix with
$(\Sigma_{1})_{(i,j),(i,j)}=\frac{1-\mathrm{e}^{-2t}}{n}.$ (5.7)
Therefore, using Lemma A.5, the analogue of (3.3) reads
$\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GUE}_{n})=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{n^{2}/2}}\exp\left(n|h_{0}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right).$
(5.8)
where
$|h_{0}|^{2}=\sum_{1\leq i,j\leq n}\pi_{i,j}(h_{0})^{2}=\sum_{1\leq i,j\leq
n}|(h_{0})_{i,j}|^{2}=\mathrm{Tr}(h_{0}^{2})=|x_{0}^{n}|^{2}.$ (5.9)
Taking now $c_{n}:=\log(\sqrt{n}|x_{0}^{n}|)\vee\log(\sqrt{n})$, for any
$\varepsilon\in(0,1)$, we get
$\chi^{2}(\mathrm{Law}(X_{(1+\varepsilon)c_{n}}^{n})\mid
P_{n}^{\beta})\leq\chi^{2}(\mathrm{Law}(H_{(1+\varepsilon)c_{n}})\mid\mathrm{GUE}_{n})\underset{n\to\infty}{\longrightarrow}0.$
(5.10)
In the right-hand side of (5.8), the factor $n^{2}$ is the dimension of the
$\mathbb{R}^{n^{2}}$ to which $\mathrm{Herm}_{n}$ is identified, while the
factor $n$ in the first term is due to the $1/n$ scaling in the stochastic
differential equation of the process. This explains the difference with the
analogue (3.3) in dimension $n$.
From the comparison between TV, Hellinger, Kullback and $\chi^{2}$ stated in
Lemma A.1, we easily deduce that the previous convergence remains true upon
replacing $\chi^{2}$ by $\mathrm{TV}$, $\mathrm{Hellinger}$ or
$\mathrm{Kullback}$.
It remains to cover the upper bound for the Wasserstein distance. This
distance is more sensitive to contraction arguments: according to Lemma A.2,
one needs to control the Lipschitz norm of the “contraction map” at stake. It
happens that the spectral map, restricted to the set $\mathrm{Herm}_{n}$ of
$n\times n$ Hermitian matrices, is $1$-Lipschitz: more precisely, the
Hoffman–Wielandt inequality, see [43] and [45, Th. 6.3.5], asserts that for
any two such matrices $A$ and $B$, denoting
$\Lambda(A)=(\lambda_{i}(A))_{1\leq i\leq n}$ and
$\Lambda(B)=(\lambda_{i}(B))_{1\leq i\leq n}$ the ordered sequences of their
eigenvalues, we have
$\sum_{i=1}^{n}|\lambda_{i}(A)-\lambda_{i}(B)|^{2}\leq\sum_{i,j}|A_{i,j}-B_{i,j}|^{2}.$
Applying Lemma A.2, we thus deduce that
$\mathrm{Wasserstein}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})\leq\mathrm{Wasserstein}(\mathrm{Law}(H_{t}),\mathrm{GUE}_{n}).$
(5.11)
Following the Gaussian computations in the proof of Theorem 1.2, we obtain
$\mathrm{Wasserstein}^{2}(\mathrm{Law}(H_{t}),\mathrm{GUE}_{n})=|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$
(5.12)
Set $c_{n}:=\log(|x_{0}^{n}|)$. If $c_{n}\to\infty$ as $n\to\infty$ then for
all $\varepsilon\in(0,1)$ we find
$\mathrm{Wasserstein}(\mathrm{Law}(X_{(1+\varepsilon)c_{n}}^{n}),P_{n}^{\beta})\underset{n\to\infty}{\longrightarrow}0.$
This completes the proof of Theorem 1.5.
### 5.2. Symmetric case ($\beta=1$)
The method is similar to the case $\beta=2$. Let us focus only on the
differences. Let $\mathrm{Sym}_{n}$ be the set of $n\times n$ real symmetric
matrices, namely the set of $s\in\mathcal{M}_{n,n}(\mathbb{R})$ with
$s_{i,j}=s_{j,i}$ for all $1\leq i,j\leq n$. An element $s\in\mathrm{Sym}_{n}$
is parametrized by the $n+\frac{n^{2}-n}{2}=\frac{n(n+1)}{2}$ real variables
$(s_{i,j})_{1\leq i\leq j\leq n}$. We define, for $s\in\mathrm{Sym}_{n}$ and
$1\leq i\leq j\leq n$,
$\pi_{i,j}(s)=\begin{cases}s_{i,i}&\text{if $i=j$}\\\
\sqrt{2}\,s_{i,j}&\text{if $i<j$}\end{cases}.$ (5.13)
Note that
$\mathrm{Tr}(s^{2})=\sum_{i,j=1}^{n}s_{i,j}^{2}=\sum_{i=1}^{n}s_{i,i}^{2}+2\sum_{i<j}s_{i,j}^{2}=\sum_{1\leq
i\leq j\leq n}\pi_{i,j}(s)^{2}.$
We thus identify isometrically $\mathrm{Sym}_{n}$, endowed with the norm
$\sqrt{\mathrm{Tr}(h^{2})}$, with
$\mathbb{R}^{n}\times\mathbb{R}^{\frac{n^{2}-n}{2}}=\mathbb{R}^{\frac{n(n+1)}{2}}$
endowed with the Euclidean norm.
The Gaussian Orthogonal Ensemble $\mathrm{GOE}_{n}$ is the Gaussian law on
$\mathrm{Sym}_{n}$ with density
$s\in\mathrm{Sym}_{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(s^{2})}}{C_{n}}\quad\text{where}\quad
C_{n}:=\int_{\mathbb{R}^{\frac{n(n+1)}{2}}}\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(s^{2})}\prod_{1\leq
i\leq j\leq n}\mathrm{d}s_{i,j}.$ (5.14)
If $S$ is a random $n\times n$ real symmetric matrix then
$S\sim\mathrm{GOE}_{n}$ if and only if the $\frac{n(n+1)}{2}$ real random
variables $\pi_{i,j}(S)$, $1\leq i\leq j\leq n$, are independent Gaussian
random variables with
$\pi_{i,j}(S)\sim\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)},\quad 1\leq i\leq
j\leq n.$ (5.15)
The law $\mathrm{GOE}_{n}$ is the unique invariant law of the real symmetric
matrix OU process ${(S_{t})}_{t\geq 0}$ on $\mathrm{Sym}_{n}$ solution of the
stochastic differential equation
$S_{0}=s_{0}\in\mathrm{Sym}_{n},\quad\mathrm{d}S_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-S_{t}\mathrm{d}t$
(5.16)
where $B={(B_{t})}_{t\geq 0}$ is a Brownian motion on $\mathrm{Sym}_{n}$, in
the sense that the stochastic processes $(\pi_{i,j}(B_{t}))_{t\geq 0}$, $1\leq
i\leq j\leq n$, are independent standard one-dimensional BM. The coordinates
stochastic processes ${(\pi_{i,j}(S_{t}))}_{t\geq 0}$, $1\leq i\leq j\leq n$,
are independent real OU processes.
For any $s$ in $\mathrm{Sym}_{n}$, we denote by $\Lambda(s)$ the vector of the
eigenvalues of $s$ ordered in non-decreasing order. Lemma 5.2 below is the
real symmetric analogue of Lemma 5.1.
###### Lemma 5.2 (From matrix OU to DOU).
The image of $\mathrm{GOE}_{n}$ by the map $\Lambda$ is the Coulomb gas
$P_{n}^{\beta}$ given by (1.6) with $\beta=1$. Moreover the stochastic process
$X^{n}={(X^{n}_{t})}_{t\geq 0}={(\Lambda(S_{t}))}_{t\geq 0}$ is well-defined
and solves the stochastic differential equation (1.3) with $\beta=1$ and
$x_{0}^{n}=\Lambda(s_{0})$.
As for the case $\beta=2$, the idea now is that the DOU process is sandwiched
between a real OU process and a matrix OU process.
By similar computations to the case $\beta=2$, the analogue of (5.8) becomes
$\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GOE}_{n})=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{\frac{(n(n+1))^{2}}{8}}}\exp\left(n|h_{0}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right).$
(5.17)
This allows to deduce the upper bound for TV, Hellinger, Kullback and
$\chi^{2}$. Regarding the Wasserstein distance, the analogue of (5.12) reads
$\mathrm{Wasserstein}^{2}(\mathrm{Law}(S_{t}),\mathrm{GOE}_{n})=|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$
(5.18)
If $\lim_{n\to\infty}\log(|x_{0}^{n}|)=\infty$ then we deduce the asserted
result, concluding the proof of Theorem 1.5.
### 5.3. Proof of Corollary 1.6
Let $\beta\in\\{1,2\\}$. Recall the definitions of $a_{n}$ and $c_{n}$ from
the statement. Take $x_{0}^{n,i}=a_{n}$ for all $i$, and note that
$\pi(x_{0}^{n})=na_{n}$. Given our assumptions on $a_{n}$, Corollary 1.4
yields for this particular choice of initial condition and for any
$\varepsilon\in(0,1)$
$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=\max.$
On the other hand, in the proof of Theorem 1.5 we saw that
$\chi^{2}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq-1+\frac{1}{(1-\mathrm{e}^{-4t})^{b_{n}/2}}\exp\left(n|x_{0}^{n}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right),$
where $b_{n}=n^{2}$ for $\beta=2$ and $b_{n}=(n(n+1)/2)^{2}$ for $\beta=1$.
Since $|x_{0}^{n}|\leq\sqrt{n}a_{n}$ for all $x_{0}^{n}\in[-a_{n},a_{n}]^{n}$,
and given the comparison between TV, Hellinger, Kullback and $\chi^{2}$ stated
in Lemma A.1 we obtain for
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$
and for all $\varepsilon\in(0,1)$
$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0,$
thus concluding the proof of Corollary 1.6 regarding theses distances.
Concerning Wasserstein, the proof of Theorem 1.5 shows that for any
$x_{0}^{n}\in[-a_{n},a_{n}]^{n}$ we have
$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})$
$\displaystyle\leq|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}$
$\displaystyle\leq
na_{n}^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$
If $\sqrt{n}a_{n}\to\infty$, then for $c_{n}=\log(\sqrt{n}a_{n})$ we deduce
that for all $\varepsilon\in(0,1)$
$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$
## 6\. Cutoff phenomenon for the DOU in TV and Hellinger
In this section, we prove Theorem 1.7 and Corollary 1.8 for the TV and
Hellinger distances. We only consider the case $\beta\geq 1$, although the
arguments could be adapted _mutatis mutandis_ to cover the case $\beta=0$:
note that the result of Theorem 1.7 and Corollary 1.8 for $\beta=0$ can be
deduced from Theorem 1.2. At the end of this section, we also provide the
proof of Theorem 1.10.
### 6.1. Proof of Theorem 1.7 in TV and Hellinger
By the comparison between TV and Hellinger stated in Lemma A.1, it suffices to
prove the result for the TV distance, so we concentrate on this distance until
the end of this section. Our proof is based on the exponential decay of the
relative entropy at an explicit rate given by the optimal logarithmic Sobolev
constant. However, this requires the relative entropy of the initial condition
to be _finite_. Consequently, we proceed in three steps. First, given an
arbitrary initial condition $x_{0}^{n}\in\overline{D}_{n}$, we build an
absolutely continuous probability measure $\mu_{x_{0}^{n}}$ on $D_{n}$ that
approximates $\delta_{x_{0}^{n}}$ and whose relative entropy is not too large.
Second, we derive a decay estimate starting from this regularized initial
condition. Third, we control the total variation distance between the two
processes starting respectively from $\delta_{x_{0}^{n}}$ and
$\mu_{x_{0}^{n}}$.
#### 6.1.1. Regularization
In order to have a finite relative entropy at time $0$, we first regularize
the initial condition by smearing out each particle in a ball of radius
bounded below by $n^{-(\kappa+1)}$, for some $\kappa>0$. Let us first
introduce the regularization at scale $\eta$ of a Dirac distribution
$\delta_{z}$, $z\in\mathbb{R}$ by
$\delta_{z}^{(\eta)}(\mathrm{d}u)=\mathrm{Uniform}([z,z+\eta])(\mathrm{d}u)=\eta^{-1}\mathbf{1}_{[z,z+\eta]}\mathrm{d}u.$
Given $x\in\overline{D}_{n}$ and $\kappa>0$, we define a regularized version
of $\delta_{x}$ at scale $n^{-\kappa}$, that we denote $\mu_{x}$, by setting
$\mu_{x}=\otimes_{i=1}^{n}\delta_{x_{i}+3i\eta}^{(\eta)},$ (6.1)
where $\eta:=n^{-(\kappa+1)}$. The parameters have been tuned in such a way
that, independently of the choice of $x\in\overline{D}_{n}$, the following
properties hold. The supports of the Dirac masses
$\delta_{x_{i}+3i\eta}^{(\eta)}$, $i\in\\{1,\ldots,n\\}$, lie at distance at
least $\eta$ from each other. The volume of the support of $\mu_{x}$ is equal
to $\eta^{n}$, and therefore the relative entropy of $\mu_{x}$ with respect to
the Lebesgue measure is not too large. Finally, provided $X_{0}^{n}=x$ and
$Y_{0}^{n}$ is distributed according to $\mu_{x}$, almost surely
$|X_{0}^{n}-Y_{0}^{n}|_{\infty}\leq(3n+1)\eta$.
#### 6.1.2. Convergence of the regularized process to equilibrium
###### Lemma 6.1 (Convergence of regularized process).
Let $(Y_{t}^{n})_{t\geq 0}$ be a DOU process solution of (1.3), $\beta\geq 1$,
and let $P_{n}^{\beta}$ be its invariant law. Assume that
$\mathrm{Law}(Y^{n}_{0})$ is the regularized measure $\mu_{x_{0}^{n}}$ in
(6.1) associated to some initial condition $x_{0}^{n}\in\overline{D}_{n}$.
Then there exists a constant $C>0$, only depending on $\kappa$, such that for
all $t\geq 0$, all $n\geq 2$ and all $x_{0}^{n}\in\overline{D}_{n}$
$\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid P_{n}^{\beta})\leq
C(n|x_{0}^{n}|^{2}+n^{2}\log(n))\mathrm{e}^{-2t}.$
###### Proof of Lemma 6.1.
By Lemma B.2 and since $\mathrm{Law}(Y^{n}_{0})=\mu_{x_{0}^{n}}$, for all
$t\geq 0$, there holds
$\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid
P_{n}^{\beta})\leq\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})\mathrm{e}^{-2t}.$ (6.2)
Now we have
$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})=\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\log\frac{\mathrm{d}\mu_{x_{0}^{n}}}{\mathrm{d}P_{n}^{\beta}}\right].$
Recall the definition of $S$ in (1.15). As $P_{n}^{\beta}$ has density
$\frac{\mathrm{e}^{-E}}{C_{n}^{\beta}}$, we may re-write this as
$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})=S(\mu_{x_{0}^{n}})+\mathbb{E}_{\mu_{x_{0}^{n}}}[E]+\log
C_{n}^{\beta}.$ (6.3)
Recall the partition function $C_{*n}^{\beta}=n!C_{n}^{\beta}$ from Subsection
2.2. It is proved in [12], using explicit expressions involving Gamma
functions via a Selberg integral, that for some constant $C>0$
$\log C_{n}^{\beta}\leq\log C_{*n}^{\beta}\leq Cn^{2}.$ (6.4)
Next, we claim that $S(\mu_{x_{0}^{n}})\leq n\log(n^{1+\kappa})$. Indeed since
$\mu_{x_{0}^{n}}$ is a product measure, the tensorization property of entropy
recalled in Lemma A.4 gives
$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid\mathrm{d}x)=\sum_{i=1}^{n}\mathrm{Kullback}(\delta_{0}^{(\eta)}\mid\mathrm{d}x).$
Moreover an immediate computation yields
$\mathrm{Kullback}(\delta_{0}^{(\eta)}\mid\mathrm{d}x)=\log(\eta^{-1})$ so
that, given the definition of $\eta$, we get
$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid\mathrm{d}x)=n\log(n^{\kappa+1}).$ (6.5)
We turn to the estimation of the term $\mathbb{E}_{\mu_{x_{0}^{n}}}[E].$ The
confinement term can be easily bounded:
$\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\frac{n}{2}\sum_{i=1}^{n}{x_{i}^{2}}\right]\leq(n|x_{0}^{n}|^{2}+n^{2}\eta^{2}).$
Let us now estimate the logarithmic energy of $\mu_{x_{0}^{n}}$. Using the
fact that the logarithmic function is increasing, together with the fact the
supports of $\delta_{x_{i}+3i\eta}^{(\eta)}$ lie at distance at least $\eta$
from each other, we notice that for any $i>j$ there holds
$\displaystyle\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\log|x_{i}-x_{j}|\right]$
$\displaystyle=\iint\log|x-y|\delta_{x_{i}+3i\eta}^{(\eta)}(\mathrm{d}x)\delta_{x_{j}+3j\eta}^{(\eta)}(\mathrm{d}y)$
$\displaystyle\geq\iint\log|x-y|\delta_{3\eta}^{(\eta)}(\mathrm{d}x)\delta_{0}^{(\eta)}(\mathrm{d}y)$
$\displaystyle\geq\log\eta.$
It follows that the initial logarithmic energy cannot be much larger than
$n^{2}\log n$:
$\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}\right]\leq\frac{n(n-1)}{2}\log
n^{\kappa+1}.$
This implies that there exists a constant $C>0$, only depending on $\kappa$,
such that for all $n\geq 2$
$\mathbb{E}_{\mu_{x_{0}^{n}}}[E]=\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\frac{n}{2}\sum_{i=1}^{n}|x_{i}|^{2}+{\beta}\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}\right]\leq
C\big{(}n|x_{0}^{n}|^{2}+n^{2}\log n\big{)}.$ (6.6)
Inserting (6.4), (6.5) and (6.6) into (6.3) we obtain (for a different
constant $C>0$)
$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid P_{n}^{\beta})\leq
C\big{(}n|x_{0}^{n}|^{2}+n^{2}\log n\big{)}.$
This bound, combined with (6.2), concludes the proof of Lemma 6.1. ∎
#### 6.1.3. Convergence to the regularized process in total variation
distance
Let $(X_{t}^{n})_{t\geq 0}$ and $(Y_{t}^{n})_{t\geq 0}$ be two DOU processes
with $X_{0}^{n}=x_{0}^{n}$ and $\mathrm{Law}(Y_{0}^{n})=\mu_{x_{0}^{n}}$,
where the measure $\mu_{x_{0}^{n}}$ is defined in (6.1). Below we prove that,
as soon as the parameter $\kappa$ is large enough, the total variation
distance between $\mathrm{Law}(X_{t}^{n})$ and $\mathrm{Law}(Y_{t}^{n})$ tends
to $0$, for any fixed $t>0$.
Note that at time $0$, almost surely, there holds $X_{0}^{n,i}\leq
Y_{0}^{n,i}$, for every $i\in\\{1,\ldots,n\\}$. We now introduce a coupling of
the processes $(X_{t}^{n})_{t\geq 0}$ and $(Y_{t}^{n})_{t\geq 0}$ that
preserves this ordering at all times. Consider two independent standard BM
$B^{n}$ and $W^{n}$ in $\mathbb{R}^{n}$. Let $X^{n}$ be the solution of (1.3)
driven by $B^{n}$, and let $Y^{n}$ be the solution of
$\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2}{n}}\Big{(}\mathbf{1}_{\\{Y_{t}^{n,i}\neq
X_{t}^{n,i}\\}}\mathrm{d}W^{i}_{t}+\mathbf{1}_{\\{Y_{t}^{n,i}=X_{t}^{n,i}\\}}\mathrm{d}B^{i}_{t}\Big{)}-Y_{t}^{n,i}\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n.$
We denote by $\mathbb{P}$ the probability measure under which these two
processes are coupled. Let us comment on the driving noise in the equation
satisfied by $Y^{n}$. When the $i$-th coordinates of $X^{n}$ and $Y^{n}$
equal, we take the same driving Brownian motion and the difference
$Y^{n,i}-X^{n,i}$ remains non-negative due to the convexity of $-\log$ defined
in (1.5), see the monotoncity result stated in Lemma B.4. On the other hand,
when these two coordinates differ, we take independent driving Brownian
motions in order for their difference to have non-zero quadratic variation
(this allows to increase their merging probability). Under this coupling, the
ordering of $X^{n}$ and $Y^{n}$ is thus preserved at all times, and if
$X_{s}^{n}=Y_{s}^{n}$ for some $s\geq 0$, then it remains true at all times
$t\geq s$. Note however that if $X_{s}^{n,i}=Y_{s}^{n,i}$, then this equality
does not remain true at all times except if all the coordinates match.
As in (A.7), the total variation distance between the laws of $X_{t}^{n}$ and
$Y_{t}^{n}$ may be bounded by
$\|\mathrm{Law}(Y_{t}^{n})-\mathrm{Law}(X_{t}^{n})\|_{\mathrm{TV}}\leq\mathbb{P}(X_{t}^{n}\neq
Y_{t}^{n}),$
for all $t\geq 0$. We wish to establish that for any given $t>0$,
$\lim_{n\to\infty}\mathbb{P}(X_{t}^{n}\neq Y_{t}^{n})=0.$
To do so, we work with the area between the two processes $X^{n}$ and $Y^{n}$,
defined by
$A^{n}_{t}:=\sum_{i=1}^{n}\big{(}Y^{n,i}_{t}-X^{n,i}_{t}\big{)}=\pi(Y^{n}_{t})-\pi(X^{n}_{t}),\quad
t\geq 0.$
As the two processes are ordered at any time, this is nothing but the
geometric area between the two discrete interfaces $i\mapsto X_{t}^{n,i}$ and
$i\mapsto Y_{t}^{n,i}$ associated to the configurations $X_{t}^{n}$ and
$Y_{t}^{n}$. We deduce that the merging time of the two processes coincide
with the hitting time of $0$ by this area, that we denote by
$\tau=\inf\\{t\geq 0:A_{t}^{n}=0\\}$.
The process $A^{n}$ has a very simple structure: it is a semimartingale that
behaves like an OU process with a randomly varying quadratic variation. Let
$N_{t}$ be the number of coordinates that do not coincide at time $t$, that is
$N_{t}:=\\#\big{\\{}i\in\\{1,\ldots,n\\}:X^{n,i}_{t}\neq
Y^{n,i}_{t}\big{\\}}.$
Then $A^{n}$ satisfies
$\mathrm{d}A^{n}_{t}=-A^{n}_{t}\mathrm{d}t+\mathrm{d}M_{t},$
where $M$ is a centered martingale with quadratic variation
$\mathrm{d}\langle M\rangle_{t}=\frac{2}{n}N_{t}\mathrm{d}t.$ (6.7)
Note that whenever $t<\tau$ we have
$\mathrm{d}\langle M\rangle_{t}\geq\frac{2}{n}.$
This _a priori_ lower bound on the quadratic variation of $M$, combined with
the Dubins–Schwarz theorem, allows to check that $\tau<\infty$ almost surely.
Note that in view of the coupling between $X_{t}^{n}$ and $Y_{t}^{n}$, we have
$X_{t}^{n}=Y_{t}^{n}$ for all $t\geq\tau$.
Recall the following informal fact: with large probability, a Brownian motion
starting from $a$ hits $b$ by a time of order $(a-b)^{2}$. For a continuous
martingale, this becomes: with large probability, a continuous martingale
starting from $a$ accumulates a quadratic variation of order $(a-b)^{2}$ up to
its first hitting time of $b$. Our next lemma states such a bound on the
supermartingale $A^{n}$.
###### Lemma 6.2.
Let $a>b\geq 0$. Let $\tau_{b}=\inf\\{t>0:A_{t}=b\\}<\infty$ almost surely.
Then, for all $u\geq 1$,
$\mathbb{P}(\langle A\rangle_{\tau_{b}}\geq(a-b)^{2}u\mid A_{0}=a)\leq
4u^{-1/2}.$
###### Proof.
Without loss of generality one can assume that $A_{0}=a$ almost surely.
By Itô’s formula, for all $\lambda\geq 0$, the process
$S_{t}=\exp\Bigr{(}-\lambda A_{t}-\frac{\lambda^{2}}{2}\langle
A\rangle_{t}\Bigr{)},$
defines a submartingale (taking its values in $[0,1]$). Doob’s stopping
theorem yields
$\mathbb{E}[\mathrm{e}^{-\frac{\lambda^{2}}{2}\langle
A\rangle_{\tau_{b}}}]=\mathrm{e}^{\lambda
b}\mathbb{E}[S_{\tau_{b}}]\geq\mathrm{e}^{\lambda
b}\mathbb{E}[S_{0}]=\mathrm{e}^{-\lambda(a-b)}.$
On the other hand, for $\lambda=2(a-b)^{-1}u^{-1/2}$, there holds
$\displaystyle\mathbb{E}[\mathrm{e}^{-\frac{\lambda^{2}}{2}\langle
A\rangle_{\tau_{b}}}]$ $\displaystyle\leq\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}<(a-b)^{2}u\big{)}+\mathrm{e}^{-\frac{\lambda^{2}}{2}(a-b)^{2}u}\,\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}$ $\displaystyle\leq
1-(1-\mathrm{e}^{-\frac{\lambda^{2}}{2}(a-b)^{2}u})\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}$ $\displaystyle\leq
1-\frac{1}{2}\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}.$
Consequently one deduces that
$\mathbb{P}(\langle A\rangle_{\tau_{b}}\geq(a-b)^{2}u)\leq
2(1-\mathrm{e}^{-\lambda(a-b)})\leq 4u^{-1/2}.$
∎
We are now ready to prove the following lemma:
###### Lemma 6.3.
If $\kappa>\frac{3}{2}$, then for every sequence of times ${(t_{n})}_{n}$ with
$\varliminf_{n\to\infty}t_{n}>0$, we have
$\lim_{n\to\infty}\sup_{x_{0}^{n}\in\overline{D}_{n}}\|\mathrm{Law}(Y_{t_{n}}^{n})-\mathrm{Law}(X_{t_{n}}^{n})\|_{\mathrm{TV}}=0.$
###### Proof of Lemma 6.3.
Let ${(t_{n})}_{n}$ be a sequence of times such that
$\varliminf_{n\to\infty}t_{n}>0$. In view of the definition of
$\mu_{{x_{0}^{n}}}$ and $\eta$, the initial area satisfies almost surely
$A_{0}^{n}\leq 4n^{1-\kappa}.$
According to Lemma 6.2, with a probability that goes to $1$, one has
$\langle A^{n}\rangle_{\tau}-\langle A^{n}\rangle_{0}<16n^{2-2\kappa}\log n.$
On the other hand, by (6.7), we have the following control on the quadratic
variation:
$\langle A\rangle_{\tau}-\langle A\rangle_{0}\geq\frac{2}{n}\tau.$
One deduces that, with a probability that goes to $1$,
$\tau\leq\frac{16}{2}n^{3-2\kappa}\log n,$
and this quantity goes to $0$ as $n\to\infty$, whenever $\kappa>\frac{3}{2}$.
Therefore for $\kappa>\frac{3}{2}$, there holds
$\lim_{n\to\infty}\sup_{x_{0}^{n}\in\overline{D}_{n}}\mathbb{P}(X_{t_{n}}^{n}\neq
Y_{t_{n}}^{n})=0,$
thus concluding the proof of Lemma 6.3. ∎
###### Proof of Theorem 1.7 in TV and Hellinger.
Let $\kappa>\frac{3}{2}$ and fix some initial condition
$x_{0}^{n}\in\overline{D}_{n}$. By the triangle inequality for $\mathrm{TV}$,
there holds
$\|\mathrm{Law}(X_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}\leq\|\mathrm{Law}(Y_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}+\|\mathrm{Law}(X_{t}^{n})-\mathrm{Law}(Y_{t}^{n})\|_{\mathrm{TV}}.$
(6.8)
Taking $t=t_{n}(1+\varepsilon)$ with
$t_{n}=\log(\sqrt{n}|x_{0}^{n}|)\vee\log(n)$, one deduces from Lemma 6.1 and
the Pinsker inequality stated in Lemma A.1 that the first term in the right-
hand side of (6.8) vanishes as $n$ tends to infinity. Meanwhile Lemma 6.3
guaranties that the second term tends to $0$ as $n$ tends to infinity. We also
conclude using the comparison between TV and Hellinger (see Lemma A.1) that
$\lim_{n\to\infty}\mathrm{Hellinger}(\mathrm{Law}(X_{t_{n}}^{n}),P_{n}^{\beta})=0.$
∎
### 6.2. Proof of Corollary 1.8 in TV and Hellinger
###### Proof of Corollary 1.8 in TV and Hellinger.
By Lemma A.1 and the triangle inequality for $\mathrm{TV}$, we have
$\displaystyle\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\|\mathrm{Law}(X_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}$
$\displaystyle\leq\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\|\mathrm{Law}(Y_{t}^{n})-\mathrm{Law}(X_{t}^{n})\|_{\mathrm{TV}}$
$\displaystyle\quad+\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\sqrt{2\,\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid
P_{n}^{\beta})}.$
Take $t=(1+\varepsilon)c_{n}$ with $c_{n}=\log(na_{n})$. Lemmas 6.1 and 6.3,
combined with the assumption made on $(a_{n})$, show that the two terms on the
right-hand side vanish as $n\to\infty$. Using Lemma A.1, the same result holds
for $\mathrm{Hellinger}$.
On the other hand, take $x_{0}^{n,i}=a_{n}$ for all $i$ and note that
$\pi(x_{0}^{n})=na_{n}$ goes to $+\infty$ as $n\to\infty$. By Corollary 1.4 we
find
$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=1\;$
whenever $\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$. ∎
### 6.3. Proof of Theorem 1.10
###### Proof of Theorem 1.10.
_Lower bound._ The contraction property provided by Lemma A.2 gives
$\mathrm{Kullback}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\geq\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1}).$
By Theorem 1.3 $P_{n}\circ\pi^{-1}=\mathcal{N}(0,1)$ and $Y=\pi(X^{n})$ is an
OU process weak solution of $Y_{0}=\pi(X^{n}_{0})$ and
$\mathrm{d}Y_{t}=\sqrt{2}\mathrm{d}B_{t}-Y_{t}\mathrm{d}t$. In particular for
all $t\geq 0$, $\mathrm{Law}(Y_{t})$ is a mixture of Gaussian laws in the
sense that for any measurable test function $g$ with polynomial growth,
$\mathbb{E}_{\mathrm{Law}(Y_{t})}[g]=\mathbb{E}[g(Y_{t})]=\mathbb{E}[G_{t}(Y_{0})]\quad\text{where}\quad
G_{t}(y)=\mathbb{E}_{\mathcal{N}(y\mathrm{e}^{-t},1-\mathrm{e}^{-2t})}[g].$
Now we use (again) the variational formula used in the proof of Lemma A.2 to
get
$\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1})=\sup_{g}\\{\mathbb{E}_{\mathrm{Law}(\pi(X_{t}^{n}))}[g]-\log\mathbb{E}_{\mathcal{N}(0,1)}[\mathrm{e}^{g}]\\},$
and taking for $g$ the linear function defined by $g(x)=\lambda x$ for all
$x\in\mathbb{R}$ and for some $\lambda\neq 0$ yields
$\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1})\geq\lambda\mathrm{e}^{-t}\sum_{i=1}^{n}\int
x\mu_{i}(\mathrm{d}x)-\frac{\lambda^{2}}{2}.$
Finally, by using the assumption on first moment and taking $\lambda$ small
enough we get, for all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(\pi(X_{(1-\varepsilon)\log(n)}^{n})\mid
P_{n}^{\beta}\circ\pi^{-1})=+\infty,$
_Upper bound._ From Lemma B.2 we have, for all $t\geq 0$,
$\mathrm{Kullback}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid
P_{n}^{\beta})\mathrm{e}^{-2t}.$
Arguing like in the proof of Lemma 6.1 and using the contraction property of
$\mathrm{Kullback}$ provided by Lemma A.2 for the map $\Psi$ defined in
(1.17), we can write the following decomposition
$\displaystyle\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid P_{n}^{\beta})$
$\displaystyle\leq\mathrm{Kullback}(\otimes_{i=1}^{n}\mu_{i}\mid
P_{*n}^{\beta})$
$\displaystyle=S(\otimes_{i=1}^{n}\mu_{i})+\mathbb{E}_{\otimes_{i=1}^{n}\mu_{i}}[E]+\log
C_{*n}^{\beta}$ $\displaystyle\leq\sum_{i=1}^{n}S(\mu_{i})+\sum_{i\neq
j}\iint\Phi\mathrm{d}\mu_{i}\otimes\mathrm{d}\mu_{j}+Cn^{2}.$
Combining (6.4) with the assumptions on the $\mu_{i}$’s yields for some
constant $C>0$
$\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid P_{n}^{\beta})\leq Cn^{2}$
and it follows finally that for all $\varepsilon\in(0,1)$,
$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1+\varepsilon)\log(n)})\mid
P_{n}^{\beta})=0.$
∎
## 7\. Cutoff phenomenon for the DOU in Wasserstein
### 7.1. Proofs of Theorem 1.7 and Corollary 1.8 in Wasserstein
Let ${(X_{t})}_{t\geq 0}$ be the DOU process. By Lemma B.2, for all $t\geq 0$
and all initial conditions $X_{0}\in\overline{D}_{n}$,
$\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}),P_{n}^{\beta})\leq\mathrm{e}^{-2t}\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{0}),P_{n}^{\beta}).$
Suppose now that $\mathrm{Law}(X^{n}_{0})=\delta_{x^{n}_{0}}$. Then the
triangle inequality for the Wasserstein distance gives
$\mathrm{Wasserstein}^{2}(\delta_{x^{n}_{0}},P_{n}^{\beta})=\int\left|x^{n}_{0}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)\leq
2|x^{n}_{0}|^{2}+2\int\left|x\right|^{2}P_{n}^{\beta}(\mathrm{d}x).$
By Theorem 1.3, the mean at equilibrium of $|X_{t}^{n}|^{2}$ equals
$1+\frac{\beta}{2}(n-1)$ and therefore
$\int\left|x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=1+\frac{\beta}{2}(n-1).$
We thus get
$\mathrm{Wasserstein}^{2}(\mathrm{Law}(X^{n}_{t}),P_{n}^{\beta})\leq
2(|x^{n}_{0}|^{2}+1+\frac{\beta}{2}(n-1))\mathrm{e}^{-2t}.$
Set $c_{n}:=\log(|x_{0}^{n}|)\vee\log(\sqrt{n})$. For any
$\varepsilon\in(0,1)$, we have
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}}),P_{n}^{\beta})=0$
and this concludes the proof of Theorem 1.7 in the Wasserstein distance.
Regarding the proof of Corollary 1.8, if $x_{0}^{n}\in[-a_{n},a_{n}]^{n}$ then
$|x_{0}^{n}|\leq\sqrt{n}a_{n}$. Therefore if $\inf_{n}a_{n}>0$, setting
$c_{n}=\log(\sqrt{n}a_{n})$ we find, as required,
$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}}),P_{n}^{\beta})=0.$
### 7.2. Proof of Theorem 1.9
This is an adaptation of the previous proof. We compute
$\displaystyle\mathrm{Wasserstein}^{2}(\delta_{x_{0}^{n}},P_{n}^{\beta})$
$\displaystyle=\int\left|x^{n}_{0}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)$
$\displaystyle\leq
2\left|x^{n}_{0}-\rho_{n}\right|^{2}+2\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x),$
where $\rho_{n}\in D_{n}$ is the vector of the quantiles of order $1/n$ of the
semi-circle law as in (1.14). The rigidity estimates established in [17, Th.
2.4] justify that
$\lim_{n\to\infty}\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=0.$
If $|x^{n}_{0}-\rho_{n}|$ diverges with $n$, we deduce that for all
$\varepsilon\in(0,1)$, with $t_{n}=\log(|x^{n}_{0}-\rho_{n}|)$,
$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)t_{n}}),P_{n}^{\beta})=0.$
On the other hand, if $|x^{n}_{0}-\rho_{n}|$ converges to some limit $\alpha$
then we easily get, for any $t\geq 0$,
$\varlimsup_{n\to\infty}\mathrm{Wasserstein}^{2}(\mathrm{Law}(X^{n}_{t}),P_{n}^{\beta})\leq\alpha^{2}\mathrm{e}^{-2t}.$
###### Remark 7.1 (High-dimensional phenomena).
With $X_{n}\sim P_{n}^{\beta}$, in the bias-variance decomposition
$\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=|\mathbb{E}X_{n}-\rho_{n}|^{2}+\mathbb{E}(|X_{n}-\mathbb{E}X_{n}|^{2}),$
the second term of the right hand side is a variance term that measures the
concentration of the log-concave random vector $X_{n}$ around its mean
$\mathbb{E}X_{n}$, while the first term in the right hand side is a bias term
that measures the distance of the mean $\mathbb{E}X_{n}$ to the mean-field
limit $\rho_{n}$. Note also that
$\mathbb{E}(|X_{n}-\mathbb{E}X_{n}|^{2})=\mathbb{E}(|X_{n}|^{2})-|\mathbb{E}X_{n}|^{2}=1+\frac{\beta}{2}(n-1)-|\mathbb{E}X_{n}|^{2}$,
|
L>0$, and thus $N^{(k_{\ell})}_{1}(u)>N^{(k_{\ell})}_{2}(u)>0$ for all
$u\in[t,t+\epsilon]$, for all $\ell$ large enough. It follows that $X(j)=0$
for all $j=\lfloor k_{\ell}t\rfloor+1,\dots,\lfloor
k_{\ell}(t+\epsilon)\rfloor$, and that the minima in equations (27) and (28)
are never attained in the third case (otherwise we would have
$N^{(k_{\ell})}_{2}(u)=0$ for some $u\in[t,t+\epsilon]$). Therefore, we have
$\displaystyle M_{1}(j)$
$\displaystyle=\min\Big{\\{}Q_{2}(j)-D_{2}(j)+S_{2}(j),\,\,Q_{1}(j)-D_{1}(j)+S_{1}(j)\Big{\\}},$
$\displaystyle M_{2}(j)$
$\displaystyle=\min\Big{\\{}Q_{3}(j)-D_{3}(j)+S_{3}(j)-M_{1}(j),\,\,Q_{2}(j)-D_{2}(j)+S_{2}(j)\Big{\\}},$
where $Q(\cdot)$ is defined recursively as
$\displaystyle Q_{1}(u+1)$ $\displaystyle=Q_{1}(u)-D_{1}(u)+S_{1}(u)-M_{1}(u)$
$\displaystyle Q_{2}(u+1)$
$\displaystyle=Q_{2}(u)-D_{2}(u)+S_{2}(u)-M_{1}(u)-M_{2}(u)$ $\displaystyle
Q_{3}(u+1)$ $\displaystyle=Q_{3}(u)-D_{3}(u)+S_{3}(u)-M_{2}(u).$
Note that this corresponds to Case 1 in Lemma 8. Therefore, since
$\omega\in\mathcal{C}$, Lemma 8 implies
$\displaystyle r_{1}(t+\epsilon)-r_{1}(t)$
$\displaystyle=\lim\limits_{\ell\to\infty}R^{(k_{\ell})}_{1}(t+\epsilon)-R^{(k_{\ell})}_{1}(t)$
$\displaystyle=\epsilon.\,\overline{C}_{1},$
and
$\displaystyle r_{2}(t+\epsilon)-r_{2}(t)$
$\displaystyle=\lim\limits_{\ell\to\infty}R^{(k_{\ell})}_{2}(t+\epsilon)-R^{(k_{\ell})}_{2}(t)$
$\displaystyle=\epsilon.\,\underline{C}_{2}.$
Dividing by $\epsilon$ and taking the limit as $\epsilon$ goes to zero we
conclude that, when $n_{1}(t)>n_{2}(t)>0$, we have
$\displaystyle\frac{dn_{1}(t)}{dt}$
$\displaystyle=\lambda_{1}-\overline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(t)}{dt}=\lambda_{2}-\underline{C}_{2}.$
Case (b): $n_{1}(t)=n_{2}(t)>0$
First note that, if $n_{1}(t)>0$ and $n_{2}(t)>0$, the same argument as in the
previous case gives us
$r_{i}(t+\epsilon)-r_{i}(t)\in\big{[}\epsilon\underline{C}_{i},\,\epsilon\overline{C}_{i}\big{]},$
(30)
and
$r_{1}(t+\epsilon)+r_{2}(t+\epsilon)-r_{1}(t)-r_{2}(t)=\epsilon C_{1,2},$ (31)
for all sufficiently small $\epsilon$.
Suppose that
$\lambda_{1}-\overline{C}_{1}>\lambda_{2}-\underline{C}_{2}.$
Combining this with Equation (30), it follows that
$n_{1}(u)>n_{2}(u)$
for all $u\in(t,t+\epsilon]$. Therefore
$\displaystyle\frac{dn_{1}(u)}{du}$
$\displaystyle=\lambda_{1}-\overline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(u)}{du}=\lambda_{2}-\underline{C}_{2},$
for all $u\in(t,t+\epsilon]$. Since $t$ is a regular time, we also have
$\displaystyle\frac{dn_{1}(t)}{dt}$
$\displaystyle=\lambda_{1}-\overline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(t)}{dt}=\lambda_{2}-\underline{C}_{2}.$
Analogously, if
$\lambda_{2}-\overline{C}_{2}>\lambda_{1}-\underline{C}_{1},$
we have
$\displaystyle\frac{dn_{1}(t)}{dt}$
$\displaystyle=\lambda_{1}-\underline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(t)}{dt}=\lambda_{2}-\overline{C}_{2}.$
Finally, suppose that
$\frac{\lambda_{1}+\lambda_{2}-C_{1,2}}{2}\geq\max\left\\{\lambda_{1}-\overline{C}_{1},\,\,\lambda_{2}-\overline{C}_{2}\right\\}.$
In particular, this means that
$\lambda_{1}-\overline{C}_{1}\leq\lambda_{2}-\underline{C}_{2}\qquad\text{and}\qquad\lambda_{2}-\overline{C}_{2}\leq\lambda_{1}-\underline{C}_{1}.$
Therefore, we have
$\frac{dn_{1}(u)}{du}-\frac{dn_{2}(u)}{du}\leq 0$
when $n_{1}(u)>n_{2}(u)$, and
$\frac{dn_{1}(u)}{du}-\frac{dn_{2}(u)}{du}\geq 0$
when $n_{1}(u)<n_{2}(u)$.
###### Lemma 10.
We have
$\frac{dn_{1}(t)}{dt}-\frac{dn_{2}(t)}{dt}=0,$
and
$\frac{dn_{1}(t)}{dt}=\frac{dn_{2}(t)}{dt}=\frac{\lambda_{1}+\lambda_{2}-C_{1,2}}{2}.$
###### Proof.
Proof: We prove this by contradiction. Suppose that
$\frac{dn_{1}(t)}{dt}-\frac{dn_{2}(t)}{dt}>0.$
Since $n_{1}(t)=n_{2}(t)$ and $t$ is a regular point, we must have
$n_{1}(u)>n_{2}(u)$ for all sufficiently small $u>t$. However, we have
$\frac{dn_{1}(u)}{du}-\frac{dn_{2}(u)}{du}\leq 0$
for all sufficiently small $u>t$, which contradicts the regularity of $t$.
Assuming
$\frac{dn_{1}(t)}{dt}-\frac{dn_{2}(t)}{dt}<0$
yields the same contradiction. Therefore, we must have
$\frac{dn_{1}(t)}{dt}-\frac{dn_{2}(t)}{dt}=0.$
Combining this with Equation (31), we obtain
$\frac{dn_{1}(t)}{dt}=\frac{dn_{2}(t)}{dt}=\frac{\lambda_{1}+\lambda_{2}-C_{1,2}}{2}.$
$\square$
∎
Case (c): $n_{1}(t)>n_{2}(t)=0$
Suppose that $\lambda_{2}<\underline{C}_{2}$. Then, using the same argument as
in Case (a) but now with only $N_{1}(\cdot)$ being infinitely backlogged and
$N_{2}(\cdot)$ being stable (Lemma 3), we conclude that
$\displaystyle\frac{dn_{1}(t)}{dt}$
$\displaystyle=\lambda_{1}-C_{1}(\lambda_{2})\qquad\text{and}\qquad\frac{dn_{2}(t)}{dt}=0.$
Suppose that $\lambda_{2}>\underline{C}_{2}$. Since $n_{2}(u)<n_{1}(u)$ for
all sufficiently small $u>t$, we have
$\displaystyle\frac{dr_{2}(t)}{dt}$ $\displaystyle\leq\underline{C}_{2}.$ (32)
Therefore, we have $n_{2}(u)>0$ for all sufficiently small $u>t$. By Case (a),
we have
$\displaystyle\frac{dn_{1}(u)}{du}$
$\displaystyle=\lambda_{1}-\overline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(u)}{du}=\lambda_{2}-\underline{C}_{2}$
for all sufficiently small $u>t$. Using once more that $t$ is a regular time,
it follows that
$\displaystyle\frac{dn_{1}(t)}{dt}$
$\displaystyle=\lambda_{1}-\overline{C}_{1}\qquad\text{and}\qquad\frac{dn_{2}(t)}{dt}=\lambda_{2}-\underline{C}_{2}.$
Finally, suppose that $\lambda_{2}=\underline{C}_{2}$. Using the same argument
as in Lemma 10, it can be checked that
$\displaystyle\frac{dn_{2}(t)}{dt}$ $\displaystyle=0.$
Case (d): $n_{1}(t)=n_{2}(t)=0$
Since $t$ is a regular point, $n_{i}(0)>0$, and cases (a), (b), and (c) imply
that $n_{i}(\cdot)$ can only become $0$ at a non-regular point, we must have
$n_{1}(u)=n_{2}(u)=0$ for all sufficiently large $u<t$. It follows that
$\frac{dn_{1}(u)}{du}=\frac{dn_{2}(u)}{du}=0$
for all sufficiently large $u<t$. Using once again that $t$ is a regular point
yields
$\frac{dn_{1}(t)}{dt}=\frac{dn_{2}(t)}{dt}=0.$
$\square$
∎
### F.3 Completing the proof of Theorem 3
For every sample path in $\mathcal{C}$, we have established the following.
Proposition 2 implies the existence of limit points of the sequence of
processes $\left\\{N^{(k)}(\cdot)\right\\}_{k=1}^{\infty}$. Furthermore,
according to Proposition 3 these limit points verify the differential
equations of the fluid model. Combining this with the fact that the
trajectories are Lipschitz continuous, and thus they are differentiable almost
everywhere, the limit points are fluid solutions. In particular, this means
that fluid solutions exist.
Finally, since the trajectories are piece-wise linear and any component that
hits zero stays at zero forever, the uniqueness of solutions follows from the
uniqueness of the pieces that comprise them. In particular, this implies that
all limit points are the same, and therefore the limit converges.
## Appendix G Proof of Theorem 2
* (i)
When $\lambda_{1}+\lambda_{2}<C_{1,2}$, $\lambda_{1}<C_{1}(\lambda_{2})$, and
$\lambda_{2}<C_{2}(\lambda_{1})$, theorems 3 and 4 imply that there exists
$\delta>0$ such that, for any $n^{0}\in\mathbb{R}_{+}^{2}$ with
$\|n^{0}\|_{1}>0$, if $N(0)=kn^{0}$, we have
$\lim\limits_{k\to\infty}\frac{N_{i}(kt)}{k}=0,\qquad a.s.,$
for all $t\geq\delta\|n^{0}\|_{1}$. Moreover, Equation (13) implies that, for
any $q^{0}\in\mathbb{R}_{+}^{3}$ with $\|q^{0}\|_{1}>0$, if $Q(0)=kq^{0}$, we
have
$\lim\limits_{k\to\infty}\frac{Q_{j}(kt)}{k}=0,\qquad a.s.,$
for all $t>0$. Therefore, the positive recurrence of $\big{(}{\bf
N}(\cdot),{\bf Q}(\cdot)\big{)}$ when $\lambda_{1}+\lambda_{2}<C_{1,2}$,
$\lambda_{1}<C_{1}(\lambda_{2})$, and $\lambda_{2}<C_{2}(\lambda_{1})$ follows
from these finite-time convergences to $0$, and [11, Theorem 6.2].
* (ii)
We first consider a coupled process $\left(\underline{N}(\cdot),\underline{\bf
Q}(\cdot)\right)$, where the coupling with the original processes is done in a
way so that $\underline{N}(0)={\bf N}_{1}(0)+{\bf N}_{2}(0)$, $\underline{\bf
Q}_{1}(0)={\bf Q}_{1}(0)+{\bf Q}_{3}(0)$, and $\underline{\bf Q}_{2}(0)={\bf
Q}_{2}(0)$ and so that they have the same arrival processes $A_{1}(\cdot)$,
$A_{2}(\cdot)$, $S_{1}(\cdot)$, $S_{2}(\cdot)$, and $S_{3}(\cdot)$, and the
same abandonment primitives $Z_{\ell}^{(i)}(\cdot)$. These new processes are
defined recursively as
$\displaystyle\underline{N}(t+1)$
$\displaystyle=\underline{N}(t)+A_{1}(t)+A_{2}(t)-\sum\limits_{\ell=1}^{\overline{\overline{M}}(t)}Y_{\ell}(t)$
$\displaystyle\underline{Q}_{1}(t+1)$
$\displaystyle=\underline{Q}_{1}(t)-\underline{D}_{1}(t)+S_{1}(t)+S_{3}(t)-\overline{\overline{M}}(t)$
$\displaystyle\underline{Q}_{2}(t+1)$
$\displaystyle=\underline{Q}_{2}(t)-\underline{D}_{2}(t)+S_{2}(t)-\overline{\overline{M}}(t),$
where
$\overline{\overline{M}}(t)=\min\left\\{\underline{Q}_{1}(t)-\underline{D}_{1}(t)+S_{1}(t)+S_{3}(t),\,\,\underline{Q}_{2}(t)-\underline{D}_{2}(t)+S_{2}(t)\right\\}.$
This corresponds to a system where three-way matchings are always attempted if
there are enough qubits, regardless of the requests available. Then,
$\underline{N}(\cdot)$ keeps track of the difference between the total number
of requests that arrived to the system, and the total number of successful
three-way matchings. This can be negative, and it can be checked that
$\underline{N}(t)\leq{\bf N}_{1}(t)+{\bf N}_{2}(t)$, for all $t\geq 0$, for
any non-idling policy. Moreover, since $\underline{Q}_{1}(\cdot)$ and
$\underline{Q}_{2}(\cdot)$ behave as two-sided queues with abandonments, we
have
$\lim\limits_{t\to\infty}\frac{1}{t}\sum\limits_{s=0}^{t}\sum\limits_{\ell=1}^{\overline{\overline{M}}(s)}Y_{\ell}(s)=C_{1,2}.$
Therefore, since $\lambda_{1}+\lambda_{2}>C_{1,2}$, we have that
$\underline{N}(t)$ diverges to $+\infty$ almost surely and, since
$\underline{N}(t)\leq{\bf N}_{1}(t)+{\bf N}_{2}(t)$ for all $t\geq 0$, the
process $\big{(}{\bf N}(\cdot),{\bf Q}(\cdot)\big{)}$ is transient.
On the other hand, suppose that $\lambda_{1}+\lambda_{2}\leq C_{1,2}$, and
that $\lambda_{j}<\underline{C}_{j}$ and $\lambda_{i}>C_{i}(\lambda_{j})$.
Analogously to the previous case, we can construct a coupled process
$\underline{N}_{i}(\cdot)$ such that $N_{i}(t)\geq\underline{N}_{i}(t)$ for
all $t\geq 0$, and such that its throughput is $C_{i}(\lambda_{j})$. Then,
since $\lambda_{i}>C_{i}(\lambda_{j})$, we have that
$\underline{N}_{i}(\cdot)$ diverges to $+\infty$ almost surely and, since
$N_{i}(t)\geq\underline{N}_{i}(t)$ for all $t\geq 0$, the process $\big{(}{\bf
N}(\cdot),{\bf Q}(\cdot)\big{)}$ is transient.
|
# Artificial-Noise-Aided Secure MIMO Wireless Communications via Intelligent
Reflecting Surface
Sheng Hong, Cunhua Pan, Hong Ren, Kezhi Wang, and Arumugam Nallanathan This
work was supported by the National Natural Science Foundation of China
(61661032), the Young Natural Science Foundation of Jiangxi Province
(20181BAB202002), the China Postdoctoral Science Foundation (2017M622102), the
Foundation from China Scholarship Council (201906825071).S. Hong is with
Information Engineering School of Nanchang University, Nanchang 330031, China.
(email: [email protected]). C. Pan, H. Ren, and A. Nallanathan are with the
School of Electronic Engineering and Computer Science at Queen Mary University
of London, London E1 4NS, U.K. (e-mail:{c.pan, h.ren,
a.nallanathan}@qmul.ac.uk). K. Wang is with Department of Computer and
Information Sciences, Northumbria University, UK. (email:
[email protected]).
###### Abstract
This paper considers a MIMO secure wireless communication system aided by the
physical layer security technique of sending artificial noise (AN). To further
enhance the system security performance, the advanced intelligent reflecting
surface (IRS) is invoked in the AN-aided communication system, where the base
station (BS), legitimate information receiver (IR) and eavesdropper (Eve) are
equipped with multiple antennas. With the aim for maximizing the secrecy rate
(SR), the transmit precoding (TPC) matrix at the BS, covariance matrix of AN
and phase shifts at the IRS are jointly optimized subject to constrains of
transmit power limit and unit modulus of IRS phase shifts. Then, the secrecy
rate maximization (SRM) problem is formulated, which is a non-convex problem
with multiple coupled variables. To tackle it, we propose to utilize the block
coordinate descent (BCD) algorithm to alternately update the TPC matrix, AN
covariance matrix, and phase shifts while keeping SR non-decreasing.
Specifically, the optimal TPC matrix and AN covariance matrix are derived by
Lagrangian multiplier method, and the optimal phase shifts are obtained by
Majorization-Minimization (MM) algorithm. Since all variables can be
calculated in closed form, the proposed algorithm is very efficient. We also
extend the SRM problem to the more general multiple-IRs scenario and propose a
BCD algorithm to solve it. Finally, simulation results validate the
effectiveness of system security enhancement via an IRS.
###### Index Terms:
Intelligent Reflecting Surface (IRS), Reconfigurable Intelligent Surfaces,
Secure Communication, Physical Layer Security, Artificial Noise (AN), MIMO.
## I Introduction
The next-generation (i.e, 6G) communication is expected to be a sustainable
green, cost-effective and secure communication system [1]. In particular,
secure communication is crucially important in 6G communication networks since
communication environment becomes increasingly complicated and the security of
private information is imperative [2]. The information security using
crytographic encryption (in the network layer) is a conventional secure
communication technique, which suffers from the vulnerabilities, such as
secret key distribution, protection and management [3]. Unlike this network
layer security approach, the physical layer security can guarantee good
security performance bypassing the relevant manipulations on the secret key,
thus is more attractive for the academia and industry [4]. There are various
physical-layer secrecy scenarios. The first one is the classical physical-
layer secrecy setting where there is one legitimate information receiver (IR)
and one eavesdropper (Eve) operating over a single-input-single-output (SISO)
channel (i.e., the so-called three-terminal SISO Gaussian wiretap channel) [5,
6]. The second one considers the physical-layer secrecy with an IR and Eve
operating over a multiple-input-single-output (MISO) channel, which is called
as three-terminal MISO Gaussian wiretap channel. The third one is a renewed
and timely scenario with one IR and one Eve operating over a multiple-input-
multiple-output (MIMO) channel, which is named as three-terminal MIMO Gaussian
wiretap channel[7, 8] and is the focus of this paper. For MIMO systems, a
novel idea in physical-layer security is to transmit artificial noise (AN)
from the base station (BS) to contaminate the Eve’s received signal [9, 10,
11]. For these AN-aided methods, a portion of transmit power is assigned to
the artificially generated noise to interfere the Eve, which should be
carefully designed. For AN-aided secrecy systems, while most of the existing
AN-aided design papers focused on the MISO wiretap channel and null-space AN
[7, 12], designing the transmit precoding (TPC) matrix together with AN
covariance matrix for the MIMO wiretap channel is more challenging [13].
In general, the secrecy rate (SR) achieved by the mutual information
difference between the legitimate IR and the Eve is limited by the channel
difference between the BS-IR link and the BS-Eve link. The AN-aided method can
further improve the SR, but it consumes the transmit power destined for the
legitimate IR. When the transmit power is confined, the performance bottleneck
always exists for the AN-aided secure communication. To conquer the dilemma,
the recently proposed intelligent reflecting surface (IRS) technique can be
exploited. Since higher SR can be achieved by enhancing the channel quality in
the BS-IR link and degrading the channel condition in the BS-Eve link, the IRS
can serve as a powerful complement to AN-aided secure communication due to its
capability of reconfiguring the wireless propagation environment.
The IRS technique has been regarded as a revolutionary technique to control
and reconfigure the wireless environment [14, 15],[16]. An IRS comprises an
array of reflecting elements, which can reflect the incident electromagnetic
(EM) wave passively, and the complex reflection coefficient contains the phase
shift and amplitude. In practical applications, the phase shifts of the
reflection coefficients are discrete due to the manufacturing cost[17].
However, many works on IRS aided wireless communications are based on the
assumption of continuous phase shifts [18],[19]. To investigate the potential
effect of IRS on the secure communication, we also assume continuous phase
shifts to simplify the problem. We evaluate its impact on the system
performance in the simulation section. Theoretically, the reflection amplitude
of each IRS element can be adjusted for different purpose [20]. However,
considering the hardware cost, the reflection amplitude is usually assumed to
be 1 for simplicity. Hence, by smartly tuning the phase shifts with a
preprogrammed controller, the direct signals from the BS and the reflected
signals from the IRS can be combined constructively or destructively according
to different requirements. In comparison to the existing related techniques
which the IRS resembles, such as active intelligent surface [21], traditional
reflecting surfaces[22], backscatter communication [23] and amplify-and-
forward (AF) relay [24], the IRSs have the advantages of flexible
reconfiguration on the phase shifts in real time, minor additional power
consumption, easy installation with many reflecting elements, etc.
Furthermore, due to the light weight and compact size, the IRS can be
integrated into the traditional communication systems with minor modifications
[25]. Because of these appealing virtues, IRS has introduced into various
wireless communication systems, including the single-user case [26, 27], the
downlink multiuser case [28, 18, 29, 30, 31], mobile edge computing [32],
wireless information and power transfer design [33], and the physical layer
security design [34, 35, 36, 37].
IRS is promising to strengthen the system security of wireless communication.
In [34, 36, 38], the authors investigated the problem of maximizing the
achievable SR in a secure MISO communication system aided by IRS, where both
the legitimate user and eavesdropper are equipped with a single antenna. The
TPC matrix at the BS and the phase shifts at the IRS were optimized by an
alternate optimization (AO) strategy. To handle the nonconvex unit modulus
constraint, the semidefinite relaxation (SDR) [39], majorization-minimization
(MM) [18, 40], complex circle manifold (CCM) [41] techniques were proposed to
optimize phase shifts. An IRS-assisted MISO secure communication with a single
IR and single Eve was also considered in [35], but it was limited to a special
scenario, where the Eve has a stronger channel than the IR, and the two
channels from BS to Eve and IR are highly correlated. Under this assumption,
the transmit beamforming and the IRS reflection beamforming are jointly
optimized to improve the SR. Similarly, a secure IRS-assisted downlink MISO
broadcast system was considered in [37], and it assumes that multiple
legitimate IRs and multiple Eves are in the same directions to the BS, which
implies that the IR channels are highly correlated with the Eve channels. [42]
considered the transmission design for an IRS-aided secure MISO communication
with a single IR and single Eve, in which the system energy consumption is
minimized under two assumptions that the channels of access point (AP)-IRS
links are rank-one and full-rank. An IRS-assisted MISO network with
cooperative jamming was investigated in [2]. The physical layer security in a
simultaneous wireless information and power transfer (SWIPT) system was
considered with the aid of IRS [43]. However, there are a paucity of papers
considering the IRS-assisted secure communication with AN. A secure MISO
communication system aided by the transmit jamming and AN was considered in
[44], where a large number of Eves exist, and the AN beamforming vector and
jamming vector were optimized to reap the additional degrees of freedom (DoF)
brought by the IRS. [45] investigated the resource allocation problem in an
IRS-assisted MISO communication by jointly optimizing the beamforming vectors,
the phase shifts of the IRS, and AN covariance matrix for secrecy rate
maximization (SRM), but the direct BS-IRs links and direct BS-Eves link are
assumed to be blocked.
Although a few papers have studied security enhancement for an AN-aided system
through the IRS, the existing papers related to this topic either only studied
the MISO scenario or assumed special settings to the channels. The
investigation on the MIMO scenario with general channel settings is absent in
the existing literature. Hence, we investigate this problem in this paper by
employing an IRS in an AN-aided MIMO communication system for the physical
layer security enhancement. Specifically, by carefully designing the phase
shifts of the IRS, the reflected signals are combined with the direct signals
constructively for enhancing the data rate at the IR and destructively for
decreasing the rate at the Eve. As a result, the TPC matrix and AN covariance
matrix at the BS can be designed flexibly with a higher DoF than the case
without IRS. In this work, the TPC matrix, AN covariance matrix and the phase
shift matrix are jointly optimized. Since these optimization variables are
highly coupled, an efficient algorithm based on the block coordinate descent
(BCD) and MM techniques for solving the problem is proposed.
We summarize our main contributions as follows:
1. 1.
This is the first research on exploiting an IRS to enhance security in AN-
aided MIMO communication systems. Specifically, an SRM problem is formulated
by jointly optimizing the TPC matrix and AN covariance matrix at the BS,
together with the phase shifts of the IRS subject to maximum transmit power
limit and the unit modulus constraint of the phase shifters. The objective
function (OF) of this problem is the difference of two Shannon capacity
expressions, thus is not jointly concave over the three highly-coupled
variables. To handle it, the popular minimum mean-square error (MMSE)
algorithm is used to reformulate the SRM problem.
2. 2.
The BCD algorithm is exploited to optimize the variables alternately. Firstly,
given the phase shifts of IRS, the optimal TPC matrix and AN covariance matrix
are obtained in closed form by utilizing the Lagrangian multiplier method.
Then, given the TPC matrix and AN covariance matrix, the optimization problem
for IRS phase shifts is transformed by sophisticated matrix manipulations into
a quadratically constrained quadratic program (QCQP) problem subject to unit
modulus constraints. To solve it, the MM algorithm is utilized, where the
phase shifts are derived in closed form iteratively. Based on the BCD-MM
algorithm, the original formulated SRM problem can be solved efficiently.
3. 3.
The SRM problem is also extended to the more general scenario of multiple
legitimate IRs. A new BCD algorithm is proposed to solve it, where the optimal
TPC matrix and AN covariance matrix are obtained by solving a QCQP problem,
and the unit modulus constraint is handled by the penalty convex-concave
procedure (CCP) method.
4. 4.
The simulation results confirm that on the one hand, the IRS can greatly
enhance the security of an AN-aided MIMO communication system; on the other
hand, the phase shifts of IRS should be properly optimized. Simulation results
also show that larger IRS element number and more transmit power is beneficial
to the security. Moreover, properly-selected IRS location and good channel
states of the IRS-related links are important to realize the full potential of
IRS.
This paper is organized as follows. Section II provides the signal model of an
AN-aided MIMO communication system assisted by an IRS, and the SRM problem
formulation. The SRM problem is reformulated in Section III, where the BCD-MM
algorithm is proposed to optimize the TPC matrix, AN covariance matrix and
phase shifts of IRS. Section IV extends the SRM problem to a more general
scenario of multiple IRs. In Section V, numerical simulations are given to
validate the algorithm efficiency and security enhancement. Section VI
concludes this paper.
_Notations_ : Throughout this paper, boldface lower case, boldface upper case
and regular letters are used to denote vectors, matrices, and scalars
respectively. ${\bf{X}}\odot{\bf{Y}}$ is the Hadamard product of $\bf X$ and
$\bf Y$. ${\rm{Tr}}\left({\bf{X}}\right)$ and $\left|{\bf{X}}\right|$ denote
the trace and determinant of ${\bf{X}}$ respectively. ${{\mathbb{C}}^{M\times
N}}$ denotes the space of $M\times N$ complex matrices. ${\rm{Re}}\\{\cdot\\}$
and $\arg\\{\cdot\\}$ denote the real part of a complex value and the
extraction of phase information respectively. ${\rm{diag}}\\{\cdot\\}$ is the
operator for diagonalization. ${\cal C}{\cal N}({\bm{\mu}},{\bf{Z}})$
represents a circularly symmetric complex gaussian (CSCG) random vector with
mean ${\bm{\mu}}$ and covariance matrix ${\bf{Z}}$.
${\left(\cdot\right)^{\rm{T}}}$, ${\left(\cdot\right)^{\rm{H}}}$ and
${\left(\cdot\right)^{\rm{\ast}}}$ denote the transpose, Hermitian and
conjugate operators respectively. $(\cdot)^{\star}$ stands for the optimal
value, and $(\cdot)^{{\dagger}}$ means the pseudo-inverse. $[\cdot]^{+}$ is
the projection onto the non-negative number, i.e, if $y=[x]^{+}$, then
$y=\rm{max}\\{0,x\\}$.
## II Signal Model and Problem Formulation
### II-A Signal Model
We consider an IRS-aided communication network shown in Fig. 1 that consists
of a BS, a legitimate IR and an Eve, all of which are equipped with multiple
antennas. The number of transmit antennas at the BS is ${{N}_{T}}\geq 1$, and
the numbers of receive antennas at the legitimate IR and Eve are
${{N}_{I}}\geq 1$ and ${{N}_{E}}\geq 1$ respectively. To ensure secure
transmission from the BS to the IR, the AN is sent from the BS to interfere
the eavesdropper to achieve strong secrecy.
Figure 1: An AN-aided MIMO secure communication system with IRS.
With above assumptions, the BS employed the TPC matrix to transmit data
streams with AN. The transmitted signal can be modeled as
$\displaystyle{\bf{x}}={\bf{Vs}}+{\bf{n}},$ (1)
where ${\bf{V}}\in{{\mathbb{C}}^{{{N}_{T}}\times d}}$ is the TPC matrix; the
number of data streams is $d\leq\min({{N}_{T}},{{N}_{I}})$; the transmitted
data towards the IR is
$\mathbf{s}\sim\mathcal{C}\mathcal{N}(0,{\mathbf{I}_{d}})$; and
$\mathbf{n}\in{\cal C}{\cal N}({\bm{0}},{\bf{Z}})$ represents the AN random
vector with zero mean and covariance matrix $\mathbf{Z}$.
Assuming that the wireless signals are propagated in a non-dispersive and
narrow-band way, we model the equivalent channels of the BS-IRS link, the BS-
IR link, the BS-Eve link, the IRS-IR link, the IRS-Eve link by the matrices
$\mathbf{G}\in{{\mathbb{C}}^{M\times{{N}_{T}}}}$,
${{\bf{H}}_{b,I}}\in{{\mathbb{C}}^{{{N}_{I}}\times{{N}_{T}}}}$,
${{\bf{H}}_{b,E}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{T}}}}$,
${{\bf{H}}_{R,I}}\in{{\mathbb{C}}^{{{N}_{I}}\times
M}}$,${{\bf{H}}_{R,E}}\in{{\mathbb{C}}^{{{N}_{E}}\times M}}$, respectively.
The phase shift coefficients of IRS are collected in a diagonal matrix defined
by ${\bf{\Phi}}=\text{diag }\\!\\!\\{\\!\\!\text{
}{{\phi}_{1}},\cdots,{{\phi}_{m}},\cdots,{{\phi}_{M}}\text{
}\\!\\!\\}\\!\\!\text{ }$ and ${{\phi}_{m}}={{e}^{j{{\theta}_{m}}}}$, where
${{\theta}_{m}}\in[0,2\pi]$ denotes the phase shift of the $m$-th reflection
element. The multi-path signals that have been reflected by multiple times are
considered to be absorbed and diffracted, then the signal received at the
legitimate IR is given by
${\bf{y}}_{I}=({\bf{H}}_{b,I}+{\bf{H}}_{R,I}{\bf{\Phi}}{\bf{G}}){\bf{x}}+{\bf{n}}_{I},$
(2)
where ${{\bf{n}}_{I}}$ is the random noise vector at IR obeying the
distribution
${{\bf{n}}_{I}}\sim\mathcal{C}\mathcal{N}({\bf{0}},\sigma_{I}^{2}{\bf{I}}_{{{N}_{I}}})$.
The signal received at the Eve is
$\displaystyle{\bf{y}}_{E}=({\bf{H}}_{b,E}+{\bf{H}}_{R,E}{\bf{\Phi}}{\bf{G}}){\bf{x}}+{{\bf{n}}_{E}},$
(3)
where ${\bf{n}}_{E}$ is the Eve’s noise vector following the distribution
${\bf{n}}_{E}\sim\mathcal{C}\mathcal{N}({\bf{0}},\sigma_{E}^{2}{\bf{I}}_{{{N}_{E}}})$.
Assume that the BS has acquired the prior information of all the channel state
information (CSI). Then the BS is responsible for optimizing the IRS phase
shifts and sending them back to the IRS controller through a separate low-rate
communication link such as wireless links [20],[17] or wired lines [46]. The
assumption of perfect CSI knowledge is idealistic, since the CSI estimation
for IRS networks is challenging. However, the algorithms developed allow us to
derive the relevant performance upper bounds for realistic scenarios in the
presence of realistic CSI errors. Recently, we have investigated the design of
robust and secure transmission in IRS-aided MISO wireless communication
systems in [47] by considering the statistical CSI error model associated with
the cascaded channels for the eavesdropper. Its extension to the MIMO scenario
will be studied in our future work.
Upon substituting $\bf{x}$ into (2), ${\bf{y}}_{I}$ can be rewirtten as
$\displaystyle{\bf{y}}_{I}={\hat{\bf{H}}_{I}}({\bf{V}}{\bf{s}}+{\bf{n}})+{{\bf{n}}_{I}}{\rm{=}}{\hat{\bf{H}}_{I}}{\bf{V}}{\bf{s}}+{\hat{\bf{H}}_{I}}{\bf{n}}+{{\bf{n}}_{I}},$
(4)
where
${{\hat{\bf{H}}}_{I}}\overset{\triangle}{=}{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi}}{\bf{G}}$
is defined as the equivalent channel spanning from the BS to the legitimate
IR. Then, the data rate (bit/s/Hz) achieved by the legitimate IR is given by
$\displaystyle{R_{I}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{\bf{J}}_{I}^{-1}}\right|,$
(5)
where ${{\bf{J}}_{I}}$ is the interference-plus-noise covariance matrix given
by
${{\bf{J}}_{I}}\overset{\triangle}{=}{{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}$.
Upon substituting $\bf{x}$ into (3), ${{\bf{y}}_{E}}$ can be rewritten as
$\displaystyle{\bf{y}}_{E}={\hat{\bf{H}}_{E}}({\bf{Vs}}+{\bf{n}})+{\bf{n}}_{E}={\hat{\bf{H}}_{E}}{\bf{Vs}}+{\hat{\bf{H}}_{E}}{\bf{n}}+{\bf{n}}_{E},$
(6)
where
${{\hat{\bf{H}}}_{E}}\overset{\triangle}{=}{{\bf{H}}_{b,E}}+{{\bf{H}}_{R,E}}{\bf{\Phi}}{\bf{G}}$
is defined as the equivalent channel spanning from the BS to the Eve. Then,
the data rate (bit/s/Hz) achieved by the Eve is given by
$\displaystyle{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{\bf{J}}_{E}^{-1}}\right|,$
(7)
where ${{\bf{J}}_{E}}$ is the interference-plus-noise covariance matrix given
by
${{\bf{J}}_{E}}\overset{\triangle}{=}{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{{N}_{E}}}}$.
The achievable secrecy rate is given by
$\displaystyle{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{\bf{\Phi}},{\bf{Z}})$
$\displaystyle{\rm{=}}[{R_{I}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})-{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}}){]^{+}}$
$\displaystyle={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{\bf{J}}_{I}^{-1}}\right|-{\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{\bf{J}}_{E}^{-1}}\right|$
$\displaystyle={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|$
$\displaystyle\quad-{\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{{({{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})}^{-1}}}\right|.$
(8)
### II-B Problem Formulation
With the aim for maximizing SR, the TPC matrix ${\bf{V}}$ at the BS, the AN
covariance matrix ${\bf{Z}}$ at the BS, and the phase shift matrix
${\bf{\Phi}}$ at the IRS should be optimized jointly subject to the
constraints of the maximum transmit power and unit modulus of phase shifts.
Hence, we formulate the SRM problem as
$\displaystyle\
\underset{{\bf{V}},{\bf{\Phi}},{\bf{Z}}}{\mathop{\text{missing}}{max}}\ \
{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{\bf{\Phi}},{\bf{Z}}{\rm{)}}$ (9a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{\bf{Z}}{\rm{)}}\leq{P_{T}},$ (9b)
$\displaystyle\quad\quad\quad{\bf{Z}}\succeq 0,$ (9c)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(9d)
where ${P_{T}}$ is the maximum transmit power limit. The optimal value of SR
in (9) is always non-negative, which can be proved by using contradiction.
Assume that the optimal value of SR is negative, then we can simply set the
TPC matrix ${\bf{V}}$ to zero matrix, and the resulted SR will be equal to
zero, which is larger than a negative SR.
By variable substitution ${\bf{Z}}={{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}$, where
${{\bf{V}}_{E}}\in{{\mathbb{C}}^{{{N}_{T}}\times{{N}_{T}}}}$, Problem (9) is
equivalent to
$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{max}}\
\ {{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ (10a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq{P_{T}},$
(10b)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(10c)
where the OF of (10a) is obtained by substituting
${\bf{Z}}={{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}$ into (II-A). In (10a), the
expression of OF is difficult to tackle, and the variables of ${\bf{V}}$,
${\bf{V}}_{E}$ and ${\bf{\Phi}}$ are coupled with each other, which make
Problem (10) difficult to solve. In addition, the unit modulus constraint
imposed on the phase shifts in (10c) aggravates the difficulty. In the
following, we provide a low-complexity algorithm to solve this problem.
## III A Low-Complexity Algorithm of BCD-MM
Firstly, the OF of Problem (10) is reformulated into a more tractable
expression equivalently. Then, the BCD-MM method is proposed for optimizing
the TPC matrix ${\bf{V}}$, ${\bf{V}}_{E}$, and the phase shift matrix
${\bf{\Phi}}$ alternately.
### III-A Reformulation of the Original Problem
Firstly, the achievable SR
${{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ in (II-A)
can be further simplified as
$\displaystyle{{\rm{C}}_{AN}}\rm{(}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}\rm{)}$
$\displaystyle{\rm{=log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|{\rm{+log}}\left|{{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}}\right|$
$\displaystyle\quad-{\rm{log}}\left|{{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}}\right|$
$\displaystyle=\underbrace{{\rm{log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|}_{{f_{1}}}$
$\displaystyle\quad{\rm{+}}\underbrace{{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+{{\hat{\bf{H}}}_{E}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{{\hat{\bf{H}}}_{E}}^{H}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})^{-1}}\right|}_{{f_{2}}}$
$\displaystyle\quad\underbrace{-{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}})\hat{\bf{H}}_{E}^{H}}\right|}_{{f_{3}}}.$
(11)
The expression in $f_{1}$ represents the data rate of the legitimate IR, which
can be reformulated by exploiting the relationship between the data rate and
the mean-square error (MSE) for the optimal decoding matrix. Specifically, the
linear decoding matrix ${{\bf{U}}_{I}}\in{{\mathbb{C}}^{{{N}_{T}}\times{d}}}$
is applied to estimate the signal vector $\hat{{\bf{s}}}$ for the legitimate
IR, and the MSE matrix of the legitimate IR is given by
$\displaystyle{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}})$
$\displaystyle\buildrel\Delta\over{=}{{\mathbb{E}}_{{\bf{s}},{\bf{n}},{{\bf{n}}_{I}}}}\left[{(\hat{\bf{s}}-{\bf{s}}){{(\hat{\bf{s}}-{\bf{s}})}^{H}}}\right]$
$\displaystyle{\rm{=}}({{\bf{U}}_{I}}^{H}{{\hat{\bf{H}}}_{I}}{\bf{V}}-{\bf{I}}_{d}){({{\bf{U}}_{I}}^{H}{{\hat{\bf{H}}}_{I}}{\bf{V}}-{\bf{I}}_{d})^{H}}+{{\bf{U}}_{I}}^{H}({{\hat{\bf{H}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{\hat{\bf{H}}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}_{I}}.$
(12)
By introducing an auxiliary matrix ${{\bf{W}}_{I}}\succeq 0$,
${{\bf{W}}_{I}}\in{{\mathbb{C}}^{{d}\times{d}}}$ and exploiting the fact 3) of
Lemma 4.1 in [48], we have
$\displaystyle f_{1}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$
$\displaystyle\buildrel\Delta\over{=}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0}\log\left|{{{\bf{W}}_{I}}}\right|-{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))+d.$
(13)
$h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$ is concave with
respect to (w.r.t.) each matrix of the matrices
${{\bf{U}}_{I}}$,${\bf{V}}$,${{\bf{V}}_{E}}$,${{\bf{W}}_{I}}$ by fixing the
other three matrices. According to the facts 1) and 2) of Lemma 4.1 in [48],
the optimal ${{\bf{U}}^{\star}_{I}}$, ${{\bf{W}}^{\star}_{I}}$ to achieve the
maximum value of
$h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$ is given by
$\displaystyle{{\bf{U}}^{\star}_{I}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}}}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}}){\rm{=}}({\hat{\bf{H}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\hat{\bf{H}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}{\rm{+}}{\hat{\bf{H}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{\hat{\bf{H}}_{I}}^{H})^{-1}{\hat{\bf{H}}_{I}}{\bf{V}},$
(14)
$\displaystyle{{\bf{W}}^{\star}_{I}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{I}}\succeq
0}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}}){\rm{=[}}{{\bf{E}}^{\star}_{I}}({{\bf{U}}^{\star}_{I}},{\bf{V}},{{\bf{V}}_{E}}){]^{-1}},$
(15)
where ${{\bf{E}}^{\star}_{I}}$ is obtained by plugging the expression of
${{\bf{U}}^{\star}_{I}}$ into
${{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}})$ as
$\displaystyle{{\bf{E}}^{\star}_{I}}({{\bf{U}}^{\star}_{I}},{\bf{V}},{{\bf{V}}_{E}}){\rm{=}}({{\bf{U}}^{{\star}H}_{I}}{\hat{\bf{H}}_{I}}{\bf{V}}-{\bf{I}}_{d}){({{\bf{U}}^{{\star}H}_{I}}{\hat{\bf{H}}_{I}}{\bf{V}}-{\bf{I}}_{d})^{H}}+{{\bf{U}}^{{\star}H}_{I}}({\hat{\bf{H}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\hat{\bf{H}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}^{{\star}}_{I}}{{\rm{}}}.$
(16)
Similarly, by introducing the auxiliary variables ${{\bf{W}}_{E}}\succeq 0$,
${{\bf{W}}_{E}}\in{{\mathbb{C}}^{{{N}_{T}}\times{{N}_{T}}}}$,
${{\bf{U}}_{E}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{T}}}}$, and exploiting
the fact 3) of Lemma 4.1 in [48], we have
$\displaystyle f_{2}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle\buildrel\Delta\over{=}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}\log\left|{{{\bf{W}}_{E}}}\right|-{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))+{N}_{t},$
(17)
$h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ is concave w.r.t each
matrix of the matrices ${{\bf{U}}_{E}}$,${{\bf{V}}_{E}}$,${{\bf{W}}_{E}}$ when
the other two matrices are given. According to the facts 1) and 2) of Lemma
4.1 in [48], the optimal ${{\bf{U}}^{\star}_{E}}$, ${{\bf{W}}^{\star}_{E}}$ to
achieve the maximum value of
$h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ is given by
$\displaystyle{{\bf{U}}^{\star}_{E}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}}}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}}){\rm{=}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}{\rm{+}}{\hat{\bf{H}}_{E}}{{\bf{V}}_{E}}{{{\bf{V}}^{H}_{E}}}{\hat{\bf{H}}_{E}}^{H})^{-1}{\hat{\bf{H}}_{E}}{{\bf{V}}_{E}},$
(18)
$\displaystyle{{\bf{W}}^{\star}_{E}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}}){\rm{=[}}{{\bf{E}}^{\star}_{E}}({{\bf{U}}^{\star}_{E}},{{\bf{V}}_{E}}){]^{-1}},$
(19)
where ${{\bf{E}}^{\star}_{E}}$ is obtained by plugging the expression of
${{\bf{U}}^{\star}_{E}}$ into ${{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}})$
as
$\displaystyle\begin{array}[]{l}{{\bf{E}}^{\star}_{E}}({{\bf{U}}^{\star}_{E}},{{\bf{V}}_{E}})=({{\bf{U}}_{E}}^{{\star}H}{{\hat{\bf{H}}}_{E}}{\bf{V}}_{E}-{\bf{I}}_{N_{T}}){({{\bf{U}}^{{\star}H}_{E}}{{\hat{\bf{H}}}_{E}}{\bf{V}}_{E}-{\bf{I}}_{N_{T}})^{H}}+{{\bf{U}}^{{\star}H}_{E}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}){{\bf{U}}^{\star}_{E}}.\end{array}$
(21)
By using the Lemma 1 in [13], we have
$\displaystyle f_{3}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}\log\left|{{{\bf{W}}_{X}}}\right|-{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))+{N}_{E},$
(22)
where ${{\bf{W}}_{X}}\succeq 0$,
${{\bf{W}}_{X}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{E}}}}$ are the
introduced auxiliary variable, and
$\displaystyle\begin{array}[]{l}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}})\buildrel\Delta\over{=}{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}})\hat{\bf{H}}_{E}^{H}}.\end{array}$
(24)
$h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ is concave w.r.t each matrix
of the matrices ${{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}$ when the other two
matrices are given. The optimal ${{\bf{W}}^{\star}_{X}}$ to achieve the
maximum value of $h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ is
$\displaystyle{{\bf{W}}^{\star}_{X}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}){\rm{=[}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}){]^{-1}}.$
(25)
By substituting (III-A), (III-A), (III-A) into (III-A), we have
$\displaystyle{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}}{\rm{)}}$
$\displaystyle=\mathop{\text{missing}}{argmax}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}}{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}}),$
(26)
where
$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}})\buildrel\Delta\over{=}$
$\displaystyle
h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle+h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}).$ (27)
Obviously,
${{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}})$
is a concave function for each of the matrices
${{\bf{U}}_{I}}$,${{\bf{W}}_{I}}$,${{\bf{U}}_{E}}$,${{\bf{W}}_{E}}$,${{\bf{W}}_{X}}$,${\bf{V}}$,${{\bf{V}}_{E}}$
when the other six matrices are given. By substituting (26) into Problem (10),
we have the following equivalent problem:
$\displaystyle\ \underset{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0,{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0,{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{max}}\ \
{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})$
(28a) $\displaystyle\quad\quad\quad\quad\quad\quad\text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{\rm{)}}\leq{P_{T}},$
(28b) $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\ \
\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M.$ (28c)
To solve Problem (28), we apply the BCD method, each iteration of which
consists the following two sub-iterations. Firstly, with given
${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$, update
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$
by using (14), (15), (18), (19), (25) respectively. Secondly, with given
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$,
update ${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$ by solving the following
subproblem:
$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{min}}\
\
-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}^{H}_{E}}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(29a) $\displaystyle\ \
\text{s.t.}\quad{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq
P_{T},$ (29b)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(29c)
where
$\displaystyle{{\mathbf{H}}_{V}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{\mathbf{\hat{H}}}_{I}}+\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(30)
$\displaystyle{{\mathbf{H}}_{VE}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{\mathbf{\hat{H}}}_{I}}+{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(31)
Problem (29) is obtained from Problem (28) by taking the
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$
as constant values, and the specific derivations are given in Appendix A.
It is obvious that Problem (29) is much easier to tackle than Problem (10) due
to the convex quadratic OF in (29a). Now, we devote to solve Problem (29)
equivalently instead of Problem (10), and the matrices ${\bf{V}}$,
${\bf{V}}_{E}$, and phase shift matrix $\mathbf{\Phi}$ will be optimized.
### III-B Optimizing the Matrices ${\bf{V}}$ and ${\bf{V}}_{E}$
In this subsection, the TPC matrix ${\bf{V}}$ and matrix ${\bf{V}}_{E}$ are
optimized by fixing $\mathbf{\Phi}$. Specifically, the unit modulus constraint
on the phase shifts $\mathbf{\Phi}$ is removed, and the updated optimization
problem reduced from Problem (29) is given by
$\displaystyle\ \
\underset{{\bf{V}},{{\bf{V}}_{E}}}{\mathop{\text{missing}}{min}}\ \
-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}^{H}_{E}}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(32a) $\displaystyle\ \
\text{s.t.}\quad{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq
P_{T}.$ (32b)
The above problem is a convex QCQP problem, and the standard optimization
packages, such as CVX [49] can be exploited to solve it. However, the
calculation burden is heavy. To reduce the complexity, the near-optimal closed
form expressions of the TPC matrix and AN covariance matrix are provided by
applying the Lagrangian multiplier method.
Since Problem (32) is a convex problem, the Slater’s condition is satisfied,
where the duality gap between Problem (32) and its dual problem is zero. Thus,
Problem (32) can be solved by addressing its dual problem if the dual problem
is easier. For this purpose, by introducing Lagrange multiplier $\lambda$ to
combine the the constraint and OF of Problem (32), the Lagrangian function of
Problem (32) is obtained as
$\displaystyle\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$
$\displaystyle\\!\buildrel\Delta\over{=}\\!-{\rm{Tr}}\left({{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)\\!+\\!{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)$
$\displaystyle\quad-{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)+\lambda[{{\rm{Tr}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right)}-P_{T}]$
$\displaystyle=-{\rm{Tr}}\left({{{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}}\right)-{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)+{\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}}\right]$
$\displaystyle\quad-{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)\\!+\\!{\rm{Tr}}\left[{{\bf{V}}_{E}^{H}\left({{{\bf{H}}_{VE}}\\!+\\!\lambda{\bf{I}}}\right){{\bf{V}}_{E}}}\right]\\!-\\!\lambda{P_{T}}.$
(33)
Then the dual problem of Problem (32) is
$\displaystyle\mathop{\max}\limits_{\lambda}\quad\quad{\rm{}}h\left(\lambda\right)$
(34a) $\displaystyle\text{s.t.}\quad\quad{\rm{}}\lambda\geq 0,$ (34b)
where $h\left(\lambda\right)$ is the dual function given by
$\displaystyle
h\left(\lambda\right)\buildrel\Delta\over{=}\mathop{\min}\limits_{{\bf{V}},{{\bf{V}}_{E}}}{\rm{}}\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right).$
(35)
Note that Problem (35) is a convex quadratic optimization problem with no
constraint, which can be solved in closed form. The optimal solution
${\bf{V}^{\star}},{{\bf{V}^{{\star}}}_{E}}$ for Problem (35) is
$\displaystyle[{\bf{V}^{\star}},{{\bf{V}^{{\star}}}_{E}}]=\text{arg}\mathop{\min}\limits_{{\bf{V}},{{\bf{V}}_{E}}}{\rm{}}\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right).$
(36)
By setting the first-order derivative of
$\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$ w.r.t.
${{{\bf{V}}}}$ to zero matrix, we can obtain the optimal solution of
${\bf{V}}$ as follows:
$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{\bf{V}}}}=\bf{0},$
(37a)
$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{{\bf{V}}_{E}}}}=\bf{0}.$
(37b)
The left hand side of Equation (37a) can be expanded as
$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{\bf{V}}}}$
$\displaystyle=\frac{{\partial{\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}}\right]}}{{\partial{\bf{V}}}}-\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}}\right)^{H}-\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right)$
$\displaystyle=2\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}-2\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(38)
The equation (37a) becomes
$\displaystyle\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}=\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(39)
Then the optimal solution ${\bf{V}^{\star}}$ for Problem (36) is
$\displaystyle{{\bf{V}}^{\star}}$
$\displaystyle=\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right)$
$\displaystyle\buildrel\Delta\over{=}{{\bf{\Theta}}_{V}}\left(\lambda\right)\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(40)
Similarly, we solve Problem (36) by setting the first-order derivative of
$\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$ w.r.t.
${{{\bf{V}}_{E}}}$ to zero matrix, which becomes
$\displaystyle
2\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right){{\bf{V}}_{E}}-2{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}=\bf{0}.$
(41)
Then the optimal solution ${\bf{V}}_{E}^{\star}$ for Problem (36) is
$\displaystyle{\bf{V}}_{E}^{\star}$
$\displaystyle=\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}$
$\displaystyle\buildrel\Delta\over{=}{{\bf{\Theta}}_{VE}}\left(\lambda\right){\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}.$
(42)
Once the optimal solution ${\lambda}^{\star}$ for Problem (34) is found, the
final optimal ${\bf{V}^{\star}},{\bf{V}}_{E}^{\star}$ can be obtained. The
value of ${\lambda}^{\star}$ should be chosen in order to guarantee the
complementary slackness condition as
$\displaystyle\lambda[{\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{+}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}}-P_{T}]=0.$
(43)
We define
$\displaystyle P(\lambda)$
$\displaystyle\buildrel\Delta\over{=}{\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{+}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}}={\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{)}}+{\rm{Tr(}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}},$
(44)
where
$\displaystyle{\rm{Tr}}\left({{\bf{V}}^{\star}{{\bf{V}}^{{\star}H}}}\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}_{V}}\left(\lambda\right)({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}{{\bf{\Theta}}^{H}_{V}}\left(\lambda\right)\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}^{H}_{V}}\left(\lambda\right){{\bf{\Theta}}_{V}}\left(\lambda\right)({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}\right),$
(45)
$\displaystyle{\rm{Tr}}\left({{{\bf{V}}_{E}^{{\star}H}}{\bf{V}}_{E}^{\star}}\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}_{VE}}\left(\lambda\right)({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}{{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right)\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right){{\bf{\Theta}}_{VE}}\left(\lambda\right)({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}\right).$
(46)
Then $P(\lambda)$ becomes
$\displaystyle
P(\lambda)={\rm{Tr}}\left({{{\bf{\Theta}}^{n}_{V}}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}}\right)+{\rm{Tr}}\left({{{\bf{\Theta}}^{n}_{VE}}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}}\right),$
(47)
where
$\displaystyle{{\bf{\Theta}}^{n}_{V}}$
$\displaystyle={{\bf{\Theta}}^{H}_{V}}\left(\lambda\right){{\bf{\Theta}}_{V}}\left(\lambda\right)=\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}H}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}},$
(48) $\displaystyle{{\bf{\Theta}}^{n}_{VE}}$
$\displaystyle={{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right){{\bf{\Theta}}_{VE}}\left(\lambda\right)=\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}H}\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}}.$
(49)
To find the optimal ${\lambda^{\star}}\geq 0$, we first check whether
$\lambda=0$ is the optimal solution or not. If
$\displaystyle
P(0)={\rm{Tr}}\left({{{\bf{V}}^{{\star}H}}(0){\bf{V}}^{\star}(0)}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{{\star}H}(0){{\bf{V}}_{E}}^{\star}(0)}\right)\leq{P_{T}},$
(50)
then the optimal solutions are given by ${{\bf{V}}^{\star}}={{\bf{V}}}(0)$ and
${{\bf{V}}_{E}^{\star}}={{\bf{V}}_{E}}(0)$. Otherwise, the optimal
$\lambda^{\star}>0$ is the solution of the equation $P(\lambda)=0$.
It is ready to verify that ${{\bf{H}}_{V}}$ and ${{\bf{H}}_{VE}}$ is a
positive semidefinite matrix. Let us define the rank of ${{\bf{H}}_{V}}$ and
${{\bf{H}}_{VE}}$ as $r_{V}={\rm{rank}}({\bf{H}}_{V})\leq N_{T}$ and
$r_{VE}={\rm{rank}}({\bf{H}}_{VE})\leq N_{T}$ respectively. By decomposing
${{\bf{H}}_{V}}$ and ${{\bf{H}}_{VE}}$ by using the singular value
decomposition (SVD), we have
${{\bf{H}}_{V}}=\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]{{\bf{\Sigma}}_{V}}{\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]^{\rm{H}}},{{\bf{H}}_{VE}}=\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]{{\bf{\Sigma}}_{VE}}{\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]^{\rm{H}}},$
(51)
where ${\bf{P}}_{V,1}$ comprises the first $r_{V}$ singular vectors associated
with the $r_{V}$ positive eigenvalues of ${{\bf{H}}_{V}}$, and
${\bf{P}}_{V,2}$ includes the last $N_{T}-r_{V}$ singular vectors associated
with the $N_{T}-r_{V}$ zero-valued eigenvalues of ${{\bf{H}}_{V}}$,
${{\bm{\Sigma}}_{V}}={\rm{diag}}\left\\{{{{\bm{\Sigma}}_{V,1}},{{\bf{0}}_{\left({{N_{T}}-{r_{V}}}\right)\times\left({{N_{T}}-{r_{V}}}\right)}}}\right\\}$
with ${\bm{\Sigma}}_{V,1}$ representing the diagonal submatrix collecting the
first $r_{V}$ positive eigenvalues. Similarly, the first $r_{VE}$ singular
vectors corresponding to the $r_{VE}$ positive eigenvalues of
${{\bf{H}}_{VE}}$ are contained in ${\bf{P}}_{VE,1}$, while the last
$N_{T}-r_{VE}$ singular vectors corresponding to the $N_{T}-r_{VE}$ zero-
valued eigenvalues of ${{\bf{H}}_{VE}}$ are held in ${\bf{P}}_{VE,2}$.
${{\bm{\Sigma}}_{VE}}={\rm{diag}}\left\\{{{{\bm{\Sigma}}_{{VE},1}},{{\bf{0}}_{\left({{N_{T}}-{r_{VE}}}\right)\times\left({{N_{T}}-{r_{VE}}}\right)}}}\right\\}$
is a diagonal matrix with ${\bm{\Sigma}}_{{VE},1}$ representing the diagonal
submatrix gathering the first $r_{VE}$ positive eigenvalues. By defining
${{\bf{P}}_{V}}\buildrel\Delta\over{=}\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]$
and
${{\bf{P}}_{VE}}\buildrel\Delta\over{=}\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]$,
and substituting (51) into (48) and (49), $P(\lambda)$ becomes
$\displaystyle
P(\lambda)={\rm{Tr}}\left({[{\left({{{\bf{P}}_{V}}{{\bf{\Sigma}}_{V}}{\bf{P}}_{V}^{H}+\lambda{{\bf{P}}_{V}}{\bf{P}}_{V}^{H}}\right)^{-1}}{\left({{{\bf{P}}_{V}}{{\bf{\Sigma}}_{V}}{\bf{P}}_{V}^{H}+\lambda{{\bf{P}}_{V}}{\bf{P}}_{V}^{H}}\right)^{-1}}]({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}}\right)$
$\displaystyle\\!\\!+\\!\\!{\rm{Tr}}\\!\left(\\!{[\\!{\left(\\!{{{\bf{P}}_{VE}}{{\bf{\Sigma}}_{VE}}{\bf{P}}_{VE}^{H}\\!+\\!\lambda{{\bf{P}}_{VE}}{\bf{P}}_{VE}^{H}}\\!\right)^{-1}}\\!\\!{\left({{{\bf{P}}_{VE}}{{\bf{\Sigma}}_{VE}}{\bf{P}}_{VE}^{H}\\!+\\!\lambda{{\bf{P}}_{VE}}{\bf{P}}_{VE}^{H}}\right)^{-1}}\\!]\\!(\\!{{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}}\\!)\\!({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}}\\!\right)$
$\displaystyle={\rm{Tr}}\left({[{\left({{{\bf{\Sigma}}_{V}}+\lambda{\bf{I}}}\right)^{-2}}]{\bf{Z}}_{V}}\right)+{\rm{Tr}}\left({[{\left({{{\bf{\Sigma}}_{VE}}+\lambda{\bf{I}}}\right)^{-2}}]{\bf{Z}}_{VE}}\right)$
$\displaystyle{=}\sum\limits_{i=1}^{r_{V}}\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({{{\left[{{\Sigma}_{V}}\right]}_{i,i}}\\!+\\!\lambda}\right)}^{2}}}}\right]+\sum\limits_{i=1}^{r_{VE}}\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({{{\left[{{\Sigma}_{VE}}\right]}_{i,i}}\\!+\\!\lambda}\right)}^{2}}}}\right]+\sum\limits_{i={r_{V}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({\lambda}\right)}^{2}}}}\right]}\\!+\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({\lambda}\right)}^{2}}}}\right]},$
(52)
where
${{\bf{Z}}_{V}}={\bf{P}}_{V}^{H}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}{{\bf{P}}_{V}}$
and
${{\bf{Z}}_{VE}}={\bf{P}}_{VE}^{H}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}{{\bf{P}}_{VE}}$.
${\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}$,
${\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}$,
${\left[{{{\Sigma}}_{V}}\right]}_{i,i}$, and
${\left[{{{\Sigma}}_{VE}}\right]}_{i,i}$ represent the $i$th diagonal element
of matrices ${{{\bf{Z}}_{V}}}$, ${{\bf{Z}}_{VE}}$, ${{{\Sigma}}_{V}}$, and
${{{\Sigma}}_{VE}}$, respectively. The first line of (III-B) is obtained by
substituting (51) into the expression of $P({\lambda})$ in (47). It can be
verified from the last line of (III-B) that $P({\lambda})$ is a monotonically
decreasing function.
Then, the optimal $\lambda^{\star}$ can be obtained by solving the following
equation,
$\displaystyle\sum\limits_{i=1}^{r_{V}}\\!\left[{\frac{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}{{{{\left({{{\left[{{{\Sigma}}_{V}}\right]}_{i,i}}+\lambda}\right)}^{2}}}}}\right]\\!+\\!\\!\sum\limits_{i=1}^{r_{VE}}\\!\left[{\frac{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}{{{{\left({{{\left[{{{\Sigma}}_{VE}}\right]}_{i,i}}+\lambda}\right)}^{2}}}}}\right]\\!+\\!\\!\sum\limits_{i={r_{V}}+1}^{{N_{T}}}\\!{\left[{\frac{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}{{{{\left({\lambda}\right)}^{2}}}}}\right]}\\!+\\!\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}\\!{\left[{\frac{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}{{{{\left({\lambda}\right)}^{2}}}}}\right]}=P_{T}.$
(53)
To solve it, the bisection search method is utilized. Since $P(\infty)=0$, the
solution to Equation (53) must exist. The lower bound of $\lambda^{\star}$ is
a positive value approaching zero, while the upper bound of $\lambda^{\star}$
is given by
${\lambda^{\star}}<\sqrt{\frac{{\sum\limits_{i=1}^{{N_{T}}}{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}+{\sum\limits_{i=1}^{{N_{T}}}{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}}{{{P_{T}}}}}\buildrel\Delta\over{=}\lambda^{{\rm{ub}}}.$
(54)
which can be proved as
$\displaystyle{P}(\lambda^{{\rm{ub}}})$
$\displaystyle=\sum\limits_{i=1}^{r_{V}}{\frac{{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}{{{{\left({{{\left[{{{\Sigma}}_{V}}\right]}_{i,i}}+{\lambda^{{\rm{ub}}}}}\right)}^{2}}}}}+\sum\limits_{i=1}^{r_{VE}}{\frac{{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}{{{{\left({{{\left[{{{\Sigma}}_{VE}}\right]}_{i,i}}+{\lambda^{{\rm{ub}}}}}\right)}^{2}}}}}+\sum\limits_{i={r_{V}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}\right]}\\!+\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}\right]}$
$\displaystyle<\sum\limits_{i=1}^{{N_{T}}}{\frac{{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}{{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}}+\sum\limits_{i=1}^{{N_{T}}}{\frac{{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}{{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}}={P_{T}}.$
(55)
When the optimal $\lambda^{{\star}}$ is found, the optimal matrices
${{\bf{V}}^{{\star}}}$ and ${{\bf{V}}_{E}^{{\star}}}$ can be obtained by
substituting $\lambda^{\star}$ into (III-B) and (III-B).
### III-C Optimizing the Phase Shifts $\mathbf{\Phi}$
In this subsection, the phase shift matrix $\mathbf{\Phi}$ is optimized by
fixing ${{\bf{V}}}$ and ${{\bf{V}}_{E}}$. The transmit power constraint in
Problem (29) is only related with ${{\bf{V}}}$ and ${{\bf{V}}_{E}}$, thus is
removed. Then, the optimization problem for $\mathbf{\Phi}$ reduced from
Problem (29) is formulated as
$\displaystyle\ \ \underset{{\bf{\Phi}}}{\mathop{\text{missing}}{min}}\ \
{g_{0}}(\mathbf{\Phi})\buildrel\Delta\over{=}-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad\quad\quad\
-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(56a) $\displaystyle\ \
\text{s.t.}\quad\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M.$ (56b)
By the aid of complex mathematical manipulations, which are given in details
in Appendix B, Problem (56) can be transformed into a form that can facilitate
the MM algorithm. Based on the derivations in Appendix B, the OF
${g_{0}}(\mathbf{\Phi})$ can be equivalently transformed into
$\displaystyle{g_{0}}(\mathbf{\Phi})={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}{\bf{\Phi}}{{\bf{C}}_{VE}}}\right]+{\rm{Tr}}\left({{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}{\bf{\Phi}}{{\bf{C}}_{V}}}\right)+C_{t},$
(57)
where $C_{t}$, ${\bf{D}}$, ${{\bf{C}}_{VE}}$, ${{\bf{C}}_{V}}$,
${{\bf{B}}_{VE}}$ and ${{\bf{B}}_{V}}$ are constants for ${\bf{\Phi}}$, and
are given in Appendix B.
By exploiting the matrix properties in [50, Eq. (1.10.6)], the trace operators
can be removed, and the third and fourth terms in (57) become as
$\displaystyle{\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{\bf{B}}_{VE}{\bm{\Phi}}{\bf{C}}_{VE}}\right)={{\bm{\phi}}^{\rm{H}}}\left({{\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}}\right){\bm{\phi}},$
(58a)
$\displaystyle{\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{\bf{B}}_{V}{\bm{\Phi}}{\bf{C}}_{V}}\right)={{\bm{\phi}}^{\rm{H}}}\left({{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}}\right){\bm{\phi}},$
(58b)
where
${\bm{\phi}}\buildrel\Delta\over{=}{\left[{{e^{j{\theta_{1}}}},\cdots,{e^{j{\theta_{m}}}},\cdots,{e^{j{\theta_{M}}}}}\right]^{\rm{T}}}$
is a vector holding the diagonal elements of ${\bm{\Phi}}$.
Similarly, the trace operators can be removed for the first and second terms
in (57) as
${\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{{\bf{D}}^{\rm{H}}}}\right)={{\bf{d}}^{\rm{H}}}({{\bm{\phi}}}^{*}),{\rm{Tr}}\left({{\bm{\Phi}}{\bf{D}}}\right)={\bm{\phi}}^{\rm{T}}{\bf{d}},$
(59)
where
${\bf{d}}={\left[{{{\left[{\bf{D}}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}\right]}_{M,M}}}\right]^{\rm{T}}}$
is a vector gathering the diagonal elements of matrix ${\bf{D}}$.
Hence, Problem (56) can be rewritten as
$\displaystyle{\mathop{\min}\limits_{{\bm{\phi}}}\quad{{\bm{\phi}}^{\rm{H}}}{\bm{\Xi}}{\bm{\phi}}+{\bm{\phi}}^{\rm{T}}{\bf{d}}+{{\bf{d}}^{\rm{H}}}({{\bm{\phi}}}^{*})}$
(60a)
$\displaystyle\textrm{s.t.}\quad\left|{{\phi_{m}}}\right|=1,m=1,\cdots,M,$
(60b)
where
$\bm{\Xi}={\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}+{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}$.
$\bm{\Xi}$ is a semidefinite matrix, because it is a sum of two semidefinite
matrices, both of which are Hadamard products of two semidefinite matrices. It
is observed that ${\bf{B}}_{VE}$, ${{\bf{C}}_{VE}^{\rm{T}}}$, ${\bf{B}}_{V}$
and ${{\bf{C}}_{V}^{\rm{T}}}$ are semidefinite matrices. Then, the Hadamard
products of ${{\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}}$ and
${{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}}$ are semidefinite according to the
Property (9) on Page 104 of [50]. Problem (60) can be further simplified as
$\displaystyle{\mathop{\min}\limits_{\bm{\phi}}\quad
f({\bm{\phi}})\buildrel\Delta\over{=}{{\bm{\phi}}^{\rm{H}}}{\bm{\Xi}}{\bm{\phi}}+2{\rm{Re}}\left\\{{{{\bm{\phi}}^{\rm{H}}}({{\bf{d}}}^{*})}\right\\}}$
(61a)
$\displaystyle\textrm{s.t.}\quad\left|{{\phi_{m}}}\right|=1,m=1,\cdots,M.$
(61b)
The Problem (61) can be solved by the SDR technique [28] by transforming the
unimodulus constraint into a rank-one constraint, however, the rank-one
solution cannot always be obtained and the computation complexity is heavy for
the SDR method. Thus, we propose to solve Problem (61) efficiently by the MM
algorithm as [25], where the closed-form solution can be obtained in each
iteration. Details are omitted for simplicity.
### III-D Overall Algorithm to Solve Problem (10)
To sum up, the detailed execution of the overall BCD-MM algorithm proposed for
solving Problem (10) is provided in Algorithm 1. The MM algorithm is exploited
for solving the optimal phase shifts ${\bf{\Phi}}^{(n+1)}$ of Problem (61) in
Step 5. The iteration process in MM algorithm ensures that the OF value of
Problem (61) decreases monotonically. Moreover, the BCD algorithm also
guarantees that the OF value of Problem (29) monotonically decreases in each
step and each iteration of Algorithm 1. Since the OF value in (29a) has a
lower bound with the power limit, the convergence of Algorithm 1 is
guaranteed.
Algorithm 1 BCD-MM Algorithm
1: Parameter Setting. Set the maximum number of iterations $n_{\rm{max}}$ and
the first iterative number $n=1$; Give the error tolerance $\varepsilon$.
2: Variables Initialization. Initialize the variables ${\bf{V}}^{(1)}$,
${\bf{V}}_{E}^{(1)}$ and ${\bf{\Phi}}^{(1)}$ in the feasible region; Compute
the OF value of Problem (10) as
${\rm{OF(}}{{\bf{V}}^{(1)}},{{\bf{V}}^{(1)}_{E}},{\bf{\Phi}}^{(1)}{\rm{)}}$;
3: Auxiliary Variables Calculation. Given
${\bf{V}}^{(n)},{{\bf{V}}^{(n)}_{E}}$, ${\bf{\Phi}}^{(n)}$, compute the
optimal matrices
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$
according to (14), (15), (18), (19), (25) respectively;
4: Matrices Optimization. Given
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$,
solve the optimal TPC matrix ${\bf{V}}^{(n+1)}$ and equivalent AN covariance
matrix ${{\bf{V}}^{(n+1)}_{E}}$ of Problem (36) with the Lagrangian multiplier
method;
5: Phase Shifts Optimization. Given
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$
and ${\bf{V}}^{(n+1)},{{\bf{V}}^{(n+1)}_{E}}$, solve the optimal phase shifts
${\bf{\Phi}}^{(n+1)}$ of Problem (61) with the MM algorithm;
6: Termination Check. If
${{\left|\\!{{\rm{OF}}\\!(\\!{\bf{V}}^{(n+1)}\\!,\\!{{\bf{V}}^{(n+1)}_{E}}\\!,\\!{\bf{\Phi}}^{(n+1)}\\!)\\!\\!-\\!\\!{\rm{OF}}\\!(\\!{\bf{V}}^{(n)}\\!,\\!{{\bf{V}}^{(n)}_{E}}\\!,\\!{\bf{\Phi}}^{(n)}\\!)}\\!\right|}/\\!{{\rm{OF}}\\!(\\!{\bf{V}}^{(n+1)}\\!,\\!{{\bf{V}}^{(n+1)}_{E}}\\!,\\!{\bf{\Phi}}^{(n+1)}\\!)}}\\!<\varepsilon$
or $n\geq n_{\rm{max}}$, terminate. Otherwise, update $n\leftarrow n+1$ and
jump to step 2.
Based on the algorithm description, the complexity analysis of the proposed
BCD-MM algorithm is performed. In Step 3, computing the decoding matrices
${{\bf{U}}^{(n)}_{I}}$ and ${{\bf{U}}^{(n)}_{E}}$ costs the complexity of
${\cal O}(N_{I}^{3})+{\cal O}(N_{E}^{3})$, while calculating the auxiliary
matrices ${{\bf{W}}^{(n)}_{I}}$, ${{\bf{W}}^{(n)}_{E}}$, and
${{\bf{W}}^{(n)}_{X}}$ consumes the complexity of ${\cal O}(d^{3})+{\cal
O}(N_{T}^{3})+{\cal O}(N_{E}^{3})$. The complexity of calculating the TPC
matrix ${\bf{V}}^{(n+1)}$ and AN covariance matrix ${{\bf{V}}^{(n+1)}_{E}}$ in
Step 4 can be analyzed according to the specific process of Lagrangian
multiplier method based on the fact that the complexity of computing product
${\bf{XY}}$ of complex matrices ${\bf{X}}\in{{\mathbb{C}}^{m\times n}}$ and
${\bf{Y}}\in{{\mathbb{C}}^{n\times p}}$ is ${\cal O}\left({mnp}\right)$. By
assuming that $N_{T}>N_{I}({\rm{or\ }}N_{E})>d$, the complexity of computing
the matrices $\\{{{\mathbf{H}}_{V}},{{\mathbf{H}}_{VE}}\\}$ in (30) and (31)
is ${\cal O}(N_{T}^{3})+{\cal O}(2N_{T}^{2}d)+{\cal O}(2N_{T}^{2}N_{E})$;
while the complexity of calculating ${\bf{V}}^{*}$, ${\bf{V}}_{E}^{*}$ in
(III-B) and (III-B) is ${\cal O}(2N_{T}^{3})$. The SVD decomposition of
$\\{{{\mathbf{H}}_{V}},{{\mathbf{H}}_{VE}}\\}$ requires the computation
complexity of ${\cal O}(2N_{T}^{3})$, while calculating $\\{{\bf{Z}}_{V}\\}$
and $\\{{\bf{Z}}_{VE}\\}$ requires the complexity of ${\cal
O}(N_{T}^{2}N_{I})+{\cal O}(2N_{T}^{3})$. The complexity of finding the
Lagrangian multipliers $\\{\lambda\\}$ is negligible. Thus, the overall
complexity for ${\bf{V}}^{(n+1)}$, ${\bf{V}}_{E}^{(n+1)}$ is about ${\cal
O}({\rm{max}}\\{2N_{T}^{3},2N_{T}^{2}N_{E}\\})$. In step 5, obtaining optimal
${\bf{\Phi}}^{(n+1)}$ by the MM algorithm need a complexity of $C_{MM}={\cal
O}(M^{3}+T_{MM}M^{2})$, where $T_{MM}$ is the iteration number for
convergence. Based on the complexity required in Step 3, 4 and 5, the overall
complexity $C_{\rm{BCD-MM}}$ of the BCD-MM algorithm can be evaluated by
$C_{\rm{BCD-MM}}={\cal O}({\rm{max}}\\{2N_{T}^{3},2N_{T}^{2}N_{E},C_{MM}\\}).$
(62)
## IV Extension to the Multiple-IRs Scenario
### IV-A Problem Formulation
Consider a multicast extension where there are $L\geq 2$ legitimate IRs, and
they all intend to receive the same message. The signal model for the MIMO
multi-IRs wiretap channel scenario is
$\displaystyle{\bf{y}}_{I,l}={\hat{{\bf{H}}}_{I,l}}({\bf{V}}{\bf{s}}+{\bf{n}})+{{\bf{n}}_{I,l}},l=1,\cdots,L,$
(63)
where
${{\hat{\bf{H}}}_{I,l}}\overset{\triangle}{=}{{\bf{H}}_{b,I,l}}+{{\bf{H}}_{R,I,l}}{\bf{\Phi}}{\bf{G}}$.
The subscript $l$ indicates the $l$th IR, and the other notations are the same
as (4) and (6). Under these settings, the achievable SR is given by [51]
$\displaystyle R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{{R_{I,l}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})-{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})\\},$
(64)
where
${R_{I,l}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}}{{\bf{V}}^{H}}\hat{{\bf{H}}}_{I,l}^{H}{\bf{J}}_{I,l}^{-1}}\right|$
and
${{\bf{J}}_{I,l}}\overset{\triangle}{=}{{\hat{{\bf{H}}}}_{I,l}}{\bf{Z}}{{\hat{{\bf{H}}}}_{I,l}}^{H}+\sigma_{I,l}^{2}{{\bf{I}}_{{{N}_{I}}}}$.
Then the multicast counterpart of the AN-aided SRM problem (10) is formulated
as
$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{max}}}\ \
{R_{s}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ (65a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(65b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(65c)
The objective function of Problem (65) can be rewritten as
$\displaystyle R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{\underbrace{{\rm{log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}}{{\bf{V}}^{H}}\hat{{\bf{H}}}_{I,l}^{H}{{({{\hat{{\bf{H}}}}_{I,l}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}+\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|}_{{f_{1,l}}}\\}$
$\displaystyle\quad{\rm{+}}\underbrace{{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+{{\hat{{\bf{H}}}}_{E}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{{\hat{{\bf{H}}}}_{E}}^{H}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})^{-1}}\right|}_{{f_{2}}}$
$\displaystyle\quad\underbrace{-{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{{\bf{H}}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}})\hat{{\bf{H}}}_{E}^{H}}\right|}_{{f_{3}}},$
(66a)
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}+\mathop{\text{max}}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle\quad+\mathop{\text{max}}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}).$ (66b)
The lower bound to the first term of (66b) can be found as
$\displaystyle\underset{l=1,\cdots,L}{\min}\\{\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}$ (67a)
$\displaystyle\geq\mathop{\text{max}}\limits_{\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L}}\\{\underset{l=1,\cdots,L}{\min}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\},$
(67b)
where (67b) holds due to the fact that $\underset{x}{\min}\
\underset{y}{\max}f(x,y)\geq\underset{y}{\max}\ \underset{x}{\min}f(x,y)$ for
any function $f(x,y)$. Here by exchanging the positions of
$\mathop{\text{max}}\limits_{\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L}}$ and $\underset{l=1,\cdots,L}{\min}$ in (67a), we can find a
lower bound to $R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ as
$\displaystyle
f_{ms}({\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}})$
$\displaystyle\triangleq\mathop{\text{max}}\limits_{{\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0}\\{\underset{l=1,\cdots,L}{\min}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})$
$\displaystyle\quad+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})+h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})\\}.$
(68)
We simplify Problem (65) by maximizing a lower bound to its original objective
as follows,
$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0}{\mathop{\text{max}}}f_{ms}({\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}})$
(69a) $\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(69b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(69c)
To solve the multicast AN-aided SRM problem in (69), a BCD-QCQP-CCP algorithm
is proposed.
### IV-B BCD Iterations for Problem (69)
The equivalent SRM problem in (69) provides a desirable formulation for BCD
algorithm. In particular, one can show that problem (69) is convex w.r.t.
either ${\bf{V}},{{\bf{V}}_{E}}$ or ${\bf{\Phi}}$ or $\\{{{\bf{U}}_{I,l}}$,
${{\bf{W}}_{I,l}}\\}_{l=1}^{L}$, ${{\bf{U}}_{E}}$, ${{\bf{W}}_{E}}$,
${{\bf{W}}_{X}}$. By fixing ${\bf{\Phi}}$, the iteration process for problem
(69) is as follows. Let ${\bf{V}}^{n}$, ${{\bf{V}}_{E}^{n}}$,
$\\{{{\bf{U}}_{I,l}^{n}},{{\bf{W}}_{I,l}^{n}}\\}_{l=1}^{L},{{\bf{U}}_{E}^{n}},{{\bf{W}}_{E}^{n}},{{\bf{W}}_{X}^{n}}$
denote the BCD iterate at the $n$th iteration. The BCD iterates are generated
via
$\displaystyle{{\bf{U}}_{I,l}^{n}}=\text{arg}\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}}}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{I,l}^{n}})$
$\displaystyle\qquad\>\>{\rm{=}}({\hat{{\bf{H}}}_{I,l}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H}{\rm{+}}\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}}{\rm{+}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}{{\bf{V}}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H})^{-1}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n},$
(70a)
$\displaystyle{{\bf{W}}_{I,l}^{n}}=\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}^{n}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{I,l}}){\rm{=[}}{{\bf{E}}_{I,l}}({{\bf{U}}_{I,l}^{n}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle{\rm{=}}\text{[}({{\bf{U}}_{I,l}^{{n}H}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}-{\bf{I}}_{d}){({{\bf{U}}_{I,l}^{{n}H}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}-{\bf{I}}_{d})^{H}}+{{\bf{U}}_{I,l}^{{n}H}}({\hat{{\bf{H}}}_{I,l}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H}{\rm{+}}\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}_{I,l}^{{n}}}]^{-1},$
(70b)
$\displaystyle{{\bf{U}}_{E}^{n}}{\rm{=}}\text{arg}\mathop{\text{max}}\limits_{{{\bf{U}}_{E}}}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{E}^{n}})$
$\displaystyle\qquad\>\>{\rm{=}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}{\rm{+}}{\hat{{\bf{H}}}_{E}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{E}}^{H})^{-1}{\hat{{\bf{H}}}_{E}}{{\bf{V}}_{E}^{n}},$
(70c)
$\displaystyle{{\bf{W}}_{E}^{n}}{=}\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}^{n}},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{E}}){\rm{=[}}{{\bf{E}}_{E}}({{\bf{U}}_{E}^{n}},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle\qquad\>\>{\rm{=}}[({{\bf{U}}_{E}}^{{n}H}{{\hat{{\bf{H}}}}_{E}}{\bf{V}}_{E}^{n}-{\bf{I}}_{N_{T}}){({{\bf{U}}_{E}^{{n}H}}{{\hat{{\bf{H}}}}_{E}}{\bf{V}}_{E}^{n}-{\bf{I}}_{N_{T}})^{H}}+{{\bf{U}}_{E}^{{n}H}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}){{\bf{U}}_{E}^{n}}]^{-1},$
(70d)
$\displaystyle{{\bf{W}}_{X}^{n}}{\rm{=}}\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{X}}){\rm{=[}}{{\bf{E}}_{X}}({\bf{V}}^{n},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle\qquad\>\>{\rm{=}}[{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{{\bf{H}}}}_{E}}({\bf{V}}^{n}{{\bf{V}}^{nH}}+{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}})\hat{{\bf{H}}}_{E}^{H}}]^{-1}.$
(70e)
The parameters ${\bf{V}}^{n}$, ${{\bf{V}}_{E}^{n}}$, ${\bf{\Phi}}^{n}$ are
obtained by solving the following problem
$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{max}}}\ \
\underset{l=1,\cdots,L}{\min}\\{h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})+h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$
(71a) $\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(71b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(71c)
### IV-C Optimizing the Matrices ${\bf{V}}$ and ${{\bf{V}}_{E}}$
By fixing ${\bf{\Phi}}$, Problem (71) can be written more compactly as
$\displaystyle\ \underset{{\bf{V}},{{\bf{V}}_{E}}}{\mathop{\text{min}}}\ \
\underset{l=1,\cdots,L}{\max}\\{-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{V}}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}{{\bf{U}}_{I,l}})-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V,l}^{i}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE,l}^{i}}{{\mathbf{V}}_{E}})-{C}_{l}\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}{{\mathbf{V}}_{E}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}^{e}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE}^{e}}{{\mathbf{V}}_{E}})$
(72a) $\displaystyle\ \ \ \ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(72b)
where
$\displaystyle{{\mathbf{H}}_{V}^{e}}\text{(}\mathbf{\Phi}\text{)}=\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(73a)
$\displaystyle{{\mathbf{H}}_{VE}^{e}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(73b)
$\displaystyle{{\mathbf{H}}_{V,l}^{i}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{I,l}}^{H}{{\mathbf{U}}_{I,l}}{{\mathbf{W}}_{I,l}}{{\mathbf{U}}_{I,l}^{H}}{{\mathbf{\hat{H}}}_{I,l}},$
(73c)
$\displaystyle{{\mathbf{H}}_{VE,l}^{i}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{I,l}}^{H}{{\mathbf{U}}_{I,l}}{{\mathbf{W}}_{I,l}}{{\mathbf{U}}_{I,l}^{H}}{{\mathbf{\hat{H}}}_{I,l}},$
(73d)
$\displaystyle{C}_{l}=\log\left|{{\bf{W}}_{I,l}}\right|+d-\text{Tr}({{\bf{W}}_{I,l}}+\sigma_{I,l}^{2}{{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\bf{U}}_{I,l}}).$
(73e)
The Problem (72) is a convex QCQP problem, we can obtain its optimal solution
using a general-purpose convex optimization solver.
### IV-D Optimizing the Phase Shifts $\mathbf{\Phi}$
By fixing ${\bf{V}},{{\bf{V}}_{E}}$, the optimization problem for the phase
shift matrix $\mathbf{\Phi}$ reduced from Problem (72) is formulated as
$\displaystyle\ \underset{\mathbf{\Phi}}{\mathop{\text{min}}}\ \
{g_{0}}(\mathbf{\Phi})\triangleq\underset{l=1,\cdots,L}{\max}\\{-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{V}}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}{{\bf{U}}_{I,l}})-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V,l}^{i}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE,l}^{i}}{{\mathbf{V}}_{E}})-{C}_{l}\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}{{\mathbf{V}}_{E}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}^{e}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE}^{e}}{{\mathbf{V}}_{E}})$
(74a) $\displaystyle\ \ \ \ \ \text{s.t.}\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$ (74b)
By complex mathematical manipulations, the OF ${g_{0}}(\mathbf{\Phi})$ can be
equivalently transformed into
$\displaystyle{g_{0}}(\mathbf{\Phi})$
$\displaystyle\triangleq\underset{l=1,\cdots,L}{\max}\\{{g_{0,l}^{i}}(\mathbf{\Phi})\\}+{g_{0}^{e}}(\mathbf{\Phi}),$
(75)
where
$\displaystyle{g_{0,l}^{i}}(\mathbf{\Phi})$
$\displaystyle={\rm{Tr}}\left({{\bf{\Phi}}^{H}{\bf{D}}_{l}^{iH}}\right)+{\rm{Tr}}\left({\bf{\Phi}}{\bf{D}}_{l}^{i}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE,l}^{i}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V,l}^{i}}}\right)+C_{l}^{i},$
(76a) $\displaystyle{g_{0}^{e}}(\mathbf{\Phi})$
$\displaystyle={\rm{Tr}}\left({{\bf{\Phi}}^{H}{\bf{D}}^{eH}}\right)+{\rm{Tr}}\left({\bf{\Phi}}{\bf{D}}^{e}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}^{e}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}^{e}}}\right)+C^{e},$
(76b)
and
$\displaystyle{\bf{D}}_{l}^{i}$
$\displaystyle={\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I,l}^{H}{{\bf{M}}_{I,l}}{{\bf{H}}_{R,I,l}}-{\bf{GV}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}},$
(77a) $\displaystyle{{\bf{C}}_{VE}}$
$\displaystyle={\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}},$ (77b)
$\displaystyle{{\bf{C}}_{V}}$
$\displaystyle={\bf{G}}{\bf{V}}{\bf{V}}^{H}{{\bf{G}}^{H}},$ (77c)
$\displaystyle{{\bf{B}}_{VE,l}^{i}}$
$\displaystyle=\left({{\bf{H}}_{R,I,l}^{H}{{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}}}\right),$
(77d) $\displaystyle{{\bf{B}}_{V,l}^{i}}$
$\displaystyle=\left({{\bf{H}}_{R,I,l}^{H}{{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}}}\right),$
(77e) $\displaystyle C_{l}^{i}$
$\displaystyle={\rm{Tr}}\left[{{\bf{H}}_{b,I,l}}{{\bf{V}}_{X}}{\bf{H}}_{b,I,l}^{H}{{\bf{M}}_{I,l}}\right]\\!+\\!{\rm{Tr}}\left[{{{\bf{U}}_{I,l}}{\bf{W}}_{I,l}^{H}{{\bf{V}}^{H}}{\bf{H}}_{b,I,l}^{H}}\right]\\!+\\!{\rm{Tr}}\left[{{{\bf{H}}_{b,I,l}}{\bf{V}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}}\right]-{C}_{l},$
(77f) $\displaystyle{\bf{M}}_{I,l}$
$\displaystyle={{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H},$ (77g)
$\displaystyle{\bf{D}}^{e}$
$\displaystyle=\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}},$
(77h) $\displaystyle{{\bf{B}}_{VE}^{e}}$
$\displaystyle=\left({\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right),$
(77i) $\displaystyle{{\bf{B}}_{V}^{e}}$
$\displaystyle=\left({\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right),$
(77j) $\displaystyle C^{e}$
$\displaystyle=\sigma_{E}^{-2}{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}\right]+{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}\right]$
$\displaystyle+{\rm{Tr}}\left[{{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}}\right]+{\rm{Tr}}\left[{{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right].$
(77k)
Similarly, Problem (74) can be further simplified as
$\displaystyle\underset{{\bm{\phi}}}{\mathop{\text{min}}}$
$\displaystyle\underset{l=1,\cdots,L}{\max}\\{{{\bm{\phi}}^{{\rm{H}}}}{\bm{\Xi}_{l}^{i}}{\bm{\phi}}+2\textrm{Re}[{\bm{\phi}}^{{\rm{H}}}{\bf{d}}_{l}^{i*}]+C_{l}^{i}\\}+{{\bm{\phi}}^{{\rm{H}}}}{\bm{\Xi}^{e}}{\bm{\phi}}+2\textrm{Re}[{\bm{\phi}}^{{\rm{H}}}{\bf{d}}^{e*}]+C^{e}$
(78a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (78b)
where
$\displaystyle{\bm{\Xi}_{l}^{i}}$
$\displaystyle={\bf{B}}_{VE,l}^{i}\odot{{\bf{C}}_{VE}^{{\rm{T}}}}+{\bf{B}}_{V,l}^{i}\odot{{\bf{C}}_{V}^{{\rm{T}}}},$
(79a) $\displaystyle{\bm{\Xi}^{e}}$
$\displaystyle={\bf{B}}_{VE}^{e}\odot{{\bf{C}}_{VE}^{{\rm{T}}}}+{\bf{B}}_{V}^{e}\odot{{\bf{C}}_{V}^{{\rm{T}}}},$
(79b) $\displaystyle{\bf{d}}_{l}^{i}$
$\displaystyle={\left[{{{\left[{\bf{D}}_{l}^{i}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}_{l}^{i}\right]}_{M,M}}}\right]^{{\rm{T}}}},$
(79c) $\displaystyle{\bf{d}}^{e}$
$\displaystyle={\left[{{{\left[{\bf{D}}^{e}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}^{e}\right]}_{M,M}}}\right]^{{\rm{T}}}}.$
(79d)
By using the Lemma 1 in [25], Problem (78) will be recast as
$\displaystyle\underset{{\bm{\phi}}}{\mathop{\max}}$
$\displaystyle\underset{l=1,\cdots,L}{\min}\\{2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{i,t}]-C_{q,l}^{i,t}\\}+2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}^{e,t}]-C_{q}^{e,t}$
(80a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (80b)
where
$\displaystyle{\bf{q}}_{l}^{i,t}$
$\displaystyle=\left({{\lambda_{l,{\rm{\max}}}^{i}}{{\bf{I}}_{M}}-{\bm{\Xi}_{l}^{i}}}\right){{\bm{\phi}}^{t}}-{\bf{d}}_{l}^{i*},$
(81a) $\displaystyle C_{q,l}^{i,t}$
$\displaystyle=2M{\lambda_{l,{\rm{\max}}}^{i}}-{\left({{\bm{\phi}}^{t}}\right)^{{\rm{H}}}}\left({\bm{\Xi}_{l}^{i}}\right){{\bm{\phi}}^{t}}+C_{l}^{i},$
(81b) $\displaystyle{\bf{q}}^{e,t}$
$\displaystyle=\left({{\lambda_{{\rm{\max}}}^{e}}{{\bf{I}}_{M}}-{\bm{\Xi}^{e}}}\right){{\bm{\phi}}^{t}}-{\bf{d}}^{e*},$
(81c) $\displaystyle C_{q}^{e,t}$
$\displaystyle=2M{\lambda_{{\rm{\max}}}^{e}}-{\left({{\bm{\phi}}^{t}}\right)^{{\rm{H}}}}\left({\bm{\Xi}^{e}}\right){{\bm{\phi}}^{t}}+C^{e},$
(81d)
and ${\lambda_{l,{\rm{\max}}}^{i}}$ is the the maximum eigenvalue of
${\bm{\Xi}_{l}^{i}}$, and ${\lambda_{{\rm{\max}}}^{e}}$ is the maximum
eigenvalue of ${\bm{\Xi}^{e}}$. By defining
${\bf{q}}_{l}^{ie,t}\triangleq{\bf{q}}_{l}^{i,t}+{\bf{q}}^{e,t}$ and
$C_{q,l}^{ie,t}\triangleq C_{q,l}^{i,t}+C_{q}^{e,t}$, Problem (80) can be
rewritten as
$\displaystyle\underset{{\bm{\phi}}}{\mathop{\max}}$
$\displaystyle\underset{l=1,\cdots,L}{\min}\\{2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\\}$
(82a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (82b)
which is equivalent to the following problem
$\displaystyle\underset{{\bm{\phi}},z}{\mathop{\max}}$ $\displaystyle\quad z$
(83a) s.t. $\displaystyle
2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\geq
z,l=1,\cdots,L,$ (83b)
$\displaystyle\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$ (83c)
We note that the above problem is still non-convex due to the unit-modulus
constraints. To deal with the non-convex constraints, the penalty CCP method
is applied. Following the penalty CCP framework, the constraint (83c) are
firstly equivalently rewritten as $-1\leq\left|{{\phi}_{m}}\right|^{2}\leq
1,m=1,\cdots,M$. The non-convex parts of the resulting constraints are then
linearized by
$\left|{{\phi}_{m}^{\text{(n)}}}\right|^{2}-2\textrm{Re}({{\phi}_{m}^{\text{(*)}}}{{\phi}_{m}^{\text{(n)}}})\leq-1,m=1,\cdots,M$.
We finally have the following convex subproblem of ${\bm{\phi}}$ as
$\displaystyle\underset{{\bm{\phi}},z}{\mathop{\max}}$ $\displaystyle\quad
z-\lambda^{(t)}\left\|\mathbf{b}\right\|_{1}$ (84a) s.t. $\displaystyle\
2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\geq
z,$ (84b)
$\displaystyle\left|{{\phi}_{m}^{\text{(n)}}}\right|^{2}-2\textrm{Re}({{\phi}_{m}^{\text{(*)}}}{{\phi}_{m}^{\text{(n)}}})\leq
b_{m}-1,m=1,\cdots,M,$ (84c) $\displaystyle\left|{{\phi}_{m}}\right|^{2}\leq
1+b_{M+m},m=1,\cdots,M.$ (84d)
where $\mathbf{b}=[b_{1},\cdots,b_{2M}]^{T}$ are slack variables imposed over
the equivalent linear constraints of the unit-modulus constraints, and
$\left\|\mathbf{b}\right\|_{1}$ is the penalty term in the OF.
$\left\|\mathbf{b}\right\|_{1}$ is scaled by the regularization factor
$\lambda^{(t)}$ to control the feasibility of the constraints. The specific
steps of the penalty CCP method can be referred in [52].
## V Simulation Results
In this section, numerical simulations are carried out to evaluate the
assistance of the IRS on the AN-aided MIMO secure communication system. We
focus on the scenario of the standard three-terminal MIMO Guassian wiretap
channel shown in Fig. 2, where there are one BS, one legitimate IR and one
Eve, all with multiple antennas. The distance from the BS to the IRS is
$d_{BR}=50$ m. We assume that the line connecting the IR and Eve is parallel
to the line connecting the BS and the IRS, and that the vertical distance
between them is $d_{v}=2$ m.
Figure 2: The three-terminal MIMO communication scenario in simulation.
The large-scale path loss is modeled as
${\rm{PL=}}{{\rm{P}}{{\rm{L}}_{0}}-10\alpha{{\log}_{10}}\left({\frac{d}{{{d_{0}}}}}\right)}$,
where ${\rm{P}}{{\rm{L}}_{0}}$ is the path loss at the reference distance
$d_{0}=1$ m, $\alpha$ is the path loss exponent, $d$ is the link distance. In
our simulations, we set ${\rm{P}}{{\rm{L}}_{0}}=-30$ dB. The path loss
exponents of the links from BS to Eve, from BS to IR, from IRS to Eve and from
IRS to IR are ${\alpha_{{\rm{BE}}}}=3.5$, ${\alpha_{{\rm{BI}}}}=3.5$,
${\alpha_{{\rm{RE}}}}=2.5$ and ${\alpha_{{\rm{RI}}}}=2.5$ respectively. The
path-loss exponents of the link from BS to IRS is set to be
${\alpha_{{\rm{BR}}}}=2.2$, which means that the IRS is well-located, and the
path loss is negligible in this link.
For the direct channels from the BSs to the Eve and IR, the small-scale fading
is assumed to be Rayleigh fading due to extensive scatters. However, for the
IRS-related channels, the small-scale fading is assumed to be Rician fading.
Specifically, the small-scale channel can be modeled as
$\displaystyle\tilde{\mathbf{H}}$
$\displaystyle=\left(\sqrt{\frac{\beta}{1+\beta}}\tilde{\mathbf{H}}^{LOS}+\sqrt{\frac{1}{1+\beta}}\tilde{\mathbf{H}}^{NLOS}\right),$
(85)
where $\beta$ is the Rican factor, $\tilde{\mathbf{H}}^{LOS}$ denotes the
deterministic line of sight (LoS) component of the IRS-related channel, and
$\tilde{\mathbf{H}}^{NLOS}$ denotes the non-LoS (NLoS) component of the IRS-
related channel, which is modeled as Rayleigh fading. By assuming the antennas
at the BS, IRS, Eve and IR are arranged in a uniform linear array (ULA), the
$\tilde{\mathbf{H}}^{LOS}$ can be modeled as
$\tilde{\mathbf{H}}^{LOS}=\mathbf{a}_{r}\mathbf{a}_{t}^{H}$, where
$\mathbf{a}_{t}$ and $\mathbf{a}_{r}$ are the steering vectors of the
transmitting and receiving arrays respectively. The $\mathbf{a}_{t}$ and
$\mathbf{a}_{r}$ are defined as,
$\displaystyle\mathbf{a}_{t}$
$\displaystyle=\left[\begin{array}[]{cccc}1,&\exp(j2\pi\frac{d_{t}}{\lambda}\sin\varphi_{t}),&\cdots,&\exp(j2\pi\frac{d_{t}}{\lambda}(N_{t}-1)\sin\varphi_{t})\end{array}\right]^{T},$
(86b) $\displaystyle\mathbf{a}_{r}$
$\displaystyle=\left[\begin{array}[]{cccc}1,&\exp(j2\pi\frac{d_{r}}{\lambda}\sin\varphi_{r}),&\cdots,&\exp(j2\pi\frac{d_{r}}{\lambda}(N_{r}-1)\sin\varphi_{r})\end{array}\right]^{T}.$
(86d)
In (86), $\lambda$ is the wavelength; $d_{t}$ and $d_{r}$ are the element
intervals of the transmitting and receiving array; $\varphi_{t}$ and
$\varphi_{r}$ are the angle of departure and the angle of arrival; $N_{t}$ and
$N_{r}$ are the number of antennas/elements at the transmitter and receiver,
respectively. We set $\frac{d_{t}}{\lambda}=\frac{d_{r}}{\lambda}=0.5$, and
$\varphi_{t}=\tan^{-1}(\frac{y_{r}-y_{t}}{x_{r}-x_{t}})$,
$\varphi_{r}=\pi-\varphi_{t}$, where $(x_{t},y_{t})$ is the location of the
transmitter, and $(x_{r},y_{r})$ is the location of the receiver.
If not specified, the simulation parameters are set as follows. The IR’s noise
power and the Eve’s noise power are $\sigma_{I}^{2}=-75$ dBm and
$\sigma_{E}^{2}=-75$ dBm. The numbers of BS antennas, IR antennas, and Eve
antennas are $N_{T}=4$, $N_{I}=2$, and $N_{E}=2$ respectively. There are $d=2$
data streams and $M=50$ IRS reflection elements. The transmit power limit is
$P_{T}=15$ dBm, and the error tolerance is $\varepsilon=10^{-6}$. The
horizontal distance between the BS and the Eve is $d_{BE}=44$ m. The
horizontal distance between the BS and the IR is selected from $d_{BI}=[10\
\text{m},70\ \text{m}]$. The channels are realized 200 times independently to
average the simulation results.
### V-A Convergence Analysis
The convergence performance of the proposed BCD-MM algorithm is investigated.
The iterations of the BCD algorithm are termed as outer-layer iterations,
while the iterations of the MM algorithm are termed as the inner-layer
iterations. Fig. 4 shows three examples of convergence behaviour for $M=$10,
20 and 40 phase shifts of IRS. In Fig. 4, the SR increases versus the
iteration number, and finally reaches a stable value. It is shown that the
algorithm converges quickly, almost with 20 iterations, which demonstrates the
efficiency of the proposed algorithm. Moreover, a larger converged SR value is
reached with a larger $M$, which means that better security can be obtained by
using more IRS elements. However, more IRS elements bring a heavier
computation, which is demonstrated in Fig. 4 in the form of a slower
convergence speed with more phase shifters.
Figure 3: Convergence behaviour of the BCD algorithm.
Figure 4: Convergence behaviour of the MM algorithm.
Specifically, we evaluate the convergence performance of the MM algorithm used
for solving the optimal IRS phase shifts. The inner-layer iterative process of
the MM algorithm in the first iteration of the BCD algorithm is shown in Fig.
4. The SR value increases as the iteration number increases, and finally
converges to a stable value. According with the convergency performance in the
out-layer iteration, similar conclusions can be drawn for the inner-layer
iteration, which is that a higher converged SR value can be obtained with more
phase shifts but at the cost of lower convergence speed. The reason for the
lower convergence speed with larger $M$ value is that more optimization
variables are introduced, which require more computation complexity.
### V-B Performance Evaluation
In this subsection, our proposed algorithm is evaluated by comparing the
simulation results to three schemes of
1. 1.
RandPhase: The phase shifts of the IRS are randomly selected from $[0,2\pi]$.
In this scheme, the MM algorithm is skipped, and only the TPC matrix and AN
covariance matrix are optimized.
2. 2.
No-IRS: Without the IRS, the channel matrices of IRS related links become zero
matrices, which is ${\bf{H}}_{R,I}={\bf{0}}$, ${\bf{H}}_{R,E}={\bf{0}}$ and
${{\bf{G}}}={\bf{0}}$. This scheme results a conventional AN-aided
communication system, and only the TPC matrix and AN covariance matrix need to
be optimized.
3. 3.
BCD-QCQP-SDR: The BCD algorithm is utilized. However, the TPC matrix and the
AN covariance matrix is optimized by tackling Problem (32) as a QCQP problem,
which is solved by the general CVX solvers, e.g. Sedumi or Mosek. The phase
shifts of IRS are optimized by solving Problem (61) with the SDR technique.
#### V-B1 Impact of Transmit Power
Figure 5: Achievable SR versus the transmit power limit.
Figure 6: Achievable SR versus the number of phase shifts $M$.
To evaluate the impact of the transmit power limit $P_{T}$, the average SRs
versus the transmit power limit for various schemes are given in Fig. 6, which
demonstrates that the achieved SRs of three schemes increase as the power
limit $P_{T}$ increases. It is observed that the BCD-MM algorithm
significantly outperforms the other three benchmark schemes over the entire
range of transmit power limits. By comparing the RandPhase scheme to the No-
IRS scheme, we find that the RandPhase scheme is better than the No-IRS scheme
for obtaining higher SR, and that the SR gap increases with the power limit
$P_{T}$. The reason is that, for the RandPhase scheme, the IR is closer to the
IRS than the Eve is, and more signal power from the IRS can be acquired by the
IR than that by the Eve, while for the No-IRS scheme, the IR is further from
the BS than the Eve is, and less signal power from the BS can be acquired by
the IR than that by the Eve. This comparison result signifies that even the
phase shifts of IRS are random, the IRS can enhance the system security. In
comparison to the no-IRS scheme, the SR gain achieved by the proposed
algorithm is very obvious, and increases greatly with the power limit $P_{T}$,
which confirms the effectiveness and benefits of employing the IRS. By
comparing the proposed scheme and the RandPhase scheme, we find that the
security gain obtained for the proposed scheme is much greater than that for
the RandPhase scheme. That’s because the phase shifts of IRS are properly
designed to enhance the signal received at the IR more constructively, and
weaken the signal received at the Eve more destructively. This comparison
result emphasizes that optimizing the phase shifts of IRS is important and
necessary. By comparing the proposed BCD-MM algorithm and the BCD-QCQP-SDR
algorithm, we observe that in terms of the SR performance, the proposed BCD-MM
algorithm is better BCD-QCQP-SDR, and the performance gain increases with the
$P_{T}$. Moreover, the proposed BCD-MM algorithm is much more efficient than
the BCD-QCQP-SDR algorithm. The superiority of the proposed algorithm is
further validated.
#### V-B2 Impact of the Phase Shifts Number
The averaged SR performance of four schemes with various phase shifts number
$M$ is shown in Fig. 6, which demonstrates that the proposed BCD-MM algorithm
is significantly superior to the other three schemes. We observe that the SR
achieved by the BCD-MM scheme obviously increases with $M$, while the
RandPhase scheme only shows a slight improvement as $M$ increases, and the No-
IRS scheme has very low SRs irrelative with $M$. Larger the element number $M$
of IRS is, more significant the performance gain obtained by the proposed
algorithm is. For example, when $M$ is small as $M=10$, the SR gain of the
BCD-MM relative to the No-IRS is only 1.3 bit/s/Hz, while this SR gain becomes
9.5 bit/s/Hz when $M$ increases to $M=100$. The performance gain for the
proposed algorithm originates from two perspectives. On the one hand, a higher
array gain can be obtained by increasing $M$, since more signal power can be
received at the IRS with larger $M$. On the other hand, a higher reflecting
beamforming gain can be obtained by increasing $M$, which means that the sum
of coherently adding the reflected signals at the IRS elements increases with
$M$ by appropriately designing the phase shifts. However, only the array gain
can be exploited by the RandPhase scheme, thus the SRs for it increase very
slowly, and remain at much lower values than those for the proposed algorithm.
These results further confirm that more security improvements can be archived
by using a large IRS with more reflect elements and optimizing the phase
shifts properly, however there may bring the computation complexity problem.
In comparison to the BCD-QCQP-SDR algorithm, the proposed BCD-MM algorithm can
achieve the higher SR, and the SR performance gap increases with $M$.
#### V-B3 Impact of the relative location of IRS
Figure 7: Achievable SR versus the location of the IR $d_{\rm{BI}}$.
Figure 8: Achievable SR versus the path loss exponent of IRS-related links.
Fig. 8 illustrates the achieved SRs for four schemes with various BS-IR
horizontal distance $d_{BI}$, where the BS-Eve distance is fixed to be
$d_{BE}=44$ m. It is observed that the proposed BCD-MM algorithm is the best
among the four schemes for obtaining the highest SR value. When the IR moves
far away from the BS, the SRs decrease for the four schemes, however, the SRs
achieved for the RandPhase, the proposed BCD-MM algorithm and the BCD-QCQP-SDR
algorithm increase greatly when the IR approaches the IRS. The achieved SRs at
different BS-IR distances of the RandPhase scheme and the no-IRS scheme are
almost the same, except for $d_{BI}\in[40\text{m},50\text{m}]$, in which case
the IRS brings a prominent security enhancement when IR becomes close to it
even with random IRS phase shifts. Similarly, the proposed BCD-MM algorithm
and the BCD-QCQP-SDR algorithm can achieve almost the same SRs, except for
$d_{BI}\in[40\text{m},50\text{m}]$, in which case the IR is close to the IRS,
and the proposed BCD-MM algorithm is superior to the BCD-QCQP-SDR algorithm.
For other BS-IR distances where the IR is far from the IRS, the SRs of
RandPhase scheme are similar with those of the No-IRS scheme due to the not
fully explored potential of IRS. By optimizing the phase shifts of IRS, the
SRs are enhanced at different BS-IRS distances. And the SR gain of the
proposed BCD-MM algorithm over the RandPhase scheme increases when the IR
moves close to the IRS ($d_{BI}\in[40\text{m},50\text{m}]$). This signifies
that as long as the IRS is deployed close to the IR, significant security
enhancement can be achieved by the IRS in an AN-aided MIMO communication
system. Moreover, it is highly recommended that the IRS phase shifts should be
optimized to prevent the system security degrading into the level of No-IRS
scheme.
#### V-B4 Impact of the Path Loss Exponent of IRS-related Links
In the above simulations, the path loss exponents of the IRS-related links
(including the BS-IRS link, IRS-IR link and IRS-Eve link) are set to be low by
assuming that the IRS is properly located to obtain clean channels without
heavy blockage. Practically, such kind of settings may not always be sensible
due to the real-field environment. Thus, it is necessary to investigate the
security gain brought by the IRS and our proposed algorithm with higher values
of IRS-related path loss exponents. For the sake of analysis, we assume the
path-loss exponents of the links from BS to IRS, from IRS to IR and from IRS
to Eve are the same as
${\alpha_{{\rm{BR}}}}={\alpha_{{\rm{RI}}}}={\alpha_{{\rm{RE}}}}\buildrel\Delta\over{=}{\alpha_{{\rm{IRS}}}}$.
Then, the achieved SRs versus the path-loss exponent ${\alpha_{{\rm{IRS}}}}$
of IRS-related links are shown in Fig. 8, which demonstrates that the SR
obtained by the BCD-MM algorithm decreases as ${\alpha_{{\rm{IRS}}}}$
increases, and finally drops to the same SR value which is achieved by the
RandPhase, BCD-QCQP-SDR and No-IRS schemes. The reason is that larger
${\alpha_{{\rm{IRS}}}}$ means more severe signal attenuation in the IRS-
related links, and more weakened signal received and reflected at the IRS. In
comparison to the BCD-QCQP-SDR algorithm, the proposed BCD-MM algorithm can
achieve the higher SR when the channel state of the IRS related channels is
good, i.e., ${\alpha_{{\rm{IRS}}}}$ is low, and achieve almost the same SR
when ${\alpha_{{\rm{IRS}}}}$ is large. Similarly, the performance gains
brought by our proposed algorithm over the RandPhase and No-IRS schemes is
significant with a small ${\alpha_{{\rm{IRS}}}}$. Specifically, for
${\alpha_{{\rm{IRS}}}}=2$ (almost ideal channels), the security gain is up to
9.6 bit/s/Hz over the No-IRS scheme, and 6.8 bit/s/Hz over the RandPhase
scheme. Therefore, the security gain of IRS-assisted systems depends on the
channel conditions of the IRS-related links. This suggests that it is much
preferred to deploy the IRS with fewer obstacles, in which case, the
performance gain brought by the IRS can be explored thoroughly. Fig. 8 also
shows that when ${\alpha_{{\rm{IRS}}}}$ is small, the RandPhase scheme can
obtain security gain over the No-IRS scheme, but this security gain decreases
to zero when ${\alpha_{{\rm{IRS}}}}$ becomes large. However, the SR gain of
the RandPhase scheme over the No-IRS scheme is almost negligible in comparison
to the SR gain of the proposed scheme over the No-IRS scheme, which
demonstrates that the necessity of jointly optimizing the TPC matrix, AN
covariance matrix and the phase shifts at the IRS.
#### V-B5 Impact of the Number of Data Streams
Compared with the MISO scenario, a significant advantage of the MIMO scenario
is that multiple data streams can be transmitted to the users. To evaluate the
impact of the number of data streams on the SR, the average SRs versus the
transmit power limit for various numbers of data streams are given in Fig. 10.
Figure 9: Achievable SR versus the transmit power limit for various numbers of
data streams.
Figure 10: Achievable SR versus the reflection amplitude $\eta$.
The number of transmit antennas is $N_{T}=4$. The Rician fading channels are
utilized. The path loss exponents are ${\alpha_{{\rm{BR}}}}=2.2$,
${\alpha_{{\rm{BE}}}}=3.5$, ${\alpha_{{\rm{BI}}}}=2.5$,
${\alpha_{{\rm{RE}}}}=3.5$ and ${\alpha_{{\rm{RI}}}}=2.5$ respectively. The
Rician factor is $\beta=3$. The number of phase shifts is $M=50$. As shown in
Fig. 10, the SR increases with the transmit power limit and larger number of
data streams result the higher SR. When the transmit power limit is low,
marginal performance gains are achieved by increasing the number $d$ of data
streams. When the transmit power limit is high, significant performance gains
can be achieved by increasing the number $d$ of data streams. This means that
a greater number of data streams ensure the higher SR, and the performance
gain expands with the transmit power limit. For the case of $d=1$, the SR
performance of $N_{I}=N_{E}=4$ and $N_{I}=N_{E}=1$ is further compared. It is
revealed that the SR obtained by four receiving antennas is higher than the SR
obtained by one single receiving antenna when the transmit power limit is
relatively low. With the increase of transmit power limit, the SR performance
gain brought by multiple receiving antennas decreases. When the transmit power
limit is high enough, the SR performance is saturated, and the SR performance
of the multiple receiving antennas and single receiving antenna becomes the
same.
#### V-B6 Impact of the Reflection Amplitude
Due to the manufactural and hardware reasons, the signals reflected by the IRS
may be attenuated. Then, in Fig. 10, we study the impact of the reflection
amplitude on the security performance. The transmit power limit is 10dBm. We
assume that the reflection amplitudes of all the IRS elements are same as
$\eta$, and that the phase shift matrix of the IRS is rewritten as
${\bf{\Phi}}=\eta\text{diag }\\!\\!\\{\\!\\!\text{
}{{\phi}_{1}},\cdots,{{\phi}_{m}},\cdots,{{\phi}_{M}}\text{
}\\!\\!\\}\\!\\!\text{ }$. As expected, the SR achieved by the IRS-aided
scheme increases with $\eta$ due to less power loss. As $\eta$ increases, the
superiority of the proposed BCD-MM algorithm over the other algorithms becomes
more obvious. The reflection amplitude has a great impact on the security
performance. Specifically, when $\eta$ increases from 0.2 to 1, the SR
increases over 3.6 bit/s/Hz for the proposed BCD-MM algorithm.
#### V-B7 Impact of the Discrete Phase Shifts
In practice, it is difficult to realize continuous phase shifts at the
reflecting elements of the IRS due to the high manufacturing cost. It is more
cost-effective to implement only discrete phase shifts with a small number of
control bits for each element, e.g., 1-bit for two-level (0 or $\pi$) phase
shifts. Thus, the impact of the controlling bits $b$ of the discrete phase
shifts on the security performance is investigated in Fig. 12. The transmit
power limit is 10dBm. It is shown that the SR with continuous phase shifts of
the IRS is higher than those with discrete phase shifts. The limited discrete
phase shifts inevitably cause SR performance degradation. The SR of the IRS
with discrete phase shifts increases with the number of controlling bits $b$,
and becomes saturated when $b\geq 4$, which means that the SR loss is
inevitable even when the number of controlling bits $b$ is high. For the
proposed BCD-MM algorithm, the maximum SR gap between the continuous phase
shifts and the discrete phase shifts is 1.4 bit/s/Hz.
Figure 11: Achievable SR versus the discrete phase bits $b$.
Figure 12: Achievable SR versus the transmit power limit for multiple IRs.
#### V-B8 Multiple IRs Scenario
Finally, we consider the multiple IRs scenario to investigate the security
enhancement brought by the IRS on the AN-aided MIMO communication systems. The
horizontal distances between the BS and the two IRs are selected as
$d_{BI,1}=47$m and $d_{BI,2}=49$m. Considering the heavy computational load,
the element number of the IRS is assumed to be $M=20$. The proposed BCD-QCQP-
CCP algorithm is utilized to perform the joint optimization of the TPC matrix,
AN covariance matrix and the phase shifts of the IRS. The achieved SRs for the
proposed algorithm, the random IRS scheme and the No-IRS scheme are shown in
Fig. 12. By comparing with the Random IRS scheme and the No-IRS scheme, the
proposed BCD-QCQP-CCP algorithm can optimize the phase shifts of the IRS, thus
achieve the higher SR. The SR gain increases with the power limit $P_{T}$.
However, the performance gain is not as much as that in Fig. 6. On the one
hand, the element number of the IRS is set to be lower than that in Fig. 6 due
to the heavy computational load. On the other hand, it is more difficult for
the IRS to adjust the phase shifts to guarantee higher SRs for more legitimate
IRs.
## VI Conclusions
In this paper, we propose to enhance the security of AN-aided MIMO secure
communication systems by exploiting an IRS. To exploit the IRS efficiently, we
formulate a SRM problem by jointly optimizing the TPC matrix at the BS, the
covariance matrix of AN and phase shifts at the IRS with the constraints of
transmit power limit and unit-modulus of phase shifts. To solve this non-
convex problem, we propose to use the BCD algorithm to decouple the
optimization variables, and optimize them iteratively. The optimal TPC matrix
and AN covariance matrix were obtained in semi-closed form by the Lagrange
multiplier method, and the phase shifts at the IRS were obtained in closed
form by an efficient MM algorithm. Various simulations validated that
significant security gains can be achieved by the proposed algorithm with IRS.
Furthermore, useful suggestions for choosing and deploying the IRS are
provided.
## Appendix A Derivation of the Problem (29)
By substituting $h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$
of (III-A), $h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ of (III-A)
and $h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ of (III-A) into (III-A),
we have
$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=\log\left|{{{\bf{W}}_{I}}}\right|-{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))+\log\left|{{{\bf{W}}_{E}}}\right|$
$\displaystyle\qquad\qquad\qquad-{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))+\log\left|{{{\bf{W}}_{X}}}\right|-{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))+d+N_{t}+N_{E}$
$\displaystyle\qquad=C_{g_{0}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))}_{g_{1}}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))}_{g_{2}}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))}_{g_{3}}},$
(87)
where
$C_{g_{0}}\buildrel\Delta\over{=}\log\left|{{{\bf{W}}_{I}}}\right|+\log\left|{{{\bf{W}}_{E}}}\right|+\log\left|{{{\bf{W}}_{X}}}\right|+d+N_{t}+N_{E}$.
$C_{g_{0}}$ contains the constant terms irrelated with
${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$. By putting matrix functions
${{\bf{E}}_{I}}$, ${{\bf{E}}_{E}}$ and ${{\bf{E}}_{X}}$ into (A), we expand
${g_{1}}$, ${g_{2}}$, and ${g_{3}}$ respectively as follows.
(1) ${{g}_{1}}$ can be reformulated as
$\displaystyle{{g}_{1}}$
$\displaystyle=\text{Tr}({{\bf{W}}_{I}}[({\bf{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}){{({\bf{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})}^{H}}+{{\bf{U}}_{I}}^{H}({{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}){{\bf{U}}_{I}}])$
$\displaystyle=\text{Tr}({{\bf{W}}_{I}}[({\bf{I}}-{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}+{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}})$
$\displaystyle\quad+({{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}\text{+}{{\bf{U}}_{I}}^{H}\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}{{\bf{U}}_{I}})]).$
(88)
By gathering the constant terms related with ${{\bf{W}}_{I}},{{\bf{U}}_{I}}$
in ${{C}}_{g_{1}}$, ${{g}_{1}}$ can be simplified as
$\displaystyle{{g}_{1}}$
$\displaystyle=-\text{Tr}({{\bf{W}}_{I}}{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}})-\text{Tr}({{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})+\text{Tr}({{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})$
$\displaystyle\quad+\text{Tr}({{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}})+{{C}}_{g_{1}},$
(89)
where
${{C}}_{g_{1}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{I}}+\sigma_{I}^{2}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{\bf{U}}_{I}})$.
(2) ${{g}_{2}}$ can be reformulated as
$\displaystyle{{g}_{2}}$
$\displaystyle=\text{Tr}({{\mathbf{W}}_{E}}[(\mathbf{I}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}){{(\mathbf{I}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})}^{H}}+\sigma_{E}^{2}{{\mathbf{U}}_{E}}^{H}{{\mathbf{U}}_{E}}])$
$\displaystyle=\text{Tr}({{\mathbf{W}}_{E}}[(\mathbf{I}-{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}+{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}$
$\displaystyle\quad+\sigma_{E}^{2}{{\mathbf{U}}_{E}}^{H}{{\mathbf{U}}_{E}}]).$
(90)
By gathering the constant terms related with
${{\mathbf{W}}_{E}},{{\mathbf{U}}_{E}}$ in ${{C}}_{g_{2}}$, ${{g}_{2}}$ can be
simplified as
$\displaystyle{{g}_{2}}=-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+{{C}}_{g_{2}},$
(91)
where
${{C}}_{g_{2}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{E}}+\sigma_{E}^{2}{{\bf{W}}_{E}}{{\bf{U}}_{E}}^{H}{{\bf{U}}_{E}})$.
(3) ${{g}_{3}}$ can be reformulated as
$\displaystyle{{g}_{3}}=\text{Tr}({{\mathbf{W}}_{X}}({{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H})\hat{\bf{H}}_{E}^{H}})).$
(92)
By gathering the constant terms related with ${{\mathbf{W}}_{X}}$ in
${{C}}_{g_{3}}$, ${{g}_{3}}$ can be simplified as
$\displaystyle{{g}_{3}}=\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}^{H}}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}\mathbf{V})+\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+{{C}}_{g_{3}},$
(93)
where ${{C}}_{g_{3}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{X}})$.
By substituting (89), (91) and (93) into (A), we have
$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})$
$\displaystyle-\text{Tr}({{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})$
$\displaystyle+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}}_{E}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}\mathbf{V})$
$\displaystyle-\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+C_{g},$
(94)
where $C_{g}\buildrel\Delta\over{=}C_{g_{0}}-C_{g_{1}}-C_{g_{2}}-C_{g_{3}}$.
Equation (A) can be rewritten more compactly as
$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=C_{g}+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})$
$\displaystyle\quad-\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}}).$
(95)
where
$\displaystyle{{\mathbf{H}}_{V}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{\mathbf{\hat{H}}}_{I}}+\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(96)
$\displaystyle{{\mathbf{H}}_{VE}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{\mathbf{\hat{H}}}_{I}}+{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(97)
By substituting (95) into Problem (28), and removing the constant term
$C_{g}$, we arrive at the Problem (29).
## Appendix B Derivation of the new OF form in (57)
The objective function of Problem (56) is
$\displaystyle{g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})=$
$\displaystyle-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}}).$
(98)
The third term of (B) is
$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)$
$\displaystyle={\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}+\sigma_{E}^{-2}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}{{{\bf{\hat{H}}}}_{E}}}\right){\bf{V}}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right].$
(99)
The sixth term of (B) is
$\displaystyle{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)=$
$\displaystyle{\rm{Tr}}\left[{{\bf{V}}_{E}^{H}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}+{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}+\sigma_{E}^{-2}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}{{{\bf{\hat{H}}}}_{E}}}\right){{\bf{V}}_{E}}}\right]$
$\displaystyle=$
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]+{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]$
$\displaystyle+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right].$
(100)
The summation of Equation (B) and Equation (B) is
$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)=$
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right]$
$\displaystyle+{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right].$
(101)
By defining
${{\bf{V}}_{X}}=\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right)$
and ${{\bf{M}}_{I}}={{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}$, the first
part of (B) can be derived as
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{{\bf{V}}_{X}}{\bf{\hat{H}}}_{I}^{H}{{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}}\right){{\bf{V}}_{X}}\left({{\bf{H}}_{b,I}^{H}+{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}}\right){{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}+{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}}\right){{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}[{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}+{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}$
$\displaystyle\quad+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}].$
(102)
The derivation in (B) can be used for the second and third parts of (B).
Based on the derivation in (B), it is obvious that the second part of (B) can
be derived as
$\displaystyle\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right]$
$\displaystyle=\sigma_{E}^{-2}{\rm{Tr}}[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}+{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}$
$\displaystyle\quad+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}].$
(103)
Based on the derivation in (B) and by defining
${{\bf{M}}_{E}}={{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}$, it is obvious
that the third part of (B) can be derived as
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{M}}_{E}}}\right]$
$\displaystyle={\rm{Tr}}[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}+{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}$
$\displaystyle\quad+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}].$
(104)
By adding (B), (B) and (B), and gathering constant terms irreverent with
${\bf{\Phi}}$, Equation (B) becomes
$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left({{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right]+C_{{t}_{1}},$
(105)
where
$\displaystyle
C_{{t}_{1}}={\rm{Tr}}\left[{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}\right]+\sigma_{E}^{-2}{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}\right]+{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}\right].$
(106)
The first term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as
$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}}\right)\\!=\\!{\rm{Tr}}\left({{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}}\right)\\!=\\!\underbrace{{\rm{Tr}}\left[{{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{\bf{H}}_{b,I}^{H}}\right]}_{C_{{t}_{2}}(\text{constant
for
}\mathbf{\Phi})}\\!+\\!{\rm{Tr}}\left[{{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}}\right].$
(107)
The second term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as
$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)={\rm{Tr}}\left({{{{\bf{\hat{H}}}}_{I}}{\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right)={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}}\right){\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{H}}_{b,I}}{\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]}_{C_{{t}_{3}}(\text{constant
for }\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}}\right].$ (108)
The fourth term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as
$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)={\rm{Tr}}\left({{{\bf{U}}_{E}}{{\bf{W}}_{E}^{H}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}}\right)$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}}\right]}_{C_{{t}_{4}}(\text{constant
for
}\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}}\right].$
(109)
The fifth term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as
$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)={\rm{Tr}}\left({{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right)$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]}_{C_{{t}_{5}}(\text{constant
for }\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right].$
(110)
By including the first term in (107), the second term in (B), the fourth term
in (B), the fifth term in (B), and the sum of the third and six terms in (B)
of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ and gathering
constant terms irreverent with ${\bf{\Phi}}$, we have
$\displaystyle{g_{0}}(\mathbf{\Phi})=-\rm{Equation\
}\eqref{firstoffirstconst}-\rm{Equation\
}\eqref{secondoffirstconst}-\rm{Equation\
}\eqref{fourthoffirstconst}-\rm{Equation\
}\eqref{fifthoffirstconst}+\rm{Equation\ }\eqref{third_sixoffirstconst_form2}$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left(\begin{array}[]{l}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\\\
-{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}-{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\end{array}\right)}\right]$
(112)
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left(\begin{array}[]{l}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}\\\
-{\bf{GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}\end{array}\right)}\right]$
(114) $\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]+C_{t}$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left(\begin{array}[]{l}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\\\
-{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}-{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\end{array}\right)}\right]$
(116)
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left(\begin{array}[]{l}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}\\\
-{\bf{GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}\end{array}\right)}\right]$
(118) $\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]+C_{t},$
(119)
where
$\displaystyle
C_{t}=C_{{t}_{1}}+C_{{t}_{2}}+C_{{t}_{3}}+C_{{t}_{4}}+C_{{t}_{5}}.$ (120)
Then ${g_{0}}(\mathbf{\Phi})$ becomes
$\displaystyle{g_{0}}(\mathbf{\Phi})={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}}\right)+C_{t}$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}{\bf{\Phi}}{{\bf{C}}_{VE}}}\right]+{\rm{Tr}}\left({{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}{\bf{\Phi}}{{\bf{C}}_{V}}}\right)+C_{t},$
(121)
where
|
* Bédard et al. [2024] Bédard A., Blouin S., Cheng S. (2024): Buoyant crystals halt the cooling of white dwarf stars. Nature 627(8003):286–288. 10.1038/s41586-024-07102-y
* Benvenuto & Althaus [1997] Benvenuto O. G., Althaus L. G. (1997): DB white dwarf evolution in the frame of the full spectrum turbulence theory. MNRAS 288(4):1004–1014. 10.1093/mnras/288.4.1004
* Bergeron & Liebert [2002] Bergeron P., Liebert J. (2002): Spectroscopic Analysis of the DAB White Dwarf PG 1115+166: An Unresolved DA+DB Degenerate Binary. ApJ 566(2):1091–1094. 10.1086/338279, arXiv:astro-ph/0110549 [astro-ph]
* Bergeron et al. [1992] Bergeron P., Saffer R. A., Liebert J. (1992): A Spectroscopic Determination of the Mass Distribution of DA White Dwarfs. ApJ 394:228. 10.1086/171575
* Bergeron et al. [1993] Bergeron P., Wesemael F., Lamontagne R., et al. (1993): Metal-Line Blanketing and the Peculiar H beta Line Profile in the DAO Star Feige 55. ApJ 407:L85. 10.1086/186812
* Bergeron et al. [1994] Bergeron P., Wesemael F., Beauchamp A., et al. (1994): A Spectroscopic Analysis of DAO and Hot DA White Dwarfs: The Implications of the Presence of Helium and the Nature of DAO Stars. ApJ 432:305. 10.1086/174571
* Bergeron et al. [1997] Bergeron P., Ruiz M. T., Leggett S. K. (1997): The Chemical Evolution of Cool White Dwarfs and the Age of the Local Galactic Disk. ApJS 108(1):339–387. 10.1086/312955
* Bergeron et al. [2001] Bergeron P., Leggett S. K., Ruiz M. T. (2001): Photometric and Spectroscopic Analysis of Cool White Dwarfs with Trigonometric Parallax Measurements. ApJS 133(2):413–449. 10.1086/320356, arXiv:astro-ph/0011286 [astro-ph]
* Bergeron et al. [2011] Bergeron P., Wesemael F., Dufour P., et al. (2011): A Comprehensive Spectroscopic Analysis of DB White Dwarfs. ApJ 737(1):28. 10.1088/0004-637X/737/1/28, arXiv:1105.5433 [astro-ph.SR]
* Bergeron et al. [2019] Bergeron P., Dufour P., Fontaine G., et al. (2019): On the Measurement of Fundamental Parameters of White Dwarfs in the Gaia Era. ApJ 876(1):67. 10.3847/1538-4357/ab153a, arXiv:1904.02022 [astro-ph.SR]
* Bergeron et al. [2022] Bergeron P., Kilic M., Blouin S., et al. (2022): On the Nature of Ultracool White Dwarfs: Not so Cool after All. ApJ 934(1):36. 10.3847/1538-4357/ac76c7, arXiv:2206.03174 [astro-ph.SR]
* Blöcker [2001] Blöcker T. (2001): Evolution on the AGB and beyond: on the formation of H-deficient post-AGB stars. Ap&SS 275:1–14. 10.1023/A:1002777931450, arXiv:astro-ph/0102135 [astro-ph]
* Blouin [2022] Blouin S. (2022): Missing metals in DQ stars: A simple explanation. A&A 666:L7. 10.1051/0004-6361/202244944, arXiv:2209.11626 [astro-ph.SR]
* Blouin & Dufour [2019] Blouin S., Dufour P. (2019): The evolution of carbon-polluted white dwarfs at low effective temperatures. MNRAS 490(3):4166–4174. 10.1093/mnras/stz2915, arXiv:1910.06168 [astro-ph.SR]
* Blouin & Xu [2022] Blouin S., Xu S. (2022): No evidence for a strong decrease of planetesimal accretion in old white dwarfs. MNRAS 510(1):1059–1067. 10.1093/mnras/stab3446, arXiv:2111.12152 [astro-ph.SR]
* Blouin et al. [2019a] Blouin S., Dufour P., Allard N. F., et al. (2019a): A New Generation of Cool White Dwarf Atmosphere Models. III. WD J2356-209: Accretion of a Planetesimal with an Unusual Composition. ApJ 872(2):188. 10.3847/1538-4357/ab0081, arXiv:1902.03219 [astro-ph.SR]
* Blouin et al. [2019b] Blouin S., Dufour P., Thibeault C., et al. (2019b): A New Generation of Cool White Dwarf Atmosphere Models. IV. Revisiting the Spectral Evolution of Cool White Dwarfs. ApJ 878(1):63. 10.3847/1538-4357/ab1f82, arXiv:1905.02174 [astro-ph.SR]
* Blouin et al. [2021] Blouin S., Daligault J., Saumon D. (2021): 22Ne Phase Separation as a Solution to the Ultramassive White Dwarf Cooling Anomaly. ApJ 911(1):L5. 10.3847/2041-8213/abf14b, arXiv:2103.12892 [astro-ph.SR]
* Blouin et al. [2023a] Blouin S., Bédard A., Tremblay P.-E. (2023a): Carbon dredge-up required to explain the Gaia white dwarf colour-magnitude bifurcation. MNRAS 523(3):3363–3375. 10.1093/mnras/stad1574, arXiv:2305.02827 [astro-ph.SR]
* Blouin et al. [2023b] Blouin S., Kilic M., Bédard A., et al. (2023b): The ubiquity of carbon dredge-up in hydrogen-deficient white dwarfs as revealed by GALEX. MNRAS 525(1):L112–L116. 10.1093/mnrasl/slad105, arXiv:2307.14295 [astro-ph.SR]
* Bonsor et al. [2020] Bonsor A., Carter P. J., Hollands M., et al. (2020): Are exoplanetesimals differentiated? MNRAS 492(2):2683–2697. 10.1093/mnras/stz3603, arXiv:2001.04499 [astro-ph.EP]
* Brassard et al. [2007] Brassard P., Fontaine G., Dufour P., et al. (2007): The Origin and Evolution of DQ White Dwarfs: The Carbon Pollution Problem Revisited. In: Napiwotzki R., Burleigh M. R. (eds.) ASP Conf. Ser. 372: 15th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 19
* Buchan et al. [2022] Buchan A. M., Bonsor A., Shorttle O., et al. (2022): Planets or asteroids? A geochemical method to constrain the masses of White Dwarf pollutants. MNRAS 510(3):3512–3530. 10.1093/mnras/stab3624, arXiv:2111.08779 [astro-ph.EP]
* Caiazzo et al. [2023] Caiazzo I., Burdge K. B., Tremblay P.-E., et al. (2023): A rotating white dwarf shows different compositions on its opposite faces. Nature 620(7972):61–66. 10.1038/s41586-023-06171-9, arXiv:2308.07430 [astro-ph.SR]
* Camisassa et al. [2023] Camisassa M., Torres S., Hollands M., et al. (2023): A hidden population of white dwarfs with atmospheric carbon traces in the Gaia bifurcation. A&A 674:A213. 10.1051/0004-6361/202346628, arXiv:2305.02110 [astro-ph.SR]
* Camisassa et al. [2016] Camisassa M. E., Althaus L. G., Córsico A. H., et al. (2016): The Effect of 22NE Diffusion in the Evolution and Pulsational Properties of White Dwarfs with Solar Metallicity Progenitors. ApJ 823(2):158. 10.3847/0004-637X/823/2/158, arXiv:1604.01744 [astro-ph.SR]
* Camisassa et al. [2017] Camisassa M. E., Althaus L. G., Rohrmann R. D., et al. (2017): Updated Evolutionary Sequences for Hydrogen-deficient White Dwarfs. ApJ 839(1):11. 10.3847/1538-4357/aa6797, arXiv:1703.05340 [astro-ph.SR]
* Caron et al. [2023] Caron A., Bergeron P., Blouin S., et al. (2023): A spectrophotometric analysis of cool white dwarfs in the Gaia and pan-STARRS footprint. MNRAS 519(3):4529–4549. 10.1093/mnras/stac3733, arXiv:2212.08014 [astro-ph.SR]
* Cenarro et al. [2019] Cenarro A. J., Moles M., Cristóbal-Hornillos D., et al. (2019): J-PLUS: The Javalambre Photometric Local Universe Survey. A&A 622:A176. 10.1051/0004-6361/201833036, arXiv:1804.02667 [astro-ph.GA]
* Chambers et al. [2016] Chambers K. C., Magnier E. A., Metcalfe N., et al. (2016): The Pan-STARRS1 Surveys. arXiv e-prints arXiv:1612.05560. 10.48550/arXiv.1612.05560, arXiv:1612.05560 [astro-ph.IM]
* Chayer [2014] Chayer P. (2014): Radiative levitation of silicon in the atmospheres of two Hyades DA white dwarfs. MNRAS 437(1):L95–L99. 10.1093/mnrasl/slt149, arXiv:1310.6245 [astro-ph.SR]
* Chayer et al. [1995a] Chayer P., Fontaine G., Wesemael F. (1995a): Radiative Levitation in Hot White Dwarfs: Equilibrium Theory. ApJS 99:189. 10.1086/192184
* Chayer et al. [1995b] Chayer P., Vennes S., Pradhan A. K., et al. (1995b): Improved Calculations of the Equilibrium Abundances of Heavy Elements Supported by Radiative Levitation in the Atmospheres of Hot DA White Dwarfs. ApJ 454:429. 10.1086/176494
* Chayer et al. [2005] Chayer P., Vennes S., Dupuis J., et al. (2005): Abundance of Elements beyond the Iron Group in Cool DO White Dwarfs. ApJ 630(2):L169–L172. 10.1086/491699
* Chayer et al. [2023] Chayer P., Mendoza C., Meléndez M., et al. (2023): Detection of cesium in the atmosphere of the hot He-rich white dwarf HD 149499B. MNRAS 518(1):368–381. 10.1093/mnras/stac3138, arXiv:2211.01868 [astro-ph.SR]
* Chen & Hansen [2011] Chen E. Y., Hansen B. M. S. (2011): Cooling curves and chemical evolution curves of convective mixing white dwarf stars. MNRAS 413(4):2827–2837. 10.1111/j.1365-2966.2011.18355.x
* Chen & Hansen [2012] Chen E. Y., Hansen B. M. S. (2012): The Spectral Evolution of Convective Mixing White Dwarfs, the Non-DA Gap, and White Dwarf Cosmochronology. ApJ 753(1):L16. 10.1088/2041-8205/753/1/L16, arXiv:1205.7068 [astro-ph.SR]
* Chen et al. [2021] Chen J., Ferraro F. R., Cadelano M., et al. (2021): Slowly cooling white dwarfs in M13 from stable hydrogen burning. Nature Astronomy 5:1170–1177. 10.1038/s41550-021-01445-6, arXiv:2109.02306 [astro-ph.GA]
* Cheng et al. [2019] Cheng S., Cummings J. D., Ménard B. (2019): A Cooling Anomaly of High-mass White Dwarfs. ApJ 886(2):100. 10.3847/1538-4357/ab4989, arXiv:1905.12710 [astro-ph.SR]
* Coutu et al. [2019] Coutu S., Dufour P., Bergeron P., et al. (2019): Analysis of Helium-rich White Dwarfs Polluted by Heavy Elements in the Gaia Era. ApJ 885(1):74. 10.3847/1538-4357/ab46b9, arXiv:1907.05932 [astro-ph.SR]
* Cukanovaite et al. [2018] Cukanovaite E., Tremblay P. E., Freytag B., et al. (2018): Pure-helium 3D model atmospheres of white dwarfs. MNRAS 481(2):1522–1537. 10.1093/mnras/sty2383, arXiv:1809.00590 [astro-ph.SR]
* Cukanovaite et al. [2019] Cukanovaite E., Tremblay P. E., Freytag B., et al. (2019): Calibration of the mixing-length theory for structures of helium-dominated atmosphere white dwarfs. MNRAS 490(1):1010–1025. 10.1093/mnras/stz2656, arXiv:1909.10532 [astro-ph.SR]
* Cukanovaite et al. [2021] Cukanovaite E., Tremblay P.-E., Bergeron P., et al. (2021): 3D spectroscopic analysis of helium-line white dwarfs. MNRAS 501(4):5274–5293. 10.1093/mnras/staa3684, arXiv:2011.12693 [astro-ph.SR]
* Cukanovaite et al. [2023] Cukanovaite E., Tremblay P. E., Toonen S., et al. (2023): Local stellar formation history from the 40 pc white dwarf sample. MNRAS 522(2):1643–1661. 10.1093/mnras/stad1020, arXiv:2209.13919 [astro-ph.SR]
* Cummings et al. [2018] Cummings J. D., Kalirai J. S., Tremblay P. E., et al. (2018): The White Dwarf Initial-Final Mass Relation for Progenitor Stars from 0.85 to 7.5 M ⊙. ApJ 866(1):21. 10.3847/1538-4357/aadfd6, arXiv:1809.01673 [astro-ph.SR]
* Cunningham et al. [2019] Cunningham T., Tremblay P.-E., Freytag B., et al. (2019): Convective overshoot and macroscopic diffusion in pure-hydrogen-atmosphere white dwarfs. MNRAS 488(2):2503–2522. 10.1093/mnras/stz1759, arXiv:1906.11252 [astro-ph.SR]
* Cunningham et al. [2020] Cunningham T., Tremblay P.-E., Gentile Fusillo N. P., et al. (2020): From hydrogen to helium: the spectral evolution of white dwarfs as evidence for convective mixing. MNRAS 492(3):3540–3552. 10.1093/mnras/stz3638, arXiv:1911.00014 [astro-ph.SR]
* Dalton et al. [2012] Dalton G., Trager S. C., Abrams D. C., et al. (2012): WEAVE: the next generation wide-field spectroscopy facility for the William Herschel Telescope. In: McLean I. S., Ramsay S. K., Takami H. (eds.) Ground-based and Airborne Instrumentation for Astronomy IV, vol. 8446. Society of Photo-Optical Instrumentation Engineers, p. 84460P
* D’Antona & Mazzitelli [1989] D’Antona F., Mazzitelli I. (1989): The Fastest Evolving White Dwarfs. ApJ 347:934. 10.1086/168185
* de Jong et al. [2014] de Jong R. S., Barden S., Bellido-Tirado O., et al. (2014): 4MOST: 4-metre Multi-Object Spectroscopic Telescope. In: Ramsay S. K., McLean I. S., Takami H. (eds.) Ground-based and Airborne Instrumentation for Astronomy V, vol. 9147. Society of Photo-Optical Instrumentation Engineers, p. 91470M
* Deal et al. [2013] Deal M., Deheuvels S., Vauclair G., et al. (2013): Accretion from debris disks onto white dwarfs. Fingering (thermohaline) instability and derived accretion rates. A&A 557:L12. 10.1051/0004-6361/201322206, arXiv:1308.5406 [astro-ph.SR]
* Debes & Sigurdsson [2002] Debes J. H., Sigurdsson S. (2002): Are There Unstable Planetary Systems around White Dwarfs? ApJ 572(1):556–565. 10.1086/340291, arXiv:astro-ph/0202273 [astro-ph]
* Dehner & Kawaler [1995] Dehner B. T., Kawaler S. D. (1995): Thick to Thin: The Evolutionary Connection between PG 1159 Stars and the Thin Helium–enveloped Pulsating White Dwarf GD 358. ApJ 445:L141. 10.1086/187909, arXiv:astro-ph/9503099 [astro-ph]
* DESI Collaboration et al. [2016] DESI Collaboration, Aghamousa A., Aguilar J., et al. (2016): The DESI Experiment Part I: Science,Targeting, and Survey Design. arXiv e-prints arXiv:1611.00036. arXiv:1611.00036 [astro-ph.IM]
* Doyle et al. [2023] Doyle A. E., Klein B. L., Dufour P., et al. (2023): New Chondritic Bodies Identified in Eight Oxygen-bearing White Dwarfs. ApJ 950(2):93. 10.3847/1538-4357/acbd44, arXiv:2303.00063 [astro-ph.SR]
* Dreizler [1999] Dreizler S. (1999): Hubble Space Telescope spectroscopy of hot helium-rich white dwarfs: metal abundances along the cooling sequence. A&A 352:632–644
* Dreizler & Heber [1998] Dreizler S., Heber U. (1998): Spectral analyses of PG 1159 star: constraints on the GW Virginis pulsations from HST observations. A&A 334:618–632
* Dreizler & Werner [1996] Dreizler S., Werner K. (1996): Spectral analysis of hot helium-rich white dwarfs. A&A 314:217–232
* Dreizler & Wolff [1999] Dreizler S., Wolff B. (1999): Analysis of ultraviolet and extreme-ultraviolet spectra of the DA white dwarf G 191-B2B using self-consistent diffusion models. A&A 348:189–197
* Dufour [2011] Dufour P. (2011): Stars with Unusual Compositions: Carbon and Oxygen in Cool White Dwarfs. In: Hoard D. W. (ed.) White Dwarf Atmospheres and Circumstellar Environments. New York: Wiley, p. 53–88
* Dufour et al. [2005] Dufour P., Bergeron P., Fontaine G. (2005): Detailed Spectroscopic and Photometric Analysis of DQ White Dwarfs. ApJ 627(1):404–417. 10.1086/430373, arXiv:astro-ph/0503112 [astro-ph]
* Dufour et al. [2007a] Dufour P., Bergeron P., Liebert J., et al. (2007a): On the Spectral Evolution of Cool, Helium-Atmosphere White Dwarfs: Detailed Spectroscopic and Photometric Analysis of DZ Stars. ApJ 663(2):1291–1308. 10.1086/518468, arXiv:astro-ph/0703758 [astro-ph]
* Dufour et al. [2007b] Dufour P., Liebert J., Fontaine G., et al. (2007b): White dwarf stars with carbon atmospheres. Nature 450(7169):522–524. 10.1038/nature06318, arXiv:0711.3227 [astro-ph]
* Dufour et al. [2008] Dufour P., Fontaine G., Liebert J., et al. (2008): Hot DQ White Dwarfs: Something Different. ApJ 683(2):978–989. 10.1086/589855, arXiv:0805.0331 [astro-ph]
* Dufour et al. [2010a] Dufour P., Desharnais S., Wesemael F., et al. (2010a): Multiwavelength Observations of the Hot DB Star PG 0112+104. ApJ 718(2):647–656. 10.1088/0004-637X/718/2/647, arXiv:1006.0365 [astro-ph.SR]
* Dufour et al. [2010b] Dufour P., Kilic M., Fontaine G., et al. (2010b): The Discovery of the Most Metal-rich White Dwarf: Composition of a Tidally Disrupted Extrasolar Dwarf Planet. ApJ 719(1):803–809. 10.1088/0004-637X/719/1/803, arXiv:1006.3710 [astro-ph.SR]
* Dufour et al. [2012] Dufour P., Kilic M., Fontaine G., et al. (2012): Detailed Compositional Analysis of the Heavily Polluted DBZ White Dwarf SDSS J073842.56+183509.06: A Window on Planet Formation? ApJ 749(1):6. 10.1088/0004-637X/749/1/6, arXiv:1201.6252 [astro-ph.SR]
* Dufour et al. [2013] Dufour P., Vornanen T., Bergeron P., et al. (2013): White Dwarfs with Carbon Dominated Atmosphere: New Observations and Analysis. In: Krzesiński J., Stachowski G., Moskalik P., et al. (eds.) ASP Conf. Ser. 469: 18th European White Dwarf Workshop. San Francisco: Astronomical Society of the Pacific, p. 167
* Dufour et al. [2017] Dufour P., Blouin S., Coutu S., et al. (2017): The Montreal White Dwarf Database: A Tool for the Community. In: Tremblay P. E., Gänsicke B., Marsh T. (eds.) ASP Conf. Ser. 509: 20th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 3, 1610.00986
* Dunlap & Clemens [2015] Dunlap B. H., Clemens J. C. (2015): Hot DQ White Dwarf Stars as Failed Type Ia Supernovae. In: Dufour P., Bergeron P., Fontaine G. (eds.) ASP Conf. Ser. 493: 19th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 547
* Dupuis et al. [1992] Dupuis J., Fontaine G., Pelletier C., et al. (1992): A Study of Metal Abundance Patterns in Cool White Dwarfs. I. Time-dependent Calculations of Gravitational Settling. ApJS 82:505. 10.1086/191728
* Dupuis et al. [1993a] Dupuis J., Fontaine G., Pelletier C., et al. (1993a): A Study of Metal Abundance Patterns in Cool White Dwarfs. II. Simulations of Accretion Episodes. ApJS 84:73. 10.1086/191746
* Dupuis et al. [1993b] Dupuis J., Fontaine G., Wesemael F. (1993b): A Study of Metal Abundance Patterns in Cool White Dwarfs. III. Comparison of the Predictions of the Two-Phase Accretion Model with the Observations. ApJS 87:345. 10.1086/191808
* Dwomoh & Bauer [2023] Dwomoh A. M., Bauer E. B. (2023): Reinterpreting the Polluted White Dwarf SDSS J122859.93+104032.9 in Light of Thermohaline Mixing Models: More Polluting Material from a Larger Orbiting Solid Body. ApJ 952(2):95. 10.3847/1538-4357/acdb69, arXiv:2306.03864 [astro-ph.SR]
* Eisenstein et al. [2006] Eisenstein D. J., Liebert J., Koester D., et al. (2006): Hot DB White Dwarfs from the Sloan Digital Sky Survey. AJ 132(2):676–691. 10.1086/504424, arXiv:astro-ph/0606702 [astro-ph]
* El-Badry et al. [2018] El-Badry K., Rix H.-W., Weisz D. R. (2018): An Empirical Measurement of the Initial-Final Mass Relation with Gaia White Dwarfs. ApJ 860(2):L17. 10.3847/2041-8213/aaca9c, arXiv:1805.05849 [astro-ph.SR]
* Elms et al. [2022] Elms A. K., Tremblay P.-E., Gänsicke B. T., et al. (2022): Spectral analysis of ultra-cool white dwarfs polluted by planetary debris. MNRAS 517(3):4557–4574. 10.1093/mnras/stac2908, arXiv:2206.05258 [astro-ph.SR]
* Fantin et al. [2019] Fantin N. J., Côté P., McConnachie A. W., et al. (2019): The Canada-France Imaging Survey: Reconstructing the Milky Way Star Formation History from Its White Dwarf Population. ApJ 887(2):148. 10.3847/1538-4357/ab5521, arXiv:1911.02576 [astro-ph.GA]
* Farihi [2016] Farihi J. (2016): Circumstellar debris and pollution at white dwarf stars. New A Rev. 71:9–34. 10.1016/j.newar.2016.03.001, arXiv:1604.03092 [astro-ph.EP]
* Farihi et al. [2010] Farihi J., Barstow M. A., Redfield S., et al. (2010): Rocky planetesimals as the origin of metals in DZ stars. MNRAS 404(4):2123–2135. 10.1111/j.1365-2966.2010.16426.x, arXiv:1001.5025 [astro-ph.EP]
* Farihi et al. [2011] Farihi J., Brinkworth C. S., Gänsicke B. T., et al. (2011): Possible Signs of Water and Differentiation in a Rocky Exoplanetary Body. ApJ 728(1):L8. 10.1088/2041-8205/728/1/L8, arXiv:1101.0158 [astro-ph.EP]
* Farihi et al. [2013] Farihi J., Gänsicke B. T., Koester D. (2013): Evidence for Water in the Rocky Debris of a Disrupted Extrasolar Minor Planet. Science 342(6155):218–220. 10.1126/science.1239447, arXiv:1310.3269 [astro-ph.EP]
* Farihi et al. [2016] Farihi J., Koester D., Zuckerman B., et al. (2016): Solar abundances of rock-forming elements, extreme oxygen and hydrogen in a young polluted white dwarf. MNRAS 463(3):3186–3192. 10.1093/mnras/stw2182, arXiv:1608.07278 [astro-ph.EP]
* Farihi et al. [2022] Farihi J., Dufour P., Wilson T. G. (2022): Missing Metals in DQ Stars; a Compelling Clue to their Origin. arXiv e-prints arXiv:2208.05990. 10.48550/arXiv.2208.05990, arXiv:2208.05990 [astro-ph.SR]
* Finley et al. [1997] Finley D. S., Koester D., Basri G. (1997): The Temperature Scale and Mass Distribution of Hot DA White Dwarfs. ApJ 488(1):375–396. 10.1086/304668
* Fleming et al. [1986] Fleming T. A., Liebert J., Green R. F. (1986): The Luminosity Function of DA White Dwarfs. ApJ 308:176. 10.1086/164488
* Fontaine & Brassard [2002] Fontaine G., Brassard P. (2002): Can White Dwarf Asteroseismology Really Constrain the 12C($\alpha$, $\gamma$)16O Reaction Rate? ApJ 581(1):L33–L37. 10.1086/345787
* Fontaine & Brassard [2005] Fontaine G., Brassard P. (2005): Carbon in Hot DB White Dwarfs: A New Challenge for the Theory of the Spectral Evolution of White Dwarfs. In: Koester D., Moehler S. (eds.) ASP Conf. Ser. 334: 14th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 49
* Fontaine & Michaud [1979] Fontaine G., Michaud G. (1979): Diffusion time scales in white dwarfs. ApJ 231:826–840. 10.1086/157247
* Fontaine & van Horn [1976] Fontaine G., van Horn H. M. (1976): Convective white-dwarf envelope model grids for H-, He-, and C-rich compositions. ApJS 31:467–487. 10.1086/190388
* Fontaine & Wesemael [1987] Fontaine G., Wesemael F. (1987): Recent advances in the theory of white dwarf spectral evolution. In: Philip A. G. D., Hayes D. S., Liebert J. W. (eds.) IAU Colloq. 95: Second Conference on Faint Blue Stars. Schenectady: Davis Press, pp. 319–326
* Fontaine et al. [1984] Fontaine G., Villeneuve B., Wesemael F., et al. (1984): Carbon in the cool DC and C2 white dwarfs - Dredge-up in compositionally stratified envelopes. ApJ 277:L61–L64. 10.1086/184203
* Fontaine et al. [2001] Fontaine G., Brassard P., Bergeron P. (2001): The Potential of White Dwarf Cosmochronology. PASP 113(782):409–435. 10.1086/319535
* Fortin-Archambault et al. [2020] Fortin-Archambault M., Dufour P., Xu S. (2020): Modeling of the Variable Circumstellar Absorption Features of WD 1145+017. ApJ 888(1):47. 10.3847/1538-4357/ab585a, arXiv:1911.05690 [astro-ph.SR]
* Gaia Collaboration et al. [2016] Gaia Collaboration, Prusti T., de Bruijne J. H. J., et al. (2016): The Gaia mission. A&A 595:A1. 10.1051/0004-6361/201629272, arXiv:1609.04153 [astro-ph.IM]
* Gaia Collaboration et al. [2018a] Gaia Collaboration, Babusiaux C., van Leeuwen F., et al. (2018a): Gaia Data Release 2. Observational Hertzsprung-Russell diagrams. A&A 616:A10. 10.1051/0004-6361/201832843, arXiv:1804.09378 [astro-ph.SR]
* Gaia Collaboration et al. [2018b] Gaia Collaboration, Brown A. G. A., Vallenari A., et al. (2018b): Gaia Data Release 2. Summary of the contents and survey properties. A&A 616:A1. 10.1051/0004-6361/201833051, arXiv:1804.09365 [astro-ph.GA]
* Gaia Collaboration et al. [2021] Gaia Collaboration, Brown A. G. A., Vallenari A., et al. (2021): Gaia Early Data Release 3. Summary of the contents and survey properties. A&A 649:A1. 10.1051/0004-6361/202039657, arXiv:2012.01533 [astro-ph.GA]
* Gaia Collaboration et al. [2023a] Gaia Collaboration, Montegriffo P., Bellazzini M., et al. (2023a): Gaia Data Release 3. The Galaxy in your preferred colours: Synthetic photometry from Gaia low-resolution spectra. A&A 674:A33. 10.1051/0004-6361/202243709, arXiv:2206.06215 [astro-ph.SR]
* Gaia Collaboration et al. [2023b] Gaia Collaboration, Vallenari A., Brown A. G. A., et al. (2023b): Gaia Data Release 3. Summary of the content and survey properties. A&A 674:A1. 10.1051/0004-6361/202243940, arXiv:2208.00211 [astro-ph.GA]
* Gänsicke et al. [2012] Gänsicke B. T., Koester D., Farihi J., et al. (2012): The chemical diversity of exo-terrestrial planetary debris around white dwarfs. MNRAS 424(1):333–347. 10.1111/j.1365-2966.2012.21201.x, arXiv:1205.0167 [astro-ph.EP]
* García-Berro et al. [2010] García-Berro E., Torres S., Althaus L. G., et al. (2010): A white dwarf cooling age of 8Gyr for NGC 6791 from physical separation processes. Nature 465(7295):194–196. 10.1038/nature09045, arXiv:1005.2272 [astro-ph.SR]
* García-Zamora et al. [2023] García-Zamora E. M., Torres S., Rebassa-Mansergas A. (2023): White dwarf Random Forest classification through Gaia spectral coefficients. A&A 679:A127. 10.1051/0004-6361/202347601, arXiv:2308.07090 [astro-ph.SR]
* Genest-Beaulieu & Bergeron [2019a] Genest-Beaulieu C., Bergeron P. (2019a): A Comprehensive Spectroscopic and Photometric Analysis of DA and DB White Dwarfs from SDSS and Gaia. ApJ 871(2):169. 10.3847/1538-4357/aafac6
* Genest-Beaulieu & Bergeron [2019b] Genest-Beaulieu C., Bergeron P. (2019b): A Photometric and Spectroscopic Investigation of the DB White Dwarf Population Using SDSS and Gaia Data. ApJ 882(2):106. 10.3847/1538-4357/ab379e, arXiv:1908.01728 [astro-ph.SR]
* Gentile Fusillo et al. [2017] Gentile Fusillo N. P., Gänsicke B. T., Farihi J., et al. (2017): Trace hydrogen in helium atmosphere white dwarfs as a possible signature of water accretion. MNRAS 468(1):971–980. 10.1093/mnras/stx468, arXiv:1702.06542 [astro-ph.SR]
* Gentile Fusillo et al. [2019] Gentile Fusillo N. P., Tremblay P.-E., Gänsicke B. T., et al. (2019): A Gaia Data Release 2 catalogue of white dwarfs and a comparison with SDSS. MNRAS 482(4):4570–4591. 10.1093/mnras/sty3016, arXiv:1807.03315 [astro-ph.SR]
* Gentile Fusillo et al. [2020] Gentile Fusillo N. P., Tremblay P.-E., Bohlin R. C., et al. (2020): Cool white dwarfs as standards for infrared observations. MNRAS 491(3):3613–3623. 10.1093/mnras/stz2984, arXiv:1910.08087 [astro-ph.SR]
* Gentile Fusillo et al. [2021] Gentile Fusillo N. P., Tremblay P. E., Cukanovaite E., et al. (2021): A catalogue of white dwarfs in Gaia EDR3. MNRAS 508(3):3877–3896. 10.1093/mnras/stab2672, arXiv:2106.07669 [astro-ph.SR]
* Giammichele et al. [2012] Giammichele N., Bergeron P., Dufour P. (2012): Know Your Neighborhood: A Detailed Model Atmosphere Analysis of Nearby White Dwarfs. ApJS 199(2):29. 10.1088/0067-0049/199/2/29, arXiv:1202.5581 [astro-ph.SR]
* Gianninas et al. [2010] Gianninas A., Bergeron P., Dupuis J., et al. (2010): Spectroscopic Analysis of Hot, Hydrogen-rich White Dwarfs: The Presence of Metals and the Balmer-line Problem. ApJ 720(1):581–602. 10.1088/0004-637X/720/1/581
* Gianninas et al. [2011] Gianninas A., Bergeron P., Ruiz M. T. (2011): A Spectroscopic Survey and Analysis of Bright, Hydrogen-rich White Dwarfs. ApJ 743(2):138. 10.1088/0004-637X/743/2/138, arXiv:1109.3171 [astro-ph.SR]
* Good et al. [2004] Good S. A., Barstow M. A., Holberg J. B., et al. (2004): Comparison of the effective temperatures, gravities and helium abundances of DAO white dwarfs from Balmer and Lyman line studies. MNRAS 355(3):1031–1040. 10.1111/j.1365-2966.2004.08406.x
* Good et al. [2005] Good S. A., Barstow M. A., Burleigh M. R., et al. (2005): Heavy element abundances in DAO white dwarfs measured from FUSE data. MNRAS 363(1):183–196. 10.1111/j.1365-2966.2005.09428.x, arXiv:astro-ph/0507341 [astro-ph]
* Greenstein [1986] Greenstein J. L. (1986): The Frequency of Hydrogen White Dwarfs as Observed at High Signal to Noise. ApJ 304:334. 10.1086/164168
* Hansen et al. [2007] Hansen B. M. S., Anderson J., Brewer J., et al. (2007): The White Dwarf Cooling Sequence of NGC 6397. ApJ 671(1):380–401. 10.1086/522567, arXiv:astro-ph/0701738 [astro-ph]
* Harrison et al. [2018] Harrison J. H. D., Bonsor A., Madhusudhan N. (2018): Polluted white dwarfs: constraints on the origin and geology of exoplanetary material. MNRAS 479(3):3814–3841. 10.1093/mnras/sty1700, arXiv:1806.09917 [astro-ph.EP]
* Harrison et al. [2021] Harrison J. H. D., Bonsor A., Kama M., et al. (2021): Bayesian constraints on the origin and geology of exoplanetary material using a population of externally polluted white dwarfs. MNRAS 504(2):2853–2867. 10.1093/mnras/stab736, arXiv:2103.05713 [astro-ph.EP]
* Heber et al. [1997] Heber U., Napiwotzki R., Lemke M., et al. (1997): Helium line profile variations in the DAB white dwarf HS 0209+0832. A&A 324:L53–L56
* Heinonen et al. [2020] Heinonen R. A., Saumon D., Daligault J., et al. (2020): Diffusion Coefficients in the Envelopes of White Dwarfs. ApJ 896(1):2. 10.3847/1538-4357/ab91ad, arXiv:2005.05891 [astro-ph.SR]
* Heintz et al. [2022] Heintz T. M., Hermes J. J., El-Badry K., et al. (2022): Testing White Dwarf Age Estimates Using Wide Double White Dwarf Binaries from Gaia EDR3. ApJ 934(2):148. 10.3847/1538-4357/ac78d9, arXiv:2206.00025 [astro-ph.SR]
* Herwig et al. [1999] Herwig F., Blöcker T., Langer N., et al. (1999): On the formation of hydrogen-deficient post-AGB stars. A&A 349:L5–L8. arXiv:astro-ph/9908108 [astro-ph]
* Holberg et al. [1993] Holberg J. B., Barstow M. A., Buckley D. A. H., et al. (1993): Two New Extremely Iron-rich Hot DA White Dwarfs and the Nature of the EUV Opacity. ApJ 416:806. 10.1086/173278
* Hollands et al. [2017] Hollands M. A., Koester D., Alekseev V., et al. (2017): Cool DZ white dwarfs - I. Identification and spectral analysis. MNRAS 467(4):4970–5000. 10.1093/mnras/stx250, arXiv:1701.07827 [astro-ph.SR]
* Hollands et al. [2018a] Hollands M. A., Gänsicke B. T., Koester D. (2018a): Cool DZ white dwarfs II: compositions and evolution of old remnant planetary systems. MNRAS 477(1):93–111. 10.1093/mnras/sty592, arXiv:1801.07714 [astro-ph.SR]
* Hollands et al. [2018b] Hollands M. A., Tremblay P. E., Gänsicke B. T., et al. (2018b): The Gaia 20 pc white dwarf sample. MNRAS 480(3):3942–3961. 10.1093/mnras/sty2057, arXiv:1805.12590 [astro-ph.SR]
* Hollands et al. [2020] Hollands M. A., Tremblay P. E., Gänsicke B. T., et al. (2020): An ultra-massive white dwarf with a mixed hydrogen-carbon atmosphere as a likely merger remnant. Nature Astronomy 4:663–669. 10.1038/s41550-020-1028-0, arXiv:2003.00028 [astro-ph.SR]
* Hollands et al. [2021] Hollands M. A., Tremblay P.-E., Gänsicke B. T., et al. (2021): Alkali metals in white dwarf atmospheres as tracers of ancient planetary crusts. Nature Astronomy 5:451–459. 10.1038/s41550-020-01296-7, arXiv:2101.01225 [astro-ph.EP]
* Hollands et al. [2022] Hollands M. A., Tremblay P. E., Gänsicke B. T., et al. (2022): Spectral analysis of cool white dwarfs accreting from planetary systems: from the ultraviolet to the optical. MNRAS 511(1):71–82. 10.1093/mnras/stab3696, arXiv:2112.08887 [astro-ph.SR]
* Hoskin et al. [2020] Hoskin M. J., Toloza O., Gänsicke B. T., et al. (2020): White dwarf pollution by hydrated planetary remnants: hydrogen and metals in WD J204713.76-125908.9. MNRAS 499(1):171–182. 10.1093/mnras/staa2717, arXiv:2009.05053 [astro-ph.EP]
* Hoyer et al. [2017] Hoyer D., Rauch T., Werner K., et al. (2017): Complete spectral energy distribution of the hot, helium-rich white dwarf RX J0503.9-2854. A&A 598:A135. 10.1051/0004-6361/201629869, arXiv:1610.09177 [astro-ph.SR]
* Hoyer et al. [2018] Hoyer D., Rauch T., Werner K., et al. (2018): Search for trans-iron elements in hot, helium-rich white dwarfs with the HST Cosmic Origins Spectrograph. A&A 612:A62. 10.1051/0004-6361/201732401, arXiv:1801.02414 [astro-ph.SR]
* Hügelmeyer et al. [2005] Hügelmeyer S. D., Dreizler S., Werner K., et al. (2005): Spectral analyses of DO white dwarfs and PG 1159 stars from the Sloan Digital Sky Survey. A&A 442(1):309–314. 10.1051/0004-6361:20053280, arXiv:astro-ph/0508101 [astro-ph]
* Hügelmeyer et al. [2006] Hügelmeyer S. D., Dreizler S., Homeier D., et al. (2006): Spectral analyses of eighteen hot H-deficient (pre-) white dwarfs from the Sloan Digital Sky Survey Data Release 4. A&A 454(2):617–624. 10.1051/0004-6361:20064869, arXiv:astro-ph/0605551 [astro-ph]
* Iben & Tutukov [1984] Iben J.I., Tutukov A. V. (1984): Cooling of low-mass carbon-oxygen dwarfs from the planetary nucleus stage through the crystallization stage. ApJ 282:615–630. 10.1086/162241
* Iben et al. [1983] Iben J.I., Kaler J. B., Truran J. W., et al. (1983): On the evolution of those nuclei of planetary nebulae that experiencea final helium shell flash. ApJ 264:605–612. 10.1086/160631
* Isern [2019] Isern J. (2019): The Star Formation History in the Solar Neighborhood as Told by Massive White Dwarfs. ApJ 878(1):L11. 10.3847/2041-8213/ab238e, arXiv:1905.10779 [astro-ph.GA]
* Izquierdo et al. [2021] Izquierdo P., Toloza O., Gänsicke B. T., et al. (2021): GD 424 - a helium-atmosphere white dwarf with a large amount of trace hydrogen in the process of digesting a rocky planetesimal. MNRAS 501(3):4276–4288. 10.1093/mnras/staa3987, arXiv:2012.12957 [astro-ph.EP]
* Jiménez-Esteban et al. [2018] Jiménez-Esteban F. M., Torres S., Rebassa-Mansergas A., et al. (2018): A white dwarf catalogue from Gaia-DR2 and the Virtual Observatory. MNRAS 480(4):4505–4518. 10.1093/mnras/sty2120, arXiv:1807.02559 [astro-ph.SR]
* Jiménez-Esteban et al. [2023] Jiménez-Esteban F. M., Torres S., Rebassa-Mansergas A., et al. (2023): Spectral classification of the 100 pc white dwarf population from Gaia-DR3 and the virtual observatory. MNRAS 518(4):5106–5122. 10.1093/mnras/stac3382, arXiv:2211.08852 [astro-ph.SR]
* Johnson et al. [2022] Johnson T. M., Klein B. L., Koester D., et al. (2022): Unusual Abundances from Planetary System Material Polluting the White Dwarf G238-44. ApJ 941(2):113. 10.3847/1538-4357/aca089, arXiv:2211.02673 [astro-ph.EP]
* Jordan & Koester [1986] Jordan S., Koester D. (1986): Model atmospheres and synthetic spectra for white dwarfs with chemically stratified atmospheres. A&AS 65:367–377
* Jura & Xu [2010] Jura M., Xu S. (2010): The Survival of Water Within Extrasolar Minor Planets. AJ 140(5):1129–1136. 10.1088/0004-6256/140/5/1129, arXiv:1001.2595 [astro-ph.EP]
* Jura & Young [2014] Jura M., Young E. D. (2014): Extrasolar Cosmochemistry. Ann. Rev. Earth Planet. Sci. 42(1):45–67. 10.1146/annurev-earth-060313-054740
* Jura et al. [2012] Jura M., Xu S., Klein B., et al. (2012): Two Extrasolar Asteroids with Low Volatile-element Mass Fractions. ApJ 750(1):69. 10.1088/0004-637X/750/1/69, arXiv:1203.2885 [astro-ph.EP]
* Kaiser et al. [2021] Kaiser B. C., Clemens J. C., Blouin S., et al. (2021): Lithium pollution of a white dwarf records the accretion of an extrasolar planetesimal. Science 371(6525):168–172. 10.1126/science.abd1714, arXiv:2012.12900 [astro-ph.EP]
* Kalirai [2012] Kalirai J. S. (2012): The age of the Milky Way inner halo. Nature 486(7401):90–92. 10.1038/nature11062, arXiv:1205.6802 [astro-ph.GA]
* Kawka et al. [2023] Kawka A., Ferrario L., Vennes S. (2023): The non-explosive stellar merging origin of the ultra-massive carbon-rich white dwarfs. MNRAS 520(4):6299–6311. 10.1093/mnras/stad553, arXiv:2302.11118 [astro-ph.SR]
* Kepler et al. [2019] Kepler S. O., Pelisoli I., Koester D., et al. (2019): White dwarf and subdwarf stars in the Sloan Digital Sky Survey Data Release 14. MNRAS 486(2):2169–2183. 10.1093/mnras/stz960, arXiv:1904.01626 [astro-ph.SR]
* Kepler et al. [2021] Kepler S. O., Koester D., Pelisoli I., et al. (2021): White dwarf and subdwarf stars in the Sloan Digital Sky Survey Data Release 16. MNRAS 507(3):4646–4660. 10.1093/mnras/stab2411, arXiv:2108.10915 [astro-ph.SR]
* Kilic et al. [2017] Kilic M., Munn J. A., Harris H. C., et al. (2017): The Ages of the Thin Disk, Thick Disk, and the Halo from Nearby White Dwarfs. ApJ 837(2):162. 10.3847/1538-4357/aa62a5, arXiv:1702.06984 [astro-ph.SR]
* Kilic et al. [2018] Kilic M., Hambly N. C., Bergeron P., et al. (2018): Gaia reveals evidence for merged white dwarfs. MNRAS 479(1):L113–L117. 10.1093/mnrasl/sly110, arXiv:1805.01227 [astro-ph.SR]
* Kilic et al. [2020] Kilic M., Bergeron P., Kosakowski A., et al. (2020): The 100 pc White Dwarf Sample in the SDSS Footprint. ApJ 898(1):84. 10.3847/1538-4357/ab9b8d, arXiv:2006.00323 [astro-ph.SR]
* Kilic et al. [2024] Kilic M., Bergeron P., Blouin S., et al. (2024): White Dwarf Merger Remnants: The DAQ Subclass. arXiv e-prints arXiv:2403.08878. 10.48550/arXiv.2403.08878, arXiv:2403.08878 [astro-ph.SR]
* Klein et al. [2010] Klein B., Jura M., Koester D., et al. (2010): Chemical Abundances in the Externally Polluted White Dwarf GD 40: Evidence of a Rocky Extrasolar Minor Planet. ApJ 709(2):950–962. 10.1088/0004-637X/709/2/950, arXiv:0912.1422 [astro-ph.EP]
* Klein et al. [2011] Klein B., Jura M., Koester D., et al. (2011): Rocky Extrasolar Planetary Compositions Derived from Externally Polluted White Dwarfs. ApJ 741(1):64. 10.1088/0004-637X/741/1/64, arXiv:1108.1565 [astro-ph.EP]
* Klein et al. [2021] Klein B. L., Doyle A. E., Zuckerman B., et al. (2021): Discovery of Beryllium in White Dwarfs Polluted by Planetesimal Accretion. ApJ 914(1):61. 10.3847/1538-4357/abe40b, arXiv:2102.01834 [astro-ph.SR]
* Koester [1976] Koester D. (1976): Convective Mixing and Accretion in White Dwarfs. A&A 52:415
* Koester [2009] Koester D. (2009): Accretion and diffusion in white dwarfs. New diffusion timescales and applications to GD 362 and G 29-38. A&A 498(2):517–525. 10.1051/0004-6361/200811468, arXiv:0903.1499 [astro-ph.SR]
* Koester [2015] Koester D. (2015): On Thermohaline Mixing in Accreting White Dwarfs. In: Dufour P., Bergeron P., Fontaine G. (eds.) ASP Conf. Ser. 493: 19th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 129, 10.48550/arXiv.1408.6934, 1408.6934
* Koester & Kepler [2015] Koester D., Kepler S. O. (2015): DB white dwarfs in the Sloan Digital Sky Survey data release 10 and 12. A&A 583:A86. 10.1051/0004-6361/201527169, arXiv:1509.08244 [astro-ph.SR]
* Koester & Kepler [2019] Koester D., Kepler S. O. (2019): Carbon-rich (DQ) white dwarfs in the Sloan Digital Sky Survey. A&A 628:A102. 10.1051/0004-6361/201935946, arXiv:1905.11174 [astro-ph.SR]
* Koester & Knist [2006] Koester D., Knist S. (2006): New DQ white dwarfs in the Sloan Digital Sky Survey DR4: confirmation of two sequences. A&A 454(3):951–956. 10.1051/0004-6361:20065287, arXiv:astro-ph/0603734 [astro-ph]
* Koester & Wilken [2006] Koester D., Wilken D. (2006): The accretion-diffusion scenario for metals in cool white dwarfs. A&A 453(3):1051–1057. 10.1051/0004-6361:20064843, arXiv:astro-ph/0603185 [astro-ph]
* Koester et al. [1982] Koester D., Weidemann V., Zeidler E. M. (1982): Atmospheric parameters and carbon abundance of white dwarfs of spectral types C2 and DC. A&A 116:147–157
* Koester et al. [1994] Koester D., Liebert J., Saffer R. A. (1994): GD 323: New Observations and Analysis of the Prototype DAB White Dwarf. ApJ 422:783. 10.1086/173770
* Koester et al. [2005] Koester D., Rollenhagen K., Napiwotzki R., et al. (2005): Metal traces in white dwarfs of the SPY (ESO Supernova Ia Progenitor Survey) sample. A&A 432(3):1025–1032. 10.1051/0004-6361:20041927
* Koester et al. [2011] Koester D., Girven J., Gänsicke B. T., et al. (2011): Cool DZ white dwarfs in the SDSS. A&A 530:A114. 10.1051/0004-6361/201116816, arXiv:1105.0268 [astro-ph.SR]
* Koester et al. [2014a] Koester D., Gänsicke B. T., Farihi J. (2014a): The frequency of planetary debris around young white dwarfs. A&A 566:A34. 10.1051/0004-6361/201423691, arXiv:1404.2617 [astro-ph.SR]
* Koester et al. [2014b] Koester D., Provencal J., Gänsicke B. T. (2014b): Atmospheric parameters and carbon abundance for hot DB white dwarfs. A&A 568:A118. 10.1051/0004-6361/201424231, arXiv:1407.6157 [astro-ph.SR]
* Koester et al. [2020] Koester D., Kepler S. O., Irwin A. W. (2020): New white dwarf envelope models and diffusion. Application to DQ white dwarfs. A&A 635:A103. 10.1051/0004-6361/202037530, arXiv:2002.10170 [astro-ph.SR]
* Kollmeier et al. [2017] Kollmeier J. A., Zasowski G., Rix H.-W., et al. (2017): SDSS-V: Pioneering Panoptic Spectroscopy. arXiv e-prints arXiv:1711.03234. 10.48550/arXiv.1711.03234, arXiv:1711.03234 [astro-ph.GA]
* Krzesinski et al. [2009] Krzesinski J., Kleinman S. J., Nitta A., et al. (2009): A hot white dwarf luminosity function from the Sloan Digital Sky Survey. A&A 508(1):339–344. 10.1051/0004-6361/200912094
* Kudritzki & Puls [2000] Kudritzki R.-P., Puls J. (2000): Winds from Hot Stars. ARA&A 38:613–666. 10.1146/annurev.astro.38.1.613
* Kupka et al. [2018] Kupka F., Zaussinger F., Montgomery M. H. (2018): Mixing and overshooting in surface convection zones of DA white dwarfs: first results from ANTARES. MNRAS 474(4):4660–4671. 10.1093/mnras/stx3119, arXiv:1712.00641 [astro-ph.SR]
* Lawlor & MacDonald [2006] Lawlor T. M., MacDonald J. (2006): The mass of helium in white dwarf stars and the formation and evolution of hydrogen-deficient post-AGB stars. MNRAS 371(1):263–282. 10.1111/j.1365-2966.2006.10641.x, arXiv:astro-ph/0605747 [astro-ph]
* Leggett et al. [1998] Leggett S. K., Ruiz M. T., Bergeron P. (1998): The Cool White Dwarf Luminosity Function and the Age of the Galactic Disk. ApJ 497(1):294–302. 10.1086/305463
* Liebert [1983] Liebert J. (1983): G35-26: carbon in a peculiar DA white dwarf. PASP 95:878–882. 10.1086/131264
* Liebert et al. [2005] Liebert J., Bergeron P., Holberg J. B. (2005): The Formation Rate and Mass and Luminosity Functions of DA White Dwarfs from the Palomar Green Survey. ApJS 156(1):47–68. 10.1086/425738, arXiv:astro-ph/0406657 [astro-ph]
* Limoges & Bergeron [2010] Limoges M. M., Bergeron P. (2010): A Spectroscopic Analysis of White Dwarfs in the Kiso Survey. ApJ 714(2):1037–1051. 10.1088/0004-637X/714/2/1037, arXiv:1003.4313 [astro-ph.SR]
* Limoges et al. [2009] Limoges M. M., Bergeron P., Dufour P. (2009): Spectroscopic Analysis of the White Dwarf KUV 02196+2816: A New Unresolved DA+DB Degenerate Binary. ApJ 696(2):1461–1465. 10.1088/0004-637X/696/2/1461, arXiv:0902.3640 [astro-ph.SR]
* Limoges et al. [2015] Limoges M. M., Bergeron P., Lépine S. (2015): Physical Properties of the Current Census of Northern White Dwarfs within 40 pc of the Sun. ApJS 219(2):19. 10.1088/0067-0049/219/2/19, arXiv:1505.02297 [astro-ph.SR]
* Löbling et al. [2020] Löbling L., Maney M. A., Rauch T., et al. (2020): First discovery of trans-iron elements in a DAO-type white dwarf (BD-22∘3467). MNRAS 492(1):528–548. 10.1093/mnras/stz3247, arXiv:1911.09573 [astro-ph.SR]
* López-Sanjuan et al. [2022] López-Sanjuan C., Tremblay P. E., Ederoclite A., et al. (2022): J-PLUS: Spectral evolution of white dwarfs by PDF analysis. A&A 658:A79. 10.1051/0004-6361/202141746, arXiv:2110.14421 [astro-ph.SR]
* MacDonald & Vennes [1991] MacDonald J., Vennes S. (1991): How Much Hydrogen Is There in a White Dwarf? ApJ 371:719. 10.1086/169937
* MacDonald et al. [1998] MacDonald J., Hernanz M., Jose J. (1998): Evolutionary calculations of carbon dredge-up in helium envelope white dwarfs. MNRAS 296(3):523–530. 10.1046/j.1365-8711.1998.01392.x, arXiv:astro-ph/9803121 [astro-ph]
* Macfarlane et al. [2017] Macfarlane S. A., Woudt P. A., Dufour P., et al. (2017): The OmegaWhite Survey for short-period variable stars - IV. Discovery of the warm DQ white dwarf OW J175358.85-310728.9. MNRAS 470(1):732–741. 10.1093/mnras/stx741, arXiv:1703.08122 [astro-ph.SR]
* Manseau et al. [2016] Manseau P. M., Bergeron P., Green E. M. (2016): A Spectroscopic Search for Chemically Stratified White Dwarfs in the Sloan Digital Sky Survey. ApJ 833(2):127. 10.3847/1538-4357/833/2/127
* Manser et al. [2024] Manser C. J., Gänsicke B. T., Izquierdo P., et al. (2024): The frequency of metal-enrichment of cool helium-atmosphere white dwarfs using the DESI Early Data Release. MNRAS 10.1093/mnrasl/slae026, arXiv:2402.18644 [astro-ph.EP]
* Martin et al. [2005] Martin D. C., Fanson J., Schiminovich D., et al. (2005): The Galaxy Evolution Explorer: A Space Ultraviolet Survey Mission. ApJ 619(1):L1–L6. 10.1086/426387, arXiv:astro-ph/0411302 [astro-ph]
* McCleery et al. [2020] McCleery J., Tremblay P.-E., Gentile Fusillo N. P., et al. (2020): Gaia white dwarfs within 40 pc II: the volume-limited Northern hemisphere sample. MNRAS 499(2):1890–1908. 10.1093/mnras/staa2030, arXiv:2006.00874 [astro-ph.SR]
* Melis et al. [2011] Melis C., Farihi J., Dufour P., et al. (2011): Accretion of a Terrestrial-like Minor Planet by a White Dwarf. ApJ 732(2):90. 10.1088/0004-637X/732/2/90, arXiv:1102.0311 [astro-ph.SR]
* Michaud et al. [2015] Michaud G., Alecian G., Richer J. (2015): Atomic Diffusion in Stars. Cham: Springer International
* Miller Bertolami & Althaus [2006] Miller Bertolami M. M., Althaus L. G. (2006): Full evolutionary models for PG 1159 stars. Implications for the helium-rich O(He) stars. A&A 454(3):845–854. 10.1051/0004-6361:20054723, arXiv:astro-ph/0603846 [astro-ph]
* Miller Bertolami et al. [2006] Miller Bertolami M. M., Althaus L. G., Serenelli A. M., et al. (2006): New evolutionary calculations for the born again scenario. A&A 449(1):313–326. 10.1051/0004-6361:20053804, arXiv:astro-ph/0511406 [astro-ph]
* Miller Bertolami et al. [2017] Miller Bertolami M. M., Althaus L. G., Córsico A. H. (2017): On the Formation of DA White Dwarfs with low Hydrogen Contents: Preliminary Results. In: Tremblay P. E., Gänsicke B., Marsh T. (eds.) ASP Conf. Ser. 509: 20th European Workshop on White Dwarfs. San Francisco: Astronomical Society of the Pacific, p. 435, 1609.08683
* Moss et al. [2024] Moss A., Bergeron P., Kilic M., et al. (2024): Discovery of a magnetic double-faced DBA white dwarf. MNRAS 527(4):10111–10122. 10.1093/mnras/stad3825
* Napiwotzki [1992] Napiwotzki R. (1992): Analysis of central stars of old planetary nebulae: Problems with the Balmer lines. In: Heber U., Jeffery C. S. (eds.) The Atmospheres of Early-Type Stars, vol. 401. Berlin: Springer, p. 310, 10.1007/3-540-55256-1_328
* Napiwotzki [1999] Napiwotzki R. (1999): Spectroscopic investigation of old planetaries. IV. Model atmosphere analysis. A&A 350:101–119. arXiv:astro-ph/9908181 [astro-ph]
* Napiwotzki & Rauch [1994] Napiwotzki R., Rauch T. (1994): The Balmer line problem of hot stars and the impact of ion-dynamical effects on the Stark broadening of HI and HeII lines. A&A 285:603–608
* O’Brien et al. [2023] O’Brien M. W., Tremblay P. E., Gentile Fusillo N. P., et al. (2023): Gaia white dwarfs within 40 pc - III. Spectroscopic observations of new candidates in the Southern hemisphere. MNRAS 518(2):3055–3073. 10.1093/mnras/stac3303, arXiv:2210.01608 [astro-ph.SR]
* O’Brien et al. [2024] O’Brien M. W., Tremblay P. E., Klein B. L., et al. (2024): The 40 pc sample of white dwarfs from Gaia. MNRAS 527(3):8687–8705. 10.1093/mnras/stad3773, arXiv:2312.02735 [astro-ph.SR]
* Oswalt et al. [1996] Oswalt T. D., Smith J. A., Wood M. A., et al. (1996): A lower limit of 9.5 Gyr on the age of the Galactic disk from the oldest white dwarf stars. Nature 382(6593):692–694. 10.1038/382692a0
* Ourique et al. [2019] Ourique G., Romero A. D., Kepler S. O., et al. (2019): A study of cool white dwarfs in the Sloan Digital Sky Survey Data Release 12. MNRAS 482(1):649–657. 10.1093/mnras/sty2751, arXiv:1810.03554 [astro-ph.SR]
* Ourique et al. [2020] Ourique G., Kepler S. O., Romero A. D., et al. (2020): Evidence of spectral evolution on the white dwarf sample from the Gaia mission. MNRAS 492(4):5003–5010. 10.1093/mnras/staa120, arXiv:2001.04378 [astro-ph.SR]
* Paquette et al. [1986] Paquette C., Pelletier C., Fontaine G., et al. (1986): Diffusion in White Dwarfs: New Results and Comparative Study. ApJS 61:197. 10.1086/191112
* Paxton et al. [2011] Paxton B., Bildsten L., Dotter A., et al. (2011): Modules for Experiments in Stellar Astrophysics (MESA). ApJS 192(1):3. 10.1088/0067-0049/192/1/3, arXiv:1009.1622 [astro-ph.SR]
* Pelletier et al. [1986] Pelletier C., Fontaine G., Wesemael F., et al. (1986): Carbon Pollution in Helium-rich White Dwarf Atmospheres: Time-dependent Calculations of the Dredge-up Process. ApJ 307:242. 10.1086/164410
* Pereira et al. [2005] Pereira C., Bergeron P., Wesemael F. (2005): Discovery of Spectroscopic Variations in the DAB White Dwarf GD 323. ApJ 623(2):1076–1082. 10.1086/429219, arXiv:astro-ph/0501620 [astro-ph]
* Petitclerc et al. [2005] Petitclerc N., Wesemael F., Kruk J. W., et al. (2005): FUSE Observations of DB White Dwarfs. ApJ 624(1):317–330. 10.1086/428750
* Preval et al. [2013] Preval S. P., Barstow M. A., Holberg J. B., et al. (2013): A comprehensive near- and far-ultraviolet spectroscopic study of the hot DA white dwarf G191-B2B. MNRAS 436(1):659–674. 10.1093/mnras/stt1604, arXiv:1308.4825 [astro-ph.SR]
* Preval et al. [2019] Preval S. P., Barstow M. A., Bainbridge M., et al. (2019): A far-UV survey of three hot, metal-polluted white dwarf stars: WD0455-282, WD0621-376, and WD2211-495. MNRAS 487(3):3470–3487. 10.1093/mnras/stz1506, arXiv:1905.12350 [astro-ph.SR]
* Provencal et al. [2000] Provencal J. L., Shipman H. L., Thejll P., et al. (2000): Carbon and Hydrogen in Hot DB White Dwarfs. ApJ 542(2):1041–1056. 10.1086/317030
* Quirion et al. [2012] Quirion P. O., Fontaine G., Brassard P. (2012): Wind Competing Against Settling: A Coherent Model of the GW Virginis Instability Domain. ApJ 755(2):128. 10.1088/0004-637X/755/2/128
* Raddi et al. [2015] Raddi R., Gänsicke B. T., Koester D., et al. (2015): Likely detection of water-rich asteroid debris in a metal-polluted white dwarf. MNRAS 450(2):2083–2093. 10.1093/mnras/stv701, arXiv:1503.07864 [astro-ph.SR]
* Rauch et al. [1998] Rauch T., Dreizler S., Wolff B. (1998): Spectral analysis of O(He)-type post-AGB stars. A&A 338:651–660
* Rauch et al. [2013] Rauch T., Werner K., Bohlin R., et al. (2013): The virtual observatory service TheoSSA: Establishing a database of synthetic stellar flux standards. I. NLTE spectral analysis of the DA-type white dwarf G191-B2B. A&A 560:A106. 10.1051/0004-6361/201322336, arXiv:1308.6450 [astro-ph.SR]
* Rauch et al. [2014a] Rauch T., Werner K., Quinet P., et al. (2014a): Stellar laboratories. II. New Zn iv and Zn v oscillator strengths and their validation in the hot white dwarfs G191-B2B and RE 0503-289. A&A 564:A41. 10.1051/0004-6361/201423491, arXiv:1403.2183 [astro-ph.SR]
* Rauch et al. [2014b] Rauch T., Werner K., Quinet P., et al. (2014b): Stellar laboratories. III. New Ba v, Ba vi, and Ba vii oscillator strengths and the barium abundance in the hot white dwarfs G191-B2B and RE 0503-289. A&A 566:A10. 10.1051/0004-6361/201423878, arXiv:1404.6094 [astro-ph.SR]
* Rauch et al. [2015a] Rauch T., Hoyer D., Quinet P., et al. (2015a): Stellar laboratories. V. The Xe vi ultraviolet spectrum and the xenon abundance in the hot DO-type white dwarf RE 0503-289. A&A 577:A88. 10.1051/0004-6361/201526078, arXiv:1504.01991 [astro-ph.SR]
* Rauch et al. [2015b] Rauch T., Werner K., Quinet P., et al. (2015b): Stellar laboratories. IV. New Ga iv, Ga v, and Ga vi oscillator strengths and the gallium abundance in the hot white dwarfs G191-B2B and RE 0503-289. A&A 577:A6. 10.1051/0004-6361/201425326, arXiv:1501.07751 [astro-ph.SR]
* Rauch et al. [2016a] Rauch T., Quinet P., Hoyer D., et al. (2016a): Stellar laboratories. VI. New Mo iv-vii oscillator strengths and the molybdenum abundance in the hot white dwarfs G191-B2B and RE 0503-289. A&A 587:A39. 10.1051/0004-6361/201527324, arXiv:1512.07525 [astro-ph.SR]
* Rauch et al. [2016b] Rauch T., Quinet P., Hoyer D., et al. (2016b): Stellar laboratories. VII. New Kr iv - vii oscillator strengths and an improved spectral analysis of the hot, hydrogen-deficient DO-type white dwarf RE 0503-289. A&A 590:A128. 10.1051/0004-6361/201628131, arXiv:1603.00701 [astro-ph.SR]
* Rauch et al. [2017a] Rauch T., Gamrath S., Quinet P., et al. (2017a): Stellar laboratories . VIII. New Zr iv-vii, Xe iv-v, and Xe vii oscillator strengths and the Al, Zr, and Xe abundances in the hot white dwarfs G191-B2B and RE 0503-289. A&A 599:A142. 10.1051/0004-6361/201629794, arXiv:1611.07364 [physics.atom-ph]
* Rauch et al. [2017b] Rauch T., Quinet P., Knörzer M., et al. (2017b): Stellar laboratories . IX. New Se v, Sr iv-vii, Te vi, and I vi oscillator strengths and the Se, Sr, Te, and I abundances in the hot white dwarfs G191-B2B and RE 0503-289. A&A 606:A105. 10.1051/0004-6361/201730383, arXiv:1706.09215 [astro-ph.SR]
* Rauch et al. [2020] Rauch T., Gamrath S., Quinet P., et al. (2020): Stellar laboratories. X. New Cu IV-VII oscillator strengths and the first detection of copper and indium in hot white dwarfs. A&A 637:A4. 10.1051/0004-6361/201936620, arXiv:2004.01633 [astro-ph.SR]
* Reindl et al. [2014a] Reindl N., Rauch T., Werner K., et al. (2014a): Analysis of cool DO-type white dwarfs from the Sloan Digital Sky Survey data release 10. A&A 572:A117. 10.1051/0004-6361/201424861, arXiv:1410.7666 [astro-ph.SR]
* Reindl et al. [2014b] Reindl N., Rauch T., Werner K., et al. (2014b): On helium-dominated stellar evolution: the mysterious role of the O(He)-type stars. A&A 566:A116. 10.1051/0004-6361/201423498, arXiv:1405.1589 [astro-ph.SR]
* Reindl et al. [2023] Reindl N., Islami R., Werner K., et al. (2023): The bright blue side of the night sky: Spectroscopic survey of bright and hot (pre-) white dwarfs. A&A 677:A29. 10.1051/0004-6361/202346865, arXiv:2307.03721 [astro-ph.SR]
* Renedo et al. [2010] Renedo I., Althaus L. G., Miller Bertolami M. M., et al. (2010): New Cooling Sequences for Old White Dwarfs. ApJ 717(1):183–195. 10.1088/0004-637X/717/1/183, arXiv:1005.2170 [astro-ph.SR]
* Rogers et al. [2024] Rogers L. K., Bonsor A., Xu S., et al. (2024): Seven white dwarfs with circumstellar gas discs I: white dwarf parameters and accreted planetary abundances. MNRAS 527(3):6038–6054. 10.1093/mnras/stad3557, arXiv:2311.14048 [astro-ph.EP]
* Rolland et al. [2018] Rolland B., Bergeron P., Fontaine G. (2018): On the Spectral Evolution of Helium-atmosphere White Dwarfs Showing Traces of Hydrogen. ApJ 857(1):56. 10.3847/1538-4357/aab713, arXiv:1803.05965 [astro-ph.SR]
* Rolland et al. [2020] Rolland B., Bergeron P., Fontaine G. (2020): A Convective Dredge-up Model as the Origin of Hydrogen in DBA White Dwarfs. ApJ 889(2):87. 10.3847/1538-4357/ab6602, arXiv:2001.01085 [astro-ph.SR]
* Romero et al. [2012] Romero A. D., Córsico A. H., Althaus L. G., et al. (2012): Toward ensemble asteroseismology of ZZ Ceti stars with fully evolutionary models. MNRAS 420(2):1462–1480. 10.1111/j.1365-2966.2011.20134.x, arXiv:1109.6682 [astro-ph.SR]
* Salaris & Cassisi [2017] Salaris M., Cassisi S. (2017): Chemical element transport in stellar evolution models. RSOS 4(8):170192. 10.1098/rsos.170192, arXiv:1707.07454 [astro-ph.SR]
* Salaris et al. [2022] Salaris M., Cassisi S., Pietrinferni A., et al. (2022): The updated BASTI stellar evolution models and isochrones - III. White dwarfs. MNRAS 509(4):5197–5208. 10.1093/mnras/stab3359, arXiv:2111.09285 [astro-ph.SR]
* Saumon et al. [2022] Saumon D., Blouin S., Tremblay P.-E. (2022): Current challenges in the physics of white dwarf stars. Phys. Rep. 988:1–63. 10.1016/j.physrep.2022.09.001, arXiv:2209.02846 [astro-ph.SR]
* Schreiber et al. [2019] Schreiber M. R., Gänsicke B. T., Toloza O., et al. (2019): Cold Giant Planets Evaporated by Hot White Dwarfs. ApJ 887(1):L4. 10.3847/2041-8213/ab42e2, arXiv:1912.02345 [astro-ph.SR]
* Schuh et al. [2002] Schuh S. L., Dreizler S., Wolff B. (2002): Equilibrium abundances in hot DA white dwarfs as derived from self-consistent diffusion models. I. Analysis of spectroscopic EUVE data. A&A 382:164–173. 10.1051/0004-6361:20011588, arXiv:astro-ph/0111245 [astro-ph]
* Scóccola et al. [2006] Scóccola C. G., Althaus L. G., Serenelli A. M., et al. (2006): DQ white-dwarf stars with low C abundance: possible progenitors. A&A 451(1):147–155. 10.1051/0004-6361:20053769, arXiv:astro-ph/0602196 [astro-ph]
* Sion [1984] Sion E. M. (1984): Implications of the absolute magnitude distribution functions of DA and non-DA white dwarfs. ApJ 282:612–614. 10.1086/162240
* Sion [1999] Sion E. M. (1999): White Dwarfs in Cataclysmic Variables. PASP 111(759):532–555. 10.1086/316361
* Sion et al. [1983] Sion E. M., Greenstein J. L., Landstreet J. D., et al. (1983): A proposed new white dwarf spectral classification system. ApJ 269:253–257. 10.1086/161036
* Subasavage et al. [2017] Subasavage J. P., Jao W.-C., Henry T. J., et al. (2017): The Solar Neighborhood. XXXIX. Parallax Results from the CTIOPI and NOFS Programs: 50 New Members of the 25 parsec White Dwarf Sample. AJ 154(1):32. 10.3847/1538-3881/aa76e0, arXiv:1706.00709 [astro-ph.SR]
* Swan et al. [2019] Swan A., Farihi J., Koester D., et al. (2019): Interpretation and diversity of exoplanetary material orbiting white dwarfs. MNRAS 490(1):202–218. 10.1093/mnras/stz2337, arXiv:1908.08047 [astro-ph.EP]
* Swan et al. [2023] Swan A., Farihi J., Melis C., et al. (2023): Planetesimals at DZ stars - I. Chondritic compositions and a massive accretion event. MNRAS 526(3):3815–3831. 10.1093/mnras/stad2867, arXiv:2309.06467 [astro-ph.EP]
* Tassoul et al. [1990] Tassoul M., Fontaine G., Winget D. E. (1990): Evolutionary Models for Pulsation Studies of White Dwarfs. ApJS 72:335. 10.1086/191420
* Torres et al. [2023] Torres S., Cruz P., Murillo-Ojeda R., et al. (2023): White dwarf spectral type-temperature distribution from Gaia DR3 and the Virtual Observatory. A&A 677:A159. 10.1051/0004-6361/202346977, arXiv:2307.13629 [astro-ph.SR]
* Tremblay & Bergeron [2008] Tremblay P. E., Bergeron P. (2008): The Ratio of Helium- to Hydrogen-Atmosphere White Dwarfs: Direct Evidence for Convective Mixing. ApJ 672(2):1144–1152. 10.1086/524134, arXiv:0710.1073 [astro-ph]
* Tremblay et al. [2011] Tremblay P. E., Bergeron P., Gianninas A. (2011): An Improved Spectroscopic Analysis of DA White Dwarfs from the Sloan Digital Sky Survey Data Release 4. ApJ 730(2):128. 10.1088/0004-637X/730/2/128, arXiv:1102.0056 [astro-ph.SR]
* Tremblay et al. [2013] Tremblay P. E., Ludwig H. G., Steffen M., et al. (2013): Pure-hydrogen 3D model atmospheres of cool white dwarfs. A&A 552:A13. 10.1051/0004-6361/201220813, arXiv:1302.2013 [astro-ph.SR]
* Tremblay et al. [2014] Tremblay P. E., Kalirai J. S., Soderblom D. R., et al. (2014): White Dwarf Cosmochronology in the Solar Neighborhood. ApJ 791(2):92. 10.1088/0004-637X/791/2/92, arXiv:1406.5173 [astro-ph.SR]
* Tremblay et al. [2015] Tremblay P. E., Ludwig H. G., Freytag B., et al. (2015): Calibration of the Mixing-length Theory for Convective White Dwarf Envelopes. ApJ 799(2):142. 10.1088/0004-637X/799/2/142, arXiv:1412.1789 [astro-ph.SR]
* Tremblay et al. [2019] Tremblay P. E., Cukanovaite E., Gentile Fusillo N. P., et al. (2019): Fundamental parameter accuracy of DA and DB white dwarfs in Gaia Data Release 2. MNRAS 482(4):5222–5232. 10.1093/mnras/sty3067, arXiv:1811.03084 [astro-ph.SR]
* Unglaub [2008] Unglaub K. (2008): Mass-loss and diffusion in subdwarf B stars and hot white dwarfs: do weak winds exist? A&A 486(3):923–940. 10.1051/0004-6361:20078019, arXiv:0808.1072 [astro-ph]
* Unglaub & Bues [1998] Unglaub K., Bues I. (1998): The effect of diffusion and mass loss on the helium abundance in hot white dwarfs and subdwarfs. A&A 338:75–84
* Unglaub & Bues [2000] Unglaub K., Bues I. (2000): The chemical evolution of hot white dwarfs in the presence of diffusion and mass loss. A&A 359:1042–1058
* Vennes & Fontaine [1992] Vennes S., Fontaine G. (1992): An Interpretation of the Spectral Properties of Hot Hydrogen-rich White Dwarfs with Stratified H/He Model Atmospheres. ApJ 401:288. 10.1086/172060
* Vennes et al. [1988] Vennes S., Pelletier C., Fontaine G., et al. (1988): The Presence of Helium in Hot DA White Dwarfs: The Role of Radiative Levitation and the Case for Stratified Atmospheres. ApJ 331:876. 10.1086/166606
* Vennes et al. [2005] Vennes S., Chayer P., Dupuis J. (2005): Discovery of Photospheric Germanium in Hot DA White Dwarfs. ApJ 622(2):L121–L124. 10.1086/429667
* Vennes et al. [2006] Vennes S., Chayer P., Dupuis J., et al. (2006): Iron in Hot DA White Dwarfs. ApJ 652(2):1554–1562. 10.1086/508509, arXiv:astro-ph/0608416 [astro-ph]
* Vennes et al. [2024] Vennes S., Kawka A., Klein B. L., et al. (2024): A cool, magnetic white dwarf accreting planetary debris. MNRAS 527(2):3122–3138. 10.1093/mnras/stad3370, arXiv:2311.07937 [astro-ph.SR]
* Veras [2021] Veras D. (2021): Planetary Systems Around White Dwarfs. In: Oxford Research Encyclopedia of Planetary Science. Oxford: Oxford University Press, p. 1, 10.1093/acrefore/9780190647926.013.238
* Veras et al. [2014] Veras D., Shannon A., Gänsicke B. T. (2014): Hydrogen delivery onto white dwarfs from remnant exo-Oort cloud comets. MNRAS 445(4):4175–4185. 10.1093/mnras/stu2026, arXiv:1409.7691 [astro-ph.SR]
* Vincent et al. [2024] Vincent O., Barstow M. A., Jordan S., et al. (2024): Classification and parameterization of a large Gaia sample of white dwarfs using XP spectra. A&A 682:A5. 10.1051/0004-6361/202347694, arXiv:2308.05572 [astro-ph.SR]
* Voss et al. [2007] Voss B., Koester D., Napiwotzki R., et al. (2007): High-resolution UVES/VLT spectra of white dwarfs observed for the ESO SN Ia progenitor survey. II. DB and DBA stars. A&A 470(3):1079–1088. 10.1051/0004-6361:20077285
* Wachlin et al. [2017] Wachlin F. C., Vauclair G., Vauclair S., et al. (2017): Importance of fingering convection for accreting white dwarfs in the framework of full evolutionary calculations: the case of the hydrogen-rich white dwarfs GD 133 and G 29-38. A&A 601:A13. 10.1051/0004-6361/201630094, arXiv:1612.09320 [astro-ph.SR]
* Wachlin et al. [2022] Wachlin F. C., Vauclair G., Vauclair S., et al. (2022): New simulations of accreting DA white dwarfs: Inferring accretion rates from the surface contamination. A&A 660:A30. 10.1051/0004-6361/202142289, arXiv:2109.11370 [astro-ph.SR]
* Weidemann & Koester [1995] Weidemann V., Koester D. (1995): Surface carbon abundances and compositional stratification of cool helium-rich white dwarfs. A&A 297:216–222
* Werner [1996a] Werner K. (1996a): On the Balmer Line Problem. ApJ 457:L39. 10.1086/309889
* Werner [1996b] Werner K. (1996b): Search for trace amounts of hydrogen in hot DO white dwarfs. A&A 309:861–866
* Werner & Dreizler [1994] Werner K., Dreizler S. (1994): On the nickel abundance in hot hydrogen-rich white dwarfs. A&A 286:L31–L34
* Werner & Herwig [2006] Werner K., Herwig F. (2006): The Elemental Abundances in Bare Planetary Nebula Central Stars and the Shell Burning in AGB Stars. PASP 118(840):183–204. 10.1086/500443, arXiv:astro-ph/0512320 [astro-ph]
* Werner & Rauch [2014] Werner K., Rauch T. (2014): Weak metal lines in optical high-resolution Very Large Telescope and Keck spectra of “cool” PG 1159 stars. A&A 569:A99. 10.1051/0004-6361/201424051
* Werner et al. [1991] Werner K., Heber U., Hunger K. (1991): Non-LTE analysis of four PG1159 stars. A&A 244:437
* Werner et al. [2007] Werner K., Rauch T., Kruk J. W. (2007): Discovery of photospheric argon in very hot central stars of planetary nebulae and white dwarfs. A&A 466(1):317–322. 10.1051/0004-6361:20077101, arXiv:astro-ph/0702387 [astro-ph]
* Werner et al. [2012] Werner K., Rauch T., Ringat E., et al. (2012): First Detection of Krypton and Xenon in a White Dwarf. ApJ 753(1):L7. 10.1088/2041-8205/753/1/L7
* Werner et al. [2014] Werner K., Rauch T., Kepler S. O. (2014): New hydrogen-deficient (pre-) white dwarfs in the Sloan Digital Sky Survey Data Release 10. A&A 564:A53. 10.1051/0004-6361/201423441
* Werner et al. [2017] Werner K., Rauch T., Kruk J. W. (2017): Far-UV spectroscopy of two extremely hot, helium-rich white dwarfs. A&A 601:A8. 10.1051/0004-6361/201630266
* Werner et al. [2018a] Werner K., Rauch T., Knörzer M., et al. (2018a): First detection of bromine and antimony in hot stars. A&A 614:A96. 10.1051/0004-6361/201832723, arXiv:1803.04809 [astro-ph.SR]
* Werner et al. [2018b] Werner K., Rauch T., Kruk J. W. (2018b): Metal abundances in hot white dwarfs with signatures of a superionized wind. A&A 609:A107. 10.1051/0004-6361/201731740, arXiv:1711.04138 [astro-ph.SR]
* Werner et al. [2019] Werner K., Rauch T., Reindl N. (2019): Spectral analysis of the extremely hot DA white dwarf PG 0948+534. MNRAS 483(4):5291–5300. 10.1093/mnras/sty3408, arXiv:1812.07486 [astro-ph.SR]
* Wesemael et al. [1993] Wesemael F., Greenstein J. L., Liebert J., et al. (1993): An Atlas of Optical Spectra of White-Dwarf Stars. PASP 105:761. 10.1086/133228
* Wesemael et al. [1994] Wesemael F., Bergeron P., Lamontagne R. L., et al. (1994): Hot Degenerates in the Montreal-Cambridge-Tololo Survey. II. Two New Hybrid White Dwarfs, MCT 0128-3846 and MCT 0453-2933, and the Nature of the DAB Stars. ApJ 429:369. 10.1086/174326
* Williams et al. [2013] Williams K. A., Winget D. E., Montgomery M. H., et al. (2013): Photometric Variability in a Warm, Strongly Magnetic DQ White Dwarf, SDSS J103655.39+652252.2. ApJ 769(2):123. 10.1088/0004-637X/769/2/123, arXiv:1304.3165 [astro-ph.SR]
* Williams et al. [2016] Williams K. A., Montgomery M. H., Winget D. E., et al. (2016): Variability in Hot Carbon-dominated Atmosphere (Hot DQ) White Dwarfs: Rapid Rotation? ApJ 817(1):27. 10.3847/0004-637X/817/1/27, arXiv:1511.08834 [astro-ph.SR]
* Wilson et al. [2015] Wilson D. J., Gänsicke B. T., Koester D., et al. (2015): The composition of a disrupted extrasolar planetesimal at SDSS J0845+2257 (Ton 345). MNRAS 451(3):3237–3248. 10.1093/mnras/stv1201, arXiv:1505.07466 [astro-ph.EP]
* Winget et al. [1987] Winget D. E., Hansen C. J., Liebert J., et al. (1987): An Independent Method for Determining the Age of the Universe. ApJ 315:L77. 10.1086/184864
* Wolff et al. [2000] Wolff B., Jordan S., Koester D., et al. (2000): The nature of the DAB white dwarf HS 0209+0832. A&A 361:629–640
* Xu et al. [2013] Xu S., Jura M., Klein B., et al. (2013): Two Beyond-primitive Extrasolar Planetesimals. ApJ 766(2):132. 10.1088/0004-637X/766/2/132, arXiv:1302.4799 [astro-ph.EP]
* Xu et al. [2014] Xu S., Jura M., Koester D., et al. (2014): Elemental Compositions of Two Extrasolar Rocky Planetesimals. ApJ 783(2):79. 10.1088/0004-637X/783/2/79, arXiv:1401.4252 [astro-ph.EP]
* Xu et al. [2017] Xu S., Zuckerman B., Dufour P., et al. (2017): The Chemical Composition of an Extrasolar Kuiper-Belt-Object. ApJ 836(1):L7. 10.3847/2041-8213/836/1/L7, arXiv:1702.02868 [astro-ph.EP]
* Xu et al. [2019] Xu S., Dufour P., Klein B., et al. (2019): Compositions of Planetary Debris around Dusty White Dwarfs. AJ 158(6):242. 10.3847/1538-3881/ab4cee, arXiv:1910.07197 [astro-ph.SR]
* York et al. [2000] York D. G., Adelman J., Anderson J.John E., et al. (2000): The Sloan Digital Sky Survey: Technical Summary. AJ 120(3):1579–1587. 10.1086/301513, arXiv:astro-ph/0006396 [astro-ph]
* Zuckerman & Reid [1998] Zuckerman B., Reid I. N. (1998): Metals in Cool DA White Dwarfs. ApJ 505(2):L143–L146. 10.1086/311608
* Zuckerman et al. [2003] Zuckerman B., Koester D., Reid I. N., et al. (2003): Metal Lines in DA White Dwarfs. ApJ 596(1):477–495. 10.1086/377492
* Zuckerman et al. [2007] Zuckerman B., Koester D., Melis C., et al. (2007): The Chemical Composition of an Extrasolar Minor Planet. ApJ 671(1):872–877. 10.1086/522223, arXiv:0708.0198 [astro-ph]
* Zuckerman et al. [2010] Zuckerman B., Melis C., Klein B., et al. (2010): Ancient Planetary Systems are Orbiting a Large Fraction of White Dwarf Stars. ApJ 722(1):725–736. 10.1088/0004-637X/722/1/725, arXiv:1007.2252 [astro-ph.SR]
* Zuckerman et al. [2011] Zuckerman B., Koester D., Dufour P., et al. (2011): An Aluminum/Calcium-rich, Iron-poor, White Dwarf Star: Evidence for an Extrasolar Planetary Lithosphere? ApJ 739(2):101. 10.1088/0004-637X/739/2/101, arXiv:1107.2167 [astro-ph.SR]
|
# Think about it! Improving defeasible reasoning by first modeling the
question scenario
Aman Madaan, Niket Tandon†, Dheeraj Rajagopal, Peter Clark†,
Yiming Yang, Eduard Hovy
Language Technologies Institute, Carnegie Mellon University, Pittsburgh, PA,
USA
† Allen Institute for Artificial Intelligence, Seattle, WA, USA
<EMAIL_ADDRESS>
{nikett<EMAIL_ADDRESS>
###### Abstract
Defeasible reasoning is the mode of reasoning where conclusions can be
overturned by taking into account new evidence. Existing cognitive science
literature on defeasible reasoning suggests that a person forms a mental model
of the problem scenario before answering questions. Our research goal asks
whether neural models can similarly benefit from envisioning the question
scenario before answering a defeasible query. Our approach is, given a
question, to have a model first create a graph of relevant influences, and
then leverage that graph as an additional input when answering the question.
Our system, curious, achieves a new state-of-the-art on three different
defeasible reasoning datasets. This result is significant as it illustrates
that performance can be improved by guiding a system to “think about” a
question and explicitly model the scenario, rather than answering
reflexively.111Code and data located at https://github.com/madaan/thinkaboutit
## 1 Introduction
Defeasible inference is a mode of reasoning where additional information can
modify conclusions Koons (2017). Here we consider the specific formulation and
challenge in Rudinger et al. (2020): Given that some premise $\mathbf{P}$
plausibly implies a hypothesis $\mathbf{H}$, does new information that the
situation is $\mathbf{S}$ weaken or strengthen the conclusion $\mathbf{H}$?
For example, consider the premise “The drinking glass fell” with a possible
implication “The glass broke”. New information that “The glass fell on a
pillow” here weakens the implication.
Figure 1: curious improves defeasible reasoning by modeling the question
scenario with an inference graph $G$ adapted from cognitive science
literature. The graph is encoded judiciously using our graph encoder $h(.)$,
resulting in end task performance improvement.
We borrow ideas from the cognitive science literature that supports defeasible
reasoning for humans with an _inference graph_ (Pollock, 2009, 1987).
Inference graphs formulation in Madaan et al. (2021), which we use in this
paper, draws connections between the $\mathbf{P}$, $\mathbf{H}$, and
$\mathbf{S}$ through mediating events. This can be seen as a _mental model_ of
the question scenario before answering the question Johnson-Laird (1983). This
paper asks the natural question: can modeling the question scenario with
inference graphs help machines in defeasible reasoning?
Our approach is as follows. First, given a question, generate an inference
graph describing important influences between question elements. Then, use
that graph as an additional input when answering the defeasible reasoning
query. Our proposed system, curious, comprises a graph generation module and a
graph encoding module to use the generated graph for the query (Figure 2).
To generate inference graphs, we build upon past work that uses a sequence to
sequence approach Madaan et al. (2021). However, our analysis revealed that
the graphs can often be erroneous, and curious also includes an error
correction module to generate higher quality inference graphs. This was
important because we found that better graphs are more helpful in the
downstream QA task.
Figure 2: An overview of curious
The generated inference graph is then used for the QA task on three existing
defeasible inference datasets from diverse domains, viz., $\delta$-snli
(natural language inference) Bowman et al. (2015), $\delta$-social (reasoning
about social norms) Forbes et al. (2020), and $\delta$-atomic (commonsense
reasoning) Sap et al. (2019). We show that the way the graph is encoded for
input is important. If we simply augment the question with the generated
graphs, there are some gains on all datasets. However, the accuracy improves
substantially across all datasets with a more judicious encoding of the graph-
augmented question that accounts for interactions between the graph nodes. To
achieve this, we use the mixture of experts approach Jacobs et al. (1991) to
include a mixture of experts layers during encoding, enabling the ability to
attend to specific nodes while capturing their interactions selectively.
In summary, our contribution is in drawing on the idea of an inference graph
from cognitive science to show benefits in a defeasible inference QA task.
Using an error correction module in the graph generation process, and a
judicious encoding of the graph augmented question, curious achieves a new
state-of-the-art over three defeasible datasets. This result is significant
also because our work illustrates that guiding a system to “think about" a
question before answering can improve performance.
Figure 3: An overview of our method to perform graph-augmented defeasible
reasoning using a hierarchical mixture of experts. First, moe-v selectively
pools the node representations to generate a representation
$\mathbf{h}_{\mathbf{G}}$ of the inference graph. Then, moe-gx pools the query
representation $\mathbf{h}_{\mathbf{x}}$ and the graph representation
generated by moe-v to pass to the upstream classifier.
## 2 Task
We use the defeasible inference task and datasets defined in Rudinger et al.
(2020), namely given an input $\mathbf{x}$ =
($\mathbf{P}$,$\mathbf{H}$,$\mathbf{S}$), predict the output
$\mathbf{y}\in\\{strengthens,weakens\\}$, where $\mathbf{P}$, $\mathbf{H}$,
and $\mathbf{S}$ are sentences describing a premise, hypothesis, and scenario
respectively, and $y$ denotes whether $S$ strengthens/weakens the plausible
conclusion that $\mathbf{H}$ follows from $\mathbf{P}$, as described in
Section 1.
## 3 Approach
Inspired by past results Madaan et al. (2021) that humans found inference
graphs useful for defeasible inference, we investigate whether neural models
can benefit from envisioning the question scenario using an inference graph
before answering a defeasible inference query.
##### Inference graphs
As inference graphs are central to our work, we give a brief description of
their structure next. Inference graphs were introduced in philosophy by
Pollock (2009) to aid defeasible reasoning for humans, and in NLP by Tandon et
al. (2019) for a counterfactual reasoning task. We interpret the inference
graphs as having four kinds of nodes Pollock (2009); Madaan et al. (2021):
* i.
Contextualizers (C-, C+): these nodes set the context around a situation and
connect to the $\mathbf{P}$.
* ii.
Situations (S, S-): these nodes are new situations that emerge which might
overturn an inference.
* iii.
Hypothesis (H-, H+): Hypothesis nodes describe the outcome/conclusion of the
situation.
* iv.
Mediators (M-, M+): Mediators are nodes that help bridge the knowledge gap
between a situation and a hypothesis node by explaining their connection
explicitly. These node can either act as a _weakener_ or _strengthener_.
Each node in an influence graph is labeled with an event (a sentence or a
phrase). The signs - and + capture the nature of the influence event node.
Concrete examples are present in Figures 1, 4, and in Appendix §D.
### 3.1 Overview of curious
Our system, curious, comprises three components, (i) a graph generator
$\textsc{gen}_{\text{init}}$, (ii) a graph corrector
$\textsc{gen}_{\text{corr}}$, (iii) a graph encoder (Figure 1).
$\textsc{gen}_{\text{init}}$ generates an inference graph from the input
$\mathbf{x}$. We borrow the sequence to sequence approach of
$\textsc{gen}_{\text{init}}$ from Madaan et al. (2021) without any
architectural changes. However, we found that the resulting graphs can often
be erroneous (which hurts task performance), so curious includes an error
correction module $\textsc{gen}_{\text{corr}}$ to generate higher quality
inference graphs that are then judiciously encoded using the graph encoder.
This encoded representation is then passed through a classifier to generate an
end task label. The overall architecture is shown in Figure 2.
### 3.2 Graph generator
Figure 4: The graphs generated by $\textsc{gen}_{\text{init}}$.The input graph
has repetitions for nodes $\\{C{-},C{+}\\}$ and $\\{S,S{-}\\}$. The corrected
graph generated by $\textsc{gen}_{\text{corr}}$ replaces the repetitions with
meaningful labels.
As the initial graph generator, we use the method described in Madaan et al.
(2021) ($\textsc{gen}_{\text{init}}$) to generate inference graphs for
defeasible reasoning.222We use their publicly available code and data Their
approach involves first training a graph-generation module, and then
performing zero-shot inference on a defeasible query to obtain an inference
graph. They obtain training data for the graph-generation module from wiqa
dataset Tandon et al. (2019). wiqa is a dataset of 2107
$(\mathbf{T}_{i},\mathbf{G}_{i})$ tuples, where $\mathbf{T}_{i}$ is the
passage text that describes a process (e.g., waves hitting a beach), and
$\mathbf{G}_{i}$ is the corresponding influence graph. The graph generator
$\textsc{gen}_{\text{init}}$ is trained as a seq2seq model, by setting
$\texttt{input}=\text{[Premise] }\mathbf{T}_{i}\mid\text{[Situation]
}\mathbf{S}_{i}\mid\text{[Hypothesis] }\mathbf{H}_{i}$, and
$\texttt{output}=\mathbf{G}_{i}$. Note that $\mathbf{S}_{i}$ and
$\mathbf{H}_{i}$ are nodes in the influence graph $\mathbf{G}_{i}$, allowing
grounded generation. [Premise], [Situation], [Hypothesis] are special tokens
used to demarcate the input.
### 3.3 Graph corrector
We found that 70% of the randomly sampled 100 graphs produced by
$\textsc{gen}_{\text{init}}$ (undesirably) had repeated nodes (an example of
repeated nodes is in Figure 4). Repeated nodes introduce noise because they
violate the semantic structure of a graph, e.g., in Figure 4, nodes C+ and C-
are repeated, although they are expected to have opposite semantics. Higher
graph quality yields better end task performance when using inference graphs
(as we will show in §4.3.1)
To repair such problems, we train a graph corrector,
$\textsc{gen}_{\text{corr}}$, that takes as input $\mathbf{G}^{\prime}$, and
as output it gives a graph $\mathbf{G}^{*}$, with repetitions fixed. To train
the model, we require ($\mathbf{G}^{\prime}$, $\mathbf{G}^{*}$) examples,
which we generate using a data augmentation technique described in the
Appendix §A. Because the nodes in the graph are from an open vocabulary, we
then train a T5 sequence-to-sequence model Raffel et al. (2020) with input =
$\mathbf{G}^{\prime}$ and output = $\mathbf{G}^{*}$. In summary, given a
defeasible query $\mathbf{PHS}$, we generate a potentially incorrect initial
graph $\mathbf{G}^{\prime}$ using $\textsc{gen}_{\text{init}}$. We then feed
$\mathbf{G}^{\prime}$ to $\textsc{gen}_{\text{corr}}$ to obtain an improved
graph $\mathbf{G}$.
### 3.4 Graph Encoder
For each defeasible query ($\mathbf{P}$, $\mathbf{H}$, $\mathbf{S}$), we add
the inference graph $\mathbf{G}$ from curious (the corrected graph from §3.3),
to provide additional context for the query, as we now describe.
We concatenate the components ($\mathbf{P}$, $\mathbf{H}$, $\mathbf{S}$) of
the defeasible query into a single sequence of tokens
$\mathbf{x}=(\mathbf{P}\|\mathbf{H}\|\mathbf{S})$, where $\|$ denotes
concatenation. Thus, each sample of our graph-augmented binary-classification
task takes the form $((\mathbf{x},\mathbf{G}),\mathbf{y})$, where
$\mathbf{y}\in\\{\text{{strengthener}, {weakener}}\\}$. Following Madaan et
al. (2021), we do not use edge labels and treat all the graphs as undirected
graphs.
##### Overview:
We first use a language model $\mathcal{L}$ to obtain a dense representation
$\mathbf{h}_{\mathbf{x}}$ for the defeasible query $\mathbf{x}$, and a dense
representation $\mathbf{h}_{\mathbf{v}}$ for each node
$\mathbf{v}\in\mathbf{G}$. The node representations $\mathbf{h}_{\mathbf{v}}$
are then pooled using a hierarchical mixture of experts (MoE) to obtain a
graph representation $\mathbf{h}_{\mathbf{G}}$. The query representation
$\mathbf{h}_{\mathbf{x}}$ and the graph representation
$\mathbf{h}_{\mathbf{G}}$ are combined to solve the defeasible task. We now
provide details on obtaining $\mathbf{h}_{\mathbf{x}}$,
$\mathbf{h}_{\mathbf{v}}$, $\mathbf{h}_{\mathbf{G}}$.
#### 3.4.1 Encoding the query and nodes
Let $\mathcal{L}$ be a pre-trained language model (in our case RoBERTa Liu et
al. (2019)). We use
$\mathbf{h}_{\mathbf{S}}=\mathcal{L}(\mathbf{S})\in\mathbb{R}^{d}$ to denote
the dense representation of sequence of tokens $\mathbf{S}$ returned by the
language model $\mathcal{L}$. Specifically, we use the pooled representation
of the beginning-of-sequence token <s> as the sequence representation.
We encode the defeasible query $\mathbf{x}$ and the nodes of the graph using
$\mathcal{L}$. Query representation is computed as
$\mathbf{h}_{\mathbf{x}}=\mathcal{L}(\mathbf{x})$, and we similarly obtain a
matrix of node representations $\mathbf{h}_{\mathbf{V}}$ by encoding each node
$\mathbf{v}$ in $\mathbf{G}$ with $\mathcal{L}$ as follows:
$\displaystyle\mathbf{h}_{\mathbf{V}}=[\mathbf{h}_{v_{1}};\mathbf{h}_{v_{2}};\ldots;\mathbf{h}_{|\mathbf{V}|}]$
(1)
where $\mathbf{h}_{v_{i}}\in\mathbb{R}^{d}$ refers to the dense representation
for the $i^{th}$ node of $\mathbf{G}$ derived from $\mathcal{L}$ (i.e.,
$\mathbf{h}_{v_{i}}=\mathcal{L}(v_{i})$), and
$\mathbf{h}_{\mathbf{V}}\in\mathbb{R}^{|\mathbf{V}|\times d}$ to refer to the
matrix of node representations.
#### 3.4.2 Graph representations using MoE
Recently, mixture-of-experts Jacobs et al. (1991); Shazeer et al. (2017);
Fedus et al. (2021) has emerged as a promising method of combining multiple
feature types. Mixture-of-experts (MoE) is especially useful when the input
consists of multiple facets, where each facet has a specific semantic meaning.
Previously, Gu et al. (2018); Chen et al. (2019) have used the ability of MoE
to pool disparate features on low-resource and cross-lingual language tasks.
Since each node in the inference graphs used by us plays a specific role in
defeasible reasoning (contextualizer, situation node, or mediator), we take
inspiration from these works to design a hierarchical MoE model Jordan and Xu
(1995) to pool node representations $\mathbf{h}_{\mathbf{V}}$ into a graph
representation $\mathbf{h}_{\mathbf{G}}$.
An MoE consists of $n$ expert networks
$\mathbf{E_{1}},\mathbf{E_{2}},\ldots,\mathbf{E_{n}}$ and a gating network
$\mathbf{M}$. Given an input $\mathbf{x}\in\mathbb{R}^{d}$, each expert
network $\mathbf{E_{i}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ learns a
transform over the input. The gating network
$\mathbf{M}:\mathbb{R}^{d}\rightarrow\Delta^{d}$ gives the weights
$\mathbf{p}=\\{p_{1},p_{2},\ldots,p_{n}\\}$ to combine the expert outputs for
input $\mathbf{x}$. Finally, the output $\mathbf{y}$ is returned as a convex
combination of the expert outputs:
$\displaystyle\mathbf{p}$ $\displaystyle=\mathbf{M}(\mathbf{x})$
$\displaystyle\mathbf{y}$ $\displaystyle=\sum_{i=1}^{n}p_{i}\mathbf{E_{i}(x)}$
(2)
The output $\mathbf{y}$ can either be the logits for an end task Shazeer et
al. (2017); Fedus et al. (2021) or pooled features that are passed to a
downstream learner Chen et al. (2019); Gu et al. (2018). The gating network
$\mathbf{M}$ and the expert networks
$\mathbf{E_{1}},\mathbf{E_{2}},\ldots,\mathbf{E_{n}}$ are trained end-to-end.
During learning, the gradients to $\mathbf{M}$ train it to generate a
distribution over the experts that favors the best expert for a given input.
Appendix §B presents a further discussion on our MoE formulation and an
analysis of the gradients.
##### Hierarchical MoE for defeasible reasoning
Different parts of the inference graphs might help answer a query to a
different degree. Further, for certain queries, graphs might not be helpful
(and could even be distracting), and the model could rely primarily on the
input query alone. This motivates a two-level architecture that can: (i)
select a subset of the nodes in the graph and ii) selectively reason across
the query and the graph to varying degrees.
Given these requirements, a hierarchical MoE Jordan and Jacobs (1994) model
presents itself as a natural choice to model this task. The first MoE (moe-v)
creates a graph representation by taking a convex combination of the node
representations. The second MoE (moe-gx) then takes a convex-combination of
the graph representation returned by moe-v and query representation and passes
it to an MLP for the downstream task.
* $\bullet$
moe-v consists of five node-experts and gating network to selectively pool
node representations $\mathbf{h}_{\mathbf{v}}$ to graph representation
$\mathbf{h}_{\mathbf{G}}$:
$\displaystyle\mathbf{p}$ $\displaystyle=\mathbf{M}(\mathbf{h}_{\mathbf{V}})$
$\displaystyle\mathbf{h}_{\mathbf{G}}$
$\displaystyle=\sum_{v\in\mathbf{V}}p_{v}E_{v}(v)$ (3)
* $\bullet$
moe-gx contains two experts (graph expert $E_{\mathbf{G}}$ and question expert
$E_{\mathbf{Q}}$) and a gating network to combine the graph representation
$\mathbf{h}_{\mathbf{G}}$ returned by moe-gx and the query representation
$\mathbf{h}_{\mathbf{x}}$:
$\displaystyle\mathbf{p}$
$\displaystyle=\mathbf{M}([\mathbf{h}_{\mathbf{G}};\mathbf{h}_{\mathbf{Q}}])$
$\displaystyle\mathbf{h}_{\mathbf{y}}$
$\displaystyle=E_{\mathbf{G}}(\mathbf{h}_{\mathbf{G}})+E_{\mathbf{Q}}(\mathbf{h}_{\mathbf{Q}})$
(4)
$\mathbf{h}_{\mathbf{y}}$ is then passed to a 1-layer MLP to perform
classification. The gates and the experts in our MoE model are single-layer
MLPs, with equal input and output dimensions for the experts.
## 4 Experiments
In this section, we empirically investigate if curious can improve defeasible
inference by first modeling the question scenario using inference graphs. We
also study the reasons for any improvements.
### 4.1 Experimental setup
Dataset | Split | # Samples | Total
---|---|---|---
$\delta$-atomic | train | 35,001 | 42,977
test | 4137
dev | 3839
$\delta$-social | train | 88,675 | 92,295
test | 1836
dev | 1784
$\delta$-snli | train | 77,015 | 95,795
test | 9438
dev | 9342
Table 1: Number of samples in each dataset by split.
##### Datasets
Our end task performance is measured on the three existing datasets for
defeasible inference provided by Rudinger et al. (2020):333
github.com/rudinger/defeasible-nli $\delta$-atomic, $\delta$-snli,
$\delta$-social (Table 1). These datasets exhibit substantial diversity
because of their domains: $\delta$-snli (natural language inference),
$\delta$-social (reasoning about social norms), and $\delta$-atomic
(commonsense reasoning). Thus, it would require a general model to perform
well across these diverse datasets.
##### Baselines and setup
The previous state-of-the-art (SOTA) is the RoBERTa Liu et al. (2019) model
presented in Rudinger et al. (2020), and we report the published numbers for
this baseline. For an additional baseline, we directly use the initial
inference graph $\mathbf{G}^{\prime}$ generated by
$\textsc{gen}_{\text{init}}$, and provide it to the model simply as a string
(i.e., sequence of tokens; a simple, often-used approach). This baseline is
called e2e-STR. We use the same hyperparameters as Rudinger et al. (2020), and
add a detailed description of the hyperparameters in Appendix §C. For all the
qa experiments, we report the accuracy on the test set using the checkpoint
with the highest accuracy on the development set. We use the McNemar’s test
McNemar (1947); Dror et al. (2018) and use $p<0.05$ as a threshold for
statistical significance. All the p-values are reported in Appendix §G.
### 4.2 Results
Table 2 compares QA accuracy on these datasets without and with modeling the
question scenario. The results suggest that we get consistent gains across all
datasets, with $\delta$-snli gaining about 4 points. curious achieves a new
state-of-the-art across three datasets, as well as now producing
justifications for its answers with inference graphs.
| $\delta$-atomic | $\delta$-snli | $\delta$-social
---|---|---|---
Prev-SOTA | 78.3 | 81.6 | 86.2
e2e-STR | 78.8 | 82.2 | 86.7
curious | 80.2* | 85.6* | 88.6*
Table 2: curious is better across all the datasets. This demonstrates that
understanding the question scenario through generating an inference graph
helps. * indicates statistical significance.
### 4.3 Understanding curious gains
In this section, we study the contribution of the main components of the
curious pipeline.
#### 4.3.1 Impact of graph corrector
We ablate the graph corrector module $\textsc{gen}_{\text{corr}}$ in curious
by directly supplying the output from $\textsc{gen}_{\text{init}}$ to the
graph encoder. Table 3 shows that this ablation consistently hurts across all
the datasets. $\textsc{gen}_{\text{corr}}$ provides 2 points gain across
datasets. This indicates that better graphs lead to better task performance,
assuming that $\textsc{gen}_{\text{corr}}$ actually reduces the noise. Next,
we investigate if $\textsc{gen}_{\text{corr}}$ can produce more informative
graphs.
| $\delta$-atomic | $\delta$-snli | $\delta$-social
---|---|---|---
$\mathbf{G}^{\prime}$ | 78.5 | 83.8 | 88.2
$G$ | 80.2* | 85.6* | 88.6
Table 3: Performance w.r.t. the graph used. $\mathbf{G}^{\prime}$ is the
initial graph from $\textsc{gen}_{\text{init}}$, $G$ is the corrected graph
from $\textsc{gen}_{\text{corr}}$. Better graphs lead to better task
performance. * indicates statistical significance.
##### Do graphs corrected by $\textsc{gen}_{\text{corr}}$ show fewer
repetitions?
We evaluate the repetitions in the graphs produced by
$\textsc{gen}_{\text{init}}$ and $\textsc{gen}_{\text{corr}}$ using two
metrics: (i) repetitions per graph: average number of repeated nodes in a
graph. (ii) % with repetitions: % of graphs with at least one repeated node.
| Repetitions | $\textsc{gen}_{\text{init}}$ | $\textsc{gen}_{\text{corr}}$
---|---|---|---
$\delta$-atomic | per graph | 2.05 | 1.26
| % graphs | 72 | 48
$\delta$-snli | per graph | 2.09 | 1.18
| % graphs | 73 | 46
$\delta$-social | per graph | 2.2 | 1.32
| % graphs | 75 | 49
overall | per graph | | $\Delta$ -40%
| % graphs | | $\Delta$ -25.7%
Table 4: $\textsc{gen}_{\text{corr}}$ reduces the inconsistencies in graphs.
The number of repetitions per graph and percentage of graphs with some
repetition, both improve.
Table 4 shows $\textsc{gen}_{\text{corr}}$ does reduce repetitions by
approximately 40% (2.11 to 1.25) per graph across all datasets, and also
reduces the fraction of graphs with at least one repetition by 25.7% across.
#### 4.3.2 Impact of graph encoder
We experiment with two alternative approaches to graph encoding to compare our
MoE approach by using the graphs generated by $\textsc{gen}_{\text{corr}}$:
1\. Graph convolutional networks: We follow the approach of Lv et al. (2020)
who use gcn Kipf and Welling (2017) to learn rich node representations from
graphs. Broadly, node representations are initialized by $\mathcal{L}$ and
then refined using a gcn. Finally, multi-headed attention Vaswani et al.
(2017) between question representation $\mathbf{h}_{\mathbf{x}}$ and the node
representations is used to yield $\mathbf{h}_{\mathbf{G}}$. We add a detailed
description of this method in Appendix §H.
2\. String based representation: Another popular approach Sakaguchi et al.
(2021) is to concatenate the string representation of the nodes, and then
using $\mathcal{L}$ to obtain the graph representation
$\mathbf{h}_{\mathbf{G}}$ $=\mathcal{L}(v_{1}\|v_{2}\|..)$ where $\|$ denotes
string concatenation.
Table 5 shows that MoE graph encoder improves end task performance
significantly compared to the baseline.444Appendix §E provides an analysis on
time and memory requirements. In the following analysis, we study the reasons
for these gains in-depth.
We hypothesize that gcn is less resistant to noise than MoE in our setting,
thus causing a lower performance. The graphs augmented with each query are not
human-curated and are instead generated by a language model in a zero-shot
inference setting. Thus, the gcn style message passing might amplify the noise
in graph representations. On the other hand, moe-v first selects the most
useful nodes to answer the query to form the graph representation
$\mathbf{h}_{\mathbf{G}}$. Further, moe-gx can also decide to completely
discard the graph representations, as it does in many cases where the true
answer for the defeasible query is weakens.
To further establish the possibility of message passing hampering the
downstream task performance, we experiment with a gcn-MoE hybrid, wherein we
first refine the node representations using a 2-layer gcn as used by Lv et al.
(2020), and then pool the node representations using an MoE. We found the
results to be about the same as ones we obtained with gcn (3rd-row Table 5),
indicating that bad node representations are indeed the root cause for the bad
performance of gcn. This is also supported by Shi et al. (2019) who found that
noise propagation directly deteriorates network embedding and gcn is sensitive
to noise.
Interestingly, graphs help the end-task even when encoded using a relatively
simple str based encoding scheme, further establishing their utility.
| $\delta$-atomic | $\delta$-snli | $\delta$-social
---|---|---|---
str | 79.5 | 83.1 | 87.2
gcn | 78.9 | 82.4 | 88.1
gcn \+ MoE | 78.7 | 84.3 | 87.8
MoE | 80.2 | 85.6 | 88.6
Table 5: Contribution of MoE-based graph encoding compared with alternative
graph encoding methods. The gains of MoE over gcn are statistically
significant for all the datasets, and the gains over str are significant for
$\delta$-snli and $\delta$-social.
#### 4.3.3 Detailed MoE analysis
We now analyze the two MoEs used in curious: (i) the MoE over the nodes
(moe-v), and (ii) the MoE over $\mathbf{G}$ and input $x$ (moe-gx).
Figure 5: moe-gx gate values for the classes strengthens and weakens, averaged
over the three datasets.
##### moe-gx performs better for $y=\text{strengthens:}$
Figure 5 shows that the graph makes a stronger contribution than the input,
when the label is strengthens. In instances where the label is weakens, the
gate of moe-gx gives a higher weight to the question. This trend was present
across all the datasets. We conjecture that this happens because language
models are tuned to generate events that happen rather than events that do
not. In the case of a weakener, the nodes must be of the type event1 leads to
less of event2, whereas language models are naturally trained for event1 leads
to event2. Understanding this in-depth requires further investigation in the
future.
##### moe-v relies more on specific nodes:
We study the distribution over the types of nodes and their contribution to
moe-v. Recall from Figure 3 that C- and C+ nodes are contextualizers that
provide more background context to the question, and S- node is typically an
inverse situation (i.e., inverse $\mathbf{S}$), while M- and M+ are the
mediator nodes leading to the hypothesis. Figure 6 shows that the situation
node S- was the most important, followed by the contextualizer and the
mediator. Notably, our analysis shows that mediators are less important for
machines than they were for humans in the experiments conducted by Madaan et
al. (2021). This is probably because humans and machines use different pieces
of information. As our error analysis shows in §5, the mediators can be
redundant given the query $\mathbf{x}$. Humans might have used the redundancy
to reinforce their beliefs, whereas machines leverage the unique signals
present in S- and the contextualizers.
Figure 6: moe-v gate values for the three datasets.
##### moe-v, moe-gx have a peaky distribution:
A peaky distribution over the gate values implies that the network is
judiciously selecting the right expert for a given input. We compute the
average entropy of moe-v and moe-gx and found the entropy values to be 0.52
(max 1.61) for moe-v, and 0.08 (max 0.69) for moe-gx. The distribution of the
gate values of moe-v is relatively flat, indicating that specialization of the
node experts might have some room for improvement (additional discussion in
Appendix §B). Analogous to scene graphs-based explanations in visual QA Ghosh
et al. (2019), peaky distributions over nodes can be considered as an
explanation through supporting nodes.
##### moe-v learns the node semantics:
The network learned the semantic grouping of the nodes (contextualizers,
situation, mediators), which became evident when plotting the correlation
between the gate weights. As Figure 7 shows, there is a strong negative
correlation between the situation nodes and the context nodes, indicating that
only one of them is activated at a time.
Figure 7: Correlation between probability assigned to each semantic type of
the node by moe-v
## 5 Error analysis
| |
---|---|---
| now fail | now succ
prev. fail | $\delta$-atomic 615 $\delta$-snli 197 $\delta$-social 772 | $\delta$-atomic 294 $\delta$-snli 124 $\delta$-social 398
prev. succ | $\delta$-atomic 207 $\delta$-snli 68 $\delta$-social 302 | $\delta$-atomic 3022 $\delta$-snli 1448 $\delta$-social 7967
Table 6: Confusion matrix: can curious fix previously failing or successful
examples?
Table 6 shows that curious is able to correct several previously wrong
examples. When curious corrected previously failing cases, the moe-v relied
more on mediators, as the average mediator probabilities go up from 0.09 to
0.13 averaged over the datasets. curious still fails, and more concerning are
the cases when previously successful examples now fail. To study this, we
annotate 50 random dev samples over the three datasets (26/24 examples for
weakens/strengthens label). For each sample, a human-annotated if the graph
had errors. We observe the following error categories:555Concrete examples in
Appendix §F
* $\bullet$
All nodes off-topic (4%): The graph nodes were not on topic. This (rarely)
happens when curious cannot distinguish the sense of a word in the input
question. For instance, $\mathbf{S}$ = there is a water fountain in the center
– curious generated based on an incorrect word sense of natural water spring.
* $\bullet$
Repeated nodes (20%): These may be exact or near-exact matches. Node pairs
with similar effects tend to be repeated in some samples. E.g., the S- node is
often repeated with contextualizer C- perhaps because these nodes indirectly
affect graph nodes in a similar way.
* $\bullet$
Mediators are uninformative (34%): The mediating nodes are not correct or
informative. One source of these errors is when the $\mathbf{H}$ and
$\mathbf{S}$ are nearly connected by a single hop, e.g., $\mathbf{H}$ =
personX pees, and $\mathbf{S}$ = personX drank a lot of water previously.
* $\bullet$
Good graphs are ineffective (42%): These graphs contained the information
required to answer the question, but the gating MOE mostly ignored this graph.
This could be attributed in part to the observation in the histogram in Figure
5, that samples with weakens label disproportionately ignore the graph.
In accordance with the findings of Rudinger et al. (2020), the maximum
percentage of errors was in $\delta$-atomic, in part due to low question
quality.
## 6 Explainability
In this section, we analyze the explainability of curious model. Jacovi and
Goldberg (2020) note that an explanation should aim towards two complementary
goals: i) Plausibility: provide an interpretation of system outputs that is
convincing for humans, and ii) Faithfulness: capture the actual reasoning
process of a model. We discuss how our approach takes a step towards
addressing these goals.
##### Plausibility
In a prior work, Madaan et al. (2021) show that human annotators selectively
picked and chose parts of the graph that explained a model decision and
enabled them in improving on the task of defeasible reasoning. We show in
§4.3.3, the MoE gate values gives insights into the part of the graph
(contextualizer, mediator, situation node) that the model leveraged to answer
a query. Our model thus produces a reasoning chain that is similar to the
explanation that humans understand, providing a step towards building
inherently plausible models, while also achieving better performance.
##### Measuring faithfulness w.r.t. graphs
Since faithfulness is a widely debated term, we restrict its definition to
measure faithfulness w.r.t to the reasoning graph. This can be measured by the
correlation between the model performance and graph correctness. A high
correlation implies that the model uses both the graph and query to generate
an answer and thus is faithful to the stated reasoning mechanism (i.e., graphs
used to answer a question). Our analysis reveals this to be the case: in cases
where the model answers incorrectly, 42% of the graphs were entirely correct
(§5). In contrast, when the model answers correctly, 82% of the graphs are
correct. In summary, we hope that curious serves as a step towards building
reasoning models that are both plausible and faithful.
## 7 Related work
##### Mental Models
Cognitive science has long promoted mental models - coherent, constructed
representations of the world - as central to understanding, communication, and
problem-solving Johnson-Laird (1983); Gentner and Stevens (1983); Hilton
(1996). Our work draws on these ideas, using inference graphs to represent the
machine’s “mental model” of the problem at hand. Building the inference graph
can be viewed as first asking clarification questions about the context before
answering. This is similar to self-talk Shwartz et al. (2020) but directed
towards eliciting chains of influence.
##### Injecting Commonsense Knowledge
Many prior systems use commonsense knowledge to aid question-answering, e.g.,
using sentences retrieved from a corpus Yang et al. (2019); Guu et al. (2020),
or with knowledge generated from a separate source Shwartz et al. (2020);
Bosselut et al. (2019); and injected either as extra sentences fed directly to
the model Clark et al. (2020), via the loss function Tandon et al. (2018), or
via attention Ma et al. (2019). Unlike prior work, we use conditional language
generation techniques to create graphs that are relevant to answering a
question.
##### Encoding Graph Representations
Several existing methods use graphs as an additional input for commonsense
reasoning Sun et al. (2018); Lin et al. (2019); Lv et al. (2020); Feng et al.
(2020); Bosselut et al. (2021); Ma et al. (2021); Kapanipathi et al. (2020).
These methods first retrieve a graph relevant to a question using information
retrieval techniques and then encode the graph using graph representation
techniques like gcn Kipf and Welling (2017) and graph attention Velickovic et
al. (2018). Different from these works, we use a graph generated from the
query for answering the commonsense question. The graphs consumed by these
works contain entities grounded in knowledge graphs like ConceptNet Speer et
al. (2017), whereas we perform reasoning over event inference graphs where
each node describes an event. Our best model uses a mixture-of-experts (MoE)
Jacobs et al. (1991) model to pool multi-faceted input. Prior work has shown
the effectiveness of using MoE for graph classification Zhou and Luo (2019);
Hu et al. (2021), cross-lingual language learning Chen et al. (2019); Gu et
al. (2018), and model ensemble learning Fedus et al. (2021); Shazeer et al.
(2017). To the best of our knowledge, we are the first to use MoE for learning
and pooling graph representations for qa task.
## 8 Summary and Conclusion
Cognitive science suggests that people form “mental models” of a situation to
answer questions about it. Drawing on those ideas, we have presented a simple
instantiation in which the situational model is an inference graph. Different
from gcn-based models popular in graph learning, we use mixture-of-experts to
pool graph representations. Our experiments show that MoE-based pooling can be
a strong (both in terms of performance and explainability) alternative to gcn
for graph-based learning for reasoning tasks. Our method establishes a new
state-of-the-art on three defeasible reasoning datasets. Overall, our method
shows that performance can be improved by guiding a system to “think about” a
question and explicitly model the scenario, rather than answering reflexively.
## Acknowledgments
We thank the anonymous reviewers for their feedback. Special thanks to
reviewer 2 for their insightful comments on our MoE formulation. This material
is partly based on research sponsored in part by the Air Force Research
Laboratory under agreement number FA8750-19-2-0200. The U.S. Government is
authorized to reproduce and distribute reprints for Governmental purposes
notwithstanding any copyright notation thereon. The views and conclusions
contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies or endorsements, either
expressed or implied, of the Air Force Research Laboratory or the U.S.
Government.
## References
* Bosselut et al. (2021) Antoine Bosselut, Ronan Le Bras, and Yejin Choi. 2021. Dynamic neuro-symbolic knowledge graph construction for zero-shot commonsense question answering. In _Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI)_.
* Bosselut et al. (2019) Antoine Bosselut, Hannah Rashkin, Maarten Sap, Chaitanya Malaviya, Asli Celikyilmaz, and Yejin Choi. 2019. COMET: Commonsense transformers for automatic knowledge graph construction. In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 4762–4779, Florence, Italy. Association for Computational Linguistics.
* Bowman et al. (2015) Samuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015\. A large annotated corpus for learning natural language inference. In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 632–642, Lisbon, Portugal. Association for Computational Linguistics.
* Chen et al. (2019) Xilun Chen, Ahmed Hassan Awadallah, Hany Hassan, Wei Wang, and Claire Cardie. 2019\. Multi-source cross-lingual model transfer: Learning what to share. In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3098–3112, Florence, Italy. Association for Computational Linguistics.
* Clark et al. (2020) P. Clark, Oren Etzioni, Daniel Khashabi, Tushar Khot, B. D. Mishra, Kyle Richardson, Ashish Sabharwal, Carissa Schoenick, Oyvind Tafjord, Niket Tandon, Sumithra Bhakthavatsalam, Dirk Groeneveld, Michal Guerquin, and Michael Schmitz. 2020. From ’f’ to ’a’ on the n.y. regents science exams: An overview of the aristo project. _AI Mag._ , 41:39–53.
* Dror et al. (2018) Rotem Dror, Gili Baumer, Segev Shlomov, and Roi Reichart. 2018. The hitchhiker’s guide to testing statistical significance in natural language processing. In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1383–1392, Melbourne, Australia. Association for Computational Linguistics.
* Falcon (2019) et al. Falcon, WA. 2019. Pytorch lightning. _GitHub. Note: https://github.com/PyTorchLightning/pytorch-lightning_ , 3.
* Fedus et al. (2021) William Fedus, Barret Zoph, and Noam Shazeer. 2021. Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity. _arXiv preprint arXiv:2101.03961_.
* Feng et al. (2020) Yanlin Feng, Xinyue Chen, Bill Yuchen Lin, Peifeng Wang, Jun Yan, and Xiang Ren. 2020. Scalable multi-hop relational reasoning for knowledge-aware question answering. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1295–1309, Online. Association for Computational Linguistics.
* Forbes et al. (2020) Maxwell Forbes, Jena D. Hwang, Vered Shwartz, Maarten Sap, and Yejin Choi. 2020\. Social chemistry 101: Learning to reason about social and moral norms. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 653–670, Online. Association for Computational Linguistics.
* Gentner and Stevens (1983) Dedre Gentner and Albert L. Stevens. 1983. _Mental Models_. Lawrence Erlbaum Associates.
* Ghosh et al. (2019) S. Ghosh, Giedrius Burachas, Arijit Ray, and Avi Ziskind. 2019. Generating natural language explanations for visual question answering using scene graphs and visual attention. _ArXiv_ , abs/1902.05715.
* Gu et al. (2018) Jiatao Gu, Hany Hassan, Jacob Devlin, and Victor O.K. Li. 2018. Universal neural machine translation for extremely low resource languages. In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ , pages 344–354, New Orleans, Louisiana. Association for Computational Linguistics.
* Guu et al. (2020) Kelvin Guu, Kenton Lee, Zora Tung, Panupong Pasupat, and Mingwei Chang. 2020. Retrieval augmented language model pre-training. In _International Conference on Machine Learning_ , pages 3929–3938. PMLR.
* Hilton (1996) D. Hilton. 1996. Mental models and causal explanation: Judgements of probable cause and explanatory relevance. _Thinking & Reasoning_, 2:273–308.
* Hu et al. (2021) Fenyu Hu, Liping Wang, Shu Wu, Liang Wang, and Tieniu Tan. 2021. Graph classification by mixture of diverse experts. _arXiv preprint arXiv:2103.15622_.
* Jacobs et al. (1991) Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. 1991\. Adaptive mixtures of local experts. _Neural computation_ , 3(1):79–87.
* Jacovi and Goldberg (2020) Alon Jacovi and Yoav Goldberg. 2020. Towards faithfully interpretable NLP systems: How should we define and evaluate faithfulness? In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 4198–4205, Online. Association for Computational Linguistics.
* Johnson-Laird (1983) P. Johnson-Laird. 1983. _Mental Models : Towards a Cognitive Science of Language_. Harvard University Press.
* Jordan and Jacobs (1994) Michael I Jordan and Robert A Jacobs. 1994. Hierarchical mixtures of experts and the em algorithm. _Neural computation_ , 6(2):181–214.
* Jordan and Xu (1995) Michael I Jordan and Lei Xu. 1995. Convergence results for the em approach to mixtures of experts architectures. _Neural networks_ , 8(9):1409–1431.
* Kapanipathi et al. (2020) Pavan Kapanipathi, Veronika Thost, Siva Sankalp Patel, Spencer Whitehead, Ibrahim Abdelaziz, Avinash Balakrishnan, Maria Chang, Kshitij P. Fadnis, R. Chulaka Gunasekara, Bassem Makni, Nicholas Mattei, Kartik Talamadupula, and Achille Fokoue. 2020. Infusing knowledge into the textual entailment task using graph convolutional networks. In _The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020_ , pages 8074–8081. AAAI Press.
* Kipf and Welling (2017) Thomas N. Kipf and Max Welling. 2017. Semi-supervised classification with graph convolutional networks. In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_. OpenReview.net.
* Koons (2017) Robert Koons. 2017. Defeasible Reasoning. In Edward N. Zalta, editor, _The Stanford Encyclopedia of Philosophy_ , Winter 2017 edition. Metaphysics Research Lab, Stanford University.
* Lin et al. (2019) Bill Yuchen Lin, Xinyue Chen, Jamin Chen, and Xiang Ren. 2019. KagNet: Knowledge-aware graph networks for commonsense reasoning. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 2829–2839, Hong Kong, China. Association for Computational Linguistics.
* Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. Roberta: A robustly optimized bert pretraining approach. _arXiv preprint arXiv:1907.11692_.
* Lv et al. (2020) Shangwen Lv, Daya Guo, Jingjing Xu, Duyu Tang, Nan Duan, Ming Gong, Linjun Shou, Daxin Jiang, Guihong Cao, and Songlin Hu. 2020. Graph-based reasoning over heterogeneous external knowledge for commonsense question answering. In _The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020_ , pages 8449–8456. AAAI Press.
* Ma et al. (2019) Kaixin Ma, Jonathan Francis, Quanyang Lu, Eric Nyberg, and Alessandro Oltramari. 2019. Towards generalizable neuro-symbolic systems for commonsense question answering. In _Proceedings of the First Workshop on Commonsense Inference in Natural Language Processing_ , pages 22–32, Hong Kong, China. Association for Computational Linguistics.
* Ma et al. (2021) Kaixin Ma, Filip Ilievski, Jonathan Francis, Yonatan Bisk, Eric Nyberg, and Alessandro Oltramari. 2021. Knowledge-driven data construction for zero-shot evaluation in commonsense question answering. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pages 13507–13515.
* Madaan et al. (2021) Aman Madaan, Dheeraj Rajagopal, Niket Tandon, Yiming Yang, and Eduard Hovy. 2021\. Could you give me a hint ? generating inference graphs for defeasible reasoning. In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 5138–5147, Online. Association for Computational Linguistics.
* McNemar (1947) Quinn McNemar. 1947. Note on the sampling error of the difference between correlated proportions or percentages. _Psychometrika_ , 12(2):153–157.
* Paszke et al. (2017) Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. 2017. Automatic differentiation in pytorch. _NIPS 2017 Workshop Autodiff Submission_.
* Pollock (1987) J. Pollock. 1987. Defeasible reasoning. _Cogn. Sci._ , 11:481–518.
* Pollock (2009) J. Pollock. 2009. A recursive semantics for defeasible reasoning. In _Argumentation in Artificial Intelligence_.
* Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2020. Exploring the limits of transfer learning with a unified text-to-text transformer. _Journal of Machine Learning Research_ , 21:1–67.
* Rudinger et al. (2020) Rachel Rudinger, Vered Shwartz, Jena D. Hwang, Chandra Bhagavatula, Maxwell Forbes, Ronan Le Bras, Noah A. Smith, and Yejin Choi. 2020. Thinking like a skeptic: Defeasible inference in natural language. In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 4661–4675, Online. Association for Computational Linguistics.
* Sakaguchi et al. (2021) Keisuke Sakaguchi, Chandra Bhagavatula, Ronan Le Bras, Niket Tandon, Peter Clark, and Yejin Choi. 2021. proscript: Partially ordered scripts generation via pre-trained language models. _arxiv_.
* Sap et al. (2019) Maarten Sap, Ronan Le Bras, Emily Allaway, Chandra Bhagavatula, Nicholas Lourie, Hannah Rashkin, Brendan Roof, Noah A. Smith, and Yejin Choi. 2019. ATOMIC: an atlas of machine commonsense for if-then reasoning. In _The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019_ , pages 3027–3035. AAAI Press.
* Shazeer et al. (2017) Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc V. Le, Geoffrey E. Hinton, and Jeff Dean. 2017. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_. OpenReview.net.
* Shi et al. (2019) M. Shi, Yufei Tang, Xingquan Zhu, and J. Liu. 2019. Feature-attention graph convolutional networks for noise resilient learning. _ArXiv_ , abs/1912.11755.
* Shwartz et al. (2020) Vered Shwartz, Peter West, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi. 2020\. Unsupervised commonsense question answering with self-talk. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 4615–4629, Online. Association for Computational Linguistics.
* Speer et al. (2017) Robyn Speer, Joshua Chin, and Catherine Havasi. 2017. Conceptnet 5.5: An open multilingual graph of general knowledge. In _Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4-9, 2017, San Francisco, California, USA_ , pages 4444–4451. AAAI Press.
* Sun et al. (2018) Haitian Sun, Bhuwan Dhingra, Manzil Zaheer, Kathryn Mazaitis, Ruslan Salakhutdinov, and William Cohen. 2018. Open domain question answering using early fusion of knowledge bases and text. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 4231–4242, Brussels, Belgium. Association for Computational Linguistics.
* Tandon et al. (2018) Niket Tandon, Bhavana Dalvi, Joel Grus, Wen-tau Yih, Antoine Bosselut, and Peter Clark. 2018. Reasoning about actions and state changes by injecting commonsense knowledge. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 57–66, Brussels, Belgium. Association for Computational Linguistics.
* Tandon et al. (2019) Niket Tandon, Bhavana Dalvi, Keisuke Sakaguchi, Peter Clark, and Antoine Bosselut. 2019. WIQA: A dataset for “what if…” reasoning over procedural text. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 6076–6085, Hong Kong, China. Association for Computational Linguistics.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In _Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA_ , pages 5998–6008.
* Velickovic et al. (2018) Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2018. Graph attention networks. In _6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings_. OpenReview.net.
* Wolf et al. (2019) Thomas Wolf, L Debut, V Sanh, J Chaumond, C Delangue, A Moi, P Cistac, T Rault, R Louf, M Funtowicz, et al. 2019. Huggingface’s transformers: State-of-the-art natural language processing. _ArXiv, abs/1910.03771_.
* Yang et al. (2019) Wei Yang, Yuqing Xie, Aileen Lin, Xingyu Li, Luchen Tan, Kun Xiong, Ming Li, and Jimmy Lin. 2019. End-to-end open-domain question answering with BERTserini. In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics (Demonstrations)_ , pages 72–77, Minneapolis, Minnesota. Association for Computational Linguistics.
* Yang and Liu (1999) Yiming Yang and Xin Liu. 1999. A re-examination of text categorization methods. In _Proceedings of the 22nd annual international ACM SIGIR conference on Research and development in information retrieval_ , pages 42–49.
* Zhou and Luo (2019) Xuanyu Zhou and Yuanhang Luo. 2019. Explore mixture of experts in graph neural networks. _Stanford CS224W_.
## Appendix A Training graph corrector
As mentioned in Section §3.2, the graph generator $\textsc{gen}_{\text{init}}$
is trained as a seq2seq model from wiqa with $\texttt{input}=\text{[Premise]
}\mathbf{T}_{i}\mid\text{[Situation] }\mathbf{S}_{i}\mid\text{[Hypothesis]
}\mathbf{H}_{i}$, and $\texttt{output}=\mathbf{G}_{i}$. Graphs in wiqa
additionally capture the influence that the situation has on the hypothesis.
Denoting this influence label by $y_{i}$ can be either helps or hurts
From our experiments, we observe that appending $y_{i}$ to the training data
(from $\texttt{input}=\text{[Premise] }\mathbf{T}_{i}\mid\text{[Situation]
}\mathbf{S}_{i}\mid\text{[Hypothesis] }\mathbf{H}_{i}$ to
$\texttt{input}=\text{[Premise] }\mathbf{T}_{i}\mid\text{[Situation]
}\mathbf{S}_{i}\mid\text{[Hypothesis] }\mathbf{H}_{i}\mid y_{i}$) reduces
repetitions by 13%.
We refer to this data generator as $\textsc{gen}_{\text{init}}^{*}$, and the
graphs produced by it as $\mathbf{G}^{*}$. However, we do not have access to
$y$ during test time, and thus $\textsc{gen}_{\text{init}}^{*}$ cannot be used
directly to produce $\mathbf{G}^{*}$ for defeasible queries. We circumvent
this by using $\textsc{gen}_{\text{init}}^{*}$ to train a graph-to-graph
generation model, that takes as input $\mathbf{G}^{\prime}$ and generates
$\mathbf{G}^{*}$ as output ($\mathbf{G}^{\prime}\rightarrow\mathbf{G}^{*}$).
We call this system $\textsc{gen}_{\text{corr}}$. We give an overview of the
process in Figure 8. In Figure 9, we give examples of an intial graph produced
by $\textsc{gen}_{\text{init}}$, the corresponding graph produced by
$\textsc{gen}_{\text{init}}^{*}$, and the graph produced by
$\textsc{gen}_{\text{corr}}$.
Figure 8: Training data generation to train $\textsc{gen}_{\text{corr}}$.
Figure 9: The graphs generated by $\textsc{gen}_{\text{init}}$ (left),
$\textsc{gen}_{\text{init}}^{*}$ (middle), and $\textsc{gen}_{\text{corr}}$
(right).The input graph has repetitions for nodes $\\{C{-},S{-}\\}$,
$\\{C{+},H{+}\\}$, and $\\{M{-},M{+}\\}$. The corrected graph replaces the
repetitions with meaningful labels.
## Appendix B MoE gradient analysis
Figure 10: MoE gradient analysis setup: we consider a simple setting where the
weighted output of the experts (using the expert weights $p$) is directly fed
to a softmax and is used for generating class probabilities $\hat{y}$.
We restate Equation 2 for quick reference:
$\displaystyle\mathbf{p}$ $\displaystyle=\mathbf{M}(\mathbf{x})$
$\displaystyle\mathbf{o}$ $\displaystyle=\sum_{i=1}^{n}p_{i}\mathbf{E_{i}(x)}$
where we have changed the notation slightly to use $\mathbf{o}$ as the MoE
output instead of $\mathbf{y}$. We also refer to $\mathbf{E_{i}}(x)$ as
$\mathbf{E_{i}}$. Further, $o_{j}=\sum_{i=1}^{n}p_{i}E_{ij}$. We present the
analysis for a generic multi-class classification setting with $k$ classes,
with training done using a cross-entropy loss $\mathcal{L}$ (Figure 10) Let
$\hat{y}_{c}$ be the normalized probability of the correct class $c$
calculated using softmax:
$\displaystyle\hat{y}_{c}$
$\displaystyle=\frac{\exp(o_{c})}{\sum_{j=1}^{k}\exp(o_{j})}$
$\displaystyle=\frac{\exp(\sum_{i=1}^{n}p_{i}E_{ic})}{\sum_{j=1}^{k}\exp(\sum_{i=1}^{n}p_{i}E_{ij})}$
Let $\mathcal{L}$ be the cross-entropy loss:
$\displaystyle\mathcal{L}$
$\displaystyle=-\log\hat{y}_{c}=-o_{c}+\log{\sum_{j=1}^{k}\exp(o_{j})}$
$\displaystyle=-\sum_{i=1}^{n}p_{i}E_{ic}+\log{\sum_{j=1}^{k}\exp(\sum_{i=1}^{n}p_{i}E_{ij})}$
##### Evaluating $\frac{\partial\mathcal{L}}{\partial p_{m}}$
The derivatives w.r.t. the $m^{th}$ expert gate probability $p_{m}$ is given
by:
$\displaystyle\frac{\partial\mathcal{L}}{\partial p_{m}}$
$\displaystyle=-E_{mc}+\frac{\sum_{j=1}^{k}E_{mj}\exp(\sum_{i=1}^{n}p_{i}E_{ij})}{\sum_{j=1}^{k}\exp(\sum_{i=1}^{n}p_{i}E_{ij})}$
$\displaystyle=-E_{mc}+\sum_{j=1}^{k}\hat{y}_{j}E_{mj}$
$\displaystyle=-E_{mc}(1-\hat{y}_{c})+\sum_{j=1,j\not=c}^{k}\hat{y}_{j}E_{mj}$
(5)
##### Evaluating $\frac{\partial\mathcal{L}}{\partial E_{mc}}$
the derivatives w.r.t. the logits $E_{mc}$ (logit for the correct class by
$m^{th}$ expert) is given by:
$\displaystyle\frac{\partial\mathcal{L}}{\partial E_{mc}}$
$\displaystyle=-p_{m}+\frac{\exp(o_{c})p_{m}}{\sum_{j=1}^{k}\exp o_{j}}$
$\displaystyle=-p_{m}(1-\hat{y}_{c})$ (6)
Equations 5 and 6 have natural interpretations: the gradient on both the
mixture probability $p_{m}$ and the logits $E_{mc}$ will be 0 (note that for
Equation 5, $\mathbf{y}^{c}=1\implies\mathbf{y}^{j}=0$ for $j\not=c$) when the
network makes perfect predictions ($\hat{y}_{c}=1$). As noted by Jacobs et al.
(1991) (Section 1), this might cause the network to specialize slower, as the
gradient will be small for experts that are helping in making the correct
prediction. They suggest a different loss function that promotes faster
specialization by redefining the error function in terms of a mixture
distribution, with the mixture weights supplied by the $p_{i}$ terms.
Analyzing the effect of loss function for applications where the MoE is used
to pool representations remains an interesting future work.
## Appendix C Hyperparameters
##### Training details
All of our experiments were done on a single Nvidia GeForceRTX 2080 Ti. We
base our implementation on PyTorch Paszke et al. (2017) and also use PyTorch
Lightning Falcon (2019) and Huggingface Wolf et al. (2019). The gates and the
experts in our MoE model were a single layer MLP. For the experts, we set the
input size set to be the same as output size. Table 7 shows the parameters
shared by all the methods, and 8 shows the hyperparameters applicable to gcn
encoder.
Hyperparameter | Value
---|---
Pre-trained model | RoBERTa-base
Learning rate | 2e-5
Gradient accumulation batches | 2
Num epochs | 30
Optimizer | AdamW
Dropout | 0.1
Learning rate scheduling | linear
Warmup | 3 epochs
Batch size | 16
Weight decay | 0.01
Gradient clipping | 1.0
Table 7: General hyperparameters used by all the models. Hyperparameter | Value
---|---
# Layers | 2
Layer dropout | 0.1
Number of attention heads | 1
Attention dimension | 256
Table 8: Hyperparameters specific to gcn.
## Appendix D Schema of an influence graph
Figure 11 shows the skeleton of an influence graph.
Figure 11: Schema of an inference graph.
## Appendix E Runtime Analysis
Finally, we discuss the cost-performance tradeoffs for various encoding
mechanisms (Table 9). As Table 9 shows, both gcn and MoE take about 7% more
number of parameters than the str encoding scheme and have about 2x the
runtime. Further, as we use one expert per node, the number of parameters
scales linearly with the number of nodes. While this is not prohibitive in our
setting (each graph has a small number of nodes), our analysis shows that the
behavior of the nodes that have similar semantics is correlated, indicating
that the experts for those nodes can share parameters. Alternatively, MoE with
more than two layers Jordan and Xu (1995) can also help in scaling the number
of parameters only logarithmically with the number of nodes.
Method | str | gcn | MoE
---|---|---|---
#Params | 124M | 131M | 133M
Runtime | 0.17 | 0.47 | 0.40
Table 9: Number of parameters in the different encoding methods. Runtime
reports the number of seconds to process one training example.
## Appendix F Error Analysis Examples
We show three examples with different types of errors. These examples are
taken from Dev set, and these are for the cases where curious introduced a
wrong answer, while baseline answered this correctly without the graph.
* $\bullet$
Figure 12 shows a failure case when a good graph is unused. Example from
$\delta$-atomic dev set.
* $\bullet$
Figure 13 shows a failure case when an off topic graph is produced due to
confusion in the sense of water fountain. Example from $\delta$-snli dev set.
* $\bullet$
Figure 14 shows a failure case when the mediator is wrong. Example from
$\delta$-social dev set.
Figure 12: Example of a failure case: A good graph is unused. Example from
$\delta$-atomic dev set. Figure 13: Example of a failure case: The generated
graph is off topic (wrong sense of water fountain is used). Example from
$\delta$-snli dev set. Figure 14: Example of a failure case: The mediator
nodes (second last level in the graph) are unhelpful. Example from
$\delta$-social dev set.
## Appendix G Significance Tests
We perform two statistical tests for verifying our results: i) The micro-sign
test (s-test) Yang and Liu (1999), and ii) McNermar’s test Dror et al. (2018).
Dataset | s-test | McNemar’s test
---|---|---
$\delta$-atomic | 5.07e-05 | 1.1e-04
$\delta$-snli | 2.65e-05 | 6.5e-05
$\delta$-social | 1.4e-04 | 3.2e-04
Table 10: p-values for the three datasets and two different statistical tests while comparing the results with and without graphs (Table 2). As the p-values show, the results in Table 2 are highly significant Dataset | s-test | McNemar’s test
---|---|---
$\delta$-atomic | 0.001 | 0.003676
$\delta$-snli | 0.01 | 0.026556
$\delta$-social | 0.06 | 0.146536
Table 11: p-values for the three datasets and two different statistical tests while comparing the results with noisy vs. cleaned graphs (Table 3). | $\delta$-atomic | $\delta$-snli | $\delta$-social
---|---|---|---
str | 0.13 | 1.8e-06 | 8.7e-06
gcn | 0.006 | 1.31e-05 | 0.03
Table 12: p-values for the s-test for Table 5. | $\delta$-atomic | $\delta$-snli | $\delta$-social
---|---|---|---
str | 0.28 | 4e-06 | 2e-05
gcn | 0.015127 | 3.2e-05 | 0.06
Table 13: p-values for the McNemar’s for Table 5.
## Appendix H Description of gcn encoder
We now describe our adaptation of the method by Lv et al. (2020) to pool
$\mathbf{h}_{\mathbf{V}}$ into $\mathbf{h}_{\mathbf{G}}$ using gcn. Figure 15
captures the overall design.
##### Refining node representations
The representation for each node $v\in\mathbf{V}$ is first initialized using:
$\mathbf{h}_{\mathbf{v}}^{0}=\mathbf{W}^{0}\mathbf{h}_{\mathbf{v}}$
Where $\mathbf{h}_{\mathbf{v}}\in\mathbb{R}^{d}$ is the node representation
returned by the $\mathcal{L}$, and $\mathbf{W}^{0}\in\mathbb{R}^{d\times k}$.
This initial representation is then refined by running L-layers of a GCN Kipf
and Welling (2017), where each layer $l+1$ is updated by using representations
from the $l^{th}$ layer as follows:
$\displaystyle\mathbf{h}_{v}^{(l+1)}$
$\displaystyle=\sigma\left(\frac{1}{|\mathbf{A}(v)|}{\sum_{w\in\mathbf{A}(v)}\mathbf{W}^{l}\mathbf{h}_{w}^{l}}+\mathbf{W}^{l}\mathbf{h}_{v}^{l}\right)$
$\displaystyle\mathbf{H}^{L}$
$\displaystyle=[\mathbf{h}_{0}^{L};\mathbf{h}_{1}^{L};\ldots;\mathbf{h}_{|\mathbf{V}|-1}^{L}]$
(7)
Where $\sigma$ is a non-linear activation function,
$\mathbf{W}^{l}\in\mathbb{R}^{k\times k}$ is the GCN weight matrix for the
$l^{th}$ layer, $\mathbf{A}(v)$ is the list of neighbors of a vertex $v$, and
$\mathbf{H}^{L}\in\mathbb{R}^{|\mathbf{V}|\times k}$ is a matrix of the
$L^{th}$ layer representations the $|\mathbf{V}|$ nodes such that
$\mathbf{H}^{L}_{i}=\mathbf{h}_{i}^{L}$.
##### Learning graph representation
We use multi-headed attention Vaswani et al. (2017) to combine the query
representation $\mathbf{h}_{\mathbf{Q}}$ and the nodes representations
$\mathbf{H}^{L}$ to learn a graph representation $\mathbf{h}_{\mathbf{G}}$.
The multiheaded attention operation is defined as follows:
$\displaystyle\mathbf{a}_{i}$
$\displaystyle=\text{softmax}\left(\frac{(\mathbf{W^{q}_{i}}\mathbf{h}_{\mathbf{Q}})(\mathbf{W^{k}_{i}}\mathbf{H}^{L})^{T}}{\sqrt{d}}\right)$
$\displaystyle\text{head}_{i}$
$\displaystyle=\mathbf{a}_{i}(\mathbf{W^{v}_{i}}\mathbf{H}^{L})$
$\displaystyle\mathbf{h}_{\mathbf{G}}$
$\displaystyle=Concat(head_{1},\ldots,head_{h})\mathbf{W}^{O}$
$\displaystyle=\text{MultiHead}(\mathbf{h}_{Q},\mathbf{H}^{L})$ (9)
Where $h$ is the number of attention heads,
$\mathbf{W}^{q}_{i},\mathbf{W}^{k}_{i},\mathbf{W}^{v}_{i}\in\mathbb{R}^{k\times
d}$ and $\mathbf{W}^{O}\in\mathbb{R}^{hd\times d}$.
Finally, the graph representation generated by the the MultiHead attention
$\mathbf{h}_{\mathbf{G}}\in\mathbb{R}^{n}$ is concatenated with with the
question representation $\mathbf{h}_{\mathbf{Q}}$ to get the prediction:
$\hat{y}=\text{softmax}([\mathbf{h}_{\mathbf{G}},\mathbf{h}_{\mathbf{Q}}]\mathbf{W}_{out})$
where $\mathbf{W}_{out}\in\mathbb{R}^{d\times 2}$ is a single linear layer
MLP.
## Appendix I All results
Our experiments span two types of graphs ($\mathbf{G}^{\prime}$, $G$), three
datasets ($\delta$-snli, $\delta$-social, $\delta$-atomic), and three graph
encoding schemes (str, gcn, MoE). Table 14 above shows the results on all 18
combinations of $\\{\text{graph
types}\\}\times\\{\text{datasets}\\}\times\\{\text{graph encoding schemes}\\}$
Dataset | Encoder | Graph Type | Accuracy
---|---|---|---
$\delta$-atomic | | n/a |
str | $\mathbf{G}^{\prime}$ | 78.78
str | $G$ | 79.48
gcn | $\mathbf{G}^{\prime}$ | 78.25
gcn | $G$ | 78.85
MoE | $\mathbf{G}^{\prime}$ | 78.83
MoE | $G$ | 80.15
$\delta$-snli | | n/a |
str | $\mathbf{G}^{\prime}$ | 82.16
str | $G$ | 83.11
gcn | $\mathbf{G}^{\prime}$ | 82.63
gcn | $G$ | 83.09
MoE | $\mathbf{G}^{\prime}$ | 83.83
MoE | $G$ | 85.59
$\delta$-social | | n/a | 87.6
str | $\mathbf{G}^{\prime}$ | 86.75
str | $G$ | 87.24
gcn | $\mathbf{G}^{\prime}$ | 87.92
gcn | $G$ | 88.12
MoE | $\mathbf{G}^{\prime}$ | 88.45
MoE | $G$ | 88.62
Table 14: Results for different combinations of graph encoder, graph type.
## Appendix J Graph-augmented defeasible reasoning algorithm
In Algorithm 1, we outline our graph-augmented defeasible learning process.
Given: A language model $\mathcal{L}$, defeasible query with graph
$(\mathbf{x},\mathbf{G})$.
Result: Result for the query.
// Encode query
$\mathbf{h}_{\mathbf{Q}}\leftarrow\mathcal{L}(\mathbf{x})$;
// encode nodes of $\mathbf{G}$
$\mathbf{h}_{\mathbf{V}}\leftarrow\mathcal{L}(\mathbf{v}\in\mathbf{G})$ ;
// MOE1: Combine nodes
$\mathbf{h}_{\mathbf{G}}\leftarrow$ Equation 3;
// MOE2: Combine $Q$, $G$
$\mathbf{h}_{\mathbf{y}}\leftarrow$ Equation 4;
return _softmax(MLP($\mathbf{h}_{\mathbf{y}}$))_
Algorithm 1 Graph-augmented defeasible reasoning using MoE. Figure 15:
Overview of the gcn encoder.
|
# Multi-clue Consistency Learning to Bridge Gaps Between General and Oriented
Object in Semi-supervised Detection
Chenxu Wang Nanjing University of Science and TechnologyNanjingChina
<EMAIL_ADDRESS>, Chunyan Xu Nanjing University of Science and
TechnologyNanjingChina , Ziqi Gu Nanjing University of Science and
TechnologyNanjingChina and Zhen Cui Nanjing University of Science and
TechnologyNanjingChina
###### Abstract.
While existing semi-supervised object detection (SSOD) methods perform well in
general scenes, they encounter challenges in handling oriented objects in
aerial images. We experimentally find three gaps between general and oriented
object detection in semi-supervised learning: 1) Sampling inconsistency: the
common center sampling is not suitable for oriented objects with larger aspect
ratios when selecting positive labels from labeled data. 2) Assignment
inconsistency: balancing the precision and localization quality of oriented
pseudo-boxes poses greater challenges which introduces more noise when
selecting positive labels from unlabeled data. 3) Confidence inconsistency:
there exists more mismatch between the predicted classification and
localization qualities when considering oriented objects, affecting the
selection of pseudo-labels. Therefore, we propose a Multi-clue Consistency
Learning (MCL) framework to bridge gaps between general and oriented objects
in semi-supervised detection. Specifically, considering various shapes of
rotated objects, the Gaussian Center Assignment is specially designed to
select the pixel-level positive labels from labeled data. We then introduce
the Scale-aware Label Assignment to select pixel-level pseudo-labels instead
of unreliable pseudo-boxes, which is a divide-and-rule strategy suited for
objects with various scales. The Consistent Confidence Soft Label is adopted
to further boost the detector by maintaining the alignment of the predicted
results. Comprehensive experiments on DOTA-v1.5 and DOTA-v1.0 benchmarks
demonstrate that our proposed MCL can achieve state-of-the-art performance in
the semi-supervised oriented object detection task. The code will be available
at https://github.com/facias914/sood-mcl
## 1\. Introduction
The fully-supervised object detection in aerial images (Xia et al., 2018; Han
et al., 2021; Xie et al., 2021; Li et al., 2023) has achieved remarkable
success in the past few years. However, labeling a large amount of accurate
annotations is labour-consuming, especially for oriented dense objects. To
reduce the annotation burden, the semi-supervised object detection (SSOD)
attempts to leverage both labeled data and unlabeled data for boosting the
detector performance. Existing advanced SSOD methods, which mainly adopt the
self-training framework (Liu et al., 2021; Li et al., 2022a; Liu et al., 2022;
Wang et al., 2023) to employ these unlabeled data, have achieved significant
performance in general scenes. However, they encounter various challenges in
handling the oriented object detection task due to the tricky characteristics
such as arbitrary rotation angles, large aspect ratio changes and dense
arrangement. This prompts us to study the performance gaps between general and
oriented object detection when performing semi-supervised learning.
With comprehensive investigations in Section 3.2, we figure out that the
performance of SOOD is significantly hindered by three gaps and inconsistency
problems. Specifically, the sampling inconsistency represents that the
positive labels selected from labeled data are inconsistent with the
supervision information required by the oriented objects. It is caused by the
sub-optimal center-based label assignment and the gap in aspect ratio
distribution between general and oriented objects. That is to say, oriented
objects usually have more extreme aspect ratio distributions than general
ones, while the center-sampling strategy only focuses on the center region,
which results in the discriminative information of large aspect ratio objects
distributed at the edge being assigned as background, affecting the
optimization of the detector.
Figure 1. The pipeline of our MCL. To address the sampling inconsistency
issue, the Gaussian Center Assignment is introduced to select more accurate
pixel-level positive labels from labeled data. The Scale-aware Label
Assignment is proposed to select pixel-level pseudo-labels for objects with
various scales. The Consistent Confidence Soft Label is adopted to mitigate
the mismatch problem between classification and localization qualities through
maintaining the alignment of the predicted results.
In addition, the assignment inconsistency denotes that the predicted oriented
pseudo-boxes from unlabeled data are difficult to guide the semi-supervised
learning process, resulting in noisy label assignment. In general, most
advanced SSOD methods (Li et al., 2022a; Wang et al., 2023; Zhang et al.,
2023) follow the pseudo-boxes paradigm (Sohn et al., 2020) that using the
pseudo-boxes predicted by teacher model act as the supervised signals to
optimize the student model. However, the introduction of angle parameters in
rotated pseudo-boxes makes their localization quality less controllable than
that of horizontal ones. Under the guidance of low-quality rotated pseudo-
boxes, noisy labels are inevitably introduced during the label assignment
process. Moreover, even with the aid of NMS (Non-Maximum Suppression) and
threshold filtering, it is impossible to guarantee there are not redundant
boxes which will bring additional noisy labels.
On the other hand, the confidence inconsistency indicates that there exists
the mismatch between the predicted classification and localization qualities,
affecting the selection of pseudo-labels. Prior researches (Sun et al., 2021;
Xu et al., 2021; Li et al., 2022a) have validated that the classification
score is unable to measure the localization quality, and the lack of
interaction between the two confidences causes an inconsistency in
predictions, leading to sub-optimal pseudo-label selection results. Moreover,
while numerous researches have studied the proxy localization quality of
horizontal box, there is a lack of research for oriented boxes. Our
experiments show that there remains a gap in the representation of
localization quality between horizontal and oriented boxes, while sub-optimal
proxy localization quality also affects the performance of oriented object
detection.
Based on the above observations, we propose a Multi-clue Consistency Learning
(MCL) framework to boost the performance of the SOOD task through bridging
multi-gaps between general and oriented object detection in semi-supervised
learning process. Figure 1 illustrates the pipeline of MCL. Specifically, to
address the sampling inconsistency problem, we introduce the Gaussian Center
Assignment (GCA) strategy to select more accurate positive labels from the
limited annotated data. Here each oriented object region can be represented as
a $2D$ Gaussian Distribution (Yang et al., 2022) according to its
corresponding ground truth box, which can be adopted to guide the pixel-level
positive label selection. In addition, the normalized distance to the center
of the object (termed centerness) can be formulated as a $2D$ Gaussian
distribution, which is used to better measure the localization quality of each
pixel-level positive label. Compared to other label assignment strategies
(Ming et al., 2021; Hou et al., 2022; Xu et al., 2022, 2023) for fully-
supervised detection, our designed GCA can not only extract more comprehensive
target information from limited labeled data but also avoid the performance
degradation caused by the sampling inconsistent supervision of large-ratio
aerial objects.
To mine accurate potential information from a large amount of unlabeled data,
we would resist the assignment and confidence inconsistency. As for the
assignment inconsistency, we propose the Scale-aware Label Assignment (SLA)
method to select pixel-level pseudo-labels instead of unreliable pseudo-boxes,
which adopts a divide-and-rule strategy to develop different pseudo-labels
selection rules for objects with different scales. For small objects, a
coarse-to-fine pseudo-label selection is used to filter high quality pseudo
labels from dense features, which can preserve only a few candidates with both
high classification and localization qualities. For large-scale objects with
sparse features, we just use a score threshold filtering mechanism to maintain
scale balance. Different from the previous dense pseudo-labels assignments
(Zhou et al., 2022; Liu et al., 2023a) used in general scenes, our proposed
SLA is more suitable for various scale objects in oriented object detection
and then select more reliable pixel-level pseudo-labels from the unlabeled
data.
To address the confidence inconsistency problem, we employ the soft label (Li
et al., 2020) that the value at the ground-truth category index indicates its
corresponding localization quality, replacing the one-hot label to supervise
the classification branch learning. However, there is a gap in the proxy
representations of localization quality between horizontal and oriented boxes.
Through experimental analysis in Section 3.2, we find that centerness (Tian et
al., 2019) can effectively represent the localization quality of the oriented
box and outperforms the predicted IoU (Intersection over Union) used on the
horizontal box (Li et al., 2020; Liu et al., 2023a). Thereby we propose the
Consistent Confidence Soft Label (CCSL) based on centerness to further promote
the selection of pseudo-labels through mitigating the inconsistency problem.
Prior to this, we have not seen any work on the localization quality
representation of oriented box. In conclusion, our contributions are
summarized as follows:
* •
We look deep into the gaps between general and oriented object detection in
semi-supervised learning, and then propose a multi-clue consistency learning
framework to improve the performance of SOOD task.
* •
To address three inconsistency problems, we specially design the Gaussian
Center Assignment, the Scale-aware Label Assignment, and the Consistent
Confidence Soft Label methods to mine more instructive supervised signals from
the annotated data and unlabeled training data.
* •
Comprehensive evaluations and comparisons demonstrate that our MCL achieves
state-of-the-art performance on DOTA-v1.5 and DOTA-v1.0 benchmark (Xia et al.,
2018).
## 2\. RELATED WORK
Semi-supervised Object Detection. Most of previous methods (Liu et al., 2021;
Yang et al., 2021; Li et al., 2022c, d; Liu et al., 2023b; Zhang et al., 2023;
Nie et al., 2023) are inherited from the Mean Teacher paradigm (Tarvainen and
Valpola, 2017), where the teacher model is updated via Exponential Moving
Average (EMA) of the student weights and produce pseudo labels over unlabeled
data as ground truth for training the student model. Under this scheme, the
quality and precision of pseudo labels play a substantial role. ISMT (Yang et
al., 2021) maintains a memory bank to ensure the consistency of pseudo labels
across various iteration stages. Unbiased Teacher (Liu et al., 2021) applies
Focal Loss (Lin et al., 2017b) to tackle the class-imbalance pseudo labels
issues. Soft Teacher (Xu et al., 2021) employs a box jittering argumentation
to select reliable pseudo labels. For misleading instances, Unbiased Teacher
v2 (Liu et al., 2022) selects pseudo labels through utilizing uncertainty
predictions. Consistent Teacher (Wang et al., 2023) dynamically adjusting the
threshold for pseudo labels filtering. Mix Teacher (Liu et al., 2023b)
enhances the quality of pseudo labels through mixed-scale prediction.
Different from these, some methods (Zhou et al., 2022; Li et al., 2022b; Liu
et al., 2023a) directly use the dense predictions of the teacher model as
pixel-level pseudo-labels for semi-supervised learning. However, since the
usage scenarios of oriented object detection are more complex, there still
exist the performance gaps when applying these SSOD methods in semi-supervised
oriented object detection.
Semi-supervised Oriented Object Detection. Oriented object detection requires
predicting rotated bounding boxes by adding an angle parameter in the
regression task, posing significant challenges for controlling the quality and
accuracy of pseudo boxes. SOOD (Hua et al., 2023) uses the absolute distance
of the predicted angle between the teacher and student models to weight the
regression loss. Additionally, it introduces the optimal transport cost
(Monge, 1781) to evaluate the global similarity of layouts between the
predictions of teacher and student. However, SOOD fails to recognize the root
issue hindering semi-supervised learning in oriented object detection, which
lies in how to eliminate inconsistency issues caused by the divergence of
general and oriented object detection. In this work, we dive into the
performance gaps between SSOD and SOOD methods, proposing a multi-clue
consistency learning method for semi-supervised learning in oriented object
detection.
Label Assignment in Oriented Object Detection. Previous works have (Zhang et
al., 2020; Ge et al., 2021; Kim and Lee, 2020) revealed that label assignment
plays a crucial role in object detection. For oriented object detection, DAL
(Ming et al., 2021) introduce matching degree to select anchors with high
prior and regression qualities. SASM (Hou et al., 2022) set dynamic IoU
thresholds decided by object shapes to filter prior anchors. In order to
achieve scale-balance learning for tiny objects, RFLA (Xu et al.,
2022)replaces the IoU-based or center sampling assignment with hierarchical
label assignment built on gaussian respective field, while DCFL (Xu et al.,
2023) develops the dynamic priors instead of static priors. Although these
assignment methods are effective, they are limited to fully-supervised
settings. With such over-designed label assignment strategies, the model is
insufficient to learn perfect target information from sparse labeled data. In
this work, we show that a simple changed sampling prior significantly improves
the performance of SOOD task.
Representation of Localization Quality. The representation of localization
quality has attracted increasing attention in general object detection task.
FCOS (Tian et al., 2019) utilize a separate branch to perform localization
quality estimation in the form of centerness. After that, GFL (Li et al.,
2020) points out that centerness is easier to get very small values in object
edge by its definition, making it perform consistently worse than IoU. In SSOD
task, several researches (Liu et al., 2022; Li et al., 2022a; Liu et al.,
2023a) also focus on accurately measuring the localization quality of pseudo
labels. For instance, PseCo (Li et al., 2022a) measures the localization
quality of pseudo labels by estimating the consistency of predicted proposals
(Ren et al., 2015) for two-stage detectors. Unbiased Teacher v2 (Liu et al.,
2022) reveals that the centerness is not reliable for reflecting whether a
prediction is a positive sample due to lack of background supervision. Thereby
it uses the predicted localization uncertainty of four box boundaries instead.
ARSL(Liu et al., 2023a) also replaces the centerness branch with the IoU
branch following the VFNet(Zhang et al., 2021) and GFL(Li et al., 2020).
However, none of these works are designed for rotating bounding boxes in
oriented object detection. In this work, we figure out that centerness is
still the best choice as the proxy localization quality and we will verify
this in Section 3.2.
## 3\. METHODOLOGY
To study the gaps between the general and oriented object detection in the
semi-supervised learning, we start from the classic SSOD framework (Tarvainen
and Valpola, 2017), as introduced in Section 3.1. We then experimentally
analyze the three inconsistency issues (Section 3.2). To mitigate the
inconsistency and bridge the gaps, we subsequently propose a Multi-clue
Consistency Learning (MCL) framework for the SOOD task. It mainly comprises
three modules: Gaussian Center Assignment (Section 3.3), Scale-aware Label
Assignment (Section 3.4), and Consistent Confidence Soft Label (Section 3.5).
### 3.1. The Basic SSOD Framework
We introduce the classic Pseudo-labeling SSOD framework inherited from Mean
Teacher (Tarvainen and Valpola, 2017) as our baseline, which adopts a rotated
FCOS version (Tian et al., 2019) as the detector. In general, the SSOD
framework comprises two branches: one dedicated to the supervised learning
branch and the other to the unsupervised learning branch. In the burn-in
stage, the student model undergoes pre-training with the limited annotated
data, where the supervised loss $\mathcal{L}^{s}$ is minimized by combining
classification, centerness, and regression. Simultaneously, the learned
weights of the student model are mirrored onto the teacher model. In the self-
training stage, the unlabeled training data with weak data augmentation is fed
into the teacher model to generate pseudo-labels. Subsequently, the student
model is further optimized with these generated pseudo-labels (i.e.,
supervised by $\mathcal{L}^{u}$), while the input comprises unlabeled data
augmented more aggressively. Meanwhile, the supervised learning with
$\mathcal{L}^{s}$ is maintained on the student network. The overall loss
function is thus formulated as
$\mathcal{L}=\mathcal{L}^{s}+\beta\mathcal{L}^{u}$, where $\beta$ is a hyper-
parameter regulating the impact of unsupervised learning. The network
parameters of the student model are continually updated onto the teacher
network through an exponential moving average (EMA) mechanism at each
iteration.
### 3.2. Inconsistency Analysis
Figure 2. Analysis of the sampling inconsistency on the general COCO and
aerial DOTA-v1.5 datasets. (a) Statistics of object aspect ratio distribution
on both datasets. (b) The class activation mapping of an aerial object with
large aspect ratio. (c) The positive label selection of an aerial object by
the common center sampling strategy, where the red points and blue points
represent negatives and positives, respectively.
Since oriented objects in aerial images have some specific characteristics
(e.g., the bird’s-eye view perspective, the complex environment, and the
variant scales of objects), the standard SSOD framework would encounter
various challenging problems when directly applied to the SOOD task. We
attempt to analyze the core components of the SSOD framework (including
positive label assignment and pseudo-label selection), and then find some
gaps/inconsistencies between general and oriented object detection in the
semi-supervised learning. Specifically, we conduct a series of experimental
analysis on two datasets representing general scenes and aerial perspectives,
namely COCO (Lin et al., 2014) and DOTA-v1.5 (Xia et al., 2018), respectively.
To ensure a fair comparison, we use the FCOS as the object detector on both
datasets.
Sampling Inconsistency: When employing these limited annotated data, the
common center sampling strategy is adopted to select positive labels for
general objects, but it is not suitable for oriented objects with larger
aspect ratios. For an intuitive understanding, we statistically analyze the
aspect ratio distributions of targets on the COCO and DOTA-v1.5 datasets, as
shown in Figure 2(a). More than 90% of objects are with the aspect ratios
exceeding 0.5 on the COCO dataset, thus the center sampling strategy can be
adopted to select positive points of the general objects, thereby providing
sufficient supervision signals. However, more than 70% of objects exhibit
aspect ratios less than 0.5 on the DOTA-v1.5 dataset, especially for some
objects with extreme aspect ratios (seen in Figure 2(b)) in the bird’s-eye
view perspective. As seen in Figure 2(c), only these blue points can be
selected as pixel-level positive points with the common center sampling
strategy, while these unselective red points would be seen as negative
background information, leading to incorrect guidance for the model
optimization. Therefore, the inconsistency between the shape-agnostic center
sampling and objects with large aspect ratios would cause the performance
degradation in the SOOD task.
Assignment Inconsistency. Compared with the horizontal object detection in the
general images, the localization of oriented objects in aerial images is more
challenging due to their arbitrary rotation angles. To elucidate this, we
conduct evaluations with the FCOS detector on the DOTA-v1.5 and COCO datasets,
and then utilize the Intersection over Union (IoU) metric between predicted
boxes and the corresponding ground-truth boxes to assess the localization
quality of the pseudo-boxes. As shown in Figure 3(a), we use different IoU
thresholds to obtain the pseudo-box precision under the same recall rate on
the DOTA-v1.5 and COCO datasets, respectively. With a lower IoU threshold, we
can achieve a higher precision on the DOTA-v1.5 dataset compared to the COCO
dataset, but the pseudo-boxes with the poor localization quality would affect
the detection optimization. Under the guidance of low-quality rotated pseudo-
boxes, noisy labels are inevitably introduced during the label assignment
process. When improving the requirement of localization quality by increasing
the IoU threshold (from 0.5 to 0.9) on both datasets, the detection precision
would severely drop from 84% to 17% on the DOTA-v1.5 dataset, while it will be
also reduced from 76% to 54% on the COCO dataset. A large number of redundant
noisy boxes caused by reduced precision can also introduce the noisy labels.
Hence, it’s hard to achieve a trade-off between the precision and localization
quality of rotated pseudo-boxes, and the assignment inconsistency problem
should be addressed to guide the semi-supervised learning process.
Figure 3. Investigation on the assignment and confidence inconsistency
problems. (a) The pseudo-box precision of the DOTA-v1.5 dataset and COCO
dataset under different IoU thresholds. (b) The top heat-map illustrates the
consistency of centerness and IoU between ground truth boxes and their
corresponding true positive boxes on the DOTA-v1.5 dataset, while the bottom
one is on the COCO dataset.
Confidence Inconsistency: Prior research (Li et al., 2020; Liu et al., 2023a)
has validated that the confidence inconsistency problem is existing between
the predicted classification and localization qualities. To enhance the
correlation between them, most previous horizontal object detectors (Li et
al., 2020; Liu et al., 2023a) use the IoU-based soft label to supervise the
classification branch. However, predicting a confidence IoU value is difficult
for the oriented objects due to the increased uncertainty introduced by the
rotation angle parameter. Meanwhile, we discover that centerness can well
represent the localization quality of oriented bounding boxes. As in Figure
3(b), we statically analyze the IoU and centerness between all ground truth
boxes and their corresponding true positive boxes on the DOTA-v1.5 dataset
(the top subfigure) and the COCO dataset ((the bottom subfigure). Compared to
the COCO dataset, the correlation between the IoU and centerness values is
significantly higher on the DOTA-v1.5 dataset. Therefore, to address the
confidence inconsistency problem, we utilize the predicted centerness value to
measure the localization quality of oriented objects.
### 3.3. Gaussian Center Assignment
To deal with the sampling inconsistency problem, we attempt to select more
suitable positive labels from the following aspects: 1) making full use of
sparse annotated data; 2) taking the shape information of oriented objects
into consideration. For the first condition, the optimal solution is to
leverage these sparse annotations to assign more positive samples, hence some
carefully designed label assignment strategies are excluded. Also considering
the second condition, we propose a shape-aware method named Gaussian Center
Assignment (GCA) to selectpixel-level positive labels for oriented object
detection in semi-supervised learning.
Concretely, We model the object as a $2D$ Gaussian distribution that can well
reflect the shape and direction of the aerial object. We denote a rotated
bounding box as $B=\left(x_{c},y_{c},w,h,\theta\right)$, where $\theta$
represents the angle parameter; $(x_{c},y_{c})$, $w$, and $h$ represent the
center coordinates, width, and height of the oriented box, respectively. The
coordinate of the center point $(x_{c},y_{c})$ serves as the mean vector of
$2D$ Gaussian distribution $\boldsymbol{\mu}_{g}=(x_{c},y_{c})^{\top}$, and
the co-variance matrix $\boldsymbol{\Sigma}_{g}$ can be formulated as:
(1)
$\boldsymbol{\Sigma}_{g}=\left[\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right]\left[\begin{array}[]{cc}\frac{w^{2}}{4}&0\\\
0&\frac{h^{2}}{4}\end{array}\right]\left[\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right].$
We then define a point coordinate as $\boldsymbol{P}=(x,y)^{\top}$ and
determine whether it is a positive label based on the following formula:
(2) $\text{ label }=\left\\{\begin{array}[]{ll}1,&\text{ if
}(\boldsymbol{P}-\boldsymbol{\mu}_{g})^{\top}\boldsymbol{\Sigma}_{g}^{-1}(\boldsymbol{P}-\boldsymbol{\mu}_{g})\leq
1\\\ 0,&\text{ otherwise }.\end{array}\right.$
This label assignment ensures that positive labels cover not only the central
region but also the edge region of the target, thereby avoiding the omission
of discriminative features for aerial objects with large aspect ratios.
Moreover, a large number of sub-optimal positive samples (pixels near the
boundaries) also provide sufficient supervision information for the detector
optimization. For the oriented object, the new expression of centerness is
formulated as follows:
(3) $\text{ centerness
}^{*}=1-(\boldsymbol{P}-\boldsymbol{\mu}_{g})^{\top}\boldsymbol{\Sigma}_{g}^{-1}(\boldsymbol{P}-\boldsymbol{\mu}_{g}).$
According to the prior that pixels located around the center of the object are
more representative than those near the object boundaries, we establish a new
normalized distinction in the object region to represent the pixel-wise
localization quality. The Gaussian Center Assignment (GCA) can select
comprehensive supervision signals from the sparse annotated data to guide the
learing, and further improve the performance for objects with large aspect
ratios.
### 3.4. Scale-aware Label Assignment
Figure 4. Analysis on difference feature maps and the object centerness. (a)
Score imbalance between feature maps at different levels. (b) For centerness-
based soft label, condition 1 introduce ambiguities while condition 2 is more
suitable.
To tackle the assignment inconsistency problem, we conduct the pixel-level
pseudo-label selection based on the feature level predictions of the teacher
model rather than the post-processed pseudo-boxes. According to multi-level
prediction with FPN (Lin et al., 2017a), different sizes of objects can be
detected on different levels of feature maps (i.e.,
$P_{3},P_{4},P_{5},P_{6},P_{7}$). However, as shown in Figure 4(a), there is a
score imbalance problem between feature maps at different levels, where the
higher level feature map tend to predict the lower scores. To balance the
allocation of pseudo-labels for oriented targets with different scales, we
propose Scale-aware Label Assignment (SLA) to improve the quality of pseudo-
labels by employing different sampling rules for different level feature maps.
For the low-level feature maps (i.e., $P_{3},P_{4}$) with dense features, we
design a coarse-to-fine rule to predict these small-scale objects with two
following stages. In the coarse selection stage, the joint confidence (denoted
as the multiplication of score and centerness) is used to rank all pixels and
the top-k sorted pixels will be selected as candidate pseudo-labels. The joint
confidence is expected to select samples with the high classification and
localization qualities. As has been proven that the joint confidence is
dominated by centerness due to its higher value than classification score (Liu
et al., 2022), a high centerness value is unreliable to determine whether a
pixel is positive since the centerness branch lacks supervision from negative
instances. Therefore, we further filter these candidates through a score
threshold to ensure the precision.
For the high-level feature maps (i.e., $P_{5},P_{6},P_{7}$) with the lower
predicted scores, we only use the score threshold to filter high-confidence
pseudo-labels, thereby avoiding the unbalanced pseudo-labels allocation across
various scales. Here the unsupervised loss consists of three parts:
(4)
$\mathcal{L}_{u}=\frac{1}{N_{all}}\sum_{i}^{N_{all}}\mathcal{L}_{i}^{cls}+\frac{\alpha}{N_{pos}}\sum_{j}^{N_{pos}}w_{j}(\mathcal{L}_{j}^{cen}+\mathcal{L}_{j}^{reg})$
where $\alpha$ is a weighting parameter, $N_{all}$ represents the number of
pixels across five level feature maps, $N_{pos}$ denotes the number of
selected pseudo-labels. We apply the quality focal loss (Li et al., 2020), the
binary cross-entropy loss and the smooth-l1 Loss for guiding the
classification, centerness and regression branches, respectively. Besides, the
localization weights $w$ are determined by the following formula:
(5) $w_{j}=\left\\{\begin{array}[]{ll}s_{j}\times c_{j}&j\in[P_{3},P_{4}]\\\
s_{j}&j\in[P_{5},P_{6},P_{7}]\end{array}\right.$
where $s$ and $c$ denote the score and centerness, respectively. Since the
pseudo-labels selection is based on model predictions, the existence of false
positive labels is inevitable and the localization task is more sensitive to
such noisy samples. Through employing the weight $w$, we reduce the
contribution of low-confidence samples to the overall loss.
### 3.5. Consistent Confidence Soft Label
In this part, we focus on alleviating the confidence inconsistency issue.
Based on the analysis in Section 3.2, we design a centerness-based soft label
to supervise the classification branch. The soft label have to satisfy two
conditions: (1) being positively correlated with centerness; (2) the score
variance among all points within the target should not be too large,
especially for small targets. Therefore, we propose the Consistent Confidence
Soft Label (CCSL) strategy to align the classification and localization
confidences.
The core idea of CCSL is to replace the value of one-hot label at
corresponding category index with a float value $y\in[0,1]$, which satisfies
the condition that the covariance calculated with centerness equal to 1. The
soft label $y\in[0,1]$ is given as follows:
(6) $\text{ y
}=[1-(\boldsymbol{P}-\boldsymbol{\mu}_{g})^{\top}\boldsymbol{\Sigma}_{g}^{-1}(\boldsymbol{P}-\boldsymbol{\mu}_{g})]^{\gamma}$
where $\gamma$ is a scale factor designed for the second condition. As shown
in Figure 4(b), all points belonging to a small object have smaller stride,
resulting in very similar point features. We hope that the score values of all
points within the target are very close. Therefore, we weight them with the
scale factor $\gamma$, which is formulated as $\sqrt[\beta]{(h\times
w)/(H\times W)}$. $h$ and $w$ represent the height and width of the oriented
box, $H$ and $W$ denote the height and width of the whole image, while $\beta$
controls the smoothing degree of $\gamma$. $\gamma$ can help prevent confusion
during model training caused by excessive differences in point scores within
the target. The proposed CCSL can bridge the gap between horizontal and
oriented boxes in localization quality, and effectively mitigate the mismatch
between classification and localization confidences, thereby bolstering the
semi-supervised learning performance.
## 4\. Experiment
### 4.1. Experimental Setup
Dataset: The experiments are conducted on DOTA-v1.5 (Xia et al., 2018) and
DOTA-v1.0 (Xia et al., 2018). DOTA-v1.5 comprises 16 categories which contains
400$k$ annotated oriented instances. DOTA-v1.5-train, DOTA-v1.5-val, and
DOTAv1.5-test contain 1411, 458, and 937 images, respectively. DOTA-v1.0 uses
the same images as DOTA-v1.5 and the number of annotated instances is half the
DOTA-v1.5’s, with targets having a pixel area smaller than 10 being ignored.
We include three evaluation protocols: 1) DOTA1.5-Partial. Following SOOD
method (Hua et al., 2023), we randomly sample10%, 20%, and 30% images from
DOTA-v1.5-train as labeled data and set the remaining images as unlabeled
data. 2) DOTA1.5-Full. We set DOTA-v1.5-train as labeled data, DOTA-v1.5-test
as unlabeled data and perform the evaluation on the DOTA-v1.5-val. 3)
DOTA1.0-Full. The DOTA-v1.0-train and DOTA-v1.0-test are employed as labeled
data and unlabeled data respectively, while the DOTA-v1.0-val is set as the
evaluation dataset. For all evaluation protocols, the mAP (Xia et al., 2018)
is adopted as the evaluation metric.
Implementation Details: We adopt the FCOS (Tian et al., 2019) as the detector
and ResNet-50 (He et al., 2016) pretrained on ImageNet (Deng et al., 2009) as
the backbone. Following the previous works (Li et al., 2023; Xie et al., 2021;
Han et al., 2021), we crop the original images into patches of size 1024
$\times$ 1024, with an overlap region of 200 pixels between adjacent patches.
To ensure a fair comparison, the MCL is trained on 2 RTX4090 GPUs with 3
images per GPU (2 labeled images and 1 unlabeled image). The SGD optimizer is
applied with an initial learning rate of 0.0025, a momentum of 0.9, and a
weight decay of 0.0001. For 30% partial and full setting experiments, the MCL
is trained for 180$k$ iterations, and the learning rate is divided by 10 at
120k and 160k. For 10% partial and 20% partial setting experiments, we train
the MCL for 120$k$ iterations, while the learning rate is divided by 10 at 80k
and 110k. We follow the same pipeline in (Hua et al., 2023; Zhou et al.,
2022), employing a burn-in strategy to initialize the parameters of the
teacher model and applying the EMA strategy to update its parameters. The weak
data augmentation strategy for the teacher model and strong data augmentation
strategy for the student model are also consistent with them (Hua et al.,
2023; Zhou et al., 2022).
Table 1. Performance comparison with other state-of-the-art methods on the DOTA-v1.5-Partial dataset. ${\circ}$ and ${\dagger}$ denotes the base detector is Faster RCNN and FCOS respectively. | | mAP(%) $\uparrow$
---|---|---
Task | Method | 10% | 20% | 30%
OD | Faster RCNN (Girshick, 2015) | 43.43 | 51.32 | 53.14
FCOS (Tian et al., 2019) | 42.78 | 50.11 | 54.79
SSOD | Unbaised Teacher∘ (Liu et al., 2021) | 44.51 | 52.80 | 53.33
Soft Teacher∘ (Xu et al., 2021) | 48.46 | 54.89 | 57.83
Dense Teacher† (Zhou et al., 2022) | 46.90 | 53.93 | 57.86
PseCo∘ (Li et al., 2022a) | 48.04 | 55.28 | 58.03
DualPolish∘ (Zhang et al., 2023) | 49.02 | 55.17 | 58.44
ARSL† (Liu et al., 2023a) | 48.17 | 55.34 | 59.02
SOOD | SOOD*† (Hua et al., 2023) | 48.63 | 55.58 | 59.23
PST∘ (Wu et al., 2024) | 49.63 | 57.39 | 60.40
MCL†(Ours) | 52.98 | 59.63 | 62.63
Table 2. Performance comparison on DOTA-v1.5-Full and DOTA-v1.0-Full, while numbers in font of the arrow indicate the result of supervised baseline Method | DOTA-v1.5-Full | DOTA-v1.0-Full
---|---|---
Unbiased Teacher (Liu et al., 2021) | 66.12 $\xrightarrow{-1.27}$ 64.85 | -
Soft Teacher (Xu et al., 2021) | 66.12 $\xrightarrow{+0.28}$ 66.40 | -
Dense Teacher (Zhou et al., 2022) | 65.46 $\xrightarrow{+0.92}$ 66.38 | 66.72 $\xrightarrow{+3.58}$ 70.30
SOOD* (Hua et al., 2023) | 65.46 $\xrightarrow{+2.24}$ 67.70 | 66.72 $\xrightarrow{+4.09}$ 70.81
MCL(Ours) | $\textbf{65.46}\xrightarrow{\textbf{+3.62}}\textbf{69.08}$ | $\textbf{66.72}\xrightarrow{\textbf{+6.91}}\textbf{73.63}$
### 4.2. Comparison with State-of-the-Arts
To validate the effectiveness of MCL, we compare it with previous methods
under DOTA1.5-Partial, DOTA1.5-Full and DOTA1.0-Full evaluation protocols. For
distinguishing with SOOD task, the SOOD (Hua et al., 2023) method is termed as
SOOD*. Moreover, some SSOD methods with their oriented version also
participate in comparison.
DOTA-v1.5-Partial: Experiment results under the DOTA-v1.5-Partial protocol are
presented in Table 1. In SOOD task, SSOD methods typically underperform
compared to SOOD methods, which exhibits the necessity of bridging the gap
between horizontal boxes and oriented boxes in semi-supervised detection.
SOOD* (Hua et al., 2023) and PST (Wu et al., 2024) are specifically designed
for the characteristics of oriented boxes and remote sensing images, yet the
performance gains are marginal. By bridging the gaps and addressing the
concerns of inconsistency, our MCL achieves remarkable enhancements over the
supervised baseline FCOS and outperforms all SOOD methods under all
proportions. It scores 52.98, 59.63 and 62.63 mAP on 10% 20% 30% experiments,
respectively, largely surpassing the state-of-the-arts method PST by 2 $\sim$
3.5 mAP, validating the necessity of our analyses and methods.
DOTA-v1.5-Full and DOTA-v1.0-Full: Table 2 gives the comparison results in
terms of mAP on the DOTA-v1.5-Full and DOTA-v1.0-Full configurations. Our MCL
demonstrates remarkable advancements over existing methods on the
DOTA-v1.5-Full and DOTA-v1.0-Full experiment settings. Compared to the state-
of-the-art SOOD* using the same detector, the MCL can achieve the notable
improvement: 1.38% on the the DOTA-v1.5-Full setting, and 2.82% on the
DOTA-v1.0-Full setting, exhibit its effectiveness.
Figure 5. Some visualization results from the DOTA-v1.5 dataset. The first and
the last rows are the results of Dense Teacher and MCL respectively. True
Positive, False Negative, and False Positive predictions are marked in green,
red, and blue respectively.
Visualization Analysis: The visualization results under the 30% setting of
DOTA-v1.5-Partial are shown in Figure 5. Compared with the Dense Teacher (Zhou
et al., 2022), MCL shows an enhancement in the detection of targets with large
aspect ratios aided by the GCA module. Additionally, supported by the SLA
module, the MCL demonstrates an improved capability in handling targets of
various sizes in remote sensing scenarios. Moreover, the CCSL module assists
the MCL in identifying high-quality pseudo-labels in densely populated target
scenes, effectively reducing the incidence of False Negative predictions. This
collective integration of GCA, SLA, and CCSL within the MCL not only refines
its detection precision but also establishes a robust approach for enhancing
object detection in complex remote sensing images.
### 4.3. Ablation Studies
Subsequently, we delve into the detailed analysis of each module. All the
ablation experiments are performed on the 30% setting of DOTA-v1.5-Partial
without special instructions. We set the Dense Teacher (Zhou et al., 2022) as
the baseline. It also uses pixel-level pseudo labels by sorting all pixels
according to classification scores and selects the top ratio(%) ones. The
original parameter setting sets ratio to 1%, and we set ratio to 3% by default
to maximize the performance.
Table 3. Performance comparison with different modules. | | | | mAP(%) $\uparrow$
---|---|---|---|---
| GCA | SLA | CCSL | 10% | 20% | 30%
baseline | $\times$ | $\times$ | $\times$ | 49.71 | 55.64 | 59.65
MCL | $\checkmark$ | $\times$ | $\times$ | 51.12 | 56.74 | 60.62
$\times$ | $\checkmark$ | $\times$ | 51.84 | 57.16 | 61.32
$\checkmark$ | $\checkmark$ | $\times$ | 52.02 | 57.28 | 61.46
$\checkmark$ | $\checkmark$ | $\checkmark$ | 52.98 | 59.63 | 62.63
The effect of each module: We study the effectiveness of our proposed three
modules (GCA, SLA, CCSL) under different experiment settings (i.e., 10% 20%
30% ). The results are shown in Table 3. After comprehensively extracting
supervision information from annotated data, GCA facilitates a performance
enhancement for MCL. Furthermore, SLA enhances the quality of unsupervised
information extracted from unlabeled data and also contributes to performance
improvement. The CCSL can further improve the precision of pseudo-labels by
mitigating the inconsistency between classification and localization
confidence, thereby the participation of the CCSL leads to additional
performance improvement for MCL.
All Sampling vs GCA: To evaluate the effective of the GCA in SOOD task, we
compare GCA with the center sampling strategy used in FCOS (Tian et al., 2019)
and the all sampling strategy that labels all pixels insides ground-truth
boxes as positives and the remaining pixels as negatives. The detailed
comparable results are shown in Table 5. Although the aim is to fully utilize
labeled data to extract more positive supervision information, all sampling
strategy still shows performance degradation compared to center sampling. We
speculate that this is due to all sampling introducing excessive background at
the target’s edges. In contrast, GCA can provide sufficient positive
supervision information while accounting for the target’s aspect ratio without
introducing excessive noise, thereby improving the performance.
Table 4. Performance comparison between GCA and All Sampling strategy.
Sampling strategies | mAP
---|---
Center Sampling | 59.65
All Sampling | 58.49
GCA | 60.62
Table 5. Performance comparison between Mean Teacher and Dense Teacher.
Method | mAP
---|---
FCOS(base) | 54.79
Mean Teacher | 26.02 (-28.77)
Dense Teacher | 59.65 (+4.86)
Table 6. Ablation studies on SLA. | ratio | thr | topk | Recall | Precision | mAP
---|---|---|---|---|---|---
score | 1% | - | - | 40.08 | 30.76 | 57.86
3% | - | - | 69.44 | 17.76 | 59.65
score $\times$ center | 3% | - | - | 70.38 | 18.01 | 60.45
SLA | - | 0.01 | 1500 | 90.35 | 12.74 | 59.97
0.02 | 88.76 | 19.50 | 60.61
0.03 | 87.15 | 24.03 | 60.59
- | 0.01 | 2000 | 93.72 | 12.39 | 60.34
0.02 | 91.89 | 19.13 | 61.32
0.03 | 90.04 | 23.69 | 60.89
The analysis of SLA: To verify the impact of SLA hyper-parameters and
demonstrate the advantages of SLA over other selection strategies, we conduct
the experiments with the pixel-level recall and precision as the metrics. In
detail, we denote the selected points from the ground truth boxes as ground
truth points. The selected pseudo-labels that coincide with ground truth
points are denoted as true positives and the remaining ones are false
positives. The computation forms of pixel-level recall and precision are
consistent with the previous work (Lin et al., 2014). The experimentation is
shown in Table 6. It can be observed that the performance of semi-supervised
learning improves 1.79% mAP when recall increases from 40.08% to 69.44%,
demonstrating that maintaining high recall of pseudo-labels is crucial for
ensuring the performance of SOOD task. In addition, replacing score selection
used in Dense Teacher (Zhou et al., 2022) with joint confidence selection can
slightly enhances both recall and precision, thereby improving the performance
from 59.65 mAP to 60.45 mAP. However, it is still not the optimal solution. By
considering the imbalance between scores of objects of different scales and
the disparity between scores and centerness values, SLA significantly enhances
recall while maintaining satisfactory precision, thereby markedly improving
SOOD performance.
The analysis of CCSL: To validate our analysis in Section 3.2 that predicted
centerness is a more suitable proxy localization quality of oriented box than
predicted IoU, we perform the comparative experiment between them. As shown in
Table 7, compared to predicting the uncontrollable IoU, predicting the
centerness with considering the scale information improves the SOOD
performance. In addition, the scale factor $\gamma$ is also a critical
component. The central and edge features of extremely small objects are very
similar, leading to significant differences in the predicted centerness values
confuse the algorithm’s training. By combining the scale factor $\gamma$, the
variance of the target internal center distribution can be dynamically
adjusted according to the target scale. Therefore, compared to vanilla
centerness, centerness with the scale factor can better enhance the detection
performance.
Table 7. Ablation studies on CCSL. Localization Quality | $\beta$ | mAP
---|---|---
IoU | - | 60.98
Centerness | 0 | 60.87
0.1 | 61.38
0.2 | 62.63
0.25 | 62.26
Table 8. The impacts of GCA on objects with large aspect ratios. | HB | BD | GTF | mAP
---|---|---|---|---
| Recall | mAP | Recall | mAP | Recall | mAP
FCOS | 81.7 | 71.8 | 60.8 | 40.1 | 72.6 | 55.2 | 65.5
+GCA | 84.6 | 74.1 | 66.5 | 45.7 | 83.0 | 62.5 | 67.2
### 4.4. Inconsistency Mitigation
Sampling Inconsistency: We conduct additional experiments to demonstrate the
effectiveness of GCA in sampling inconsistency mitigation. The metric is the
detection performance for objects with large aspect ratios. The models are
trained on the DOTA-v1.5-train and evaluated on the DOTA-v1.5-val. We extract
the three categories with the highest aspect ratio from the 16 categories in
DOTA-v1.5, namely HB (harbor), BD (bridge), and GTF (ground-track-field). The
experimental results are shown in Table 8. With the support of GCA, there is a
notable enhancement in both recall and mAP for these objects with the large
aspect ratios, which verifies that GCA can effectively alleviating the
sampling inconsistency, enhancing the performance of large aspect ratio object
detection.
Assignment Inconsistency: To verify the necessity of using pixel-level pseudo-
labels to eliminate assignment inconsistency as discussed in Section 3.2, we
compare the performance of the pseudo-boxes-based method Mean Teacher(we
follow the Consistent Teacher (Wang et al., 2023) that call the vanilla
pseudo-boxes framework Mean Teacher) and the pixel-level pseudo-labels-based
method Dense Teacher (Zhou et al., 2022) on FCOS (Tian et al., 2019). As shown
in Table 5, Mean Teacher exhibits a significant performance drop due to the
fact that dense object detectors adopt a binary label assignment strategy,
making them highly sensitive to the overall quality of the oriented pseudo-
boxes which is difficult to guarantee. In contrast, pixel-level pseudo-labels
greatly alleviate the assignment inconsistency introduced during label
assignment, thereby improving the performance.
Confidence Inconsistency: To assess the effectiveness of CCSL in eliminate
confidence inconsistency, we train two FCOS detectors on DOTA-v1.5-train, one
of which is supervised with one-hot label as usual and the other with CCSL. We
visualize the score-centerness heatmaps of all ground truth points on the
DOAT-v1.5-val set, which is shown in Figure 6. As highlighted in the blue
squares, with the incorporation of CCSL, there is an improvement in the
positive correlation between classification scores and centerness that
represents the localization quality, which demonstrates that the CCSL is
effective in eliminating the inconsistency between classification and
localization qualities.
## 5\. Conclusion
In this work, we propose a Multi-clue Consistency Learning (MCL) framework to
boost the performance of semi-supervised oriented object detection. We attempt
to bridge the gaps between general and oriented object detection in semi-
supervised learning through investigating three inconsistency issues. To
mitigate the sampling inconsistency, we design the Gaussian Center Assignment
to consider various shapes of aerial objects by employing the limited
annotation data. The proposed Scale-aware Label Assignment and Consistent
Confidence Soft Label strategies to mine more accurate supervised information
from a large amount of unlabeled data, which can further promote the semi-
supervised learning. Extensive experimental results on the public DOTA-v1.0
and DOTA-v1.0 datasets clearly demonstrate our proposed MCL is effective to
bridge the gaps between the general and oriented object detection in the semi-
supervised learning.
Figure 6. Illustrated analysis of CCSL. (a) The heatmap of the scores and
centerness of positive samples trained based on CCSL. (b) The heatmap of the
scores and centerness of positive samples trained based on one-hot label.
## References
* (1)
* Deng et al. (2009) Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet: A large-scale hierarchical image database. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. Ieee, 248–255.
* Ge et al. (2021) Zheng Ge, Songtao Liu, Zeming Li, Osamu Yoshie, and Jian Sun. 2021. Ota: Optimal transport assignment for object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 303–312.
* Girshick (2015) Ross Girshick. 2015. Fast r-cnn. In _Proceedings of the IEEE International Conference on Computer Vision_. 1440–1448.
* Han et al. (2021) Jiaming Han, Jian Ding, Nan Xue, and Gui-Song Xia. 2021. Redet: A rotation-equivariant detector for aerial object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 2786–2795.
* He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep residual learning for image recognition. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 770–778.
* Hou et al. (2022) Liping Hou, Ke Lu, Jian Xue, and Yuqiu Li. 2022. Shape-adaptive selection and measurement for oriented object detection. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 36. 923–932.
* Hua et al. (2023) Wei Hua, Dingkang Liang, Jingyu Li, Xiaolong Liu, Zhikang Zou, Xiaoqing Ye, and Xiang Bai. 2023. SOOD: Towards semi-supervised oriented object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 15558–15567.
* Kim and Lee (2020) Kang Kim and Hee Seok Lee. 2020. Probabilistic anchor assignment with iou prediction for object detection. In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXV 16_. Springer, 355–371.
* Li et al. (2022d) Aoxue Li, Peng Yuan, and Zhenguo Li. 2022d. Semi-supervised object detection via multi-instance alignment with global class prototypes. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 9809–9818.
* Li et al. (2022a) Gang Li, Xiang Li, Yujie Wang, Yichao Wu, Ding Liang, and Shanshan Zhang. 2022a. Pseco: Pseudo labeling and consistency training for semi-supervised object detection. In _European Conference on Computer Vision_. 457–472.
* Li et al. (2022b) Gang Li, Xiang Li, Yujie Wang, Wu Yichao, Ding Liang, and Shanshan Zhang. 2022b. Dtg-ssod: Dense teacher guidance for semi-supervised object detection. _Advances in Neural Information Processing Systems_ 35 (2022), 8840–8852.
* Li et al. (2022c) Hengduo Li, Zuxuan Wu, Abhinav Shrivastava, and Larry S Davis. 2022c. Rethinking pseudo labels for semi-supervised object detection. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 36. 1314–1322.
* Li et al. (2020) Xiang Li, Wenhai Wang, Lijun Wu, Shuo Chen, Xiaolin Hu, Jun Li, Jinhui Tang, and Jian Yang. 2020. Generalized focal loss: Learning qualified and distributed bounding boxes for dense object detection. _Advances in Neural Information Processing Systems_ 33 (2020), 21002–21012.
* Li et al. (2023) Yuxuan Li, Qibin Hou, Zhaohui Zheng, Ming-Ming Cheng, Jian Yang, and Xiang Li. 2023. Large selective kernel network for remote sensing object detection. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_. 16794–16805.
* Lin et al. (2017a) Tsung-Yi Lin, Piotr Dollár, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie. 2017a. Feature pyramid networks for object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 2117–2125.
* Lin et al. (2017b) Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár. 2017b. Focal loss for dense object detection. In _Proceedings of the IEEE International Conference on Computer Vision_. 2980–2988.
* Lin et al. (2014) Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. 2014. Microsoft coco: Common objects in context. In _European Conference Computer Vision_. 740–755.
* Liu et al. (2023a) Chang Liu, Weiming Zhang, Xiangru Lin, Wei Zhang, Xiao Tan, Junyu Han, Xiaomao Li, Errui Ding, and Jingdong Wang. 2023a. Ambiguity-resistant semi-supervised learning for dense object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 15579–15588.
* Liu et al. (2023b) Liang Liu, Boshen Zhang, Jiangning Zhang, Wuhao Zhang, Zhenye Gan, Guanzhong Tian, Wenbing Zhu, Yabiao Wang, and Chengjie Wang. 2023b. Mixteacher: Mining promising labels with mixed scale teacher for semi-supervised object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7370–7379.
* Liu et al. (2021) Yen-Cheng Liu, Chih-Yao Ma, Zijian He, Chia-Wen Kuo, Kan Chen, Peizhao Zhang, Bichen Wu, Zsolt Kira, and Peter Vajda. 2021. Unbiased teacher for semi-supervised object detection. _arXiv preprint arXiv:2102.09480_ (2021).
* Liu et al. (2022) Yen-Cheng Liu, Chih-Yao Ma, and Zsolt Kira. 2022. Unbiased teacher v2: Semi-supervised object detection for anchor-free and anchor-based detectors. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 9819–9828.
* Ming et al. (2021) Qi Ming, Zhiqiang Zhou, Lingjuan Miao, Hongwei Zhang, and Linhao Li. 2021. Dynamic anchor learning for arbitrary-oriented object detection. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 35. 2355–2363.
* Monge (1781) Gaspard Monge. 1781. Mémoire sur la théorie des déblais et des remblais. _Mem. Math. Phys. Acad. Royale Sci._ (1781), 666–704.
* Nie et al. (2023) Yuxiang Nie, Chaowei Fang, Lechao Cheng, Liang Lin, and Guanbin Li. 2023. Adapting object size variance and class imbalance for semi-supervised object detection. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 37. 1966–1974.
* Ren et al. (2015) Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. 2015. Faster r-cnn: Towards real-time object detection with region proposal networks. _Advances in Neural Information Processing Systems_ 28 (2015).
* Sohn et al. (2020) Kihyuk Sohn, Zizhao Zhang, Chun-Liang Li, Han Zhang, Chen-Yu Lee, and Tomas Pfister. 2020. A simple semi-supervised learning framework for object detection. _arXiv preprint arXiv:2005.04757_ (2020).
* Sun et al. (2021) Peize Sun, Yi Jiang, Enze Xie, Wenqi Shao, Zehuan Yuan, Changhu Wang, and Ping Luo. 2021. What makes for end-to-end object detection?. In _International Conference on Machine Learning_. PMLR, 9934–9944.
* Tarvainen and Valpola (2017) Antti Tarvainen and Harri Valpola. 2017. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. _Advances in Neural Information Processing Systems_ 30 (2017).
* Tian et al. (2019) Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. 2019. Fcos: Fully convolutional one-stage object detection. In _Proceedings of the IEEE International Conference on Computer Vision_. 9627–9636.
* Wang et al. (2023) Xinjiang Wang, Xingyi Yang, Shilong Zhang, Yijiang Li, Litong Feng, Shijie Fang, Chengqi Lyu, Kai Chen, and Wayne Zhang. 2023. Consistent-Teacher: Towards Reducing Inconsistent Pseudo-Targets in Semi-Supervised Object Detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 3240–3249.
* Wu et al. (2024) Wenhao Wu, Hau-San Wong, and Si Wu. 2024. Pseudo-Siamese Teacher for Semi-Supervised Oriented Object Detection. _IEEE Transactions on Geoscience and Remote Sensing_ (2024).
* Xia et al. (2018) Gui-Song Xia, Xiang Bai, Jian Ding, Zhen Zhu, Serge Belongie, Jiebo Luo, Mihai Datcu, Marcello Pelillo, and Liangpei Zhang. 2018. DOTA: A large-scale dataset for object detection in aerial images. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 3974–3983.
* Xie et al. (2021) Xingxing Xie, Gong Cheng, Jiabao Wang, Xiwen Yao, and Junwei Han. 2021. Oriented R-CNN for object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 3520–3529.
* Xu et al. (2023) Chang Xu, Jian Ding, Jinwang Wang, Wen Yang, Huai Yu, Lei Yu, and Gui-Song Xia. 2023. Dynamic coarse-to-fine learning for oriented tiny object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7318–7328.
* Xu et al. (2022) Chang Xu, Jinwang Wang, Wen Yang, Huai Yu, Lei Yu, and Gui-Song Xia. 2022. RFLA: Gaussian receptive field based label assignment for tiny object detection. In _European Conference on Computer Vision_. Springer, 526–543.
* Xu et al. (2021) Mengde Xu, Zheng Zhang, Han Hu, Jianfeng Wang, Lijuan Wang, Fangyun Wei, Xiang Bai, and Zicheng Liu. 2021. End-to-end semi-supervised object detection with soft teacher. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 3060–3069.
* Yang et al. (2021) Qize Yang, Xihan Wei, Biao Wang, Xian-Sheng Hua, and Lei Zhang. 2021. Interactive self-training with mean teachers for semi-supervised object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 5941–5950.
* Yang et al. (2022) Xue Yang, Gefan Zhang, Xiaojiang Yang, Yue Zhou, Wentao Wang, Jin Tang, Tao He, and Junchi Yan. 2022. Detecting rotated objects as gaussian distributions and its 3-d generalization. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ 45, 4 (2022), 4335–4354.
* Zhang et al. (2021) Haoyang Zhang, Ying Wang, Feras Dayoub, and Niko Sunderhauf. 2021. Varifocalnet: An iou-aware dense object detector. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 8514–8523.
* Zhang et al. (2023) Lei Zhang, Yuxuan Sun, and Wei Wei. 2023. Mind the gap: Polishing pseudo labels for accurate semi-supervised object detection. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 37. 3463–3471.
* Zhang et al. (2020) Shifeng Zhang, Cheng Chi, Yongqiang Yao, Zhen Lei, and Stan Z Li. 2020. Bridging the gap between anchor-based and anchor-free detection via adaptive training sample selection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 9759–9768.
* Zhou et al. (2022) Hongyu Zhou, Zheng Ge, Songtao Liu, Weixin Mao, Zeming Li, Haiyan Yu, and Jian Sun. 2022. Dense teacher: Dense pseudo-labels for semi-supervised object detection. In _European Conference on Computer Vision_. 35–50.
|
# Using Graph Algorithms to Pretrain Graph Completion Transformers
Jonathan Pilault1 , Michael Galkin1, Bahare Fatemi2∗,
Perouz Taslakian3∗, David Vasquez4, Christopher Pal1,4,5
1Polytechnique Montreal & Mila, 2 Google Research,
3Samsung AI, 4ServiceNow Research, 5Canada CIFAR AI Chair
<EMAIL_ADDRESS>Work performed
at ServiceNow Research.
###### Abstract
Recent work on Graph Neural Networks has demonstrated that self-supervised
pretraining can further enhance performance on downstream graph, link, and
node classification tasks. However, the efficacy of pretraining tasks has not
been fully investigated for downstream large knowledge graph completion tasks.
Using a contextualized knowledge graph embedding approach, we investigate five
different pretraining signals, constructed using several graph algorithms and
_no external data_ , as well as their combination. We leverage the versatility
of our Transformer-based model to explore _graph structure generation_
pretraining tasks, typically inapplicable to most graph embedding methods. We
further propose a new path-finding algorithm guided by information gain and
find that it is the best-performing pretraining task across three downstream
knowledge graph completion datasets. In a multitask setting that combines all
pretraining tasks, our method surpasses some of the latest and strong
performing knowledge graph embedding methods on all metrics for fb15k-237, on
Hit@1 for wn18rr and on MRR and hit@10 for jf17k (a knowledge hypergraph
dataset).
## 1 Introduction
Transfer learning has recently emerged as a powerful technique in several
domains Ramachandran et al. (2017); Pennington et al. (2014); Tomar et al.
(2017); Subramanian et al. (2018); Oquab et al. (2014); Yosinski et al.
(2014); Trinh et al. (2019), in which a model is first pretrained on relevant
tasks before being fine-tuned on a downstream task. In most modern vision and
NLP applications, such pretraining is often based on the versatile Transformer
model Vaswani et al. (2017) using self-supervised learning on unlabeled data
Devlin et al. (2019); Raffel et al. (2020); Bao et al. (2022). Transformer-
based and self-supervised pretraining have also been applied in the graph
representation learning scenarios. These studies, however, focus mostly on
non-relational graphs Thakoor et al. (2022); Fatemi et al. (2021) or small
molecular graphs Ying et al. (2021), leaving pretraining approaches for
Knowledge Graphs (KG) relatively unexplored.
Many Graph Neural Network (GNN) techniques use positional embeddings of the
entities in the relation along with entity/relation embeddings to represent
tuples You et al. (2019). Transformer Language Models Vaswani et al. (2017)
use positional embeddings in a similar way. Transformers can perform
_contextualized_ link prediction by masking out one of the entities in an
input tuple. In an arity-2 graph, a KG can be viewed as a set of triplets
$\mathcal{G}=\\{(h_{i},r_{i},t_{i})\\}$, $i\in\\{1,\dotsc,T\\}$, where $h_{i}$
is the head entity, $t_{i}$ is the tail entity, and $r$ is the relation
between $h_{i}$ and $t_{i}$. We can feed the sequence $h_{i}$, $r_{i}$, [mask]
or [mask], $r_{i}$, $t_{i}$, where [mask] represents the entity to predict.
Since Transformers can process sequences of arbitrary length, we can go beyond
triple-based KGs, and consider the more general case of _Knowledge
Hypergraphs_ (KHGs, Fatemi et al. (2020)), where edges are composed of tuples
with an arbitrary number of entities. With the versatility of our base model,
we can also perform other tasks, such as autoregressive path generation or
query conditioned neighborhood prediction, that GNNs or KG embeddings cannot.
In this paper, we study the effectiveness of various graph-based pretraining
schemes that _do not use external data_ applied to a Transformer model,
trained either separately or jointly, and evaluated on the downstream task of
KG completion (link prediction). Not using external data is instrumental in
situations where data is scarce, confidential, or classified, for example,
medical records, crime networks, or terrorist activity. Our techniques can be
applied to a generalization of KGs, i.e. KHGs. Our contributions are as
follows: (i) formalizing and introducing five pretraining objectives suitable
for sequence-based KG Transformers; (ii) introducing a new path-finding
algorithm that is guided by information gain; (iii) evaluating the effect of
pretraining strategies on triple-based KGs as well as on KHGs.
## 2 Related Works
Our method is a _Contextualized KG Embedding_ method Wang et al. (2019) that
uses _Self-Supervised Graph Representation_ pretraining.
#### Knowledge Graph Embedding
methods encode distributed representations of entities and relations using a
continuous vector. The representation is learned via a scoring function that
measures the plausibility that a given triple $\\{(h_{i},r_{i},t_{i})\\}$
exists. Such embeddings maximize the plausibility of observed triples, which
may not be predictive enough for downstream tasks Wang et al. (2015); Wei et
al. (2015). To make the embeddings more transferable, researchers have
incorporated other types of information. For example, _external_ data sources
can complement a KG embedding by also encoding entity types Guo et al. (2015);
Xie et al. (2016b) or textual descriptions Xie et al. (2016a); Wang and Li
(2016). As explored here, we can also enhance KG embedding by _only_ using
structural features from the graph. For example, relational paths and multi-
hop relationships between entities (see section A for more details), are a
type of structural information that has proven useful Lin et al. (2015);
Toutanova et al. (2016); Das et al. (2017) for KG completion.
#### Self-Supervised Graph Representation
learning techniques broadly fall into three categories and include methods
that: (1) use random walk procedures to encode diverse neighborhoods Perozzi
et al. (2014); Grover and Leskovec (2016); Hamilton et al. (2017a); (2)
reconstruct a graph’s adjacency matrix Kipf and Welling (2016); Hasanzadeh et
al. (2019); Fatemi et al. (2021); (3) maximize mutual information between
local node representations and global graph representations Veličković et al.
(2019); You et al. (2020); Liang et al. (2021). For applications in chemistry
and biology, most pre-training techniques also use additional datasets and
signals like MAE loss Hu et al. (2020a); Ying et al. (2021), however, our
technique does not use _external_ data. With the exception of Deep Relational
Graph Infomax (drgi) Liang et al. (2021), most self-supervised pretraining
schemes based on graph structure have never been applied to solve large KG
completion tasks. Other techniques such as kg-bert Yao et al. (2019) use a
bert-based Devlin et al. (2019) pretrained language models with entity
descriptions for KG link prediction. Again, such work relies on external data
and additional information (descriptions) while our pretraining signals are
self-contained.
## 3 Methodology
In this section, we provide an overview of the five different graph algorithms
that we use to create pretraining tasks for our experiments: (1) Relational
Shortest Path sequence generation (sp); (2) Information gain Path sequence
generation (ip); (3) K-Hop Neighbor prediction (khn); (4) Invariant Adjacency
matrix classification (iva); (5) Local Clustering Coefficient estimation
(lcc). Please see Appendix A for more details on definitions and notation used
in this section. When jointly trained (all), we use all pretraining tasks and
prepend a task token to delineate each task. Note that for all, the total is
the unweighted sum of each pretraining task’s loss. An overview of the task
objectives is outlined in Table 1.
Task | Input | Target | Objective | Type
---|---|---|---|---
Path-based
sp | $\\{e_{i},r,e_{j}\\}$ | $\\{P_{rel}\\}_{r}$ | $\prod_{k}^{n}f(p_{k}|p_{<k},e_{i},r,e_{j})$ | SG
ip
Neighborhood Based
khn | $\\{e_{i},E_{\mathcal{N}_{k-1}}\\}$ | $\mathbf{O}_{\mathcal{N}_{k}}$ | $\text{KL}(\mathbf{O}_{\mathcal{N}_{k}}||f(e_{i},E_{\mathcal{N}_{k-1}}))$ | MP
iva | $\\{\mathbf{A},\widetilde{\mathbf{A}}|\mathbf{A}^{\prime}\\}$ | $\\{1|0\\}$ | $y_{i}log(f(\mathbf{A},\widetilde{\mathbf{A}}|\mathbf{A}^{\prime}))$ | BC
lcc | $\\{e_{i},E_{\mathcal{N}_{k-1}}\\}$ | $c_{e_{i}}$ | $(f(e_{i})-c_{e_{i}})_{k}^{2}$ | R
Table 1: Overview of Graph Algorithm Pretraining Tasks. Rel=Uses Relations;
SG=Sequence Generation; BC=Binary Classification; MP=Multi-label Prediction;
R=Regression.
### 3.1 Relational Shortest Paths (sp)
The set of relational paths $\\{P_{rel}\\}_{r}$ as defined in Section A may
yield an exponential number of choices. In practice, we would like to limit
the number of such paths and hence, we need a way to select a subset of
$\\{P_{rel}\\}_{r}$. There are a few possible heuristics for selecting a
subset of such paths. For example, we can find shortest paths based on
Dijkstra’s algorithm and then just keep the sequences of relations joining two
entities $\\{e_{i},r,e_{j}\\}$. As specified in Table 1, we condition our path
sequence generation with $\\{e_{i},r,e_{j}\\}$. The next token generated is
the first relation on a sampled path $\\{P_{rel}\\}_{r}$. Our path-based
pretraining algorithms allow the model to explore ${\mathcal{G}}$ beyond the
tuples included in $\mathcal{G^{\prime}}$ since ${\mathcal{G}}$ includes
relational paths between entities that may not appear in any of the train,
validation, or test sets. A masked query “[Spain] [form of government] [mask]”
can have many relational paths linking the [Spain] entity to a target entity
that we want to predict. For example, a relational path may be “[nominated
for], [film/country]” that will link [Spain] to the target entity to be
predicted. When no path exist between $\\{e_{i},e_{j}\\}$, the model target is
the [no_path] token.
### 3.2 Information gain Paths (ip)
Input: Training set $\mathcal{G^{\prime}}$; query relation $r$; $k$ number of
top paths; max relational paths length (max hops) $l$
Output: at most $n=k^{l}$ relational paths $\\{P_{rel}\\}_{r}$;
1
$R^{\prime}\leftarrow\text{FindIncidentRelations}(r,\emptyset)$ // Algorithm 2
$R^{\prime}\leftarrow\text{TopEntropy}(k,R^{\prime})$ // Algorithm 3
2 $\\{P_{rel}\\}_{r}\leftarrow R^{\prime}$
3 for _$i\in\\{1,\dotsc,l\\}$_ do
4 for _$r^{\prime}\in R^{\prime}$ ; $p\in\\{P_{rel}\\}_{r}$_ do
5 $R^{\prime\prime}\leftarrow\text{FindIncidentRelations}(r^{\prime})$
$R^{\prime\prime}\leftarrow\text{BottomCondEntropy}(k,R^{\prime\prime},r^{\prime})$
// Algorithm 3
6
7 for _$r^{\prime\prime}\in R^{\prime\prime}$_ do
8 if _$r^{\prime}$ is the last element in $p$_ then
9 $\\{P_{rel}\\}_{r}\leftarrow\\{p\\}\ \cup\\{r^{\prime\prime}\\}$
10 end if
11
12 end for
13
14 end for
15 $R^{\prime}\leftarrow R^{\prime\prime}$
16 end for
Return: $\\{P_{rel}\\}_{r}$
Algo 1 Top_k Path Information Gain
While sp provides a measure of subgraph distance and connectivity, i.e. the
number of elements between entities $\\{e_{i},e_{j}\\}$ in the graph, it may
often yield reasoning chains having little semantic overlap with the original
query. For example, a masked query “[Spain] [form of government] [mask]”
produces many sp such as “[nominated for], [film/country]”, which is a
sequence of relations unrelated to “[form of government]”. Further, sp
algorithms select paths that minimize the distance (number of hops) between
two entities. It is therefore formed from shortest paths of the same length,
which may limit subgraph exploration. We propose _Information Gain Paths_ in
Algorithm 1. ip has the same initial condition $\\{e_{i},r,e_{j}\\}$ to start
generating the paths and has the same learning objective as sp. ip also helps
us reduce the number of paths by selecting the ones that constitute a
beneficial context for answering the query based on a measure of information
gain. Given a query tuple $Q_{r}$ (having relation $r$), the algorithm
progressively builds relational paths $\\{P_{rel}\\}_{r}$ starting from a
relation $r$ such that at each step, it selects the (at most) $k$ _incident_
relations that would yield the $k$ highest information gain for the paths
constructed so far.
With infinite $k$, the algorithm is identical to the breadth-first search as
no relations are ignored. Algorithm 1 bears some similarities with beam search
Goldberg and Reddy (1977). There are however a few differences: (1) for $l$
hops and $k$ beam size, we obtain a maximum of $k^{l}$ paths; (2) a relation
already in $P_{rel}$ cannot be used to form the next hop; (3) the paths are
formed via a back-chaining process starting from the last relation on path
$P_{rel}$ that connects $e_{i}$ and $e_{j}$; (4) an extra step is performed to
select paths in $\\{P_{rel}\\}_{r}$ linking entities in $Q_{r}$.
We let $\text{IG}({P_{rel}})$ denote the _Information Gain_ of relational path
$P_{rel}:r_{1},\dotsc,r_{l}\in{\mathcal{G}}^{\prime}$ having length $l$ as
$\text{IG}({P_{rel}})=H(r_{l})-\sum\limits_{i=1}^{l-1}H(r_{l-i}|r_{l-i+1})$.
We define the _Information Entropy_ $H(r_{l})$ for a relation $r_{l}$, and
entities $E(r_{l})$ that appear in a tuple in ${\mathcal{G}}^{\prime}$ having
relation $r_{l}$ as:
$\scriptstyle
H(r_{l})=-\left(\frac{|E(r_{l})|}{|{\mathcal{E}}|}\log(\frac{|E(r_{l})|}{|{\mathcal{E}}|})+\frac{|{\mathcal{E}}\setminus
E(r_{l})|}{|{\mathcal{E}}|}\log(\frac{|{\mathcal{E}}\setminus
E(r_{l})|}{|{\mathcal{E}}|})\right),$ (1)
where ${\mathcal{E}}$ is all entities in ${\mathcal{G}}^{\prime}$.
The _Conditional Information Entropy_ $H(r_{i-1}|r_{i})$ of two consecutive
relations is defined as:
$\displaystyle H(r_{i-1}|r_{i})=$
$\displaystyle-\frac{|E(r_{i-1})|}{|{\mathcal{E}}|}\biggl{(}\mathrm{U}\log(\mathrm{U})+\mathrm{V}\log(\mathrm{V})\biggr{)}$
(2)
where $\mathrm{U}=\frac{|E(r_{i-1})\cup E(r_{i})\setminus E(r_{i-1})\cap
E(r_{i})|}{|E(r_{i-1})|}$ and $\mathrm{V}=\frac{|E(r_{i-1})\cap
E(r_{i})|}{|E(r_{i-1})|}$.
| Pre | fb15k-237 | wn18rr | jf17k
---|---|---|---|---
Trained | MRR | H@1 | H@3 | H@10 | MRR | H@1 | H@3 | H@10 | MRR | H@1 | H@3 | H@10
Contextualized KG Embedding Results
kg-trsf 1 | $\times$ | 0.364 | 0.272 | 0.400 | 0.549 | 0.484 | 0.450 | 0.496 | 0.553 | 0.532 | 0.445 | 0.561 | 0.687
kg-trsf sp | ✓ | 0.363 | 0.272 | 0.403 | 0.559 | 0.473 | 0.438 | 0.488 | 0.549 | 0.537 | 0.450 | 0.569 | 0.703
kg-trsf ip | ✓ | 0.370 | 0.273 | 0.408 | 0.571 | 0.489 | 0.454 | 0.505 | 0.573 | 0.549 | 0.457 | 0.582 | 0.723
kg-trsf khn | ✓ | 0.361 | 0.266 | 0.398 | 0.558 | 0.479 | 0.444 | 0.491 | 0.556 | 0.536 | 0.450 | 0.565 | 0.701
kg-trsf lcc | ✓ | 0.350 | 0.258 | 0.386 | 0.535 | 0.477 | 0.442 | 0.488 | 0.544 | 0.532 | 0.445 | 0.561 | 0.679
kg-trsf iva | ✓ | 0.358 | 0.264 | 0.397 | 0.552 | 0.479 | 0.445 | 0.494 | 0.556 | 0.538 | 0.453 | 0.568 | 0.705
kg-trsf all | ✓ | 0.380 | 0.277 | 0.422 | 0.591 | 0.499 | 0.456 | 0.511 | 0.588 | 0.554 | 0.465 | 0.594 | 0.715
SOTA KG Embedding Results
boxe | ✓ | 0.337 | 0.238 | 0.374 | 0.538 | 0.451 | 0.400 | 0.472 | 0.541 | 0.553 | 0.467 | 0.596 | 0.711
drgi | ✓ | 0.362 | 0.270 | 0.399 | 0.549 | 0.479 | 0.445 | 0.496 | 0.543 | — | — | — | —
star* | ✓ | 0.358 | 0.205 | 0.322 | 0.482 | 0.401 | 0.243 | 0.491 | 0.675 | — | — | — | —
lp-bert* | ✓ | 0.310 | 0.223 | 0.336 | 0.490 | 0.482 | 0.343 | 0.563 | 0.752 | — | — | — | —
Table 2: Link prediction test results . H<EMAIL_ADDRESS>Results from: 1Wang et al.
(2019). In the table, bold is best and underline is second best. _*Uses
external data and is a contextualized KG embedding method_.
### 3.3 k-Hop Neighborhood prediction (khn)
Our next graph algorithm to create our pretraining data is based on the
observation that the representation of nodes in a neighborhood
$\mathcal{N}_{k}(e_{i})$ encodes subgraph structure and provides a more
powerful representation of $e_{i}$ Hamilton et al. (2017b). To coerce a
Transformer model to understand local structure up to $k$ hops, we ask the
model to generate $\mathbf{O}_{\mathcal{N}_{k}}$ of the next hop from Eq. 5,
given $E_{\mathcal{N}_{k-1}}$ and $e_{i}$. We input an arbitrarily ordered set
of entities $E_{\mathcal{N}_{k-1}}$ to condition our prediction. Note that
typically entities are ordered according to their appearance in the original
datasets. The loss to learn output entity occurrence probability of entities
for hops up to $k=3$ is
$L_{\textsl{{khn}}}=\sum^{k}_{i=1}\big{(}\text{KL}(\mathbf{O}_{\mathcal{N}_{k}}||f(e_{i},E_{\mathcal{N}_{k-1}}))\big{)}$,
where $E_{\mathcal{N}_{0}}=\\{\\}$ and KL is the Kullback–Leibler divergence
Kullback and Leibler (1951). Please note that we directly chose $k=3$ since it
typically performs better than $k=2$ in a variety of settings Nikolentzos et
al. (2020).
### 3.4 Invariant Adjacency matrix (iva)
One of the key issues with previous heuristics in Section 3.3 is that a
certain order is assumed. However, graph representations are more powerful
when they are order invariant Khasanova and Frossard (2017). To allow order
invariance, we propose a binary classification task based on adjacency
matrices $E_{\mathcal{N}_{k}}$ of the local neighborhood subgraph. The label
is $1$ if the Transformer is given inputs
$\\{\mathbf{A},\widetilde{\mathbf{A}}\\}$, where $\widetilde{\mathbf{A}}$ is
the equivalent adjacency matrix to $\mathbf{A}$ where the entity order is
randomly permuted. The label is $0$ if the Transformer is given inputs
$\\{\mathbf{A},\mathbf{A}^{\prime}\\}$, where $\mathbf{A}^{\prime}$ is a
corrupted adjacency matrix. We corrupt the adjacency matrix by randomly
swapping the columns or by randomly assigning different adjacency matrix
values. Since $\mathbf{A}$ is a symmetric matrix, the input to the Transformer
is a flattened sequence of the upper triangular matrix that contains (1)
column entities and (2)
$\text{row}_{1}(\mathbf{A}),\dots,\text{row}_{|E_{\mathcal{N}_{k}}|}(\mathbf{A})$.
For example, $\mathbf{A}=\begin{bmatrix}1&2&1\\\ 2&0&3\\\ 1&3&1\\\
\end{bmatrix}$ with columns $[e_{1},e_{2},e_{3}]$, will produce the flattened
sequence $[e_{1},e_{2},e_{3},1,2,1,0,3,1]$.
### 3.5 Local Clustering Coefficient (lcc)
For lcc, we use the same inputs as with khn in section 3.3. However, we are
trying to estimate the local clustering coefficient of a subgraph $k$-hops
away from $e_{i}$. The loss for $k=3$ is therefore
$L_{\textsl{{lcc}}}=\sum_{i=1}^{k}(f(e_{i})-c_{e_{i}})_{k}^{2}$.
## 4 Experiments
Experimental set-up and training details are left to Appendix C. We compare
our results to several strong baselines described in Appendix D. Our results
are presented in Table 2. Our baseline model, kg-trsf or “CoKe” Wang et al.
(2019), is a Transformer model that is trained from scratch on link
prediction.
### 4.1 Results and Discussion
Our new algorithm ip provides benefit over kg-trsf across _all_ datasets.
Further, we see that H@10 is typically better for all pretraining signals
compared to kg-trsf. Moreover, all pretraining signals (except lcc) provide
gains on the hypergraph jf17k compared to kg-trsf. We then see that using all
pretraining tasks jointly provides $5$% relative MRR score increase over the
unpretrained baseline, and often surpassing the competitive performances of
state-of-the-art models. The all results are in-line with recent graph
pretraining methods Hu et al. (2020a, b) that show that multitask pretraining
performs better than any individual task. Interestingly, our path-based
algorithm ip provides the largest single task increase in performance over the
unpretrained baseline.
Except for our wn18rr results, path-based pretraining surpasses neighborhood-
based pretraining. Compared to sp, we hypothesize that ip provides a higher
quality pretraining signal for two reasons: (1) paths are more diverse and (2)
paths are semantically closer to a query $Q_{r}$. sp produces paths that have
often redundant first 2-hop relations on a k-hop path $\\{P_{rel}\\}_{r}$.
Further, all the paths have the same minimum length and often transit through
high degree entities. As seen in Section 3.1, high degree entities do not
necessarily provide the most meaningful reasoning chains; though the ip-based
reasoning chains seem more semantically relevant. For example, for a masked
query “Spain”, “form of government” and [mask]”, ip paths are [“military
conflict/combatants”, “international organization/member states”], [“adjoining
relationship/adjoins”, “continents/countries within”] or [“organization
member/member of”, “international organization/member states”].
### 4.2 Analysis and Ablation
In this section, we discuss the choice and combination of tasks. An ablation
study is presented in table 3 that allows us to compare various multitask
combinations. We first notice that given sp, ip or sp \+ ip relational path-
based pretraining schemes, adding any other pretraining signal (khn, lcc or
iva) typically results in an improvement. khn provides the largest MRR
increase when combined with relational path-based signals, with sp \+ ip \+
khn providing a 0.06 improvement in MRR. iva comes in as the second best
signal to combine and lcc only improving MRR slightly.
Pretraining Signal Combination (MRR)
---
sp | 0.363 | ip | 0.370 | sp \+ ip | 0.372
\+ khn | 0.367 | \+ khn | 0.373 | \+ khn | 0.378
\+ lcc | 0.364 | \+ lcc | 0.370 | \+ lcc | 0.374
\+ iva | 0.366 | \+ iva | 0.373 | \+ iva | 0.375
Table 3: Task combination ablation study on fb15k-237 using MRR. Adding
individual signals to sp, ip or sp \+ ip pretraining schemes.
## 5 Conclusion
We have investigated the effectiveness of five graph algorithmic pretraining
schemes that do not rely on external data sources. On three downstream KG
completion tasks, we found that kg-trsf: (1) multitask pretraining results in
performance increases, (2) generally, pretrained models exhibit better
improvements in recall (H@10) rather than precision (H@1), and (3) the ip
pretraining tasks work best. Our study has important implications since it
shows that using various graph structural signals that do not rely on external
data can outperform strong baselines pretrained with external data.
## Limitations.
Our technique has a few limitations. First, the pretraining signals that we
used in this paper require a highly adaptive model. Out of the five
pretraining schemes, two include the task of relational path sequence
generation (sp and ip) and another is a local clustering coefficient
regression task (lcc). Such tasks typically cannot be performed by most Graph
Neural Networks or Graph Embedding methods. However, the versatility of our
Transformer-based technique also means that our model can be pretrained on
multiple modalities Mustafa et al. (2022); Lee et al. (2022). For example,
pretraining with text and KGs has already proven very powerful in language
generation tasks Agarwal et al. (2021). The point of our study was to show
that we can surpass other methods such as drgi and lp-bert (see Appendix D for
an overview of baselines) that use pretrained bert models and entity
descriptions. We suspect that enhancing our entity and relation
representations with text will only make the model even stronger at KG
completion. Finally, we notice that jointly training on all pretraining tasks
yields the best results. However, since it is possible that negative task
interference occurs (a negative side effect of multitask learning Zhang et al.
(2020); Pilault et al. (2021), a more throughout study of task combinations
can help unlock even better performances for each specific dataset.
## References
* Abboud et al. (2020) Ralph Abboud, Ismail Ceylan, Thomas Lukasiewicz, and Tommaso Salvatori. 2020. Boxe: A box embedding model for knowledge base completion. In _Advances in Neural Information Processing Systems_ , volume 33, pages 9649–9661. Curran Associates, Inc.
* Agarwal et al. (2021) Oshin Agarwal, Heming Ge, Siamak Shakeri, and Rami Al-Rfou. 2021. Knowledge graph based synthetic corpus generation for knowledge-enhanced language model pre-training. In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 3554–3565, Online. Association for Computational Linguistics.
* Bao et al. (2022) Hangbo Bao, Li Dong, Songhao Piao, and Furu Wei. 2022. BEit: BERT pre-training of image transformers. In _International Conference on Learning Representations_.
* Bollacker et al. (2008) Kurt D. Bollacker, Colin Evans, Praveen K. Paritosh, Tim Sturge, and Jamie Taylor. 2008. Freebase: a collaboratively created graph database for structuring human knowledge. In _SIGMOD Conference_.
* Bordes et al. (2013) Antoine Bordes, Nicolas Usunier, Alberto Garcia-Durán, Jason Weston, and Oksana Yakhnenko. 2013. Translating embeddings for modeling multi-relational data. In _Proceedings of the 26th International Conference on Neural Information Processing Systems - Volume 2_ , NIPS’13, page 2787–2795, Red Hook, NY, USA. Curran Associates Inc.
* Das et al. (2017) Rajarshi Das, Arvind Neelakantan, David Belanger, and Andrew McCallum. 2017. Chains of reasoning over entities, relations, and text using recurrent neural networks. In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 1, Long Papers_ , pages 132–141, Valencia, Spain. Association for Computational Linguistics.
* Dettmers et al. (2018) Tim Dettmers, Pasquale Minervini, Pontus Stenetorp, and Sebastian Riedel. 2018. Convolutional 2d knowledge graph embeddings. In _Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018_ , pages 1811–1818. AAAI Press.
* Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of deep bidirectional transformers for language understanding. In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
* Dong et al. (2019) Li Dong, Nan Yang, Wenhui Wang, Furu Wei, Xiaodong Liu, Yu Wang, Jianfeng Gao, Ming Zhou, and Hsiao-Wuen Hon. 2019. Unified language model pre-training for natural language understanding and generation. In _Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada_ , pages 13042–13054.
* Fatemi et al. (2021) Bahare Fatemi, Layla El Asri, and Seyed Mehran Kazemi. 2021. Slaps: Self-supervision improves structure learning for graph neural networks. _Advances in Neural Information Processing Systems_ , 34:22667–22681.
* Fatemi et al. (2020) Bahare Fatemi, Perouz Taslakian, David Vazquez, and David Poole. 2020. Knowledge hypergraphs: Prediction beyond binary relations. In _Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20_ , pages 2191–2197. International Joint Conferences on Artificial Intelligence Organization. Main track.
* Goldberg and Reddy (1977) H. G. Goldberg and R. Reddy. 1977. Speech understanding systems. Summary of results of the five-year research effort at Carnegie-Mellon University. Interim Report Carnegie-Mellon Univ.
* Grover and Leskovec (2016) Aditya Grover and Jure Leskovec. 2016. node2vec: Scalable feature learning for networks. In _Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining_ , pages 855–864.
* Guo et al. (2015) Shu Guo, Quan Wang, Bin Wang, Lihong Wang, and Li Guo. 2015. Semantically smooth knowledge graph embedding. In _Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 84–94, Beijing, China. Association for Computational Linguistics.
* Hamilton et al. (2017a) Will Hamilton, Zhitao Ying, and Jure Leskovec. 2017a. Inductive representation learning on large graphs. In _Advances in Neural Information Processing Systems_ , volume 30. Curran Associates, Inc.
* Hamilton et al. (2017b) William L. Hamilton, Rex Ying, and Jure Leskovec. 2017b. Representation learning on graphs: Methods and applications. _IEEE Data Eng. Bull._ , 40(3):52–74.
* Hasanzadeh et al. (2019) Arman Hasanzadeh, Ehsan Hajiramezanali, Krishna Narayanan, Nick Duffield, Mingyuan Zhou, and Xiaoning Qian. 2019. Semi-implicit graph variational auto-encoders. In _Advances in Neural Information Processing Systems_ , volume 32. Curran Associates, Inc.
* Hu et al. (2020a) Weihua Hu, Bowen Liu*, Joseph Gomes, Marinka Zitnik, Percy Liang, Vijay Pande, and Jure Leskovec. 2020a. Strategies for pre-training graph neural networks. In _International Conference on Learning Representations_.
* Hu et al. (2020b) Ziniu Hu, Yuxiao Dong, Kuansan Wang, Kai-Wei Chang, and Yizhou Sun. 2020b. GPT-GNN: generative pre-training of graph neural networks. In _KDD ’20: The 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, CA, USA, August 23-27, 2020_ , pages 1857–1867. ACM.
* Khasanova and Frossard (2017) Renata Khasanova and Pascal Frossard. 2017. Graph-based isometry invariant representation learning. In _Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017_ , volume 70 of _Proceedings of Machine Learning Research_ , pages 1847–1856. PMLR.
* Kipf and Welling (2016) Thomas N. Kipf and Max Welling. 2016. Variational graph auto-encoders. _CoRR_ , abs/1611.07308.
* Kullback and Leibler (1951) S. Kullback and R. A. Leibler. 1951. On Information and Sufficiency. _The Annals of Mathematical Statistics_ , 22(1):79 – 86.
* Lee et al. (2022) Kuang-Huei Lee, Ofir Nachum, Mengjiao Yang, Lisa Lee, Daniel Freeman, Winnie Xu, Sergio Guadarrama, Ian Fischer, Eric Jang, Henryk Michalewski, and Igor Mordatch. 2022. Multi-game decision transformers.
* Li et al. (2022) Da Li, Ming Yi, and Yukai He. 2022. LP-BERT: multi-task pre-training knowledge graph BERT for link prediction. _CoRR_ , abs/2201.04843.
* Liang et al. (2021) Shuang Liang, Jie Shao, Dongyang Zhang, Jiasheng Zhang, and Bin Cui. 2021. Drgi: Deep relational graph infomax for knowledge graph completion. _IEEE Transactions on Knowledge and Data Engineering_ , pages 1–1.
* Lin et al. (2015) Yankai Lin, Zhiyuan Liu, Huanbo Luan, Maosong Sun, Siwei Rao, and Song Liu. 2015\. Modeling relation paths for representation learning of knowledge bases. In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 705–714, Lisbon, Portugal. Association for Computational Linguistics.
* Mohamed et al. (2019) Sameh K. Mohamed, Vít Novácek, Pierre-Yves Vandenbussche, and Emir Muñoz. 2019. Loss functions in knowledge graph embedding models. In _DL4KG@ESWC_.
* Mustafa et al. (2022) Basil Mustafa, Carlos Riquelme, Joan Puigcerver, Rodolphe Jenatton, and Neil Houlsby. 2022. Multimodal contrastive learning with limoe: the language-image mixture of experts.
* Nikolentzos et al. (2020) Giannis Nikolentzos, George Dasoulas, and Michalis Vazirgiannis. 2020. k-hop graph neural networks. _Neural Networks_ , 130:195–205.
* Oquab et al. (2014) Maxime Oquab, Leon Bottou, Ivan Laptev, and Josef Sivic. 2014. Learning and transferring mid-level image representations using convolutional neural networks. In _2014 IEEE Conference on Computer Vision and Pattern Recognition_ , pages 1717–1724.
* Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014. GloVe: Global vectors for word representation. In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1532–1543, Doha, Qatar. Association for Computational Linguistics.
* Perozzi et al. (2014) Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. 2014. Deepwalk: Online learning of social representations. In _Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , KDD ’14, page 701–710, New York, NY, USA. Association for Computing Machinery.
* Pilault et al. (2021) Jonathan Pilault, Amine El hattami, and Christopher Pal. 2021. Conditionally adaptive multi-task learning: Improving transfer learning in NLP using fewer parameters & less data. In _International Conference on Learning Representations_.
* Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020. Exploring the limits of transfer learning with a unified text-to-text transformer. _Journal of Machine Learning Research_ , 21(140):1–67.
* Ramachandran et al. (2017) Prajit Ramachandran, Peter Liu, and Quoc Le. 2017. Unsupervised pretraining for sequence to sequence learning. In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 383–391, Copenhagen, Denmark. Association for Computational Linguistics.
* Subramanian et al. (2018) Sandeep Subramanian, Adam Trischler, Yoshua Bengio, and Christopher J Pal. 2018\. Learning general purpose distributed sentence representations via large scale multi-task learning. In _International Conference on Learning Representations_.
* Thakoor et al. (2022) Shantanu Thakoor, Corentin Tallec, Mohammad Gheshlaghi Azar, Mehdi Azabou, Eva L Dyer, Remi Munos, Petar Veličković, and Michal Valko. 2022. Large-scale representation learning on graphs via bootstrapping. In _International Conference on Learning Representations_.
* Tomar et al. (2017) Gaurav Singh Tomar, Thyago Duque, Oscar Täckström, Jakob Uszkoreit, and Dipanjan Das. 2017. Neural paraphrase identification of questions with noisy pretraining. In _Proceedings of the First Workshop on Subword and Character Level Models in NLP_ , pages 142–147, Copenhagen, Denmark. Association for Computational Linguistics.
* Toutanova et al. (2016) Kristina Toutanova, Victoria Lin, Wen-tau Yih, Hoifung Poon, and Chris Quirk. 2016\. Compositional learning of embeddings for relation paths in knowledge base and text. In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1434–1444, Berlin, Germany. Association for Computational Linguistics.
* Trinh et al. (2019) Trieu H. Trinh, Minh-Thang Luong, and Quoc V. Le. 2019. Selfie: Self-supervised pretraining for image embedding. _CoRR_ , abs/1906.02940.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In _Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA_ , pages 5998–6008.
* Veličković et al. (2019) Petar Veličković, William Fedus, William L. Hamilton, Pietro Liò, Yoshua Bengio, and R Devon Hjelm. 2019. Deep graph infomax. In _International Conference on Learning Representations_.
* Wang et al. (2021) Bo Wang, Tao Shen, Guodong Long, Tianyi Zhou, Ying Wang, and Yi Chang. 2021. Structure-augmented text representation learning for efficient knowledge graph completion. In _Proceedings of the Web Conference 2021_ , WWW ’21, page 1737–1748, New York, NY, USA. Association for Computing Machinery.
* Wang et al. (2019) Quan Wang, Pingping Huang, Haifeng Wang, Songtai Dai, Wenbin Jiang, Jing Liu, Yajuan Lyu, Yong Zhu, and Hua Wu. 2019. Coke: Contextualized knowledge graph embedding. _CoRR_ , abs/1911.02168.
* Wang et al. (2015) Quan Wang, Bin Wang, and Li Guo. 2015. Knowledge base completion using embeddings and rules. In _Proceedings of the 24th International Conference on Artificial Intelligence_ , IJCAI’15, page 1859–1865. AAAI Press.
* Wang and Li (2016) Zhigang Wang and Juanzi Li. 2016. Text-enhanced representation learning for knowledge graph. In _Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence_ , IJCAI’16, page 1293–1299. AAAI Press.
* Watts and Strogatz (1998) D.J. Watts and S.H. Strogatz. 1998. Collective dynamics of ’small-world’ networks. _Nature_ , (393):440–442.
* Wei et al. (2015) Zhuoyu Wei, Jun Zhao, Kang Liu, Zhenyu Qi, Zhengya Sun, and Guanhua Tian. 2015. Large-scale knowledge base completion: Inferring via grounding network sampling over selected instances. In _Proceedings of the 24th ACM International on Conference on Information and Knowledge Management_ , CIKM ’15, page 1331–1340, New York, NY, USA. Association for Computing Machinery.
* Wen et al. (2016) Jianfeng Wen, Jianxin Li, Yongyi Mao, Shini Chen, and Richong Zhang. 2016. On the representation and embedding of knowledge bases beyond binary relations. In _Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, NY, USA, 9-15 July 2016_ , pages 1300–1307. IJCAI/AAAI Press.
* Xie et al. (2016a) Ruobing Xie, Zhiyuan Liu, Jia Jia, Huanbo Luan, and Maosong Sun. 2016a. Representation learning of knowledge graphs with entity descriptions. In _Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence_ , AAAI’16, page 2659–2665. AAAI Press.
* Xie et al. (2016b) Ruobing Xie, Zhiyuan Liu, and Maosong Sun. 2016b. Representation learning of knowledge graphs with hierarchical types. In _Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence_ , IJCAI’16, page 2965–2971. AAAI Press.
* Yao et al. (2019) Liang Yao, Chengsheng Mao, and Yuan Luo. 2019. Kg-bert: Bert for knowledge graph completion. _ArXiv_ , abs/1909.03193.
* Ying et al. (2021) Chengxuan Ying, Tianle Cai, Shengjie Luo, Shuxin Zheng, Guolin Ke, Di He, Yanming Shen, and Tie-Yan Liu. 2021. Do transformers really perform badly for graph representation? In _Advances in Neural Information Processing Systems_.
* Yosinski et al. (2014) Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. 2014. How transferable are features in deep neural networks? In _Advances in neural information processing systems_ , pages 3320–3328.
* You et al. (2019) Jiaxuan You, Rex Ying, and Jure Leskovec. 2019. Position-aware graph neural networks. In _International conference on machine learning_ , pages 7134–7143. PMLR.
* You et al. (2020) Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen. 2020. Graph contrastive learning with augmentations. In _Advances in Neural Information Processing Systems_ , volume 33, pages 5812–5823. Curran Associates, Inc.
* Zhang et al. (2020) Wen Zhang, Lingfei Deng, and Dongrui Wu. 2020. Overcoming negative transfer: A survey. _CoRR_ , abs/2009.00909.
## Appendix A Definition and Notation
Without loss of generality, in this section, we formulate our definitions and
notation based on hypergraphs. Given a finite set of entities ${\mathcal{E}}$
and a finite set of relations ${\mathcal{R}}$, a _tuple_ is an ordered set of
the form $r(e_{1},\dots,e_{n})$, where $r\in{\mathcal{R}}$,
$e_{i}\in{\mathcal{E}}$ for all $i=1,\dots,n$ and $|r|=n$ is its _arity_. Let
${\mathcal{G}}$ be the set of ground truth tuples; that is, it specifies all
of the tuples that are true so that if a tuple is not in $\mathcal{G}$, it is
false. A _knowledge hypergraph_ (KHG) consists of a subset of the tuples
$\mathcal{G^{\prime}}\subseteq{\mathcal{G}}$. A knowledge graph is a special
case of a knowledge hypergraph where all relations have arity $2$. We let
$E(r)\subseteq{\mathcal{E}}$ denote the set of all entities that appear in a
tuple in ${\mathcal{G}}^{\prime}$ having relation $r$. That is,
$E(r)=\\{e_{i}|r(\dots,e_{i},\dots)\in{\mathcal{G}}^{\prime}\\}$. Similarly,
we let $R(e_{i})\subseteq{\mathcal{R}}$ denote the set of all relations that
appear in a tuple in ${\mathcal{G}}^{\prime}$ linked to entity $e_{i}$. That
is, $R(e_{i})=\\{r|r(\dots,e_{i},\dots)\in{\mathcal{G}}^{\prime}\\}$. We say
that two tuples are _incident_ in ${\mathcal{G}}^{\prime}$ if they share at
least one entity. Two entities are _connected_ in $\mathcal{G}$’ if they
appear together in a tuple. For example, $r_{1}(e_{1},e_{2},e_{3})$ is
incident to $r_{2}(e_{4},e_{3},e_{5})$ because they share $e_{3}$. Entities
$e_{2}$ and $e_{3}$ are connected because they both appear in the first tuple.
A query for the knowledge completion task is a tuple with one missing entity
that needs to be predicted. We let
$Q_{r}=[r,e_{1},\dots,\mbox{{[mask]}},\dots,e_{n}]_{q}$ denote a query tuple
with relation $r$, where [mask] is the placeholder token (the masked-out
entity) for the entity we want to predict.
A _path_ $P$ in a KHG is a sequence of tuples where two consecutive tuples
share at least one entity. We say that $P$ _connects_ entities $e_{i}$ and
$e_{j}$ if the first tuple in $P$ contains $e_{i}$ and the last tuple contains
$e_{j}$. A _relational path_ $P_{rel}$ between $e_{i}$ and $e_{j}$ is the
sequence of relations along the edges of a path connecting $e_{i}$ and $e_{j}$
such that no relation is repeated along the path (without cycles). For
example, $e_{1}$ and $e_{6}$ are connected through path
$P:r_{1}(e_{1},e_{2},e_{3}),r_{2}(e_{3},e_{4},e_{5}),r_{3}(e_{4},e_{5},e_{6})$,
and have a relational path $P_{rel}:r_{1},r_{2},r_{3}$. Given the set of
entities in queries $\\{Q_{r}\\}_{r}$, we define the set of all possible paths
between pairwise entities as $\\{P_{rel}\\}_{r}$.
The _k-hop neighborhood_ of entity $e_{i}$, denoted $\mathcal{N}_{k}(e_{i})$,
is the unordered set of entities
$E_{\mathcal{N}_{k}}=\\{e_{k},\dots,e_{j}\\}\subseteq{\mathcal{E}}$ enclosed
in a $k$-hop radius around $e_{i}$. We denote the relation-less adjacency
matrix of entities in $\mathcal{N}_{k}(e_{i})$ as
$\mathbf{A}_{E_{\mathcal{N}_{k}}}\in\mathbb{R}^{|E_{\mathcal{N}_{k}}|\times|E_{\mathcal{N}_{k}}|}$.
For a relational graph $\mathcal{G^{\prime}}$, we define a relation-less
adjacency matrix $\mathbf{A}_{E}$ as:
$\displaystyle\mathbf{A}_{E}=\sum_{r\in{\mathcal{R}}}\mathbf{A}_{E(r)},\mathbf{A}_{E}\in\mathbb{R}^{|E_{{\mathcal{E}}}|\times|E_{{\mathcal{E}}}|}.$
(3)
Adjacency $\mathbf{A}$ has a certain order of appearance of entities in
columns and rows which is arbitrarily set by the order in which dataset tuples
are sampled. We let $\widetilde{\mathbf{A}}$ be the equivalent adjacency
matrix with a permuted order of entities in columns and rows (e.g.:
$r_{1},r_{2},r_{3},r_{4}$ can become $r_{2},r_{3},r_{1},r_{4}$). The local
clustering coefficient $c_{e_{i}}$ Watts and Strogatz (1998) of
$\mathcal{N}_{k}(e_{i})$ measures the proportion of closed triangles in the
local k-hop neighborhood of an entity such that:
$\displaystyle
c_{e_{i}}=\frac{|(e_{1},e_{2}\in{\mathcal{E}}:e_{1},e_{2}\in\mathcal{N}_{k}(e_{i})|}{\binom{d_{e_{i}}}{2}},$
(4)
where $d_{e_{i}}=\sum_{e_{j}\in\mathcal{N}_{k}(e_{i})}\mathbf{A}[e_{i},e_{j}]$
is the node degree of $e_{i}$. For the pretraining task in section 3.3, we
define the probability that an entity $e_{i}$ is found in a in $k$-hop
neighborhood (probability of occurrence) $O(e_{i})$ as:
$\displaystyle O(e_{i})=\frac{d_{e_{i}}}{\sum_{e_{j}\in
E_{\mathcal{N}_{k}}}d_{e_{j}}},\text{ and
}\mathbf{O}_{\mathcal{N}_{k}}=\begin{bmatrix}O(e_{1})\\\ \vdots\\\
O(e_{|{\mathcal{E}}|})\end{bmatrix}$ (5)
## Appendix B More details on ip Algorithms
1 Function _FindIncidentRelations(_$r$ , $P_{rel}$_)_:
2 $R^{\prime}\leftarrow\emptyset$
3 for _$r^{\prime}\in{\mathcal{R}}\backslash\\{r\\}$_ do
4 if _$E(r^{\prime})\cap E(r)\neq\emptyset\textrm{{ and }}r^{\prime}\notin
P_{rel}$_ then
5 $R^{\prime}\leftarrow r^{\prime}$
6
7
8 return $R^{\prime}$
9End Function
Algo 2 Finding all relations incident to $r$ in ${\mathcal{G}}^{\prime}$.
1 Function _TopEntropy(_$k$ , $R$_)_:
2 Compute $H(r)$ for all the relations $r\in R$ (Eq. (1)) return set ${r\in
R}$ with top-k highest $H(r)$
3End Function
4 Function _BottomCondEntropy(_$k$ , $R^{\prime}$, $r$_)_:
5 Compute $H(r|r^{\prime})$ for all $r^{\prime}\in R$ (Eq. (2)) return set
${r^{\prime}}\in R^{\prime}$ with k lowest $H(r|r^{\prime})$
6End Function
Algo 3 Finding $r\in R$ with top $k$ entropy $H(r)$ or with with bottom $k$
conditional entropy $H(r|r^{\prime})$.
## Appendix C More details on Experimental Set-Up
We use fb15k-237 Bollacker et al. (2008) (extracted from Freebase) and wn18rr
Dettmers et al. (2018). We also test our method on a hypergraph link
prediction task based off the jf17k Wen et al. (2016) dataset. Table 4
summarizes dataset properties.
Dataset | $|\mathcal{E}|$ | $|\mathcal{R}|$ | #train | #valid | #test | density
---|---|---|---|---|---|---
wn18rr | 40,943 | 11 | 86,835 | 3,034 | 3,134 | 2.1
fb15k-237 | 14,541 | 237 | 272,115 | 17,535 | 20,466 | 18.7
jf17k | 29,177 | 327 | 61,911 | 15,822 | 24,915 | 35.9
Table 4: Dataset Statistics.
The individual pretraining signals seem to lower precision (H@1) and increase
recall (H@10) compared to kg-trsf from scratch. Further, individual
pretraining signals have a larger positive effect as graph density increases.
Individual pretraining signals typically show MRR improvements for jf17k, the
densest of the KGs.
#### Training and Evaluation
MRR is the Mean Reciprocal Rank and H@(1,3,10) are HIT@ measures all commonly
used in link prediction Mohamed et al. (2019). For all tasks, we use the same
autoregressive Transformer model that applies a transformation $f$ on the
input and that is optimized with different loss functions. Our Transformer
model is a single monolithic architecture that uses a masking scheme similar
to UniLM Dong et al. (2019), allowing the model to play a variety of roles
(encoder-only, decoder-only, or encoder-decoder) and tackle a variety of
objectives (classification, regression or sequence generation). In our
experiments, we have used $L=12$ Transformer layers, with $D=256$ hidden
dimension, $A=12$ self-attention heads, an intermediate layer of size $2D$.
For path algorithms, we truncate our path sequences (max hops) to $l=4$, and,
for neighborhood-based algorithms, we limit $k=3$. The pretraining data
generated from each dataset is applied to separate model instances. For each
dataset, we save the weights of the pretrained model that performs best on the
evaluation set of the link prediction task. During link prediction finetuning,
we use a dropout rate $\rho\in\\{0.2,0.3,0.5\\}$, a label smoothing rate
$\zeta\in\\{0.6,0.7,0.8,0.9\\}$ and a learning rate of $\eta=5^{-4}$. In our
multitask setting (all), each task is sampled from the uniform distribution
$\frac{1}{|D_{t}|}$, for dataset $D_{t}$ of a pretraining task $t$. A batch
may therefore contain multiple tasks. We weight the losses of tasks from
$\tau=\\{\textsl{{sp}},\textsl{{ip}},\textsl{{khn}},\textsl{{lcc}},\textsl{{iva}}\\}$
according the available pretraining dataset size such that
$L_{\textsl{{all}}}=\sum_{t\in\tau}\alpha_{t}L_{t}$, where the
$\alpha_{t}=|D_{t}|/\sum_{t^{\prime}\in\tau}|D_{t}^{\prime}|$. For khn, note
that if $|E_{\mathcal{N}_{k}-1}|>1024$, we clip the token sequence for a
maximum length of 1024. At inference, each entity in $Q_{r}$ is masked and
evaluated once. We use two evaluation metrics: HIT@$n$ and Mean Reciprocal
Rank (MRR).
## Appendix D More details on Baselines
In table 2, we compare our method against strong and recent KG Embedding
methods. boxe Abboud et al. (2020) is a spatio-translational graph embedding
model that uses logical rules (similar to paths) and that can support
hierarchical inference patterns and higher-arity relations (knowledge
hypergraphs or KHG). This is one of the rare methods that can be applied to
both graphs and hypergraphs. boxe achieves state-of-the-art results on jf17k
while remaining competitive on fb15k-237 and wn18rr. It is unclear however if
boxe is pretrainable111In table 2, we wrote “N/A” since we are not certain if
pretraining is applicable.. To our knowledge, drgi Liang et al. (2021) is the
only other KG completion method that was pretrained using only signals from
the graph structure. Similarly to Deep Graph Infomax Veličković et al. (2019),
drgi is pretrained on artifacts of the graph structure by maximizing the
mutual information between local and global graph representations. We were not
able to ascertain at the moment if drgi is applicable to KHGs. Both lp-bert Li
et al. (2022) and star Wang et al. (2021) are Transformer-based Contextualized
KG Embedding methods. Both techniques are improvements over kg-bert Yao et al.
(2019) that uses textual descriptions of entities and relations on top of a
bert model, which is pretrained on a large text corpus222The text pretraining
data is external data and not self-contained to the KG data. lp-bert also uses
a multitask pretraining step by jointly training on three denoising tasks:
random word masking in the descriptions, entity masking, and relation masking.
star combines kg-bert and TransE Bordes et al. (2013). star contextualizes the
translation function by embedding the head and relation descriptions with
bert. The authors claim that the technique is structure-aware since
translation-based graph embedding approaches conduct structure learning by
measuring spatial distance. Note that KHG tuples in jf17k can have up to 6
entities. In most cases, the description of all entities in a tuple exceeds
the standard bert maximum sequence length of 1024. For this reason, we were
not able to apply lp-bert and star to jf17k.
|
# Binding energies and thermal width $\Gamma$ of the QGP at finite quark
chemical potential
Siddhartha Solanki, Manohar Lal, and Vineet Kumar Agotiya Department of
Physics, Central University of Jharkhand, Ranchi, India, 835 222.
Corresponding author id-<EMAIL_ADDRESS>
###### Abstract
The present work have investigated the properties of the heavy quarkonia in
the presence of finite quark chemical potential for different number of
flavors by using the quasi particle approach. The effect of the finite quark
chemical potential has been incorporated through the quasi-particle Debye mass
to examine the binding energies of the quarkonium states. From the imaginary
part of the potential we have calculated the thermal width of the ground state
of the quarkonia and found that the thermal width increases with finite quark
chemical potential. The dissociation temperature $(T_{D})$ of the $J/\psi$ and
$\Upsilon$ have been calculated in the presence of finite quark chemical
potential for different flavors. The effect of the finite quark chemical
potential on the mass spectra of the quarkonium states has been also studied.
qusi-particle Debye mass, Heavy quark complex potential, Number of flavors,
quark chemical potential, thermal width, dissociation temperature.
## I Introduction
The ongoing heavy ion collision experiment at the relativistic heavy ion
collider (RHIC) accelerator in BNL, USA and the LHC collider at CERN,
Switzerland has their great importance for exploring the phase diagram of the
Quantum Chromo-dynamics (QCD) after the discovery of the Quark Gluon Plasma
(QGP), the fourth state of matter. The study of the fundamental forces between
quark and gluon is essential for the understanding of QCD and it has been
expected that at varying temperature, finite quark chemical potential scales
(low or high) and different phases contribute in the T-$\mu$ plane. For
instance, at small or vanishing temperature quarks and gluons are limited by
the strong force while at high temperature asymptotic freedom suggests a
somewhat different QCD medium which consists of rather weakly coupled
deconfined quarks and gluons, the so-called QGP. The anomalous suppression of
the $J/\psi$ production in heavy-ion collisions which has been observed
experimentally M.C.Abreu in the depletion of the dilepton multiplicity in the
region of invariant mass which corresponds to the $J/\psi$ meson was proposed
long back as a possibly unmistakable sign of the beginning of deconfinement.
Matsui and Satz T.Matsui argued that charmonium states generated before the
formation of a thermalized QGP would tend to melt in their way through the
deconfined medium, because the binding coulomb potential is screened by the
large number of color charges, thereby producing an anomalous drop in the
$J/\psi$ yields. The pair develops into the physical resonance during
formation time and passes through the plasma and the hadronic matter before
they leave the interacting system to decay into a dilepton to be detected.
Figure 1: The variation of potential with the distance ’r’ for the quasi-
particle, leading order and lattice parameterized Debye mass at $N_{f}$= $0$,
$2$, $3$ has been shown.
Figure 2: Shows the variation of quasi-particle Debye mass with finite quark
chemical potential at different values of temperature 2(a) and with
temperature at different values of finite quark chemical potential 2(b)
respectively.
Figure 3: Shows the variation of binding energy of $J/\psi$ and $\Upsilon$
with the temperature in figures 3(a) and 3(b) for different flavors at $N_{f}$
= $2$ and $3$.
This long ’trek’ within the interacting system is fairly dangerous for the
$J/\psi$. Even before the resonance occurs, it may be absorbed by the nucleons
streaming past it C.Gerschel . By the time the quarkonium resonance is formed,
the screening of the color forces in the plasma may be sufficient to inhibit a
binding of the charmonium T.Matsui or an energetic gluon X.M.Xu or a
comoving hadron could dissociate the resonance. Quarkonia at finite
temperature is an important tool for the studying QGP formation in heavy-ion
collisions. Many effort have been devoted to determine the $T_{D}$ of
quarkonium state in the deconfined medium, using either lattice calculations
of $Q\bar{Q}$ spectral function or non-relativistic calculations based upon
some effective screened potential. Lattice studies are directly based on QCD
and these studies answer most of the questions that arises while studying the
QCD phase diagram. However, in lattice studies the spectral function must be
extracted using rather limited sets of data from the Euclidian (imaginary
time) correlators, which are directly based on the lattice. This along with
the intrinsic technical difficulties of lattice calculations, limit the
reliability of the results obtained so far, and also their scope is
essentially limited to the mass of the ground states in each quarkonium
channel. Potential models, on the other hand, provide a simple and intuitive
framework to study the properties of quarkonium at finite temperature, from
which quantities can be calculated that are beyond the current possibilities
for lattice studies. Umeda and Alberico T.Umeda ; W.M.Alberico(2008) have
shown that the lattice computations of mesons correlators at finite
temperature contain a constant contribution, due to the presence of zero modes
in the spectral functions. The problem of dissociation of bound states in a
hot QCD medium is of great importance in heavy-ion collisions as it provides
evidence for the creation of the QGP Leitch . The physical understanding of
the quarkonium dissociation within a deconfined medium has undergone several
refinements in the last couple of years Laine:2008cf ; BKP:2000 ; BKP:2001 ;
BKP:2002 ; BKP:2004 . As the heavy quark and anti-quark in a quarkonia state
are bound together by almost static (off-shell) gluons, therefore, the issue
of their dissociation boils down to how the gluon self-energy behaves at high
temperature. It has been noticed that the gluon self-energy has both real and
imaginary parts laine . Note that the real part lead to the Debye screening,
while the imaginary part leads to Landau damping and give rise the thermal
width to the quarkonia.
It indeed provides a useful way to examine quarkonium binding energies,
quarkonium wave functions, reaction rates, transition rates, and decay widths.
It further allows the extrapolation to the region of high temperatures by
expressing screening effects reflecting on the temperature dependence of the
potential and finite quark chemical potential. The effects of dynamics of
quarks on the stability of quarkonia can be studied by using finite quark
chemical potential extracted from thermodynamic quantities that are computed
in full QCD. At high temperatures, the deconfined phase of QCD exhibits
screening of static color-electric fields E.V.Shuryak ; GPY ; it is,
therefore, expected that the screening will lead to the dissociation of
quarkonium states. After the success at zero temperature while predicting
hadronic mass spectra, potential model descriptions have been also applied to
understand quarkonium properties at finite temperature and finite quark
chemical potential. It is well known that the production of $J/\psi$ and
$\Upsilon$ mesons in hadronic reactions occur in part via the production of
higher excited $c\bar{c}$ (or $b\bar{b}$) states and they decay into the
respective ground state. Since the lifetime of different quarkonium state is
much larger than the typical lifetime of the medium produced in nucleus-
nucleus collisions; their decay occurs almost completely outside the produced
medium lain ; he1 . Since the produced medium can be probed not only by the
ground state quarkonium but also by different excited quarkonium states. So,
the potential model in this context could be helpful in predicting for the
binding energies of various quarkonia state by setting up and solving
appropriate Schrodinger equation in the hot QCD medium. The first step towards
this is to model an appropriate medium dependent interquark interaction
potential at finite temperature and finite quark chemical potential.
Thereafter the dissociation of excited states of quarkonia has been studied.
The present work is different from the work U.Kakade in which the authors
have studied the effect of baryonic chemical potential on the Quarkonium
states using leading order Debye mass. In the current work we have started
with the modified form of the potential V.Agotiya:2009 in which the effect of
the finite quark chemical potential has been incorporated through the quasi-
particle Debye mass. The energy eigen values of the ground states of the
charmonium and bottomonium for different flavors has been obtained by solving
the Schrodinger equation. Further we have studied the effect of the finite
quark chemical potential on the thermal width and the dissociation temperature
of the quarkonium states.
The present manuscript is organized in the following manner: a brief
description about the real and the imaginary part of the heavy quark potential
is given in section-II. Whereas in section-III the quark finite quark chemical
potential is introduced into the the quasi-particle Debye mass. In Section-IV,
we examine the binding energy and $T_{D}$ of various quarkonium states at
different values of flavor and the finite quark chemical potential. The effect
of the finite quark chemical potential on the mass spectra of the quarkonium
states is evaluated in the section-V. In section-VI we discuss the result of
present work. Finally, in section-VII, we conclude with the future prospects
of the present work.
Figure 4: Shows the variation of binding energy of $J/\psi$ and$\Upsilon$ with
the temperature in the figures 4(a) and 4(b) for different values of the
finite quark chemical potential at $N_{f}=3$.
Figure 5: Shows the variation of Binding energy of $\psi^{\prime}$ and
$\Upsilon^{\prime}$ with the temperature in the figures 5(a) and 5(b) for
different values of the finite quark chemical potential at $N_{f}$= $3$.
Figure 6: Mass spectra of $J/\psi$ and $\Upsilon$ with the temperature in the
figures 6(a) and 6(b) for different masses has been shown.
## II Heavy quark complex potential
### II.1 Real Part of the potential
Quantitative understanding of the bound state properties needs the exact
potential at finite-temperature which should be derived directly from QCD,
like the Cornell potential at zero temperature has been derived from pNRQCD
from the zeroth-order matching coefficient. Such derivations at finite
temperature for weakly-coupled plasma have been recently come up in the
literature Brambilla05 ; Brambilla08 but they are plagued by the existence of
temperature-driven hard as well as soft scales, $T$, $gT$, $g^{2}T$,
respectively. Due to these difficulties in finite temperature extension in
effective field theories, the lattice-based potentials become popular.
However, neither the free energy nor the internal energy can be directly used
as the potential. What kind of screened potentials should be used in the
Schrödinger equation which well describes is the bound states at finite
temperature is still an open question. The potential model based phenomenology
as well as the lattice QCD approaches infers that the quark-antiquark
interaction potential in presence of medium plays a crucial role in
understanding the nature of quark-antiquark bound state in the hot QCD/QGP
medium. The potential employed is commonly screening columns (Yukawa form)
Brambilla05 ; L.Kluberg . In case of finite temperature QCD we employ the
ansatz that the medium modification enter in the Fourier transform of heavy
quark potential V(k) as V.Agotiya:2009 .
$\tilde{V(k)}=\frac{V(k)}{\varepsilon(k)}$ (1)
Where, $\varepsilon(k)$ is dielectric permitivity which is obtain from the
static limit of the longitudnal part of the gluon self energy R.A.Schneider ;
H.A.Weldon .
$\varepsilon(k)=\left(1+\frac{\pi_{L}(0,k,T)}{k^{2}}\right)\equiv\left(1+\frac{m^{2}_{D}(T,\mu)}{k^{2}}\right)$
(2)
V(k) is the Fourier transform of the Cornell potential given as,
$\mbox{\boldmath$V$}(k)=-\sqrt{\frac{2}{\pi}}\frac{\alpha}{k^{2}}\\\
-\frac{4\sigma}{\sqrt{2\pi}k^{4}}$ (3)
Substituting the value of Eq.(2) and Eq.(3) in equation Eq.(1), and solving
using inverse Fourier transform, we get the medium modified potential
depending upon ’r’ V.Chandra:2007 ; R.A.Schneider ; A.Ranjan .
$\mbox{\boldmath$V$}(r,T,\mu)=\left(\frac{2\sigma}{m^{2}_{D}(T,\mu)}-\alpha\right)\frac{exp(-m_{D}(T,\mu)r)}{r}\\\
-\frac{2\sigma}{m^{2}_{D}(T,\mu)r}+\frac{2\sigma}{m_{D}(T,\mu)}-\alpha
m_{D}(T,\mu)$ (4)
In addition to standard Yukawa potential, this potential
$\mbox{\boldmath$V$}(r,T,\mu)$ given by Eq.(4) has also long-range coulomb
tail. The constant added to yield could appear naturally while performing the
basic computation of the real time static potential in the hot QCD, these
constant are required to yield the correct limit of
$\mbox{\boldmath$V$}(r,T,\mu)$ M.Laine as T$\longrightarrow$0 and can also
found from the real and imaginary time correlators in thermal QCD medium
A.Beraudo . The potential in the Eq.(4) will look like, in the limiting case
$r\gg\frac{1}{m_{D}}$ as,
$V(r,T,\mu)\approx-\frac{2\sigma}{m^{2}_{D}(T,\mu)r}-\alpha m_{D}(T,\mu)$ (5)
Figure 7: Shows variation of the $J/\psi$ and $\Upsilon$ thermal width
$\Gamma$ with the temperature in the figures 7(a) and 7(b) for different quark
finite quark chemical potential.
Figure 8: Shows the real and imaginary binding energies of the $J/\psi$ and
$\Upsilon$ for different finite quark chemical potential at $N_{f}$= $0$ in
the figures 8(a) and 8(b).
### II.2 Imaginary Part of the potential
Heavy quark complex potential contains both the real and the imaginary part
which were obtained by using the gluon self energy which in turn responsible
for both the Debye screening and the Landau damping respectively. However,
Debye screening is obtained by using both the retard and self-energy
propagator whereas the static limit of the symmetric self-energy has been used
for calculating the imaginary part of the potential. The imaginary part of the
potential has its importance while studying the threshold enhancement of the
bound state or the thermal width of the charmonium and bottomonium resonances.
This thermal width in spectral function is further used to determine the
$T_{D}$ of the quarkonium resonances. From the previous studies AgnesMocsy ;
M.LaineJHEP2007 is it clear that, the dissociation of the quarkonium states
takes place when thermal width becomes equal to the twice of the binding
energy.
Figure 9: Shows the real and imaginary binding energies of the $J/\psi$ and
$\Upsilon$ for different finite quark chemical potential at $N_{f}$= $2$ in
the figures 9(a) and 9(b).
Figure 10: Shows the real and imaginary binding energies of the $J/\psi$ and
$\Upsilon$ for different finite quark chemical potential at $N_{f}$= $3$ in
the figures 10(a) and 10(b)
The symmetric propagator for the imaginary part is given below:
$ImD_{F}^{00}(0,p)=\frac{-2\pi{T}{m_{D}^{2}}}{p\left(p^{2}+m_{D}^{2}\right)^{2}}$
(6)
Eq.6 gives the imaginary part of the dielectric function as;
$\epsilon^{-1}(p)=-\pi{T}{m_{D}^{2}}\
\frac{p^{2}}{p\left(p^{2}+m_{D}^{2}\right)^{2}}$ (7)
Thus, we obtained the imaginary part of the potential using the following
equation:
$V(r,T,\mu)=\int{\frac{d^{3}k}{\left(2\pi\right)^{\frac{3}{2}}}\left(e^{i.pr}-1\right)\frac{V(p)}{\epsilon(p)}}$
(8)
This implies
$ImV\left(r,T,\mu\right)=\int{\frac{d^{3}k}{\left(2\pi\right)^{\frac{3}{2}}}\left(e^{i.pr}-1\right)\left(-\sqrt{\frac{2}{\pi}}\frac{\alpha}{p^{2}}-\frac{4\sigma}{\sqrt{\pi}{p^{4}}}\right)}\\\
\times
p^{2}\left({\frac{-\pi^{2}{T}{m_{D}^{2}}}{p\left(p^{2}+m_{D}^{2}\right)^{2}}}\right)$
(9)
After performing the integration of the above Eq.9, the contribution due to
coulomb and the string term becomes:
$ImV\left(r,T,\mu\right)=-2{\alpha}{T}\int_{0}^{\infty}\\\
{\frac{dz}{{(z^{2}+1)}^{2}}\left(1-\frac{\sin{z}}{z\hat{r}}\right)+\frac{4\sigma{T}}{m_{D}^{2}}\int_{0}^{\infty}{\frac{dz}{{(z^{2}+1)}^{2}}\left(1-\frac{\sin{z\hat{r}}}{z\hat{r}}\right)}}$
(10)
Where z = $\frac{P}{m_{D}}$. This can be further simplified as:
$ImV\left(r,T,\mu\right)=\\-{\alpha}{T}\phi_{0}\left(\hat{r}\right)+\frac{2\sigma{T}}{m_{D}^{2}}\psi_{0}\left(\hat{r}\right)$
(11)
Where
$\phi_{0}\left(\hat{r}\right)=-{\alpha}{T}\left(\frac{{\hat{r}}^{2}}{9}\left(-4+3\gamma_{E}+3log(\hat{r})\right)\right)$
and
$\psi_{0}\left(\hat{r}\right)=-\frac{{\hat{r}}^{2}}{6}\
+\left(\frac{-107+60\gamma_{E}+60log(\hat{r})}{3600}\right){\hat{r}}^{4}+O({\hat{r}}^{5})$
For the limit $\hat{r}<<1$, we have
$ImV\left(r,T,\mu\right)=-T\left(\frac{\alpha{\hat{r}}^{2}}{3}+\frac{\sigma{\hat{r}}^{4}}{30m_{D}^{2}}\right)log\left(\frac{1}{\hat{r}}\right)$
(12)
Now, by using Eq.12 we calculate the thermal width $\Gamma$ of the resonance
state(1S). This can be done by folding with unperturbed (1S) Coulomb wave
function and is given by prd2016_vin ,
$\Gamma=\left(\frac{4T}{\alpha{m_{Q}^{2}}}+\frac{12\sigma{T}}{\alpha^{2}m_{Q}^{4}}\right)m_{D}^{2}log\frac{\alpha{m_{Q}}}{2m_{D}}$
(13)
## III Quasi-particle Debye mass in the presence of finite quark chemical
potential
The perturbative nature of the leading order Debye mass in QCD coupling at
high temperature is known for a long time A.Rebhan . In QCD the Debye mass is
non-perturbative and gauge independent defined in E.Braaten . Debye mass is
also calculated for the two polyakov loops by Braaten and Nicto at high
temperature Y.Burnier . The basic definition of the Debye mass itself creates
a difficulty because of the gauge variant nature of electric correlators in
K.Kajantie . To overcome this problem many approaches have been purposed so
far K.Kajantie ; Anbazavov ; S.Nadkarni . Keeping in view of all the
interaction between the quasiparticle because of the quasi-parton, a number of
attempts have been made so far such as, effective mass model V.Goloviznin ;
A.Peshier , the effective mass with Polyakov loop M.D.Elia , model based on
PNJL and NJL A.Dumitru , effective fugacity model V.Chandra:2007 ;
V.Chandra:2009 . Quasi-particle model is important to describe the non-ideal
behavior of the QGP and the masses arise because of the surrounding matter in
the medium around the parton and this quasi-parton acquired the same quantum
number as the real particle i.e quarks and gluon P.K.Srivastava . Here we use
quasi-particle EoS M.Cheng . The quasi-particle Debye mass ${m_{D}}$ in terms
of the temperature and finite quark chemical potential for full QCD case is
given as,
Table 1: The dissociation temperature of the ${J/\psi}$ and ${\Upsilon}$ at $N_{f}=0$ has been calculated for the different values of finite quark chemical potential when the thermal width $\Gamma=2B.E$. $State$ | $\mu=300MeV$ | $\mu=325MeV$ | $\mu=350MeV$
---|---|---|---
$J/\psi$ | 0.8628 | 0.8571 | 0.8234
$\Upsilon$ | 0.9307 | 0.9209 | 0.9124
Table 2: The dissociation temperature of the ${J/\psi}$ and ${\Upsilon}$ at $N_{f}=2$ has been calculated for the different values of finite quark chemical potential when the thermal width $\Gamma=2B.E$. $State$ | $\mu=300MeV$ | $\mu=325MeV$ | $\mu=350MeV$
---|---|---|---
$J/\psi$ | 0.8652 | 0.8596 | 0.8546
$\Upsilon$ | 0.9354 | 0.9299 | 0.9248
$\displaystyle m^{2}_{D}\left(T,\mu\right)$ $\displaystyle=$ $\displaystyle
g^{2}(T)T^{2}\bigg{[}\bigg{(}\frac{N_{c}}{3}\times\frac{6PolyLog[2,z_{g}]}{\pi^{2}}\bigg{)}$
(14)
$\displaystyle+{\bigg{(}\frac{\hat{N_{f}}}{6}\times\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}\bigg{]}}$
and the value of $\hat{N_{f}}$ is,
$\displaystyle\hat{N_{f}}$ $\displaystyle=$
$\displaystyle\bigg{(}N_{f}+\frac{3}{\pi^{2}}\bigg{(}\frac{\mu^{2}}{T^{2}}\bigg{)}\bigg{)}$
(15)
and
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\frac{\mu_{b}}{3}$ (16)
U.Kakade , where $(\mu)$ defined the finite quark chemical potential and
$(\mu_{b})$ is finite baryonic chemical potential. After introducing the value
of $\hat{N_{f}}$ in the Eq.(14), we get the full expression of quasi-particle
debye mass in terms of temperature and finite quark chemical potential as,
$m^{2}_{D}\left(T,\mu\right)=T^{2}\left\\{\bigg{(}\frac{N_{c}}{3}Q^{2}_{g}\bigg{)}+\bigg{[}\frac{N_{f}}{6}+\frac{1}{2\pi^{2}}\bigg{(}\frac{\mu^{2}}{T^{2}}\bigg{)}\bigg{]}Q^{2}_{q}\right\\}$
(17)
Where, $Q_{g}$ and $Q_{q}$ are the effective charges given by the equations:
$z_{g,q}=a_{q,g}\exp\bigg{(}-\frac{b_{g,q}}{x^{2}}-\frac{c_{g,q}}{x^{4}}-\frac{d_{g,q}}{x^{6}}\bigg{)}.$
(18)
Here $x=T/T_{c}$ and $a$, $b$ and $c$ and $d$ are fitting parameters, for
equation of state in the quasi-particle description V.Chandra:2007 ;
V.Chandra:2009 respectively.
$\displaystyle Q^{2}_{g}$ $\displaystyle=$ $\displaystyle
g^{2}(T)\frac{6PolyLog[2,z_{g}]}{\pi^{2}}$ $\displaystyle Q^{2}_{q}$
$\displaystyle=$ $\displaystyle g^{2}(T)\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}$
(19)
Here, $g(T)$ is the QCD running coupling constant, $N_{c}=3$ ($SU(3)$) and
$N_{f}$ is the number of flavor, the function $PolyLog[2,z]$ having form,
$PolyLog[2,z]=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{2}}$ and $z_{g}$ is the
quasi-gluon effective fugacity and $z_{q}$ is quasi-quark effective fugacity.
These distribution functions are isotropic in nature. Both $z_{g}$ and $z_{q}$
have a very complicated temperature dependence and asymptotically reach to the
ideal value unity. In our present analysis we have used the quasi-particle
Debye mass, $m_{D}^{QP}$ depending upon the temperature and finite quark
chemical potential for different number of flavors.
## IV Binding energy and dissociation temperature
In this section, we have studied the effect of the finite quark chemical
potential on the binding energies and the dissociation temperature of
c$\bar{c}$ and b$\bar{b}$ resonances with different flavors number. To reach
this end we solve the Schrodinger equation for the complete understanding of
the quarkonia in the hot QGP medium. The binding energy of charmonium and
bottomonium state at $T$=$0$ is defined by the difference of energy between
the $m_{Q}$ (mass of quarkonia) and the bottom/open charm threshold. But
distance between the continuum threshold and the peak position is defined the
binding energy at finite value of temperature AgnesMocsy . Hence, the solution
of Schrodinger equation for the potential given by Eq.(5) gives the energy
eigen values for the ground and excited states of charmonium and bottomonium
i.e, $J/\psi$, $\psi^{\prime}$, $\Upsilon$ and $\Upsilon^{\prime}$ as:
$E_{n}=-\frac{1}{n^{2}}\frac{m_{Q}\sigma^{2}}{m^{4}_{D}}$ (20)
It has been observed that the binding energy decreases with the temperature
and also with finite quark chemical potential which is shown in the figures,
3, 4, 5 and 6. The binding energy of charmonium and bottomonium state at
particular values of temperature becomes smaller or equal to the value of mean
thermal energy; i.e, the state of quarkonia is said to be dissociated at that
given value of temperature. The dissociation temperature for the states of
charmonium and bottomonium is also discussed in P.Sandin ; prd2016_vin ;
prd2018_vin . The dissociation temperature of $J/\psi$ and $\Upsilon$ with
different number of flavors at different values of the finite quark chemical
potential has been obtained by using the condition: $\Gamma$ = 2B.E. The
intersecting point between the binding energy curve and the thermal width of a
quarkonium states ($J/\psi$ and $\Upsilon$) is taken as dissociation point of
such states.
Table 3: The dissociation temperature of the ${J/\psi}$ and ${\Upsilon}$ at $N_{f}=3$ has been calculated for the different values of finite quark chemical potential when the thermal width $\Gamma=2B.E$. $State$ | $\mu=300MeV$ | $\mu=325MeV$ | $\mu=350MeV$
---|---|---|---
$J/\psi$ | 0.8653 | 0.8597 | 0.8547
$\Upsilon$ | 0.9350 | 0.9247 | 0.9258
Table 4: Comparison of the mass spectra for $J/\psi$ and $\Upsilon$ obtained in the present work with the theoretical and experimental data. $State$ | present work | Exp.massTanabashi | Error [in GeV]
---|---|---|---
$J/\psi$ | 3.1 | 3.096 | 0.00129
$\Upsilon$ | 9.5 | 9.460 | 0.00421
## V Mass Spectra of Quarkonium state
For calculating the mass spectra of heavy quarkonia the relation is:
$M=2m_{Q}+B.E$ (21)
Here, mass spectra is equal to the sum of the exact formula of energy and
twice the quark-mass. Now, we substitute the values of $E_{nl}^{n}$ (B.E) in
the above equation,
$M=2m_{Q}+\frac{1}{n^{2}}\frac{m_{Q}\sigma^{2}}{m^{4}_{D}}$ (22)
Where, $m_{Q}$ is used for the mass of quarkonium state, such as charmonium
and bottomonium mass and $n$ is used for the states of quarkonium (n = $1$ for
ground states and n = $2$ for excited states).
## VI Results and Discussion
In the present work we have investigated the propeties of the charmonium and
the bottomonium at finite temperature and quark chemical potential using quasi
particle approach. Figure 1 shows the variation of the potential with distance
(r) for different form of the Debye mass (e.g. Quasi-Particle, Leading order
and Lattice parametrized) at finite quark chemical potential $\mu$ = $300$ MeV
and $T$ = $150$ MeV with different number of flavours. It has been found that
the variation of potential increases with the distance (r). This potential is
not similar to the lattice free energy of heavy-quark in the deconfined phase,
which is a well known coulomb potential H.Satz , the Cornell potential is
solvable by one-dimensional Fourier-transform (FT) in the hot QCD medium and
has a similar form that has been used for the study of quarkonium properties
(which is consider like the color flux tube structure).
The variation of quasi-particle Debye mass with finite quark chemical
potential at different values of temperature (i.e, $150$, $160$ and $170$ MeV)
is shown in figure 2 (a), and the variation of quasi-particle Debye mass with
temperature at different values of finite quark chemical potential (i.e,
$150$, $200$ and $250$ MeV) is shown in the figure 2 (b) respectively. The
Debye screening mass at baryon density and temperature has been studied by the
lattice Taylor expansion method M.Doring . If we increase the value of finite
quark chemical potential, Debye mass increases for different temperatures is
shown in the figure 2 (a). But in the figure 2 (b), Debye mass increases with
the temperature for different quark chemical potential. The variation of the
binding energy for the $J/\psi$ and $\Upsilon$ with the temperature for
different number of the flavors $N_{f}$ = $2$, $3$ has been shown in figure 3
(a) and in figure 3 (b) respectively. When we increase the value of number of
flavors in the figure 3, the variation of binding energy decreases at $\mu$ =
$100$ MeV.
Figures 4 and 5 shows the variation of binding energy of $J/\psi$ and
$\psi^{\prime}$ in the figure 4 (a) and 5 (a), $\Upsilon$ and
$\Upsilon^{\prime}$ in the figure 4 (b) and 5 (b) respectively, with the
temperature at $N_{f}$ = $3$ for different values of the finite quark chemical
potential ($\mu$ = $100$, $110$, $120$ MeV). From these figures, we notice
that the binding energy $J/\psi$, $\psi^{\prime}$, $\Upsilon$ and
$\Upsilon^{\prime}$ decreases with the temperature as we increase the value of
finite quark chemical potential. So, it become an interesting fact about the
rate of exponential decay of binding energy. This behavior of binding energy
could be understood by the strongerness of Debye screening with increasing
values of the finite quark chemical potential $(\mu)$ and also the strength of
inter-quark potential, which is weaker as compared to the zero finite quark
chemical potential $\mu$ = $0$. The binding energy at finite value of
temperature and quark chemical potential gives information about the
dissociation of the quarkonium states (charmonium and bottomonium states).
Since, it is known that the binding energy is directly proportional to the
mass of quarkonia so, if the value of quarkonia mass increases, in other words
we turns towards higher masses i.e. from mass of $J/\psi$ ($1.5$ GeV) to mass
of $\Upsilon$ ($4.5$ GeV), we notice that the binding energy increases. The
mass spectra of the quarkonia states in the presence of finite quark chemical
potential has been also studied and its variation with the temperature has
been shown in the figures 6. It has been observed that the mass spectra of the
$J/\psi$ in the figure 6 (a) and $\Upsilon$ in the figure 6 (b) increases as
the mass of the quarkonia increaes. We have also made a comparison of the
masses of the $J/\psi$ and $\Upsilon$ with the experimental data Tanabashi
and this has been given in the Table-IV. The variation of the thermal width of
$J/\psi$ in the figure 7 (a) and $\Upsilon$ in the figure 7 (b) with the
temperature for different values of the finite quark chemical potential (i.e,
$200$, $250$ and $300$ MeV) has been shown. It has been seen that the thermal
width of the quarkonium states with the finite quark chemical potential
increase. The real and the imaginary part of the binding energies of the
$J/\psi$ in the figures 8 (a), 9 (a) and 10 (a) and $\Upsilon$ in the figures
8 (b), 9 (b) and 10 (b) with the $T/T_{c}$ for $N_{f}$ = $0$, $2$, $3$ at
$\mu$ = $300$, $325$, $350$ MeV has been shown respectively. From these
figures we have examined the dissociation temperature (By the intersection
point of the thermal width and two times of the binding energy of the
quarkonia) of the $J/\psi$ and $\Upsilon$ for different flavors $N_{f}$= $0$,
$2$, $3$ and the obtained values of the dissociation temperatures, given in
the Table-I, II and III respectively. The dissociation temperature of the
$J/\psi$ and $\Upsilon$ at $N_{f}$ = $0$ for the finite quark chemical
potential $\mu$ = $300$, $325$, $350$ MeV are found $0.8628$, $0.8571$,
$0.8234$ and $0.9307$, $0.9209$, $0.9124$ respectively (in terms of $T_{c}$).
Similarly at $N_{f}$ = $2$ the dissociation temperature of the $J/\psi$ and
$\Upsilon$ for different finite quark chemical potential $\mu$ = $300$, $325$,
$350$ MeV are $0.8652$, $0.8596$, $0.8546$ and $0.9354$, $0.9299$, $0.9248$
respectively (in unit of $T_{c}$). Finally, the dissociation point of the
$J/\psi$ and $\Upsilon$ for different values of the finite quark chemical
potential $\mu$ = $300$, $325$, $350$ MeV was found as $0.8653$, $0.85970$,
$8547$ and $0.9350$, $0.9247$, $0.9258$ respectively (in unit of $T_{c}$) when
the number of flavor was taken as $N_{f}$ = $3$.
## VII Conclusion and future outcomes
The current work have determined the quarkonium dissociation behavior in the
hot and dense QGP medium, in the presence of the finite quark chemical
potential. It has been observed that the Cornell potential with distance ’r’,
increases for the different Debye screening. The behavior of binding energy
with temperature has also seen by the earlier studies prd2016_vin and with
anisotropic parameter at zero finite quark chemical potential in prd2018_vin .
Here we consider the finite values of the quark chemical potential. It is
noticed that the binding energy for the different quarkonium states decreases
with the temperature as we increase the value of finite quark chemical
potential. The dissociation temperature of the quarkonium states, on the other
hand, increases with the increasing values of the finite quark chemical
potential. This can be seen from the Table-I, II and III for number of flavor
$N_{f}$= $0$, $2$, $3$ respectively. The maximum error in the mass spectra of
the $J/\psi$ and $\Upsilon$ deduced by using Chi square function is $0.0019$
and $0.00421$ (in GeV) respectively.
The present work might be helpful in exploring the studies of the compact
objects like neutron stars. Since the Compressed Baryonic Matter (CBM)
experiment at FAIR is exploring the quark gluon plasma at higher baryon
densities, so such type of theoretical studies may contribute to the physics
of compact bodies with high baryon densities.
## VIII Acknowledgments
One of the author, V.K.Agotiya acknowledge the Science and Engineering
Research Board (SERB) Project No. EEQ/2018/000181 New Delhi for the research
support in basic sciences.
## IX Declarations
### IX.1 Funding
No any other funding.
### IX.2 Conflict of interest
The author declare no competing interests.
## References
* (1) M. C. Abreu et al, Physics Letters B 477 no.1-3 pp.28-36 (2000).
* (2) T. Matsui and H. Satz, Physics Letters B 178 no.4 pp.416-422 (1986).
* (3) C. Gerschel and J. Hufner, Physics Letters B 207 pp.253 (1988).
* (4) X. M. Xu, D. Kharzeev, H. Satz and X. N. Wang, Physical Review C 53 no.6 pp.3051 (1996).
* (5) T. Umeda, Physical Review D 75 pp.094502 (2007).
* (6) W. M. Alberico, A. Beraudo, A. De. Pace and A. Molinari, Physical Review D 77 pp.017502 (2008).
* (7) M. J. Leitch [PHENIX Collaboration] arXiv: 0806.1244 [nucl-ex].
* (8) M. Laine, Nuclear Physics A 820 pp.25C (2009).
* (9) D. Pal, B. K. Patra and D. K. Srivastava, European Physics Journal C 17, pp.179-186, (2000).
* (10) Binoy Krishna Patra and D. K. Srivastava, Physics Letters B 505 no.1-4 pp.113-118 (2001).
* (11) Binoy Krishna Patra, V. J. Menon, Nuclear Physics A 708 pp.353-364 (2002).
* (12) Binoy Krishna Patra and V. J. Menon, The European Physical Journal C 37 pp.115-121 (2004).
* (13) Y. Burnier, M. Laine, M. Vepsáláinen, Physics Letters B 678 no.1 pp.86-89 (2009).
* (14) E. V. Shuryak, Physics Reports 61 no.2 pp.71-158 (1980).
* (15) D. J. Gross, R. D. Pisarki and L. G. Yaffe, Reviews of Modern Physics 53 pp.43 (1981).
* (16) M. Laine, Journal of High Energy Physics 04 pp.124 (2011).
* (17) M. He, R. J. Fries and R. Rapp, Physics Letters B 701 pp.445 (2011).
* (18) N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Reviews of Modern Physics 77 no.4 pp.1423-1496 (2005).
* (19) N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Physical Review D 78 no.01 pp.4017 (2008).
* (20) L. Kluberg and H. Satz, https://arXiv:hep-ph/0901.3831.
* (21) V. Agotiya et al, Physical Review C 80 no.2 id-025210 (2009).
* (22) R. A. Schneider, Physical Review D 66 no.3 id-036003 (2002).
* (23) H. A. Weldon, Physical Review D 26 no.6 pp.1394-1407 (1982).
* (24) V. Chandra, R. Kumar and V. Ravishankar, Physical Review C 76 no.6 id-054909 (2007).
* (25) A. Ranjan and V. Ravishankar, https://arXiv:0707.3697.
* (26) M. Laine, O. Philipsen, M. Tassler, and P. Romatschke, Journal of High Energy Physics 03 pp.054 (2007).
* (27) A. Beraudo, J. P. Blaizot and C. Ratti, Nuclear Physics A 806 no.1-4 pp.312-338 (2008).
* (28) M. Laine, O. Philipsen and M. Tassler, Journal of High Energy Physics 09,pp.066 (2007).
* (29) A. Rebhan, Physical Review D 48 no.9 pp.R3967 (1993).
* (30) E. Braaten and A. Nieto, Physical Review Letters 73 no.18 pp.2402 (1994).
* (31) Y. Burnier and A. Rothkopf, Physics Letters B 753 pp.232-236 (2016).
* (32) K. Kajantie, M. Laine, J. Peisa, A. Rajantie, K. Rummukainen and M. E. Shaposhnikov, Physical Review Letters 79 no.17 pp.3130 (1997).
* (33) Anbazavov, F. Kirsch, P. Petrezky and S. Mukherjee, Physical Review D 91 no.5 id-054503 (2015).
* (34) S. Nadkarni, Physical Review D 33 no.12 pp.3738 (1986).
* (35) V. Goloviznin and H. Satz, Zeitschrift fur Physik C particlse and fields 57 pp.671-675 (1993).
* (36) A. Peshier, B. Kampfer, O. P. Pavlenko, and G. Soff, Physical Review D 54 no.3 pp.2399-2402 (1996).
* (37) M. D’Elia, A. Di. Giacomo, and E. Meggiolaro, Physical Review D 67 no.11 id-114504 (2003).
* (38) A. Dumitru and R. D. Pisarski, Physics Letters B 525 no.1-2 pp.95-100 (2002).
* (39) V. Chandra, V. K. Agotiya and B. K. Patra, https://arXiv:0901.2084v1 (2009).
* (40) P. K. Srivastava, S. K. Tiwari and C. P. Singh, Physical Review D 82 no.01 id-014023 (2010).
* (41) M. Cheng et al, Physical Review D 77 no.01 id-014511 (2008).
* (42) U. Kakade and B. K. Patra, Physical Review C 92 no.2 id-024901 (2015).
* (43) A. Mocsy and P. Petreczky, Physics Review Letters 99 no.21 id-211602 (2007).
* (44) M. Margotta, K. McCarty, C. McGahan, M. Strick-land and D. Y. Elorriaga, Physical Review D 83 no.10 id-105019 (2011).
* (45) L. Thakur, U. Kakade and B. K. Patra, Physical Review D 89 id-094020 (2014).
* (46) P. Sandin, M. Ogren and M. Gulliksson, Physical Review E 93 no.3 id-033301 (2016).
* (47) V. K. Agotiya, V. Chandra, M. Y. JamaL and I. Nilima, Physical Review D 94 no.9 id-094006 (2016).
* (48) M Y Jamal, I Nilima, V. Chandra and V. K. Agotiya, https://arXiv:1805.04763 (2018).
* (49) H. Satz, Reports on Progress in Physics 63 pp-1511 https://arXiv:hep-ph/0007069 (2000).
* (50) M. Doring, S. Ejiri, O. Kaczmarek, F. Karsch and E. Laermann, Proceedings of Science (XXIIIrd international symposium on lattice field theory) LAT2005 pp.193 (2005).
* (51) M. Tanabashi, C. Carone, T. G. Trippe and C. G. Wohl, Phys. Rev. D 98 546-548 (2018).
|
$\displaystyle+\ln\left(1+\exp\left(y\right)\right)-\ln\left(1+\exp\left(e\left(\boldsymbol{\theta}_{2}\right)\right)\right),$
and
$h\left(\tau,\boldsymbol{\theta}\right):=\tau\left[FZ_{\tau}^{sp}\left(q\left(\boldsymbol{\theta}_{1}\right),e\left(\boldsymbol{\theta}_{2}\right),Y\right)-\ln\left(1+\exp\left(Y\right)\right)\right].$
Let $\mathcal{T}:=\left(0,1\right)$ and
$\displaystyle B_{1}\left(V\right)$ $\displaystyle:=$
$\displaystyle\sup_{\boldsymbol{\theta}_{1}\in\boldsymbol{\Theta}_{1}}\left|q\left(\boldsymbol{\theta}_{1}\right)\right|,$
$\displaystyle B_{2}\left(V\right)$ $\displaystyle:=$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\left\|q\left(\boldsymbol{\theta}_{1}\right)\nabla_{\boldsymbol{\theta}_{2}}e\left(\boldsymbol{\theta}_{2}\right)\right\|,$
$\displaystyle B_{3}\left(V\right)$ $\displaystyle:=$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}_{1}}\left\|\nabla_{\boldsymbol{\theta}_{1}}q\left(\boldsymbol{\theta}_{1}\right)\right\|,$
$\displaystyle B_{4}\left(V\right)$ $\displaystyle:=$
$\displaystyle\sup_{\boldsymbol{\theta}_{2}\in\boldsymbol{\Theta}_{2}}\left\|Y\nabla_{\boldsymbol{\theta}_{2}}e\left(\boldsymbol{\theta}_{2}\right)\right\|,$
$\displaystyle B_{5}\left(V\right)$ $\displaystyle:=$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\left\|\left(e\left(\boldsymbol{\theta}_{2}\right)-q\left(\boldsymbol{\theta}_{1}\right)\right)\nabla_{\boldsymbol{\theta}_{2}}e\left(\boldsymbol{\theta}_{2}\right)\right\|.$
Let
$\boldsymbol{\theta}^{1}:=\left(\boldsymbol{\theta}_{1}^{1},\boldsymbol{\theta}_{2}^{1}\right)$
and
$\boldsymbol{\theta}^{2}:=\left(\boldsymbol{\theta}_{1}^{2},\boldsymbol{\theta}_{2}^{2}\right)$
be two vectors of parameter values.
###### Lemma 3 (Global Lipschitz continuity)
If Assumption 2.4 holds, for all
$\left(\tau_{1},\tau_{2}\right)\in\mathcal{T}$,
$\left(\boldsymbol{\theta}_{1}^{1},\boldsymbol{\theta}_{1}^{2}\right)\in\boldsymbol{\Theta}_{1}$
and
$\left(\boldsymbol{\theta}_{2}^{1},\boldsymbol{\theta}_{2}^{2}\right)\in\boldsymbol{\Theta}_{2}$,
given $V=\left(Y,W\right)$, there exists a constant
$B_{\tau}^{*}\left(V\right)$ such that
$\left|h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)\right|\leq
B_{\tau}^{*}\left(V\right)\left(\left\|\tau_{1}-\tau_{2}\right\|+\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{2}\right\|\right).$
Proof. We use similar notations and strategy as in proving Lemma 2. At first
note that $h\left(\tau,\boldsymbol{\theta}\right)$ can be rewritten as
$\displaystyle h\left(\tau,\boldsymbol{\theta}\right)$ $\displaystyle=$
$\displaystyle\frac{\exp\left(e\left(\boldsymbol{\theta}_{2}\right)\right)}{1+\exp\left(e\left(\boldsymbol{\theta}_{2}\right)\right)}\left\\{0.5\left[\left|q\left(\boldsymbol{\theta}_{1}\right)-Y\right|+q\left(\boldsymbol{\theta}_{1}\right)-Y\right]+\tau\left(e\left(\boldsymbol{\theta}_{2}\right)-q\left(\boldsymbol{\theta}_{1}\right)\right)\right\\}$
$\displaystyle-\tau\ln\left(1+\exp\left(e\left(\boldsymbol{\theta}_{2}\right)\right)\right).$
Then
$\displaystyle
h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{0.5\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left[\left|q_{1}-Y\right|+q_{1}-Y\right]-\frac{0.5\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left[\left|q_{2}-Y\right|+q_{2}-Y\right]$
$\displaystyle+\left.\left[\frac{\tau_{1}\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left(e_{1}-q_{1}\right)-\frac{\tau_{2}\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left(e_{2}-q_{2}\right)\right.\right.$
$\displaystyle\left.-\left(\tau_{1}\ln\left(1+\exp\left(e_{1}\right)\right)-\tau_{2}\ln\left(1+\exp\left(e_{2}\right)\right)\right)\right]$
We assume $q_{1}\geq q_{2}$ and $\tau_{1}\geq\tau_{2}$. The results for
$q_{1}<q_{2}$ and $\tau_{1}<\tau_{2}$ can be proved in a similar manner. We
separate the proof into three cases.
Cases 1. Suppose $q_{1}\geq q_{2}>Y$. It can be shown that
$\displaystyle\left|h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}q_{1}-\frac{\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}q_{2}\right|$
$\displaystyle+\left|Y\frac{\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}-Y\frac{\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\right|$
$\displaystyle+\left|\frac{\tau_{1}\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left(e_{1}-q_{1}\right)-\frac{\tau_{2}\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left(e_{2}-q_{2}\right)\right.$
$\displaystyle\left.-\left(\tau_{1}\ln\left(1+\exp\left(e_{1}\right)\right)-\tau_{2}\ln\left(1+\exp\left(e_{2}\right)\right)\right)\right|$
Case 2. Suppose $q_{1}\geq Y\geq q_{2}$. It can be shown that
$\displaystyle\left|h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}q_{1}-\frac{\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}q_{2}\right|$
$\displaystyle+\left|\frac{\tau_{1}\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left(e_{1}-q_{1}\right)-\frac{\tau_{2}\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left(e_{2}-q_{2}\right)\right.$
$\displaystyle\left.-\left(\tau_{1}\ln\left(1+\exp\left(e_{1}\right)\right)-\tau_{2}\ln\left(1+\exp\left(e_{2}\right)\right)\right)\right|$
Case 3. Suppose $Y>q_{1}\geq q_{2}$. It can be shown that
$\displaystyle\left|h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)\right|$
$\displaystyle\leq$
$\displaystyle+\left|\frac{\tau_{1}\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left(e_{1}-q_{1}\right)-\frac{\tau_{2}\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left(e_{2}-q_{2}\right)\right.$
$\displaystyle\left.-\left(\tau_{1}\ln\left(1+\exp\left(e_{1}\right)\right)-\tau_{2}\ln\left(1+\exp\left(e_{2}\right)\right)\right)\right|$
Let $\left(\bar{\tau},\bar{q},\bar{e}\right)$ be some middle point between
$\left(\tau_{1},q_{1},e_{1}\right)$ and $\left(\tau_{2},q_{2},e_{2}\right)$.
Using the mean value theorem, it is straightforward to show that
$\displaystyle\left|\frac{\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}q_{1}-\frac{\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}q_{2}\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\right|\left|q_{1}-q_{2}\right|$
(50)
$\displaystyle+\left|\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\left(1-\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\right)\right|\left|\bar{q}\left(e_{1}-e_{2}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|q_{1}-q_{2}\right|+\left|\bar{q}\left(e_{1}-e_{2}\right)\right|,$
$\displaystyle\left|Y\frac{\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}-Y\frac{\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\left(1-\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\right)\right|\left|Y\left(e_{1}-e_{2}\right)\right|$
(51) $\displaystyle\leq$
$\displaystyle\left|Y\left(e_{1}-e_{2}\right)\right|,$
and
$\displaystyle\left|\frac{\tau_{1}\exp\left(e_{1}\right)}{1+\exp\left(e_{1}\right)}\left(e_{1}-q_{1}\right)-\frac{\tau_{2}\exp\left(e_{2}\right)}{1+\exp\left(e_{2}\right)}\left(e_{2}-q_{2}\right)\right.$
$\displaystyle\left.-\left(\tau_{1}\ln\left(1+\exp\left(e_{1}\right)\right)-\tau_{2}\ln\left(1+\exp\left(e_{2}\right)\right)\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\bar{e}-\ln\left(1+\exp\left(\bar{e}\right)\right)\right|\left|\tau_{1}-\tau_{2}\right|$
$\displaystyle+\left|\frac{\exp\left(\bar{e}\right)\bar{q}}{1+\exp\left(\bar{e}\right)}\right|\left|\tau_{1}-\tau_{2}\right|+\left|\frac{\bar{\tau}\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\right|\left|q_{1}-q_{2}\right|$
$\displaystyle+\left|\frac{\bar{\tau}\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\left(1-\frac{\exp\left(\bar{e}\right)}{1+\exp\left(\bar{e}\right)}\right)\right|$
$\displaystyle\times\left|\left(\bar{e}-\bar{q}\right)\left(e_{1}-e_{2}\right)\right|$
$\displaystyle\leq$ $\displaystyle\left(\ln
2+\left|\bar{q}\right|\right)\left|\tau_{1}-\tau_{2}\right|+\left|q_{1}-q_{2}\right|$
$\displaystyle+\left|\left(\bar{e}-\bar{q}\right)\left(e_{1}-e_{2}\right)\right|$
To see why (A.4) holds, let
$\varpi\left(x\right)=\exp\left(x\right)\left(1+\exp\left(x\right)\right)^{-1}x-\ln\left(1+\exp\left(x\right)\right)$.
It can be shown that $\varpi\left(x\right)$ is monotonically decreasing for
$x<0$ and monotonically increasing for $x\geq 0$ and
$\varpi\left(0\right)=-\ln 2$,
$\lim_{x\rightarrow\infty}\varpi\left(x\right)=0$, and
$\lim_{x\rightarrow-\infty}\varpi\left(x\right)=0$. Therefore we can conclude
that
$\left|\varpi\left(x\right)\right|\leq\ln 2$
for all $x\in\mathbb{R}$. If Assumption 2.4 holds, it can be shown for (50),
$\displaystyle\left|q_{1}-q_{2}\right|+\left|\bar{q}\left(e_{1}-e_{2}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\nabla_{\boldsymbol{\theta}_{1}}q\left(\bar{\boldsymbol{\theta}}_{1}\right)\right|\left\|\boldsymbol{\theta}_{1}^{1}-\boldsymbol{\theta}_{1}^{2}\right\|+\left|q\left(\boldsymbol{\theta}_{1}\right)\nabla_{\boldsymbol{\theta}_{2}}e\left(\boldsymbol{\theta}_{2}\right)\right|\left\|\boldsymbol{\theta}_{2}^{1}-\boldsymbol{\theta}_{2}^{2}\right\|$
$\displaystyle\leq$ $\displaystyle
B_{3}\left(V\right)\left\|\boldsymbol{\theta}_{1}^{1}-\boldsymbol{\theta}_{1}^{2}\right\|+B_{2}\left(V\right)\left\|\boldsymbol{\theta}_{2}^{1}-\boldsymbol{\theta}_{2}^{2}\right\|,$
for (51),
$\displaystyle\left|Y\left(e_{1}-e_{2}\right)\right|$ $\displaystyle\leq$
$\displaystyle
B_{4}\left(V\right)\left\|\boldsymbol{\theta}_{2}^{1}-\boldsymbol{\theta}_{2}^{2}\right\|,$
and for (A.4),
$\displaystyle\left(\ln
2+\left|\bar{q}\right|\right)\left|\tau_{1}-\tau_{2}\right|+\left|q_{1}-q_{2}\right|+\left|\left(\bar{e}-\bar{q}\right)\left(e_{1}-e_{2}\right)\right|$
$\displaystyle\leq$ $\displaystyle\left(\ln
2+B_{1}\left(V\right)\right)\left|\tau_{1}-\tau_{2}\right|+B_{3}\left(V\right)\left\|\boldsymbol{\theta}_{1}^{1}-\boldsymbol{\theta}_{1}^{2}\right\|$
$\displaystyle+B_{5}\left(V\right)\left\|\boldsymbol{\theta}_{2}^{1}-\boldsymbol{\theta}_{2}^{2}\right\|.$
Summing the above inequalities together, given $V=\left(Y,W\right)$, we can
conclude that
$\left|h\left(\tau_{1},\boldsymbol{\theta}^{1}\right)-h\left(\tau_{2},\boldsymbol{\theta}^{2}\right)\right|\leq
B^{*}\left(V\right)\left(\left\|\tau_{1}-\tau_{2}\right\|+\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{2}\right\|\right),$
where $B^{*}\left(V\right)$ is a constant and is determined by a sum of $\ln
2$ and constants $B_{i}\left(V\right)$, $i=1,\ldots,5$.
### A.5 Constructing the simultaneous confidence bands (scb) with the
bootstrap
We use the models of linear in parameters of (6) and (7) as an example to
illustrate the bootstrap procedures for constructing the simultaneous
confidence bands. Let
$\boldsymbol{\theta}_{\tau}=\left(\alpha_{1,\tau},\boldsymbol{\beta}_{1,\tau}^{\top},\alpha_{2,\tau},\boldsymbol{\beta}_{2,\tau}^{\top}\right)^{\top}$
be a vector for parameters at the $\tau$-quantile. The bootstrap procedures
are summarized as follows.
1. 1.
Draw a bootstrap sample of size $n$: $W_{1}^{*},\ldots,W_{n}^{*}$ with
replacement, where
$W_{i}^{*}=\left(Y_{i}^{*},D_{i}^{*},X_{i}^{*\top},Z_{i}^{*}\right)^{\top},$
$i=1,\ldots,n$, is a vector for the $i$th random draw sample.
2. 2.
Re-estimate the weight $\bar{K}$ with the bootstrap sample. Let
$\hat{\tilde{K}}_{i}^{*}$, $i=1,\ldots,n$ denote the bootstrap estimated
truncated weight evaluated with $\left(Y_{i}^{*},D_{i}^{*},X_{i}^{*}\right)$.
3. 3.
With the bootstrap estimated truncated weight $\hat{\bar{K}}_{i}^{*}$, obtain
the bootstrap estimated parameters
$\hat{\boldsymbol{\theta}}_{\tau,b}^{*}=\left(\hat{\alpha}_{1,\tau,b}^{*},\hat{\boldsymbol{\beta}}_{1,\tau,b}^{*\top},\hat{\alpha}_{2,\tau,b}^{*},\hat{\boldsymbol{\beta}}_{2,\tau,b}^{*\top}\right)^{\top}$
for $\tau\in\left(0,1\right)$.
4. 4.
Repeat procedures 1 to 3 $B$ times to obtain $B$ bootstrap estimated
parameters
$\hat{\boldsymbol{\theta}}_{\tau,1}^{*},\ldots,\hat{\boldsymbol{\theta}}_{\tau,B}^{*}$.
With the bootstrap estimated parameters, we use $\hat{\alpha}_{2,\tau}$ as an
example to illustrate the procedures to construct the scb as follows.
1. 1.
Calculate the bootstrap standard deviation of $\hat{\alpha}_{2,\tau}$,
$\hat{\sigma}_{\hat{\alpha}_{2,\tau}}^{*}$ with the $B$ bootstrap estimates
$\left(\hat{\alpha}_{2,\tau,1}^{*},\ldots,\hat{\alpha}_{2,\tau,B}^{*}\right)$.
2. 2.
Calculate the bootstrap $t$ statistic of $\hat{\alpha}_{2,\tau}$:
$\hat{t}_{\hat{\alpha}_{2,\tau},b}^{*}=\frac{\sqrt{n}\left(\hat{\alpha}_{2,\tau,b}^{*}-\hat{\alpha}_{2,\tau}\right)}{\hat{\sigma}^{*}_{\hat{\alpha}_{2,\tau}}},$
$b=1,\ldots,B$.
3. 3.
For each $b=1,\ldots,B$, calculate the maximal absolute bootstrap $t$
statistic for $\tau\in\left(0,1\right)$:
$\hat{t}_{\hat{\alpha}_{2},b}^{*}=\sup_{\tau\in\left(0,1\right)}\left|\hat{t}_{\hat{\alpha}_{2,\tau},b}\right|.$
4. 4.
Let $\hat{t}_{\hat{\alpha}_{2}}^{*(1-g)}$ be the $\left(1-g\right)$th sample
quantile of
$\hat{t}_{\hat{\alpha}_{2},1}^{*},\ldots,\hat{t}_{\hat{\alpha}_{2},B}^{*}$.
The $1-g$ scb of $\hat{\alpha}_{2,\tau}$ is
$\left[\hat{\alpha}_{2,\tau}-\hat{t}_{\hat{\alpha}_{2}}^{*(1-g)}\frac{\hat{\sigma}_{\hat{\alpha}_{2,\tau}}^{*}}{\sqrt{n}},\hat{\alpha}_{2,\tau}+\hat{t}_{\hat{\alpha}_{2}}^{*(1-g)}\frac{\hat{\sigma}_{\hat{\alpha}_{2,\tau}}^{*}}{\sqrt{n}}\right].$
(53)
## References
* Abadie (2003) Abadie, A. (2003): “Semiparametric instrumental variable estimation of treatment response models,” _Journal of Econometrics_ , 113, 231–263.
* Abadie et al. (2002) Abadie, A., J. Angrist, and G. Imbens (2002): “Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings,” _Econometrica_ , 70, 91–117.
* Angrist et al. (2006) Angrist, J., V. Chernozhukov, and I. Fernández-Val (2006): “Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure,” _Econometrica_ , 74, 539–563.
* Angrist et al. (1996) Angrist, J. D., G. W. Imbens, and D. B. Rubin (1996): “Identification of Causal Effects Using Instrumental Variables,” _Journal of the American Statistical Association_ , 91, 444–455.
* Belloni et al. (2017) Belloni, A., V. Chernozhukov, I. Fernández-Val, and C. Hansen (2017): “Program Evaluation and Causal Inference With High-Dimensional Data,” _Econometrica_ , 85, 233–298.
* Chen et al. (2020) Chen, Y.-T., Y.-C. Hsu, and H.-J. Wang (2020): “A Stochastic Frontier Model with Endogenous Treatment Status and Mediator,” _Journal of Business & Economic Statistics_, 38, 243–256.
* Chernozhukov et al. (2013) Chernozhukov, V., I. Fernández-Val, and B. Melly (2013): “Inference on Counterfactual Distributions,” _Econometrica_ , 81, 2205–2268.
* Chernozhukov and Hansen (2005) Chernozhukov, V. and C. Hansen (2005): “An IV Model of Quantile Treatment Effects,” _Econometrica_ , 73, 245–261.
* Chernozhukov and Hansen (2006) ——— (2006): “Instrumental quantile regression inference for structural and treatment effect models,” _Journal of Econometrics_ , 132, 491–525.
* Chernozhukov and Hansen (2008) ——— (2008): “Instrumental variable quantile regression: A robust inference approach,” _Journal of Econometrics_ , 142, 379–398.
* Chernozhukov et al. (2007) Chernozhukov, V., C. Hansen, and M. Jansson (2007): “Inference approaches for instrumental variable quantile regression,” _Economics Letters_ , 95, 272–277.
* Chernozhukov et al. (2009) ——— (2009): “Finite sample inference for quantile regression models,” _Journal of Econometrics_ , 152, 93–103.
* Chou et al. (2020) Chou, R. Y., T.-J. Yen, and Y.-M. Yen (2020): “Forecasting Expected Shortfall and Value-at-Risk with the FZ Loss and Realized Variance Measures,” Available at SSRN: https://ssrn.com/abstract=3448882.
* Dimitriadis and Bayer (2019) Dimitriadis, T. and S. Bayer (2019): “A joint quantile and expected shortfall regression framework,” _Electron. J. Statist._ , 13, 1823–1871.
* Engle and Manganelli (2004) Engle, R. F. and S. Manganelli (2004): “CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles,” _Journal of Business & Economic Statistics_, 22, 367–381.
* Fissler and Ziegel (2016) Fissler, T. and J. F. Ziegel (2016): “Higher order elicitability and Osband’s principle,” _The Annals of Statistics_ , 44, 1680–1707.
* Fricke et al. (2020) Fricke, H., M. Frölich, M. Huber, and M. Lechner (2020): “Endogeneity and non-response bias in treatment evaluation – nonparametric identification of causal effects by instruments,” _Journal of Applied Econometrics_ , 35, 481–504.
* Frölich and Huber (2017) Frölich, M. and M. Huber (2017): “Direct and indirect treatment effects–causal chains and mediation analysis with instrumental variables,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 79, 1645–1666.
* Frölich and Melly (2013) Frölich, M. and B. Melly (2013): “Unconditional Quantile Treatment Effects Under Endogeneity,” _Journal of Business and Economic Statistics_ , 31, 346–357.
* Heckman et al. (1997) Heckman, J. J., J. Smith, and N. Clements (1997): “Making The Most Out Of Programme Evaluations and Social Experiments: Accounting For Heterogeneity in Programme Impacts,” _The Review of Economic Studies_ , 64, 487–535.
* Hill (2015) Hill, J. B. (2015): “Expected Shortfall Estimation and Gaussian Inference for Infinite Variance Time Series,” _Journal of Financial Econometrics_ , 13, 1–44.
* Imbens and Angrist (1994) Imbens, G. and J. Angrist (1994): “Identification and Estimation of Local Average Treatment Effects,” _Econometrica_ , 62, 467–75.
* Imbens and Rubin (2015) Imbens, G. W. and D. B. Rubin (2015): _Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction_ , Cambridge University Press.
* Kianian et al. (2021) Kianian, B., J. I. Kim, J. P. Fine, and L. Peng (2021): “Causal Proportional Hazards Estimation with a Binary Instrumental Variable,” _Statistica Sinica_ , 31, 673–699.
* Koenker (2005) Koenker, R. (2005): _Quantile Regression_ , Econometric Society Monographs, Cambridge University Press.
* Koenker and Bassett (1978) Koenker, R. and G. Bassett (1978): “Regression Quantiles,” _Econometrica_ , 46, 33–50.
* Levy (2016) Levy, H. (2016): _Stochastic Dominance-Investment Decision Making under Uncertainty_ , Springer, 3 ed.
* Linton and Xiao (2013) Linton, O. and Z. Xiao (2013): “Estimation of and Inference about the expected shortfall for time series with infinite variance,” _Econometric Theory_ , 29, 771–807.
* Meng and Taylor (2020) Meng, X. and J. W. Taylor (2020): “Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data,” _European Journal of Operational Research_ , 280, 191 – 202.
* Newey (1991) Newey, W. K. (1991): “Uniform Convergence in Probability and Stochastic Equicontinuity,” _Econometrica_ , 59, 1161–1167.
* Newey (1997) ——— (1997): “Convergence rates and asymptotic normality for series estimators,” _Journal of Econometrics_ , 79, 147 – 168.
* Newey and McFadden (1994) Newey, W. K. and D. McFadden (1994): “Chapter 36: Large sample estimation and hypothesis testing,” in _Handbook of Econometrics_ , Elsevier, vol. 4, 2111–2245.
* Patton et al. (2019) Patton, A. J., J. F. Ziegel, and R. Chen (2019): “Dynamic semiparametric models for expected shortfall (and Value-at-Risk),” _Journal of Econometrics_ , 211, 388 – 413.
* Powell (1984) Powell, J. L. (1984): “Least absolute deviations estimation for the censored regression model,” _Journal of Econometrics_ , 25, 303–325.
* Taylor (2019) Taylor, J. W. (2019): “Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution,” _Journal of Business & Economic Statistics_, 37, 121–133.
* van der Vaart (1998) van der Vaart, A. W. (1998): _Asymptotic Statistics_ , Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.
* Wei et al. (2021) Wei, B., L. Peng, M.-J. Zhang, and J. P. Fine (2021): “Estimation of causal quantile effects with a binary instrumental variable and censored data,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , forthcoming.
* Wüthrich (2020) Wüthrich, K. (2020): “A Comparison of Two Quantile Models With Endogeneity,” _Journal of Business & Economic Statistics_, 38, 443–456.
Figure 1: Bias, variance and MSE of the estimated conditional CTATE for the
compliers under different quantile levels when $\rho=0$.
Figure 2: Bias, variance and MSE of the estimated conditional QTE for the
compliers under different quantile levels when $\rho=0$.
Figure 3: Bias, variance and MSE of the estimated conditional CTATE for the
compliers under different quantile levels when $\rho=0.5$.
Figure 4: Bias, variance and MSE of the estimated conditional QTE for the
compliers under different quantile levels when $\rho=0.5$.
Figure 5: CTATE, QTE and the corresponding 95% pointwise and simultaneous
confidence bands: Adult men’s earnings.
Figure 6: CTATE, QTE and the corresponding 95% pointwise and simultaneous
confidence bands: Adult women’s earnings.
Figure 7: IQATE and 95% pointwise confidence bands. Upper panel: Adult men’s
earnings. Lower panel: Adult women’s earnings.
|
Universal Source Separation with Weakly Labelled Data
M. Shell was with the Department
of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta,
GA, 30332.
E-mail: see http://www.michaelshell.org/contact.html
J. Doe and J. Doe are with Anonymous University.
Manuscript received April 19, 2005; revised August 26, 2015.
Journal of Class Files, Vol. 14, No. 8, August 2015
Shell et al.: Bare Demo of IEEEtran.cls for Computer Society Journals
Universal source separation (USS) is a fundamental research task for computational auditory scene analysis (CASA), which aims to separate mono recordings into individual source tracks. From the perspective of system designs, there are two potential challenges awaiting the solution to the audio source separation task. First, previous audio source separation systems mainly focus on separating one or a limited number of specific sources, such as speech, singing, or musical instruments. However, there is a lack of research on building a unified system that can separate arbitrary sources via a single model. Second, most previous systems require strongly labeled data, i.e., clean source and mixture tracks to train a separator. Nonetheless, strongly labeled data is scarce. There should be a potential usage of large-scale weakly labeled/unlabeled audio data for audio source separation. To combat those problems, we propose a universal audio source separation framework containing: 1) an audio tagging model trained on weakly labeled data only as a query net for source separation; and 2) a conditional source separation model that can separate arbitrary sound sources according to the mixture and the query. We investigate various query nets, source separation models, and different training strategies to verify how they affect the performance in the universal source separation framework. We propose a hierarchy USS strategy to automatically detect and separate sound classes from the AudioSet ontology. Our separation system is recognized with a strong generalization ability. By solely leveraging AudioSet with 2 million weakly labeled audio event samples for the model training, our USS system could achieve compatible performance on various source separation benchmarks, such as sound event separation, music source separation, and speech enhancement. Namely, an average signal-to-distortion ratio improvement (SDRi) of 5.57 dB over AudioSet's 527 sound classes; an SDRi of 10.57 dB on the DCASE 2018 Task 2 dataset; an SDRi of 8.12 dB on the MUDDB18 dataset; an SDRi of 7.28 dB on the Slakh2100 dataset; and an SSNR of 9.00 on the voicebank-demand dataset. We release the source code at <https://github.com/bytedance/audioset_source_separation>.
Universal source separation, weakly labelled data, AudioSet, computational auditory scene analysis.
§ INTRODUCTION
Source separation is a task to separate mono audio recordings into individual source tracks. An audio recording may consist of several sound events and acoustic scenes.
Source separation has been researched for several years and has a wide range of applications, including speech enhancement <cit.>, music source separation <cit.>, and sound events separation<cit.>.
Universal source separation (USS) is closely related to the well-known cocktail party problem <cit.>, where sounds from different sources in the world mix in the air before arriving at the ear, requiring the brain to estimate individual sources from the received mixture. Humans can focus on a particular sound source and discriminate it from others, alternatively described as selective hearing. As a study of auditory scene analysis by computational means, computational auditory scene analysis (CASA) <cit.> systems are machine listening systems that aim to separate mixtures of sound sources in the same way that human listeners do. Different from specified source separation tasks such as speech enhancement or music source separation, a universal source separation system aims to separate the tracks of arbitrary sound sources from a mixture, regarded as a unified system to separate all different sounds.
However, many previous works mainly focus on specific source separation that only separate one or a few sources, such as the speech enhancement <cit.> and music source separation <cit.> tasks. USS remains a challenging problem. One difficulty is that there are hundreds of different sounds in the world, and it is difficult to separate all sounds using a unified model. Recently, the USS problem has attracted several interests. A system <cit.> was proposed to separate arbitrary sounds by predicting the masks of sounds. Unsupervised USS systems <cit.> were proposed to separate sounds by mixture invariant training, where training examples are constructed by mixing together existing mixtures, and the model separates them into a variable number of latent sources. A Free Universal Sound Separation (FUSS) dataset and a time-domain convolutional network (TDCN++) were proposed to separate a mixture into up to 4 separate sources. Some other methods use sound embeddings as conditions to separate sources from a mixture <cit.>. A sound selector applies one-hot encodings of sound classes as conditions to separate corresponding sources <cit.>. Weak supervision methods were proposed in <cit.>. SuDoRM-RM systems <cit.> were proposed to separate sound signals. Language-based source separation was proposed in <cit.>. A class conditions system <cit.> was proposed for music source separation.
Those systems are mainly trained on small datasets and do not scale to hundreds of sound classes.
The standard architecture of deep-learning-based audio source separation model. Left: two types of time-domain separation model. Right: the general type of frequency-domain separation model.
Another challenge of universal source separation is that there is a lack of clean source data for a wide range of sound events.
For the source separation problem, we define strongly labelled data as audio segments that only contain target sources without other sources. We can mix those clean sources as audio mixtures and train the separators. However, the collection of clean data is time-consuming. For sound classes such as speech, clean data can be recorded in the laboratory. But for many other sounds, the collection of clean data is difficult. Recently, the concept of weakly labelled data <cit.> was used in audio signal processing. In contrast to strongly labelled data, weakly labelled data are only labelled with one or multiple tags, while the time information of tags is unknown. For example, a 10-second audio track is labelled as “thunder” and “rain”, but when these two events exactly appear within this 10-second is not provided. Sometimes, different audio events may interfere with each other. Weakly labelled data has been widely used in audio tagging <cit.> and sound event detection <cit.>. There is a strong potential usage of it in the audio source separation task.
To address the challenge of lacking strongly labelled data to separate hundreds of sound classes. In this work, we propose a USS framework that can be trained with weakly labelled data only. This work is an extension of our previously proposed source separation systems <cit.>. The whole system contains: 1) pretrained audio tagging models to provide both the anchor segments of events in weakly labelled data and latent embeddings for source separation controls; 2) a query-based separation model for conditional audio source separation via the given source queries; and 3) a pipeline to train USS systems with weakly labelled data. The contribution of this work includes:
* We are the first to propose USS systems trained only on weakly labelled data only, without using any strongly labelled data. We train the USS system on AudioSet <cit.>.
* We propose an anchor segment mining strategy to detect short segments from long weakly labelled audio recordings. We show that anchor segments are the core parts to constitute mixtures to build the USS system.
* We propose several latent condition strategies and show that pretrained audio tagging system is essential to build USS systems.
* Our proposed USS systems are the first to separate 527 sound classes. We propose a hierarchy USS strategy to separate sources of different AudioSet ontology levels. Our proposed USS system is able to automatically detect and separate active sound classes of an audio recording without the need to specify the sound classes to separate.
* We show that a single USS system is able to achieve compatible performance to various source separation benchmarks, including sound events separation, music source separation, and speech enhancement tasks. We conduct comprehensive ablation studies to investigate how different factors in our system affect the separation performance.
This article is organized as follows. Section 2 introduces neural network-based source separation systems. Section 3 introduces our proposed weakly labelled source separation framework. Section 4 shows experiments. Section 5 concludes this work.
§ SOURCE SEPARATION VIA NEURAL NETWORKS
Deep learning methods for audio source separation has already outperformed traditional methods such as Non-negative Matrix Factorization <cit.>. Fig. <ref> shows the source separation methods in the time-domain (left) and in the frequency-domain (right). Here, we introduce the basic methodology of those separation models.
§.§ Time-domain Separation Models
A time-domain separation model $f$ via neural networks is typically constructed as an encoder-decoder architecture, as shown in the left of Fig. <ref>. Formally, given a mono audio clip $x \in \mathbb{R}^{L}$ and a separation target $s \in \mathbb{R}^{L}$, where $L$ is sample length, the separator $f$ performs two methods: synthesis separation and mask separation.
Synthesis separation systems such as Demucs V2 <cit.> and Wave-U-Net <cit.> $f$ directly estimates the final target samples of separation: $\hat{s} = f(x)$. The encoder is designed to extract the key feature for the separated source, and the decoder is designed to reconstruct the feature back to the separation samples together with the input features. Mask separation systems such as TasNet <cit.> and ConvTasNet <cit.> predict masks in the latent space. Then, decoders are designed to reconstruct the separated waveform from the masked latent feature.
§.§ Frequency-domain Separation Models
Different from time-domain models, frequency-domain models leverages the spectrogram such as short-time Fourier transform (STFT) to facilitate the separation process. The harmonic features are more discriminative in the frequency domain than those in the time domain, which might help improve the separation performance in specific source separation tasks, such as music source separation and environmental sound separation.
Formally, given a mono audio clip $x$. We denote the STFT of $x$ as $X \in \mathbb{C}^{T \times F}$, where $T$ is the number of time frames and $F$ is the number of frequency bins. The STFT spectrogram $X$ is in the complex domain. We denote the magnitude and the phase of $ X $ as $|X|$ and $\angle S_X$, respectively. The right part of Fig. <ref> shows the frequency-domain separation system $f$ predict the mask $ M $, where $ M $ can be an ideal ratio mask (IRM) $M \in \mathbb{R}^{T \times F}$ or a complex mask $M \in \mathbb{C}^{T \times F}$. When using the complex mask $ M $, the complex STFT of the separated source $ \hat{S} \in \mathbb{C}^{T \times F} $ can be calculated by:
\begin{align}
\hat{S}=M \odot X
\end{align}
Then, the separated source $ \hat{s} \in \mathbb{R}^{L} $ can be obtained by applying an inverse STFT on $ \hat{S} $.
Frequency domain models include fully connected neural networks <cit.>, recurrent neural networks (RNNs) <cit.>, and convolutional neural networks (CNNs) <cit.>. UNets <cit.> are variants of CNN that contain encoder and decoder layers for source separation. Band-split RNN (BSRNN) <cit.> proposes to apply RNNs along both time and frequency axes to capture time and frequency domain dependencies. There are also researches such as hybrid Demucs <cit.> combines time and frequency domain systems to build source separation systems.
§.§ Challenges of Source Separation Models
As mentioned above, many previous source separation systems are trained with strongly labelled data, which requires clean source tracks. However, the collection of strongly labelled data is difficult and time-consuming. Table <ref> summarizes the datasets that can be used for source separation. On one hand, previous strongly labelled datasets have durations of around tens of hours. On the other hand, weakly labelled datasets are usually larger than strongly labelled datasets and clean datasets. AudioSet <cit.> is a representative of weakly labelled datasets containing over 5,800 hours of 10-second audio clips and is larger in both size and sound class numbers than strongly labelled datasets. AudioSet has an ontology of 527 sound classes in its released version. The ontology of AudioSet has a tree structure. Each audio clip may contain multiple tags.
In this work, we use the weakly labelled AudioSet dataset containing 5,800 hours to train the USS system that can separate hundreds of sound classes as described in the next section.
Source separation datasets
Dataset Dur. (h) Classes Labels
Voicebank-Demand 18.9 1 Strong
MUSDB18 <cit.> 6.0 4 Strong
UrbanSound8K 9.7 10 Strong
FSDKaggle 2018 41 Strong
FUSS 357 Strong
AudioCaps — Language
AudioSet 5800 527 Weak
Left: Strongly labelled data of sound class “Flute”. Right: Weakly labelled data of sound class “Air horn, truck horn” which only occur between 2.5s - 4.0s.
The architecture of our proposed query-based audio source separation pipeline trained from weakly-labeld data, including datasets, sampling strategies, audio tagging model, and conditional audio source separation models.
Two audio tagging models for audio classification, sound event detection, and latent feature production. Left: Pretrained Audio Neural Networks (PANN) in CNN-14 architecture. Right: Hierarchical Token-Semantic Transformer (HTS-AT) in 4-block architecture.
§ USS WITH WEAKLY LABELLED DATA
§.§ Weakly Labelled Data
Different from strongly labelled data containing clean sources of sound classes, weakly labelled data only contains the labels of what sound classes are present in an audio recording. Weakly labelled data may contain interference sounds. There are no time stamps information of sound classes nor clean sources. We denote a weakly labelled audio clip as $ x $ and its tags as $ y \in \{0, 1\}^{K} $, where $ K $ is the number of sound classes. The value $ 1 $ indicates the presence of sound class and the value $ 0 $ indicates the absence of the sound class in the audio clip. We denote a weakly labelled dataset as $ D = \{x_{n}, y_{n}\}_{n=1}^{N} $, where $ N $ is the number of training samples in the dataset. The left part of Fig. <ref> shows a strongly labelled audio clip containing the clean waveform of “Flute”. The right part of Fig. <ref> shows a weakly labelled audio clip containing a target sound class “Air horn, truck horn” which only occur between 2.5 s and 4.0 s. The weakly labelled audio recording also contain unknown interference sounds.
The goal of a weakly labelled USS system is to separate arbitrary sounds via a unified model.
Fig. <ref> depicts the architecture of our proposed system, containing four steps:
* Apply a sampling strategy to sample audio clips from a weakly labelled audio dataset, such as AudioSet.
* Apply an anchor segment mining algorithm to localize the occurrence of events/tags in the weakly labelled audio tracks.
* Calculate the segment predictions or latent embeddings of anchor segments using pretrained audio tagging models. Mix anchor segments as input mixtures.
* Train a query-based separation network to separate the mixture into target sources queried by the source embedding.
§.§ Audio Clips Sampling Strategies
For a large-scale unbalanced dataset, we apply two sampling strategies: 1) random sampling: randomly sample audio clips from the dataset to constitute a mini-batch; and 2) balanced sampling: sample audio clips from different sound classes to constitute a mini-batch to ensure the mixture of them contain as many as independent sounds. AudioSet is highly unbalanced. Sound classes such as “Speech” and “Music” have almost 1 million audio clips, while sound classes such as “tooth breath” has only tens of training samples. Without balanced sampling, “tooth breath” are less likely to be selected for training. Following the training scheme of the audio classification systems <cit.>, we apply the balanced sampling strategy to retrieve audio data from AudioSet. We denote a mini-batch of sampled audio clips as $ \{ x_{1}, ..., x_{B} \} $, where $ B $ is the mini-batch size.
The anchor segment mining algorithm involves an audio tagging model to localize the sound events in the given audio track. Since the whole pipeline requires only weakly labelled data, it is better that we could leverage audio tagging models that require only the weakly-labelled data for training, but is able to localize the occurrence (i.e., time stamps) of the sound classes. Recently, audio tagging systems trained with the weakly labelled AudioSet <cit.> have outperformed systems trained with strongly labelled data. We apply Pretrained Audio Neural Networks (PANN) <cit.> and Hierarchical Token-semantic Audio Transformer (HTS-AT) <cit.> as our audio tagging models to perform the anchor segment mining procedure. Such models are able to extract audio clips, with relatively clean sound sources, from weakly labelled audio samples. In section <ref>, we introduce the detail of audio tagging models and the mining algorithm.
To extract the embedding from audio clips, we have different formulations. The most intuitive embedding can be the one-hot class vector, and the class probability vector or the latent embedding from audio tagging models could be a potential choice. Together with the query-based source separation model, we demonstrate how it works through this separation pipeline in section
<ref> and <ref>.
§.§ Anchor Segments Mining
Anchor segment mining is the core part of USS systems with weakly labelled data. Since the weakly labeled audio track does not always contain the labeled sound class throughout its timeline, we need to extract a short audio segment inside this track to create a relative cleaner source data for training the separation model. Formally, given an audio clip $x \in \mathbb{R}^L$, an anchor segment mining algorithm extracts an anchor segment $s \in \mathbb{R}^{L'}$ from $x$, where $L' < L$ is the samples number of the anchor segment. The center of $s$ is the time stamp where the sound class label is most likely to occur. As shown in Fig <ref> and illustrated in Algorithm <ref>, an anchor mining algorithm is performed as three steps:
* By the sampling strategy, we sample $n$ audio tracks $\{x_1, x_2, ..., x_n\}$ with the corresponding tags $\{e_1,e_2, ..., e_n\}$ as one input for our system.
* Two anchor mining algorithms can be performed to extract the audio clip $s_{i}$ from $x_{i}$: i) random method: randomly select an audio clip from the audio track; or ii) sound event detection (SED) method: apply an audio tagging model to localize the labeled sound class as an anchor position $t$ in the audio track, then extract an audio clip whose center is the anchor position.
* To further verify the localization, an optional step can be performed to filter the audio clips from duplicate sound classes, ensuring that the mixture does not come from the same sound classes.
The detail demonstration of each step is illustrated in Algorithm <ref>.
For the SED method of anchor mining, in this paper, we leverage two audio tagging models, Pretrain Audio Neural Networks (PANN) <cit.> and Hierarchical Token-Semantic Audio Transformer (HTS-AT) <cit.>, to perform the sound event detection of the audio track. These models are also applied into the construction of query embeddings during the separation process.
Formally, given an audio track $x$ as input, the audio tagging model produces two main outputs: 1) event classification vector $v_e \in \mathbb{R}^K$; and 2) framewise event presence map $m_e \in \mathbb{R}^{T \times K}$, where $T$ the size of time frames and $K$ the number of sound classes. The framewise event presence map is usually connected to the last neural network layer by taking the average value along the time axis to produce the event classification vector. As mentioned in <ref>, both PANN and HTS-AT are trained with the weakly labeled data by minimizing the binary cross entropy loss between the label vector $y \in [0,1]^K$ and $v_e$:
\begin{equation} \label{eq:aggregate_max}
l = - \sum_{k=1}^{K} y(k) \ln v_e(k) + (1 - y(k)) \ln (1 - v_e(k)).
\end{equation}
where $y(k)$ and $v_e(k)$ denote the $k$-th component of $y$ and $v_e$.
After training the audio tagging model, we can not only use it to predict the event of the given audio track, but also weakly predict the event of each time frame via framewise event presence map $m_e$ (i.e., sound event detection). We denote $m_e(t,k)$ as the probability value of the $k$-th sound class at the $t$-th time frame. Then the anchor segment $t_{anchor}$ for the labeled sound class $k$ is obtain by:
\begin{align} \label{eq:sed_area}
q_{k}(t) = \sum_{t - \tau / 2}^{t + \tau / 2} m_e(t, k) \\
t_{\text{anchor}} = \underset{t}{\text{argmax}} \ q_{k}(t).
\end{align}
where $ \tau $ is the duration of anchor segments. We apply the sound event detection algorithm to the $n$-track audio input and get $n$ audio clips $\{s_{1}, ..., s_{n}\}$ as anchor segments (if Step-3 in Algorithm <ref> is applied, the number of anchor segments will be smaller than the number of input audio tracks). Finally, we mix these segments as the input mixture of the separation model.
Fig. <ref> shows the model architectures of both PANN and HTS-AT as two audio tagging models we employed. PANN contains VGG-like CNNs to convert an audio mel-spectrogram into a $(T, K)$ featuremap. The model averages the featuremap over the time axis to obtain a final event classification vector $(1, K)$ and computes the binary cross-entropy loss between it and the groudtruth label. Since CNNs can capture the information in each time window, the featuremap $(T, K)$ is empirically regarded as a framewise presence probability map of each sound event at each time frame. Additionally, the penultimate layer's output $(T, H)$ can be used to obtain its averaged vector $(1, H)$ as a latent source embedding for the query-based source separation in our system.
HTS-AT is a hierarchical token-semantic transformer for audio classification. It applies swin-transformer <cit.> into the audio classification task. In the right of Figure <ref>, a mel-spectrogram is cut into different patch tokens with a patch-embed CNN and sent into the transformer in order. The time and frequency lengths of the patch is equal as $P \times P$. To better capture the relationship between frequency bins of the same time frame, HTS-AT first splits the mel-spectrogram into windows $w_1, w_2, ..., w_n$ and then splits the patches in each window. The order of tokens $Q$ follows time$\to$frequency$\to$window.
The patch tokens pass through several network groups, each of which contains several transformer-encoder blocks. Between every two groups, a patch-merge layer is applied to reduce the number of tokens to construct a hierarchical representation. Each transformer-encoder block is a swin-transformer block with the shifted window attention module <cit.>, a modified self-attention module to improve the training efficiency.
Then, HTS-AT applies a token-semantic 2D-CNN to further process the reshaped output $(\frac{T}{8P}, \frac{F}{8P}, H)$ into the framewise event presence map $(T,K)$, then average it as the event classification vector $(1,K)$. The latent embedding, at the same time, is produced by averaging the reshaped output into a $H$-dimension vector with an average-pooling layer.
In the following sections, we will evaluate how these two models perform in our universal source separation system.
Anchor segment mining.
[1]
Input: dataset $ D $.
Step 1 - Balanced Sampling: Sample $ n $ sound classes from $ \{1, ..., K\} $ without replacement.
For each selected sound class, sample one audio track $x_{i} \in \mathbb{R}^L$ to constitute $ \{x_{1}, ..., x_{n}\} $.
Step 2 - Random Method: for each audio track $x_{i}$, randomly select an audio clip $s_{i} \in \mathbb{R}^{L'} (L' < L)$ from the audio track.
Step 2 - Sound Event Detection:
Apply an audio tagging model to detect an anchor position $t_{i}$ throughout the audio track $x_{i}$, which is the most probable occurrence of the labeled sound class. Then extract the audio clip $s_{i} \in \mathbb{R}^{L'} (L' < L)$ whose center is $t_{i}$.
Step 3 (optional): Use the same audio tagging model to obtain the event presence probability of each extracted audio clip $\{s_1, s_2, ..., s_n\}$ and apply thresholds $ \theta \in [0, 1]^{K} $ to get binary results $ r \in \{0, 1\}^{K} $. Then, given audio clips $S=\{s_1, s_2, ..., s_n\}$:
each $ r_{i} $
$ r_{i} \cap tr = \o $
$tr = tr \lor r_{i} $
Remove $s_i$ from $S$
Output: A set $S$ of anchor segments .
§.§ Query-based Source Separation
A typical source separator is a single-input-single-output (SISO) model that deals with one specific source, such as vocal, drum, bass, and others. To enable the model to separator arbitrary sound sources, we apply a query-based source separator by adding conditional embeddings into the source separation backbone – ResUNet <cit.>.
As mentioned in <ref>, the input to the ResUNet separator is a mixture audio segment. First, we apply short-time Fourier transform (STFT) on the waveform to extract the complex spectrum $ X = \mathbb{C}^{T \times F} $. Then, we follow the same setting of <cit.> to construct an encoder-decoder network to process the magnitude spectrogram $ |X| $. The ResUNet encoder-decoder consists of 6 encoder blocks, 4 bottleneck blocks, and 6 decoder blocks. Each encoder block consists of 4 residual convolutional blocks to downsample the spectrogram into a bottleneck feature, and each decoder block consists of 4 residual deconvolutional blocks to upsample the feature back to separation components. The skip-connection is applied from each encoder block to the corresponding decoder block of the same downsampling/upsampling rate. For the residual block, it contains 2 CNN/DCNN layers, 2 batch normalization <ref> layers and 2 Leaky-ReLU activation layers. An additional residual shortcut is added between the input and the output of each residual block. The detail of the model architecture can be found at <cit.>.
The ResUNet separator outputs the magnitude and the phases of the complex mask $ M \in \mathbb{C}^{T \times F} $. The separated complex spectrum can be obtained by:
\begin{equation} \label{eq:mask_mul}
\begin{split}
\hat{S} &= M \odot X. \\
& = |M| \odot |X|e^{j (\angle M + \angle X)}.
\end{split}
\end{equation}
where both $|M|$ and $\angle M$ are the output of the separator. The separated source can be obtained by multiplying the STFT of mixture and the complex ideal ratio mask (cIRM). The multiplication can be also decoupled into a magnitude and a phase part. The magnitude $ |M| $ controls how much the magnitude of $ |X| $ should be scaled, and the angle $ \angle M $ controls how much the angle of $ X $ should be rotated.
Based on the ResUNet separator, we adopt a feature-wise linear modulation (FiLM) <cit.> method to construct convolutional blocks within the separator. Similar to voice filtering work <cit.> and multi-task source separation work <cit.>, for each encoder and decoder block, we incorporate the conditional embedding as:
\begin{equation} \label{eq:FiLM}
h = W * (\sigma(\text{bn}(x) + Vc))
\end{equation}
where $x$ is the output feature of the encoder and decoder block, $ V $ is a embedding layer to map the conditional vector $ c $ into an embedding space. Then, this embedding is regarded as a bias added to the the linear convolutional operation after batch normalization (BN).
A ReLU activation layer is applied to increase the non-linearity of the neural network. The final output feature $h$ will be sent to the next encoder and decoder block.
Let $f(s,c)$ denote the query-based source separator, the complete algorithm of our weakly labeled universal source separation system is illustrated in Algorithm <ref>.
§.§ Source Query Embeddings
The query-based source separator extracts pre-defined representations <cit.> or learnable representations <cit.> used to control what sources to separate from a mixture. To determine the choice of the source embedding $c$ for query-based source separation in our system, we investigate three types as follows.
§.§.§ Hard One-Hot Condition
The first type is the one-hot vector of sound classes. Each anchor segment $ s $ has a corresponding tag $ y \in [0,1]^K$. Then, the source query embedding is $ c = y $ in both training and inference stages.
§.§.§ Soft Probability Condition
The second type of source query embeddings is to use pretrained audio tagging models, such as PANN and HTS-AT, to obtain its final output, the event classification probability vector $v_e$, as the query embedding. Since the audio tagging model $ f_{\text{AT}} $ is trained on weakly labelled data with the binary cross entropy loss, which indicates that each audio clip $s$ can be categorized into each sound class with at some probabilities. Compared with the one-hot vector The event classification probability vector $v_e$ is a good description of the audio clip $ s $.
§.§.§ Latent Embedding Condition
As mentioned in <ref>, the latent embedding $(1,H)$ from pretrained audio tagging models can be applied as the source query embedding. The advantage of using the latent embedding is that the separation are not limited to the given $ K $ sound classes. In the experiment section, we investigate a variety of PANNs, including CNN5, CNN10, CNN14, and a HTS-AT model to extract the latent embeddings and we evaluate their efficiency on the separation performance. Apart from AudioSet tagging embeddings, our conditional source separation system also supports to use automatic speech recognition (ASR) embeddings (e.g., wav2vec <cit.>) and speaker embeddings <cit.>. We also compare them with our PANN and HST-AT latent embeddings.
Furthermore, the latent embedding can be learnable during the training of our universal source separation system. One method is to train both audio tagging model and source separator from scratch. Another method is to use the pretrained audio tagging model but add a learnable shortcut to build a joint representation: $c = $ . Or, the pretrained audio tagging model can connect to a fully connected layer as an adaptor by $c=f_{\text{ada}}(f_{\text{AT}}(s))$. We will investigate how these variations of latent embeddings contribute to the source separation performance.
Source separation
[1]
Dataset $D$: AudioSet.
loss function $l$ does not converge
Sample $n$ audio tracks $\{x_1,x_2,...,x_n\}$ from the balanced sampling strategy.
Obtain $n$ audio clips $\{\s_1, s_2, ..., s_n\\}$ using the anchor segment mining algorithm <ref>.
Create the mixture $x=\sum_i s_i$ .
Obtain the source query embedding $ \{c_1,c_2,...,c_n\} $.
each $s_i$
Obtain the separation $\hat{s_{i}}=f(x,c_i)$
Calculate loss by $ l(\hat{s_{i}}, s_{i}) $
§.§ Data augmentation
AudioSet is highly unbalanced. Some sound classes only have tens number of training samples. Furthermore, when constituting the mixture with $ s_{1} $ and $ s_{2} $, the loudness of $ s_{1} $ and $ s_{2} $ can be different. To address those problems, we propose a loudness augmentation method. That is, we first calculate the energy of a signal $ s $ by $ E = || s ||_{2}^{2} $. We denote the energy of $ s_{1} $ and $ s_{2} $ as $ E_{1} $ and $ E_{2} $. We apply a scaling factor $ \alpha = \sqrt{E_{1} / E_{2}} $ to $ s_{2} $ so that both anchor segments have the same energy. That is:
\begin{equation} \label{eq:scale_augmentation}
x = s_{1} + \alpha s_{2}.
\end{equation}
On one hand, we match the energy of anchor segments to let the neural network to learn to separate the sound classes. On the other hand, the loudness diversity of sound classes are increased. We will show this loudness augmentation is beneficial in our experiments.
§.§ Loss functions
We propose to use loss functions to train the end-to-end universal source separation system. Several loss functions have been discussed in <cit.> for source separation. The most commonly used loss function is the spectrogram domain L2 loss. However, the spectrogram based loss function does not consider the phase information of the separated source. In this work, we apply a L1 loss in the time-domain waveform between ground truth source and separated source:
\begin{equation} \label{eq:loss}
l = ||s - \hat{s}||_{1},
\end{equation}
where $ l $ is the loss function to train the neural network. Lower loss in (<ref>) indicates that the separated signal $ \hat{s} $ is close to the ground truth signal $ s $. In training, the gradients of parameters are calculated by $ \partial l / \partial \theta $, where $ \theta $ are the parameters of a neural network.
§.§ Inference
In inference, the oracle embedding is unavailable due to the clean sources are unknown. Instead, we calculate $ c $ from the training dataset by:
\begin{equation} \label{eq:avg_emb}
c = \frac{1}{N}\sum_{n=1}^{N}f_{\text{emb}}(s_{n}).
\end{equation}
where $ \{s_{n}\}_{n=1}^{N} $ are query samples of one sound class and $ N $ is the number of query samples. To explain, we average all conditional embeddings of query samples from a same sound class to constitute $ c $. The $ f_{\text{emb}} $ can be either segment prediction or the latent embedding prediction.
Automatic Filtering and Separation
[1]
Inputs: audio $ x $, source separation system $ f_{\text{ss}} $, audio tagging system $ f_{\text{at}} $.
Outputs: Separated sources $ D = \{\hat{s}_{1}, ..., \hat{s}_{I}\} $.
Split $ x $ into 2-second segments. Apply $ f_{\text{at}} $ on all segments to obtain $ P(t, k) $.
# Calculate ontology predictions.
hierarchy separation
$ Q(t, m) = \text{HierarchyOntologyGrouping}(P(t, k), l) $
$ Q(t, m) = P(t, k) $
# Detect sound event candidates.
$ C = \{\} $
$m=1, ..., M$
$ \text{max}_{t}Q(t, m) > \delta$:
$ C = C \cup \{m\} $
# Do separation.
$ D = \{\} $
$ m \in C $
$t=1, ..., T$
$ Q(t, m) > \delta$:
Get condition $ c $ by (<ref>).
$ \hat{s}_{t} = f_{\text{ss}}(x, c) $
$ \hat{s}_{t} = \textbf{0} $
$\hat{s_{k}} = \{\hat{s}_{1}, ..., \hat{s}_{T}\}$
$ D = D \cup \{\hat{s}_{k}\} $
Hierarchy Ontology Grouping.
[1]
Inputs: segment-wise prediction $ P(t, k) $, hierarchy level $ l $.
Outputs: $ Q(t, m) $
Denote sound classes from AudioSet ontology with level $ l $ as a set $ C = \{ c_{1}, ..., c_{M} \} $.
$ m = 1, ..., M $
$ Q(t, m) = \text{max}\{P(t, c')\}_{c' \in \text{child}(c_{m})} $
# $\text{child}(c_{m})$ is all children sound classes of $ c_{m} $ including $ c_{m} $.
§.§ Inference with Hierarchy AudioSet Ontology
For the CASA problem, usually it is unknown how many and what sound classes are present and to separate in an audio clip. To address this problem, we propose a hierarchy sound class detection strategy to detect the present sound classes. Then, we separate the present sound classes by using the trained USS system.
We first split long audio recordings into short segments. Algorithm <ref> shows the automatic sound classes detection and separation steps. The input to the USS system is an audio clip $ x $. We first split the audio clip into 2-second segments and apply an audio tagging system $ f_{\text{at}} $ to calculate the segment-wise prediction $ P(t, k) $ with a shape of $ T \times K $, where $ T $ is the number of segments. The AudioSet ontology has a tree structure. The first level of the tree structure contains seven sound classes, including “Human sounds”, “Animal”, “Music”, “Source-ambiguous sounds”, “Sounds of things”, “Nature sounds”, and “Channel, environment and background”. Each root category contains several sub-level sound classes. The second level contains 41 sound classes. The third level contains 251 sound classes. The tree structure have a maximum depth of six levels.
In inference, the USS system supports hierarchy source separation with different levels. To separate sources of level $ l $, we denote the sound classes of level $ l $ as $ C = \{c_{1}, ..., c_{M}\} $, where $ M $ is the number of sound classes in the level $ l $. For a sound class $ m $, we denote the set of its all children sound classes including itself as $ \text{child}(m) $. For example, for the human sounds class 0, $ \text{children}(0) = \{0, 1, ..., 72\} $. For each sound class $ c_{m} $ we set its hierarchy score $ Q(t, m) $ as the maximum probability of its all children sound classes including itself: $\text{max}\{P(t, c')\}_{c' \in \text{child}(c_{m})}$. The hierarchy ontology grouping is described in Algorithm <ref>.
\begin{equation} \label{eq:inference_hierarchy}
\left\{\begin{matrix}
c_{j}=f_{\text{AT}}(x),j \in \text{child}(c)\\
c_{j}=0,j \notin \text{child}(c).
\end{matrix}\right.
\end{equation}
Then, we detect active sound classes $ C = \{m\} $ if $ \text{max}_{t}Q(t, m) $ is larger than a threshold $\delta$. Then, for each acitve sound class $m$, we detect it active segments if $ \text{max}_{t}Q(t, m) $ is larger than a threshold $ \delta $. We set separated segments to silence if $ Q(t, m) $ is smaller than $ \theta $. Then, we apply the conditional source separation system by using (<ref>) as the condition.
§ EXPERIMENTS
In this section, we investigate our proposed universal source separation system on several tasks, including AudioSet separation <cit.>, sound events separation <cit.>, music source separation <cit.>, and speech enhancement <cit.>. Our USS system is only trained on the large-scale weakly labelled AudioSet <cit.> without using any clean training data, which is a major difference from the previous source separation systems that are trained on specific datasets with clean sources <cit.>. The trained USS system can address a wide range of source separation tasks without being finetuned.
§.§ Training Dataset
AudioSet is a large-scale weakly labelled audio dataset containing 2 million 10-second audio clips sourced from the YouTube website[<https://www.youtube.com/>]. Audio clips are only labelled with the presence or absence of sound classes, without knowing when sound events occur. There are 527 sound classes in its released version, covering a wide range of sound classes in the world, such as “Human sounds”, “Animal”, etc. The training set consists of 2,063,839 audio clips, including a balanced subset of 22,160 audio clips. There are at least 50 audio clips for each sound class in the balanced training set.
Due to some audio links are no longer available, we successfully downloaded 1,934,187 (94%) audio clips from the full training set.
All audio clips are padded with silence or truncated into 10 seconds. Due to the fact that a large amount of audio recordings from YouTube have sampling rates lower than 32 kHz, we resample all audio recordings into mono and 32 kHz which will remain the information of most audio clips.
§.§ Training Details
We select anchor segments as described in <ref> and mix two anchor segments to constitute a mixture $ x $. The duration of each anchor segment is 2 seconds. We apply matching energy data augmentation to scale two anchor segments to have the same energy. and We extract the short-time Fourier transform (STFT) feature $ X $ from $ x $ with a Hann window size 1024 and a hop size 320. This hop size leads to 100 frames in a second consistent to the audio tagging systems in PANNs <cit.> and HTS-AT <cit.>.
The query net is a Cnn14 of PANNs or HTS-AT. The query net is pretrained on the AudioSet tagging task <cit.> and the parameters are frozen during the training of the USS system. The prediction and the embedding layers of the query net have dimensions of 527 and 2048, respectively. Either the prediction or the embedding layer is connected to fully connected layers and input to all layers of the source separation branch as FiLMs. We adopt ResUNet <cit.> as the source separation branch. The 30-layer ResUNet consists of 6 encoder and 6 decoder blocks. Each encoder block consists of two convolutional layers with kernel sizes of $ 3 \times 3 $. Following the pre-activation strategy <cit.>, we apply batch normalization <cit.> and leaky ReLU <cit.> before each convolutional layer. The FiLM is added to each convolutional layer as described in (<ref>). The number of output feature maps of the encoder blocks are 32, 64, 128, 256, 512, and 1024, respectively. The decoder blocks are symmetric to the encoder blocks. We apply an Adam optimizer <cit.> with a learning rate of $ 10^{-3} $ to train the system. A batch size of 16 is used to train the USS system. The total training steps is 600 k trained for 3 days on a single Tesla V100 GPU card.
§.§ Conditional Embedding Calculation
For AudioSet source separation, the oracle embedding is calculated by:
\begin{equation} \label{eq:ora_emb}
c = f_{\text{emb}}(s),
\end{equation}
where $ s $ is the clean source. Using oracle embedding as condition indicates the upper bound of the universal source separation system. We calculate the conditional embeddings by (<ref>) from the training set of the AudioSet, FSD50Kaggle2018, FSD50k, Musdb18, Slakkh2100, and VoicebankDemand datasets to evaluate on those datasets, respectively.
Confusion matrix of event based accuracy in Task 1.
Learning curve.
USS results with different conditional embedding types
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
wav2vec (46c) 8.87 4.30 8.95 8.91 4.52 4.70 1.90 7.03 2.96 8.37 -1.08 6.66 2.11 6.02
speaker (46d) 8.87 2.82 6.69 6.65 3.00 3.03 1.52 6.85 2.48 7.94 0.18 7.92 2.13 4.72
Cnn6 (45a2) 8.68 5.30 10.36 10.31 5.25 5.50 3.05 8.43 3.94 9.42 -0.37 7.37 2.27 9.39
Cnn10 (45a3) 8.35 5.36 9.95 9.90 5.19 5.43 2.87 8.10 4.11 9.34 -0.27 7.47 2.27 8.68
+Cnn14 (44a) 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
HTSAT (45c) 9.38 3.78 7.95 7.91 3.38 3.51 2.83 8.48 3.77 9.36 0.81 8.55 2.23 8.78
Segment prediction condition and embedding condition
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
Segment prediction (dim=527) (46b) 7.80 6.42 11.22 11.18 6.60 6.92 2.48 7.26 3.58 8.80 -1.69 6.05 2.20 8.30
+Embedding (dim=2048) 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
§.§ Evaluation Datasets
§.§.§ AudioSet
The evaluation set of AudioSet <cit.> contains 20,317 audio clips with 527 sound classes. We successfully downloaded 18,887 audio clips from the evaluation set (93%) out of 20,317 audio clips. The AudioSet source separation is a challenging problem due to the USS need to separate 527 sound classes using a single model. We are the first to propose using AudioSet <cit.> to evaluate USS. To create evaluation data, similar to Section <ref>, we first apply a sound event detection system to each 10-second audio clip to detect anchor segments. Then, select two anchor segments from different sound classes and sum them as a mixture. We constitute 100 mixtures for each sound class, leading to 52,700 mixtures for all sound classes in total.
§.§.§ FSDKaggle2018
The FSDKaggle2018 <cit.> is a general-purpose audio tagging dataset containing 41 sound classes ranging from musical instruments, human sounds, domestic sounds, and animals, etc. The duration of audio clips ranges from 300 ms to 30 s. Each audio clips contain a unique audio tag. The test set is composed of 1,600 audio clips with manually-verified annotations. We pad or truncate each audio clip into 2-second segments from the start due to sound events usually occur in the start of audio clips. We mix two segments from different sound classes to consist a pair. We constitute 100 mixtures for each sound class. This leads to in total 4,100 evaluation pairs.
§.§.§ FSD50K dataset
The Freesound Dataset 50k (FSD50K) dataset <cit.> contains 51,197 training clips distributed in 200 sound classes from the AudioSet ontology. Different from the FSDKaggle2018 dataset, each audio clip may contain multiple tags with a hierarchy architecture. There are in average 2.79 tags in each audio clip. All audio clips are sourced from Freesound[<https://freesound.org/>]. There are 10,231 audio clips distributed in 195 sound classes in the test set. Audio clips have variable durations between 0.3s to 30s. the average duration of audio clips are 7.1 seconds. We mix two segments from different sound classes to consist a pair. We constitute 100 mixtures for each sound class. This leads to in total 19,500 evaluation pairs.
§.§.§ MUSDB18
The MUSDB18 dataset <cit.> is designed for the music source separation task. The test set of the MUSDB18 dataset contains 50 songs with four types of stems, including vocals, bass, drums, and other. We linearly sum all stems to constitute mixtures as input to the USS system. We use the museval toolkit[https://github.com/sigsep/sigsep-mus-eval] to evaluate the SDR metrics.
§.§.§ Slakh2100
The Slakh2100 dataset <cit.> is a multiple-instrument dataset for music source separation and transcription. The test of the Slakh2100 dataset contains 225 songs. The sound of different instruments are rendered by 167 different types of plugins. We filtered 151 non-silent plugin types for evaluation. Different from the MUSDB18 dataset, there can be over 10 instruments in a song, leading to the Slakh2100 instrument separation a challenging problem.
§.§.§ Voicebank-Demand
The Voicebank-Demand <cit.> dataset is designed for the enhancement task. The Voicebank dataset <cit.> contains clean speech. The Demand dataset <cit.> contains multiple different background sounds that are used to create mixtures. The noisy utterances are created by mixing the VoiceBank dataset and the Demand dataset under under signal-to-noise ratios of 15, 10, 5, and 0 dB. The test set of the Voicebank-Demand dataset contains in total 824 utterances.
§.§ Evaluation Metrics
We use the signal-to-distortion ratio (SDR) <cit.> and SDR improvement (SDRi) <cit.> to evaluate the source separation performance. The SDR is defined as:
\begin{equation} \label{eq:sdr}
\text{SDR}(s, \hat{s}) = 10 \text{log}_{10} \left ( \frac{||s||^2}{||s - \hat{s}||^2} \right ),
\end{equation}
where $ s $ and $ \hat{s} $ are the target source and estimated source, respectively. Larger SDR indicates better separation performance. The SDRi is proposed to evaluate how much SDR a USS system improves compared to without separation:
\begin{equation} \label{eq:sdr}
\text{SDRi} = \text{SDR}(s, \hat{s}) - \text{SDR}(s, x),
\end{equation}
where $ x $ is the mixture signal. For the speech enhancement task, we apply the Perceptual evaluation of speech quality (PESQ) <cit.> and segmental signal-to-ratio noise (SSNR) <cit.> for evaluation.
Freeze, finetune, and adapt conditional embeddings
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
Cnn14 (random weights) (45b) 8.51 2.82 5.96 5.91 2.82 2.82 -0.48 4.59 1.97 7.08 -1.34 6.40 2.28 6.96
Cnn14 (scratch) (47b) 2.38 2.38 2.46 2.41 2.22 2.15 0.71 5.78 1.16 6.30 -1.20 6.54 1.62 -0.28
Cnn14 (finetune) (47b2) 9.83 1.96 3.42 3.38 1.50 1.40 2.10 7.77 3.10 8.52 1.39 9.12 1.77 3.52
+Cnn14 (freeze) (44a) 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
Cnn14 + shortcut. (50b) 6.95 4.57 9.29 9.25 4.74 4.94 1.84 7.05 3.40 8.78 -1.44 6.30 2.06 8.91
Cnn14 + adaptor (49a2) 8.01 5.81 11.00 10.96 5.79 6.07 2.95 7.96 3.90 9.24 -0.87 6.87 2.30 9.60
Universal source separation models
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
open-unmix (55c) 3.96 3.39 3.90 3.86 2.96 2.92 0.40 5.50 2.13 7.42 -1.09 6.65 2.40 2.26
ConvTasNet (56a) 6.96 5.00 9.49 9.45 5.31 5.54 0.61 5.61 2.64 8.10 -2.96 4.77 1.87 6.46
UNet (55a) 8.14 5.50 10.83 10.79 5.49 5.75 2.49 7.78 3.70 9.17 -0.45 7.29 2.12 8.47
ResUNet30 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
ResUNet60 (55b) 7.97 5.70 11.34 11.30 6.04 6.32 1.71 6.81 3.64 9.01 -2.77 4.97 2.40 9.35
§.§ Results Analysis
§.§.§ Conditional Embedding Types
The default configuration of our USS system is a 30-layer ResUNet30 trained on the balanced set of AudioSet. Table <ref> shows the USS system results trained with different conditional embedding types including wav2vec <cit.>, speaker embeddings[https://github.com/RF5/simple-speaker-embedding], Cnn6, Cnn10, Cnn14 from PANNs <cit.>, and HTS-AT <cit.>. The wav2vec embedding is trained using unsupervised contrastive learning on 960 hours of speech data. The wav2vec embeddings are averaged along the time axis to a single embedding with a dimension of 512. The speaker embedding is a gated recurrent unit (GRU) with three recurrent layers operates on log mel-spectrogram and has output has a shape of 256. The Cnn6 and the Cnn10 have dimensions of 512. The Cnn14 and the HTS-AT have dimensions of 2048. The oracle embedding (ora emb) shows the results using (<ref>) as condition. The average embedding (avg emb) shows the results using (<ref>) as condition.
Table <ref> shows that the Cnn6, Cnn10, Cnn14 embeddings achieve AudioSet SDR between 5.30 dB and 5.57 dB using the average embedding, outperforming the wav2vec of 4.30 dB and the speaker embedding of 2.82 dB. One explanation is that both wav2vec and the speaker embeddings are trained on speech data only, so that they are not comparable to PANNs and HTS-AT trained for general audio tagging. The wav2vec embedding slightly outperforms the speaker embedding on FSDKaggle2018, FSD50k, and Musdb18 separation, indicating that the unsupervised learned ASR embeddings are more suitable for universal source separation. The HTS-AT achieves the highest oracle embedding SDR among all systems. All of Cnn6, Cnn10, Cnn14, and HTS-AT outperform the wav2vec embedding and the speaker embedding in AudioSet, FSDKaggle2018, FSD50k, Musdb18, Slakh2100, and Voicebank-Demand datasets by a large margin. The Cnn14 slightly outperforms Cnn6 and Cnn10. In the following experiments, we use Cnn14 as the default conditional embedding.
Table <ref> shows the comparison between using the Cnn14 segment prediction with a dimension of 527 and the Cnn14 embedding condition with a dimension of 2048 to build the USS system. On one hand, the segment prediction embedding achieves an SDR of 7.80 dB on AudioSet, outperforming the embedding condition of 5.57 dB. The segment prediction also achieves higher SDRis than the embedding condition on the FSDKaggle2018, and the FSD50k dataset datasets. An explaination is that the sound classes of all of the AudioSet, FSDKaggle2018, and the FSD50k datasets are sub-classes of the AudioSet. The segment prediction performs better than embedding condition in in-vocabulary sound classes separation. On the other hand, the embedding condition achieves higher SDRs than the segment prediction on the Musdb18 and the Slakh2100 dataset. This result indicates that the embedding condition perform better than segment prediction in new-vocabulary sound classes separation.
Fig. <ref> in the end of this paper shows the classwise SDRi results of AudioSet separation including 527 sound classes. The dashed lines show the SDRi with oracle segment prediction or embedding as conditions. The solid lines show the SDRi with averaged segment prediction or embedding calculated from the anchor segments mined from the balanced training subset. Fig. <ref> shows that sound classes such as busy signal, sine wave, bicycle bell achieve the highest SDRi over 15 dB. We discovered that clear defined sound classes such as instruments can achieve high SDRi scores. Most of sound classes achieve positive SDRi scores. The tendency of using segment prediction and embedding as conditions are the same, although the segment prediction outperform the embedding and vice versa in some sound classes.
§.§.§ Freeze, Finetune, and Adapt Conditional Embeddings
Table <ref> shows the comparison of using random, frozen, finetuned, and adapted conditional embeddings to build the USS system. All the variations of the conditional embeddings extractors are based on the Cnn14 architecture. Using random weights to extract conditional embeddings achieves an SDRi of 2.82 dB on AudioSet, compared to use pretrained Cnn14 to extract conditional embeddings achieves an SDR 5.57 dB. We show that using random weights to extract conditional embeddings work for all USS tasks, such as achieves an SDRi of 5.91 dB on the FSDKaggle2018 dataset compared to the pretrained Cnn14 embedding extractor of 10.57 dB.
Next, we experiment with learning the parameters of the conditional embedding extractor from scratch or finetune the weights from pretrained models. Table <ref> shows that neither the learning from scratch nor the finetuning approach improves the USS system performance. The learning from scratch approach and the finetuning approaches achieve SDRi of 2.41 dB and 3.38 dB on the FSDKaggle2018 dataset, even underperform the random weights of 5.91 dB. One explanation is that the parameters of the conditional embedding branch and the source separation branch are difficult to be jointly optimized when both of them are deep. The training falls to a collapse mode. Using the pretrained frozen Cnn14 system as conditional embedding extractor significantly improves the SDRi to 10.57 dB on the FSDKaggle2018 dataset.
Based on the pretrained frozen Cnn14 conditional embedding extractor, we propose to add a learnable shortcut or add an learnable adaptor on top of the Cnn14 system. The learnable short cut has a Cnn14 architecture with learnable parameters. Table <ref> shows that the learnable shortcut conditional embedding extractor achieves an SDR of 9.29 dB on FSDKaggle2018, less than using the pretrained frozen Cnn14 conditional embedding extractor of 10.57 dB. One explanation is that the learnable shortcut destorys the embedding information for source separation. The adaptor is a 2-layer fully connected neural network on top of the pretrained frozen Cnn14 conditional embedding extractor. With the adaptor, we achieve an SDR of 11.10 dB and outperforms the Cnn14 system. This result indicates that the adaptor is beneficial for the USS task.
USS results trained with different anchor segment durations.
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
0.5 s (52a) 4.07 2.86 4.51 4.47 2.55 2.51 0.97 0.78 2.61 2.43 -0.79 6.95 1.57 5.96
1s (52a2) 7.50 4.99 9.45 9.41 4.81 5.00 0.18 -0.02 2.54 2.50 -1.66 6.08 2.17 8.55
+2s 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
4s (52a3) 7.39 5.21 10.22 10.17 5.38 5.60 1.83 6.79 3.38 8.68 -2.62 5.12 -2.62 5.12
6s (52a4) 6.39 4.68 9.20 9.16 5.05 5.24 0.00 4.98 2.70 7.97 -4.26 3.48 2.21 2.56
8s (52a5) 6.26 4.48 8.85 8.80 4.77 4.94 -3.67 -4.00 1.60 1.50 -5.68 2.06 2.24 2.35
10s (52a6) 6.29 4.47 9.11 9.07 4.80 4.98 -2.68 -2.79 1.56 1.53 -5.07 2.67 2.13 2.14
USS results with different anchor mining strategies
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
mining 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
in-clip random (53b) 4.89 3.94 5.53 5.49 3.63 3.66 1.10 6.05 2.36 7.79 -1.72 6.01 2.21 5.69
out-clip random (53a) 8.19 5.90 11.06 11.01 6.04 6.34 2.57 7.68 3.81 9.17 -1.08 6.66 2.39 9.48
§.§.§ Separation Architectures
Table <ref> shows the results of building USS systems with different source separation backbones. The open-unmix system <cit.> is a 3-layer bidirectional long short term memory (BLSTM) system. The BLSTM is applied on the mixture spectrogram to output the estimated clean spectrogram. The open-unmix system achieves an SDR of 3.39 dB on AudioSet separation and achieves a PESQ of 2.40 on the Voicebank-Demand speech enhancement task, indicating that the BLSTM backbone performs well for speech enhancement. The open-unmix system underperforms other backbone source separation systems in FSDKaggle2018, FSD50k, Musdb18, and Slakh2100 separation, indicating that the capacity of the open-unmix system is not large enough to separate a wide range of sound classes. The ConvTasNet <cit.> is a time-domain source separation system consists of one-dimensional convolutional encoder and decoder layers. The ConvTasNet achieves an SDRi of 5.00 dB on AudioSet separation and outperforms the open-unmix system.
Our proposed UNet30 <cit.> is an encoder-decoder convolutional architecture consists of 30 convolutional layers. The ResUNet30 <cit.> adds residual shortcuts in the encoder and decoder blocks in UNet30. The UNet30 and the ResUNet30 systesm achieve SDRis of 5.50 dB and 5.57 dB on AudioSet, outperforming the ConvTasNet by around 1 dB in all source separation tasks. We extend ResUNet30 to a deeper system ResUNet60 with 60 convolutiona layers. Table <ref> shows that ResUNet60 outperforms ResUNet30 by around 0.5 dB in all USS tasks. This result indicates that deeper architectures are beneficial for USS.
§.§.§ Different Anchor Segment Durations
Table <ref> shows the results of USS systems trained with different anchor segment durations ranging from 0.5 s to 10 s. The anchor segments are mined by a pretrained SED system as described in Section <ref>. On one hand, Table <ref> shows that the separation scores increase with anchor segment durations increase from 0.5 s to 2 s and achieves the best SDRi of 5.57 dB at anchor segments of 2 s on AudioSet separation. This result shows that the anchor segment should be long enough to contain sufficient context information to build the USS system. On the other hand, the separation scores decrease with anchor segment durations decrease from 2 s to 10 s on all tasks. One explanation is that long anchor segments contain undesired interfering sounds that will impair the training of the USS system. Therefore, we use 2-second anchor segments in all other experiments.
§.§.§ Different Anchor Segment Mining Strategies
Table <ref> shows the results of different anchor mining strategies. The in-clip random strategy randomly select two anchor segments from a same 10-second audio clip which significantly underperform the SED mining strategy in all of the source separation tasks. The out-clip random strategy randomly select two anchor segments from two different 10-second audio clips. The out-clip random strategy achieves an SDRi of 5.90 dB on AudioSet, outperforms the SED mining of 5.57 dB. On one hand, the out-clip random strategy also outperforms the SED mining strategy in FSDKaggle2018 and the FSD50k dataset. On the other hand, the SED mining strategy outperforms the out-clip random strategy in Musdb18 and Slakh2100 source separation. Both the out-clip and the SED mining strategies outperform the in-clip random strategy.
USS results with different sources number
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
2 srcs to 1 src 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
3 srcs to 1-2 srcs (54a) 7.37 4.71 8.30 8.26 4.36 4.52 2.43 8.08 3.56 8.69 -0.48 7.26 2.37 8.34
4 srcs to 1-3 srcs (54a2) 7.03 4.38 7.49 7.45 3.99 4.10 2.43 7.99 3.51 8.98 0.70 8.44 2.38 7.78
USS results with different data augmentation
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
no aug (51a) 7.11 3.81 7.19 7.14 3.27 3.35 1.78 7.22 3.09 8.74 0.69 8.43 2.39 6.36
+- 20 dB (51a2) 5.51 3.62 5.77 5.73 2.93 2.94 1.69 7.02 2.51 8.03 -0.34 7.40 2.22 5.34
+match energy 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
Training with balanced and full AudioSet
2cAudioSet (SDRi) 2cFSDK2018 2cFSD50k 4cMusdb18 2cSlakh2100 2cVoicebankDemand
(lr)2-3 (lr)4-5 (lr)6-7(lr)8-11(lr)12-13(lr)14-15
ora. emb avg. emb SDR SDRi SDR SDRi SDR SDRi wSDR wSDRi SDR SDRi PESQ SSNR
+Balanced set, 44a 8.26 5.57 10.61 10.57 5.54 5.79 3.08 8.12 4.02 9.22 -0.46 7.28 2.18 9.00
Full set, 44b 8.21 6.14 10.34 10.30 5.45 5.71 2.31 7.20 3.60 8.92 -1.29 6.45 2.40 9.45
Full set (long train), 44b2 9.60 6.76 13.07 13.03 6.52 6.85 1.77 7.00 3.51 9.00 -2.62 5.12 2.45 10.00
§.§.§ Sources number to mix during training
Table <ref> shows the USS results trained with different number of sources $ J $ to constitute a mixture. Table <ref> shows that $ J=2 $ performs the best on AudioSet with an SDRi of 5.57 dB and also performs the best on the FSDKaggle2018, FSD50k, and on Musdb18 datasets. This result shows that mixing two sources is sufficient for those source separation tasks. By using $ J=4 $ the USS system perform the beston the Slakh2100 dataset. An explanation is that the Slakh2100 contains audio clips contain multiple instruments being played simultaneously. Using more sources to constitute a mixture perform better than using fewer sources.
§.§.§ Data augmentation
Table <ref> shows the USS results with different augmentation strategies applied to sources to create a mixture. First, we do not apply any data augmentation to create a mixture. Second, we randomly scale the volume of each source by $ \pm 20 $ dB. Third, we propose a matching energy data augmentation to scale the volume of sources to create a mixture to ensure the sources have the same energy. <ref> shows that the matching energy data augmentation significantly outperform the systems trained without data augmentation and random volume scale augmentation, with an SDRi of 5.57 dB compared to 3.81 dB and 3.63 dB on AudioSet separation. The matching energy data augmentation also outperform no data augmentation and random volume augmentation on all the other tasks.
§.§.§ USS results Trained with balanced and full AudioSet
Table <ref> shows the results of training the USS systems with the balanced and the full AudioSet, respectively. The full training data is 100 times larger than the balanced data. We also experiment with training the USS system with 4 GPUs and a larger batch size of 64. Fig. <ref> shows the oracle embedding SDRi on the test set of AudioSet. The USS system trained on the full AudioSet outperforms the USS system trained on the balanced set after trained 1 million steps. Fig. <ref> also shows that a large batch sizes of 64 is beneficial to train the USS system. Table <ref> shows that training on the full AudioSet with a batch size of 64 achieves an SDRi of 6.76 dB, outperforming training on the balanced set of 5.57 dB.
§.§.§ Visualization of Hierarchy Separation
One application of the hierarchy separation is to separate arbitrary audio recordings into individual sources with AudioSet ontology. For example, the USS system can separate the sound in a movie into different tracks. One challenge of the hierarchy separation is the number of present sources are unknown. We use the methods in Section <ref> to detect and separate the present sound events.
Fig. <ref> shows the automatically detected and separated waveforms of a movie clip Harry Potter and the Sorcerer's Stone from ontology levels 1 to 3 by using Algorithm <ref>. Level 1 indicates coarse sound classes and level 3 indicates fine sound classes. In level 1, the USS system successfully separate human sounds, music and sounds of things. In level 2, the USS system further separate human group actions, vehicle, and animals. In level 3, the USS system separate fine-grained sound classes such as bell, bird, crowd, and scary music.
Confusion matrix of event based accuracy in Task 1.
§ CONCLUSION
In this paper, we propose universal source separation (USS) systems trained on the large-scale weakly labelled AudioSet. The USS systems can separate hundreds of sound classes using a single model. The separation system can achieve zero-shot separation by using the embedding calculated from query examples as condition. In training, we first apply a sound event detection (SED) system to detect the anchor segments that are most likely to contain sound events. We constitute a mixture by mixing several anchor segments. Then, we use a pretrained audio tagging system to calculate the segment prediction probability or the embedding vector as the condition of the target anchor segment. The USS system takes the mixture and the condition as input to output the desired anchor segment waveform. In inference, we propose both zero-shot separation and hierarchy separation with an AudioSet ontology. We evaluated our proposed USS systems on a wide range of separation tasks, including AudioSet separation, FSDKaggle2018 and FSD50k general sound separation, Musdb18 and Slakh2100 music instruments separation, and Voicebank-Demand speech enhancement without training on those datasets. We show the USS system is an approach to address the computation auditory scene analysis (CASA) problem. In future, we will improve the quality of the separated waveforms of the weakly labelled USS systems.
AudioSet source separation result.
Confusion matrix of event based accuracy in Task 1.
Confusion matrix of event based accuracy in Task 1.
|
# Fully Connected Reconfigurable Intelligent Surface Aided Rate-Splitting
Multiple Access for Multi-User Multi-Antenna Transmission
Tianyu Fang∗†‡ , Yijie Mao∗, Shanpu Shen§, Zhencai Zhu‡, Bruno Clerckx¶
This work was sponsored by Shanghai Sailing Program under Grant 22YF1428400.
∗School of Information Science and Technology, ShanghaiTech University,
Shanghai, China
†University of Chinese Academy of Sciences, Beijing, China
‡Innovation Academy for Microsatellites, Chinese Academy of Sciences,
Shanghai, China
§Department of Electronic and Computer Engineering, The Hong Kong University
of Science and Technology, Hong Kong
¶Department of Electrical and Electronic Engineering, Imperial College London,
United Kingdom
Email: {fangty<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Rate-splitting multiple access (RSMA) has been recognized as a promising and
powerful multiple access (MA) scheme, non-orthogonal transmission framework
and interference management strategy for 6G. Inspired by the appealing
spectral efficiency gain achieved by RSMA over conventional MA schemes in
multi-user multi-antenna transmission, in this paper we introduce RSMA to
reconfigurable intelligent surface (RIS)-aided multiple-input single-out
(MISO) broadcast channel (BC). To further enhance the spectral efficiency, a
more generalized RIS architecture called fully connected RIS is considered. By
jointly optimizing the scattering matrix of the fully connected RIS and the
transmit beamformers to maximize the sum-rate, we show that the proposed fully
connected RIS aided RSMA transmission scheme significantly improves the
spectral efficiency compared with the conventional single connected RIS
schemes and the schemes without RIS. It acts as a new benchmark for linearly
precoded multi-user multi-antenna networks.
###### Index Terms:
fully connected, multiple-user multi-antenna network, rate-splitting multiple
access, reconfigurable intelligent surface, spectral efficiency.
## I Introduction
The past few years have witnessed the development of rate-splitting multiple
access (RSMA) for multi-antenna networks. It has been recognized as a
promising physical-layer non-orthogonal transmission strategy, a powerful
interference management approach, and a candidate of the multiple access
technique for the sixth generation wireless network (6G) [1, 2, 3]. By
splitting the user messages into common and private parts, encoding the common
parts into the common streams to be decoded by multiple users, and encoding
the private parts respectively into the private streams to be decoded by the
corresponding users only, RSMA enables a flexible interference management
capability of partially decoding the interference and partially treating the
interference as noise [4, 5]. Existing works have shown that RSMA outperforms
other multiple access techniques (including linearly precoded space division
multiple access–SDMA, power-domain non-orthogonal multiple access–NOMA,
orthogonal multiple access-OMA, and multicasting) in terms of spectral
efficiency [4, 6, 5, 7, 8], energy efficiency [9, 7], max-min fairness [8,
10], and robustness towards inaccuracies of channel state information at the
transmitter (CSIT) [6, 11].
Reconfigurable intelligent surfaces (RISs), consisting of a large number of
reconfigurable scattering elements, is another promising technology for 6G
[12]. RISs can smartly reconfigurable the wireless propagation environment so
as to effectively enhance the spectral and energy efficiency. Most of existing
RIS research relies on using a simple architecture referred to as single
connected RIS, which is characterized by a diagonal matrix with constant
modulus entries [13, 14, 15, 12]. To enhance the RIS performance, a more
general architecture referred to as fully connected RIS has recently been
proposed in [16]. The fully connected RIS is characterized by a complex
symmetric unitary matrix, which achieves a better performance than the single
connected RIS.
The appealing performance benefits of RSMA and RIS have motivated the study on
the integration of them [17, 18, 19, 20, 21]. The interplay of RSMA and RIS is
first investigated in [17] for multi-user multi-antenna networks, where RIS
aided RSMA transmission model achieves higher energy efficiency than NOMA and
orthogonal frequency division multiple access (OFDMA). RIS aided RSMA
transmission is further shown to enhance the fairness among users [18], reduce
the transmit power [19], and achieve superior outage performance [20, 21] over
conventional RIS-aided networks. However, all the above works only use the
single connected RIS architecture. Therefore, we would like to further enhance
the performance by leveraging the more advanced fully connected RIS
architecture. To the best of our knowledge, there is no existing work that
investigates the sum-rate achieved by the fully connected RIS aided RSMA
transmission networks.
In this work, we propose a fully connected RIS aided RSMA downlink multi-
antenna multi-user transmission model. The scattering matrix of RIS and the
transmit beamformers of RSMA are jointly optimized in order to maximize the
sum-rate. To solve the problem, we propose an alternative optimization
framework where the scattering matrix of RIS and the beamformers of RSMA are
iteratively optimized. Numerical results show that by synergizing RSMA and
fully connected RIS, the proposed scheme significantly improves the spectral
efficiency in multi-user multiple-input single-output (MU-MISO) MU-MISO
networks. The proposed scheme explores a larger achievable sum-rate than the
conventional single connected RIS aided schemes and the schemes without RIS.
It acts as a new benchmark for linearly precoded multi-user multi-antenna
networks.
Notations: Vectors and matrices are denoted by bold lower and upper case
letters. $\mathbb{R}^{m\times n}$ and $\mathbb{C}^{m\times n}$ represent the
real-valued and complex-valued spaces with dimension $m\times n$. $|x|$
indicates the magnitude of a complex number $x$ and $\Re(x)$ denotes its real
part. For a vector $\mathbf{x}$, $\|\mathbf{x}\|$ denotes its Euclidean norm.
$\mathbb{E}\\{\cdot\\}$ denotes the statistical expectation operator for a
random variable. $(\cdot)^{H}$, $(\cdot)^{T}$ and $\mathrm{tr}(\cdot)$
respectively denote the conjugate transpose, transpose and trace operators.
$\mathrm{diag}\left(x_{1},x_{2},\cdots,x_{n}\right)$ is a diagonal matrix with
$(x_{1},x_{2},\cdots,x_{n})$ being its diagonal elements. $\mathbf{I}$ is the
identity matrix. $\mathcal{CN}(\mu,\sigma^{2})$ denotes the circularly
symmetric complex Gaussian (CSCG) distribution with mean $\mu$ and variance
$\sigma^{2}$.
## II System Model and Problem Formulation
We consider a MU-MISO communication network consisting of one base station
(BS) equipped with $M$ antennas, one RIS with a set of $N$ passive reflecting
elements indexed by $\mathcal{N}=\\{1,2,\ldots,N\\}$, and a set of $K$ single-
antenna users indexed by $\mathcal{K}=\\{1,2,\ldots,K\\}$. As illustrated in
Fig. 1, the BS simultaneously serves the $K$ users with the assistance of a
fully-connected RIS (as in Fig. 1 (a)). The reconfigurable impedance network
of RIS is adjusted and determined by a smart controller attached to the RIS,
which also acts as a gateway to exchange the information between the BS and
the RIS. The channels from the BS to the users, from the RIS to the users, and
from the BS to the RIS are denoted as $\mathbf{g}_{k}\in\mathbb{C}^{M\times
1}$, $\mathbf{h}_{k}\in\mathbb{C}^{N\times 1}$, $k\in\mathcal{K}$ and
$\mathbf{G}\in\mathbb{C}^{N\times M}$, respectively. All channels are assumed
to be invariant during one transmission block and perfect CSI is available at
the BS. Although the assumption of perfect CSI is ideal, the proposed scheme
explores a larger achievable sum-rate than the conventional schemes, which
therefore acts as a new benchmark for multi-user multi-antenna networks as
well as future study for the corresponding imperfect CSIT settings. At the BS,
message $W_{k}$ intends to user $k$ is split into a common part $W_{c,k}$ and
a private part $W_{p,k}$. The common parts of all users are combined and
encoded into a common stream $s_{0}$ while the private parts are independently
encoded into the private streams $s_{1},\cdots,s_{K}$. Denote
$\mathbf{s}=[s_{0},s_{1},\cdots,s_{K}]^{T}$ and
$\mathbf{W}=[\mathbf{w}_{0},\mathbf{w}_{1},\cdots,\mathbf{w}_{K}]\in\mathbb{C}^{M\times(K+1)}$
as the data stream vector and beamforming matrix for all streams,
respectively. We assume that each stream $s_{k},$ $k\in\mathcal{K}\cup\\{0\\}$
has zero mean and unit variance, i.e.,
$\mathbb{E}\\{\mathbf{s}\mathbf{s}^{H}\\}=\mathbf{I}$. The transmitted signal
at the BS is
$\mathbf{x}=\sum\limits_{k=0}^{K}\mathbf{w}_{k}s_{k},$ (1)
and the transmit power constraint is
$\mathrm{tr}(\mathbf{WW}^{H})\leq P_{t},$ (2)
where $P_{t}$ refers to the maximum transmit power of the BS. The signal is
transmitted through the direct signal path from the BS to the users as well as
the RIS-aided path. At user $k$, the total received signal is
$y_{k}=(\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G})\sum\limits_{i=0}^{K}\mathbf{w}_{i}s_{i}+z_{k},$
(3)
where $\mathbf{\Theta}\in\mathbb{C}^{N\times N}$ refers to the scattering
matrix of the $N$-port reconfigurable impedance network in the $N$-element
RIS, and $z_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the additive white
Gaussian noise (AWGN).
Figure 1: A multi-antenna multi-user transmission network with the assistance
of a (a) fully connected RIS, (b) single connected RIS.
### II-A Fully Connected Reconfigurable Intelligent Surface
In this work, we focus on using a fully connected RIS [16] to enhance the
spectral efficiency. An example of a 4-element fully connected RIS is
illustrated in Fig. 1 (a). In the reconfigurable impedance network of fully
connected RIS, each port is connected with other ports through a
reconfigurable reactance. Accordingly, the scattering matrix of a fully
connected RIS $\mathbf{\Theta}$ satisfies the constraints
$\displaystyle\mathbf{\Theta}^{H}\bm{\Theta}=\mathbf{I},$ (4a)
$\displaystyle\mathbf{\Theta}=\mathbf{\Theta}^{T}.$ (4b)
As per [16, 22], constraint (4) is equivalent to
$\displaystyle\mathbf{\Theta}$
$\displaystyle=(j\mathbf{X}+Z_{0}\mathbf{I})^{-1}(j\mathbf{X}-Z_{0}\mathbf{I}),$
(5a) $\displaystyle\mathbf{X}$ $\displaystyle=\mathbf{X}^{T},$ (5b)
where $Z_{0}$ refers to the reference impedance and $\mathbf{X}$ is a
symmetric real matrix referring to the reactance matrix of the reconfigurable
impedance network in RIS. With constraint (5), a closed-form expression for
scattering matrix $\mathbf{\Theta}$ satisfying constraint (4) is obtained by
introducing an unconstrained symmetrical real matrix $\mathbf{X}$.
Remark 1. When each port is disconnected with other ports in the
reconfigurable impedance network, the fully connected RIS reduces to the
single connected RIS [23] as illustrated in Fig. 1 (b). The single connected
RIS has been widely used in existing works [17, 18, 20, 19, 21], where the
scattering matrix $\mathbf{\Theta}$ satisfies the constraint
$\bm{\Theta}=\mathrm{diag}\left(e^{j\theta_{1}},e^{j\theta_{2}},\cdots,e^{j\theta_{N}}\right),$
(6)
where $\theta_{n}\in[0,2\pi)$ denotes the phase of the scattering parameter of
the $n$-th port in reconfigurable impedance network. Accordingly, it can be
also equivalently transformed to
$\displaystyle\bm{\Theta}$
$\displaystyle=(j\mathbf{X}+Z_{0}\mathbf{I})^{-1}(j\mathbf{X}-Z_{0}\mathbf{I}),$
(7a) $\displaystyle\mathbf{X}$
$\displaystyle=\mathrm{diag}\left(x_{1},x_{2},\cdots,x_{N}\right),$ (7b)
where $x_{n}\in\mathbb{R}$ is the reconfigurable reactance component connected
to the $n$-th port.
It should be noted that a $N$-port fully connected RIS given in (5) requires
to tune $N(N+1)/2$ scattering parameters while a $N$-port single connected RIS
given in (7) only requires to tune $N$ scattering parameters. The fully
connected RIS therefore brings a larger searching space for the optimal RIS
design.
### II-B Problem Formulation
At user sides, each user first decodes the common stream by treating all
private streams as interference. Thus, the signal-to-interference-plus-noise
ratio (SINR) of $s_{0}$ at user $k$ is
$\gamma_{0,k}=\frac{|(\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G})\mathbf{w}_{0}|^{2}}{\sum\limits_{i=1}^{K}|(\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G})\mathbf{w}_{i}|^{2}+\sigma_{k}^{2}},$
(8)
and the corresponding transmission rate is
$r_{0,k}=\log_{2}\left(1+\gamma_{0,k}\right)$. To ensure common message
$s_{0}$ is successfully decoded by all users, the achievable rate of $s_{0}$
should satisfy $r_{0}=\min_{k\in\mathcal{K}}r_{0,k}$. After decoding the
common stream $s_{0}$, each user employs SIC to remove the common stream from
the received signal, and then decodes the intended private stream with the
SINR
$\gamma_{k}=\frac{|(\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G})\mathbf{w}_{k}|^{2}}{\sum\limits_{i=1,i\neq
k}^{K}|(\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G})\mathbf{w}_{i}|^{2}+\sigma_{k}^{2}}.$
(9)
The rate of decoding private message is
$r_{k}=\log_{2}\left(1+\gamma_{k}\right)$. User $k$ then reconstructs its
message by combining the submessages $W_{c,k}$ and $W_{p,k}$ respectively
decoded from the common and private streams [4].
In this work, we aim at jointly optimizing the scattering matrix of RIS
$\mathbf{\Theta}$ and the beamforming matrix $\mathbf{W}$ of RSMA to maximize
the sum-rate of the system. The sum-rate problem for the downlink fully
connected RIS aided RSMA network can be formulated as:
$\displaystyle(\mathcal{P}_{1})\,\,\max\limits_{\bm{\Theta},\mathbf{W}}\,\,$
$\displaystyle\sum\limits_{k=0}^{K}r_{k}$ (10a) s.t.
$\displaystyle\mathrm{tr}(\mathbf{WW}^{H})\leq P_{t},$ (10b)
$\displaystyle\bm{\Theta}^{H}\bm{\Theta}=\bm{I},$ (10c)
$\displaystyle\bm{\Theta}=\bm{\Theta}^{T}.$ (10d)
Constraint (10b) is the transmit power constraint at the BS. (10c) and (10d)
show that the reconfigurable impedance network in fully connected RIS is a
lossless and reciprocal circuit network.
When constraints (10c) and (10d) for scattering matrix $\mathbf{\Theta}$ are
replaced by constraint (6), $\mathcal{P}_{1}$ reduces to the sum-rate problem
of the single connected RIS aided RSMA [24]. When the power allocated to
$\mathbf{w}_{0}$ is fixed to zero, the problem reduces to the sum-rate problem
for the conventional single connected RIS aided SDMA [14].
## III Alternative Optimization Framework
Problem $\mathcal{P}_{1}$ is a joint beamforming matrix and RIS scattering
matrix optimization problem. It is non-convex and the beamformers are coupled
with the scattering matrix in multiple fractional SINR expressions. Following
existing works [13, 14, 17, 18, 20, 19, 21], we propose an alternative
optimization (AO) framework to solve $\mathcal{P}_{1}$. Specifically, the
problem is first decomposed into the subproblems of beamforming design and
scattering matrix design. The former is solved by a weighted minimum mean
square error (WMMSE)-based approach while the latter is solved by the quasi-
Newton algorithm. The two subproblems are solved iteratively until
convergence. In the following subsections, the proposed optimization algorithm
is delineated.
### III-A Beamforming Optimization
With a given scattering matrix $\bm{\Theta}$, the channel responses from the
RIS to the users are fixed. To ease notations, we denote the effective channel
from the BS and the RIS to user $k$ as
$\widetilde{\mathbf{g}}_{k}^{H}=\mathbf{g}_{k}^{H}+\mathbf{h}_{k}^{H}\mathbf{\Theta}\mathbf{G}.$
(11)
And $\mathcal{P}_{1}$ reduces to
$\displaystyle(\mathcal{P}_{2})\,\,\max\limits_{\mathbf{W}}\,\,$
$\displaystyle\sum\limits_{k=0}^{K}r_{k}$ (12a) s.t.
$\displaystyle\mathrm{tr}(\mathbf{WW}^{H})\leq P_{t},$ (12b)
which can be solved by the WMMSE algorithm [6] as briefly introduced below.
Denote the equalizers to estimate $s_{0}$ and $s_{k}$ as $e_{0,k}$ and
$e_{k}$, respectively. $\hat{s}_{0,k}=e_{0,k}y_{k}$ is the estimate of
$s_{0}$, and
$\hat{s}_{k}=e_{k}(y_{k}-\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{0}\hat{s}_{0,k})$
is the estimate of $s_{k}$ at user $k$. The mean square errors (MSEs) of
decoding $s_{0}$ and $s_{k}$ are calculated as
$\begin{split}\varepsilon_{0,k}&\triangleq\mathbb{E}\\{|\hat{s}_{0,k}-s_{0,k}|^{2}\\}=|e_{0,k}|^{2}T_{0,k}-2\Re\\{e_{0,k}\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{0}\\}+1,\\\
\varepsilon_{k}&\triangleq\mathbb{E}\\{|\hat{s}_{k}-s_{k}|^{2}\\}=|e_{k}|^{2}T_{k}-2\Re\\{e_{k}\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{k}\\}+1,\end{split}$
(13)
where
$T_{0,k}=\textstyle\sum_{i=0}^{K}|\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{i}|^{2}+\sigma_{k}^{2}$,
and
$T_{k}=\textstyle\sum_{i=1}^{K}|\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{i}|^{2}+\sigma_{k}^{2}$
are the average power of the received signal and the signal after removing the
common stream, respectively. By setting $\partial\varepsilon_{0,k}/\partial
e_{0,k}$ and $\partial\varepsilon_{k}/\partial e_{k}$ to zero respectively,
the minimum MSE (MMSE) equalizers are given by
$e_{0,k}^{\text{MMSE}}=\mathbf{w}_{0}^{H}\widetilde{\mathbf{g}}_{k}T_{0,k}^{-1},\,\,e_{k}^{\text{MMSE}}=\mathbf{w}_{k}^{H}\widetilde{\mathbf{g}}_{k}T_{k}^{-1}.$
(14)
Substituting (14) into (13), the MMSEs are given by
$\varepsilon_{0,k}^{\text{MMSE}}=(T_{0,k}-|\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{0}|^{2})T_{0,k}^{-1},\,\,\varepsilon_{k}^{\text{MMSE}}=(T_{k}-|\widetilde{\mathbf{g}}_{k}^{H}\mathbf{w}_{k}|^{2})T_{k}^{-1}.$
Then, the SINRs coresponding to $s_{0}$ and $s_{k}$ can be transformed to
$\gamma_{0,k}=1/\varepsilon_{0,k}^{\text{MMSE}}-1,\,\gamma_{k}=1/\varepsilon_{k}^{\text{MMSE}}-1.$
The rates of common and private streams become
$r_{0,k}=-\log_{2}(\varepsilon_{0,k}^{\text{MMSE}})$ and
$\,\,r_{k}=-\log_{2}(\varepsilon_{k}^{\text{MMSE}}).$ However, the logarithmic
rate-MMSE relationships above cannot be used directly for the sum-rate
problem. To tackle the issue, the augmented MMSEs are introduced as follows
$\xi_{0,k}\triangleq\lambda_{0,k}\varepsilon_{0,k}-\log_{2}(\lambda_{0,k}),\,\,\xi_{k}\triangleq\lambda_{k}\varepsilon_{k}-\log(\lambda_{k}),$
(15)
where $\lambda_{0,k}$ and $\lambda_{k}$ are auxiliary variables (also known as
weights) for the rate-WMMSE relationships of $r_{0,k}$ and $r_{k}$,
respectively. By
calculating$\frac{\partial\xi_{0,k}}{\partial\lambda_{0,k}}=0,\,\frac{\partial\xi_{k}}{\partial\lambda_{k}}=0$,
we obtain the optimum weights given by
$\lambda_{0,k}^{\text{MMSE}}=(\varepsilon_{0,k}^{\text{MMSE}})^{-1},\,\,\lambda_{k}^{\text{MMSE}}=(\varepsilon_{k}^{\text{MMSE}})^{-1}.$
(16)
Substituting (13) and (16) into (15), the rate-WMMSE relationships are
established as
$\xi_{0,k}^{\text{MMSE}}=1-r_{0,k},\,\,\xi_{k}^{\text{MMSE}}=1-r_{k}.$ With
the rate-WMMSE relationships above, $\mathcal{P}_{2}$ is equivalently
transformed into the WMMSE problem
$\displaystyle(\mathcal{P}_{3})\,\min\limits_{\mathbf{W},\bm{\lambda},\mathbf{e}}\,$
$\displaystyle\sum\limits_{k=0}^{K}\xi_{k}$ (17a) s.t.
$\displaystyle\mathrm{tr}(\mathbf{WW}^{H})\leq P_{t},$ (17b)
where
$\bm{\lambda}=[\lambda_{0,1},\cdots,\lambda_{0,K},\lambda_{1},\cdots,\lambda_{K}]^{T}$
is the weight vector and
$\mathbf{e}=[e_{0,1},\cdots,e_{0,K},e_{1},\cdots,e_{K}]^{T}$ is the equalizer
vector. $\xi_{0}=\max_{k\in\mathcal{K}}\xi_{0,k}$. $\mathcal{P}_{3}$ is still
non-convex, an AO framework is applied to decompose it into three convex
subproblems. For each block, one of $\mathbf{W},\bm{\lambda},\bm{e}$ is
optimized by fixing the other two blocks. Algorithm 1 specifies the procedure
of the WMMSE method to optimize the beamforming vectors. Readers are referred
to [25] for the details of the convergence proof for Algorithm 1.
1 Initialize: $t\leftarrow 0$, $\epsilon$, $\mathbf{W}^{[t]}$,
$\mathrm{SR}^{[t]}$;
2 repeat
3 t $\leftarrow$ t+1;
4 update $\mathbf{e}^{[t]},\mathbf{\lambda}^{[t]}$ by (14), (16);
5 update $\mathbf{W}^{[t]}$ by solving problem $\mathcal{P}_{3}$ using
$\mathbf{e}^{[t]},\mathbf{\lambda}^{[t]}$ ;
6
7until _$|\mathrm{SR}^{[t]}-\mathrm{SR}^{[t-1]}| <\epsilon$_;
Algorithm 1 WMMSE algorithm for beamforming design
### III-B Scattering Matrix Optimization
Similarly, with a given beamforming design $\mathbf{W}$, problem
$\mathcal{P}_{1}$ is simplified as
$\displaystyle(\mathcal{P}_{4})\,\,\max\limits_{\bm{\Theta}}\,\,$
$\displaystyle\sum\limits_{k=0}^{K}r_{k}$ (18a) s.t.
$\displaystyle\bm{\Theta}^{H}\bm{\Theta}=\bm{I},$ (18b)
$\displaystyle\bm{\Theta}=\bm{\Theta}^{T}.$ (18c)
However, it is challenging to transform problem $\mathcal{P}_{4}$ into a
convex problem due to the non-convex matrix equality constraints. Hence, we
apply equality (5) to equivalently reformulate $\mathcal{P}_{4}$ as
$\displaystyle(\mathcal{P}_{5})\,\max\limits_{\bm{X}}\,$
$\displaystyle\sum\limits_{k=0}^{K}r_{k}$ (19a) s.t.
$\displaystyle\mathbf{\Theta}=(j\mathbf{X}+Z_{0}\mathbf{I})^{-1}(j\mathbf{X}-Z_{0}\mathbf{I}),$
(19b) $\displaystyle\mathbf{X}=\mathbf{X}^{T}.$ (19c)
Substituting (19b) into (19a) , and removing (19c) (by defining a symmetric
matrix variable), problem $\mathcal{P}_{5}$ becomes an unconstrained
optimization problem. Moreover, the matrix variable $\mathbf{X}$ is a real
symmetric matrix in which $N(N+1)/2$ variables are adjustable. Such
unconstrained optimization problem can be directly solved by the quasi-Newton
method [16].
### III-C Alternative Optimization Algorithm
1 Initilize: $n\leftarrow 0$, $\epsilon$, $\mathbf{W}^{[n]}$,
$\mathbf{X}^{[n]}$, $\mathrm{SR}^{[n]}$;
2 repeat
3 $n\leftarrow n+1$ ;
4 calculate $\mathbf{\Theta}^{[n-1]}$ by (19b);
5 Given $\mathbf{\Theta}^{[n-1]}$, update $\mathbf{W}^{[n]}$ by solving
$\mathcal{P}_{3}$ with Algorithm 1;
6 Given $\mathbf{W}^{[n]}$, update $\mathbf{X}^{[n]}$ by solving
$\mathcal{P}_{5}$ with the quasi-Newton method using $\mathbf{X}^{[n-1]}$ as
the initial point;
7
8until _$|\mathrm{SR}^{[n]}-\mathrm{SR}^{[n-1]}| <\epsilon$_;
Algorithm 2 Alternative Optimization to solve ($\mathcal{P}_{1}$)
The proposed AO algorithm to jointly maximize the scattering matrix and the
beamforming matrix is specified in Algorithm 2. Starting with a feasible
beamforming matrix $\mathbf{W}^{[0]}$ and a symmetric reactance matrix
$\mathbf{X}^{[0]}$, in $n$-th iteration, we first calculate the scattering
matrix $\mathbf{\Theta}^{[n-1]}$ with reactance matrix $\mathbf{X}^{[n-1]}$
from the last iteration. Next, with a fixed scattering matrix
$\mathbf{\Theta}^{[n-1]}$, the beamforming matrix $\mathbf{W}^{[n]}$ is
updated by Algorithm 1. For a given $\mathbf{W}^{[n]}$, the reactance matrix
$\mathbf{X}^{[n]}$ is then updated based on the quasi-Newton method using
$\mathbf{X}^{[n-1]}$ as the initial point. The sum-rate $\mathrm{SR}^{[n]}$ is
then calculated based on the updated $\mathbf{W}^{[n]}$ and
$\mathbf{X}^{[n]}$. The process is repeated until convergence.
Convergence Analysis: In each iteration $[n]$, the solution of
$\mathcal{P}_{1}$ is also a feasible solution of $\mathcal{P}_{1}$ for the
next iteration. Hence, the sum-rate $\mathrm{SR}^{[n+1]}$ is larger than or
equal to $\mathrm{SR}^{[n]}$. Moreover, the non-decreasing sequences generated
by Algorithm 2 is bounded above by the transmit power constraint. Therefore,
the proposed AO algorithm is guaranteed to converge with a given tolerance
$\epsilon$.
## IV Numerical Results
In this section, we evaluate the performance of the proposed system model and
the proposed algorithm. The following six schemes are compared:
* •
Fully RIS RSMA: This is the scheme proposed in Section II.
* •
Fully RIS SDMA: This is a special case of the proposed scheme when the power
allocated to the common stream is fixed to zero.
* •
Single RIS RSMA: This is the single connected RIS aided RSMA scheme, as
studied in [24].
* •
Single RIS SDMA: This is the conventional single connected RIS aided SDMA
scheme, as studied in [14].
* •
no RIS RSMA: This is the conventional RSMA scheme without using RIS, as
studied in [4, 6, 5, 7, 8].
* •
no RIS SDMA: This is the conventional multi-user linearly precoded SDMA scheme
without using RIS, as studied in [4, 26].
The sum-rate maximization problems of fully/single RIS RSMA and fully/single
RIS SDMA are solved by Algorithm 2 while the corresponding problems of no RIS
RSMA and no RIS SDMA are solved by WMMSE directly. Problem $\mathcal{P}_{3}$
is solved by the CVX toolbox [27] with the interior-point method, and problem
$\mathcal{P}_{5}$ is solved by the optimization toolbox in Matlab with the
quasi-Newton method.
The setting of the simulation follows [13], which is a planar RIS-aided
network as shown in Fig. 2. The BS and RIS are located at $(0,0)$ and
$(50,50)$, respectively. In addition, there are $K=4$ users randomly generated
in a circle centered at $(150,0)$ meters with a diameter of 20 meters. The
path loss of the channels are modeled as $P(d)=L_{0}d^{-\alpha}$, where
$L_{0}=-30$ dB is the reference path loss at $d=1$ m, $d$ refers to the link
distance, and $\alpha$ denotes the path loss exponent. Assuming that the
location of RIS is chosen carefully, we set the path loss exponent of BS to
users, BS to RIS, and RIS to users are 3.5, 2, 2.2, respectively [13]. For
simplicity, the small-scale fading of all channels are modeled as Rayleigh
fading. Hence, the channels are given as
${\mathbf{g}}_{k}\sim\mathcal{CN}(0,P(d_{k}^{g})\mathbf{I})),\mathbf{h}_{k}\sim\mathcal{CN}(0,P(d_{k}^{h})\mathbf{I}),$
and $\mathbf{G}\sim\mathcal{CN}(0,P(d^{G})\mathbf{I})$, where
$d_{k}^{g},d_{k}^{h}$ and $d^{G}$ respectively denote the distance between the
BS and user $k$, the distance between the RIS and user $k$, and the distance
between the BS and RIS. Besides, the reference impedance of RIS is $Z_{0}=50$
$\Omega$, the convergence tolerance is $\epsilon=10^{-3}$, and the noise at
user $k$ is $\sigma_{k}^{2}=1$. The transmit singal-to-noise ratio (SNR)
SNR$\triangleq P_{t}/\sigma_{k}^{2}$ is therefore equal to the transmit power
numerically. All simulation results are averaged over 100 random channel
realizations.
Figure 2: The Simulated RIS-aided $K$-user MISO transmission scenario. Figure
3: Sum rate versus the transmit power, when $M=4$, $K=4$, and $N=32$.
Fig. 3 shows the sum-rate of different strategies versus the transmit power
when $M=4$, $K=4$, and $N=32$. It shows that the proposed fully connected RIS
aided RSMA scheme outperforms all other baseline schemes. The relative sum-
rate gain of fully RIS RSMA over single RIS RSMA and no RIS RSMA are at least
$4.6\%$ and $16.5\%$ respectively when SNR is $30$ dB. By using the fully
connected RIS aided RSMA model, the sum-rate of the multi-user multi-antenna
network increases significantly. Moreover, the single connected RIS aided RSMA
achieves approximately the same sum-rate as the fully connected RIS aided SDMA
in the high SNR regime.
Figure 4: Sum rate versus the number of RIS Elements, when $M=4$, $K=4$, and
SNR is $30$ dB.
Fig. 4 shows the impact of the number of passive reflecting elements at the
RIS (i.e., $N$) to the sum-rate of different strategies when $M=4$, $K=4$, and
SNR is $30$ dB. For both fully RIS RSMA and SDMA schemes, the sum-rate
increases faster than the corresponding single RIS RSMA and SDMA as $N$
increases. Particularly, the single RIS RSMA achieves a higher sum-rate than
the fully RIS SDMA when the number of elements is less than $16$. In this
regime, the gain obtained by RSMA scheme is more significant than the gain
obtained by fully connected RIS.
Figure 5: Convergence of the algorithms in one channel realization.
Fig. 5 illustrates the convergence of Algorithm 2 for the fully RIS RSMA
scheme and other baseline schemes (no RIS schemes are not included since they
only adopt Algorithm 1) when $M=4$, $K=4$, $N=32$ and SNR is $25$ dB. It can
be observed that single connected RIS aided schemes converge faster than the
fully connected RIS aided schemes due to the smaller number of variables in
the RIS scattering matrix. In general, the algorithm can converge with 100
iterations.
## V Conclusion
In this work, we propose a fully connected RIS aided RSMA downlink
transmission network. The beamforming vectors at the BS and the scattering
matrix of the fully connected RIS are jointly designed to maximize the sum-
rate of the network. To solve this problem, we propose an effective algorithm
that alternatively optimizes the beamforming and scattering matrices.
Simulation results show the outstanding spectral efficiency of the proposed
fully connected RIS aided RSMA scheme over the existing transmission schemes.
It acts as a new benchmark for linearly precoded multi-user multi-antenna
networks. Moreover, we show that the single connected RIS aided RSMA can
achieve approximately the same spectral efficiency as the fully connected RIS
aided SDMA in the high SNR regime. Therefore, we conclude that by marrying
RSMA and RIS, the spectral efficiency can be enhanced significantly.
## References
* [1] O. Dizdar, Y. Mao, Y. Xu, P. Zhu, and B. Clerckx, “Rate-splitting multiple access for enhanced URLLC and eMBB in 6G,” _Proc. Int. Symp. Wirel. Commun. Syst. (ISWCS)_ , 2021.
* [2] H. Tataria, M. Shafi, A. F. Molisch, M. Dohler, H. Sjoland, and F. Tufvesson, “6G wireless systems: Vision, requirements, challenges, insights, and opportunities,” _Proc. IEEE_ , vol. 109, no. 7, pp. 1166–1199, 2021.
* [3] Y. Mao, O. Dizdar, B. Clerckx, R. Schober, P. Popovski, and H. V. Poor, “Rate-splitting multiple access: Fundamentals, survey, and future research trends,” _arXiv:2201.03192_ , 2021.
* [4] Y. Mao, B. Clerckx, and V. O. Li, “Rate-splitting multiple access for downlink communication systems: Bridging, generalizing, and outperforming SDMA and NOMA,” _Eurasip J. Wireless Commun. Networking_ , 2018.
* [5] B. Clerckx, H. Joudeh, C. Hao, M. Dai, and B. Rassouli, “Rate splitting for MIMO wireless networks: A promising PHY-layer strategy for LTE evolution,” _IEEE Commun. Mag._ , vol. 54, no. 5, pp. 98–105, 2016.
* [6] H. Joudeh and B. Clerckx, “Sum-rate maximization for linearly precoded downlink multiuser MISO systems with partial CSIT: A rate-splitting approach,” _IEEE Trans. Commun._ , vol. 64, no. 11, pp. 4847–4861, 2016\.
* [7] Y. Mao, B. Clerckx, and V. O. K. Li, “Rate-splitting for multi-antenna non-orthogonal unicast and multicast transmission: Spectral and energy efficiency analysis,” _IEEE Trans. Commun._ , vol. 67, no. 12, pp. 8754–8770, 2019.
* [8] B. Clerckx, Y. Mao, R. Schober, E. Jorswieck, D. J. Love, J. Yuan, L. Hanzo, G. Y. Li, E. G. Larsson, and G. Caire, “Is NOMA efficient in multi-antenna networks? A critical look at next generation multiple access techniques,” _IEEE open J. Commun. Soc._ , 2021.
* [9] Y. Mao, B. Clerckx, and V. O. Li, “Energy efficiency of rate-splitting multiple access, and performance benefits over SDMA and NOMA,” in _Proc. Int. Symp. Wirel. Commun. Syst. (ISWCS)_ , 2018, pp. 1–5.
* [10] Y. Mao, B. Clerckx, J. Zhang, V. O. Li, and M. A. Arafah, “Max-min fairness of k-user cooperative rate-splitting in MISO broadcast channel with user relaying,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 10, pp. 6362–6376, 2020.
* [11] Y. Mao and B. Clerckx, “Beyond dirty paper coding for multi-antenna broadcast channel with partial CSIT: A rate-splitting approach,” _IEEE Trans. Commun._ , vol. 68, no. 11, pp. 6775–6791, 2020.
* [12] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: A tutorial,” _IEEE Trans. Commun._ , 2021.
* [13] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5394–5409, 2019.
* [14] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate maximization for reconfigurable intelligent surface aided wireless networks,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 5, pp. 3064–3076, 2020.
* [15] X. Yu, V. Jamali, D. Xu, D. W. K. Ng, and R. Schober, “Smart and reconfigurable wireless communications: From IRS modeling to algorithm design,” _arXiv preprint arXiv:2103.07046_ , 2021.
* [16] S. Shen, B. Clerckx, and R. Murch, “Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,” _IEEE Trans. Wireless Commun._ , pp. 1–1, 2021.
* [17] Z. Yang, J. Shi, Z. Li, M. Chen, W. Xu, and M. Shikh-Bahaei, “Energy efficient rate splitting multiple access (RSMA) with reconfigurable intelligent surface,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2020, pp. 1–6.
* [18] H. Fu, S. Feng, and D. W. Kwan Ng, “Resource allocation design for IRS-aided downlink MU-MISO RSMA systems,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2021, pp. 1–6.
* [19] K. Weinberger, A. A. Ahmad, and A. Sezgin, “On dynergistic benefits of tate dplitting in IRS-assisted cloud radio access networks,” in _IEEE Int. Conf. Commun. (ICC)_ , 2021, pp. 1–6.
* [20] A. Bansal, K. Singh, and C.-P. Li, “Analysis of hierarchical rate splitting for intelligent reflecting surfaces-aided downlink multiuser MISO communications,” _IEEE open J. Commun. Soc._ , vol. 2, pp. 785–798, 2021\.
* [21] A. Bansal, K. Singh, B. Clerckx, C.-P. Li, and M.-S. Alouini, “Rate-splitting multiple access for intelligent reflecting surface aided multi-user communications,” _IEEE Trans. Veh. Technol_ , vol. 70, no. 9, pp. 9217–9229, 2021.
* [22] D. M. Pozar, “Microwave engineering USA: John Wiley & Sons,” 2009.
* [23] N. K. Kundu, Z. Li, J. Rao, S. Shen, M. R. McKay, and R. Murch, “Optimal grouping strategy for reconfigurable intelligent surface assisted wireless Ccommunications,” _arXiv preprint arXiv:2111.10550_ , 2021.
* [24] A. Jolly, S. Biswas, and K. Singh, “An analysis on rate-splitting multiple access for IRS aided 6G communication,” _arXiv preprint arXiv:2106.04418_ , 2021.
* [25] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,” _IEEE Trans. Signal Processing_ , vol. 59, no. 9, pp. 4331–4340, 2011.
* [26] S. S. Christensen, R. Agarwal, E. De Carvalho, and J. M. Cioffi, “Weighted sum-rate maximization using weighted MMSE for MIMO-BC beamforming design,” _IEEE Trans. Wireless Commun._ , vol. 7, no. 12, pp. 4792–4799, 2008.
* [27] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” 2008.
|
# Non-minimal coupling inspires the Dirac cosmological model
H<EMAIL_ADDRESS>H.
<EMAIL_ADDRESS>A. H<EMAIL_ADDRESS>U. K<EMAIL_ADDRESS>1 Research Institute for Astronomy and
Astrophysics of Maragha (RIAAM), University of Maragheh, P.O. Box 55136-553,
Maragheh, Iran
2 Physics Department, Faculty of Sciences, University of Sistan and
Baluchestan, Zahedan, Iran
3 Department of Mathematics, Institute of Applied Sciences and Humanities, GLA
University, Mathura-281406, Uttar Pradesh, India
###### Abstract
In the framework of the generalized Rastall theory (GRT), we study the ability
of a non-minimal coupling between geometry and matter fields in order to
provide a setting which allows for a variable $G$ during the cosmic evolution.
In this regard, the compatibility of this theory with Dirac hypothesis on the
variations of $G$ is investigated, and additionally, the possibility of
obtaining the current accelerated universe is also addressed. In summary, our
study indicates that, in GRT, having in hand the $G$ profile, one may find the
corresponding non-minimal coupling between the energy source and geometry and
vise versa, in a compatible way with the current accelerated universe.
## I Introduction
The idea that $G$ (the Newtonian gravitational coupling) has probably
experienced diverse values during the cosmic evolution has many motivations.
It began with Dirac’s proposal dir1 ; dir2 ; dir3 which states that, the
ubiquitousness of certain large dimensionless numbers (LDN’s), arising in
combinations of physical constants and cosmological quantities WZE was not a
coincidence but an outcome of an underlying relationship between them BTB . In
his proposal, Dirac pointed out that the electrical force between proton and
electron within a hydrogen atom i.e., $F_{e}=e^{2}/4\pi\epsilon_{0}r^{2}$ is a
large number being 40 orders of magnitude greater than their gravitational
force $F_{G}=Gm_{p}m_{e}/r^{2}$, i.e.,
$\displaystyle{\rm
LN}_{1}=\frac{F_{e}}{F_{G}}=\frac{e^{2}}{4\pi\epsilon_{0}Gm_{p}m_{e}}\approx
10^{40},$ (1)
where $m_{e},e,m_{p},\epsilon_{0}$ and $G$ are the mass and charge of
electron, the proton mass, the vacuum permittivity and gravitational constant,
respectively. On the other side, the ratio of the age of the universe and the
time for light to traverse an electron is also nearly of the same size, i.e.,
$\displaystyle{\rm LN}_{2}=\frac{t}{e^{2}/4\pi\epsilon_{0}m_{e}c^{3}}\approx
10^{40}.$ (2)
Dirac then suggested that the above two quantities are equal. As a result of
such a relationship, some of the fundamental constants cannot remain constant
for ever since ${\rm LN}_{2}$ varies with the age of the universe. According
to Dirac’s hypothesis, atomic parameters cannot change with time and thus $G$
should change inversely with time, i.e., $G\propto t^{-1}$ CHK , see also
DIRACREV for recent reviews. Since the advent of this idea, it has led to
interesting implications within theoretical physics, and has attracted a great
deal of attention during the past decades ras2 ; vin ; sab ; bap ; bee ; wu ;
deg ; bar1 ; bar2 ; bar3 ; mans ; gaz ; clif ; bro ; sol ; uza1 ; uza2 ; smo ;
fri ; les . Moreover, it has even interesting power to justify baryogenesis
les , the current and primary accelerated universes ell and can support the
de Sitter spacetime uza1 ; uza2 .
In Newtonian gravity one is allowed to write an explicit time variation of $G$
without the need of satisfying any further constraint. However, the situation
is different in GR as there are further constraints to be satisfied. Consider
the Einstein field equation $G^{\mu}_{\,\nu}=8\pi GT^{\mu}_{\,\,\nu}$ with the
assumption of $G=G(t)$ and $c\equiv 1$. If one takes the covariant divergence
of this equation the left hand side vanishes as a result of Bianchi identity.
Then, if the ordinary energy-momentum conservation law (OCL) is assumed to
hold, i.e., $T^{\mu}_{\,\,\nu;\mu}=0$, one finds that $G$ must be a constant
with respect to spacetime coordinates, i.e., $\partial G/\partial x^{\mu}=0$
always. In this respect, GR does not allow for any variation in the
gravitational coupling $G$ owing to the fact that the Einstein tensor is
divergence free and the divergence of energy-momentum tensor is also zero.
Hence, in the light of Dirac’s proposal, some modifications of GR field
equation are essential. This is because, if we simply let $G$ to be a variable
then the OCL is violated CanutoAdams . In this respect, investigating the
effects of a varying $G$ can be performed only through modified field
equations along with modified conservation laws. From these arguments, one may
intuitively imagine that a varying $G$ could contribute as a new degree of
freedom within the OCL. As in GR, $G$ denotes mutual relation between geometry
and matter fields, hence, variations of $G$ together with the violation of OCL
may be considered as a signal for the idea that another relation between
geometry and matter fields may exist that connects their changes to each
other. However, there are modifications of GR with a varying $G$ that respect
the OCL such as Brans-Dicke theory, in which, the dynamical scalar field
$\phi$ can be considered as the origin of gravitational coupling and thus it
varies as $G\propto\frac{1}{\phi}$ 11 ; 12 ; 13 ; 14 ; bar2 .
OCL, as one of the cornerstones of GR pois , is not respected in all modified
gravity theories, for example, it is broken in the non-minimal curvature
matter coupling theories od1 ; all ; koi ; bert ; hark ; car ; boh ; ras1 ;
mor1 ; mora . Rastall gravity is a pioneering theory in this area ras1 in
accordance with various observations li ; raw1 ; raw2 ; raw3 ; maj ; arb ;
rah1 ; rah2 ; rah3 ; mor2 ; man ; ortiz2020 ; shabooni2020 and its
corresponding cosmology avoids the age and entropy problems arisen in the
framework of the standard cosmology fab . In fact, this theory can even
provide a better platform for describing the matter dominated era compared to
the Einstein theory raw2 . A generalized form of this theory allows us to
relate the current and primary accelerated universe to the ability of the
spacetime to couple with the energy-momentum sources, filling the background,
and in fact, introduces this coupling as a candidate for dark energy and
inflaton field mor1 .
In addition to inflationary models powered by employing varying $G$ theories
ell , there are also other models to describe inflation without considering an
inflaton field jos ; mor1 ; wat ; gam . In Ref. mor1 , it has been shown that
while the existence of an interaction between the geometry and matter fields
may model the primary and current inflationary eras, it does not necessarily
lead to the break-down of OCL. In fact, if geometry has the ability of being
non-minimally coupled with the matter fields, then this ability may support
the primary inflationary era and the current accelerated phase mor1 . To
obtain these results, authors focus on the special case of $T^{\mu\nu}_{\ \
;\mu}=0$, and find out the form of non-minimal coupling in each cosmic era.
The study of various non-minimal couplings can at least make us familiar with
their consequences and properties which may finally lead to a better
understanding of spacetime that helps us provide better predictions about its
behavior and nature. In GRT, cosmological scenarios jos ; mor1 ; das ; lin
imply the power of non-minimal coupling in $i$) providing both singular and
non-singular universes, $ii$) describing the particle production process,
$iii$) avoiding the coincidence problem, and $iv$) playing the role of dark
energy (unlike the original Rastall theory mor1 ; batis ). In this regard,
thermodynamically it has also been shown that the confusion in defining energy
and some of its outcomes which may lead to the OCL generalization (or
equivalently, the breakdown of OCL) could make the cosmos dark mor3 ; mor4 .
Since in Rastall gravity, the gravitational coupling is a constant, but
differs from those of GR and Newtonian gravity (NG) mor5 ; ras1 , Rastall
theory (and indeed a mutual non-minimal coupling between the geometry and
matter fields in the Rastall way) cannot provide a theoretical basis for the
probable variations of $G$ during the cosmic evolution. These points will be
reopened in more details in the next section.
Motivated by the above arguments, it is reasonable to $i$) examine the ability
of non-minimal coupling between geometry and matter fields in producing a non-
constant $G$, and also $ii$) study the results of a non-constant $G$ in the
framework of Rastall theory. The latter is tried to be answered by some
authors in Ref. ref , by combining Rastall and Brans-Dicke theories with each
other. In the present study, the changes in $G$ is not originated by the
Rastall theory meaning that the first part is still unsolved and debateable.
We therefore focus on GRT to describe the compatibility of a non-minimal
coupling with Dirac’s idea on evolution of $G$. Indeed, we are eager to show
that, at least phenomenologically, a non-minimal coupling may itself change
$G$ and play the role of dark energy.
The present work is then arranged as follows. In Sects. II and III, a brief
review on the Rastall theory and its generalization mor1 has been provided,
and some of their predictions about the variations of $G$ are addressed. Sect.
IV includes our survey on the possibility of explaining a well-known Dirac
cosmological model, previously introduced by other authors, within the
framework of GRT. To show the ability of non-minimal coupling in satisfying
Dirac hypothesis and describing the cosmic evolution, simultaneously, a new
model is also introduced in Sect. V. Sect. VI is devoted to concluding
remarks. Here, we use $c=\hbar=1$ units.
## II Rastall theory and a model for varying $G$
Originally, P. Rastall argued that the OCL may not be valid in a curved
spacetime leading to ras1
$\displaystyle T^{\mu\nu}_{\ \ ;\mu}\neq 0,$ (3)
in the non-flat spacetimes. From the mathematical point of view,
$T^{\mu\nu}_{\ \ ;\mu}$ is a ranked one tensor field written as $T^{\mu\nu}_{\
\ ;\mu}=Q^{,\nu}$ where $Q$ is an unknown scalar function found out from other
parts of physics, mathematics and observations ras1 . Since $Q$ is a scalar
and Rastall hypothesis admits the violation of OCL in a curved spacetime
(where Ricci scalar is not always zero), therefore Ricci scalar, R, can be
considered as a suitable suggestion for $Q$, and thus ras1
$\displaystyle T^{\mu\nu}_{\ \ ;\mu}=\lambda^{\prime}R^{;\nu},$ (4)
where $\lambda^{\prime}$ is called the Rastall constant parameter. Using the
Bianchi identity, it is easy to get
$\displaystyle
G_{\mu\nu}+\kappa^{\prime}\lambda^{\prime}g_{\mu\nu}R=\kappa^{\prime}T_{\mu\nu},$
(5)
which $\kappa^{\prime}$ is a constant ras1 called the Rastall gravitational
coupling constant. Applying the Newtonian limit on this result, we obtain ras1
$\displaystyle\frac{\kappa^{\prime}}{4\kappa^{\prime}\lambda^{\prime}-1}\left(3\kappa^{\prime}\lambda^{\prime}-\frac{1}{2}\right)=\kappa_{G},$
(6)
where $\kappa_{G}\equiv 4\pi G$. Hence, since $\kappa^{\prime}$ and
$\lambda^{\prime}$ are constants, $G$ should also be a constant as well (the
current value of $G$, namely $G_{0}$, is proper option leading to
$\kappa_{G}\equiv\kappa_{G_{0}}=4\pi G_{0}$). We therefore conclude that,
since the left hand side of (6) is a constant then a mutual non-minimal
interaction between the geometry and matter fields within the framework of
original version of Rastall gravity does not support the varying $G$ theories.
Eq. (6) also reveals that the Rastall gravitational coupling constant
($\kappa^{\prime}$) differs from that of GR ($2\kappa_{G}=8\pi G$) and only if
$\lambda^{\prime}=0$ then they will be equal.
It is also useful to note that one may use Eq. (5) in order to introduce the
generalized energy-momentum tensor
$\Theta_{\mu\nu}=T_{\mu\nu}-(\kappa^{\prime}\lambda^{\prime})/(4\kappa^{\prime}\lambda^{\prime}-1)Tg_{\mu\nu}$
which finally leads to the GR counterpart form of the Rastall field equations,
given as $G_{\mu\nu}=\kappa^{\prime}\Theta_{\mu\nu}$. In this manner, although
the obtained field equations are similar to those of GR, their solutions for
$T_{\mu\nu}$ differ in general from those of GR mor4 ; dar , a result
confirmed by various observational data, see e.g., li ; mor2 ; dar and
references therein).
One can also generalize the Rastall theory by considering
$\lambda^{\prime}\rightarrow\lambda$, where $\lambda$ is a varying parameter.
Therefore Eq. (4) is extended as follows mor1
$\displaystyle T^{\mu\nu}_{\ \ \ ;\mu}=\left(\lambda R\right)^{;\nu},$ (7)
which finally leads to
$\displaystyle G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu},$ (8)
where $\kappa$ is again a constant but $\lambda$ can change over time. Using
the trace of Eq. (8), one can also rewrite this equation as
$G_{\mu\nu}+\tau Tg_{\mu\nu}=\kappa T_{\mu\nu},$ (9)
in which
$\tau=\frac{\kappa^{2}\lambda}{4\kappa\lambda-1}.$ (10)
Now, since $\kappa$ is constant, the covariant derivative of Eq. (9) leads to
$\tau^{,\nu}T+\tau T^{,\nu}=\kappa T^{\nu\,\,\,;\mu}_{\,\,\,\mu},$ (11)
meaning that even if OCL is respected and until $\tau\neq constant$ (or
equally, $\lambda\neq constant$), the non-minimal coupling affects the
evolution of the energy-momentum source and vice versa mor1 . Therefore,
unlike the Rastall theory, OCL can be met in this framework even in the
presence of non-minimal coupling. In this regard, it is shown that, in the
framework of Eq. (8), even if OCL is met, the accelerated universe can be
explained under the shadow of $\lambda$ without resorting to a dark energy
source mor1 .
Now, considering the Newtonian limit (ignoring the pressure of $T_{\mu\nu}$
and utilizing relation $R_{00}=\nabla^{2}\phi$, in which $\phi$ denotes the
Newtonian potential mor6 ), one can easily find
$\displaystyle\frac{\kappa}{4\kappa\lambda-1}\left(3\kappa\lambda-\frac{1}{2}\right)=\kappa_{G}.$
(12)
Due to the similarity of Eqs. (8) and (5), one could expect that the Newtonian
limit of field equations (8) is obtainable by replacing $\kappa^{\prime}$ and
$\lambda^{\prime}$ with $\kappa$ and $\lambda$, respectively, in Eq. (6). Eq.
(12) also indicates that $G$ (or equally $\kappa_{G}$) does not necessarily
remain constant in this theory. Therefore, this generalization of Rastall
theory provides a basis for theories including a varying $G$ dir1 ; dir2 ; vin
; sab ; bap ; bee ; wu ; mans ; gaz ; bar1 ; bar2 ; bar3 ; clif ; uza1 ; uza2
; smo ; fri ; les . In fact, this equation tells that a non-minimal coupling
between the geometry and matter fields can make $G$ variable mot meaning that
such coupling can be considered as a theoretical basis for varying $G$
theories.
## III Newtonian limit, a model for running $G$, and the value of $\kappa$
Now, using Eq. (12), and following Ref. mor1 , in which
$\kappa\lambda\equiv\beta=[4+\theta(1+z)^{3}]^{-1}$, where $\theta$ is an
unknown constant and $z$ denotes the redshift, one can obtain
$\displaystyle\kappa_{G}=\frac{\kappa}{2}\left[1-\frac{2}{\theta(1+z)^{3}}\right],$
(13)
finally leading to
$\displaystyle\kappa_{G}=\frac{\kappa}{2}\left[1-\frac{2}{\theta}\right]\equiv\kappa_{G_{0}},$
(14)
and
$\displaystyle\kappa_{G}=\frac{\kappa}{2},$ (15)
for $z\rightarrow 0$ and $z\rightarrow\infty$, respectively. Based on Ref.
mor1 , whenever $0<\theta\leq 1/2$ (leading to $\beta>0$), the current
accelerated universe is explainable in the presence of OCL, and without
considering a dark energy-like source. Moreover, expression
$\beta=[4+\theta(1+z)^{3}]^{-1}$ is present in both of the matter dominated
era (MDE) and the current accelerated universe mor1 . Hence, Eq. (15) can be
considered as the value of $G$ at the beginning of MDE whereas the value of
$\kappa$ is obtainable by using Eq. (14)
$\displaystyle\kappa=\frac{8\pi G_{0}}{1-\frac{2}{\theta}},$ (16)
combined with Eq. (15) to see that $\kappa$, and thus $\kappa_{G}$ are
negative at the beginning of MDE. Therefore, in the model proposed in Ref.
mor1 which still respects OCL in the framework of (8), $G$ is not always
positive during the cosmic evolution. Negative values of $\kappa$ provide a
setting for baryonic matters to support traversable wormholes in the Rastall
framework mor5 . Moreover, in the framework of GRT, it has been shown that
negative values of $\kappa$ could have their own effects on matter
perturbations and formation of structures in large scale universe AHH2020 . In
this regard, overdense and underdense regions in the universe could form
periodically so that both large scale structures and voids could form as the
universe evolves from MDE to present time. Also, emergence of structures in a
class of alternative theories of gravity has been reported in Lohiya1996 ,
where the authors considered a non-minimally coupled scalar field in addition
to an induced negative gravitational constant and studied structure formation
with repulsive gravitation on the large scale. In the framework of general
scalar tensor-theories, a cosmological mechanism has been proposed in which it
is possible for $G$ to change sign from a positive branch (attracting) to a
negative branch (repulsive gravity) and vice versa Nunez2019 . It is also
worth mentioning that negative values of $G$ have previously been reported in
some other approaches studying the variations of $G$ bar2 ; uza1 ; uza2 .
Beside the effects of repulsive gravity (represented by a universal negative
coupling) on the evolution of perturbations and formation of structures, the
study of possible consequences of $\kappa<0$ on the stability of the model is
of particular importance. In this regard, from the viewpoint of perturbative
analysis, the existence of a repulsive gravity phase in the evolution of the
universe could lead to growing models with respect to scalar perturbations
producing then, large inhomogeneities. Hence a repulsive phase may destroy
homogeneity and in this sense it may be unstable Batista2001 . In Star1981 ,
it has been discussed that a transition from positive gravitational coupling
$G$ to negative one results in an instability, in such a way that, small
deviations from isotropy and homogeneity within the gravitational field will
grow unboundedly, leading to a true cosmological singularity at the boundary
between gravity and anti gravity. Also, investigating classical stability of
the model through dynamical system approach is of long-standing interest and
significance. Work along this line has been carried out for a class of GRT
models Lin2020 , where the authors have shown that the eventual fate of the
universe ends in late time attractors which are classically stable. However,
investigating these issues for the present model needs a deeper analysis with
more scrutiny and future studies will be reported elsewhere. Finally, we note
that, since $\dot{G}$ does not decrease with time for $0<\theta\leq 1/2$
($\dot{G}>0$ in this manner), this model does not respect the Dirac’s
hypothesis claiming that $G$ should decrease as a function of time dir1 ; dir2
; vin ; bap . Hence, more comprehensive non-minimal couplings are needed to
provide settings for Dirac hypothesis and also to model the cosmic evolution
without considering a mysterious fluid (dark energy), simultaneously.
### III.1 Another possibility
In Ref. das , choosing $\lambda=(1+d_{0}H)/[3\kappa(w+1)]$, in which $w\equiv
p/\rho$ (where $p$ and $\rho$ denote the pressure and energy density of the
cosmic fluid, respectively), it has been shown that non-singular cosmic
evolution is obtainable in GRT. In this case $d_{0}$ is a free parameter, and
some outcomes of this proposal in various cosmic eras have also been studied
in Ref. das . Accepting this proposal along with considering the unit
$\kappa=8\pi G_{0}$ and also assuming $G(H_{0})=G_{0}$ (which helps us in
finding $d_{0}$), one easily reaches
$\displaystyle G(H)=G_{0}\frac{3(1-w)H_{0}-6H}{(1-3w)H_{0}-4H},$ (17)
where $H_{0}$ is the current value of $H$ and use has been made of Eq. (12).
## IV Dirac cosmological model
As in the present model there is no evolution equation for the variation of
$G$ which is promoted as a dynamical field, one then has to impose a suitable
ansatz on the behavior of this parameter. Based on Dirac hypothesis, $G$
should decrease with time, i.e, $G\propto t^{-1}$ CHK . In general, one may
consider $G=G_{0}f$, in which $f$ is a decreasing function of time dir1 ; dir2
; vin ; bap ; clif ), in order to preserve Dirac hypothesis. Now, combining
Eq. (12) with $\kappa=8\pi G_{0}\alpha$ raw2 , along with Eqs. (7) and (8) for
a flat FLRW universe, one finds
$\displaystyle\gamma\equiv\lambda\kappa=\frac{f-\alpha}{4f-6\alpha},$ (18)
$\displaystyle
3\int(\rho+p)\frac{da}{a}=\frac{1}{2\alpha}\Big{[}(f-3\alpha)\rho-3(f-\alpha)p\Big{]},$
$\displaystyle H^{2}=\frac{1}{6}\Big{[}(3\alpha-f)\rho+3(f-\alpha)p\Big{]},$
$\displaystyle
q=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{3\alpha(\rho+p)}{\rho(3\alpha-f)+3(f-\alpha)p},$
whenever a fluid with energy density $\rho$ and pressure $p$ fills the
background. We note that $\gamma$ is a varying parameter and, $q$ and $a$
denote deceleration parameter and scale factor, respectively, and we also have
assumed $8\pi G_{0}=1$.
Figure 1: The evolution of $q$ and state parameter $w$ versus $z$ for
$H(z=0)=67$ dom . Upper panels are provided for the case (i) and the lower
ones are depicted for case (ii) discussed in Sect. IV. The model parameters
used to draw the curves of $w$ are the same as those of $q$ diagrams.
The case with $f=a^{-n}$ leads to a decreasing function of time whenever $n>0$
gaz ; smo . In this manner, assuming $w\equiv p/\rho=0$, together with using
Eqs. (18), one easily finds $q=(3\alpha-1)^{-1}$, and
$\rho=\rho_{0}a^{n}(1-3\alpha a)^{-(n+2)/n}$, where $\rho_{0}$ is the
integration constant. These results indicate that, at limit $a\rightarrow 1$,
the obtained pressureless fluid can accelerate the universe expansion with
$q\leq-1/2$ for $-1/3\leq\alpha<1/3$. Consequently, the non-minimal coupling
$\gamma=[(1+z)^{n}-\alpha]/[4(1+z)^{n}-6\alpha]$ allows $G$ to vary as
$G=G_{0}(1+z)^{n}$ gaz , where we used the $1+z=1/a$ relation. It is also easy
to see that the universe described by this model has begun from a primary
inflationary phase ($q=-1$) corresponding to the $a\rightarrow 0$ point. In
fact, in this limit, we also have $\gamma=1/4$, a value that supports an
inflationary phase for even an empty universe mor1 .
Now, let us consider two more comprehensive cases i.e., $i$)
$p=k\rho^{1+1/m}$, where $m$ and $k$ are unknown constants to be evaluated
later, and $ii$) $p=\sigma\rho/(a-b\rho)-c\rho^{2}$ in which $\sigma$, $a$,
$b$ and $c$ are unknown coefficients. In this manner, as it is obvious from
Fig. 1, a proper behavior is obtainable for the cosmos. Here, $w\equiv p/\rho$
denotes the equation of state of cosmic fluids. Depending on the values of
unknown parameters, the universe can also experience a transition at $z_{t}$
which can even take values smaller than $1$. Clearly, both fluids behave as
dark energy sources, and the corresponding non-minimal coupling can not be
considered as a dark energy source.
## V A new proposal for $\lambda$ parameter
Now, let us consider a flat FRW universe filled by a pressureless fluid with
energy density $\rho$ when $\lambda R=\zeta H^{n}$ in which $\zeta$ and $n$
are unknown constants. In this manner, the $\lambda$ parameter takes the form
$\displaystyle\lambda=\zeta\frac{H^{n}}{R}=\frac{\zeta}{6}\frac{H^{n}}{\dot{H}+2H^{2}},$
(19)
whence, the corresponding Friedmann equations read
$\displaystyle H^{2}-\frac{\kappa\zeta}{3}H^{n}=\frac{\kappa}{3}\rho,$
$\displaystyle H^{2}+\frac{2}{3}\dot{H}-\frac{\kappa\zeta}{3}H^{n}=0.$ (20)
Defining $\Omega=8\pi G\rho/3H^{2}$, while $\Omega_{0}$ denotes its current
value macq , the evolution of $q$ and $G/G_{0}$ have been plotted in Fig. (2).
For the employed parameters, transition redshift ($z_{t}$) lies within the
range of $0.4\leq z_{t}\leq 0.88$. The sensitivity of diagrams to the values
of $\Omega_{0}$ and $H_{0}$ is so weak compared with those of $\zeta$ and $n$
and $\kappa$. Indeed, although we only consider a baryonic source for current
density parameter $\Omega_{0}=0.049$ macq , and $H_{0}=67.66$ agh , the
obtained behaviors are also achievable for other candidates of $\Omega$ (such
as dark matter) and also the other values of $H_{0}$, reported in the
literature. Hence, suitable behavior of $q$ is obtainable by only considering
the baryonic content of the universe, meaning that the $\zeta H^{n}$ term may
play the role of the unknown parts (dark components) of cosmos. Dirac
hypothesis is also respected during the cosmic evolution. Remarkably, $G$ will
take negative values in future meaning that gravity will become repulsive
which speeds the universe expansion rate up more i.e., $q$ decreases. All
these happen under the shadow of the existence of non-minimal coupling
$\lambda$ which varies during the evolution of the universe. In Fig. (3),
$H(z)$ far and the distance modulus ama are plotted for the $\Lambda$CDM
model and also our model.
Figure 2: The evolution of $q$ and $G/G_{0}$ assuming $w=0$, for the case
discussed in Sec.V. The diagrams for $G/G_{0}$ are plotted using the same
model parameters as of $q$ diagrams.
Figure 3: The evolution of $H(z)$ and $\mu(z)$ whenever $w=0$, for the case
discussed in Sec.V. The same values of parameters as of Fig. 2 have been used.
The black dashed lines show $H(z)$ and $\mu(z)$ for $\Lambda$CDM model.
The negative value of $G$ is the direct result of the assumed $\lambda$, and
changes in the values of model parameters do not affect this result. There are
also other works that predict negative values for $G$ bar2 ; uza1 ; uza2 .
Theoretically, our model shows that a non-minimal coupling between geometry
and matter fields can accelerate the universe expansion and has an ability to
satisfy Dirac hypothesis.
## VI concluding remarks
After addressing some properties of previously cosmological models introduced
in the framework of GRT mor1 ; das , the implications of GRT on obtaining
varying $G$ has been studied through considering the Newtonian limit of the
field equations. Thereinafter, following a proposal of Dirac hypothesis
introduced in gaz ; smo , the required non-minimal coupling needed to support
Dirac model was also obtained. Our results show that the dark sectors of
cosmos can be unified into one cosmic fluid which behaves as a pressureless
fluid in high redshift limit, and also accelerates the universe in line with
the current observations (Fig. 1). We also proposed a non-minimal coupling
(Sec. V) which can play the role of dark side of cosmos satisfying Dirac
hypothesis. Indeed, the present study addresses a deep connection between non-
minimal coupling (between the matter fields and geometry) and the idea of
variable $G$. This translates into saying that one may find the footprints of
non-minimal coupling between the matter fields and geometry by having the
observationally confirmed profile of $G$ and conversely.
Although relying on the Rastall hypothesis on relation between the changes in
spacetime curvature and violation of OCL we only focused on the implications
of the violation of OCL in cosmology and its connection with Dirac hypothesis,
the OCL violation can also be allowed due to the quantum considerations such
as uncertainty principle, and in the framework of unimodular gravity producing
significant cosmological outcomes jos . Indeed, even in the framework of GR
and thanks to the Bianchi identity, OCL is violated as the result of the
existence of a non-constant $G$. In summary, it was our goal to address $i$)
probable connection between Dirac hypothesis and non-minimal couplings, and
simultaneously, $ii$) the ability of such couplings in being responsible for
the unknown parts (dark sides) of cosmos. Therefore, such couplings need to be
further studied from both of the theoretical and observational viewpoints.
Finally, we would like to mention that, though Rastall gravity and its
generalizations provide interesting results, cosmological models based on this
theory need to be accurately tested by observations. In the present model, we
tried to explore theoretical consequences of a varying G cosmology based on
GRT and also briefly examined observational aspects of the theory. However, a
full observational treatment of the present model, e.g., in light of
Akarsu2020 , needs to be done and work along this line can be considered as an
interesting subject for future studies and developments.
## References
* (1) Dirac P. A. M., Nature 139 (1937), 323
* (2) Dirac P. A. M., Proc. Roy. Soc. London, Ser. A 165 (1938), 199.
* (3) Dirac P. A. M., Nature, 139, (1937), 1001.
* (4) Weyl H. Ann. Phys., 59, 129 (1919);
Zwicky F., Phys. Rev., 55, 726 (1939);
Eddington A. S., The Mathematical Theory of Relativity, Cambridge University
Press, London (1923).
* (5) Barrow J. D., Tipler F. J., The Anthropic Cosmological Principle, Oxford University Press, Oxford, (1986);
J. D. Barrow, Varying G and Other Constants, In: S$\acute{a}$nchez N.,
Zichichi A. (eds) Current Topics in Astrofundamental Physics: Primordial
Cosmology. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol
511. Springer, Dordrecht (1998).
* (6) Chandrasekhar S., Nature 139 (1937), 757;
Kothari D. S., Nature, 142 (1938), 354.
* (7) S. Ray, U. Mukhopadhyay, S. Ray, A. Bhattacharjee, Int. Journal Mod. Phys. D 28 (2019), 1930014.
* (8) Rastall P., Can. J. Phys. 54 (1976), 66
* (9) Vinti J. P., Celestial Mechanics 16 (1977), 391
* (10) De Sabbata V., Acta Cosmologica Zesz. 9 (1980), 63
* (11) Baptista J. P., Batista A. B., Fabris J. C., Revista Brasileira de Fisica. 14 (1984), 208
* (12) Beesham A., Int. J. Theo. Phys. 25 (1986), 1295
* (13) Wu Y. S., Wang Z., Phys. Rev. Lett. 57 (1986), 16
* (14) Degl’Innocenti S. et al., A&A 312 (1996), 345
* (15) Barrow J. D., Mon. Not. R. Astron. Soc. 282 (1996), 1397
* (16) Barrow J. D., 1997, arXiv:gr-qc/9711084.
* (17) Barrow J. D., The Constants of Nature, (Vintage Books, London, 2002)
* (18) Mansouri R., Nasseri F., Khorrami M., Phys. Lett. A 259 (1999), 194
* (19) Gaztañaga E. et al., Phys. Rev. D 65 (2001), 023506
* (20) Clifton T., Mota D., Barrow J. D, Mon. Not. R. Astron. Soc. 358 (2005), 601
* (21) Bronnikov K. A., Kononogov S. A., Metrologia 43 (2006), 1
* (22) Solà J., J. Phys. A: Math. Theor. 41 (2008), 164066
* (23) Uzan J. P., Rev. Mod. Phys. 75 (2003), 403
* (24) Uzan J. P., Liv. Rev. Relativ. 14 (2011), 2
* (25) Smolin L., Class. Quantum Grav. 33 (2016), 025011
* (26) Fritzsch H., Solà J., Nunes R. C., Eur. Phys. J. C 77 (2017), 193
* (27) Leszczyńska K., Da̧browski M. P., Denkiewicz T., Eur. Phys. J. C 79 (2019), 222
* (28) Ellis G. F. R., Maartens R., Maccallum M. A. H., The Constants of Nature, (Cambridge University Press, UK, 2012).
* (29) Canuto, V., Adams, P. J., Hsieh, S. H., Tsiang, E., Phy. Rev. D 16, 6 (1977);
Wesson, P., Goodson, R. E., Observ. 101, 105 (1981).
* (30) C. Brans, R. H. Dicke, Phys. Rev. 124, 925 (1961).
* (31) R. H. Dicke, Phys. Rev. 125, 2163 (1962).
* (32) R. H. Dicke, Rev. Mod. Phys. 29, 355 (1957).
* (33) R. H. Dicke, Nature 192, 440 (1961).
* (34) Poisson E., A Relativist’s Toolkit, (Cambridge University Press, UK, 2004).
* (35) Nojiri S., Odintsov S. D., Phys. Lett. B 599 (2004), 137
* (36) Allemandi G. et al., Phys. Rev. D 72 (2005), 063505
* (37) Koivisto T.,Class. Quant. Grav. 23 (2006), 4289
* (38) Bertolami O. et al., Phys. Rev. D 75 (2007), 104016.
* (39) Harko T., Lobo F. S. N., Galaxies 2 (2014), 410.
* (40) Carloni S., Phys. Lett. B 766 (2017), 55.
* (41) Boehmer C. G., Carloni S.,Phys. Rev. D 98 (2018), 024054.
* (42) Rastall P., Phys. Rev. D 6 (1972), 3357.
* (43) Moradpour H. et al., The European Physical Journal C, 77 (2017), 259.
* (44) De Moraes W. A. G., Santos A. F., Gen. Relativ. Gravit. 51 (2019), 167.
* (45) Li R. et al., Mon. Not. R. Astron. Soc. 486 (2019), 2407.
* (46) Al-Rawaf A. S., Taha O. M., Phys. Lett. B 366 (1996), 69.
* (47) Al-Rawaf A. S., Taha O. M., Gen. Relat. Gravit. 28 (1996), 935.
* (48) Al-Rawaf A. S., Int. J. Mod. Phys. D 14 (2005), 1941.
* (49) Majernik V., Gen. Relat. Gravit. 35 (2003), 1007.
* (50) Arbab A. I., J. Cosmol. Astropart. Phys. 05 (2003), 008.
* (51) Abdel-Rahman A. M. M., Astrophys. Space. Sci. 278 (2001), 383.
* (52) Abdel-Rahman A. M. M., Hashim M. H. A., Astrophys. Space. Sci. 298 (2005), 519.
* (53) Abdel-Rahman A. M. M., Riad I. F., Astron. J. 134 (2007), 1931.
* (54) Moradpour H. et al., Phys. Rev. D. 96 (2017), 123504.
* (55) Manna T., Rahaman F., Mondal M., Mod. Phys. Lett. A 35 (2020), 2050034.
* (56) S. K. Maurya and F. T.-Ortiz, Phys. Dark Univ. 29 (2020), 100577.
* (57) H. Shabani and A. H. Ziaie, Europhysics Letters 129, (2020) 20004.
* (58) Fabris J. C., Kerner R., Tossa J., Int. J. Mod. Phys. D 9 (2000), 111.
* (59) Josset T., Perez A., Phys. Rev. Lett. 118 118 (2017), 021102.
* (60) Watson S. et al., J. Cosmol. Astropart. Phys. 07 (2017), 11.
* (61) Gamboa J. et al., Phys. Rev. D 96 (2017), 083534.
* (62) Das D., Dutta S., Chakraborty S., Eur. Phys. J. C 78 (2018), 810.
* (63) Lin K., Qian W. L., Eur. Phys. J. C 80 (2020), 561.
* (64) C. E. M. Batista, M. H. Daouda, J. C. Fabris, O. F. Piattella, D. C. Rodrigues, Phys. Rev. D 85, (2012), 084008.
* (65) Moradpour H. et al., Mod. Phys. Lett. A 32 (2017), 1750078
* (66) Moradpour H. et al., Adv. High Energy Phys. 2018 (2018), 7124730
* (67) Moradpour H., Sadeghnezhad N., Hendi S. H., Can. J. Phys. 95 (2017), 1257
* (68) T. R. P. Carames. et al., Eur. Phys. J. C74 (2014) 3145.
* (69) Darabi F. et al., Eur. Phys. J. C 78 (2018), 25
* (70) Moradpour H. et al., Mod. Phys. Lett. A 33 (2019), 1950096
* (71) Mota C.E., et al., arXiv:2007.01968.
* (72) A. H. Ziaie, H. Moradpour, H. Shabani, Eur. Phys. J. Plus 135 (2020), 916.
* (73) D. Lohiya, A. Batra, S. Mehra, S. Mahajan and A. Mukherjee, Astron. Astrophys. Trans. 14 (1997), 199.
* (74) I. Ayuso, J. P. Mimoso and N. J. Nunes, Galaxies, 7 (2019), 38.
* (75) A. B. Batista, J. C. Fabris and S. V. B. Goncalves, Class. Quant. Grav. 18 (2001), 1389.
* (76) A. A. Starobinskij, Pisma v Astronomicheskii Zhurnal, 7, (1981), 67; Soviet Astronomy Letters, 7, (1981), 36, Translation.
* (77) K. Lin and W.-L. Qian, Eur. Phys. J. C 80 (2020), 561.
* (78) Domínguez A. et al., Astrophys. J. 885 (2019), 137.
* (79) Macquart J. et al., Nature 581 (2020), 391.
* (80) Farooq O. et al., Astrophys. J. 835 (2017), 26.
* (81) Aghanim N. et al., A&A 641 (2020), A6.
* (82) Amanullah et al., Astrophys. J. 716 (2010), 712.
* (83) O. Akarsu, N. Katirci, S. Kumar, R. C. Nunes, B. Ozturk and S. Sharma, Eur. Phys. J. C 80 (2020), 1050.
|
[1]Otto Brookes [1]Majid Mirmehdi [2]Mimi Arandjelovic [2,9]Hjalmar Kühl
[1]Tilo Burghardt
[1]Department of Computer Science, University of Bristol, United Kingdom
2]Max Planck Institute for Evolutionary Anthropology, Leipzig, Germany
3]Wild Chimpanzee Foundation, Leipzig, Germany
4]Department of Human Evolutionary Biology, Harvard University, Massachusetts,
USA
5]School of Psychology & Neuroscience, University of St Andrews, St. Andrews,
Scotland
6]Faculty of Artes Liberales, University of Warsaw, Warsaw, Poland
7]Institute for Cognitive Sciences Marc Jeannerod, University of Lyon, Lyon,
France
8]Institute of Human Origins, Arizona State University, Arizona, USA
9]Senckenberg Museum of Natural History Goerlitz, Goerlitz, Germany
# PanAf20K: A Large Video Dataset for Wild Ape Detection and Behaviour
Recognition
<EMAIL_ADDRESS><EMAIL_ADDRESS>Colleen Stephens Samuel
Angedakin Katherine Corogenes Dervla Dowd Paula Dieguez Thurston C. Hicks
Sorrel Jones Kevin Lee Vera Leinert Juan Lapuente Maureen S. McCarthy
Amelia Meier Mizuki Murai Emmanuelle Normand Virginie Vergnes Erin G.
Wessling Roman M. Wittig Kevin Langergraber Nuria Maldonado Xinyu Yang
Klaus Zuberbühler Christophe Boesch<EMAIL_ADDRESS><EMAIL_ADDRESS>* [ [ [ [ [ [ [ [
###### Abstract
We present the PanAf20K dataset, the largest and most diverse open-access
annotated video dataset of great apes in their natural environment. It
comprises more than 7 million frames across $\sim$20,000 camera trap videos of
chimpanzees and gorillas collected at 14 field sites in tropical Africa as
part of the Pan African Programme: The Cultured Chimpanzee. The footage is
accompanied by a rich set of annotations and benchmarks making it suitable for
training and testing a variety of challenging and ecologically important
computer vision tasks including ape detection and behaviour recognition.
Furthering AI analysis of camera trap information is critical given the
International Union for Conservation of Nature now lists all species in the
great ape family as either Endangered or Critically Endangered. We hope the
dataset can form a solid basis for engagement of the AI community to improve
performance, efficiency, and result interpretation in order to support
assessments of great ape presence, abundance, distribution, and behaviour and
thereby aid conservation efforts. The dataset and code are available from the
project website: PanAf20K
###### keywords:
animal biometrics, video dataset, behaviour recognition, wildlife, imageomics,
conservation technology
## 1 Introduction
Figure 1: PanAf20K Visual Overview. We present the largest and most diverse
open-access video dataset of great apes in the wild. It comprises $\sim$20,000
videos and more than 7 million frames extracted from camera traps at 14 study
sites spanning 6 African countries. Shown are 25 representative still frames
from the dataset highlighting its diversity with respect to many important
aspects such as behavioural activities, species, number of apes, habitat,
day/night recordings, scene lighting, and more.
Motivation. As the biodiversity crisis intensifies, the survival of many
endangered species grows increasingly precarious, evidenced by species
diversity continuing to fall at an unprecedented rate (Vié et al, 2009,
Ceballos et al, 2020). The great ape family, whose survival is threatened by
habitat degradation and fragmentation, climate change, hunting and disease, is
a prime example (Carvalho et al, 2021). The International Union for
Conservation of Nature (IUCN) considers all three member species, that is
orangutans, gorillas, chimpanzees (including bonobos), to be either endangered
or critically endangered.
The threat to great apes has far-reaching ecological implications. Great apes
contribute to the balance of healthy ecosystems by seed dispersal, consumption
of leaves and bark, and shaping habitats by creating canopy gaps and trails
(Haurez et al, 2015, Tarszisz et al, 2018, Chappell and Thorpe, 2022). They
also form part of complex forest food webs, their removal from which would
have cascading consequences for local food chains. In addition, great apes are
our closest evolutionary relatives and a key target for anthropological
research. We share 97% of our DNA with the phylogenetically most distant
orangutans and 98.8% with the closer chimpanzees and bonobos. The study of
great apes, including their physiology, genetics, and behaviour, is essential
to addressing questions of human nature and evolution (Pollen et al, 2023).
Urgent conservation action for the protection and preservation of these
emblematic species is therefore essential.
The timely and efficient assessment of great ape presence, abundance,
distribution, and behaviour is becoming increasingly important in evaluating
the effectiveness of conservation policies and intervention measures. The
potential of exploiting camera trap imagery for conservation or biological
modelling is well recognised (Kühl and Burghardt, 2013, Tuia et al, 2022).
However, even small camera networks generate large volumes of data (Fegraus et
al, 2011) and the number and complexity of downstream processing tasks
required to perform ecological analysis is extensive. Typically, ecologists
first need to identify those videos that contain footage of the target study
species followed by further downstream analyses, such as estimating the
distance of the animals from the camera (i.e., camera trap distance sampling)
to calculate species density or identification of ecologically or
anthropologically important behaviours, such as tool use or camera reactivity
(Houa et al, 2022). Performing these tasks manually is time consuming and
limited by the availability of human resources and expertise, becoming
infeasible at large scale. This underlines the need for rapid, scalable, and
efficient deep learning methods for automating the detection and assessment of
great ape populations and analysis of their behaviours.
To facilitate the development of methods for automating the interpretation of
camera trap data, large-scale, open-access video datasets must be available to
the relevant scientific communities, whilst removing geographic details that
could potentially threaten the safety of animals (Tuia et al, 2022). Unlike
the field of human action recognition and behaviour understanding, where
several large, widely acknowledged datasets exist for benchmarking (Kuehne et
al, 2011, Soomro et al, 2012, Kay et al, 2017), the number of great ape
datasets is limited and those that are currently available lack scale,
diversity and rich annotations.
Contribution. In this study, we present the PanAf20K dataset, the largest and
most diverse open-access video dataset of great apes in the wild – ready for
AI training. The dataset comprises footage collected from 14 study sites
across 6 African countries, featuring apes in over 20 distinct habitats (i.e.,
forests, savannahs, and marshes). It displays great apes in over 100
individual locations (e.g., trails, termite mounds, and water sources)
displaying an extensive range of 18 behaviour categories. A visual overview of
the dataset is presented in Fig. 1. The footage is accompanied by a rich set
of annotations suitable for a range of ecologically important tasks such as
detection, action localisation, fine-grained and multi-label behaviour
recognition.
Paper Organisation. Following this introduction, Sec. 2 reviews existing
animal behaviour datasets and methodologies for great ape detection and
behaviour recognition. Sec. 3 describes both parts of the dataset, the
PanAf20K and the PanAf500, and details how the data was collected and
annotated. Benchmark results for several computer vision tasks are presented
in Sec 4. Sec 5 discusses the main findings as well as any limitations
alongside future research directions while Sec 6 summarises the dataset and
highlights its potential applications.
## 2 Related Work
Great Ape Video Datasets for AI Development. While there have been encouraging
trends in the creation of new animal datasets (Swanson et al, 2015, Cui et al,
2018, Van Horn et al, 2018, Beery et al, 2021), there is still only a limited
number specifically designed for great apes and even fewer suitable for
behavioural analysis. In this section, the most relevant datasets are
described.
Bain et al. (Bain et al, 2021), curated a large camera trap video dataset
($>40$ hours) with fine-grained annotations for two behaviours; buttress
drumming and nut cracking. However, the data and corresponding annotations are
not yet publicly available and the range of annotations is limited to two
audio-visually distinct behaviours. The Animal Kingdom dataset (Ng et al,
2022), created for advancing behavioural understanding, comprises footage
sourced from YouTube (50hr, 30K videos) along with annotations that cover a
wide range of actions, from eating to fighting. The MammalNet dataset (Chen et
al, 2023), which is larger and more diverse, is also composed from YouTube
footage (18K videos, 539 hours) and focuses on behavioural understanding
across species. It comprises taxonomy-guided annotations for 12 common
behaviours, identified through previous animal behaviour studies, for 173
mammal categories. While both datasets are valuable resources for the study of
animal behaviour, they contain relatively few great ape videos since these
species make up only a small proportion of the overall dataset. Animal Kingdom
spans $\sim$100 videos while MammalNet includes $\sim$1000 videos across the
whole great ape family, representing $\sim$0.5% and $\sim$5% of all videos,
respectively. Other work to curate great ape datasets has focused annotation
efforts on age, sex, facial location, and individual identification (Freytag
et al, 2016, Brookes and Burghardt, 2020, Schofield et al, 2019), rather than
behaviour.
For the study of great ape behaviour, the currently available datasets have
many limitations. First, they are too small to capture the full breadth of
behavioural diversity. This is particularly relevant for great apes, which are
a deeply complex species, displaying a range of individual, paired and group
behaviours, that are still not well understood (Tennie et al, 2016, Samuni et
al, 2021). Secondly, they are not composed of footage captured by sensors
commonly used in ecological studies, such as camera traps and drones. This
means that apes are not observed in their natural environment and the
distribution of behaviours will not be representative of the wild (i.e.,
biased towards ’interesting’ or ’entertaining’ behaviours). Additionally, the
footage may be biased towards captive or human-habituated animals which
display altered or unnatural behaviours and are unsuitable for studying their
wild counterparts (Clark, 2011, Chappell and Thorpe, 2022). All these factors
may limit the ability of trained models to generalise effectively to wild
footage of great apes where conservation efforts are most urgently needed.
This, in turn, limits their practical and immediate utility. We aim to
overcome these limitations by introducing a large scale, open-access video
dataset that enables researchers to develop models for analysing the behaviour
of great apes in the wild and evaluate them against established methods.xxx
Great Ape Detection & Individual Recognition. Yang et al. (Yang et al, 2019)
developed a multi-frame system capable of accurately detecting the full body
location of apes in challenging camera-trap footage. In more recent work, Yang
et al. developed a curriculum learning approach that enables the utilisation
of large volumes of unlabelled data to improve detection performance (Yang et
al, 2023). Several other works focus on facial detection and individual
identification. In early research, Freytag et al. (Freytag et al, 2016)
applied YOLOv2 (Redmon and Farhadi, 2017), to localise the faces of
chimpanzees. They utilised a second deep CNN for feature extraction (AlexNet
(Krizhevsky et al, 2012) and VGGFaces (Parkhi et al, 2015)), and a linear
support vector machine for identification. Later, Brust et al. (Brust et al,
2017) extended their work utilising a much larger and diverse dataset.
Schofield et al. (Schofield et al, 2019) presented a pipeline for
identification of 23 chimpanzees across a video archive spanning 14 years.
Similar to (Brust et al, 2017), they trained the single-shot object detector,
SSD (Schofield et al, 2019), to perform initial localisation, and a secondary
CNN model to perform individual classification. Brookes et al. (Brookes and
Burghardt, 2020) employed YOLOv3 (Redmon and Farhadi, 2018) to perform one-
step simultaneous facial detection and individual identification on captive
gorillas.
Great Ape Action & Behaviour Recognition. To date, three systems have
attempted automated great ape behavioural action recognition. The first (Sakib
and Burghardt, 2020) was based on the two-stream convolutional architecture by
(Simonyan and Zisserman, 2014) and uses 3D ResNet-18s for feature extraction
and LSTM-based fusion of RGB and optical flow features. They reported a strong
top-1 accuracy of 73% across the nine behavioural actions alongside a
relatively low average per class accuracy of 42%. The second, proposed by Bain
et al. (Bain et al, 2021), utilises both audio and video inputs to detect two
specific behaviours; buttress drumming and nut cracking. Their system utilises
a 3D ResNet-18 and a 2D ResNet-18 for extraction of visual and audio features,
respectively, in different streams. They achieved an average precision of 87%
for buttress drumming and 85% for nut cracking on their unpublished dataset.
Lastly, Brookes et al. (Brookes et al, 2023) introduced a triple-stream model
that utilises RGB, optical flow and DensePose within a metric learning
framework, and achieved top-1 and average per-class accuracy of 85% and 65%,
respectively.
## 3 Dataset Overview
Figure 2: Manually annotated full-body location, species and behavioural
action labels. Sample frames extracted from PanAf20K videos with species (row
1) and behavioural action annotations (row 2) displayed. Green bounding boxes
indicate the full-body location of an ape. Species and behavioural action
annotations are shown in the corresponding text.
Task-focused Data Preparation. The PanAf20K dataset consists of two distinct
parts. The first includes a large video dataset containing 19,973 videos
annotated with multi-label behavioural labels. The second part comprises 500
videos with fine-grained annotations across $\sim$180,000 frames. Videos are
recorded at 24 FPS and resolutions of $720\times 404$ for 15 seconds
($\sim$360 frames). In this section, we provide an overview of the dataset,
including how the video data was originally collected (see Sec. 3.1) and
annotated for both parts (see Sec. 3.2).
### 3.1 Data Acquisition
Camera Trapping in the Wild. The PanAf Programme: The Cultured Chimpanzee has
39 research sites and data collection has been ongoing since January 2010. The
data included in this paper samples 14 of these sites and the available data
were obtained from studies of varying duration (7–22 months). Grids comprising
20 to 96 $1\times 1$ km cells were established for the distribution of
sampling units (to cover a minimum of 20–50 km2 in rainforest and 50–100 km2
in woodland savannah). An average of 29 (range 5–41) movement-triggered
Bushnell cameras were installed per site. One camera was installed per grid
cell where possible. However, in larger grids cameras were placed in alternate
cells. If certain grid cells did not contain suitable habitat, such as
grassland in forest-savanna mosaic sites, two cameras were placed instead as
far away from each other as possible, in cells containing suitable habitat to
maximize coverage. In areas where activities of interest (e.g., termite
fishing sites) were likely to take place, a second camera was installed to
capture the same scene from a different angle. Cameras were placed approx. 1m
high above ground, in locations that were frequently used by apes (e.g.,
trail, fruit trees). This method ensured a strategic installation of cameras,
with maximal chance of capturing footage of terrestrial activity of apes. Both
GPS location and habitat type for each location was noted. Footage was
recorded for 60 seconds with a 1 second interval between triggers and cameras
were visited every 1-3 months for maintenance and to download the recorded
footage throughout the study periods.
### 3.2 Data Annotation
Figure 3: Number of Apes & Bounding Box Size Distribution in the PanAf500
Data. The top row shows the distribution of apes across frames and videos in
(a) and (b), respectively, while the distribution of bounding box sizes is
shown in (c). The middle row shows still frame examples of videos containing
one, two, four and eight apes (viewing from left to right). The bottom row
demonstrates still frames with bounding boxes of various sizes; the colour of
bounding box and associated number represent the intra-video individual IDs.
Fine-grained Annotation of PanAf500. The PanAf500 was ground-truth labelled by
users on the community science platform Chimp&See (Arandjelovic et al, 2016)
and researchers at the University of Bristol (Yang et al, 2019, Sakib and
Burghardt, 2020) (examples are shown in Fig. 2). We re-formatted the metadata
from these sources specifically for use in computer vision under reproducible
and comparable benchmarks ready for AI-use. The dataset includes frame-by-
frame annotations for full-body location, intra-video individual
identification, and nine behavioural actions (Sakib and Burghardt, 2020)
across 500 videos and $\sim$180,000 frames.
Figure 4: Behavioural Actions in the PanAf500 Data. Examples of each one of
the nine behavioural action classes (right) and their distribution (left)
across 500 videos. The total number of per-frame annotations for each
behavioural action class is shown on top of the corresponding bar.
As shown in Fig. 3, the number of individual apes varies significantly, from
one to nine, with up to eight individuals appearing together simultaneously.
Individuals and pairs occur the most frequently while groups occur less
frequently, particularly those exceeding four apes. Bounding boxes are
categorised according to the COCO dataset (Lin et al, 2014) (i.e., $>96^{2}$,
$96^{2}$ and $32^{2}$ for large, medium and small, respectively) with small
bounding boxes occurring relatively infrequently compared to large and medium
boxes.
The behavioural action annotations cover 9 basic behavioural actions; sitting,
standing, walking, running, climbing up, climbing down, hanging, sitting on
back, and camera interaction. We refer to these classes as behavioural actions
in recognition of historical traditions in biological and computer vision
disciplines, which would consider them behaviours and actions, respectively.
Fig. 4 displays the behavioural actions classes in focus together with their
per-frame distribution. The class distribution is severely imbalanced, with
the majority of samples ($>85\%$) belonging to three head classes (i.e.,
sitting, walking and standing). The remaining behavioural actions are referred
to as tail classes. The same imbalance is observed at the clip level, as shown
in Tab. 1, although the distribution of classes across clips does not match
the per-frame distribution exactly. While behavioural actions with longer
durations (i.e., sitting) have more labelled frames, this does not necessarily
translate to more clips. For example, there are more clips of walking and
standing than sitting, and more clips of climbing up than hanging, although
the latter have fewer labelled frames.
Table 1: Behavioural Action Class Statistics. The total number of clips for each behavioural action alongside the average duration in seconds and frames. Action | Clips | Time (s) | Frames
---|---|---|---
walking | 747 | $3.49\pm 2.94$ | 83.69
standing | 366 | $4.77\pm 4.59$ | 114.57
sitting | 308 | $10.30\pm 5.51$ | 247.13
climbing up | 81 | $2.11\pm 1.67$ | 50.59
hanging | 50 | $7.35\pm 5.24$ | 176.28
climbing down | 35 | $1.73\pm 1.24$ | 41.57
running | 34 | $2.61\pm 2.06$ | 62.59
camera interaction | 32 | $2.52\pm 3.77$ | 60.59
sitting on back | 26 | $3.89\pm 4.04$ | 93.46
Figure 5: PanAf20K Behaviour Examples. Triplets of example frames for six
categories (i.e., feeding, travel, camera reaction, social interaction, chimp
carrying and tool use) in the PanAf20K dataset are shown. Note that camera
reaction, social interaction and chimp carrying have been abbreviated to
reaction, social and carrying, respectively.
Figure 6: Behavioural Annotations of the PanAf20K Dataset. The distribution of
behaviour categories for the PanAf20K dataset is shown. Figures above each bar
represent the dataset proportion (%) of each class.
Figure 7: Co-occurrence of Behaviours in the PanAf20k Dataset. A co-occurrence
matrix for the PanAf20K behaviours, where each cell reflects the number of
times two behaviours occurred together. Diagonal cells are reset to aid
visibility. Figure 8: Examples of Fine-grained and Multi-label Annotations.
For videos with fine-grained annotations, full-body locations and behavioural
actions are associated with each ape on a frame-by-frame basis (left). In
contrast, multi-label behaviour annotations are provided at the video level
(right); behaviours are not localised or assigned specifically to each ape.
Multi-label Behavioural Annotation of PanAf20K. Community scientists on the
Chimp&See platform provided multi-label behavioural annotations for
$\sim$20,000 videos. They were shown 15-second clips at a time and asked to
annotate whether animals were present or whether the clip was blank. To obtain
a balance between specificity and keeping the task accessible and interesting
to a broad group of people, annotators were presented with a choice of
classification categories. These categories allowed focus to be given to
ecologically important behaviours such as tool use, camera reaction and
bipedalism. Hashtags for behaviours not listed in the classification
categories were also permitted, allowing new and interesting behaviours to be
added when they were discovered in the videos. The new behaviours were
subcategories of the existing behaviours, many of them relating to tool use
(e.g., algae scooping and termite fishing in aboreal nests).
To ensure annotation quality and consistency a video was only deemed to be
analyzed when either three volunteers marked the video as blank, unanimous
agreement between seven volunteers was observed, or 15 volunteers annotated
the video. These annotations were then extracted and expertly grouped into 18
co-occurring classes, which form the multi-label behavioural annotations
presented here. The annotations follow a multi-hot binary format that
indicates the presence of one or many behaviours. It should also be noted that
behaviours are not assigned to individual apes or temporally localised within
each video. Fig. 5 presents examples for several of the most commonly
occurring behaviours. Fig. 7 shows the full distribution of behaviours across
videos, which is highly imbalanced. Four of the most commonly occurring
classes are observed in $>60\%$ videos, while the least commonly occurring
classes are observed in $<1\%$. The relationship between behaviours is shown
in Fig. 7 which presents co-occurring classes. The behaviours differ from the
behavioural actions included in the PanAf500 dataset, corresponding to higher
order behaviours that are commonly monitored in ecological studies. For
example, instances of travel refer to videos that contain an individual or
group of apes travelling, whereas associated behavioural actions such as
walking or running may occur in many contexts (i.e., walking towards another
ape during a social interaction or while searching for a tool).
Both parts of the dataset are suitable for different computer vision tasks.
The PanAf500 supports great ape detection, tracking, action grounding, and
multi-class action recognition, while the PanAf20k supports multi-label
behaviour recognition. The difference between the two annotation types can be
observed in Fig. 8.
Machine Labels for Animal Location and IDs. We generated full-body bounding
boxes for apes present in the remaining, unlabelled videos using state-of-the-
art (SOTA) object detection models evaluated on the PanAf500 dataset (see Sec.
4). Additionally, we assigned intra-video IDs to detected apes using the
multi-object tracker, OC-SORT (Cao et al, 2023). Note that these pseudo-labels
do not yet associate behaviours with individual bounding boxes.
## 4 Experiments & Results
Table 2: Ape Detection Benchmarks. Detection performance on the PanAf500 dataset. Results are reported for the MegaDetector (Beery et al, 2019), ResNet-101 (+SCM+TCM) (Yang et al, 2019), VarifocalNet (Zhang et al, 2021), Swin Transformer (Liu et al, 2021) and ConvNeXt (Liu et al, 2022). The highest scores for each metric are shown in bold. Model | mAP (%) | Other (%)
---|---|---
All | L | M | S | Precision | Recall | F1
MegaDetector | 88.0 | 98.05 | 82.60 | 68.21 | 56.93 | 90.58 | 69.92
ResNet-101 | 81.2 | 77.60 | 88.97 | 88.84 | 42.37 | 88.93 | 57.40
VarifocalNet | 84.1 | 84.50 | 88.73 | 82.07 | 21.73 | 88.57 | 34.90
Swin Transformer | 87.2 | 82.47 | 96.86 | 88.53 | 83.66 | 92.03 | 87.65
ConvNeXt | 86.6 | 83.51 | 95.16 | 81.13 | 81.80 | 91.93 | 86.57
This section describes experiments relating to the PanAf500 and PanAf20K
datasets. For the former, we present benchmark results for great ape detection
and fine-grained action recognition. For the latter, we present benchmark
results for multi-label behavioural classification. For both sets of
experiments, several SOTA architectures are used.
### 4.1 PanAf500 Dataset
Baseline Models. We report benchmark results for ape detection and fine-
grained behavioural action recognition for the PanAf500 dataset, trained and
evaluated on SOTA architectures. For ape detection, this entails the
MegaDetector (Beery et al, 2019), ResNet-101 (+SCM+TCM) (Yang et al, 2019),
VarifocalNet (VFNet) (Zhang et al, 2021), SwinTransformer (Liu et al, 2021)
and ConvNext (Liu et al, 2022) architectures. For fine-grained action
recognition, we considered X3D (Feichtenhofer, 2020), I3D (Carreira and
Zisserman, 2017), 3D ResNet-50 (Tran et al, 2018), Timesformer (Bertasius et
al, 2021) and MViTv2 (Li et al, 2022) architectures. Action recognition models
were chosen based on SOTA performance on human action recognition datasets and
to be consistent with the best performing models on the AnimalKingdom (Ng et
al, 2022) and MammalNet datasets (Chen et al, 2023). In all cases, train-val-
test (80:05:15) splits were generated at the video-level to ensure
generalisation across video/habitat and splits remained consistent across
tasks.
Figure 9: Megadetector (Beery et al, 2019) achieves higher precision for the
majority of cases although ConvNeXt (Liu et al, 2022) and Swin Transformer
(Liu et al, 2021) achieve better precision scores at high recall rates
($R_{det}>0.84$). blahblahblahblahblahblahblah
Figure 10: R101 (+SCM +TCM) (Yang et al, 2019) and VFNet (Zhang et al, 2021)
achieve the highest true positive rates at low false positive rates
($FPR<0.15$). At higher false positive rates R101(+SCM+TCM) (Yang et al, 2019)
performs better.
Great Ape Detection. We initialised all models with pretrained feature
extractors. For all models, except the Megadetector, we utilised MS COCO (Lin
et al, 2014) pretrained weights. We use the out-of-the-box Megadetector
implementation since it is pretrained on millions of camera trap images and
provides a strong initialisation for camera trap specific detection tasks. We
then fine-tuned each model for 50 epochs using SGD with a batch size of 16.
Training was carried out using an input image resolution of $416^{2}$ and an
Intersection over Union (IoU) threshold of $0.5$ for non maximum suppression,
at an initial learning rate of $1\times 10^{-2}$ which was reduced by $10\%$
at $80\%$ and $90\%$ of the total training epochs. All ape detection models
were evaluated using the commonly used object detection metrics: mean average
precision (mAP), precision, recall and F1-scores. All metrics follow the open
images standard (Krasin et al, 2017) and are considered in combination during
evaluation. Performance is provided separately for small ($32^{2}$), medium
($96^{2}$) and large bounding boxes ($>96^{2}$), as per the COCO object
detection standard, in addition to overall performance.
Performance. Tab. 2 shows that the fine-tuned Megadetector achieves the best
mAP score overall and for large bounding boxes, although it is outperformed by
the Swin Transformer and ResNet-101 (+Cascade R-CNN+SCM+TCM) on medium and
small bounding boxes, respectively. This shows that in-domain pre-training of
the feature extractor is valuable for fine-tuning since the Megadetector is
the only model pretrained on a camera trap dataset, rather than the COCO
dataset (Lin et al, 2014). Performance across the remaining metrics,
precision, recall and F1-score, is dominated by the Swin Transformer, which
shows the importance of modelling spatial dependencies for good detection
performance.
The precision-recall (PR) curve displayed in Fig. 10 shows that most models
maintain precision of more than 90% ($P_{det}>0.9$) at lower recall rates
($R_{det}<0.80$), except ResNet-101 (+SCM+TCM) which falls below this at
recall of 78% ($R_{det}=0.78$). The fine-tuned Megadetector achieves
consistently higher precision than other models for more than 84% of cases
($R_{det}=0.84$), outperforming other models by 5% ($P_{det}=0.05$) on
average. However, at higher recall rates ($R_{det}>0.84$) ConvNeXt and
SwinTransformer achieve higher precision, with the latter achieving marginally
better performance. The ROC curve presented in Fig. 10 shows that VFNet and
ResNet-101 (+SCM+TCM) achieve higher true positive rate than all other models
at false positive rates less than 5% ($FPR<0.05$) and 40% ($FPR<0.40$),
respectively. At higher false positive rates ConvNext and SwinTransformer are
competitive with ResNet-101 (+SCM+TCM), with marginally better performance
being established by ConvNeXt at very high false positive rates. Figure 11
presents qualitative examples of success and failure cases for the best
performing model.
Figure 11: Megadetector detection examples. A sequence of frames (along each row) extracted from 3 different videos. The ground truth bounding boxes (green) are shown alongside detections (red). The first sequence (row 1) shows successful detections. The second set of sequences (row 2-4) provide examples of false positive detections. The third set of sequences (row 5-6) provide examples of false negative detections. Figure 12: Per-class Distribution vs. Behavioural Thresholds. Distribution of each behavioural action class at various behavioural thresholds. Note that tail classes are effected more significantly by longer thresholds than head classes. Table 3: Behavioural Action Recognition Benchmarks. Behavioural action recognition performance on the PanAf500 dataset. Results are reported for X3D (Feichtenhofer, 2020), I3D (Carreira and Zisserman, 2017), 3D ResNet-50 (Hara et al, 2017), MViTV2 (Li et al, 2022), and TimeSformer (Bertasius et al, 2021) models. The highest scores for top-1 and average per-class accuracy are shown in bold. Model | Top-1 (%) | C-Avg (%)
---|---|---
16 | 32 | 64 | 128 | 16 | 32 | 64 | 128
X3D | 80.00 | 80.04 | 79.40 | 74.24 | 50.35 | 56.10 | 53.02 | 40.89
I3D | 79.29 | 78.48 | 76.90 | 67.45 | 42.15 | 48.14 | 31.65 | 24.46
3D ResNet-50 | 77.45 | 76.41 | 74.02 | 73.31 | 55.17 | 33.79 | 38.72 | 36.03
MViTV2 | 78.31 | 81.09 | 79.29 | 79.19 | 40.45 | 54.91 | 48.28 | 41.11
TimeSformer | 78.53 | 79.45 | 79.26 | 78.18 | 45.05 | 56.38 | 48.27 | 41.10
Figure 13: Class-wise Performance vs. Proportion of Data. The per-class
accuracy for each behavioural action recognition model is plotted against the
proportion of data for each class. All models consistently achieve strong
performance on the head classes, whereas performance is variable across tail
classes.
Behavioural Action Recognition. We trained all models using the protocol
established by (Sakib and Burghardt, 2020). During training we imposed a
temporal behaviour threshold that ensures that only frame sequences in which a
behaviour is exhibited for $t$ consecutive frames are utilised during training
in order to retain well-defined behaviour instances. We then sub-sampled
16-frame sequences from clips that satisfy the behaviour threshold. The test
threshold is always kept consistent ($t=16$). Fig. 12 shows the effect of
different behaviour thresholds on the number of clips available for each
class. Higher behaviour thresholds have a more significant effect on
minority/tail classes since they occur more sporadically. For example, there
are no training clips available for the climbing down class where $t=128$. All
models were initialised with feature extractors pre-trained on Kinetics-400
(Kay et al, 2017) and fine-tuned for 200 epochs using the Adam optimiser and a
standard cross-entropy loss. We utilised a batch size of 32, momentum of 0.9
and performed linear warm-up followed by cosine annealing using an initial
learning rate of $1\times 10^{-5}$ that increases to $1\times 10^{-4}$ over 20
epochs. All behavioural action recognition models were evaluated using average
top-1 and average per-class accuracy (C-Avg).
Figure 14: Confusion Matrix & Example Errors. The confusion matrix (left) is
shown alongside examples of mis-classified frames (right). For mis-classified
examples, ground truth labels are shown on the y-axis (i.e., hanging, running,
sitting) and examples of the classes most likely to be incorrectly predicted
for the ground truth class are shown on the x-axis. Note that a high
proportion of errors are due to predictions made in favour of majority
classes.
Performance. Tab. 3 shows the X3D model attains the best top-1 accuracy at
behaviour thresholds $t=16$ and $t=64$, although similar performance is
achieved by MViTV2 and TimeSformer for the latter threshold. It also achieves
the best average per-class performance at $t=64$, while TimeSformer achieves
the best performance at $t=32$ and $t=128$.
The MVITV2 models realise the best top-1 accuracy at $t=32$ and $t=128$,
although they do not achieve the best average per-class performance at any
threshold. The 3D ResNet-50 achieves the best average per-class performance at
$t=16$. When considering top-1 accuracy, model performance is competitive. At
lower behavioural thresholds, i.e., $t=16$ and $t=32$, the difference in top-1
performance is 2.55% and 4.68%, respectively, between the best and worst
performing models, although this increases to 5.38% and 11.74% at $t=64$ and
$t=128$, respectively. There is greater variation in average per-class
performance and it is rare that a model achieves the best performance across
both metrics.
Although we observe strong performance with respect to top-1 accuracy, our
models exhibit relatively poor average per-class performance. Fig. 13 plots
per-class performance against class frequency and shows that the average per-
class performance is caused by poor performance on tail classes. The average
per-class accuracy across all models for the head classes is 83.22% while only
28.33% is achieved for tail classes. There is significant variation in the
performance of models; I3D performs well on hanging and climbing up but fails
to classify any of the other classes correctly. Similarly, X3D performs
extremely well on sitting on back but achieves poor results on the other
classes. None of the models except for TimeSformer correctly classify any
instances of running during testing. Fig. 14 presents the confusion matrix
calculated on validation data alongside examples of misclassified instances.
### 4.2 PanAf20K Dataset
Table 4: Multi-label Behaviour Recognition Benchmarks. Results are reported for I3D (Carreira and Zisserman, 2017), 3D ResNet-50 (Hara et al, 2017), X3D (Feichtenhofer, 2020), MViTV2 (Li et al, 2022), and TimeSformer (Bertasius et al, 2021) models with focal loss (Cui et al, 2019), logit adjustment (Menon et al, 2020), and focal loss with weight balancing (Alshammari et al, 2022). The highest scores across all metrics are shown in bold. Model | mAP (%) c | Other (%)
---|---|---
All | Head | Middle | Tail | Accuracy | Precision | Recall
I3D | 45.49 | 87.92 | 53.43 | 6.62 | 41.99 | 51.77 | 38.62
+FocalLoss | 46.65 | 87.67 | 53.51 | 10.17 | 42.51 | 60.02 | 37.46
+LogitAdjustment | 46.81 | 87.52 | 52.54 | 12.05 | 43.02 | 57.99 | 38.88
+WeightBalancing | 46.41 | 88.41 | 51.91 | 11.07 | 41.93 | 57.36 | 35.62
3D ResNet-50 | 46.03 | 86.12 | 53.22 | 9.73 | 40.76 | 54.13 | 35.87
+FocalLoss | 47.93 | 87.07 | 54.31 | 13.35 | 43.35 | 57.77 | 38.89
+LogitAdjustment | 48.04 | 87.44 | 53.91 | 13.96 | 41.15 | 58.86 | 36.27
+WeightBalancing | 46.68 | 87.09 | 54.06 | 9.90 | 41.54 | 54.05 | 40.62
X3D | 46.06 | 87.32 | 52.95 | 9.36 | 42.70 | 49.26 | 39.25
+FocalLoss | 47.19 | 89.26 | 52.75 | 11.75 | 41.67 | 50.93 | 36.99
+LogitAdjustment | 47.85 | 88.58 | 53.06 | 13.77 | 42.64 | 59.18 | 36.94
+WeightBalancing | 45.64 | 88.45 | 51.15 | 9.75 | 40.96 | 48.40 | 33.29
MViTV2 | 45.71 | 88.72 | 51.16 | 9.76 | 42.38 | 52.03 | 36.55
+FocalLoss | 45.78 | 89.27 | 50.82 | 10.05 | 42.83 | 56.54 | 36.38
+LogitAdjustment | 45.91 | 88.97 | 50.75 | 10.74 | 41.02 | 57.73 | 38.11
+WeightBalancing | 45.36 | 88.58 | 49.82 | 10.59 | 42.44 | 49.77 | 36.20
TimeSformer | 47.24 | 88.83 | 51.91 | 13.29 | 41.60 | 57.66 | 38.63
+FocalLoss | 48.82 | 88.75 | 52.65 | 17.10 | 42.70 | 68.01 | 37.37
+LogitAdjustment | 49.39 | 88.52 | 53.21 | 18.20 | 42.44 | 71.31 | 35.45
+WeightBalancing | 48.17 | 87.98 | 51.81 | 16.78 | 41.86 | 61.16 | 39.36
Data Setup. We generate train-val-test splits (70:10:20) using iterative
stratification (Sechidis et al, 2011, Szymanski and Kajdanowicz, 2019). During
training, we uniformly sub-sample $t=16$ frames from each video, equating to
$\sim$1 frame per second (i.e., a sample interval of 22.5 frames).
Baseline Models. To establish benchmark performance for multi-label behaviour
recognition, we trained the X3D, I3D, 3D ResNet-50s, and MViTv2 models. All
models were initialised with feature extractors pre-trained on Kinetics-400
(Kay et al, 2017) and fine-tuned for 200 epochs using the Adam optimiser. We
utilised a batch size of 32, momentum of 0.9 and performed linear warm-up
followed by cosine annealing using an initial learning rate of $1\times
10^{-5}$ that increases to $1\times 10^{-4}$ over 20 epochs. Models were
evaluated using mAP, subset accuracy (i.e., exact match), precision and
recall. Behaviour classes were grouped, based on class frequency, into head
($>10\%$), middle ($>1\%$) and tail ($<1\%$) segments, and mAP performance is
reported for each segment. To address the long-tailed distribution, we
substitute the standard loss for those calculated using long-tail recognition
techniques. Specifically, we implement (i) focal loss (Cui et al, 2019)
$L_{CB}$; (ii) logit adjustment (Menon et al, 2020) $L_{LA}$; and (iii) focal
loss with weight balancing via a MaxNorm constraint (Alshammari et al, 2022).
Figure 15: Class-wise Accuracy vs. Proportion of Data. The per-class average
precision for the 3D ResNet-50 (Hara et al, 2017) and 3D ResNet-50
(+LogitAdjustment) (Hara et al, 2017, Menon et al, 2020) models is plotted
against the proportion of data for each class. In general, better model
performance is achieved on classes with high data proportions and the
ResNet-50 (+LogitAdjustment) model shows improved performance on middle and
tail classes.
Multi-label Behaviour Recognition. As shown in Table 4.2, performance is
primarily dominated by the 3D ResNet-50 and TimeSformer models when coupled
with the various long-tailed recognition techniques. The TimeSformer
(+LogitAdjustment) attains the highest mAP scores for both overall and tail
classes, while the MViTV2 (+FocalLoss) and 3D ResNet-50 (+FocalLoss)
demonstrate superior performance in terms of head and middle class mAP,
respectively. The 3D ResNet-50 (+FocalLoss) and 3D ResNet-50
(+WeightBalancing) models achieve the best subset accuracy and recall,
respectively, while the highest precision is realised by the TimeSformer
(+LogitAdjustment) model. Although the 3D ResNet-50 and TimeSformer models
perform strongest, it should be noted that the difference in overall mAP
across all models is small (i.e., 4.03% between best and worst performing
models).
Figure 16: Multi-label Errors. Frames extracted from three videos exhibit
success and failure cases of the 3D ResNet-50 model. Behaviour predictions are
shown in light boxes of the first frame of each sequence; true positives are
green, false positive are blue, and false negatives are red. In the first
video (row 1), the model fails to classify feeding by the chimp visible in
frames 1 and 2 whereas in the second video (row 2), it fails to classify tool
use by the infant chimp in the final frame. Climbing is predicted incorrectly
in the final video (row 3).
As demonstrated by the head, middle and tail mAP scores, higher performance is
achieved for more frequently occurring classes with performance deteriorating
significantly for middle and tail classes. Across models, the average
difference between head and middle, and middle and tail classes is 35.68
($\pm$1.88)% and 40.55 ($\pm$3.02)%, respectively. The inclusion of long-
tailed recognition techniques results in models that consistently attain
higher tail class mAP performance than their standard counterparts (i.e.,
models that do not use long-tail recognition techniques). The logit adjustment
technique consistently results in the best tail class mAP across models,
whereas the focal loss results in the best performance on the middle classes
for all models except the X3D model. None of the standard models achieve the
best performance on any metric.
Fig. 15 plots per-class mAP performance of the 3D ResNet-50 and 3D
ResNet-50(+LogitAdjustment) models against the per-class proportion of data.
The best performance is observed for the three most commonly occurring classes
(i.e., feeding, travel, and no behaviour) whereas the worst performance is
obtained by the most infrequently occurring classes (i.e., display,
aggression, sex, bipedal, and cross species interaction) with the exception of
piloerection. It can also be observed that the ResNet-50(+LogitAdjustment)
model outperforms its standard counterpart on the majority of middle and tail
classes, although it is outperformed on tail classes. Examples of success and
failure cases by the 3D ResNet-50 model are presented in Fig. 16.
## 5 Discussion & Future Work
Results. The performance of current SOTA methods is not currently sufficient
for facilitating the large-scale, automated behavioural monitoring required to
support conservation efforts. The conclusions drawn in ecological studies rely
on the highly accurate classification of all observed behaviours by expert
primatologists. While the current methods achieve strong performance on head
classes, relatively poor performance is observed for rare classes. Our results
are consistent with recent work on similar datasets (i.e., AnimalKingdom (Ng
et al, 2022) and MammalNet (Chen et al, 2023)) which demonstrate the
significance of the long-tailed distribution that naturally recorded data
exhibits (Liu et al, 2019). Similar to (Ng et al, 2022), our experiments show
that current long-tailed recognition techniques can help to improve
performance on tail classes, although a large discrepancy between head and
middle, and head and tail classes still exists. The extent of this performance
gap (see Tab. 4.2) emphasises the difficulty of tackling long-tailed
distributions and highlights an important direction for future work (Perrett
et al, 2023). Additionally, the near perfect performance at training time
(i.e., $>95\%$ mAP) highlights the need for methods that can learn effectively
from a minimal number of examples.
Community Science & Annotation. Although behavioural annotations are provided
by non-expert community scientists, several studies have shown the
effectiveness of citizen scientists to perform complex data annotation tasks
(Danielsen et al, 2014, McCarthy et al, 2021) typically carried out by
researchers (i.e., species classification, individual identification etc.).
However, it should be noted that, as highlighted by (Cox et al, 2012),
community scientists are more prone to errors relating to rare species. In the
case of our dataset, this may translate to simple behaviours being identified
correctly (e.g., feeding and tool use) whereas more nuanced or subtle
behaviours (e.g., display and aggression) are missed or incorrectly
interpreted, amongst other problems. This may occur despite the behaviour
categories were predetermined by experts as suitable for non-expert
annotation.
The dataset’s rich annotations suit various computer vision tasks, despite key
differences from other works. Unlike similar datasets (Ng et al, 2022, Chen et
al, 2023), behaviours in the PanAf20K dataset are not temporally located
within the video. However, the videos in our dataset are relatively short
(i.e., 15 seconds) in contrast to the long form videos included in other
datasets. Therefore, the time stamping of behaviour may be less significant
considering it is possible to utilise entire videos, with a suitably fine-
grained sample interval (i.e., 0.5-1 second), as input to standard action
recognition models. With that being said, behaviours occur sporadically and
chimpanzees are often only in frame for very short periods of time. Therefore,
future work will consider augmenting the existing annotations with temporal
localisation of actions. Moreover, while our dataset comprises a wide range of
behaviour categories, many of them exhibit significant intra-class variation.
In the context of ecological/primatological studies, this variation often
necessitates the creation of separate ethograms for individual behaviours
(Nishida et al, 1999, Zamma and Matsusaka, 2015). For instance, within the
tool use behaviour category, we find subcategories like nut cracking
(utilizing rock, stone, or wood), termite fishing, and algae fishing.
Similarly, within the camera reaction category, distinct subcategories include
attraction, avoidance, and fixation. In future, we plan to extend the existing
annotations to include more granular subcategories.
Ethics Statement. All data collection, including camera trapping, was done
non-invasively, with no animal contact and no direct observation of the
animals under study. Full research approval, data collection approval and
research and sample permits of national ministries and protected area
authorities were obtained in all countries of study. Sample and data export
was also done with all necessary certificates, export and import permits. All
work conformed to the relevant regulatory standards of the Max Planck Society,
Germany. All community science work was undertaken according to the Zooniverse
User Agreement and Privacy Policy. No experiments or data collection were
undertaken with live animals.
## 6 Conclusion
We present by-far the largest open-access video dataset of wild great apes
with rich annotations and SOTA benchmarks. The dataset is directly suitable
for visual AI training and model comparison. The size of the dataset and
extent of labelling across $>$7M frames and $\sim$20K videos (lasting $>$80
hours) now offers the first comprehensive view of great ape populations and
their behaviours to AI researchers. Task-specific annotations make the data
suitable for a range of associated, challenging computer vision tasks (i.e,
animal detection, tracking, and behaviour recognition) which can facilitate
ecological analysis urgently required to support conservation efforts. We
believe that given its immediate AI compatibility, scale, diversity, and
accessibility, the PanAf20K dataset provides an unmatched opportunity for the
many communities working in the ecological, biological, and computer vision
domains to benchmark and expand great ape monitoring capabilities. We hope
that this dataset can, ultimately, be a step towards better understanding and
more effectively conserving these charismatic species.
## Data availability
All data and code will be made publicly available from the PanAf20K project
website upon publication and is available now upon request from the authors.
Acknowledgments We thank the Pan African Programme: ‘The Cultured Chimpanzee’
team and its collaborators for allowing the use of their data for this paper.
We thank Amelie Pettrich, Antonio Buzharevski, Eva Martinez Garcia, Ivana
Kirchmair, Sebastian Schütte, Linda Gerlach and Fabina Haas. We also thank
management and support staff across all sites; specifically Yasmin Moebius,
Geoffrey Muhanguzi, Martha Robbins, Henk Eshuis, Sergio Marrocoli and John
Hart. Thanks to the team at https://www.chimpandsee.org particularly Briana
Harder, Anja Landsmann, Laura K. Lynn, Zuzana Macháčková, Heidi Pfund,
Kristeena Sigler and Jane Widness. The work that allowed for the collection of
the dataset was funded by the Max Planck Society, Max Planck Society
Innovation Fund, and Heinz L. Krekeler. In this respect we would like to
thank: Ministre des Eaux et Forêts, Ministère de l’Enseignement supérieur et
de la Recherche scientifique in Côte d’Ivoire; Institut Congolais pour la
Conservation de la Nature, Ministère de la Recherche Scientifique in
Democratic Republic of Congo; Forestry Development Authority in Liberia;
Direction Des Eaux Et Forêts, Chasses Et Conservation Des Sols in Senegal;
Makerere University Biological Field Station, Uganda National Council for
Science and Technology, Uganda Wildlife Authority, National Forestry Authority
in Uganda; National Institute for Forestry Development and Protected Area
Management, Ministry of Agriculture and Forests, Ministry of Fisheries and
Environment in Equatorial Guinea. This work was supported by the UKRI CDT in
Interactive AI under grant EP/S022937/1.
## References
* * Vié et al (2009) Vié JC, Hilton-Taylor C, Stuart SN (2009) Wildlife in a changing world: an analysis of the 2008 IUCN Red List of threatened species. IUCN
* Ceballos et al (2020) Ceballos G, Ehrlich PR, Raven PH (2020) Vertebrates on the brink as indicators of biological annihilation and the sixth mass extinction. _Proceedings of the National Academy of Sciences_ 117(24):13596–13602
* Carvalho et al (2021) Carvalho JS, Graham B, Bocksberger G, et al (2021) Predicting range shifts of african apes under global change scenarios. _Diversity and Distributions_ 27(9):1663–1679
* Haurez et al (2015) Haurez B, Daïnou K, Tagg N, et al (2015) The role of great apes in seed dispersal of the tropical forest tree species dacryodes normandii (burseraceae) in gabon. _Journal of Tropical Ecology_ 31(5):395–402
* Tarszisz et al (2018) Tarszisz E, Tomlinson S, Harrison ME, et al (2018) An ecophysiologically informed model of seed dispersal by orangutans: linking animal movement with gut passage across time and space. _Conservation Physiology_ 6(1):coy013
* Chappell and Thorpe (2022) Chappell J, Thorpe SK (2022) The role of great ape behavioral ecology in one health: Implications for captive welfare and re-habilitation success. _American journal of primatology_ 84(4-5):e23328
* Pollen et al (2023) Pollen AA, Kilik U, Lowe CB, et al (2023) Human-specific genetics: New tools to explore the molecular and cellular basis of human evolution. _Nature Reviews Genetics_ pp 1–25
* Kühl and Burghardt (2013) Kühl HS, Burghardt T (2013) Animal biometrics: quantifying and detecting phenotypic appearance. _Trends in Ecology & Evolution_ 28(7):432–441
* Tuia et al (2022) Tuia D, Kellenberger B, Beery S, et al (2022) Perspectives in machine learning for wildlife conservation. _Nature Communications_ 13(1):1–15
* Fegraus et al (2011) Fegraus EH, Lin K, Ahumada JA, et al (2011) Data acquisition and management software for camera trap data: A case study from the team network. _Ecological Informatics_ 6(6):345–353
* Houa et al (2022) Houa NA, Cappelle N, Bitty EA, et al (2022) Animal reactivity to camera traps and its effects on abundance estimate using distance sampling in the taï national park, côte d’ivoire. _PeerJ_ 10:e13510
* Kuehne et al (2011) Kuehne H, Jhuang H, Garrote E, et al (2011) Hmdb: a large video database for human motion recognition. In: _2011 International conference on computer vision_ , IEEE, pp 2556–2563
* Soomro et al (2012) Soomro K, Zamir AR, Shah M (2012) Ucf101: A dataset of 101 human actions classes from videos in the wild. _arXiv preprint arXiv:12120402_
* Kay et al (2017) Kay W, Carreira J, Simonyan K, et al (2017) The kinetics human action video dataset. _arXiv preprint arXiv:170506950_
* Swanson et al (2015) Swanson A, Kosmala M, Lintott C, et al (2015) Snapshot serengeti, high-frequency annotated camera trap images of 40 mammalian species in an african savanna. _Scientific Data_ 2(1):1–14
* Cui et al (2018) Cui Y, Song Y, Sun C, et al (2018) Large scale fine-grained categorization and domain-specific transfer learning. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 4109–4118
* Van Horn et al (2018) Van Horn G, Mac Aodha O, Song Y, et al (2018) The inaturalist species classification and detection dataset. In: _Proceedings of the IEEE International Conference on Computer Vision_ , pp 8769–8778
* Beery et al (2021) Beery S, Agarwal A, Cole E, et al (2021) The iwildcam 2021 competition dataset. _arXiv preprint arXiv:210503494_
* Bain et al (2021) Bain M, Nagrani A, Schofield D, et al (2021) Automated audiovisual behavior recognition in wild primates. _Science Advances_ 7(46):eabi4883
* Ng et al (2022) Ng XL, Ong KE, Zheng Q, et al (2022) Animal kingdom: A large and diverse dataset for animal behavior understanding. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 19023–19034
* Chen et al (2023) Chen J, Hu M, Coker DJ, et al (2023) Mammalnet: A large-scale video benchmark for mammal recognition and behavior understanding. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 13052–13061
* Freytag et al (2016) Freytag A, Rodner E, Simon M, et al (2016) Chimpanzee faces in the wild: Log-euclidean cnns for predicting identities and attributes of primates. In: _German Conference on Pattern Recognition_ , Springer, pp 51–63
* Brookes and Burghardt (2020) Brookes O, Burghardt T (2020) A dataset and application for facial recognition of individual gorillas in zoo environments. In: _Workshop on the Visual Observation and Analysis of Vertebrate and Insect Behaviour_ , 2012.04689
* Schofield et al (2019) Schofield D, Nagrani A, Zisserman A, et al (2019) Chimpanzee face recognition from videos in the wild using deep learning. _Science advances_ 5(9):eaaw0736
* Tennie et al (2016) Tennie C, Jensen K, Call J (2016) The nature of prosociality in chimpanzees. _Nature communications_ 7(1):13915
* Samuni et al (2021) Samuni L, Crockford C, Wittig RM (2021) Group-level cooperation in chimpanzees is shaped by strong social ties. _Nature communications_ 12(1):539
* Clark (2011) Clark FE (2011) Great ape cognition and captive care: Can cognitive challenges enhance well-being? _Applied Animal Behaviour Science_ 135(1-2):1–12
* Yang et al (2019) Yang X, Mirmehdi M, Burghardt T (2019) Great ape detection in challenging jungle camera trap footage via attention-based spatial and temporal feature blending. In: _Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops_ , pp 0–0
* Yang et al (2023) Yang X, Burghardt T, Mirmehdi M (2023) Dynamic curriculum learning for great ape detection in the wild. _International Journal of Computer Vision_ pp 1–19
* Redmon and Farhadi (2017) Redmon J, Farhadi A (2017) Yolo9000: better, faster, stronger. In: _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp 7263–7271
* Krizhevsky et al (2012) Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. In: _Advances in Neural Information Processing Systems_ , pp 1097–1105
* Parkhi et al (2015) Parkhi O, Vedaldi A, Zisserman A (2015) Deep face recognition. In: _Proceedings of the British Machine Vision Conference_ , British Machine Vision Association
* Brust et al (2017) Brust CA, Burghardt T, Groenenberg M, et al (2017) Towards automated visual monitoring of individual gorillas in the wild. In: _Proceedings of the IEEE International Conference on Computer Vision Workshops_ , pp 2820–2830
* Redmon and Farhadi (2018) Redmon J, Farhadi A (2018) Yolov3: An incremental improvement. _arXiv preprint arXiv:180402767_
* Sakib and Burghardt (2020) Sakib F, Burghardt T (2020) Visual recognition of great ape behaviours in the wild. In: _Workshop on the Visual Observation and Analysis of Vertebrate and Insect Behaviour_ , 2011.10759
* Simonyan and Zisserman (2014) Simonyan K, Zisserman A (2014) Two-stream convolutional networks for action recognition in videos. In: _Advances in Neural Information Processing Systems_
* Brookes et al (2023) Brookes O, Mirmehdi M, Kühl HS, et al (2023) Triple-stream deep metric learning of great ape behavioural actions. In: _Proceedings of the 18th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications_ , pp 294–302
* Arandjelovic et al (2016) Arandjelovic M, Stephens CR, McCarthy MS, et al (2016) Chimp&see: An online citizen science platform for large-scale, remote video camera trap annotation of chimpanzee behaviour, demography and individual identification. _PeerJ Preprints_
* Lin et al (2014) Lin TY, Maire M, Belongie S, et al (2014) Microsoft coco: Common objects in context. In: _Proceedings of the European Conference on Computer Vision_ , Springer, pp 740–755
* Cao et al (2023) Cao J, Pang J, Weng X, et al (2023) Observation-centric sort: Rethinking sort for robust multi-object tracking. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 9686–9696
* Beery et al (2019) Beery S, Morris D, Yang S (2019) Efficient pipeline for camera trap image review. _arXiv preprint arXiv:190706772_
* Zhang et al (2021) Zhang H, Wang Y, Dayoub F, et al (2021) Varifocalnet: An iou-aware dense object detector. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 8514–8523
* Liu et al (2021) Liu Z, Lin Y, Cao Y, et al (2021) Swin transformer: Hierarchical vision transformer using shifted windows. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 10012–10022
* Liu et al (2022) Liu Z, Mao H, Wu CY, et al (2022) A convnet for the 2020s. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 11976–11986
* Feichtenhofer (2020) Feichtenhofer C (2020) X3d: Expanding architectures for efficient video recognition. In: _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp 203–213
* Carreira and Zisserman (2017) Carreira J, Zisserman A (2017) Quo vadis, action recognition? a new model and the kinetics dataset. In: _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp 6299–6308
* Tran et al (2018) Tran D, Wang H, Torresani L, et al (2018) A closer look at spatiotemporal convolutions for action recognition. In: _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp 6450–6459
* Bertasius et al (2021) Bertasius G, Wang H, Torresani L (2021) Is space-time attention all you need for video understanding? In: _Proceedings of the International Conference on Machine Learning (ICML)_
* Li et al (2022) Li Y, Wu CY, Fan H, et al (2022) Mvitv2: Improved multiscale vision transformers for classification and detection. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 4804–4814
* Krasin et al (2017) Krasin I, Duerig T, Alldrin N, et al (2017) Openimages: A public dataset for large-scale multi-label and multi-class image classification. _Dataset available from https://storagegoogleapiscom/openimages/web/indexhtml_
* Hara et al (2017) Hara K, Kataoka H, Satoh Y (2017) Learning spatio-temporal features with 3d residual networks for action recognition. In: _Proceedings of the IEEE International Conference on Computer Vision Workshops,_ , pp 3154–3160
* Cui et al (2019) Cui Y, Jia M, Lin TY, et al (2019) Class-balanced loss based on effective number of samples. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 9268–9277
* Menon et al (2020) Menon AK, Jayasumana S, Rawat AS, et al (2020) Long-tail learning via logit adjustment. In: _Proceedings of the International Conference on Learning Representations_
* Alshammari et al (2022) Alshammari S, Wang YX, Ramanan D, et al (2022) Long-tailed recognition via weight balancing. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 6897–6907
* Sechidis et al (2011) Sechidis K, Tsoumakas G, Vlahavas I (2011) On the stratification of multi-label data. In: _Machine Learning and Knowledge Discovery in Databases_ , Springer, pp 145–158
* Szymanski and Kajdanowicz (2019) Szymanski P, Kajdanowicz T (2019) Scikit-multilearn: a scikit-based python environment for performing multi-label classification. _The Journal of Machine Learning Research_ 20(1):209–230
* Liu et al (2019) Liu Z, Miao Z, Zhan X, et al (2019) Large-scale long-tailed recognition in an open world. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 2537–2546
* Perrett et al (2023) Perrett T, Sinha S, Burghardt T, et al (2023) Use your head: Improving long-tail video recognition. In: _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp 2415–2425
* Danielsen et al (2014) Danielsen F, Jensen PM, Burgess ND, et al (2014) A multicountry assessment of tropical resource monitoring by local communities. _BioScience_ 64(3):236–251
* McCarthy et al (2021) McCarthy MS, Stephens C, Dieguez P, et al (2021) Chimpanzee identification and social network construction through an online citizen science platform. _Ecology and Evolution_ 11(4):1598–1608
* Cox et al (2012) Cox T, Philippoff J, Baumgartner E, et al (2012) Expert variability provides perspective on the strengths and weaknesses of citizen-driven intertidal monitoring program. _Ecological Applications_ 22(4):1201–1212
* Nishida et al (1999) Nishida T, Kano T, Goodall J, et al (1999) Ethogram and ethnography of mahale chimpanzees. _Anthropological Science_ 107(2):141–188
* Zamma and Matsusaka (2015) Zamma K, Matsusaka T (2015) Ethograms and the diversity of behaviors, Cambridge University Press, p 510–518
|
Rota-Baxter operators on cocommutative Hopf algebras
Maxim Goncharov
###### Abstract
We generalize the notion of a Rota-Baxter operator on groups and the notion of
a Rota-Baxter operator of weight 1 on Lie algebras and define and study the
notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If
$H=F[G]$ is the group algebra of a group $G$ or $H=U(\mathfrak{g})$ the
universal enveloping algebra of a Lie algebra $\mathfrak{g}$, then we prove
that Rota-Baxter operators on $H$ are in one to one correspondence with
corresponding Rota-Baxter operators on groups or Lie algebras.
Keywords: Rota—Baxter operator, cocommutative Hopf algebra, Rota-Baxter Lie
algebra, Rota-Baxter group.
## 1 Introduction
Given an arbitrary algebra $A$ over a field $F$ and a scalar $\lambda\in F$ a
linear operator $R\colon A\rightarrow A$ is called a Rota—Baxter operator on
$A$ of weight $\lambda$ if for all $x,y\in A$:
$R(x)R(y)=R(R(x)y+xR(y)+\lambda xy)$ (1)
Then the pair $(A,R)$ is called a Rota—Baxter algebra. If $R$ is a Rota-Baxter
operator of weight $\lambda$ and $\alpha\in F$, then $\alpha R$ is a Rota-
Baxter operator of weight $\alpha\lambda$. Thus, there are two principle
cases: when $\lambda=0$ or $\lambda=1$.
Rota-Baxter operators for associative algebras first appear in the paper of G.
Baxter as a tool for studying integral operators in the theory of probability
and mathematical statistics [2].
The combinatorial properties of (commutative) Rota-Baxter algebras and
operators were studied in papers of F.V. Atkinson, P. Cartier, G.-C. Rota and
the others (see [3]-[6]). For basic results and the main properties of Rota-
Baxter algebras see [7].
Independently, in early 80-th Rota-Baxter operators on Lie algebras naturally
appear in papers of A.A. Belavin, V.G. Drinfeld [8] and M.A. Semenov-Tyan-
Shanskii [9] while studying the solutions of the classical Yang-Baxter
equation. It turns out that on quadratic Lie algebras skew-symmetric solutions
of the classical Yang-Baxter equation are in one to one correspondence with
skew-symmetric Rota-Baxter operators.
If $\mathfrak{g}$ is a simple Lie algebra then non-skew-symmetric
$\mathfrak{g}$-invariant solutions of the classical Yang-Baxter equation (that
sometimes called solutions of modified classical Yang-Baxter equation) on
$\mathfrak{g}$ are in one to one correspondence with pairs $(R,B)$, where $R$
is a Rota-Baxter operator of weight 1 satisfying $R+R^{*}+id=0$ and $B$ is a
non-degenerate symmetric bilinear form on $\mathfrak{g}$ [10]. If
$\mathfrak{g}$ is not simple, connections between non-skew-symmetric
$\mathfrak{g}$-invariant solutions of the classical Yang-Baxter equation and
Rota-Baxter operators were considered in [11]. As a consequence of these
results we can note, that every Lie biagrebra structure on a simple Lie
algebra is induced by a Rota-Baxter operators of special type.
When one considers the problem of quantization of a Lie bialgebra
$(\mathfrak{g},\delta)$, one of the first step is to extend the
comultiplication $\delta$ to a Poisson co-bracket on the universal enveloping
algebra $U(\mathfrak{g})$ that can be done uniquely. From this point of view,
it is natural to consider the question of extension of a Rota-Baxter operator
$R$ from a Lie algebra $\mathfrak{g}$ to some reasonable operator on the
universal enveloping algebra $U(\mathfrak{g})$. Unfortunately, it is not
possible to extend $R$ to a Rota-Baxter operator of the algebra
$U(\mathfrak{g})$ (that is, to a linear map $B:U(\mathfrak{g})\mapsto
U(\mathfrak{g})$ satisfying (1)).
Nevertheless, in [12] and [13] it was proved that a structure of a pre- or a
post-Lie algebra on a Lie algebra $\mathfrak{g}$ can be extended to some
reasonable product on the universal enveloping algebra $U(\mathfrak{g})$.
These results can be considered from the point of view of Rota-Baxter
operators: it turns out that a Rota-Baxter operator $R$ of weight $\lambda$ on
$\mathfrak{g}$ induces on $\mathfrak{g}$ a structure of a pre-Lie algebra (if
$\lambda=0$) or a structure of a post-Lie algebra (if $\lambda\neq 0$). Here
again we can ask if we can extend $R$ to an operator $B:U(\mathfrak{g})\mapsto
U(\mathfrak{g})$ on the universal enveloping algebra $U(\mathfrak{g})$ in such
a way that the extension of the pre-(or post-)Lie algebra structure on
$U(\mathfrak{g})$ is somehow induced by $B$.
Recently, in [1] it was introduced the notion of a Rota-Baxter operator (of
weight 1) for groups. If $G$ is a group, then a map $B:G\mapsto G$ is called a
Rota-Baxter operator on the group $G$ if for all $g,h\in G$:
$B(g)B(h)=B(gB(g)hB(g)^{-1}).$
A group $G$ with a Rota-Baxter operator $B$ is called a Rota-Baxter group. In
the same paper it was proved, that if $(G,B)$ is a Rota-Baxter Rota-Baxter Lie
group, then the tangent map of $B$ at the identity is a Rota-Baxter operator
of weight 1 on the Lie algebra of the Lie group $G$. Also, it was showed that
many results that are true for Rota-Baxter operators on algebras have
corresponding analogs for Rota-Baxter operators on groups.
Lie algebras and groups can be regarded as foundations of two principle
examples of cocommutative Hopf algebras. In this paper we in some sense
combine notions of Rota-Baxter operators of weight 1 on Lie algebras and of
Rota-Baxter operators on groups and give the definition of a Rota-Baxter
operator (of weight 1) on cocommutative Hopf algebras. Note, that there
already exist notions of Rota-Baxter of algebras and bialgebras (see [14] and
[15]). These operators are different from the definition that we give.
The paper organised as follows. In section 2 we give the definition of Rota-
Baxter operator on a cocommutative Hopf algebra and obtain some basic results
about it that are generalisations of known results for Rota-Baxter operators
(of weight 1) on groups and algebras. In section 3, we consider two principle
cases of cocommutative Hopf algebras - the universal enveloping algebra
$U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ and the group algebra of a
group $G$. We prove that Rota-Baxter operators on $U(\mathfrak{g})$ (resp. on
$F[G]$) are in one-two-one correspondence with Rota-Baxter operators of weight
1 on $\mathfrak{g}$ (resp, on $G$). Given a Rota-Baxter operator $R$ of weight
1 on a Lie algebra $\mathfrak{g}$, one can define the structure of a post-Lie
algebra on $\mathfrak{g}$. In section 4 we first show, that this extension of
a post-Lie algebra structure to the universal enveloping algebra
$U(\mathfrak{g})$ (that was found in [13]) can be defined using the Rota-
Baxter Hopf operator $B:U(\mathfrak{g})\mapsto U(\mathfrak{g})$ that is the
extension of $R$. Further, we prove that for a given arbitrary Rota-Baxter
Hopf algebra $(H,B)$ one can define new multiplication $*$ and new antipod
$S_{B}$ that define on the space $H$ a structure of a new Hopf algebra that we
call the decedent Hopf algebra.
The author is grateful to Vsevolod Gubarev for his helpful and valuable
comments and suggestions.
## 2 Basic properties of Rota-Baxter operators on cocommutative Hopf algebras
Throughout the paper the characteristic of the ground field $F$ is 0. If $A$
is a vector space over $F$ and $\Delta:A\mapsto A$ is a comultiplication on
$A$, then we will use the following sumless Sweedler notation for the image of
$a\in A$:
$\Delta(a)=a_{(1)}\otimes a_{(2)}.$
In a Hopf algebra $H=(H,\mu,\Delta,\eta,\epsilon,S)$ we use the following
notations:
\- $\mu:H\otimes H\mapsto H$ is a multiplication,
\- $\Delta:H\mapsto H\otimes H$ is a comultiplication,
\- $\eta:F\mapsto H$ is a unit,
\- $\epsilon:H\mapsto F$ is a counit,
\- $S:H\mapsto H$ is the antipode.
If $(A,\Delta,\epsilon)$ is a coalgebra, then a linear map $\varphi:A\mapsto
A$ is called a coalgebra map, if for all $x\in A$:
$\displaystyle\Delta(\varphi(x))=\varphi(x_{(1)})\otimes\varphi(x_{(2)}).$
$\displaystyle\epsilon(\varphi(x))=\epsilon(x).$
A Hopf algebra $H$ is called cocommutative if for all $x\in H$
$x_{(1)}\otimes x_{(2)}=x_{(2)}\otimes x_{(1)}.$
If $H$ is a cocommutative coalgebra, then the antipode $S:H\mapsto H$ is a
coalgebra map. Recall that in arbitrary Hops algebra the antipode $S$ is an
algebra antihomomorphism, that is, for all $a,b\in H$ $S(ab)=S(b)S(a)$.
Definition. Let $(H,\mu,\eta,\Delta,\epsilon,S)$ be a cocommutative Hopf
algebra. A coalgebra map $B:H\mapsto H$ is called a Rota-Baxter operator on
$H$ if for all $x,y\in H$:
$B(x)B(y)=B(x_{(1)}B(x_{(2)})yS(B(x_{(3)}))),$ (2)
where $\Delta(x)=x_{(1)}\otimes x_{(2)}$. By a Rota-Baxter Hopf algebra we
mean a pair $(H,B)$ of a cocommutative Hopf algebra $H$ and a Rota-Baxter
operator $B$ on $H$. As an example on a Rota-Baxter operator on arbitrary
cocommutative Hopf algebra one can consider $B=S$, the antipode (see Corollary
2 below).
Remark. Note, that if $H$ is a commutative and cocommutative Hopf algebra,
then a coalgebra map $B$ is a Rota-Baxter operator if and only if $B$ is an
algebra map, that is $B(xy)=B(x)B(y)$ for all $x,y\in H$.
Lemma 1. Let $H$ be a cocommutative Hops algebra and $B$ be a Rota-Baxter
operator on $H$. Then
(1) If $g\in H$ is a group-like element, then $B(g)$ is also a group-like
element.
(2) $B(1)=1$.
(3) If $x\in L$ is a primitive element, then $B(x)$ is also a primitive
element.
Proof. (1) Let $g\in H$ be a group-like element. Since $B$ is a coalgebra map,
we have
$\Delta(B(g))=(B\otimes B)\Delta(g)=B(g)\otimes B(g).$
And we have two options: $B(g)=0$ or $B(g)$ is a group-like element of $H$.
Since $\epsilon(B(g))=\epsilon(g)=1$, then $B(g)\neq 0$. Therefore, $B(g)$ is
a group-like element of $H$.
(2) Since 1 is a group-like element, then so is $B(1)$. Also, by (2) we have
that
$B(1)B(1)=B(1B(1)1S(B(1))=B(1).$
And since $B(1)$ is inevitable, we get that $B(1)=1$.
(3) Let $x\in H$ be a primitive element. Consider $B(x)$:
$\Delta(B(x))=(B\otimes B)\Delta(x)=1\otimes B(x)+B(x)\otimes 1.$
That’s mean that $B(x)$ is a primitive element of $H$.
It is well known that if $R:\mathfrak{g}\mapsto\mathfrak{g}$ is a Rota-Baxter
operator of weight 1 on a Lie algebra $\mathfrak{g}$, then
$-R-id:\mathfrak{g}\mapsto\mathfrak{g}$ is again a Rota-Baxter operator of
weight 1 on $\mathfrak{g}$. Similar results for groups was proved in [1]: if
$G$ is a group and $B$ is a Rota-Baxter operator on $G$, then
$\tilde{B}:G\mapsto G$ defined by $\tilde{B}(g)=g^{-1}B(g^{-1})$ is also a
Rota-Baxter operator on $G$. For cocommutative Hopf algebras we can generalise
these results:
Proposition 1. Let $H$ be a cocommutative Hopf algebra and $B$ be a Rota-
Baxter operator on $H$. Define $\tilde{B}:H\mapsto H$ as
$\tilde{B}(x)=S(x_{(1)})B(S(x_{(2)})).$
Then $\tilde{B}$ is also a Rota-Baxter operator on $H$.
Proof. Clearly, $\tilde{B}$ is a linear map. Prove that $\tilde{B}$ is a
coalgebra map. Indeed,
$\displaystyle\Delta(\tilde{B}(x))=\Delta(S(x_{(1)})B(S(x_{(2)})))=(S(x_{(2)})\otimes
S(x_{(1)}))(B(S(x_{(4)}))\otimes B(S(x_{(3)}))=$
$\displaystyle=S(x_{(1)})B(S(x_{(2)}))\otimes
S(x_{(3)})B(S(x_{(4)}))=\tilde{B}(x_{(1)})\otimes\tilde{B}(x_{(2)}).$
In order to prove that $\tilde{B}$ is a Rota-Baxter operator consider
$\displaystyle\tilde{B}(x)\tilde{B}(y)=S(x_{(1)})B(S(x_{(2)}))S(y_{(1)})B(S(y_{(2)}))=$
$\displaystyle=S(x_{(1)})\epsilon(x_{(2)})B(S(x_{(3)}))S(y_{(1)})B(S(y_{(2)}))=$
$\displaystyle=S(x_{(1)})B(S(x_{(2)}))S(y_{(1)})\epsilon(B(S(x_{(3)})))B(S(y_{(2)}))=$
$\displaystyle=S(x_{(1)})B(S(x_{(2)}))S(y_{(1)})S(B(S(x_{(3)})))B(S(x_{(4)}))B(S(y_{(2)}))=$
$\displaystyle=hB(S(x_{(3)}))B(S(y_{(2)})),$
where $h=S(x_{(1)})B(S(x_{(2)}))S(y_{(1)})S(B(S(x_{(3)})))$. For $h$ we have:
$\displaystyle
h=S(x_{(1)})B(S(x_{(2)}))S(y_{(1)})S(B(S(x_{(3)})))=\tilde{B}(x_{(1)})S(y_{(1)})S(B(S(x_{(2)})))=$
$\displaystyle=\tilde{B}(x_{(1)})S(y_{(1)})S(B(S(x_{(2)})))x_{(3)}S(x_{(4)})=$
$\displaystyle=\tilde{B}(x_{(1)})S(y_{(1)})S(S(x_{(3)})B(S(x_{(2)})))S(x_{(4)})=\tilde{B}(x_{(1)})S(y_{(1)})S(\tilde{B}(x_{(2)}))S(x_{(3)})=$
$\displaystyle=S(x_{(1)}\tilde{B}(x_{(2)})y_{(1)}S(\tilde{B}(x_{(3)}))).$
Now consider $hB(S(x_{(4)}))B(S(y_{(2)}))$. Using similar arguments as above,
we can conclude that:
$\displaystyle
hB(S(x_{(4)}))B(S(y_{(2)}))=hB(S(x_{(4)})B(S(x_{(5)}))S(y_{(2)})S(B(x_{(6)})))=$
$\displaystyle=hB(S(x_{(4)}\tilde{B}(x_{(5)})y_{(2)}S(\tilde{B}(x_{(6)}))))=\tilde{B}(x_{(1)}\tilde{B}(S(x_{(2)}))y_{(2)}S(\tilde{B}(x_{(3)})).$
And the proposition is proved.
Another well-known result says that if a Lie algebra $\mathfrak{g}$ splits
into direct sum of two subalgebras $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$:
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$, then the map $R$
defined as $R(x_{1}+x_{2})=-x_{2}$, where $x_{i}\in\mathfrak{g}_{i}$, is a
Rota-Baxter operator of weight 1. For groups similar result ([1]) says that if
a group $G$ can be presented as a product of two subgroups $G_{1}$ and $G_{2}$
$G=G_{1}G_{2}$ such that $G_{1}\cap G_{2}=\\{e\\}$, then a map $B$ defined as
$B(g_{1}g_{2})=g_{2}^{-1}$, where $g_{i}\in G_{i}$, is a Rota-Baxter operator
on $G$. Note, that unlike the Lie algebra case, the inverse of the projection
to the first factor is not a Rota-Baxter operator on $G$. We can generalise
these results for cocommutative Hopf algebras as:
Proposition 2. Let $H$ be a cocommutative Hops algebra. Suppose $H_{1}$ and
$H_{2}$ are two Hopf subalgebras of $H$ and as a Hopf algebra $H=H_{1}H_{2}$.
Suppose that the product is direct, that is, $H$ is isomorphic to
$H_{1}\otimes_{F}H_{2}$ as a vector space. Define a map $B$ as
$B(h_{1}h_{2})=\epsilon(h_{1})S(h_{2}),$
where $h_{i}\in H_{i}$. Then $B$ is a Rota-Baxter operator on $H$.
Proof. Clearly, $B$ is a well-defined linear map. First we proof that $B$ is a
coalgebra map. For $x=\sum h_{i}g_{i}$, where $h_{i}\in H_{1}$, $g_{i}\in
H_{2}$ we have:
$\displaystyle\Delta(B(x))=\sum\limits_{i}\epsilon(h_{i})\Delta(S(g_{i}))=\sum\limits_{i}\epsilon(h_{i})S(g_{i(2)})\otimes
S(g_{i(1)})=$
$\displaystyle=\sum\limits_{i}\epsilon(h_{i(1)})S(g_{i(1)})\otimes\epsilon(h_{i(2)})S(g_{i(2)})=(B\otimes
B)\Delta(x).$
In order to prove that $B$ satisfies (2) consider $x=hg\in H$, and
$y=h^{\prime}g^{\prime}$ where $h,h^{\prime}\in H_{1}$, $g,g^{\prime}\in
H_{2}$. We have
$\displaystyle B(x_{(1)}B(x_{(2)})yS(B(x_{(3)})))=$
$\displaystyle=B((h_{(1)}g_{(1)})(\epsilon(h_{(2)})S(g_{(2)}))(h^{\prime}g^{\prime})(\epsilon(h_{(3)})S(S(g_{(3)})))=$
$\displaystyle=B(\epsilon(h_{(1)})\epsilon(h_{(2)})h_{(3)}g_{(1)}S(g_{(2)})g^{\prime}g_{(3)})=B(hh^{\prime}g^{\prime}g)=$
$\displaystyle=\epsilon(hh^{\prime})S(g^{\prime}g)=\epsilon(h)\epsilon(h^{\prime})S(g)S(g^{\prime})=B(x)B(y).$
Since $H_{1}H_{2}$ is spanned by elements of the form $hg$, the equation (2)
holds for all $x,y\in H$.
Remark. Note that as in the case of groups, the operator $B^{\prime}$ defined
as $B(h_{1}h_{2})=S(h_{1})\epsilon(h_{2})$ is not a Rota-Baxter operator on
$H$ in general.
Corollary 1. If $H$ is a cocommutative Hopf algebra with the antipode $S$,
then $B=S$ is a Rota-Baxter operator on $H$.
## 3 Rota-Baxter operators on $F[G]$ and $U(L)$.
In this section we consider two principle examples of cocommutative Hopf
algebra: the group algebra of a group $G$ and the universal enveloping algebra
$U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$.
Theorem 1. Let $(G,B)$ be a Rota-Baxter group. Then $B$ can be uniquely
extended to a Rota-Baxter operator $B:F[G]\mapsto F[G]$ on the group algebra
$F[G]$. Conversely, if $B$ is a Rota-Baxter operator on $F[G]$, then
$B(G)\subset G$ and $(G,B|_{G})$ is a Rota-Baxter group, where $B|_{G}$ is the
restriction of $B$ on $G$.
Proof. Since elements of $B$ form a linear basis of $F[G]$, we can uniquely
extend $B$ on $F[G]$ as
$B(\sum\alpha_{i}g_{i})=\sum\alpha_{i}B(g_{i}).$
It is easy to see that $B$ is a coalgebra map. We need to check that
$(F[G],B)$ is a Rota-Baxter Hopf algebra. Let $x=\sum\alpha_{i}g_{i}\in F[G]$,
$y\in G$. Then
$\displaystyle
B(x)B(y)=\sum\alpha_{i}B(g_{i})B(y)=\sum\alpha_{i}B(g_{i}B(g_{i})yB(g_{i})^{-1})$
$\displaystyle=\sum\alpha_{i}B(g_{i}B(g_{i})hS(B(g_{i})))=B(x_{(1)}B(x_{(2)})yS(B(x_{(3)})).$
And since elements of $G$ form a basis of $F[G]$, the equation (2) holds for
all $x,y\in F[G]$.
Conversely, let $B$ be a Rota-Baxter operator on the Hopf algebra $F[G]$. By
Lemma 1, if $g\in G$, then $B(g)$ is a group-like element. Therefore, $B(g)\in
G$ for every $g\in G$. The rest is obvious.
Lemma 2. Let $\mathfrak{g}$ be a Lie algebra and $R$ be a Rota-Baxter operator
of weight 1 on $\mathfrak{g}$. Then the map $R$ can be extended to a linear
map $B:U(\mathfrak{g})\mapsto U(\mathfrak{g})$ such that
1\. The restriction of $B$ on $\mathfrak{g}$ is $B|_{\mathfrak{g}}=R$.
2\. $B$ satisfies (2).
Proof. Put $B(1)=1$ and if $x,x_{1},\ldots,x_{k}\in\mathfrak{g}$,
$h=x_{1}x_{2}\ldots x_{k}$, then define
$B(xh)=B(x)B(h)-B([B(x),h]).$ (3)
First we need to prove that $B$ is well-defined. Consider elements $f,g\in
U(\mathfrak{g})$, $x,y\in\mathfrak{g}$. We want to prove that $B(f(xy-
yx-[x,y])g)=0$.
If $f=1$, then by the definition
$\displaystyle
B(xyg)=B(x)B(yg)-B([B(x),yg])=B(x)B(yg)-B([B(x),y]g)-B(y[B(x),g])=$
$\displaystyle=B(x)B(y)B(g)-B(x)B([B(y),g])-B([B(x),y])B(g)+B([B([B(x),y]),g])-$
$\displaystyle-B(y)B([B(x),g])+B([B(y),[B(x),g]]).$
Similarly,
$\displaystyle
B(xyg)=B(y)B(x)B(g)-B(y)B([B(x),g])-B([B(y),x])B(g)+B([B([B(y),x]),g])-$
$\displaystyle-B(x)B([B(y),g])+B([B(x),[B(y),g]]).$
And
$\displaystyle B(xyg-yxg)=[B(x),B(y)]B(g)-B([B(x),y]+[x,B(y)])B(g)-$
$\displaystyle+B(B([B(x),y]+B(B([x,B(y)]-[[B(x),B(y)],g])=$
$\displaystyle=B([x,y])B(g)-B(B[x,y],g])=B([x,y]g).$
Suppose that $f=x_{1}\ldots x_{k}$ for some $x_{i}\in\mathfrak{g}$ and use
induction on $k$. Denote by $f_{1}=x_{2}\ldots x_{k}$. We have
$\displaystyle B(f(xy-yx-[x,y])g)=B(x_{1}f_{1}(xy-yx-[x,y])g)=$
$\displaystyle=B(x_{1})B(f_{1}(xy-yx-[x,y])g)-B([B(x_{1}),f_{1}(xy-
yx-[x,y])g]).$
By the induction hypothesis, $B(f_{1}(xy-yx-[x,y])g)=0$. Consider the second
summand.
$\displaystyle B([B(x_{1}),f_{1}(xy-yx-[x,y])g])=B([B(x_{1}),f_{1}](xy-
yx-[x,y])g)+$ $\displaystyle+B(f_{1}[B(x_{1}),xy-yx-[x,y]]g)+B(f_{1}(xy-
yx-[x,y])[B(x_{1}),g]).$
Recall, that $B(x_{1})\in\mathfrak{g}$. Then, by the induction hypothesis,
$B([B(x_{1}),f_{1}](xy-yx-[x,y])g)=B(f_{1}(xy-yx-[x,y])[B(x_{1}),g])=0.$
Note that
$\displaystyle[B(x_{1}),xy-yx-[x,y]]=([B(x_{1}),x]y-y[B(x_{1}),x]-$
$\displaystyle-[[B(x_{1}),x],y])+(x[B(x_{1}),y]-[B(x_{1}),y]x-[x,[B(x_{1}),y]])$
and here we can also use the induction hypothesis to conclude that
$B(f_{1}[B(x_{1}),xy-yx-[x,y]]g)=0$.
Therefore $B$ is well defined. Now prove that $(U(\mathfrak{g}),B)$ is a Rota-
Baxter Hopf algebra. Take $f=x_{1}\ldots,x_{k}$, $x_{i}\in\mathfrak{g}$ and
$q\in U(\mathfrak{g})$ and use induction on $k$. If $k=0$ then clearly
$B(1)B(q)=B(q)=B(1\cdot B(1)q\cdot S(B(1))).$
Suppose that $f=xh$ where $h=x_{2},\ldots,x_{k}$. First note that
$\Delta^{(2)}(xh)=\sum\limits_{(h)}xh_{(1)}\otimes h_{(2)}\otimes
h_{(3)}+h_{(1)}\otimes xh_{(2)}\otimes h_{(3)}+h_{(1)}\otimes h_{(2)}\otimes
xh_{(3)}.$
Then
$\displaystyle
B(f_{(1)}B(f_{(2)})qS(B(f_{(3)})))=B(xh_{(1)}B(h_{(2)})qS(B(h_{(3)})))+B(h_{(1)}B(xh_{(2)})qS(B(h_{(3)})))+$
$\displaystyle+B(h_{(1)}B(h_{(2)})qS(B(xh_{(3)}))).$
Consider the first term. Using 3 and the induction hypotheses, we have
$\displaystyle
B(xh_{(1)}B(h_{(2)})qS(B(h_{(3)})))=B(x)B(h_{(1)}B(h_{(2)})qS(B(h_{(3)})))-$
$\displaystyle-B([B(x),h_{(1)}B(h_{(2)})qS(B(h_{(3)}))])=B(x)B(h)B(g)-B([B(x),h_{(1)}B(h_{(2)})qS(B(h_{(3)}))])$
Similarly,
$\displaystyle B(h_{(1)}B(xh_{(2)})qS(B(h_{(3)})))=$
$\displaystyle=B(h_{(1)}B(x)B(h_{(2)})qS(B(h_{(3)})))-B(h_{(1)}B([B(x,h_{(2)}])qS(B(h_{(3)}))))$
and
$\displaystyle B(h_{(1)}B(h_{(2)})qS(B(xh_{(3)})))=$
$\displaystyle=B(h_{(1)}B(h_{(2)})qS(B(x)B(h_{(3)})))-B(h_{(1)}B(h_{(2)})qS(B([B(x),h_{(3)}])))=$
$\displaystyle=-B(h_{(1)}B(h_{(2)})qS(B(h_{(3)}))B(x))-B(h_{(1)}B(h_{(2)})qS(B([B(x),h_{(3)}])))$
Note that
$\displaystyle-B([B(x),h_{(1)}B(h_{(2)})qS(B(h_{(3)}))])+B(h_{(1)}B(x)B(h_{(2)})qS(B(h_{(3)})))-$
$\displaystyle-B(h_{(1)}B(h_{(2)})qS(B(h_{(3)}))B(x))=-B([B(x),h_{(1)}]B(h_{(2)})qS(B(h_{(3)}))).$
Summing up the obtained equations, we get
$\displaystyle B(f_{(1)}B(f_{(2)})qS(B(f_{(3)})))=$
$\displaystyle=B(x)B(h)B(q)-B([B(x),h_{(1)}]B(h_{(2)})qS(B(h_{(3)})))-$
$\displaystyle-B(h_{(1)}B([B(x,h_{(2)}])qS(B(h_{(3)}))))-B(h_{(1)}B(h_{(2)})qS(B([B(x),h_{(3)}])))$
$\displaystyle=B(x)B(h)B(q)-B([B(x),h])B(q)=B(xh)B(q).$
The lemma is proved.
Lemma 3. Let $\mathfrak{g}$ be a Lie algebra, $R$ be a Rota-Baxter operator on
$\mathfrak{g}$ of weight 1 and $B:U(\mathfrak{g})\mapsto U(\mathfrak{g})$ be
the operator from Lemma 2. Then $B$ is a coalgebra map, that is, for all $f\in
U(\mathfrak{g})$:
$\displaystyle\Delta(B(f))=(B\otimes B)\Delta(f)=B(f_{(1)})\otimes
B(f_{(2)}).$ $\displaystyle\epsilon(B(f))=\epsilon(f).$
Proof. First we proof that $B$ preserves the comultiplication. Take
$f=x_{1},\ldots x_{k}$, where $x_{i}\in\mathfrak{g}$, and use the induction on
$k$. The statement is obvious if $f=1$. If $k=1$ we have
$\Delta(B(x))=B(x)\otimes 1+1\otimes B(x)=(B\otimes B)(\Delta(x)).$
Suppose that $f=xh$, where $h=x_{2},\ldots,x_{k}$. Then
$\Delta(B(xh))=\Delta(B(x)B(h)-B([B(x),h]))=\Delta(B(x))\Delta(B(h))-\Delta(B([B(x),h])$
By the the induction hypotheses we have:
$\Delta(B(h))=B(h_{(1)})\otimes B(h_{(2)})$
and since $B(x)\in L$:
$\Delta(B([B(x),h]))=B([B(x),h_{(1)}])\otimes B(h_{(2)})+B(h_{(1)})\otimes
B([B(x),h_{(2)}]).$
Therefore,
$\displaystyle\Delta(B(x))\Delta(B(h))-\Delta(B([B(x),h]))=(B(x)B(h_{(1)}))\otimes
B(h_{(2)})+B(h_{(1)})\otimes(B(x)B(h_{(2)}))-$
$\displaystyle-B([B(x),h_{(1)}])\otimes B(h_{(2)})-B(h_{(1)})\otimes
B([B(x),h_{(2)}])=$ $\displaystyle=B(xh_{(1)})\otimes
B(h_{(2)})+B(h_{(1)})\otimes B(xh_{(2)})=(B\otimes B)(\Delta(xh)).$
In order to prove that $B$ preserves the counit one can use similar arguments:
take $f=x_{1},\ldots x_{k}$ ($x_{i}\in g$) and use the induction on $k$. The
case $k=1$ is trivial. If $f=x_{1}h$ then
$\displaystyle\epsilon(B(xh))=\epsilon(B(x)B(h))-\epsilon(B[B(x),h])=\epsilon(B(x))\epsilon(B(h))-\epsilon(B([B(x),h]))=$
$\displaystyle=\epsilon(x)\epsilon(h)-\epsilon([B(x),h])=\epsilon(xh).$
The lemma is proved.
Lemma 4. Let $U(\mathfrak{g})$ be a universal enveloping algebra of a Lie
algebra $\mathfrak{g}$, $B$ be a Rota-Baxter operator on $U(\mathfrak{g})$.
Then $B(\mathfrak{g})\subset\mathfrak{g}$ and the restriction
$R=B|_{\mathfrak{g}}$ is a Rota-Baxter operator of weight 1 on $\mathfrak{g}$.
Proof. By Lemma 1, if $x\in\mathfrak{g}$, then $B(x)\in\mathfrak{g}$. Now
consider the restriction $R=B|_{\mathfrak{g}}$. For arbitrary
$x,y\in\mathfrak{g}$ we have
$[R(x),R(y)]=[B(x),B(y)]=B(x)B(y)-B(y)B(x)=B(xy+[B(x),y])-B(yx+[B(y),x])=$
$=B([B(x),y]+[x,B(y)]+[x,y])=R([R(x),y]+[x,R(y)]+[x,y]).$
Therefore, $R$ is a Rota-Baxter operator of weight 1 on the Lie algebra
$\mathfrak{g}$.
Theorem 2. Rota-Baxter operators of weight 1 on a Lie algebra $\mathfrak{g}$
are in one to one correspondence with Rota-Baxter operators on the universal
enveloping algebra $U(\mathfrak{g})$.
Proof. Let $R$ be a Rota-Baxter operator of weight 1 on a Lie algebra
$\mathfrak{g}$. It is only left to prove that the extension of $R$ from Lemma
2 is unique. For this we note, that if $B$ is a Rota-Baxter operator on
$U(\mathfrak{g})$, then from (2) it is follows that
$B(x)B(a)=B(xa)+B([B(x),a])$
for all $x\in\mathfrak{g}$ and $a\in U(\mathfrak{g})$. The rest can be proved
using similar arguments as in Lemma 2.
Example 1. Let $\mathfrak{g}=sl_{2}(F)$, $x,h,y$ be a basis of $sl_{2}(F)$
with the following table of multiplication:
$[h,x]=2x,\quad[h,y]=-2y,\quad[x,y]=h.$
Consider a map $R$ defined as
$R(x)=0,\quad R(h)=-\frac{h}{2},\quad R(y)=-y.$
Then $R$ is a Rota-Baxter operator of weight 1 on $\mathfrak{g}$ ([10]).
Consider the extension $B$ of $R$ on $U(\mathfrak{g})$. Let $a=xb$, where
$b\in U(\mathfrak{g})$. By the definition of $B$:
$B(a)=B(xb)=B(x)B(h)-B([B(x),b])=0.$
Similarly, if $a=yb$, we have:
$B(a)=B(yb)=B(y)B(b)-B([B(y),b])=-yB(b)+B([y,b])=-yB(b)+B(yb)-B(by).$
Therefore, $B(by)=-yB(b)$. Monomials $x^{i}h^{j}y^{k}$ form a linear basis of
$U(\mathfrak{g})$. For them we proved that
$B(x^{i}h^{j}y^{k})=\left\\{\begin{matrix}0,\ \text{if}\ i>0\\\
(-1)^{j+k}\frac{y^{k}h^{j}}{2^{j}},\ \ \text{if}\ i=0.\end{matrix}\right.$
## 4 The descendent Hopf algebra
Definition [16]. Let $(\mathfrak{g},[,])$ be a Lie algebra and $\cdot$ is a
bilinear operation on L. If for all $x,y,z\in g$:
$\displaystyle[x,y]\cdot z=(y\cdot x)\cdot z-y\cdot(x\cdot z)-(x\cdot y)\cdot
z+x\cdot(y\cdot z),$ $\displaystyle x\cdot[y,z]=[x\cdot y,z]+[y,x\cdot z],$
then $(\mathfrak{g},[,],\cdot)$ is called a Post-Lie algebra.
Given a Post-Lie algebra $(\mathfrak{g},[,],\cdot)$, one can define new Lie
bracket on $\mathfrak{g}$ by the formula
$\\{x,y\\}=x\cdot y-y\cdot x+[x,y].$ (4)
If $\mathfrak{g}$ is a Lie algebra and $R$ is a Rota-Baxter operator of weight
1 on $\mathfrak{g}$, then one can define the following structure of a post-Lie
algebra:
$x\cdot y=[R(x),y]$
for all $x,y\in\mathfrak{g}$ [17]. In this case the multiplication (4) is
equal to
$\\{x,y\\}=[R(x),y]+[x,R(y)]+[x,y].$
Definition [1]. If $R$ is a Rota-Baxter operator of weight 1 on a Lie algebra
$\mathfrak{g}$, then the pair $(\mathfrak{g},\\{,\\})$ is called the
descendent Lie algebra of the Rota-Baxter Lie algebra $(\mathfrak{g},R)$.
In the same paper it was proved that
Statement 1 [1]. If $(G,B)$ is a Rota-Baxter group, then $G$ with the product
$g*h=gB(g)hB(g)^{-1}$
is again a group called the descendent group $G_{B}$ of the Rota-Baxter group
$(G,B)$. The inverse of $g\in G_{B}$ in the descendent group is equal to
$B(g)^{-1}g^{-1}B(g)$.
Let $(\mathfrak{g},[,],\cdot)$ be a post-Lie algebra. In [13] it was proved,
that there is a unique extension of the post-Lie product $\cdot$ on the
universal enveloping algebra $U(\mathfrak{g})$ given by
$\displaystyle 1\cdot f=f,$ (5) $\displaystyle xf\cdot g=x\cdot(f\cdot
g)-(x\cdot f)\cdot g,$ (6) $\displaystyle f\cdot(gh)=(f_{(1)}\cdot
g)(f_{(2)}\cdot h)$ (7)
for all $x\in\mathfrak{g}$, $f,g\in U(\mathfrak{g})$.
Define a new multiplication on $U(\mathfrak{g})$ by
$f*g=f_{(1)}(f_{(2)}\cdot g)$
where $f,g\in U(\mathfrak{g})$. In the same paper it was proved that
$(U(\mathfrak{g}),*)$ is isomorphic to the universal enveloping algebra of
$(\mathfrak{g},\\{,\\})$.
Proposition 3. Let $(\mathfrak{g},[,])$ be a Lie algebra, $R$ be a Rota-Baxter
operator on $\mathfrak{g}$ of weight 1, $(\mathfrak{g},[,],\cdot)$ be the
corresponding post-Lie algebra and $(U(\mathfrak{g}),B)$ be the enveloping
Rota-Baxter algebra of $(\mathfrak{g},R)$. Then the extension $\cdot$ of the
post-Lie product on $U(\mathfrak{g})$ can be defined as
$f\cdot g=B(f_{(1)})gS(B(f_{(2)})).$
Proof. Since the extension defined by (5)-(7) is unique, it is enough to prove
that our product satisfies (5)-(7).
Let $x\in\mathfrak{g}$, $f,g,h\in U(\mathfrak{g})$. The first equation is
obvious. Consider (6). We have
$\displaystyle x\cdot(f\cdot g)-(x\cdot f)\cdot
g=x\cdot(B(f_{(1)})gS(B(f_{(2)})))-[B(x),f]\cdot g=$
$\displaystyle=[B(x),B(f_{(1)})gS(B(f_{(2)})]-B([B(x),f_{(1)}])gS(B(f_{(2)}))-$
$\displaystyle-B(f_{(1)})gS([B(x),B(f_{(2)})])=B(xf_{(1)}gS(B(f_{(2)}))+B(f_{(1)})gS(B(xf_{(2)}))=(xf)\cdot
g.$
Consider $(\ref{e3})$. We have
$\displaystyle(f_{(1)}\cdot g)(f_{(2)}\cdot
h)=(B(f))_{(1)}gS((B(f))_{(2)})(B(f))_{(3)}hS((B(f))_{(4)})=$
$\displaystyle=(B(f))_{(1)}g\epsilon((B(f))_{(2)}hS((B(f))_{(3)}))=(B(f))_{(1)}ghS((B(f))_{(2)})=f\cdot(gh).$
The proposition is proved.
Corollary 2. Let $(\mathfrak{g},[,])$ be a Lie algebra, $R$ be a Rota-Baxter
operator of weight 1 on $\mathfrak{g}$ and $B$ be the extension of $R$ on
$U(\mathfrak{g})$ from Lemma 2. Define new multiplication
$*:U(\mathfrak{g})\otimes U(\mathfrak{g})\mapsto U(\mathfrak{g})$ as
$f*g=f_{(1)}B(f_{(2)})gS(B(f_{(3)})).$
Then $(U(\mathfrak{g}),*,\Delta,\eta,\epsilon)$ is a bialgebra isomorphic to
the universal enveloping algebra of the Lie algebra $(g,\\{,\\})$, where
product $\\{,\\}$ is defined as
$\\{x,y\\}=[R(x),y]+[x,R(y)]+[x,y]$
for all $x,y\in\mathfrak{g}$.
Remark. As we will see in Theorem 4, the antipode $S_{B}$ on
$(U(\mathfrak{g}),*,\Delta,\eta,\epsilon)$ is defined as
$S_{B}(x)=S(B(x_{(1)}))S(x_{(2)})B(x_{(3)})$
for all $x\in U(\mathfrak{g})$. Here, $S$ is the ”old” antipode of the
universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$.
We want to generalise Corollary 2 to arbitrary cocommuative Rota-Baxter Hopf
algebra. Let $(H,\mu,\Delta,\eta,\epsilon,S)$ be a cocommutative Hopf algebra
and $B$ a Rota-Baxter operator on $H$. Define new operation on $H$: for all
$x,y\in H$ put
$x*y=x_{(1)}B(x_{2})yS(B(x_{3})).$
Proposition 4. We have that
$\displaystyle\Delta(x*y)=(x_{(1)}*y_{(1)})\otimes(x_{(2)}*y_{(2)}),$
$\displaystyle\epsilon(x*y)=\epsilon(x)\epsilon(y).$
Proof. Consider $x*y=x_{(1)}B(x_{2})yS(B(x_{3}))$. Since $\Delta$ preserves
multiplication, we have
$\Delta(x*y)=\Delta(x_{(1)}B(x_{2})yS(B(x_{3})))=$ $=(x_{(1)}\otimes
x_{(2)})(B(x_{(3)})\otimes B(x_{(4)}))(y_{(1)}\otimes
y_{(2)})(S(B(x_{(6)}))\otimes S(B(x_{(5)})))$
Since the comultiplication is cocommutative, we can rewrite the last term as
$x_{(1)}B(x_{(2)})y_{(1)}S(B(x_{(3)}))\otimes
x_{(4)}B(x_{(5)})y_{(2)}S(B(x_{(6)}))=x_{(1)}*y_{(1)}\otimes x_{(2)}*y_{(2)}.$
Now consider
$\displaystyle\epsilon(x*y)=\epsilon(x_{(1)}B(x_{2})yS(B(x_{3})))=\epsilon(x_{(1)})\epsilon(B(x_{2}))\epsilon(y)\epsilon(S(B(x_{3})))=$
$\displaystyle=\epsilon(x_{(1)})\epsilon(B(x_{2})S(B(x_{3})))\epsilon(y)=\epsilon(x_{(1)})\epsilon(B(x_{(2)})\epsilon(y)=\epsilon(x_{(1)}\epsilon(x_{(2)}))\epsilon(y)=$
$\displaystyle=\epsilon(x)\epsilon(y).$
Define a linear map $S_{B}:H\mapsto H$ as
$S_{B}(x)=S(B(x_{(1)}))S(x_{(2)})B(x_{(3)})$
for all $x\in H$. We will need the following
Proposition 5. For all $x\in H$ we have
$\epsilon(x)1=B(x_{(1)})B(S_{B}(x_{(2)})).$
Proof. Indeed,
$\displaystyle
B(x_{(1)})B(S_{B}(x_{(2)}))=B(x_{(1)}B(x_{(2)})S_{B}(x_{(3)})S(B(x_{(4)})))=$
$\displaystyle=B(x_{(1)}[B(x_{(2)})S(B(x_{(3)}))]S(x_{(4)})[B(x_{(5)})S(B(x_{(6)}))])=$
$\displaystyle
B(x_{(1)}\epsilon(B(x_{(2)}))S(x_{(3)})\epsilon(B(x_{(4)})))=B(x_{(1)}S(x_{(2)}))=\epsilon(x)B(1)=\epsilon(x)1.$
Theorem 4. $H_{B}=(H,*,\Delta,\eta,\epsilon,S_{B})$ is a cocommutative Hopf
algebra.
Proof. Note that since $B$ is a Rota-Baxter operator on $H$, we have that
$B(x*y)=B(x)B(y)$
for all $x,y\in H$.
First we proof that $(H,*)$ is an associative algebra. Indeed, take $x,y,z\in
H$. Using Proposition 4 and Proposition 5, we have
$\displaystyle(x*y)*z=(x_{(1)}*y_{(1)})B(x_{(2)}*y_{(2)})zS(B(x_{(3)}*y_{(3)})=$
$\displaystyle=x_{(1)}B(x_{(2)})y_{(1)}[S(B(x_{(3)}))B(x_{(4)})]B(y_{(2)})zS(B(y_{(3)}))S(B(x_{(5)}))=$
$\displaystyle=x_{(1)}B(x_{(2)})y_{(1)}[\epsilon(B(x_{(3)}))]B(y_{(2)})zS(B(y_{(3)}))S(B(x_{(4)}))=$
$\displaystyle=x_{(1)}B(x_{(2)})y_{(1)}B(y_{(2)})zS(B(y_{(3)}))S(B(x_{(3)}))=x*(y*z).$
Since $B(1)=1$, it is easy to see that $1*x=x*1=x$ for all $x\in H$.
By Proposition 4, $(H,*,\eta,\Delta,\epsilon)$ is a bialgebra. It is left to
proof that $S_{B}$ is the antipode of $(H,*,\Delta,\eta,\epsilon)$. We need to
prove that for all $x\in H$:
$x_{(1)}*S_{B}(x_{(2)})=S_{B}(x_{(1)})*x_{(2)}=\epsilon(x)1.$
Direct computation shows
$\displaystyle
x_{(1)}*S_{B}(x_{(2)})=x_{(1)}[B(x_{(2)})S(B(x_{(3)}))]S(x_{(4)})[B(x_{(5)})S(B(x_{(6)}))]=$
$\displaystyle=x_{(1)}\epsilon(x_{(2)})S(x_{(3)})\epsilon(x_{(4)})=x_{(1)}S(x_{(2)})=\epsilon(x)1.$
Note that $B(x_{(1)}*S_{B}(x_{(2)}))=B(x_{(1)})B(S_{B}(x_{(2)}))$. Then we get
the equality
$B(x_{(1)})B(S_{B}(x_{(2)}))=\epsilon(x)1.$ (8)
Also, we have that
$\displaystyle
B(S_{B}(x_{(1)}))B(x_{(2)})=B(S_{B}(x_{(1)}))\epsilon(B(x_{(2)}))B(x_{(3)})=$
$\displaystyle=\epsilon(B(x_{(1)}))B(S_{B}(x_{(1)}))B(x_{(3)})=S(B(x_{(1)}))[B(x_{(2)})B(S_{B}(x_{(3)}))]B(x_{(4)})=$
$\displaystyle=S(B(x_{(1)}))\epsilon(x_{(2)})B(x_{(3)})=\epsilon(x)1.$
That is, we proved that
$B(S_{B}(x_{(1)}))B(x_{(2)})=\epsilon(x)1.$ (9)
Now consider the second equality. Using (8) and (9) we compute:
$\displaystyle
S_{B}(x_{(1)})*x_{(2)}=S_{B}(x_{(1)})B(S_{B}(x_{(2)}))x_{(3)}S(B(S_{B}(x_{(4))}))=$
$\displaystyle=S(B(x_{(1)}))S(x_{(2)})[B(x_{(3)})B(S_{B}(x_{(4)}))]x_{(5)}S(B(S_{B}(x_{(6))}))=$
$\displaystyle=S(B(x_{(1)}))S(x_{(2)})[\epsilon(x_{(3)})]x_{(4)}S(B(S_{B}(x_{(5))}))=$
$\displaystyle=S(B(x_{(1)}))S(x_{(2)})x_{(3)}S(B(S_{B}(x_{(4))}))=S(B(x_{(1)}))S(B(S_{B}(x_{(2))}))=$
$\displaystyle=S(B(S_{B}(x_{(2)})B(x_{(1)}))=\epsilon(x)1.$
And the theorem is proved.
Proposition 6. $B$ is a homomorphism of Hopf algebras $H$ and $H_{B}$ and is a
Rota-Baxter operator on the Hopf algebra $H_{B}$.
Proof. First we note, that by the definition of the product $*$ and by (2),
for all $x,y\in H$ we have that: $B(x*y)=B(x)B(y)$. It is left to to proof
that $B\circ S_{B}=S\circ B$. For this note that
$S(B(x_{(1)}))B(x_{(2)})=B(x_{(1)})S(B(x_{(2)}))=\epsilon(B(x))=\epsilon(x)1.$
This means that the map $S\circ B$ is the inverse for the map $B$ in
$(End(B),\star,\eta\circ\epsilon)$ where $\star:End(B)\mapsto End(B)$ is the
convolution product defined as
$(f\star g)(x)=f(x_{(1)})g(x_{(2)})$
for all $f,g\in\mathrm{End}(H),\ x\in H$.
On the other hand, for every $x\in H$:
$B(S_{B}(x_{(1)}))B(x_{(2)})=B(S_{B}(x_{(1)})*x_{(2)})=B(\epsilon(x)1)=\epsilon(x)1.$
Similarly, $B(x_{(1)})B(S_{B}(x_{(2)}))=\epsilon(x)1$ and $B\circ S_{B}$ is
also the inverse for $B$ in $(\mathrm{End}(B),\star,\eta\circ\epsilon)$.
Therefore, $B\circ S_{B}=S\circ B$ and $B$ is a homomorphism of Hopf algebras
$H$ and $H_{B}$.
For the second statement we compute:
$\displaystyle
B(x_{(1)}*B(x_{(2)})*y*S(B(x_{(3)})))=B(x_{(1)})B(B(x_{(2)}))B(y)B(S(B(x_{(3)})))=$
$\displaystyle=B(x)*B(y).$
By analogy with Lie algebras and groups, we may give the following
Definition. The Hopf algebra $H_{B}$ is called the descendent Hopf algebra of
the Rota-Baxter Hopf algebra $(H,B)$.
Remark. Corollary 2 says that any descendent Hopf algebra of the universal
enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is the
universal enveloping algebra of the corresponding descendent Lie algebra. And
it is easy to see that if $H=F[G]$ is the group algebra of a group $G$, then
any descendent Hopf algebra $H_{B}$ is the group algebra of the correspondent
descendent group $G_{B}$.
## Acknowledgements
The work was supported by Russian Scientific Fond (project N 19-11-00039).
## References
* [1] L.Guo, H. Lang, Y. Sheng, Integration and Geometrization of Rota-Baxter Lie algebras, arxiv
* [2] Baxter G., _An analytic problem whose solution follows from a simple algebraic identity_ , Pacific J. Math., 10 (1960), 731–742.
* [3] Atkinson, F.V., _Some aspects of Baxter’s functional equation_ , J. Math. Anal. Appl., 7 (1963), 1–30.
* [4] Rota G.C., _Baxter algebras and combinatorial identities I and II_ , Bull. Amer. Math. Soc., 75 (1969), 325–334.
* [5] Miller J.B., _Some properties of Baxter operators_ , Acta Math. Acad. Sci. Hungar., 17 (1966), 387–400.
* [6] Cartier P., _On the structure of free Baxter algebras_ , Adv. Math., 9 (1972), 253–265.
* [7] Guo L., _An Introduction to Rota—Baxter Algebra_ , Surveys of Modern Mathematics, Somerville, MA: International Press; Beijing: Higher education press, 4 (2012).
* [8] Belavin A.A., Drinfeld V.G., _Solutions of the classical Yang—Baxter equation for simple Lie algebras_ , Funct. Anal. Appl., 16:3 (1982), 159–180.
* [9] Semenov-Tyan-Shanskii M.A., _What a classical r-matrix is_ , Funct. Anal. Appl., 17:4 (1983), 259–272.
* [10] Goncharov M.E. On Rota-Baxter operators of non-zero weight arisen from the solutions of the classical Yang-Baxter equation, Sib. El. Math. Rep., 14 (2017) 1533-1544
* [11] Goncharov M.E. Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang-Baxter equation on quadratic Lie algebras, Sib. El. Math. Rep. 16(2019), 2098-2109.
* [12] Oudom, J.-M., Guin D., On the Lie enveloping algebra of a pre-Lie algebra, Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2 (2008), 147–167.
* [13] Ebrahimi-Fard K., Lundervold A., Munthe-Kaas H.Z., On the Lie enveloping algebra of a post-Lie algebra, Journal of Lie Theory 25 (2015), 4, 1139-1165
* [14] Jian R.Q., Zhang J. Rota–Baxter coalgebras. 2014, arXiv:1409.3052.
* [15] Ma T., Liu L., Rota–Baxter coalgebras and Rota–Baxter bialgebras. Linear and Multilinear Algebra, 64 (2016), 968 - 979.
* [16] Vallette B., Homology of generalized partition posets, J. Pure Appl. Algebra 208 (2007), 2, 699–725.
* [17] Bai C., Guo L., Ni X., Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Comm. Math. Phys. 297 (2010), 2, 553–596.
Maxim Goncharov
Novosibirsk State University
Sobolev Institute of Mathematics
Novosibirsk, Russia
e-mail<EMAIL_ADDRESS>
|
# Accounting for survey design in Bayesian disaggregation of survey-based
areal estimates of proportions: an application to the American Community
Survey
Marco H<EMAIL_ADDRESS>[ Veronica
J<EMAIL_ADDRESS>[ Roderick J.
<EMAIL_ADDRESS>[ Nationwide Children’s Hospital
Center for Injury Research and Policy
575 Children’s Crossroad
Columbus, OH 43205
Department of Statistics
School of Information and Computer Sciences
Donald Bren Hall
University of California, Irvine
Irvine, CA 92697
Department of Biostatistics
School of Public Health
1415 Washington Heights
University of Michigan
Ann Arbor, MI 48109
###### Abstract
Understanding the effects of social determinants of health on health outcomes
requires data on characteristics of the neighborhoods in which subjects live.
However, estimates of these characteristics are often aggregated over space
and time in a fashion that diminishes their utility. Take, for example,
estimates from the American Community Survey (ACS), a multi-year nationwide
survey administered by the U.S. Census Bureau: estimates for small municipal
areas are aggregated over 5-year periods, whereas 1-year estimates are only
available for municipal areas with populations $>$65,000. Researchers may wish
to use ACS estimates in studies of population health to characterize
neighborhood-level exposures. However, 5-year estimates may not properly
characterize temporal changes or align temporally with other data in the
study, while the coarse spatial resolution of the 1-year estimates diminishes
their utility in characterizing neighborhood exposure. To circumvent this
issue, in this paper we propose a modeling framework to disaggregate estimates
of proportions derived from sampling surveys which explicitly accounts for the
survey design effect. We illustrate the utility of our model by applying it to
the ACS data, generating estimates of poverty for the state of Michigan at
fine spatio-temporal resolution.
Spatio-temporal change of support problem,
Bayesian hierarchical model,
Multi-resolution approximation,
Latent spatio-temporal process,
American Community Survey,
Survey-based estimates,
###### keywords:
, and
t1Post-doctoral Scientist t2Associate Professor t3Richard D. Remington
Distinguished University Professor
## 1 Introduction
Interest and attention in the social determinants of health, that is, the
social and economic factors that characterize where and how people live, have
soared in the last 20 years (Braverman, Egerter and Williams, 2011; Marmot et
al., 2012) as awareness of health disparities within countries’ populations
has become more prevalent. Discussions on social determinants of health have
also been at the forefront of national news (TV, newspapers, magazines, etc.)
during the first months of the current COVID-19 pandemic, as various social
determinants of health – poverty, homelessness, smoke exposure, etc. – are
suspected to worsen COVID-19 outcomes (Abrams and Szefler, 2020; Rollston and
Galea, 2020; Singu et al., 2020).
A public source of information on social determinants of health is the
American Community Survey (ACS), a multi-year national survey administered by
the United States Census Bureau. Sampling annually approximately 3.5 million
Americans, including those residing in unincorporated territories (U.S. Census
Bureau, 2008), the ACS releases every year up-to-date, timely, and accurate
population and housing information to the general public and to data-users.
Due to privacy concerns and sample size limitations, often these estimates are
aggregated over space and/or time. Currently, ACS estimates for small
municipal sub-divisions, such as census tracts, are aggregated over 5-year
time periods, whereas estimates of neighborhood characteristics corresponding
to 1-year time periods are only available for municipal sub-divisions with
populations greater than 65,000.
This aggregation, if justified by privacy and statistical considerations, can
result in estimates whose spatial and/or temporal resolution is misaligned
with the target spatial and/or temporal resolution of a research study. As an
example, a researcher who wishes to incorporate an ACS estimate of poverty
(e.g. proportion of households living in poverty) in an epidemiological
analysis is typically faced with a choice: (a) utilize estimates with fine
spatial resolution whose 5-year temporal resolution is unlikely to conform to
other data sources and, in the case of longitudinal studies, fail to properly
characterize yearly changes; or (b) utilize 1-year estimates, whose
aggregation over large areal units diminishes their ability to characterize
neighborhoods in a meaningful way. Having access to estimates at fine spatial
and temporal resolution would eliminate these problems.
This is the goal of our paper. Taking the ACS as a case study, we propose a
Bayesian hierarchical spatio-temporal model that aims to generate estimates of
certain social indicators at fine spatial and temporal resolution starting
from estimates - the ACS estimates - that are either available at fine
resolution in space but not in time, or are temporally resolved but not in
space. Hence, we offer a model that solves the so-called spatio-temporal
_change of support problem_ (COSP), that is, the problem of performing
inference about a spatial or spatio-temporal process at a resolution (or
support) that differs from that of the data, in the case of multi-year
estimates of proportions derived from a complex survey. The COSP is one of the
most common problem in spatial statistics, and reviews of methods to address
it can be found in Banerjee, Carlin and Gelfand (2004) and Gotway and Young
(2002). Gelfand, Zhu and Carlin (2001) offer an extension to the space-time
setting. Our model is not the first attempt at solving the COSP for ACS data,
nor it is the first paper that models these data spatially: Bradley, Wikle and
Holan (2015), Bradley, Wikle and Holan (2016), Bradley, Holan and Wikle
(2016), Savitsky (2016), and Simpson et al. (2019) have all contributed to
this literature. In particular, Bradley, Wikle and Holan (2015) were the first
to present a statistical model that derives estimates of a socio-economic
indicator at a different spatial support than that of the ACS data, namely
over three different Native American reservations.
Our model differs from previous work in several ways. First, it deals with
spatio-temporal estimates of proportions: previous efforts considered either
variables that could be modeled using a Gaussian distribution or dealt with
estimates that referred to counts, and thus could be modeled as Poisson random
variables. As we show in Section 5.3, application and adaptation of the
aforementioned models to handle proportions, while they yield point-level
estimates that are for the most part in agreement with the ACS estimates, tend
to underrepresent the estimates’ uncertainty, leading to credible intervals
that, when validated with hold-out data, do not achieve the correct nominal
coverage.
Another key distinction of our modeling approach is that it explicitly
accounts for the survey design effect, thus merging survey methodology with
spatial statistical modeling frameworks. Although Bradley, Wikle and Holan
(2016) did account for the sampling design when modeling ACS estimates for a
given year, they only did so when specifying a COSP model for estimates of
count data: in that case, they provided a model for both the ACS estimates and
the ACS sampling-based variance, leveraging the known relationship between the
mean and the variance of a Poisson distribution. No model for the ACS
sampling-based variance was formulated in the case of Gaussian-distributed
indicators (see Bradley, Wikle and Holan (2015)): rather, the ACS variance was
taken as known and used as the variance of the normal likelihood.
Our model proposes to account for the sampling design in two ways: first, by
including the design effect (Kish, 1965), building upon the work of Korn and
Graubard (1998), Ghitza and Gelman (2013), Mercer et al. (2014), and Chen,
Wakefield and Lumley (2014), secondly by introducing random effects specified
at the spatial resolution of the sampling frame. Specifically, using both the
ACS estimates of proportions and their sampling based variance, we create two
working variables - the _effective number of cases_ (ENC) and the _effective
sample size_ (ESS) - which we use in a Binomial likelihood. Furthermore, since
the ACS sampling design uses counties as sampling frames, to account for the
fact that estimates relative to administrative areal units within the same
county might be more strongly correlated than estimates relative to areal
units that are spatially close but within different counties, our model
introduces county-level random effects.
As in Bradley, Wikle and Holan (2015) and Bradley, Wikle and Holan (2016), we
handle the COSP by assuming that the true area-level proportions result from
the aggregation of an underlying, point-referenced spatio-temporal process
over the specified area. As in Bradley, Wikle and Holan (2015) and Bradley,
Wikle and Holan (2016), such specification allows us to derive estimates over
spatio-temporal resolutions that are equal or larger than the smallest spatial
and temporal resolution for which we have data. In our application, we focus
on generating estimates at the 1-year time scale and at census tract level,
but our modeling framework could be applied to generate estimates over any
type of areal unit. To handle the large number of areal units for which we
have data, another contribution of the paper is to introduce an approximation
that alleviates computation when trying to infer upon a point-referenced,
spatio-temporal process. The approximation, called the Spatio-Temporal Multi-
Resolution Approximation (ST-MRA), is achieved through a novel basis function
expansion, which builds upon the Multi-Resolution Approximation (MRA) of a
Gaussian process presented by Katzfuss (2017).
In its focus on yielding estimates of socio-economic indicators over areal
units, our model shares similarities with other efforts within the rich small-
area estimation (SAE) literature. In particular, of the two broad classes of
methods within SAE (Pfeffermann, 2013), our model fits within the class of
model-based methods. The latter comprises statistical approaches where a
stochastic formulation is offered for the sample data, and optimal predictors,
or approximately optimal predictors, are used to derive estimates of the
quantity of interest. In using the ACS estimates as data and in specifying a
hierarchical model, we follow the same approach as Fay and Herriot (1979),
however, differently from the latter, we account directly for the spatial
dependence in the estimates. Including spatial random effects into SAE model-
based methods is not unheard of: Singh, Shukla and Kundu (2005), Pratesi and
Salvati (2008), Pereira and Coelho (2010), and Porter et al. (2014), to name a
few, have all explicitly accounted for spatial correlation in the estimates.
However, differently from us, these models do not adjust for the sampling
design, nor do they explicitly address the change of support problem in multi-
year survey estimates, which is the raison d’être of our modeling effort.
We apply our model to ACS multi-year estimates of the proportion of families
in Michigan living in poverty, and we show the ability of our model to
generate estimates with high precision, highlighting the potential for this
model to become a tool that can be used by epidemiological researchers to
derive reliable, fine-scale estimates of socioeconomic indicators. These
estimates can be subsequently incorporated into health studies examining the
role of social determinants of health on various health outcomes.
The remainder of this paper is organized as follows. Section 2 provides more
detailed background information on the ACS. Sections 3.1 to 3.5 describe our
modeling framework whereas Section 3.6 provides a succinct description of
alternative models that we apply to survey-based estimates of proportions.
Section 4 illustrates the capabilities of our model in simulation experiments,
while Section 5 presents results for the proportion of families living in
poverty in Michigan from 2006 through 2016. In both cases, the predictive
performance of our model is compared to that of alternative models. The paper
concludes with a discussion in Section 6.
## 2 Data
In this section, we provide general information on the American Community
Survey (ACS) and we present results of an exploratory data analysis performed
on the ACS estimates of the proportion of families living in poverty in
Michigan between 2006 and 2016.
### 2.1 The American Community Survey
The American Community Survey is an ongoing survey conducted by the U.S.
Census Bureau (U.S. Census Bureau, 2008, 2014). It replaced the Census long
form in the 2000 Census. It samples approximately 3.5 million households
annually, collecting data on social, housing, economic, and other community
characteristics. In contrast to the Census long form, for which data were
gathered every 10 years, the ACS surveys are administered continuously,
allowing for the timely dissemination of up-to-date community information that
are statistically representative of the time period during which the surveys
were administered.
A comprehensive report on the ACS sampling methodology is available in (U.S.
Census Bureau, 2014). Here we provide a brief overview and focus on the
sampling of housing units rather than group quarters (e.g. college dormitories
or correctional facilities). The ACS sampling procedure is broken up into two
phases: the first phase consists of the initial sample selection while the
second phase deals with follow-up surveys being sent to unmailable and non-
responding addresses. Housing units are sampled into the ACS independently for
each county in the US. To ensure that no household is selected for the ACS
more than once in a 5-year period, the sampling frame within each county is
subdivided into five disjoint sub-frames, which are rotated through every five
years. For example, ACS surveys from 2006, 2011, and 2016 are all selected
from the same sub-frame. Each year, the first phase of the ACS sampling begins
by sorting any new housing units into one of the five sub-frames.
The ACS sampling rate varies depending on the characteristics of the
neighborhood in which a housing unit resides. Housing units belong to several
municipal sub-divisions of varying sizes, or sampling entities, for example,
the unit’s city, census tract, or school district. Each of these sampling
entities is provided with a measure of size (MOS), which is approximated based
on the number of addresses contained within the entity. Blocks of housing
units are stratified based on the MOS of the smallest sampling entity that
contains that unit, which is referred to as the units’ smallest entity’s
measure of size (SEMOS). The sampling rates for the ACS are inversely
proportional to the housing units’ SEMOS. Tables 4-1 and 4-2 in (U.S. Census
Bureau, 2014) provide details on the ACS sampling rates. Once the initial
sample is selected, each address is assigned a month in which it will receive
the survey.
In the second phase of sampling, follow-up surveys are sent to a set of
randomly selected, non-responding households with higher sampling fractions
for populations with high rates of non-response.
Much like sample selection, computation of the ACS sampling weights takes
place in several stages. The first stage provides a housing unit with a so-
called basic sampling weight, which is inversely related to the unit’s
probability of selection. A series of additional calibrations then occurs,
including adjustments to ensure that the weighted estimates derived from the
ACS conform to the Census Bureau’s Population Estimates Program (PEP).
Weighted estimates of neighborhood characteristics are weighted functions of
survey responses within a neighborhood and time period. Margins of error of
are computed using successive differences replication (U.S. Census Bureau,
2014).
Given statistical accuracy, precision and privacy concerns, ACS estimates are
released with varying spatial and temporal resolution. Specifically, estimates
for small municipal subdivisions, such as census tracts, are aggregated and
provided in the form of averages over a 5-year time period, whereas yearly
estimates are provided for administrative regions that have over 65,000
inhabitants. While certain counties meet this criterion, a sizeable number of
counties in the US have less than 65,000 residents and are therefore excluded
from a dataset with a 1-year temporal resolution ACS estimates. An alternative
to using county-level estimates is to use 1-year estimates at the Public Use
Microdata Areas (PUMA) level, that is, collections of contiguous counties
and/or census tracts whose total population exceeds 100,000 people.
### 2.2 Families in poverty in Michigan
The proportion of families in poverty in an area is one of the indicators that
the US Census Bureau employs to measure poverty in the population. A family is
deemed to live in poverty if the total income of all the family members living
together is lower than a predetermined threshold. There are multiple poverty
thresholds (now, a total of 48) depending on the size of the family and the
age of the family members. Thresholds do not vary geographically across the
U.S. but are updated annually for inflation.
In this paper, we consider data on the proportion of families living in
poverty in Michigan in the period 2006-2016. Specifically, we will utilize
1-year PUMA level estimates (for a total of 68 PUMAs in Michigan) and 5-year
census tract estimates. We will use both sets of estimates to derive census
tract-level, 1-year estimates of the proportion of families living in poverty
in Michigan for every year from 2006 to 2016. Of the 2,813 census tracts in
Michigan, 84 (3%) did not have enough data to provide estimates due to a low
number of residential buildings. Hence, these census tracts were not
considered in the analysis.
Our exploratory data analysis started with an inspection of the 1-year
estimates at the PUMA level, which showed considerable spatial variability in
poverty across Michigan. While in some PUMA’s only 1.4% of the families lived
in poverty, in others that percentage raised to about 45%. However, when
averaged across Michigan, the average proportion of families living in poverty
in Michigan’s PUMAs varied between 12.1% and 12.8% in the period 2006-2016.
Investigating whether the level of poverty changed over time, we fitted a
linear mixed model to the entire times series of estimates. We considered both
a model with a linear time trend and a model with linear and quadratic terms
of time. In both models, the temporal correlation was accounted for through
the inclusion of PUMA-specific intercepts which were assumed to be
independent, identically distributed and following a common normal
distribution. The model with the quadratic trend fitted the data better and
indicated, on average, a growing level of poverty among families in Michigan
from 2006 until 2011 followed by a gradual decline.
As the ACS estimates of the proportion of families in poverty are multi-year
estimates, another goal of our exploratory data analysis was to investigate
the type of spatio-temporal dependence in the data. As we discuss in Section
3.2, our model assumes an underlying, continuous in space, discrete in time
spatio-temporal process driving the true areal proportions. Thus, to examine
the nature of the spatio-temporal dependence, we took the centroids of the
Michigan PUMA’s as observation sites, treated the data as geostatistical data,
and we used two approaches: (i) we conducted a formal Likelihood Ratio Test
(LRT) to assess separability of space and time; and (ii) we performed a more
exploratory investigation based on comparing yearly variograms. As the
boundaries of the PUMAs in Michigan changed following the 2010 Census, in
assessing space-time separability, we split the 2006-2016 data into two sets:
one consisting of data relative to the 2006-2011 pre-boundary-changes period,
and one made of estimates relative to the 2012-2016 time period. Working on
the log scale, and performing the test of separability proposed by Mitchell,
Genton and Gumpertz (2005) on the two sets of data individually, we obtained
LRT values of $7.4\times 10^{-8}$ and $2.4\times 10^{-5}$, respectively,
suggesting time and space separability. We reached a similar conclusion when
comparing the empirical semi-variograms and the associated parameters, derived
using the log of the ACS estimates of the proportion of families in poverty
for each year. Despite some annual variation, the estimates of the marginal
variance and decay parameter were generally similar over time.
In light of these results, available in the Supplementary Material, when
modeling the underlying process driving the true, areal proportion of families
living in poverty in Michigan, we decided to adopt a separable space-time
covariance function.
## 3 Modeling Approach
Our model uses both the 5-year ACS estimates at census tract level, and the
1-year ACS estimates at the PUMA level. Following Bradley, Wikle and Holan
(2015), we denote by $z_{t}^{(l)}(A)$ the estimate of a proportion
corresponding to areal unit $A$ for the $l$-year time period ending in time
$t$. Thus, $z_{t}^{(5)}(A_{ig})$ indicates the ACS estimate for the 5-year
time period ending at year $t$ for census tract $g$, $g=1,\ldots,G_{i}$,
within PUMA $i$, $i=1,\ldots,N$, whereas $z_{t}^{(1)}(A_{i})$ refers to the
1-year ACS estimate for PUMA $i$ at year $t$. We denote by
$\tau^{2(5)}_{t}(A_{ig})$ and $\tau^{2(1)}_{t}(A_{i})$ the design-based
variance of $z_{t}^{(5)}(A_{ig})$ and $z_{t}^{(1)}(A_{i})$, respectively,
derived from the margins of error provided in the ACS dataset.
### 3.1 Modeling survey-based estimates of areal proportions accounting for
the design effect
Following Bradley, Wikle and Holan (2015), we assume that the survey-based
estimate of the proportion corresponding to areal unit $A$ over the $l$-unit
time period ($l=1$ or $5$), ending at time $t$, $z_{t}^{(l)}(A)$, is related
to the true proportion, $\pi_{t}^{(l)}(A)$, through some distribution
function.
A first idea would be to model the ACS estimate $z_{t}^{(l)}(A)$ as following
a normal distribution with mean equal to the true proportion,
$\pi_{t}^{(l)}(A)$, and variance equal to the design-based variance
$\tau^{2(l)}_{t}(A)$. However, as also noted in another context by Chen,
Wakefield and Lumley (2014), such modeling choice would be inaccurate for
small samples and will not ensure that the estimated $\pi_{t}^{(l)}(A)$
belongs to the interval $[0,1]$. For this reason, building upon the work of
Korn and Graubard (1998), and following Mercer et al. (2014) and Chen,
Wakefield and Lumley (2014), we introduce a working likelihood for a random
variable $q^{*(l)}_{t}(A)$ that we construct from the ACS estimate
$z_{t}^{(l)}(A)$ and from the effective sample size $m^{*(l)}_{t}(A)$. The
latter represents the sample size that a simple random sample (SRS) should
have to yield an estimator for the proportion that has a variance that matches
the design-based variance of the ACS estimate. To derive the effective sample
size, we use the notion of design effect $d$ introduced by Kish (1995), who
calls a survey’s design effect the ratio of the variance of an estimator under
SRS to the sampling-based variance of a survey-based estimator. By setting the
design effect equal to 1 and solving the equation for the SRS sample size
$m_{t}^{(l)}(A)$, we obtain the sample size of the SRS that will yield an
estimator with variance corresponding to the ACS design-based variance. We
call this sample size the effective sample size (ESS). Including the ESS in
the distribution function that relates $q^{*(l)}_{t}(A)$ to the true
proportion $\pi_{t}^{(l)}(A)$ allows us to account for the survey’s design
effect in our modeling framework.
More specifically, in the case of $z_{t}^{(l)}(A)$, for a SRS of size
$m_{t}^{(l)}(A)$, the estimated variance of $z_{t}^{(l)}(A)$ would be equal to
$\frac{z_{t}^{(l)}(A)(1-z_{t}^{(l)}(A))}{m_{t}^{(l)}(A)}$. Setting the
survey’s design effect $d$ equal to 1, yields the following equation
$\tau^{2(l)}_{t}(A)=\frac{z_{t}^{(l)}(A)(1-z_{t}^{(l)}(A))}{m_{t}^{(l)}(A)},$
leading to the following expression for the effective sample size,
$m_{t}^{*(l)}(A)$:
$m_{t}^{*(l)}(A)=\left[\frac{z_{t}^{(l)}(A)(1-z_{t}^{(l)}(A))}{\tau^{2(l)}_{t}(A)}\right],$
(3.1)
with $\left[\cdot\right]$ denoting rounding to the nearest integer. Although
not necessarily needed in (3.1), we introduce rounding to ensure that the
effective sample size is an integer.
We then use the effective sample size $m_{t}^{*(l)}(A)$ and the ACS estimate
$z_{t}^{(l)}(A)$ for areal unit $A$ and for the $l$-year time period ending in
year $t$ to derive the effective number of cases, $q_{t}^{*(l)}(A)$, for the
same areal unit and for the same time period, that is:
$q_{t}^{*(l)}(A):=\left[m_{t}^{*(l)}(A)\cdot z_{t}^{(l)}(A)\right],$ (3.2)
again rounded to the nearest integer. This quantity represents the number of
cases that we would have observed in a SRS of size $m_{t}^{*(l)}(A)$ to obtain
an estimate of the proportion $\pi_{t}^{(l)}(A)$ that is equal to the ACS
estimate $z_{t}^{(l)}(A)$ and with the same variance as the design-based
variance $\tau^{2(l)}_{t}(A)$.
Using now the effective number of cases in our working likelihood, our
Bayesian hierarchical model specifies at the first stage a Binomial likelihood
for $q_{t}^{*(l)}(A)$ with number of trials equal to $m_{t}^{*(l)}(A)$ and
success probability equal to the true proportion, $\pi_{t}^{(l)}(A)$, our
parameter of interest, i.e.:
$q_{t}^{*(l)}(A)|\;\pi_{t}^{(l)}(A)\sim\text{Binomial}\left(m_{t}^{*(l)}(A),\pi_{t}^{(l)}(A)\right).$
As we fit our model to 5-year and 1-year ACS estimates of areal proportions,
from (3.1) and (3.2), we derive the corresponding number of cases and
effective sample sizes - $q_{t}^{(5)}(A_{ig}),m_{t}^{*(5)}(A_{ig})$ and
$q_{t}^{(1)}(A_{i}),m_{t}^{*(1)}(A_{i})$ \- which we employ in our working
likelihood, made of the following two components
$\displaystyle q_{t}^{*(5)}(A_{ig})|\;\pi_{t}^{(5)}(A_{ig})$
$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}$
$\displaystyle\text{Binomial}\left(m_{t}^{*(5)}(A_{ig}),\pi_{t}^{(5)}(A_{ig})\right)$
$\displaystyle q_{t}^{*(1)}(A_{i})|\;\pi_{t}^{(1)}(A_{i})$
$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}$
$\displaystyle\text{Binomial}\left(m_{t}^{*(1)}(A_{i}),\pi_{t}^{(1)}(A_{i})\right)$
(3.3)
with $g=1,\ldots,G_{i}$ and $i=1,\ldots,N$.
We note that in (3.3) we are following the tradition of spatial generalized
linear models (see Diggle, Tawn and Moyeed (1998) for details) where spatial
dependence in the data is accounted for by assuming that the model parameters
are spatially correlated.
To disaggregate the ACS estimates, we link $\pi_{t}^{(5)}(A_{ig})$ and
$\pi_{t}^{(1)}(A_{i})$ to $\pi_{t}^{(1)}(A_{ig})$,
$g=1,\ldots,G_{i};i=1,\ldots,N$, the true proportions at our desired spatial
and temporal resolution, via:
$\displaystyle\pi_{t}^{(5)}(A_{ig})$ $\displaystyle=$
$\displaystyle\frac{1}{5}\sum_{k=t-4}^{t}\pi_{k}^{(1)}(A_{ig})$
$\displaystyle\pi_{t}^{(1)}(A_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{t}(A_{i})}\sum_{h=1}^{G_{i}}N_{t}(A_{ih})\pi_{t}^{(1)}(A_{ih})$
(3.4)
where $N_{t}(A)$ generally denotes the number of households in areal unit $A$
at time $t$.
### 3.2 Addressing the Change of Support Problem (COSP)
In practice, we may want to infer about proportions over areal units that are
not conveniently comprised of combinations of $A_{ig}$ and/or $A_{i}$. To this
end, we further decompose $\pi_{t}^{(1)}(A_{ig})$, i.e. the true proportion at
one-year and census tract resolution. Following in the tradition of models
handling the spatial and spatio-temporal COSP, we assume that a random
variable for an areal unit and a time period can be expressed as the
aggregation over the areal unit and the time period of a point-referenced
spatio-temporal process, continuous in space and discrete in time. Because the
process is discrete in time:
$\pi_{t}^{(5)}(A_{ig})=\frac{1}{l}\sum_{k=t-l+1}^{t}\pi_{k}^{(1)}(A_{ig})\qquad
g=1,\ldots,G_{i};\;i=1,\ldots,N.$
To allow the flexibility to work over any areal unit, we link
$\pi_{t}^{(1)}(A_{ig})$ to an underlying point-referenced spatio-temporal
process $\zeta_{t}(\mathbf{s})$, $\mathbf{s}\in\mathcal{S}$, via the probit
link function, $\Phi^{-1}(\cdot)$, thus yielding
$\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)=\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\zeta_{t}(\mathbf{s})d\mathbf{s}+\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig}),$
(3.5)
with $\epsilon_{t}(A_{ig})\stackrel{{\scriptstyle
iid}}{{\sim}}N(0,\tau^{2}_{\epsilon})$ and
$\xi(C_{A_{ig}})\stackrel{{\scriptstyle iid}}{{\sim}}N(0,\tau^{2}_{C})$. In
(3.5), $\epsilon_{t}(A_{ig})$ denote i.i.d. error terms that account for model
specification error in linking $\pi^{(1)}_{t}(A_{ig})$ to the latent process
$\zeta_{t}(\mathbf{s})$, whereas $\xi(C_{A_{ig}})$ denotes a random effect
defined at the same areal unit level as the clustering units of the sampling
survey. In (3.5), $C_{A_{ig}}$ indicates the cluster that contains areal unit
$A_{ig}$. The cluster-level random effect, $\xi(C_{A_{ig}})$, is introduced to
enforce stronger dependence among certain estimates in a way that is
reflective of the survey sampling design.
To provide an interpretation of the spatio-temporal process
$\zeta_{t}(\mathbf{s})$, $\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, in (3.5), we
consider the application to the proportion of families in poverty in any areal
unit $A$. In this case, $\zeta_{t}(\mathbf{s})$ represents a function of the
likelihood that a family living at location $\mathbf{s}\in A$ is in poverty in
year $t$. Decomposing the point-referenced spatio-temporal process
$\zeta_{t}(\mathbf{s})$, $\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, into a
large-scale spatio-temporal trend, $\mu_{t}(\mathbf{s})$, representing the
mean of the process, and a spatio-temporal random effect, $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, (3.5) becomes:
$\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)=\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\left(\mu_{t}(\mathbf{s})+w_{t}(\mathbf{s})\right)d\mathbf{s}+\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig}).$
(3.6)
In light of the results of our exploratory data analysis, discussed in Section
2.2, we model the spatio-temporal random effect, $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, as a Gaussian spatio-temporal process
with a separable space-time covariance function with an AR(1) structure in
time and a spatial dependence encoded through the covariance function
$C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})$,
$\mathbf{s},\mathbf{s}^{\prime}\in\mathcal{S}$.
For computations involving a large number of areal units, we approximate the
spatio-temporal process
$w_{t}(\mathbf{s}),\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, with a linear
combination of spatial basis functions with appropriate spatio-temporal basis
function weights. Given the nested geographies of the ACS, we elect to use the
basis functions implied by the Multi-Resolution Approximation (MRA; Katzfuss
(2017)), also characterized by a nested structure.
### 3.3 The Spatio-Temporal Multi-resolution Approximation (ST-MRA)
As the MRA is defined only in a spatial context, here we extend it to the
spatio-temporal setting. Let
$w_{t}(\mathbf{s}),\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, denote a mean-zero
spatio-temporal Gaussian process defined on a spatial domain $\mathcal{S}$
with a separable space-time covariance function that invokes a first-order
autoregressive structure in time and spatial covariance function
$C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})$,
$\mathbf{s},\mathbf{s}^{\prime}\in\mathcal{S}$. As in the MRA, we start by
introducing a first set of $r$ knots on the spatial domain $\mathcal{S}$
(level 0). Then, at each level $m$ ($m=1,\ldots,M$), we recursively partition
the spatial domain $\mathcal{S}$ in $J^{m}$ non-overlapping subregions in
which we introduce $r$ knots. Let $S^{*}_{m,j}$ denote the set of $r$ knots
defined on partition $j$ of level $m$. We define the basis functions
$\mathbf{b}_{m,j}(\mathbf{s})$, for $j=1,\ldots,J^{m};m=0,\ldots,M$
recursively as:
$\displaystyle v_{0}(\mathbf{s}_{1},\mathbf{s}_{2})$ $\displaystyle=$
$\displaystyle C(\mathbf{s}_{1},\mathbf{s}_{2};\boldsymbol{\theta})$
$\displaystyle\mathbf{b}_{m,j}(\mathbf{s})$ $\displaystyle:=$ $\displaystyle
v_{m}(\mathbf{s},S^{*}_{m,j})$ $\displaystyle\mathbf{K}^{-1}_{m,j}$
$\displaystyle:=$ $\displaystyle v_{m}(S^{*}_{m,j},S^{*}_{m,j})$
$\displaystyle v_{m+1}(\mathbf{s}_{1},\mathbf{s}_{2})$ $\displaystyle=$
$\displaystyle 0\qquad\text{ if }\mathbf{s}_{1}\text{ and
}\mathbf{s}_{2}\text{ are in different regions at resolution $m$}$ (3.7)
$\displaystyle v_{m+1}(\mathbf{s}_{1},\mathbf{s}_{2})$ $\displaystyle:=$
$\displaystyle
v_{m}(\mathbf{s}_{1},\mathbf{s}_{2})-\mathbf{b}_{m,j}(\mathbf{s}_{1})^{\prime}\mathbf{K}_{m,j}\mathbf{b}_{m,j}(\mathbf{s}_{2})\qquad\text{
otherwise.}$
In the MRA construction (Katzfuss, 2017), the basis functions weights
$\boldsymbol{\eta}_{m,j}$ are specified to follow a multivariate normal
distribution $\boldsymbol{\eta}_{m,j}\sim N_{r}(\mathbf{0},\mathbf{K}_{m,j})$.
With this specification for the basis functions and the basis functions
weights, the linear combination
$\sum_{m=0}^{M}\sum_{j=1}^{J^{m}}\mathbf{b}_{m,j}(\mathbf{s})\boldsymbol{\eta}_{m,j}$
yields an $M$-level approximation to a mean-zero Gaussian process with
covariance function $C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})$.
For our spatio-temporal process $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, we let the basis function weights
$\boldsymbol{\eta}_{t,m,j}$ vary in time, modeling them with a stationary,
first-order autoregressive structure (Gelfand, Banerjee and Gamerman, 2005).
Hence, at time $t=1$ we assume that $\boldsymbol{\eta}_{1,m,j}\sim
N_{r}(\mathbf{0},\mathbf{K}_{m,j})$, while for $t=2,\dots,T$:
$\displaystyle\boldsymbol{\eta}_{t,m,j}|\boldsymbol{\eta}_{t-1,m,j},\boldsymbol{\eta}_{t-2,m,j},\ldots,\boldsymbol{\eta}_{1,m,j}$
$\displaystyle\sim$ $\displaystyle
N_{r}(\alpha\boldsymbol{\eta}_{t-1,m,j},\mathbf{U}_{m,j}),$ (3.8)
$\displaystyle\mathbf{U}_{m,j}$ $\displaystyle=$
$\displaystyle(1-\alpha^{2})\mathbf{K}_{m,j}.$
We call
$w_{t,M}(\mathbf{s}):=\sum_{m=0}^{M}\sum_{j=1}^{J^{m}}\mathbf{b}_{m,j}(\mathbf{s})\boldsymbol{\eta}_{t,m,j}$
(3.9)
with basis functions $\mathbf{b}_{m,j}(\mathbf{s})$ defined as in (3.7), the
M-level ST-MRA approximation of the separable, spatio-temporal process
$w_{t}(\mathbf{s})$, $\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, with AR(1)
dependence in time and spatial covariance function
$C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})$. Section 1 of the
Supplementary Material (Benedetti, Berrocal and Little, 2021) shows that the
above expression does indeed provide an approximation of the desired spatio-
temporal dependence structure.
### 3.4 The Bayesian spatio-temporal disaggregation model
Combining the formulations in Sections 3.2 and 3.3, we obtain:
$\displaystyle\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)$ $\displaystyle=$
$\displaystyle\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\left(\mu_{t}(\mathbf{s})+w_{t}(\mathbf{s})\right)d\mathbf{s}+\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig})$
(3.10) $\displaystyle\approx$
$\displaystyle\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\left(\mu_{t}(\mathbf{s})+w_{t,M}(\mathbf{s})\right)d\mathbf{s}+\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig})$
$\displaystyle=$ $\displaystyle\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\left(\mu_{t}(\mathbf{s})+\sum_{m=0}^{M}\sum_{j=1}^{J^{m}}\mathbf{b}_{m,j}(\mathbf{s})\boldsymbol{\eta}_{t,m,j}\right)d\mathbf{s}+\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig}),$
where the spatio-temporal random effect $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S}$, has been replaced by its $M$-level ST-MRA
approximation $w_{t,M}(\mathbf{s})$ defined in (3.9).
As the sampling frames in the ACS survey consist of counties, denoting by
$C_{A_{ig}}$ the county containing census tract $A_{ig}$, our model for
disaggregating spatially and temporally the ACS estimates of proportions, has
the following hierarchical specification:
$\displaystyle q_{t}^{*(5)}(A_{ig})|\;\pi_{t}^{(5)}(A_{ig})$
$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}$
$\displaystyle\text{Binomial}\left(m_{t}^{*(5)}(A_{ig}),\pi_{t}^{(5)}(A_{ig})\right)$
$\displaystyle q_{t}^{*(1)}(A_{i})|\;\pi_{t}^{(1)}(A_{i})$
$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}$
$\displaystyle\text{Binomial}\left(m_{t}^{*(1)}(A_{i}),\pi_{t}^{(1)}(A_{i})\right)$
$\displaystyle\pi_{t}^{(5)}(A_{ig})$ $\displaystyle=$
$\displaystyle\frac{1}{5}\sum_{k=t-4}^{t}\pi_{k}^{(1)}(A_{ig})$ (3.11)
$\displaystyle\pi_{t}^{(1)}(A_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{t}(A_{i})}\sum_{h=1}^{G_{i}}N_{t}(A_{ih})\pi_{t}^{(1)}(A_{ih})$
$\displaystyle\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)$
$\displaystyle\approx$ $\displaystyle\frac{1}{|A_{ig}|}\int_{\mathbf{s}\in
A_{ig}}\left(\mu_{t}(\mathbf{s})+\sum_{m=0}^{M}\sum_{j=1}^{J^{m}}\mathbf{b}_{m,j}(\mathbf{s})\boldsymbol{\eta}_{t,m,j}\right)d\mathbf{s}$
$\displaystyle+$ $\displaystyle\xi(C_{A_{ig}})+\epsilon_{t}(A_{ig})$
$\displaystyle\xi(C_{A_{ig}})$ $\displaystyle\stackrel{{\scriptstyle
iid}}{{\sim}}$ $\displaystyle N(0,\tau_{C}^{2})$
$\displaystyle\epsilon_{t}(A_{ig})$ $\displaystyle\stackrel{{\scriptstyle
iid}}{{\sim}}$ $\displaystyle N(0,\tau^{2}_{\epsilon})$
with $q_{t}^{*(5)}(A_{ig})$, $q_{t}^{*(1)}(A_{i})$, $m_{t}^{*(5)}(A_{ig})$,
and $m_{t}^{*(1)}(A_{i})$ defined as in (3.2) and (3.1), respectively. The
county-level random effects in the expression of
$\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)$ allow the ACS estimates for
census tracts within the same county to exhibit greater dependence with one
another than with ACS estimates for census tracts in different counties, even
when the distances between those tracts are the same. We speculate that this
will account for the fact that factors such as sampling procedure or response
rate within a county-wide sampling frame might systematically affect ACS
estimates corresponding to most or all of the census tracts within that
county.
The integral in (3.10) can be re-expressed as:
$\displaystyle\Phi^{-1}\left(\pi_{t}^{(1)}(A_{ig})\right)$
$\displaystyle\approx$
$\displaystyle\mu_{t}(A_{ig})+\sum_{m=0}^{M}\sum_{j=1}^{J^{m}}\mathbf{b}_{m,j}(A_{ig})\boldsymbol{\eta}_{t,m,j}+\xi(C_{A_{ig}})+\tilde{\epsilon}_{t}(A_{ig})$
$\displaystyle\tilde{\epsilon}_{t}(A_{ig})$
$\displaystyle\stackrel{{\scriptstyle iid}}{{\sim}}$ $\displaystyle
N(0,\tau^{2}),$ (3.12)
where $\mu_{t}(A_{ig})$ and $\mathbf{b}_{m,j}(A_{ig})$ denote the integrals of
$\mu_{t}(\mathbf{s})$ and of the basis functions
$\mathbf{b}_{m,j}(\mathbf{s})$, $m=0,\ldots,M$; $j=1,\ldots,J^{m}$, as
$\mathbf{s}$ varies in areal unit $A_{ig}$, with
$g=1,\ldots,G_{i};i=1,\ldots,N$. The term $\tilde{\epsilon}_{t}(A_{ig})$ in
(3.12) accounts for errors due to model misspecification, aggregation as well
as any error that occurs as a result of the multi-resolution space-time
approximation.
### 3.5 Prior distributions
Our Bayesian model includes Inverse Gamma prior distributions for the error
variance parameter $\tau^{2}$ in (3.12), and for the variance of the county-
level random effects, $\tau^{2}_{C}$. We assume
$\mu_{t}(\mathbf{s})\equiv\mu_{t},t=1,2,...,T$, and model these spatially-
constant temporal trend terms as independent a priori, with an improper prior
$p(\mu_{t})\propto 1,\forall t$. This modeling choice implies that $\forall
t=1,2,\ldots,T$, the spatio-temporal random effect
$w_{t}(\mathbf{s}),\mathbf{s}\in\mathcal{S}$, accounts for all the spatial
variation in the ACS estimates. We investigated whether allowing the mean
terms $\mu_{t}$ vary in space would lead to significantly different results in
terms of model fit, but we did not observe any meaningful change.
We assign a Uniform $([0,1])$ prior to the autoregressive parameter $\alpha$
of the basis functions weights in (3.8), while the definition of the ST-MRA
basis functions is determined once we choose the spatial covariance function
$C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})$. Here we take it to be
the stationary Matérn covariance function with parameters $\sigma^{2}$, $\phi$
and $\nu$
$C(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\theta})=\frac{\sigma^{2}}{2^{\nu-1}\Gamma(\nu)}\left(\frac{||\mathbf{s}-\mathbf{s}^{\prime}||}{\phi}\right)^{\nu}\mathcal{K}_{\nu}\left(\frac{||\mathbf{s}-\mathbf{s}^{\prime}||}{\phi}\right)$
(3.13)
where $\mathbf{s},\mathbf{s}^{\prime}\in\mathcal{S}$ and
$\mathcal{K}_{\nu}(\cdot)$ is the modified Bessel function of the second kind.
We specify a non-informative Inverse Gamma prior on the marginal variance
parameter $\sigma^{2}$, while we place a Gamma$(1,1)$ prior on the range
parameter $\phi$ and a Uniform$((0,2))$ prior on the smoothness parameter
$\nu$, as suggested by Finley, Banerjee and Carlin (2007). As the latter is
notoriously difficult to estimate, an alternative specification would entail
the use of penalized complexity priors as described in Simpson et al. (2017).
### 3.6 Other models
We describe succinctly alternative models that we compare with our model in
Sections 4 and 5. More details are available in Section 3 of the Supplementary
Material (Benedetti, Berrocal and Little, 2021). To evaluate the utility of
the _effective sample size_ and _effective number of cases_ , a first
competing model specifies a “standard” Binomial likelihood for the number of
cases, obtained by multiplying the ACS estimate, $z^{(l)}_{t}(A)$, by the
number of survey responses, $m_{t}^{(l)}(A)$, obtained in areal unit $A$ over
the $l$-unit time period ending in year $t$. Calling this product
$q^{(l)}_{t}(A)$, the _standard Binomial model_ for disaggregation applied to
the 1-year and 5-year ACS data assumes that
$\begin{array}[]{rcl}q_{t}^{(1)}(A_{i})|\pi^{(1)}_{t}(A_{i})&\sim&\mbox{Binomial}\left(m^{(1)}_{t}(A_{i}),\pi^{(1)}_{t}(A_{i})\right)\\\
q^{(5)}_{t}(A_{ig})|\pi^{(5)}_{t}(A_{ig})&\sim&\mbox{Binomial}\left(m^{(5)}_{t}(A_{ig}),\pi^{(5)}_{t}(A_{ig})\right).\end{array}$
We keep the other levels of this model exactly as in the Bayesian hierarchical
model in (3.11).
The second and third model we consider are extensions and adaptations of
models proposed by Bradley, Wikle and Holan (2016) and Bradley, Wikle and
Holan (2015), respectively, when analyzing Poisson spatial-only and Gaussian
space-time ACS data.
The _BWH Poisson space-time model_ extends the model for count data proposed
by Bradley, Wikle and Holan (2016) to the space-time setting. Interpreting the
counts $q^{(1)}_{t}(A_{i})$ and $q^{(5)}_{t}(A_{ig})$ as Poisson random
variables, we assume a latent process $Y_{t}(A_{ig})$, $t=1,\ldots,T$, defined
at the census tract level, such that
$\begin{array}[]{rcl}q^{(1)}_{t}(A_{i})|\left\\{Y_{t}(A_{ig});g=1,\ldots,G_{i},i=1,\ldots,N\right\\}&\sim&\mbox{Poisson}\left(\sum_{h=1}^{G_{i}}\exp\left(Y_{t}(A_{ih})\right)\right)\\\
q^{(5)}_{t}(A_{ig})|\left\\{Y_{k}(A_{ig});k=1,\ldots,T\right\\}&\sim&\mbox{Poisson}\left(\frac{1}{5}\sum_{k=t-4}^{t}\exp\left(Y_{k}(A_{ig})\right)\right)\end{array}$
for $g=1,\ldots,G_{i};i=1,\ldots,N$ and $t=1,\ldots,T$. For each
$t=1,\ldots,T$, following Bradley, Wikle and Holan (2016), $Y_{t}(A_{ig})$ is
decomposed as:
$Y_{t}(A_{ig})=\beta_{t}+\boldsymbol{\psi}\boldsymbol{\vartheta}+\varsigma_{t}(A_{ig})$
(3.14)
with $\boldsymbol{\psi}$ Moran’s basis functions, $\boldsymbol{\vartheta}$
basis functions weights defined as in Bradley, Wikle and Holan (2016), and
$\varsigma_{t}(A_{ig})$ error terms that account for aggregation and other
types of errors. Differently from Bradley, Wikle and Holan (2016), here we are
dealing with estimates over multiple years: to accommodate this added
dimension, in (3.14) we allow the intercept terms $\beta_{t}$ to vary in time,
hence representing a temporal trend. Similarly, we allow the error terms
$\varsigma_{t}(A_{ig})$ to change in time. Finally, as in Bradley, Wikle and
Holan (2016), the _BWH Poisson space-time model_ provides a stochastic
formulation for the ACS design based variances $\tau^{2(1)}_{t}(A_{i})$ and
$\tau^{2(5)}_{t}(A_{ig})$, assumed respectively to follow a lognormal
distribution:
$\begin{array}[]{rcl}\log\left(\tau^{2(1)}_{t}(A_{i})\right)&\sim&N\left(\log\left(\sum_{h=1}^{G_{i}}\exp\left(Y_{t}(A_{ih})\right)\right),\sigma^{2(1)}(A_{i})\right)\\\
\log\left(\tau^{2(5)}_{t}(A_{ig})\right)&\sim&N\left(\log\left(\frac{1}{5}\sum_{k=t-4}^{t}\exp\left(Y_{k}(A_{ig})\right)\right),\sigma^{2(5)}(A_{ig})\right).\end{array}$
The specification of the _BWH Poisson space-time model_ is completed by the
following prior distributions, which we take directly from Bradley, Wikle and
Holan (2016): $\beta_{t}\stackrel{{\scriptstyle
iid}}{{\sim}}N(0,10^{15}),\forall t=1,\ldots,T$;
$\varsigma_{t}(A_{ig})\stackrel{{\scriptstyle
iid}}{{\sim}}N(0,\sigma^{2}_{\varsigma}),\forall t=1,\ldots,T$,
$g=1,\ldots,G_{i},i=1,\ldots,N$; $\sigma^{2(1)}(A_{i})\stackrel{{\scriptstyle
iid}}{{\sim}}\mbox{Gamma}(1,1),\forall i=1,\ldots,N$;
$\sigma^{2(5)}(A_{ig})\stackrel{{\scriptstyle
iid}}{{\sim}}\mbox{Gamma}(1,1),\forall g=1,\ldots,G_{i},i=1,\ldots,N$; and
$\sigma^{2}_{\varsigma}\sim\mbox{Gamma}(1,1)$.
The last model we consider is an adaption of the spatio-temporal model
proposed by Bradley, Wikle and Holan (2015) for Gaussian-distributed ACS
variables to ACS estimates of proportions. To frame the ACS estimates of
proportions, $z^{(l)}_{t}(A)$, within a Gaussian likelihood, we apply a
logistic transformation to them, thus obtaining variables defined in
$\mathbf{R}$. As $\tau^{2(l)}_{t}(A)$ is the design-based variance of
$z^{(l)}_{t}(A)$,we employ the delta method to derive the expression of the
variance of $\log\left(\frac{z^{(l)}_{t}(A)}{(1-z^{(l)}_{t}(A))}\right)$ for
every areal unit $A$ and $l$-unit time period. Thus, the first stage of this
new Bayesian hierarchical model, which we call the _BWH Gaussian Delta method
model_ , is given by:
$\begin{array}[]{rcl}\log\left(\frac{z^{(1)}_{t}(A_{i})}{(1-z^{(1)}_{t}(A_{i}))}\right)|\pi^{(1)}_{t}(A_{i})&\sim&N\left(\log\left[\frac{\pi^{(1)}_{t}(A_{i})}{(1-\pi^{(1)}_{t}(A_{i}))}\right],\frac{\tau^{2(1)}_{t}(A_{i})}{z^{(1)}_{t}(1-z^{(1)}_{t}(A_{i}))}\right)\\\
\\\
\log\left(\frac{z^{(5)}_{t}(A_{ig})}{(1-z^{(5)}_{t}(A_{ig}))}\right)|\pi^{(5)}_{t}(A_{ig})&\sim&N\left(\log\left[\frac{\pi^{(5)}_{t}(A_{ig})}{(1-\pi^{(5)}_{t}(A_{ig}))}\right],\frac{\tau^{2(5)}_{t}(A_{ig})}{z^{(5)}_{t}(1-z^{(5)}_{t}(A_{ig}))}\right)\end{array}$
for $i=1,\ldots,N$, $g=1,\ldots,G_{i}$, $t=1,\ldots,T$.
Calling
$\tilde{y}^{(1)}_{t}(A_{i}):=\log\left[\frac{\pi^{(1)}_{t}(A_{i})}{(1-\pi^{(1)}_{t}(A_{i}))}\right]$
and
$\tilde{y}^{(5)}_{t}(A_{ig}):=\log\left[\frac{\pi^{(1)}_{t}(A_{ig})}{(1-\pi^{(1)}_{t}(A_{ig}))}\right]$,
we achieve their disaggregation in time through the following equality
$\tilde{y}^{(5)}_{t}(A_{ig})=\frac{1}{5}\sum_{k=t-4}^{t}\tilde{y}^{(1)}_{t}(A_{ig})\qquad
i=1,\ldots,N;g=1,\ldots,G_{i}$
whereas their disaggregation in space is handled, for any areal unit $A$,
through
$\tilde{y}^{(1)}_{t}(A)=\frac{1}{|A|}\int_{s\in
A}\zeta_{t}(\mathbf{s})d\mathbf{s}\qquad\forall t=1,\ldots,T$
with $\zeta_{t}(\mathbf{s})$ spatio-temporal Gaussian process. Following a
similar approach as discussed in Section 3.2, for all $t=1,\ldots,T$, we
decompose $\zeta_{t}(\mathbf{s})$ in the sum of a spatio-temporal trend term
$\mu_{t}(\mathbf{s})$ and spatio-temporal random effects $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S};t=1,\ldots,T$, and we replace
$\zeta_{t}(\mathbf{s})$ with $\mu_{t}(\mathbf{s})+w_{t}(\mathbf{s})$ under the
integral. As in Section 3.2, $w_{t}(\mathbf{s})$ assumed to be equipped with a
separable space-time covariance function.
In fitting this model to data we use a dimension reduction approach, and we
approximate the spatio-temporal random effects $w_{t}(\mathbf{s})$,
$\mathbf{s}\in\mathcal{S};t=1,\ldots,T$ using the ST-MRA approximation
discussed in Section 3.3. Also Bradley, Wikle and Holan (2015) handled the
large dimensionality of the data through an approximation that involved a
basis function expansion. However, they used bisquare basis functions rather
than the basis functions we employ here. We believe that the difference in
basis functions employed in the approximation should not result in drastic
changes in terms of model performance.
### 3.7 Computation
We fit our Bayesian hierarchical model and all the other competing models
using Markov Chain Monte Carlo (MCMC) algorithms, with Gibbs sampling and
Metropolis-Hastings steps. For our model, posterior sampling exploits the data
augmentation method of Albert and Chib (1993) to sample the MRA basis function
coefficients $\boldsymbol{\eta}_{t,m,j}$ via Gibbs sampling, whereas posterior
samples of the Matérn covariance function parameters - $\phi$ and $\nu$ \- are
generated using a Metropolis-Hastings algorithm. We assess convergence of the
MCMC algorithms both visually, by inspecting trace plots, and numerically
using Geweke’s diagnostic for Markov chains (Geweke, 1992). We run each MCMC
algorithm for a number of iterations large enough that the effective sample
size post burn-in for each model parameter exceeds 1,000. The proposal
distributions used in the Metropolis-Hastings steps are tuned during burn-in
to achieve desirable acceptance rates (Roberts, Gelman and Gilks, 1997).
## 4 Simulation Studies
We now report results for two simulation studies: Simulation study 1 evaluates
the ability of the proposed model to disaggregate spatially and temporally
areal-level estimates of proportions, even when the data are not generated
according to our model specifications. On the other hand, Simulation study 2
gauges the need to account for the design effect.
### 4.1 Generating the true proportions
In both simulation studies, we use very similar data generating mechanisms,
all very different from our modeling framework. As a spatial domain we
envision a geographical configuration that is analogous to that of census
tracts and PUMAs. Specifically, we consider a $10\times 10$ square grid made
of 100 areal units, all assumed to have the same population size. The 100
areal units are in turn grouped into 4 distinct regions, each with the same
population size and each containing 25 areal units (see Figure 1). To simulate
data, we proceed as follows: we generate the true 1-year proportions
$\pi^{(1)}_{t}(A_{ig})$ for each time $t=1,\ldots,10$ and for each subregion
$g$, $g=1,\ldots,25$, nested within region $i$, $i=1,\ldots,4$. We repeat the
procedure thirty times, yielding a total of 30 simulated datasets per
simulation setting, and we consider 4 different data generating mechanisms.
This allows us to assess the performance of our model in settings that differ
from that of our model.
Figure 1: Areal units utilized in the simulation studies.
Under each simulation setting we assume that in each subregion $A_{ig}$, there
is a latent covariate $x(A_{ig})$, not varying in time, distributed according
to a standard normal distribution. This latent covariate drives the true
proportion $\pi^{(1)}_{t}(A_{ig})$, $g=1,2,\ldots,25$; $i=1,\ldots,4$. In the
first simulation setting, for each subregion $A_{ig}$ and at each time point
$t$, the true proportion $\pi^{(1)}_{t}(A_{ig})$ is obtained by applying the
_expit_ (inverse logistic) function to the sum of the latent covariate
$x(A_{ig})$ and the randomly generated white noise, thus allowing for temporal
and spatial variability in the true proportions. Although the true proportions
$\pi^{(1)}_{t}(A_{ig})$ will not be the same across space and time, they are
independent in space and time.
To induce spatial correlation in the true proportions, in the second
simulation setting we introduce a point-referenced spatial process,
$\lambda(\mathbf{s})$, with a Matérn covariance function with unit marginal
variance (e.g. $\sigma^{2}_{\lambda}=1$), and range and smoothness parameters
($\phi_{\lambda}$ and $\nu_{\lambda}$, respectively) equal to 0.5 and 1. This
implies that the effective range of the spatial process $\lambda(\mathbf{s})$
is between 1 and 2 resulting in true proportions for neighboring subregions
that are spatially dependent. The true proportion, $\pi^{(1)}_{t}(A_{ig})$,
for areal unit $A_{ig}$ at time $t$ is obtained by applying the expit function
to the sum of the latent covariate $x(A_{ig})$, the spatial process
$\lambda(\mathbf{s}_{ig})$ evaluated at the centroid $\mathbf{s}_{ig}$ of
areal unit $A_{ig}$, and the white noise term $e_{t}(A_{ig})$. Although this
second data generating mechanism yields spatially correlated true proportions,
they are independent over time.
The third data generating mechanism allows for a temporal trend in the true
proportions by introducing a linear time trend $\alpha_{0}+\alpha_{1}t$. Thus,
the true proportion for areal unit $A_{ig}$ at time $t$ is now obtained by
applying the expit function to the sum of the latent covariate $x(A_{ig})$,
the linear trend, $\alpha_{0}+\alpha_{1}t$, and the white noise
$e_{t}(A_{ig})$. We employ $\alpha_{0}=-1.0$ and $\alpha_{1}=0.2$ in the
linear temporal trend, resulting in a noticeable increase over time of the
true proportions.
Despite the temporal dependence in the true proportions generated under the
third simulation setting, the $\pi^{(1)}_{t}(A_{ig})$’s are not spatially
correlated. To address this shortcoming, the fourth data generating mechanism
combines the second and third data generating mechanism together yielding true
proportions $\pi^{(1)}_{t}(A_{ig})$ that display a temporal trend and are
correlated in space. Thus, in short:
$\pi^{(1)}_{t}(A_{ig})=\frac{\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}}{1+\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}}\qquad
i=1,...,4;g=1,...,25\\\ $
$\begin{array}[]{lrllrrl}\mathbf{X}&=&\\{x(A_{ig})\\}_{i=1,...,4;g=1,...,25};&&x(A_{ig})&\stackrel{{\scriptstyle
iid}}{{\sim}}&N(0,1)\\\ \\\
\boldsymbol{\lambda}&=&\\{\lambda(\mathbf{s}_{ig})\\}_{i=1,...,4;g=1,...,25};&&\boldsymbol{\lambda}&\sim&\text{MVN}\left(0,\Sigma(\boldsymbol{\theta}_{\lambda})\right)\end{array}$
with $\Sigma(\boldsymbol{\theta}_{\lambda})$, 100$\times$100 covariance matrix
induced by a Matérn covariance function, e.g. by (3.13), with
$\boldsymbol{\theta}_{\lambda}=\left(\sigma^{2}_{\lambda},\phi_{\lambda},\nu_{\lambda}\right)=\left(1.0,\;0.5,\;1.0\right)^{\prime}$.
Table 1: Data generating mechanism used in each of the four simulation settings of both simulation studies. Setting | Equation
---|---
1 | $\displaystyle\pi^{(1)}_{t}(A_{ig})=\frac{\exp\\{x(A_{ig})+e_{t}(A_{ig})\\}}{1+\exp\\{x(A_{ig})+e_{t}(A_{ig})\\}},\qquad e_{t}(A_{ig})\stackrel{{\scriptstyle iid}}{{\sim}}N(0,0.2^{2})$
2 | $\displaystyle\pi^{(1)}_{t}(A_{ig})=\frac{\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+e_{t}(A_{ig})\\}}{1+\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+e_{t}(A_{ig})\\}},\qquad e_{t}(A_{ig})\stackrel{{\scriptstyle iid}}{{\sim}}N(0,0.2^{2})$
3 | $\displaystyle\pi^{(1)}_{t}(A_{ig})=\frac{\exp\\{x(A_{ig})+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}}{1+\exp\\{x(A_{ig}))+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}},\qquad e_{t}(A_{ig})\stackrel{{\scriptstyle iid}}{{\sim}}N(0,0.2^{2})$
4 | $\displaystyle\pi^{(1)}_{t}(A_{ig})=\frac{\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}}{1+\exp\\{x(A_{ig})+\lambda(\mathbf{s}_{ig})+\alpha_{0}+\alpha_{1}t+e_{t}(A_{ig})\\}},\qquad e_{t}(A_{ig})\stackrel{{\scriptstyle iid}}{{\sim}}N(0,0.2^{2})$
### 4.2 Generating the observed estimates
Having generated the true proportions $\pi^{(1)}_{t}(A_{ig})$ under the 4 data
generating mechanisms, we proceed to simulate the corresponding “ _observed_ ”
5-year and 1-year estimates, $z^{(5)}_{t}(A_{ig})$ and $z_{t}^{(1)}(A_{i})$,
respectively. These estimates play the equivalent role to the ACS estimates,
in that they represent the data to which our model is fit. They are obtained
by adding to the true proportions additional random error, which represents
the survey-based error. Specifically, we first generate the 1-year subregional
estimates $z_{t}^{(1)}(A_{ig})$ by adding error to the true proportions on the
logit scale:
$\begin{array}[]{rcl}\log\left(\frac{z^{(1)}_{t}(A_{ig})}{1-z^{(1)}_{t}(A_{ig})}\right)&=&\text{logit}\left(\pi^{(1)}_{t}(A_{ig})+\tilde{e}_{t}(A_{ig}\right)\\\
\\\ \tilde{e}_{t}(A_{ig})&\stackrel{{\scriptstyle
iid}}{{\sim}}&N(0,v_{t}(A_{ig}))\end{array}$
From these we then derive the 1-year regional and the 5-year subregional
observed estimates, $z^{(1)}_{t}(A_{i})$ and $z^{(5)}_{t}(A_{ig})$, as
follows:
$\displaystyle z_{t}^{(5)}(A_{ig})$ $\displaystyle=$
$\displaystyle\frac{1}{5}\sum_{k=t-4}^{t}z^{(1)}_{k}(A_{ig})$ $\displaystyle
z_{t}^{(1)}(A_{i})$ $\displaystyle=$
$\displaystyle\frac{1}{25}\sum_{h=1}^{25}z^{(1)}_{t}(A_{ih}).$
We use two different strategies to determine the magnitude of the variances
$v_{t}(A_{ig})$’s. In simulation study 1, the $v_{t}(A_{ig})$’s are fixed
across the four simulation settings and are chosen so that the variation in
the simulated observed estimates at adjacent time periods resembles the year-
to-year variation in the ACS estimates of the proportion of families in
poverty. This is achieved when $v_{t}(A_{ig})=0.15^{2}$ for $t=1,2,\ldots,10$;
$i=1,\ldots,4$; $g=1,2,\ldots,25$.
In simulation study 2, we derive the magnitude of the $v_{t}(A_{ig})$’s as a
function of the design effect $d$. Since the goal of this simulation study is
to evaluate the inferential gain obtained by working with the _effective
sample size_ and _effective number of cases_ , rather than with the _observed
number of cases_ and the _observed sample size_ , we let $d$ vary. This
results in different values of the $v_{t}(A_{ig})$’s. The relationship between
the $v_{t}(A_{ig})$’s and the design effect $d$ can be determined based on the
following consideration: the $v_{t}(A_{ig})$’s ought to be such that,
conditional on the true proportions $\pi^{(1)}_{t}(A_{ig})$,
$\displaystyle\text{Var}\left(z^{(1)}_{t}(A_{ig})|\pi^{(1)}_{t}(A_{ig})\right)$
$\displaystyle=$ $\displaystyle
d\cdot\mbox{Var}\left[\text{expit}\left\\{\text{logit}\left(\pi^{(1)}_{t}(A_{ig})\right)+\tilde{e}_{t}(A_{ig})\right\\}\right]$
(4.1) $\displaystyle=$ $\displaystyle d\cdot\mbox{Var}^{(1)}_{SRS,t}(A_{ig}),$
with $\mbox{Var}^{(1)}_{SRS,t}(A_{ig})$ variance of the estimator
$\hat{\pi}^{(1)}_{SRS,t}(A_{ig})$ of the true proportion
$\pi^{(1)}_{t}(A_{ig})$ based on a simple random sample (SRS) . This leads to
the following expression for $v_{t}(A_{ig})$:
$\displaystyle v^{(1)}_{t}(A_{ig})$ $\displaystyle=$ $\displaystyle
d\times\frac{\left(\exp\left\\{\text{logit}\left(\pi^{(1)}_{t}(A_{ig})\right)\right\\}+1\right)^{4}\pi^{(1)}_{t}(A_{ig})\left(1-\pi^{(1)}_{t}(A_{ig})\right)}{m^{(1)}_{t}(A_{ig})\exp\left\\{2\times\text{logit}(\pi^{(1)}_{t}(A_{ig}))\right\\}}.$
(4.2)
Letting the sample sizes $m^{(1)}_{t}(A_{ig})$ for each subregion $A_{ig}$,
$g=1,\ldots,25;i=1,\ldots,4$, be equal to 100 for each time $t$, we derive
from (4.2) the values of the $v_{t}(A_{ig})$’s.
In simulation study 1, we fit to our Bayesian hierarchical model to the
“observed” estimates $z_{t}^{(5)}(A_{ig})$ and $z_{t}^{(1)}(A_{i})$, whereas
in simulation study 2, we fit to them both our Bayesian hierarchical model and
the standard Binomial model for disaggregation. In each case, we run the MCMC
algorithms for 10,000 iterations, discarding the first 2,000 for burn-in.
### 4.3 Simulation Results
Simulation study 1. Taking the posterior means of the
$\pi^{(1)}_{t}(A_{ig})$’s as estimates of the true proportions, and denoting
them by $\hat{\pi}^{(1)}_{t}(A_{ig})$, $i=1,\ldots,4$; $g=1,2,\ldots,25$, we
summarize the performance of our model by evaluating, for each
$\pi^{(1)}_{t}(A_{ig})$, the magnitude of the errors and the empirical
coverage of both the 50% and the 95% pointwise and joint credible intervals,
respectively.
Tables 2 and 3 present results from our simulation studies, including the Mean
Squared Error (MSE) and the Mean Absolute Error (MAE). The latter are defined
as the mean squared difference and the mean absolute difference between the
$\hat{\pi}^{(1)}_{t}(A_{ig})$’s and the true values. In addition, Tables 2 and
3 present the mean squared relative error (MSRE) and the mean absolute
relative error (MARE) defined respectively as:
$\displaystyle MSRE$ $\displaystyle=$
$\displaystyle\frac{1}{100}\sum_{g=1}^{4}\sum_{i=1}^{25}\frac{(\hat{\pi}^{(1)}_{t}(A_{ig})-\pi^{(1)}_{t}(A_{ig}))^{2}}{\pi^{(1)}_{t}(A_{ig})},$
$\displaystyle MARE$ $\displaystyle=$
$\displaystyle\frac{1}{100}\sum_{g=1}^{4}\sum_{i=1}^{25}\frac{|\hat{\pi}^{(1)}_{t}(A_{ig})-\pi^{(1)}_{t}(A_{ig})|}{\pi^{(1)}_{t}(A_{ig})}.$
(4.3)
The 50% and 95% pointwise credible intervals, computed by taking the
appropriate percentiles of the posterior samples for each
$\pi^{(1)}_{t}(A_{ig})$, contain the true values between 46.1% and 53.4% of
the time, and between 90.7% and 94.8% of the time, respectively. Similarly,
the 50% and the 95% joint credible intervals, constructed using the method of
Sørbye and Rue (2011), yield nearly nominal coverage.
We observe that in both cases, the credible intervals corresponding to the
middle of the time-series ($t=3,4,5,6,7$) have the highest coverage
probabilities.
The low values for the squared and absolute errors indicate successful
recovery of the true $\pi^{(1)}_{t}(A_{ig})$’s. Figure 2 presents scatterplots
of the true proportions against the $\hat{\pi}^{(1)}_{t}(A_{ig})$’s: all plots
illustrate our model’s ability to disaggregate survey-based estimates of areal
proportions regardless of the data generating mechanism.
Table 2: Simulation study 1. Results corresponding to 30 simulated datasets
generated under the first two of the four settings described in Table 1. For
each time $t$, $t=1,\ldots,10$, the table reports: (i) the average empirical
coverage of the 95% pointwise credible interval for $\pi^{(1)}_{t}(A_{ig})$,
$i=1,\ldots,4$, $g=1,\ldots,25$ averaged across the 100 subregions $A_{ig}$;
(ii) the average empirical coverage of the 50% pointwise credible interval for
$\pi^{(1)}_{t}(A_{ig})$; (iii) the average empirical coverage of the 95%
simultaneous credible interval for
$\boldsymbol{\pi}^{(1)}(\mathcal{S})=\left\\{\pi^{(1)}_{t}(A_{ig}):i=1,\ldots,4;g=1,\ldots,25\right\\}$
averaged across the 30 simulated datasets; (iv) the average empirical coverage
of the 50% simultaneous credible interval for
$\boldsymbol{\pi}^{(1)}(\mathcal{S})$; (v) the mean squared error (MSE); (vi)
the mean absolute error (MAE); (vii) the mean squared relative error (MSRE);
and (viii) the mean absolute relative error (MARE) as defined in (4.3).
| Average | Average | Average | Average | | | |
---|---|---|---|---|---|---|---|---
| Coverage | Coverage | Coverage | Coverage | MSE | MAE | MSRE | MARE
$t$ | 95% CI pointwise | 50% CI pointwise | 95% CI joint | 50% CI joint | $\times 10^{3}$ | $\times 10^{2}$ | $\times 10^{2}$ | $\times 10^{2}$
$1$ | $91.9\%$ | 47.7% | $90.0\%$ | 46.7% | $10.3$ | $8.3$ | $4.2$ | $25.2$
$2$ | $92.9\%$ | 48.0% | $90.0\%$ | 46.7% | $8.5$ | $7.3$ | $2.6$ | $22.0$
$3$ | $92.5\%$ | 48.9% | $93.3\%$ | 50.0% | $7.3$ | $6.8$ | $2.1$ | $18.8$
$4$ | $92.6\%$ | 49.4% | $96.7\%$ | 50.0% | $6.7$ | $6.5$ | $1.9$ | $16.9$
$5$ | $93.5\%$ | 49.6% | $96.7\%$ | 50.0% | $6.4$ | $6.3$ | $1.7$ | $16.1$
$6$ | $93.5\%$ | 50.3% | $96.7\%$ | 50.0% | $6.2$ | $6.2$ | $1.7$ | $16.1$
$7$ | $92.4\%$ | 49.8% | $93.3\%$ | 50.0% | $6.6$ | $6.4$ | $1.8$ | $17.0$
$8$ | $91.8\%$ | 50.3% | $93.3\%$ | 50.0% | $7.0$ | $6.5$ | $2.0$ | $19.3$
$9$ | $91.6\%$ | 48.9% | $90.0\%$ | 46.7% | $8.1$ | $7.2$ | $2.4$ | $21.4$
$10$ | $90.7\%$ | 47.4% | $90.0\%$ | 46.7% | $11.0$ | $8.5$ | $3.8$ | $26.1$
(a) Simulation study 1, setting 1
| Average | Average | Average | Average | | | |
---|---|---|---|---|---|---|---|---
| Coverage | Coverage | Coverage | Coverage | MSE | MAE | MSRE | MARE
$t$ | 95% CI pointwise | 50% CI pointwise | 95% CI joint | 50% CI joint | $\times 10^{3}$ | $\times 10^{2}$ | $\times 10^{2}$ | $\times 10^{2}$
$1$ | $91.3\%$ | 47.7% | $90.0\%$ | 46.7% | $9.7$ | $8.1$ | $4.0$ | $24.8$
$2$ | $92.0\%$ | 49.7% | $93.3\%$ | 46.7% | $6.3$ | $6.4$ | $2.3$ | $22.3$
$3$ | $92.3\%$ | 51.9% | $93.3\%$ | 50.0% | $5.4$ | $5.8$ | $1.9$ | $21.4$
$4$ | $94.8\%$ | 52.9% | $93.3\%$ | 50.0% | $4.8$ | $5.3$ | $1.5$ | $19.0$
$5$ | $94.4\%$ | 53.4% | $96.7\%$ | 50.0% | $5.0$ | $5.4$ | $1.5$ | $17.4$
$6$ | $94.0\%$ | 53.0% | $96.7\%$ | 50.0% | $5.0$ | $5.4$ | $1.4$ | $16.2$
$7$ | $93.9\%$ | 52.8% | $93.3\%$ | 50.0% | $5.0$ | $5.5$ | $1.6$ | $17.9$
$8$ | $94.0\%$ | 51.0% | $93.3\%$ | 50.0% | $5.3$ | $5.8$ | $1.8$ | $19.4$
$9$ | $92.8\%$ | 50.4% | $93.3\%$ | 50.0% | $6.1$ | $6.3$ | $2.2$ | $20.6$
$10$ | $91.2\%$ | 47.2% | $90.0\%$ | 46.7% | $9.4$ | $8.0$ | $4.0$ | $25.6$
(b) Simulation study 1, setting 2
Table 3: Simulation study 1. Results corresponding to 30 simulated datasets
generated under the last two of the four settings described in Table 1. For
each time $t$, $t=1,\ldots,10$, the table reports: (i) the average empirical
coverage of the 95% pointwise credible interval for $\pi^{(1)}_{t}(A_{ig})$,
$i=1,\ldots,4$, $g=1,\ldots,25$ averaged across the 100 subregions $A_{ig}$;
(ii) the average empirical coverage of the 50% pointwise credible interval for
$\pi^{(1)}_{t}(A_{ig})$; (iii) the average empirical coverage of the 95%
simultaneous credible interval for
$\boldsymbol{\pi}^{(1)}(\mathcal{S})=\left\\{\pi^{(1)}_{t}(A_{ig}):i=1,\ldots,4;g=1,\ldots,25\right\\}$
averaged across the 30 simulated datasets; (iv) the average empirical coverage
of the 50% simultaneous credible interval for
$\boldsymbol{\pi}^{(1)}(\mathcal{S})$; (v) the mean squared error (MSE); (vi)
the mean absolute error (MAE); (vii) the mean squared relative error (MSRE);
and (viii) the mean absolute relative error (MARE) as defined in (4.3).
| Average | Average | Average | Average | | | |
---|---|---|---|---|---|---|---|---
| Coverage | Coverage | Coverage | Coverage | MSE | MAE | MSRE | MARE
$t$ | 95% CI pointwise | 50% CI pointwise | 95% CI joint | 50% CI joint | $\times 10^{3}$ | $\times 10^{2}$ | $\times 10^{2}$ | $\times 10^{2}$
$1$ | $91.9\%$ | 46.8% | $90.0\%$ | 46.7% | $9.2$ | $7.8$ | $5.8$ | $36.1$
$2$ | $92.5\%$ | 48.1% | $90.0\%$ | 46.7% | $7.1$ | $6.7$ | $3.0$ | $31.0$
$3$ | $91.7\%$ | 48.9% | $90.0\%$ | 46.7% | $6.4$ | $6.4$ | $2.4$ | $25.8$
$4$ | $92.9\%$ | 51.1% | $93.3\%$ | 53.3% | $6.2$ | $6.1$ | $1.9$ | $20.6$
$5$ | $92.9\%$ | 50.9% | $93.3\%$ | 53.3% | $6.0$ | $6.1$ | $1.6$ | $17.9$
$6$ | $93.0\%$ | 50.6% | $93.3\%$ | 50.0% | $6.0$ | $6.1$ | $1.6$ | $14.3$
$7$ | $93.0\%$ | 50.8% | $93.3\%$ | 50.0% | $6.2$ | $6.2$ | $1.5$ | $15.2$
$8$ | $92.6\%$ | 48.3% | $93.3\%$ | 46.7% | $6.7$ | $6.5$ | $1.6$ | $15.9$
$9$ | $91.2\%$ | 48.6% | $90.0\%$ | 46.7% | $7.2$ | $6.8$ | $1.6$ | $16.1$
$10$ | $91.3\%$ | 47.9% | $90.0\%$ | 46.7% | $8.9$ | $7.7$ | $1.8$ | $17.6$
(c) Simulation study 1, setting 3
| Average | Average | Average | Average | | | |
---|---|---|---|---|---|---|---|---
| Coverage | Coverage | Coverage | Coverage | MSE | MAE | MSRE | MARE
$t$ | 95% CI pointwise | 50% CI pointwise | 95% CI joint | 50% CI joint | $\times 10^{3}$ | $\times 10^{2}$ | $\times 10^{2}$ | $\times 10^{2}$
$1$ | $91.8\%$ | 45.3% | $90.0\%$ | 46.7% | $9.1$ | $7.7$ | $5.2$ | $34.8$
$2$ | $91.3\%$ | 49.3% | $90.0\%$ | 46.7% | $6.5$ | $6.4$ | $3.3$ | $30.9$
$3$ | $91.1\%$ | 49.8% | $90.0\%$ | 50.0% | $5.7$ | $6.0$ | $2.5$ | $25.1$
$4$ | $92.2\%$ | 51.8% | $93.3\%$ | 53.3% | $5.3$ | $5.6$ | $1.8$ | $20.1$
$5$ | $92.6\%$ | 52.0% | $93.3\%$ | 53.3% | $5.1$ | $5.5$ | $1.5$ | $16.3$
$6$ | $93.0\%$ | 51.9% | $93.3\%$ | 53.3% | $5.2$ | $5.5$ | $1.5$ | $14.7$
$7$ | $92.7\%$ | 51.7% | $93.3\%$ | 50.0% | $5.3$ | $5.6$ | $1.5$ | $15.0$
$8$ | $92.2\%$ | 49.8% | $93.3\%$ | 46.7% | $5.9$ | $6.1$ | $1.6$ | $15.3$
$9$ | $92.0\%$ | 48.7% | $93.3\%$ | 46.7% | $6.7$ | $6.5$ | $1.7$ | $16.8$
$10$ | $91.9\%$ | 46.1% | $90.0\%$ | 46.7% | $9.2$ | $7.8$ | $1.8$ | $18.1$
(d) Simulation study 1, setting 4
(e) Simulation study 1, setting 1; $t$ = 2 (f) Simulation study 1, setting 1;
$t$ = 5
(g) Simulation study 1, setting 1; $t$ = 9 (h) Simulation study 1, setting 2;
$t$ = 2
(i) Simulation study 1, setting 2; $t$ = 5 (j) Simulation study 1, setting 2;
$t$ = 9
(k) Simulation study 1, setting 3; $t$ = 2 (l) Simulation study 1, setting 3;
$t$ = 5
(m) Simulation study 1, setting 3; $t$ = 9 (n) Simulation study 1, setting 4;
$t$ = 2
(o) Simulation study 1, setting 4; $t$ = 5 (p) Simulation study 1, setting 4;
$t$ = 9
Figure 2: Simulation study 1. Scatterplots of the true
$\pi^{(1)}_{t}(A_{ig})$’s against their corresponding estimates,
$\hat{\pi}^{(1)}_{t}(A_{ig})$, at selected times, $t=2,5,9$. The simulated
values and their estimates refer to all the 30 simulated datasets generated
under one of the four different simulation settings described in Table 1.
Simulation study 2. Here we compare the performance of our proposed model to
that of the standard Binomial model for disaggregation. Taking again the
posterior means of the $\pi^{(1)}_{t}(A_{ig})$’s as our estimates,
$\hat{\pi}^{(1)}_{t}(A_{ig})$’s, $g=1,\ldots,25;i=1,\ldots,4;t=1,\ldots,10$,
Table 4 presents, for each model, the average mean squared error and the
average mean absolute error, averaged over areal units, time points, and
simulated datasets. Conversely, Table 5 provides the empirical coverage
probabilities of 50% and 95% pointwise and joint credible intervals.
We inspect the difference in inference provided by the two models as the
design effect varies. When $d=2$, there is little difference between the
standard Binomial model and our model. However, when $d=4,6$ or $8$, the
standard Binomial model has an inferior performance with respect to each of
the metrics considered, suggesting that by ignoring the design effect, the
standard Binomial model places too much certainty in the pseudo-survey-
estimates that we have generated. On the other hand, by correctly propagating
the uncertainty of the pseudo-survey-estimates through the use of the
effective sample size and the effective number of cases, our proposed model
achieves lower mean squared and lower mean absolute error, as well as near
nominal coverage probability.
Table 4: Simulation study 2. Average probability that a 95%, respectively, a 50% pointwise, respectively, joint credible interval covers the true value. Averages are taken over areal units, time points, datasets, and simulation settings for the pointwise credible intervals, whereas they are taken over time points, datasets and simulation settings for the joint credible intervals. | Coverage | Coverage | Coverage | Coverage | Coverage | Coverage | Coverage | Coverage
---|---|---|---|---|---|---|---|---
| 95% CI | 95% CI | 50% CI | 50% CI | 95% CI | 95% CI | 50% CI | 50% CI
| pointwise - | pointwise - | pointwise - | pointwise - | joint - | joint - | joint - | joint -
| Proposed | Standard | Proposed | Standard | Proposed | Standard | Proposed | Standard
$d$ | model | Binomial | model | Binomial | model | Binomial | model | Binomial
2 | 93.2% | 89.7% | 53.3% | 46.7% | 93.1% | 93.1% | 53.8% | 53.8%
4 | 95.4% | 87.4% | 53.7% | 44.7% | 92.1% | 88.7% | 51.4% | 45.2%
6 | 93.2% | 83.6% | 52.8% | 42.6% | 91.2% | 83.6% | 50.1% | 41.0%
8 | 93.9% | 80.4% | 52.3% | 40.1% | 93.1% | 79.3% | 48.8% | 39.8%
Table 5: Simulation study 2. Average mean squared error (MSE) $\times 10^{3}$ and average mean absolute error (MAE) $\times 10^{2}$ computed by taking, respectively, the squared difference between the estimated $\hat{\pi}^{(1)}_{t}(A_{ig})$ and the true value $\pi^{(1)}_{t}(A_{ig})$, and the absolute value of the difference between the estimated $\hat{\pi}^{(1)}_{t}(A_{ig})$ and the true value $\pi^{(1)}_{t}(A_{ig})$ for $i=1,\ldots,4;g=1,\ldots,25;t=1,\ldots,10$. For each modeling framework, averages are taken over areal units, time points, datasets, and simulation settings. | MSE $\times 10^{3}$ | MSE $\times 10^{3}$ | MAE $\times 10^{2}$ | MAE $\times 10^{2}$
---|---|---|---|---
$d$ | Proposed model | Standard Binomial | Proposed model | Standard Binomial
2 | 5.7 | 5.9 | 5.2 | 5.2
4 | 6.0 | 12.3 | 5.7 | 7.3
6 | 8.6 | 14.7 | 6.2 | 9.0
8 | 9.0 | 21.9 | 7.0 | 11.7
## 5 Data Analysis
### 5.1 Families in poverty
We apply the model in Sections 3.1 to 3.5 to the ACS estimates of the
proportion of families in Michigan living in poverty from 2006 to 2016, with
the goal of obtaining annual estimates at the census tract level.
We present results from our model in a variety of ways, including a comparison
of the mean and variance of our model-based estimates to those provided in the
ACS dataset. We also compare the out-of-sample predictive performance of our
model to that of the three competing models described in Section 3.6, which we
also fit to the 2006-2016 ACS data. However, our primary focus is on the
disaggregated estimates for selected neighborhoods in Detroit. Here, we chose
a set of census tracts in Midtown, a mixed-use area in Detroit located north
of downtown and comprising several business districts, Wayne State University,
and some residential neighborhoods. Some of the census tracts in Midtown have
very high poverty, while others host various sporting arenas and other
downtown attractions, and thus exhibit considerably lower poverty rate. Some
of the high-poverty tracts have been subject to gentrification and development
in recent years (Moehlman and Robins-Somerville, 2016; Aguilar, 2015). Due to
these spatial inhomogeneities and temporal changes, we believe that these
tracts illustrate the need for fine scale spatio-temporal estimates in order
to properly characterize neighborhood surroundings.
#### 5.1.1 Comparison of 5-year model-based estimates as estimated by our
model to ACS 5-year estimates
(a) Model-based vs. ACS estimates (b) Posterior SD vs. ACS SE’s
Symbol | Tract ID | ACS Estimate | ACS SE
---|---|---|---
$\blacktriangle$ | 26163517200 | 0.00 | 0.19
$\bullet$ | 26161400100 | 0.57 | 0.20
$\blacktriangle$ | 26163550800 | 0.00 | 0 .04
$\blacktriangle$ | 26073000700 | 0.63 | 0.11
$\blacktriangle$ | 26057000400 | 0.00 | 0.61
$\blacktriangle$ | 26037011200 | 0.00 | 0.17
$\blacktriangle$ | 26163500400 | 0.64 | 0.10
$\blacktriangle$ | 26163512900 | 0.72 | 0.11
(c) ACS estimates and SE’s for tracts highlighted in (a)
Figure 3: (a) Model-based estimates of 5-year average proportion of families
in poverty in Michigan at census tract level for years 2009-2013 against
corresponding ACS estimates. Census tracts deviating greatly from the identity
line are denoted by triangles. (b) Posterior standard deviation for the 5-year
average proportion of families in poverty in Michigan at census tract level as
yielded by our model against the ACS standard error of the corresponding
estimates. (c) Tabulation of Tract ID’s, ACS estimates, and ACS standard
errors for census tracts for which our model-based estimate deviates greatly
from the ACS estimate.
Our model is intended for spatial and temporal disaggregation, but we can also
generate 5-year census tract estimates, which should resemble the
corresponding estimates from ACS. Figure 3(a) shows a scatter plot comparing
these estimates. The points tend to fall around the identity line, indicating
good agreement. Figure 3(b) compares the standard errors of the ACS 5-year
census tract estimates with the posterior standard deviations from our model.
As our model borrows information from neighboring census tracts and from the
1-year PUMA-level estimates, it yields estimates with smaller posterior
standard deviation than the ACS standard errors. Many of the points that
deviate from the identity line in Figure 3(a) correspond to census tracts with
zero-valued ACS estimates, which have large ACS standard errors compared to
the average of 0.041 over all tracts (Figure 3(c)).
An example of such a census tract is displayed in panels (a) and (b) of Figure
4 along with its neighboring tracts. Census tract 26161400100 is located in
downtown Ann Arbor and, according to the ACS estimate, has an average poverty
rate of 0.59 for the 5-year time period from 2009 to 2013. This estimate
deviates greatly from that of the neighboring tracts. In addition, it has a
design-based standard error around 5 times the average standard error for ACS
estimates of poverty in Michigan. As our model borrows information from
neighboring census tracts, the model-based estimate for this tract is pushed
towards the average of the neighboring tracts more than towards the raw ACS
estimates. We observe regression of a model-based estimate towards the average
of its neighbors in situations where the design-based standard error is quite
large.
(a) ACS estimates, Ann Arbor (b) Model-based estimates, Ann Arbor
(c) ACS estimates, Romulus (d) Model-based estimates, Romulus
(e) Posterior density
Figure 4: (a) ACS estimate for the 5-year average proportion of families in
poverty in Ann Arbor census tract 26161400100 and (b) our corresponding model-
based estimate. (c) ACS estimate for the 5-year average proportion of families
in poverty in a census tract in Romulus and (d) our model-based estimate.
Here, despite the lower-poverty level in the neighboring census tracts, our
model-based estimate is not smoothed towards the poverty-level of the
neighboring tracts. (e) Posterior densities of: (i) 5-year average proportion
and (ii) 1-year proportion of families living in poverty in census tract
26163563500 according to our model, as well as the truncated normal density
function obtained using as mean and standard deviation, respectively, the
5-year ACS estimate and its corresponding standard error.
To illustrate this phenomenon, Figure 4 shows in panels (c) and (d) a census
tract in Romulus also characterized by high poverty despite being surrounded
by census tracts with lower poverty level, according to the ACS. In this case,
since the ACS standard error is much lower, our model-based estimate of
poverty still reflects the spatial heterogeneity in the ACS estimates and it
is not smoothed towards the average of the neighboring tracts.
#### 5.1.2 Posterior density of true population proportions
ACS estimates are provided with margins of error based on standard errors
derived from an iterative estimation procedure (see U.S. Census Bureau (2014)
for details) multiplied by standard normal quantiles. These may in turn be
utilized to construct confidence intervals for the estimates by adding and
subtracting their margins of error, with users being cautioned to use “logical
boundaries when creating confidence bounds from the margins of error” (U.S.
Census Bureau, 2008) (i.e. zero and one for proportions). This implies a
truncated normal distribution centered at the ACS estimate with variance
depending on the standard error.
Through our Bayesian modeling framework, we obtain the posterior distribution
of the true proportions at any spatial and temporal scale without imposing
assumptions of symmetry or truncation. For example, Figure 3(e) plots the
posterior density of the average proportion of families living in poverty in
census tract 26163563500 located in Midtown Detroit for the 5-year period
2009-2013. For this census tract, the confidence interval for the 5-year ACS
estimate for 2009-2013 would be subject to truncation at zero. Figure 3(e)
shows the truncated normal density with mean and standard deviation equal,
respectively, to the ACS estimate and its standard error. To facilitate direct
comparison to the ACS estimates, Figure 3(e) also presents the posterior
density of the 5-year average proportion of families living in poverty as
provided by our model, as well as the posterior density for the 1-year
proportions for years 2009, 2010, 2011, 2012 and 2013. As the figure shows,
thanks to the borrowing of information from neighboring census tracts, the
posterior density of the 5-year average proportion is characterized by smaller
uncertainty than the truncated normal density centered at the ACS estimate.
#### 5.1.3 Disaggregated estimates of poverty for Detroit
Disaggregating the ACS estimates allows us to examine yearly trends in poverty
for individual census tracts, as well as for combinations of census tracts
which do not form a PUMA or are not part of a highly populated county. For
both of these cases we cannot assess temporal trends using the ACS estimates
alone. Figure 5 presents various maps of the disaggregated estimates of the
proportion of families in poverty in Michigan from our model. Panel (a)
displays census tract estimates for all of Michigan for year 2010 while panel
(b) presents the same results for Wayne County, which contains areas of
extreme poverty, as well as some of the wealthiest neighborhoods in Michigan.
Panels (c)-(k) of Figure 5 present yearly results over time for a subset of
census tracts in Wayne County located in Midtown Detroit. This set of census
tracts was selected because the poverty rates exhibit spatial heterogeneity,
with certain pairs of neighboring census tracts differing by over 20%, so a
single poverty estimate for this area would not properly characterize the
neighborhood conditions of its residents. Recent changes in these census
tracts have been well-documented (Moehlman and Robins-Somerville, 2016),
particularly in the Cass Corridor, an area of downtown Detroit that has faced
high crime and poverty, but has recently experienced sudden gentrification
(Aguilar, 2015). The 5-year ACS estimates at the census tract level may not
properly characterize yearly changes in these census tracts.
(a) Model-based estimates: Michigan 2010 (b) Model-based estimates: Wayne
County 2010
(c) 2007 (d) 2008 (e) 2009 (f) 2010 (g) 2011 (h) 2012 (i) 2013 (j) 2014 (k)
2015
Figure 5: Disaggregated estimates of the proportion of families in poverty in
Midtown Detroit.
Figure 6(a) shows the changes over time of poverty rates for a set of census
tracts in the Midtown area of Detroit. Consistent with national trends, Figure
6(b) indicates that, on average, the area experienced an increase in poverty
following the 2008 financial crisis in the US, with an eventual improvement in
later years. Figure 5, panels (c)-(k), highlights a census tract of particular
interest, indicated in green in Figure 6(a), which did not experience a
decrease in poverty until 2014. Year to year, it has had among the highest
poverty rates in Detroit. However, recent developments such as the
groundbreaking of Little Caesars Arena in 2014 and an influx of newly built
restaurants and bars, might have contributed to the drop in poverty in that
census tract from 2014 to 2015 (Moehlman and Robins-Somerville, 2016).
(a) Spaghetti plot of poverty for Midtown census tracts. (b) Mean poverty for
Midtown census tracts
Figure 6: (a) Spaghetti plot displaying the estimated proportions of families
in poverty over time with highlighted the census tract shown in Figure 5 and
(b) the estimated average poverty rate across census tracts in Midtown Detroit
between 2006 and 2016.
### 5.2 Out-of-Sample Prediction
To assess our model’s out-of-sample predictive performance, we consider the
county-level proportion of families in poverty during the 3-year time periods
2010-2012 and 2011-2013. We generate 3-year county-level predictions as a
weighted average of the disaggregated estimates within each county’s census
tracts for the appropriate 3-year periods, with weights proportional to the
number of families living in each tract. Then, those yearly county estimates
are averaged over the 3-year time periods. Our “true values” are the 3-year
ACS estimates, which we did not use for model fitting. This allows us to
assess our model’s ability to predict over time periods and areal units that
are not utilized in model fitting.
Figure 7(a) compares the estimated 3-year proportion yielded by our model with
the ACS estimates. As the figure shows, the two sets of estimates tend to be
very similar, indicating the strong predictive performance of our model. The
mean squared and mean absolute prediction errors are, respectively,
4.16$\times 10^{-5}$ and 4.83$\times 10^{-3}$, whereas the mean squared
relative prediction error is 3.13$\times 10^{-4}$ and the mean absolute
relative prediction error is 4.18$\times 10^{-2}$. These values demonstrate
reduced predictive error in our modeling framework.
### 5.3 Comparison to other models
We also generate out-of-sample predictions of the proportions of families in
poverty at the 3-year county resolution for the three competing models
presented in Section 3.6. Details on how these predictions are derived are
provided in Section 3 of the Supplementary Material (Benedetti, Berrocal and
Little, 2021). We evaluate the quality of these out-of-sample predictions by
validating them against the ACS estimates: Figure 7(b), (c) and (d) show
scatter plots of the predicted proportion of families in poverty as yielded by
each of the three competing models against the ACS estimate. The standard
Binomial model and the BWH Poisson model produce estimates that are fairly in
line with the ACS values, while there is a larger discrepancy between the
estimates yielded by the BWH Gaussian Delta Method model and the ACS 3-year
estimates. Numerically, we compare the predictive performance of our proposed
model to that of the other 3 models in terms of average predictive bias, mean
squared predictive error, mean absolute predictive error, and coverage of the
50% and 95% prediction intervals. These statistics are reported in Table 6,
with the first three summary statistics all functions of the difference
between the predicted proportions and the ACS 3-year estimates.
As the table shows, our model performs almost equivalently to the standard
Binomial model in terms of predictive accuracy with our model yielding a
coverage slightly closer to the nominal level than the standard Binomial
model. This occurs for both the 50% and the 95% prediction intervals. However,
the difference is minimal: we attribute this to the large sample size and
careful sampling design of the ACS, which limits the impact of the design
effect on the model’s performance.
Moving onto the BWH Poisson space-time model, we can see that even though this
model exhibits accurate predictions, our model is slightly more accurate. The
main differences between the two models are with respect to the posterior
predictive standard deviations: the BWH Poisson model has much smaller
posterior predictive standard deviations and thus much lower coverage
probabilities than our model.
Finally, the BWH Gaussian Delta Method model offers a poorer predictive
performance than our model with respect to all metrics. We acknowledge that
the models that we have attributed to Bradley, Wikle, and Holan are not
necessarily the approaches that the authors would have taken to model the ACS
spatio-temporal estimates of proportions. Rather, they constitute our best
effort to adapt the methods presented in Bradley, Wikle and Holan (2015) and
Bradley, Wikle and Holan (2016) to model our data. While we had to modify both
models to accommodate estimates of proportions, we took care to do so in a way
that would not needlessly favor our model.
(a) Our proposed model (b) Standard binomial model (c) BWH Poisson space-time
(d) BWH Gaussian Delta Method
Figure 7: Comparison to other models. Predicted proportion of families in poverty vs. ACS 3-year estimates of the proportion of families in poverty in Michigan counties for the period 2010-2012 and 2011-2013 as yielded by: (a) our proposed model, (b) the Standard Binomial model, (c) the BWH Poisson space-time model and (d) the BWH Gaussian Delta method model. Table 6: Comparison to other models. Bias, Mean Squared Predictive Error (MSPE), Mean Absolute Predictive Error (MAPE) of the out-of-sample predictions, as well as empirical coverage of the 50% and the 95% prediction intervals (PI) for our proposed model and the three competing models. | Bias | MSPE | MAPE | Coverage | Coverage
---|---|---|---|---|---
Model | $\times 10^{2}$ | $\times 10^{5}$ | $\times 10^{3}$ | 50% PI | 95% PI
Our proposed model | 0.03 | 4.16 | 4.83 | 52.3% | 93.0%
Standard Binomial | $-$0.03 | 3.69 | 4.39 | 46.9% | 91.4%
BWH Poisson space-time | $-$0.70 | 13.91 | 9.52 | 9.4% | 23.4%
BWH Gaussian delta method | $-$1.70 | 52.49 | 18.11 | 26.6% | 52.3%
## 6 Discussion
This paper proposes a spatio-temporal Bayesian hierarchical model to
disaggregate estimates of proportions over areal units derived from sampling
surveys while accounting for the survey design. Previous to our work, Bradley,
Wikle and Holan (2016) formulated a stochastic model for ACS estimates
distributed according to a Poisson distribution. The model explicitly
accounted for the survey design, as it specified a lognormal distribution for
the ACS design-based variance; however it focused only on addressing the
change of support problem in a spatial setting for count variables. Other work
by Bradley, Wikle and Holan (2015) considered the space-time setting, but it
did not incorporate design effects as it postulated a Gaussian likelihood with
the ACS design-based variance taken as known and set equal to the variance of
the normal distribution.
The main motivation for the development of our modeling framework is the
ability to generate data on socio-economic indicators at fine spatial and
temporal resolution, thus responding to the needs of health researchers
investigating the effect of social determinants of health on health outcomes.
We have demonstrated the utility of our Bayesian hierarchical modeling
framework by applying it to the ACS estimates of families in poverty. This
application highlighted several advantages of our model, among which the fact
that it generates annual estimates at census tract spatial resolution. In
addition, due to the borrowing of information from neighboring units and from
ACS estimates at different spatial and temporal resolutions, these estimates
are characterized by smaller uncertainty. We use our disaggregated estimates
to examine trends over time of poverty in Michigan focusing on Detroit, for
which we could highlight yearly changes at small spatial scale. These changes
could not be detected easily using the 5-year ACS census tract estimates.
We recognize that a standard Binomial disaggregation model that did not
explicitly account for the design effect, applied to the same data, yields a
very comparable, or slightly better, predictive performance than our model
when evaluated in terms of Mean Squared Predictive Error, Mean Absolute
Predictive Error, and Empirical Coverage of the 50% and 95% pointwise
prediction intervals. However, our simulation study 2 also indicated that
while such a performance by the standard Binomial model is expected for small
design effect, the aforementioned model is subject to a worsening in
predictive performance as the design effect size increases. In these
situations, our model is preferable. While it is true that the ACS design
effect for the estimates of proportion of families in poverty in Michigan
during the period considered – 2006-2016 – is estimated to be around 2.5 on
average, our model has not been developed only for handling estimates
resulting from the ACS. Rather, our model has a wider applicability and it has
been formulated to disaggregate spatially and temporally any set of survey-
based multi-year estimates of proportions. Other surveys typically used in
epidemiological studies are characterized by larger design effects than the
ACS. For example, the national Behavior Risk Factor Surveillance System
(BRFSS) in 2013 had an average design effect of 4.45, with state BRFSS surveys
having design effects ranging from 1.47 to 5.16 (Iachan et al., 2016). For
estimates provided by these surveys, our model is expected to yield better
results than the standard Binomial model.
Our model is not the only one using the concept of design effect to yield
small-scale spatio-temporal estimates: Li et al. (2019) applied the model of
Mercer et al. (2014) to smooth the spatial distribution of 1-year estimates of
the under-5 mortality rate over 35 countries in Africa, producing subnational
estimates at the 1-year resolution. Although Li et al. (2019) are also
concerned with generating small scale spatio-temporal estimates, their work
did not address the spatio-temporal change of support problem in the same way
we do here. Specifically, Li et al. (2019) introduce a spatio-temporal process
that is discrete both in space and time, whereas our model employs a point-
referenced spatio-temporal process, discrete in time but continuous in space.
Additionally, our model accounts for the clustering units of the survey (e.g.
counties in our application) by including random effects specified at the
county-level spatial resolution.
To deal with the large dimensionality of the data, we approximate the latent
spatio-temporal process driving the true population proportions via a basis
function expansion. Multiple choices are available to alleviate the
computational burden associated with fitting a spatial statistical model to
large spatial data, as reviewed by Heaton et al. (2019). Here, acknowledging
the nested and multi-resolution geography of the ACS data, also noted by
Savitsky (2016), we elect to choose the Multi-Resolution Approximation (MRA)
of Katzfuss (2017), extending it to the space-time setting, an additional
contribution of our paper. However, other basis functions could be employed,
namely, wavelets, radial basis functions, and Moran’s basis functions as in
Bradley, Wikle and Holan (2016).
In our model, partly for computational considerations, we use a probit link to
relate the true areal-level proportions to the underlying Gaussian spatio-
temporal process, and we employ the data augmentation algorithm of Albert and
Chib (1993) for posterior computation. We believe that it is possible to
devise an MCMC algorithm based on the skew-normal posterior results for probit
regressions derived by Durante (2019). Additionally, we remark that one could
replace the probit link with a logit link. In this case, we encourage readers
to employ a Pólya-Gamma augmentation scheme (Polson, Scott and Windle, 2013)
for greater computational efficiency.
Much of our predictive performance evaluation is based on the empirical
probability that credible and/or prediction intervals cover the true value,
which inherently conflates a frequentist property (empirical coverage
probability) with Bayesian modeling frameworks. This type of assessment is in
line with the notion of calibrated Bayes (Little, 2006) and recommended in a
predictive context (Dawid, 1982); moreover, it is the authors’ experience that
coverage probabilities are frequently used in assessing Bayesian models,
particularly in a spatial context (see Entezari, Brown and Rosenthal (2019);
Gilani, Berrocal and Batterman (2019); Berrocal, Gelfand and Holland (2010) as
example), where prediction is the main goal.
We note a potential abuse of terminology in calling “out-of-sample validation”
the comparison of the 3-year county-level proportions yielded by our model
with the corresponding ACS estimates. Even though the 3-year county level ACS
estimates were not used in fitting the model, the microdata that is leveraged
to derive such ACS estimates is also employed to calculate the 1-year and
5-year estimates of proportions to which our model was fit. In adopting this
terminology, we follow previous examples in the literature on this topic, see
Bradley, Wikle and Holan (2015), where this type of assessment was performed
and this nomenclature was used.
Finally, a characteristic of our model is the assumption of conditional
independence between the 1-year ACS PUMA-level estimates and the 5-year census
tract estimates, conditional on the true areal proportions. Since both sets of
estimates are derived using the same microdata, it is possible that the
assumption of conditional independence is not realistic. Not having access to
the actual microdata, we have no means to determine whether this assumption is
violated. Future work could be devoted to relax the assumption of conditional
independence.
Supplementary Information In the Supplementary Material, we derive statistical
properties for the ST-MRA method, provide details on how predictions were
derived, and present results of the exploratory data analysis described in
Section 2.2. Specifically: Section 1 shows that the ST-MRA expression
presented in Section 3.3 provides an approximation to a Gaussian spatio-
temporal process with a separable covariance function, with an AR(1) structure
in time and a dependence structure in space encoded by a Matérn covariance
function. Section 2 discusses how to derive out-of-sample predictions under
the alternative models discussed in Section 3.6, while Section 3 shows results
of the exploratory data analysis that supports our modeling choices. Finally,
Section 4 concludes the Supplementary Material presenting results for the city
of Flint.
## References
* Abrams and Szefler (2020) [author] Abrams, E. M.E. M. and Szefler, S. J.S. J. (2020). COVID-19 and the impact of social determinants of health. The Lancet - Respiratory Medicine 8 659–661.
* Aguilar (2015) [author] Aguilar, L.L. (2015). Detroit’s Cass Corridor makes way for new era. The Detroit News, published April 2015.
* Albert and Chib (1993) [author] Albert, J. H.J. H. and Chib, S.S. (1993). Bayesian analysis of binary and polychotomous data. Journal of the American Statistical Association 88 669–679.
* Banerjee, Carlin and Gelfand (2004) [author] Banerjee, S.S., Carlin, B. P.B. P. and Gelfand, A. E.A. E. (2004). Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC Boca Raton, FL.
* Benedetti, Berrocal and Little (2021) [author] Benedetti, M. H.M. H., Berrocal, V. J.V. J. and Little, R.R. (2021). Supplement to “Accounting for survey design in Bayesian disaggregation of survey-based areal estimates of proportions: an application to the American Community Survey”.
* Berrocal, Gelfand and Holland (2010) [author] Berrocal, V. J.V. J., Gelfand, A. E.A. E. and Holland (2010). A bivariate space-time downscaler under space and time misalignment. The Annals of Applied Statistics 4 1942–1975.
* Bradley, Holan and Wikle (2016) [author] Bradley, J. R.J. R., Holan, S. H.S. H. and Wikle, C. K.C. K. (2016). Multivariate spatio-temporal survey fusion with application to the American Community Survey and local area unemployment statistics. Stat 5 224–233.
* Bradley, Wikle and Holan (2015) [author] Bradley, J. R.J. R., Wikle, C. K.C. K. and Holan, S. H.S. H. (2015). Spatio-temporal change of support with application to American Community Survey multi-year period estimates. Stat 4 255–270.
* Bradley, Wikle and Holan (2016) [author] Bradley, J. R.J. R., Wikle, C. K.C. K. and Holan, S. H.S. H. (2016). Bayesian spatial change of support for count-valued survey data with application to the American Community Survey. Journal of the American Statistical Association 111 472–487.
* Braverman, Egerter and Williams (2011) [author] Braverman, P.P., Egerter, S.S. and Williams, D. R.D. R. (2011). The social determinants of health: coming of age. Annual Reviews of Public Health 32 381–398.
* U.S. Census Bureau (2008) [author] U. S. Census Bureau (2008). A Compass for Understanding and Using American Community Survey Data: What General Data Users Need to Know. U.S. Government Printing Office, Washington, DC.
* U.S. Census Bureau (2014) [author] U. S. Census Bureau (2014). American Community Survey Design and Methodology. U.S. Government Printing Office, Washington, DC.
* Chen, Wakefield and Lumley (2014) [author] Chen, C.C., Wakefield, J.J. and Lumley, T.T. (2014). The use of sampling weights in Bayesian hierarchical models for small area estimation. Spatial and Spatio-temporal Epidemiology 11 33 – 43.
* Dawid (1982) [author] Dawid, A. P.A. P. (1982). The well-calibrated Bayesian. Journal of the American Statistical Association 77 605–610.
* Diggle, Tawn and Moyeed (1998) [author] Diggle, P. J.P. J., Tawn, J. A.J. A. and Moyeed, R. A.R. A. (1998). Model-based geostatistics. Journal of the Royal Statistical Society: Series C (Applied Statistics) 47 29-9-350.
* Durante (2019) [author] Durante, D.D. (2019). Conjugate Bayes for probit regression via unified skew-normal distributions. Biometrika 106 765–779.
* Entezari, Brown and Rosenthal (2019) [author] Entezari, R.R., Brown, P. E.P. E. and Rosenthal, J. S.J. S. (2019). Bayesian spatial analysis of hardwood tree counts via MCMC. Environmetrics 31 e2608.
* Fay and Herriot (1979) [author] Fay, R.R. and Herriot, R.R. (1979). Estimates of income for small places: an application of James-Stein procedure to census data. Journal of the American Statistical Association 74 269–277.
* Finley, Banerjee and Carlin (2007) [author] Finley, A.A., Banerjee, S.S. and Carlin, B. P.B. P. (2007). spBayes: an R package for univariate and multivariate hierarchical point-referenced spatial models. Journal of Statistical Software 19 1–24.
* Gelfand, Banerjee and Gamerman (2005) [author] Gelfand, A. E.A. E., Banerjee, S.S. and Gamerman, D.D. (2005). Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16 465–479.
* Gelfand, Zhu and Carlin (2001) [author] Gelfand, A. E.A. E., Zhu, L.L. and Carlin, B. P.B. P. (2001). On the change of support problem for spatio-temporal data. Biostatistics 2 31–45.
* Geweke (1992) [author] Geweke, J.J. (1992). Evaluating the accuracy of sampling-based approaches to calculate posterior moments. In Bayesian Statistics 4 (J. M.J. M. Bernardo, J. O.J. O. Berger, A. P.A. P. Dawid and A. F. M.A. F. M. Smith, eds.) 169–193. Clarendon Press.
* Ghitza and Gelman (2013) [author] Ghitza, Y.Y. and Gelman, A.A. (2013). Deep interaction with MRP: election turnout and voting patterns among small electoral subgroups. American Journal of Political Science 57 762-776.
* Gilani, Berrocal and Batterman (2019) [author] Gilani, O.O., Berrocal, V. J.V. J. and Batterman, S.S. (2019). Nonstationary spatiotemporal Bayesian data fusion for pollutants in the near-road environment. Environmetrics 30 1-19.
* Gotway and Young (2002) [author] Gotway, C. AC. A. and Young, L. JL. J. (2002). Combining incompatible spatial data. Journal of the American Statistical Association 97 632–648.
* Heaton et al. (2019) [author] Heaton, M. J.M. J., Datta, A.A., Finley, A. O.A. O., Furrer, R.R., Guinness, J.J., Guhaniyogi, R.R., Gerber, F.F., Gramacy, R. B.R. B., Hammerling, D.D., Katzfuss, M.M., Lindgren, F.F., Nychka, D. W.D. W., Sun, F.F. and Zammit-Mangion, A.A. (2019). A case study competition among methods for analyzing large spatial data. Journal of Agricultural, Biological and Environmental Statistics 24 398-425.
* Iachan et al. (2016) [author] Iachan, RonaldoR., Pierannunzi, CarolC., Healey, KristieK., Greenlund, KurtK. and Town, MachellM. (2016). National weighting of data from the Behavioral Risk Factor Surveillance System (BRFSS). BMC Medical Research Methodology 16 1–12.
* Katzfuss (2017) [author] Katzfuss, M.M. (2017). A Multi-Resolution Approximation for massive spatial datasets. Journal of the American Statistical Association 112 201–214.
* Kish (1965) [author] Kish, L.L. (1965). Survey Sampling. John Wiley & Sons, New York.
* Kish (1995) [author] Kish, L.L. (1995). Methods for Design Effects. Journal of Official Statistics 11 55–77.
* Korn and Graubard (1998) [author] Korn, E. L.E. L. and Graubard, B. I.B. I. (1998). Confidence intervals for proportions with small expected number of positive counts estimated from survey data. Survey Methodology 24 193–201.
* Li et al. (2019) [author] Li, ZehangZ., Hsiao, YuanY., Godwin, JessicaJ., Martin, Bryan D.B. D., Wakefield, JonJ. and Clark, Samuel J.S. J. (2019). Changes in the spatial distribution of the under-five mortality rate: Small-area analysis of 122 DHS surveys in 262 subregions of 35 countries in Africa. PLOS ONE 14 1-17.
* Little (2006) [author] Little, R. J.R. J. (2006). Calibrated Bayes: A Bayes/frequentist roadmap. The American Statistician 60 213 – 223.
* Marmot et al. (2012) [author] Marmot, M.M., Allen, J.J., Bell, R.R., Bloomer, E.E., Goldblatt, P. on behalf of the Consortium for the Eurpopean Review of Social Determinants of HealthP. o. b. o. t. C. f. t. E. R. o. S. D. o. H. and the Health Divide (2012). WHO European review of social determinants of health and the health divide. Lancet 380 1011–1029.
* Mercer et al. (2014) [author] Mercer, L.L., Wakefield, J.J., Chen, C.C. and Lumley, T.T. (2014). A comparison of spatial smoothing methods for small area estimation with sampling weights. Spatial Statistics 8 69–85.
* Mitchell, Genton and Gumpertz (2005) [author] Mitchell, Matthew W.M. W., Genton, Marc G.M. G. and Gumpertz, Marcia L.M. L. (2005). Testing for separability of space?time covariances. Environmetrics 16 819-831.
* Moehlman and Robins-Somerville (2016) [author] Moehlman, L.L. and Robins-Somerville, M.M. (2016). The new Detroit: How gentrification has changed Detroit’s economic landscape. Michigan Daily, published September 2016.
* Pereira and Coelho (2010) [author] Pereira, L. N.L. N. and Coelho, P.P. (2010). Small area estimation of habitation transaction using time-series and cross sectional areal-level models. Journal of Applied Statistics 37 651–666.
* Pfeffermann (2013) [author] Pfeffermann, D.D. (2013). New important developments in small area estimation. Statistical Science 28 40–68.
* Polson, Scott and Windle (2013) [author] Polson, Nicholas G.N. G., Scott, James G.J. G. and Windle, JesseJ. (2013). Bayesian inference for logistic models Using Pólya?Gamma latent variables. Journal of the American Statistical Association 108 1339–1349.
* Porter et al. (2014) [author] Porter, A. T.A. T., Holan, S. H.S. H., Wikle, C. K.C. K. and Cressie, N.N. (2014). Spatial Fay-Herriot models for small area estimation with functional covariates. Statistical Methods and Applications 10 27-42.
* Pratesi and Salvati (2008) [author] Pratesi, M.M. and Salvati, N.N. (2008). Small area estimation: the EBLUP estimator based on spatially correlated random area effects. Statistical Methods and Applications 17 113-141.
* Roberts, Gelman and Gilks (1997) [author] Roberts, G. O.G. O., Gelman, A.A. and Gilks, W. R.W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. The Annals of Applied Probability 7 110–120.
* Rollston and Galea (2020) [author] Rollston, R.R. and Galea, S.S. (2020). COVID-19 and the social determinants of health. American Journal of Health Promotion 34 687-689.
* Savitsky (2016) [author] Savitsky, T. D.T. D. (2016). Bayesian nonparametric multiresolution estimation for the American Community Survey. Annals of Applied Statistics 10 2157–2181.
* Simpson et al. (2017) [author] Simpson, D.D., Rue, H.H., Riebler, A.A., Martins, T. G.T. G. and Sørbye, S. H.S. H. (2017). Penalising model component complexity: a principled, practical approach to constructing priors. Statistical Science 32 1–28.
* Simpson et al. (2019) [author] Simpson, M.M., Holan, S. H.S. H., Wikle, C. K.C. K. and Bradley, J. R.J. R. (2019). Interpolating distributions for populations in nested geographies using public-use data with application to the American Community Survey. Preprint available at: arXiv:1802.02626.
* Singh, Shukla and Kundu (2005) [author] Singh, B.B., Shukla, G.G. and Kundu, D.D. (2005). Spatio-temporal models in small-area estimation. Survey Methodology 31 183–195.
* Singu et al. (2020) [author] Singu, S.S., Acharya, A.A., Challagundla, K.K. and Byareddy, S. B.S. B. (2020). Impact of social determinants of health on the emerging COVID-19 pandemic in the United States. Frontiers in Public Health 8 406\.
* Sørbye and Rue (2011) [author] Sørbye, Sigrunn H.S. H. and Rue, HåvardH. (2011). Simultaneous Credible Bands for Latent Gaussian Models. Scandinavian Journal of Statistics 38 712-725.
|
$\gamma\mapsto\vec{\Omega}(\gamma)\cdot\overline{\ell}$ is identically zero,
contradicting Lemma 4.2-1, since $\overline{\ell}\neq 0$.
Case (b). $(j_{m})_{m\in{\mathbb{N}}}$ is bounded and
$|j_{m}^{\prime}|\to\infty$ (or viceversa): this case is excluded by the
momentum condition $\vec{\jmath}\cdot\ell_{m}+j_{m}-j_{m}^{\prime}=0$ in
(4.22) and since $(\ell_{m})$ is bounded.
Case (c). Both $(j_{m})_{m\in{\mathbb{N}}}$,
$(j_{m}^{\prime})_{m\in{\mathbb{N}}}$ are bounded: we have definitively that
$j_{m}=\overline{\jmath}$ and $j_{m}^{\prime}=\overline{\jmath}^{\prime}$,
with $\overline{\jmath},\overline{\jmath}^{\prime}\in{\mathbb{S}}_{0}^{c}$
and, since $j_{m}\neq j_{m}^{\prime}$,
$\overline{\jmath}\neq\overline{\jmath}^{\prime}\,.$ (4.25)
Therefore (4.22) becomes, in the limit $m\rightarrow\infty$,
$\partial_{\gamma}^{n}\big{(}\vec{\Omega}(\gamma)\cdot\overline{\ell}+\Omega_{\overline{\jmath}}(\gamma)-\Omega_{\overline{\jmath}^{\prime}}(\gamma)\big{)}_{|\gamma=\overline{\gamma}}=0\,,\
\forall\,n\in{\mathbb{N}}_{0}\,,\quad\vec{\jmath}\cdot\overline{\ell}+\overline{\jmath}-\overline{\jmath}^{\prime}=0\,.$
By analyticity, we obtain that
$\vec{\Omega}(\gamma)\cdot\overline{\ell}+\Omega_{\overline{\jmath}}(\gamma)-\Omega_{\overline{\jmath}^{\prime}}(\gamma)=0\quad\forall\,\gamma\in\Gamma\,,\quad\vec{\jmath}\cdot\overline{\ell}+\overline{\jmath}-\overline{\jmath}^{\prime}=0\,.$
(4.26)
We distinguish several cases:
* •
Let $\overline{\jmath},\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$|\overline{\jmath}|\neq|\overline{\jmath}^{\prime}|$. By (4.26) the vector
$(\vec{\Omega}(\gamma),\Omega_{\overline{\jmath}}(\gamma),\Omega_{\overline{\jmath}^{\prime}}(\gamma))$
is degenerate with $c:=(\overline{\ell},1,-1)\neq 0$, contradicting Lemma
4.2-4.
* •
Let $\overline{\jmath},\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$\overline{\jmath}^{\prime}=-\overline{\jmath}$. In view of (4.1), the first
equation in (4.26) becomes
$\vec{\omega}(\gamma)\cdot\overline{\ell}+\frac{\gamma}{2}\Big{(}\sum_{a=1}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}+2\frac{G_{\overline{\jmath}}(0)}{\overline{\jmath}}\Big{)}=0\quad\forall\gamma\in\Gamma\,.$
By Lemma 4.3-1 the vector $(\vec{\omega}(\gamma),\gamma)$ is non-degenerate,
thus $\overline{\ell}=0$ and
$2\frac{G_{\overline{\jmath}}(0)}{\overline{\jmath}}=0$, which is a
contradiction.
* •
Let $\overline{\jmath}^{\prime}\notin-{\mathbb{S}}$ and
$\overline{\jmath}\in-{\mathbb{S}}$. With no loss of generality suppose
$\overline{\jmath}=-\overline{\jmath}_{1}$. In view of (4.1), the first
equation in (4.26) implies that, for any $\gamma\in\Gamma$,
$(\overline{\ell}_{1}+1)\omega_{\overline{\jmath}_{1}}(\gamma)+\sum_{a=2}^{\nu}\overline{\ell}_{a}\omega_{\overline{\jmath}_{a}}(\gamma)-\omega_{\overline{\jmath}^{\prime}}(\gamma)+\frac{\gamma}{2}\Big{(}(\overline{\ell}_{1}-1)\frac{G_{\overline{\jmath}_{1}}(0)}{\overline{\jmath}_{1}}+\sum_{a=2}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}-\frac{G_{\overline{\jmath}^{\prime}}(0)}{\overline{\jmath}^{\prime}}\Big{)}=0\,.$
By Lemma 4.3-2 the vector
$\big{(}\vec{\omega}(\gamma),\omega_{\overline{\jmath}^{\prime}}(\gamma),\gamma\big{)}$
is non-degenerate, which is a contradiction.
* •
Last, let $\overline{\jmath},\overline{\jmath}^{\prime}\in-{\mathbb{S}}$ and
$\overline{\jmath}\neq\overline{\jmath}^{\prime}$, by (4.25). With no loss of
generality suppose $\overline{\jmath}=-\overline{\jmath}_{1}$ and
$\overline{\jmath}^{\prime}=-\overline{\jmath}_{2}$. Then the first equation
in (4.26) reads, for any $\gamma\in\Gamma$,
$\displaystyle(\overline{\ell}_{1}+1)\omega_{\overline{\jmath}_{1}}(\gamma)+\left(\overline{\ell}_{2}-1\right)\omega_{\overline{\jmath}_{2}}+\sum_{a=3}^{\nu}\overline{\ell}_{a}\omega_{\overline{\jmath}_{a}}(\gamma)$
$\displaystyle\ \ \ \
+\frac{\gamma}{2}\Big{(}(\overline{\ell}_{1}-1)\frac{G_{\overline{\jmath}_{1}}(0)}{\overline{\jmath}_{1}}+(\overline{\ell}_{2}+1)\frac{G_{\overline{\jmath}_{2}}(0)}{\overline{\jmath}_{2}}+\sum_{a=3}^{\nu}\overline{\ell}_{a}\frac{G_{\overline{\jmath}_{a}}(0)}{\overline{\jmath}_{a}}\Big{)}=0\,.$
Since the vector $(\vec{\omega}(\gamma),\gamma)$ is non-degenerate by Lemma
4.3-1, it implies $\overline{\ell}_{1}=-1$, $\overline{\ell}_{2}=1$,
$\overline{\ell}_{3}=\ldots=\overline{\ell}_{\nu}=0$. Inserting these values
in the momentum condition in (4.26) we obtain
$-2\overline{\jmath}_{1}+2\overline{\jmath}_{2}=0$. This contradicts
$\overline{\jmath}\neq\overline{\jmath}^{\prime}$.
Step 2. We finally consider the case when $(\ell_{m})_{m\in{\mathbb{N}}}$ is
unbounded. Up to subsequences $\ell_{m}\rightarrow\infty$ as
$m\rightarrow\infty$ and
$\lim_{m\to\infty}\ell_{m}/\braket{\ell_{m}}=:\overline{c}\neq 0$. By (4.1),
Lemma 4.4, (4.23), we have, for any $n\geq 1$,
$\displaystyle\partial_{\gamma}^{n}\frac{1}{\braket{\ell_{m}}}\Big{(}\Omega_{j_{m}}(\gamma)-\Omega_{j_{m}^{\prime}}(\gamma)\Big{)}_{|\gamma=\gamma_{m}}$
$\displaystyle=\partial_{\gamma}^{n}\Big{(}\frac{1}{\braket{\ell_{m}}\sqrt{g}}\Big{(}\frac{c_{j_{m}}(\gamma)}{|j_{m}|^{\frac{1}{2}}}-\frac{c_{j_{m}^{\prime}}(\gamma)}{|j_{m}^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\qquad+\frac{\gamma}{2\braket{\ell_{m}}}\Big{(}\frac{G_{j_{m}}(0)}{j_{m}}-\frac{G_{j_{m}^{\prime}}(0)}{j_{m}^{\prime}}\Big{)}_{|\gamma=\gamma_{m}}\Big{)}\to
0$
as $m\to\infty$. Therefore, for any $n\geq 1$, taking $m\rightarrow\infty$ in
(4.22) we get
$\partial_{\gamma}^{n}\big{(}\vec{\Omega}(\gamma)\cdot\overline{c}\big{)}_{|\gamma=\overline{\gamma}}=0$.
By analyticity this implies
$\vec{\Omega}(\gamma)\cdot\overline{c}=\overline{d}$, for all
$\gamma\in\Gamma$, contradicting Lemma 4.2-2, since $\overline{c}\neq 0$.
Proof of (4.18). It follows as (4.17) and we omit it. ∎
###### Remark 4.6.
For the irrotational gravity water waves equations (1.3) with $\gamma=0$,
quasi-periodic traveling waves solutions exist for most values of the _depth_
${\mathtt{h}}\in[{\mathtt{h}}_{1},{\mathtt{h}}_{2}]$. In detail, the non-
degeneracy of the linear frequencies with respect to the parameter
${\mathtt{h}}$ as in Lemma 4.2 is proved precisely in Lemma 3.2 in [2],
whereas the transversality properties hold by restricting the bounds in Lemma
3.4 in [2] to the Fourier sites satisfying the momentum conditions. We are not
able to use ${\mathtt{h}}$ as a parameter for any value of $\gamma\neq 0$ (in
this case we do not know if the non-degeneracy properties of Lemma 4.2 hold
with respect to ${\mathtt{h}}$).
## 5 Proof of Theorem 1.2
Under the rescaling $(\eta,\zeta)\mapsto(\varepsilon\eta,\varepsilon\zeta)$,
the Hamiltonian system (2.10) transforms into the Hamiltonian system generated
by
${\mathcal{H}}_{\varepsilon}(\eta,\zeta):=\varepsilon^{-2}{\mathcal{H}}(\varepsilon\eta,\varepsilon\zeta)={\mathcal{H}}_{L}(\eta,\zeta)+\varepsilon
P_{\varepsilon}(\eta,\zeta)\,,$ (5.1)
where ${\mathcal{H}}$ is the water waves Hamiltonian (2.9) expressed in the
Wahlén coordinates (2.7), ${\mathcal{H}}_{L}$ is defined in (2.15) and,
denoting ${\mathcal{H}}_{\geq 3}:={\mathcal{H}}-{\mathcal{H}}_{L}$ the cubic
part of the Hamiltonian,
$P_{\varepsilon}(\eta,\zeta):=\varepsilon^{-3}{\mathcal{H}}_{\geq
3}(\varepsilon\eta,\varepsilon\zeta)\,.$
We now study the Hamiltonian system generated by the Hamiltonian
${\mathcal{H}}_{\varepsilon}(\eta,\zeta)$, in the action-angle and normal
coordinates $(\theta,I,w)$ defined in Section 2.2. Thus we consider the
Hamiltonian $H_{\varepsilon}(\theta,I,w)$ defined by
$H_{\varepsilon}:={\mathcal{H}}_{\varepsilon}\circ
A=\varepsilon^{-2}{\mathcal{H}}\circ\varepsilon A$ (5.2)
where $A$ is the map defined in (2.2). The associated symplectic form is given
in (2.43).
By (2.47) (see also (2.30), (2.38)), in the variables $(\theta,I,w)$ the
quadratic Hamiltonian ${\mathcal{H}}_{L}$ defined in (2.15) simply reads, up
to a constant,
${\mathcal{N}}:={\mathcal{H}}_{L}\circ A=\vec{\Omega}(\gamma)\cdot
I+\tfrac{1}{2}\left({\bf{\Omega}}_{W}w,w\right)_{L^{2}}$
where $\vec{\Omega}(\gamma)\in{\mathbb{R}}^{\nu}$ is defined in (1.20) and
${\bf{\Omega}}_{W}$ in (2.14). Thus the Hamiltonian $H_{\varepsilon}$ in (5.2)
is
$H_{\varepsilon}={\mathcal{N}}+\varepsilon P\qquad{\rm with}\qquad
P:=P_{\varepsilon}\circ A\,.$ (5.3)
We look for an embedded invariant torus
$i:{\mathbb{T}}^{\nu}\rightarrow{\mathbb{R}}^{\nu}\times{\mathbb{R}}^{\nu}\times\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\,,\quad{\varphi}\mapsto
i({\varphi}):=(\theta({\varphi}),I({\varphi}),w({\varphi}))\,,$
of the Hamiltonian vector field
$X_{H_{\varepsilon}}:=(\partial_{I}H_{\varepsilon},-\partial_{\theta}H_{\varepsilon},\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J\nabla_{w}H_{\varepsilon})$
filled by quasi-periodic solutions with frequency vector
$\omega\in{\mathbb{R}}^{\nu}$.
### 5.1 Nash-Moser theorem of hypothetical conjugation
Instead of looking directly for quasi-periodic solutions of
$X_{H_{\varepsilon}}$ we look for quasi-periodic solutions of the family of
modified Hamiltonians, where $\alpha\in{\mathbb{R}}^{\nu}$ are additional
parameters,
$H_{\alpha}:={\mathcal{N}}_{\alpha}+\varepsilon
P\,,\quad{\mathcal{N}}_{\alpha}:=\alpha\cdot
I+\tfrac{1}{2}\left(w,{\bf{\Omega}}_{W}w\right)_{L^{2}}\,.$ (5.4)
We consider the nonlinear operator
$\displaystyle{\mathcal{F}}(i,\alpha)$
$\displaystyle:={\mathcal{F}}(\omega,\gamma,\varepsilon;i,\alpha):=\omega\cdot\partial_{\varphi}i({\varphi})-X_{H_{\alpha}}(i({\varphi}))$
$\displaystyle=\begin{pmatrix}\omega\cdot\partial_{\varphi}\theta({\varphi})&-\alpha-\varepsilon\partial_{I}P(i({\varphi}))\\\
\omega\cdot\partial_{\varphi}I({\varphi})&+\varepsilon\partial_{\theta}P(i({\varphi}))\\\
\omega\cdot\partial_{\varphi}w({\varphi})&-\,\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J({\bf{\Omega}}_{W}w({\varphi})+\varepsilon\nabla_{w}P(i({\varphi})))\end{pmatrix}\,.$
(5.5)
If ${\mathcal{F}}(i,\alpha)=0$, then the embedding ${\varphi}\mapsto
i({\varphi})$ is an invariant torus for the Hamiltonian vector field
$X_{H_{\alpha}}$, filled with quasi-periodic solutions with frequency
$\omega$.
Each Hamiltonian $H_{\alpha}$ in (5.4) is invariant under the involution
$\vec{\mathcal{S}}$ and the translations $\vec{\tau}_{\varsigma}$,
$\varsigma\in{\mathbb{R}}$, defined respectively in (2.40) and in (2.41):
$H_{\alpha}\circ\vec{\mathcal{S}}=H_{\alpha}\,,\qquad
H_{\alpha}\circ\vec{\tau}_{\varsigma}=H_{\alpha}\,,\quad\forall\,\varsigma\in{\mathbb{R}}\,.$
(5.6)
We look for a reversible traveling torus embedding $i(\varphi)=$
$(\theta({\varphi}),I({\varphi}),w({\varphi}))$, namely satisfying
$\vec{\mathcal{S}}i({\varphi})=i(-{\varphi})\,,\qquad\vec{\tau}_{\varsigma}i({\varphi})=i({\varphi}-\vec{\jmath}\varsigma)\,,\quad\forall\,\varsigma\in{\mathbb{R}}\,.$
(5.7)
Note that, by (5.1) and (5.6), the operator ${\mathcal{F}}(\cdot,\alpha)$ maps
a reversible, respectively traveling, wave into an anti-reversible,
respectively traveling, wave variation, according to Definition 3.26.
The norm of the periodic components of the embedded torus
${\mathfrak{I}}({\varphi}):=i({\varphi})-({\varphi},0,0):=\left(\Theta({\varphi}),I({\varphi}),w({\varphi})\right)\,,\quad\Theta({\varphi}):=\theta({\varphi})-{\varphi}\,,$
(5.8)
is
$\left\|{\mathfrak{I}}\right\|_{s}^{k_{0},\upsilon}:=\left\|\Theta\right\|_{H_{\varphi}^{s}}^{k_{0},\upsilon}+\left\|I\right\|_{H_{\varphi}^{s}}^{k_{0},\upsilon}+\left\|w\right\|_{s}^{k_{0},\upsilon}$,
where
$k_{0}:=m_{0}+2$ (5.9)
and $m_{0}\in{\mathbb{N}}$ is the index of non-degeneracy provided by
Proposition 4.5, which only depends on the linear unperturbed frequencies. We
will often omit to write the dependence of the various constants with respect
to $k_{0}$, which is considered as an absolute constant. We look for quasi-
periodic solutions of frequency $\omega$ belonging to a $\delta$-neighbourhood
(independent of $\varepsilon$)
${\mathtt{\Omega}}:=\big{\\{}\omega\in{\mathbb{R}}^{\nu}\ :\
\operatorname{dist}\big{(}\omega,\vec{\Omega}[\gamma_{1},\gamma_{2}]\big{)}<\delta\big{\\}}\,,\quad\delta>0\,,$
of the curve $\vec{\Omega}[\gamma_{1},\gamma_{2}]$ defined by (1.20).
The next theorem, whose proof is based on an implicit function iterative
scheme of Nash-Moser type, provides, for $\varepsilon$ small enough, a
solution $(i_{\infty},\alpha_{\infty})(\omega,\gamma;\varepsilon)$ of the
nonlinear operator ${\cal F}(\varepsilon,\omega,\gamma;i,\alpha)=0$ for all
the values of $(\omega,\gamma)$ in the Cantor like set ${\cal
C}_{\infty}^{\upsilon}$ below.
###### Theorem 5.1.
(Theorem of hypothetical conjugation) There exist positive constants ${\rm
a_{0}},\varepsilon_{0},C$ depending on ${\mathbb{S}}$, $k_{0}$ and $\tau\geq
1$ such that, for all $\upsilon=\varepsilon^{\rm a}$, ${\rm a}\in(0,{\rm
a}_{0})$ and for all $\varepsilon\in(0,\varepsilon_{0})$, there exist
1. 1.
a $k_{0}$-times differentiable function of the form
$\alpha_{\infty}:\,{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]\mapsto{\mathbb{R}}^{\nu}$,
$\displaystyle\alpha_{\infty}(\omega,\gamma):=\omega+r_{\varepsilon}(\omega,\gamma)\quad\text{
with }\quad|r_{\varepsilon}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,;$ (5.10)
2. 2.
a family of embedded reversible traveling tori $i_{\infty}({\varphi})$ (cfr.
(5.7)), defined for all
$(\omega,\gamma)\in{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]$, satisfying
$\|i_{\infty}({\varphi})-({\varphi},0,0)\|_{s_{0}}^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,;$ (5.11)
3. 3.
a sequence of $k_{0}$-times differentiable functions
$\mu_{j}^{\infty}:{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]\rightarrow{\mathbb{R}}$,
$j\in{\mathbb{S}}_{0}^{c}={\mathbb{Z}}\,\setminus\,({\mathbb{S}}\cup\\{0\\})$,
of the form
$\mu_{j}^{\infty}(\omega,\gamma)={\mathtt{m}}_{1}^{\infty}(\omega,\gamma)j+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\omega,\gamma)\Omega_{j}(\gamma)-{\mathtt{m}}_{0}^{\infty}(\omega,\gamma){\rm
sgn}(j)+{\mathfrak{r}}_{j}^{\infty}(\omega,\gamma)\,,$ (5.12)
with $\Omega_{j}(\gamma)$ defined in (1.13), satisfying
$|{\mathtt{m}}_{1}^{\infty}|^{k_{0},\upsilon}\leq C\varepsilon\,,\
|{\mathtt{m}}_{\frac{1}{2}}^{\infty}-1|^{k_{0},\upsilon}+|{\mathtt{m}}_{0}^{\infty}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,,\quad\sup_{j\in{\mathbb{S}}_{0}^{c}}|j|^{\frac{1}{2}}|{\mathfrak{r}}_{j}^{\infty}|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-3}\,,$ (5.13)
such that, for all $(\omega,\gamma)$ in the Cantor-like set
$\displaystyle{\mathcal{C}}_{\infty}^{\upsilon}:=$
$\displaystyle\Big{\\{}(\omega,\gamma)\in{\mathtt{\Omega}}\times[\gamma_{1},\gamma_{2}]\
:\ |\omega\cdot\ell|\geq\ 8\upsilon\langle\ell\rangle^{-\tau}\,,\ \
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,,$ (5.14) $\displaystyle\
\left|\omega\cdot\ell-{\mathtt{m}}_{1}^{\infty}(\omega,\gamma)j\right|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\text{ with
}\vec{\jmath}\cdot\ell+j=0;$ (5.15) $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\left|j\right|^{\frac{1}{2}}\braket{\ell}^{-\tau}\,,\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\text{
with }\vec{\jmath}\cdot\ell+j=0\,;$ (5.16) $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)-\mu_{j^{\prime}}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\,\braket{\ell}^{-\tau}\,,$ (5.17) $\displaystyle\
\quad\quad\forall\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c},\,(\ell,j,j^{\prime})\neq(0,j,j)\text{
with }\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,$ $\displaystyle\
\left|\omega\cdot\ell+\mu_{j}^{\infty}(\omega,\gamma)+\mu_{j^{\prime}}^{\infty}(\omega,\gamma)\right|\geq
4\upsilon\,\big{(}\left|j\right|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-\tau}\,,$
(5.18) $\displaystyle\
\quad\quad\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\text{
with }\vec{\jmath}\cdot\ell+j+j^{\prime}=0\,\Big{\\}}\,,$
the function
$i_{\infty}({\varphi}):=i_{\infty}(\omega,\gamma,\varepsilon;{\varphi})$ is a
solution of
${\mathcal{F}}(\omega,\gamma,\varepsilon;(i_{\infty},\alpha_{\infty})(\omega,\gamma))=0$.
As a consequence, the embedded torus ${\varphi}\mapsto i_{\infty}({\varphi})$
is invariant for the Hamiltonian vector field
$X_{H_{\alpha_{\infty}(\omega,\gamma)}}$ as it is filled by quasi-periodic
reversible traveling wave solutions with frequency $\omega$.
Note that the Cantor-like set ${\cal C}_{\infty}^{\upsilon}$ in (5.14)-(5.18)
is defined in terms of the functions
${\mathtt{m}}_{1}^{\infty}(\omega,\gamma)$ and the “final" perturbed normal
frequencies $\mu_{j}^{\infty}(\omega,\gamma)$, $j\in{\mathbb{S}}_{0}^{c}$,
which are defined for all the values of the parameters $(\omega,\gamma)$. This
formulation completely decouples the Nash-Moser implicit function theorem
construction of $(\alpha_{\infty},i_{\infty})(\omega,\gamma)$ (in Sections
6-9) from the discussion about the measure of the parameters where all the
required “non-resonance" conditions are verified (Section 5.2). This approach
simplifies considerably the presentation because the measure estimates
required to build $(i_{\infty},\alpha_{\infty})(\omega,\gamma)$ are not
verified at each step along the Nash-Moser iteration (the set ${\cal
C}_{\infty}^{\upsilon}$ in (5.14)-(5.18) could be empty, in such a case the
functions $(\alpha_{\infty},i_{\infty})(\omega,\gamma)$ constructed in Theorem
5.1 are obtained by just finitely many sums). In order to define the extended
functions $(i_{\infty},\alpha_{\infty})$ for all the values of
$(\omega,\gamma)$, preserving the weighted norm $\|\ \|^{k_{0},\upsilon}$, we
use the Whitney extension theory reported in Section 3.
We also remind that the conditions on the indexes in (5.14)-(5.18) (where
$\vec{\jmath}\in{\mathbb{Z}}^{\nu}$ is the vector in (2.42)) are due to the
fact that we look for traveling wave solutions. These restrictions are
essential to prove the measure estimates of the next section.
###### Remark 5.2.
The Diophantine condition (5.14) could be weakened requiring only
$|\omega\cdot\ell|\geq\ \upsilon\langle\ell\rangle^{-\tau}$ for any
$\ell\cdot\vec{\jmath}=0$. In such a case the vector $\omega$ could admit one
non-trivial resonance, i.e.
$\overline{\ell}\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$ such that
$\omega\cdot\overline{\ell}=0$, thus the orbit $\\{\omega
t\\}_{t\in{\mathbb{R}}}$ would densely fill a ($\nu-1$)-dimensional torus,
orthogonal to $\overline{\ell}$. In any case
$\vec{\jmath}\cdot\overline{\ell}\neq 0$ (otherwise
$|\omega\cdot\overline{\ell}|\geq\upsilon\langle\overline{\ell}\rangle^{-\tau}>0$,
contradicting that $\omega\cdot\overline{\ell}=0$) and then the closure of the
set $\\{\omega t-\vec{\jmath}x\\}_{t\in{\mathbb{R}},x\in{\mathbb{R}}}$ is
dense in ${\mathbb{T}}^{\nu}$. This is the natural minimal requirement to look
for traveling quasi-periodic solutions $U(\omega t-\vec{\jmath}x)$ (Definition
3.1).
The next goal is to deduce Theorem 1.2 from Theorem 5.1.
### 5.2 Measure estimates: proof of Theorem 1.2
We now want to prove the existence of quasi-periodic solutions of the original
Hamiltonian system $H_{\varepsilon}$ in (5.3), which is equivalent after a
rescaling to (2.10), and not of just of the Hamiltonian system generated by
the modified Hamiltonian $H_{\alpha_{\infty}}$. We proceed as follows. By
(5.10), the function $\alpha_{\infty}(\,\cdot\,,\gamma)$ from
${\mathtt{\Omega}}$ into its image $\alpha_{\infty}({\mathtt{\Omega}},\gamma)$
is invertible and
$\displaystyle\beta=\alpha_{\infty}(\omega,\gamma)=\omega+r_{\varepsilon}(\omega,\gamma)\
\Leftrightarrow$ (5.19)
$\displaystyle\omega=\alpha_{\infty}^{-1}(\beta,\gamma)=\beta+\breve{r}_{\varepsilon}(\beta,\gamma)\,,\quad\left|\breve{r}_{\varepsilon}\right|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,.$
Then, for any $\beta\in\alpha_{\infty}({\mathcal{C}}_{\infty}^{\upsilon})$,
Theorem 5.1 proves the existence of an embedded invariant torus filled by
quasi-periodic solutions with Diophantine frequency
$\omega=\alpha_{\infty}^{-1}(\beta,\gamma)$ for the Hamiltonian
$H_{\beta}=\beta\cdot I+\tfrac{1}{2}(w,{\bf{\Omega}}_{W}w)_{L^{2}}+\varepsilon
P\,.$
Consider the curve of the unperturbed tangential frequency vector
$\vec{\Omega}(\gamma)$ in (1.20). In Theorem 5.3 below we prove that for
"most" values of $\gamma\in[\gamma_{1},\gamma_{2}]$ the vector
$(\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma),\gamma)$ is in
${\mathcal{C}}_{\infty}^{\upsilon}$, obtaining an embedded torus for the
Hamiltonian $H_{\varepsilon}$ in (5.2), filled by quasi-periodic solutions
with Diophantine frequency vector
$\omega=\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma)$, denoted
${\widetilde{\Omega}}$ in Theorem 1.2. Thus $\varepsilon
A(i_{\infty}({\widetilde{\Omega}}t))$, where $A$ is defined in (2.2), is a
quasi-periodic traveling wave solution of the water waves equations (2.10)
written in the Wahlén variables. Finally, going back to the original Zakharov
variables via (2.7) we obtain solutions of (1.3). This proves Theorem 1.2
together with the following measure estimates.
###### Theorem 5.3.
(Measure estimates) Let
$\upsilon=\varepsilon^{\rm a}\,,\quad 0<{\rm a}<\min\\{{\rm
a}_{0},1/(4m_{0}^{2})\\}\,,\quad\tau>m_{0}(2m_{0}\nu+\nu+2)\,,$ (5.20)
where $m_{0}$ is the index of non-degeneracy given in Proposition 4.5 and
$k_{0}:=m_{0}+2$. Then, for $\varepsilon\in(0,\varepsilon_{0})$ small enough,
the measure of the set
${\mathcal{G}}_{\varepsilon}:=\big{\\{}\gamma\in[\gamma_{1},\gamma_{2}]\ :\
\big{(}\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma),\gamma\big{)}\in{\mathcal{C}}_{\infty}^{\upsilon}\big{\\}}$
(5.21)
satisfies $|{\mathcal{G}}_{\varepsilon}|\rightarrow\gamma_{2}-\gamma_{1}$ as
$\varepsilon\rightarrow 0$.
The rest of this section is devoted to prove Theorem 5.3. By (5.19) we have
$\vec{\Omega}_{\varepsilon}(\gamma):=\alpha_{\infty}^{-1}(\vec{\Omega}(\gamma),\gamma)=\vec{\Omega}(\gamma)+\vec{r}_{\varepsilon}\,,$
(5.22)
where
$\vec{r}_{\varepsilon}(\gamma):=\breve{r}_{\varepsilon}(\vec{\Omega}(\gamma),\gamma)$
satisfies
$|\partial_{\gamma}^{k}{\vec{r}}_{\varepsilon}(\gamma)|\leq
C\varepsilon\upsilon^{-(1+k)}\,,\quad\forall\,\left|k\right|\leq k_{0}\,,\
\text{uniformly on }[\gamma_{1},\gamma_{2}]\,.$ (5.23)
We also denote, with a small abuse of notation, for all
$j\in{\mathbb{S}}_{0}^{c}$,
$\mu_{j}^{\infty}(\gamma):=\mu_{j}^{\infty}\big{(}\vec{\Omega}_{\varepsilon}(\gamma),\gamma\big{)}:={\mathtt{m}}_{1}^{\infty}(\gamma)j+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)\Omega_{j}(\gamma)-{\mathtt{m}}_{0}^{\infty}(\gamma){\rm
sgn}(j)+{\mathfrak{r}}_{j}^{\infty}(\gamma)\,,$ (5.24)
where
${\mathtt{m}}_{1}^{\infty}(\gamma):={\mathtt{m}}_{1}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$,
${\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma):={\mathtt{m}}_{\frac{1}{2}}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$,
${\mathtt{m}}_{0}^{\infty}(\gamma):={\mathtt{m}}_{0}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$
and
${\mathfrak{r}}_{j}^{\infty}(\gamma):={\mathfrak{r}}_{j}^{\infty}(\vec{\Omega}_{\varepsilon}(\gamma),\gamma)$.
By (5.13) and (5.23) we get the estimates
$\displaystyle|\partial_{\gamma}^{k}{\mathtt{m}}_{1}^{\infty}(\gamma)|\leq
C\varepsilon\upsilon^{-k}\,,\,\big{|}\partial_{\gamma}^{k}\big{(}{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)-1\big{)}\big{|}+|\partial_{\gamma}^{k}{\mathtt{m}}_{0}^{\infty}(\gamma)|\leq
C\varepsilon\upsilon^{-k-1},$ (5.25)
$\displaystyle\sup_{j\in{\mathbb{S}}_{0}^{c}}|j|^{\frac{1}{2}}\left|\partial_{\gamma}^{k}{\mathfrak{r}}_{j}^{\infty}(\gamma)\right|\leq
C\varepsilon\upsilon^{-3-k}\,,\quad\forall\,0\leq k\leq k_{0}\,.$ (5.26)
Recalling (5.14)-(5.18), the Cantor set in (5.21) becomes
$\displaystyle{\mathcal{G}}_{\varepsilon}:=$
$\displaystyle\Big{\\{}\gamma\in[\gamma_{1},\gamma_{2}]\ :\
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$ $\displaystyle\ \
|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq
8\upsilon\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)|\geq
4\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-\tau}\,,\
\forall\,\ell\in{\mathbb{Z}}^{\nu}\,,\,j\in{\mathbb{S}}_{0}^{c}\,,\text{ with
}\vec{\jmath}\cdot\ell+j=0\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\,\braket{\ell}^{-\tau}\,,$ $\displaystyle\ \
\forall\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c},\,(\ell,j,j^{\prime})\neq(0,j,j)\text{
with }\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,;$ $\displaystyle\ \
|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\,\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-\tau}\,,$
$\displaystyle\ \
\forall\,\ell\in{\mathbb{Z}}^{\nu},\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\text{
with }\vec{\jmath}\cdot\ell+j+j^{\prime}=0\Big{\\}}\,.$
We estimate the measure of the complementary set
$\displaystyle{\mathcal{G}}_{\varepsilon}^{c}$
$\displaystyle:=[\gamma_{1},\gamma_{2}]\setminus{\mathcal{G}}_{\varepsilon}$
(5.27) $\displaystyle=\left(\bigcup_{\ell\neq 0}R_{\ell}^{(0)}\cup
R_{\ell}^{(T)}\right)\cup\left(\bigcup_{\ell\in{\mathbb{Z}}^{\nu},\,j\in{\mathbb{S}}_{0}^{c}\atop\vec{\jmath}\cdot\ell+j=0}R_{\ell,j}^{(I)}\right)\cup\left(\bigcup_{(\ell,j,j^{\prime})\neq(0,j,j),j\neq
j^{\prime}\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right)\cup\left(\bigcup_{\ell\in{\mathbb{Z}}^{\nu},j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\atop\vec{\jmath}\cdot\ell+j+j^{\prime}=0}Q_{\ell,j,j^{\prime}}^{(II)}\right)\,,$
where the “nearly-resonant sets" are, recalling the notation
$\Gamma=[\gamma_{1},\gamma_{2}]$,
$\displaystyle R_{\ell}^{(0)}:=R_{\ell}^{(0)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|<8\upsilon\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.28) $\displaystyle R_{\ell}^{(T)}:=R_{\ell}^{(T)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|<8\upsilon\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.29) $\displaystyle R_{\ell,j}^{(I)}:=R_{\ell,j}^{(I)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)|<4\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.30) $\displaystyle
R_{\ell,j,j^{\prime}}^{(II)}:=R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau):=$
$\displaystyle\big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|<4\upsilon\,\braket{\ell}^{-\tau}\big{\\}}\,,$
(5.31) $\displaystyle
Q_{\ell,j,j^{\prime}}^{(II)}:=Q_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau):=$
$\displaystyle\Big{\\{}\gamma\in\Gamma\,:\,|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|<\frac{4\upsilon\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}}{\braket{\ell}^{\tau}}\Big{\\}}\,.$
(5.32)
Note that in the third union in (5.27) we may require $j\neq j^{\prime}$
because $R_{\ell,j,j}^{(II)}\subset R_{\ell}^{(0)}$. In the sequel we shall
always suppose the momentum conditions on the indexes $\ell,j,j^{\prime}$
written in (5.27). Some of the above sets are empty.
###### Lemma 5.4.
For $\varepsilon\in(0,\varepsilon_{0})$ small enough, if
$Q_{\ell,j,j^{\prime}}^{(II)}\neq\emptyset$ then
$|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\leq C\braket{\ell}$.
###### Proof.
If $Q_{\ell,j,j^{\prime}}^{(II)}\neq\emptyset$ then there exists
$\gamma\in[\gamma_{1},\gamma_{2}]$ such that
$\left|\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)\right|<\frac{4\upsilon\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}}{\braket{\ell}^{\tau}}+C|\ell|\,.$
(5.33)
By (5.24) we have
$\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)={\mathtt{m}}_{1}^{\infty}(\gamma)(j+j^{\prime})+{\mathtt{m}}_{\frac{1}{2}}^{\infty}(\gamma)(\Omega_{j}(\gamma)+\Omega_{j^{\prime}}(\gamma))-{\mathtt{m}}_{0}^{\infty}(\gamma)({\rm
sgn}(j)+{\rm
sgn}(j^{\prime}))+{\mathfrak{r}}_{j}^{\infty}(\gamma)+{\mathfrak{r}}_{j^{\prime}}^{\infty}(\gamma)\,.$
Then, by (5.25)-(5.26) with $k=0$, Lemma 4.4 and the momentum condition
$j+j^{\prime}=-\vec{\jmath}\cdot\ell$, we deduce, for $\varepsilon$ small
enough,
$\displaystyle|\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma)|$
$\displaystyle\geq-C\varepsilon|\ell|+\tfrac{\sqrt{g}}{2}\,\big{|}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{|}-C^{\prime}-C\varepsilon\upsilon^{-3}\,.$
(5.34)
Combining (5.33) and (5.34), we deduce
$||j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}|\leq C\braket{\ell}$, for
$\varepsilon$ small enough. ∎
In order to estimate the measure of the sets (5.28)-(5.32), the key point is
to prove that the perturbed frequencies satisfy transversality properties
similar to the ones (4.15)-(4.18) satisfied by the unperturbed frequencies. By
Proposition 4.5, (5.22), and the estimates (5.23), (5.25)-(5.26) we deduce the
following lemma (cfr. Lemma 5.5 in [7]).
###### Lemma 5.5.
(Perturbed transversality) For $\varepsilon\in(0,\varepsilon_{0})$ small
enough and for all $\gamma\in[\gamma_{1},\gamma_{2}]$,
$\displaystyle\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\quad\forall\,\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,;$
(5.35) $\displaystyle\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\quad\forall\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$
(5.36) $\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\,,\\\
\vec{\jmath}\cdot\ell+j=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j\in{\mathbb{S}}_{0}^{c}\,;\end{cases}$ (5.37)
$\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\\\
\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,,\
(\ell,j,j^{\prime})\neq(0,j,j)\,;\end{cases}$ (5.38)
$\displaystyle\begin{cases}\max_{0\leq n\leq
m_{0}}|\partial_{\gamma}^{n}(\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)+\mu_{j^{\prime}}^{\infty}(\gamma))|\geq\frac{\rho_{0}}{2}\braket{\ell}\\\
\vec{\jmath}\cdot\ell+j+j^{\prime}=0\,,\quad\ell\in{\mathbb{Z}}^{\nu}\,,\
j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\,.\end{cases}$ (5.39)
The transversality estimates (5.35)-(5.39) and an application of Rüssmann
Theorem 17.1 in [32], directly imply the following bounds for the sets in
(5.27) (cfr. Lemma 5.6 in [7]).
###### Lemma 5.6.
(Estimates of the resonant sets) The measure of the sets (5.27)- (5.32)
satisfy
$\displaystyle|R_{\ell}^{(0)}|,|R_{\ell}^{(T)}|\lesssim(\upsilon\braket{\ell}^{-(\tau+1)})^{\frac{1}{m_{0}}}\,,$
$\displaystyle\quad|R_{\ell,j}^{(I)}|\lesssim\big{(}\upsilon|j|^{\frac{1}{2}}\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,,$
$\displaystyle|R_{\ell,j,j^{\prime}}^{(II)}|\lesssim\big{(}\upsilon\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,,$
$\displaystyle\quad|Q_{\ell,j,j^{\prime}}^{(II)}|\lesssim\big{(}\upsilon\,\big{(}|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}\big{)}\braket{\ell}^{-(\tau+1)}\big{)}^{\frac{1}{m_{0}}}\,.$
We now estimate the measure of all the sets in (5.27). By Lemma 5.6, and the
choice of $\tau$ in (5.20), we have
$\displaystyle\Big{|}\bigcup_{\ell\neq 0}R^{(0)}_{\ell}\cup
R^{(T)}_{\ell}\Big{|}\leq\sum_{\ell\neq
0}|R^{(0)}_{\ell}|+|R^{(T)}_{\ell}|\lesssim\sum_{\ell\neq
0}\Big{(}\frac{\upsilon}{\braket{\ell}^{\tau+1}}\Big{)}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,,$
(5.40) $\displaystyle\left|\bigcup_{\ell\neq
0,j=-\vec{\jmath}\cdot\ell}R_{\ell,j}^{(I)}\right|\leq\sum_{\ell\neq
0}|R_{\ell,-\vec{\jmath}\cdot\ell}^{(I)}|\lesssim\sum_{\ell}\Big{(}\frac{\upsilon}{\braket{\ell}^{\tau+\frac{1}{2}}}\Big{)}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,,$
(5.41)
and using also Lemma 5.4,
$\displaystyle\left|\bigcup_{\ell,\,j,j^{\prime}\in{\mathbb{S}}_{0}^{c}\atop\vec{\jmath}\cdot\ell+j+j^{\prime}=0}Q_{\ell,j,j^{\prime}}^{(II)}\right|\leq\sum_{\ell,\left|j\right|\leq
C\braket{\ell}^{2},\atop
j^{\prime}=-\vec{\jmath}\cdot\ell-j}|Q_{\ell,j,j^{\prime}}^{(II)}|\lesssim\sum_{\ell,\left|j\right|\leq
C\braket{\ell}^{2}}\left(\frac{\upsilon}{\braket{\ell}^{\tau}}\right)^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,.$
(5.42)
We are left with estimating the measure of
$\displaystyle\bigcup_{(\ell,j,j^{\prime})\neq(0,j,j),j\neq
j^{\prime}\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}\\!\\!\\!\\!\\!\\!\\!\\!R_{\ell,j,j^{\prime}}^{(II)}$
$\displaystyle=\left(\bigcup_{j\neq j^{\prime}\,,\ j\cdot
j^{\prime}<0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right)\cup\left(\bigcup_{j\neq
j^{\prime}\,,\ j\cdot
j^{\prime}>0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}\right):={\mathtt{I}}_{1}+{\mathtt{I}}_{2}\,.$
(5.43)
We first estimate the measure of ${\mathtt{I}}_{1}$. For $j\cdot
j^{\prime}<0$, the momentum condition reads $j-j^{\prime}={\rm
sgn}(j)(|j|+|j^{\prime}|)=-\vec{\jmath}\cdot\ell$, thus $|j|,|j^{\prime}|\leq
C\left\langle\ell\right\rangle$. Hence, by Lemma 5.6 and the choice of $\tau$
in (5.20), we have
$\displaystyle|{\mathtt{I}}_{1}|\leq\sum_{\ell,|j|\leq
C\left\langle\ell\right\rangle,j^{\prime}=j+\vec{\jmath}\cdot\ell}|R_{\ell,j,j^{\prime}}^{(II)}|\lesssim\sum_{\ell,\left|j\right|\leq
C\braket{\ell}}\left(\frac{\upsilon}{\braket{\ell}^{\tau+1}}\right)^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{m_{0}}}\,.$
(5.44)
Then we estimate the measure of ${\mathtt{I}}_{2}$ in (5.43). The key step is
given in the next lemma. Remind the definition of the sets
$R_{\ell,j,j^{\prime}}^{(II)}$ and $R_{\ell}^{(T)}$ in (5.30) and (5.31).
###### Lemma 5.7.
Let $\upsilon_{0}\geq\upsilon$ and $\tau\geq\tau_{0}\geq 1$. There is a
constant $C_{1}>0$ such that, for $\varepsilon$ small enough, for any
$\vec{\jmath}\cdot\ell+j-j^{\prime}=0$, $j\cdot j^{\prime}>0$,
$\min\\{|j|,|j^{\prime}|\\}\geq
C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}\quad\Longrightarrow\quad
R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\subset\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})\,.$ (5.45)
###### Proof.
If $\gamma\in[\gamma_{1},\gamma_{2}]\setminus\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})$, then
$|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}(\gamma)\vec{\jmath})\cdot\ell|\geq
8\upsilon_{0}\braket{\ell}^{-\tau_{0}}\,,\quad\forall\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}\,.$
(5.46)
Then, by (5.24), the momentum condition $j-j^{\prime}=-\vec{\jmath}\cdot\ell$,
(5.25), (5.26), Lemma 4.4, the condition $j\cdot j^{\prime}>0$, (4.23), and
(5.46), we deduce that
$\displaystyle|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+{\mathtt{m}}_{1}^{\infty}(j-j^{\prime})|-|{\mathtt{m}}_{\frac{1}{2}}^{\infty}||\Omega_{j}(\gamma)-\Omega_{j^{\prime}}(\gamma)|-|{\mathfrak{r}}_{j}^{\infty}(\gamma)-{\mathfrak{r}}_{j^{\prime}}^{\infty}(\gamma)|$
$\displaystyle\geq|(\vec{\Omega}_{\varepsilon}(\gamma)-{\mathtt{m}}_{1}^{\infty}\vec{\jmath})\cdot\ell|-(1-C\varepsilon\upsilon^{-1})\big{|}|j|^{\frac{1}{2}}-|j^{\prime}|^{\frac{1}{2}}\big{|}-C\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}-C\frac{\varepsilon}{\upsilon^{3}}\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\geq\frac{8\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}-\frac{1}{2}\frac{|j-j^{\prime}|}{|j|^{\frac{1}{2}}+|j^{\prime}|^{\frac{1}{2}}}-C\Big{(}\frac{1}{|j|^{\frac{1}{2}}}+\frac{1}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}\geq\frac{8\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}-C\Big{(}\frac{\braket{\ell}}{|j|^{\frac{1}{2}}}+\frac{\braket{\ell}}{|j^{\prime}|^{\frac{1}{2}}}\Big{)}$
$\displaystyle\geq\frac{4\upsilon_{0}}{\braket{\ell}^{\tau_{0}}}$
for any
$|j|,|j^{\prime}|>C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}$, for
$C_{1}>C^{2}/64$. Since $\upsilon_{0}\geq\upsilon$ and $\tau\geq\tau_{0}$ we
deduce that
$|\vec{\Omega}_{\varepsilon}(\gamma)\cdot\ell+\mu_{j}^{\infty}(\gamma)-\mu_{j^{\prime}}^{\infty}(\gamma)|\geq
4\upsilon\braket{\ell}^{-\tau}\,,$
namely that $\gamma\not\in R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)$. ∎
Note that the set of indexes $(\ell,j,j^{\prime})$ such that
$\vec{\jmath}\cdot\ell+j-j^{\prime}=0$ and
$\min\\{|j|,|j^{\prime}|\\}<C_{1}\upsilon_{0}^{-2}\braket{\ell}^{2(\tau_{0}+1)}$
is included, for $\upsilon_{0}$ small enough, into the set
${\cal I}_{\ell}:=\Big{\\{}(\ell,j,j^{\prime})\
:\,\vec{\jmath}\cdot\ell+j-j^{\prime}=0\,,\
|j|,|j^{\prime}|\leq\upsilon_{0}^{-3}\langle\ell\rangle^{2(\tau_{0}+1)}\Big{\\}}$
(5.47)
because
$\max\\{|j|,|j^{\prime}|\\}\leq\min\\{|j|,|j^{\prime}|\\}+|j-j^{\prime}|<C_{1}\upsilon_{0}^{-2}\langle\ell\rangle^{2(\tau_{0}+1)}+C\langle\ell\rangle\leq\upsilon_{0}^{-3}\langle\ell\rangle^{2(\tau_{0}+1)}$.
As a consequence, by Lemma 5.7 we deduce that
$\displaystyle{\mathtt{I}}_{2}=\bigcup_{j\neq j^{\prime}\,,\ j\cdot
j^{\prime}>0\atop\vec{\jmath}\cdot\ell+j-j^{\prime}=0}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\subset\Big{(}\bigcup_{\ell\neq
0}R_{\ell}^{(T)}(\upsilon_{0},\tau_{0})\Big{)}\bigcup\Big{(}\bigcup_{(\ell,j,j^{\prime})\in{\cal
I}_{\ell}}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\Big{)}\,.$ (5.48)
###### Lemma 5.8.
Let $\tau_{0}:=m_{0}\nu$ and $\upsilon_{0}=\upsilon^{\frac{1}{4m_{0}}}$. Then
$|{\mathtt{I}}_{2}|\leq C\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$ (5.49)
###### Proof.
By (5.40) (applied with $\upsilon_{0},\tau_{0}$ instead of $\upsilon,\tau$),
and $\tau_{0}=m_{0}\nu$, the measure of
$\Big{|}\bigcup_{\ell\neq
0}R^{(T)}_{\ell}(\upsilon_{0},\tau_{0})\Big{|}\lesssim\upsilon_{0}^{\frac{1}{m_{0}}}\lesssim\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$
(5.50)
Moreover, recalling (5.47),
$\Big{|}\bigcup_{(\ell,j,j^{\prime})\in{\cal
I}_{\ell}}R_{\ell,j,j^{\prime}}^{(II)}(\upsilon,\tau)\Big{|}\lesssim\sum_{\ell\in{\mathbb{Z}}^{\nu}\atop|j|\leq
C_{1}\upsilon_{0}^{-3}\braket{\ell}^{2(\tau_{0}+1)}}\left(\frac{\upsilon}{\braket{\ell}^{\tau+1}}\right)^{\frac{1}{m_{0}}}\lesssim\sum_{\ell\in{\mathbb{Z}}^{\nu}}\frac{\upsilon^{\frac{1}{m_{0}}}\upsilon_{0}^{-3}}{\braket{\ell}^{\frac{\tau+1}{m_{0}}-2(\tau_{0}+1)}}\leq
C\upsilon^{\frac{1}{4m_{0}}}\,,$ (5.51)
by the choice of $\tau$ in (5.20) and $\upsilon_{0}$. The bound (5.49) follows
by (5.50) and (5.51). ∎
###### Proof of Theorem 5.3 completed..
By (5.27), (5.40), (5.41), (5.42), (5.43), (5.44) and (5.49) we deduce that
$\left|{\mathcal{G}}_{\varepsilon}^{c}\right|\leq
C\upsilon^{\frac{1}{4m_{0}^{2}}}\,.$
For $\upsilon=\varepsilon^{\mathtt{a}}$ as in (5.20), we get
$|{\mathcal{G}}_{\varepsilon}|\geq\gamma_{2}-\gamma_{1}-C\varepsilon^{{\mathtt{a}}/4m_{0}^{2}}$.
The proof of Theorem 5.3 is concluded. ∎
###### Remark 5.9.
We have actually imposed in Lemma 5.8 the stronger non-resonance condition
(5.15) with $\upsilon_{0}=\upsilon^{\frac{1}{4m_{0}}}>\upsilon$. Since it has
no significant importance for Lemma 7.7 we keep $\upsilon$.
## 6 Approximate inverse
In order to implement a convergent Nash-Moser scheme that leads to a solution
of ${\mathcal{F}}(i,\alpha)=0$, where ${\mathcal{F}}(i,\alpha)$ is the
nonlinear operator defined in (5.1), we construct an _almost approximate right
inverse_ of the linearized operator
${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0},\alpha_{0})[\widehat{\imath},{\widehat{\alpha}}]=\omega\cdot\partial_{\varphi}\widehat{\imath}-{\rm
d}_{i}X_{H_{\alpha}}\left(i_{0}({\varphi})\right)[\widehat{\imath}]-\left({\widehat{\alpha}},0,0\right)\,.$
Note that ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0},\alpha_{0})={\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0})$ is independent of $\alpha_{0}$. We assume
that the torus
$i_{0}({\varphi})=(\theta_{0}({\varphi}),I_{0}({\varphi}),w_{0}({\varphi}))$
is reversible and traveling, according to (5.7).
In the sequel we shall assume the smallness condition, for some
${\mathtt{k}}:={\mathtt{k}}(\tau,\nu)>0$,
$\varepsilon\upsilon^{-{\mathtt{k}}}\ll 1$.
We closely follow the strategy presented in [6] and implemented for the water
waves equations in [9, 2, 7]. As shown in [7] this construction preserves the
momentum preserving properties needed for the search of traveling waves and
the estimates are very similar. Thus we are short.
First of all, we state tame estimates for the composition operator induced by
the Hamiltonian vector field
$X_{P}=(\partial_{I}P,-\partial_{\theta}P,\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}J\nabla_{w}P)$
in (5.1) (see Lemma 6.1 of [7]).
###### Lemma 6.1.
(Estimates of the perturbation $P$) Let ${\mathfrak{I}}({\varphi})$ in (5.8)
satisfy $\left\|{\mathfrak{I}}\right\|_{3s_{0}+2k_{0}+5}^{k_{0},\upsilon}\leq
1$. Then, for any $s\geq s_{0}$,
$\left\|X_{P}(i)\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}1+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+3}^{k_{0},\upsilon}$,
and, for all
$\widehat{\imath}:=({\widehat{\theta}},{\widehat{I}},{\widehat{w}})$,
$\displaystyle\left\|{\rm
d}_{i}X_{P}(i)[\widehat{\imath}]\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+1}^{k_{0},\upsilon}+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+4}^{k_{0},\upsilon}\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon}\,,$
$\displaystyle\left\|{\rm
d}_{i}^{2}X_{P}(i)[\widehat{\imath},\widehat{\imath}]\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+1}^{k_{0},\upsilon}\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon}+\left\|{\mathfrak{I}}\right\|_{s+2s_{0}+2k_{0}+5}^{k_{0},\upsilon}(\left\|\widehat{\imath}\right\|_{s_{0}+1}^{k_{0},\upsilon})^{2}\,.$
Along this section, we assume the following hypothesis, which is verified by
the approximate solutions obtained at each step of the Nash-Moser Theorem 9.1.
* •
ANSATZ. The map
$(\omega,\gamma)\mapsto{\mathfrak{I}}_{0}(\omega,\gamma)=i_{0}({\varphi};\omega,\gamma)-({\varphi},0,0)$
is $k_{0}$-times differentiable with respect to the parameters
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and, for
some $\mu:=\mu(\tau,\nu)>0$, $\upsilon\in(0,1)$,
$\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu}^{k_{0},\upsilon}+\left|\alpha_{0}-\omega\right|^{k_{0},\upsilon}\leq
C\varepsilon\upsilon^{-1}\,.$ (6.1)
We first modify the approximate torus $i_{0}({\varphi})$ to obtain a nearby
isotropic torus $i_{\delta}({\varphi})$, namely such that the pull-back 1-form
$i_{\delta}^{*}\Lambda$ is closed, where $\Lambda$ is the Liouville 1-form
defined in (2.44). Consider the pull-back $1$-form
$\displaystyle i_{0}^{*}\Lambda$
$\displaystyle=\sum_{k=1}^{\nu}a_{k}({\varphi}){\rm d}{\varphi}_{k}\,,\quad
a_{k}({\varphi}):=-\big{(}[\partial_{\varphi}\theta_{0}({\varphi})]^{\top}I_{0}({\varphi})\big{)}_{k}+\tfrac{1}{2}\big{(}J_{\angle}^{-1}w_{0}({\varphi}),\partial_{{\varphi}_{k}}w_{0}({\varphi})\big{)}_{L^{2}}\,,$
(6.2)
and define
$A_{kj}({\varphi}):=\partial_{{\varphi}_{k}}a_{j}({\varphi})-\partial_{{\varphi}_{j}}a_{k}({\varphi})$.
The next Lemma follows as in Lemma 5.3 in [2] and Lemma 6.2 in [7]. Let
$Z({\varphi}):={\mathcal{F}}(i_{0},\alpha_{0})({\varphi})=\omega\cdot\partial_{\varphi}i_{0}({\varphi})-X_{H_{\alpha_{0}}}(i_{0}({\varphi}))$.
###### Lemma 6.2.
(Isotropic torus) The torus
$i_{\delta}({\varphi}):=(\theta_{0}({\varphi}),I_{\delta}({\varphi}),w_{0}({\varphi}))$,
defined by
$I_{\delta}({\varphi}):=I_{0}({\varphi})+[\partial_{\varphi}\theta_{0}({\varphi})]^{-\top}\rho({\varphi})\,,\quad\rho=(\rho_{j})_{j=1,\ldots,\nu}\,,\quad\rho_{j}({\varphi}):=\Delta_{\varphi}^{-1}\sum_{k=1}^{\nu}\partial_{{\varphi}_{k}}A_{kj}({\varphi})\,,$
is isotropic. Moreover, there is $\sigma:=\sigma(\nu,\tau)$ such that, for all
$s\geq s_{0}$,
$\displaystyle\left\|I_{\delta}-I_{0}\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|{\mathfrak{I}}_{0}\right\|_{s+1}^{k_{0},\upsilon}\,,\quad\left\|I_{\delta}-I_{0}\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\upsilon^{-1}\big{(}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\big{)}$
(6.3)
$\displaystyle\left\|{\mathcal{F}}(i_{\delta},\alpha_{0})\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\,,\quad\left\|{\rm
d}_{i}(i_{\delta})[\widehat{\imath}]\right\|_{s_{1}}\lesssim_{s_{1}}\left\|\widehat{\imath}\right\|_{s_{1}+1}\,,$
(6.4)
for $s_{1}\leq s_{0}+\mu$ (cfr. (6.1)). Furthermore $i_{\delta}({\varphi})$ is
a reversible and traveling torus, cfr. (5.7).
We first find an approximate inverse of the linearized operator ${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$. We introduce the symplectic
diffeomorphism $G_{\delta}:(\phi,y,{\mathtt{w}})\rightarrow(\theta,I,w)$ of
the phase space
${\mathbb{T}}^{\nu}\times{\mathbb{R}}^{\nu}\times\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$,
$\begin{pmatrix}\theta\\\ I\\\
w\end{pmatrix}:=G_{\delta}\begin{pmatrix}\phi\\\ y\\\
{\mathtt{w}}\end{pmatrix}:=\begin{pmatrix}\theta_{0}(\phi)\\\
I_{\delta}(\phi)+\left[\partial_{\phi}\theta_{0}(\phi)\right]^{-\top}y+\left[(\partial_{\theta}{\widetilde{w}}_{0})(\theta_{0}(\phi))\right]^{\top}J_{\angle}^{-1}{\mathtt{w}}\\\
w_{0}(\phi)+{\mathtt{w}}\end{pmatrix}\,,$ (6.5)
where ${\widetilde{w}}_{0}(\theta):=w_{0}(\theta_{0}^{-1}(\theta))$. It is
proved in Lemma 2 of [6] that $G_{\delta}$ is symplectic, because the torus
$i_{\delta}$ is isotropic (Lemma 6.2). In the new coordinates, $i_{\delta}$ is
the trivial embedded torus $(\phi,y,{\mathtt{w}})=(\phi,0,0)$. Moreover the
diffeomorphism $G_{\delta}$ in (6.5) is reversibility and momentum preserving,
in the sense that (Lemma 6.3 in [7]) $\vec{\mathcal{S}}\circ
G_{\delta}=G_{\delta}\circ\vec{\mathcal{S}}$, $\vec{\tau}_{\varsigma}\circ
G_{\delta}=G_{\delta}\circ\vec{\tau}_{\varsigma}$,
$\forall\,\varsigma\in{\mathbb{R}}$, where $\vec{\mathcal{S}}$ and
$\vec{\tau}_{\varsigma}$ are defined respectively in (2.40), (2.41).
Under the symplectic diffeomorphism $G_{\delta}$, the Hamiltonian vector field
$X_{H_{\alpha}}$ changes into
$X_{K_{\alpha}}=\left(DG_{\delta}\right)^{-1}X_{H_{\alpha}}\circ
G_{\delta}\qquad{\rm where}\qquad K_{\alpha}:=H_{\alpha}\circ G_{\delta}$
is reversible and momentum preserving, in the sense that
$K_{\alpha}\circ\vec{\mathcal{S}}=K_{\alpha}$,
$K_{\alpha}\circ\vec{\tau}_{\varsigma}=K_{\alpha}$,
$\forall\,\varsigma\in{\mathbb{R}}$.
The Taylor expansion of $K_{\alpha}$ at the trivial torus $(\phi,0,0)$ is
$\displaystyle K_{\alpha}(\phi,y,{\mathtt{w}})=$ $\displaystyle\
K_{00}(\phi,\alpha)+K_{10}(\phi,\alpha)\cdot
y+(K_{01}(\phi,\alpha),{\mathtt{w}})_{L^{2}}+\tfrac{1}{2}K_{20}(\phi)y\cdot y$
(6.6)
$\displaystyle+(K_{11}(\phi)y,{\mathtt{w}})_{L^{2}}+\tfrac{1}{2}(K_{02}(\phi){\mathtt{w}},{\mathtt{w}})_{L^{2}}+K_{\geq
3}(\phi,y,{\mathtt{w}})\,,$
where $K_{\geq 3}$ collects all terms at least cubic in the variables
$(y,{\mathtt{w}})$. Here $K_{00}\in{\mathbb{R}}$,
$K_{10}\in{\mathbb{R}}^{\nu}$,
$K_{01}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, whereas $K_{20}$
is a $\nu\times\nu$ symmetric matrix,
$K_{11}\in{\mathcal{L}}({\mathbb{R}}^{\nu},\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle})$
and $K_{02}$ is a self-adjoint operator acting on
$\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.
The Hamilton equations associated to (6.6) are
$\begin{cases}\dot{\phi}=K_{10}(\phi,\alpha)+K_{20}(\phi)y+[K_{11}(\phi)]^{\top}{\mathtt{w}}+\partial_{y}K_{\geq
3}(\phi,y,{\mathtt{w}})\\\
\dot{y}=-\partial_{\phi}K_{00}(\phi,\alpha)-[\partial_{\phi}K_{10}(\phi,\alpha)]^{\top}y-[\partial_{\phi}K_{01}(\phi,\alpha)]^{\top}{\mathtt{w}}\\\
\ \ \ \ \ -\partial_{\phi}\left(\tfrac{1}{2}K_{20}(\phi)y\cdot
y+\left(K_{11}(\phi)y,{\mathtt{w}}\right)_{L^{2}}+\tfrac{1}{2}\left(K_{02}(\phi){\mathtt{w}},{\mathtt{w}}\right)_{L^{2}}+K_{\geq
3}(\phi,y,{\mathtt{w}})\right)\\\
\dot{\mathtt{w}}=J_{\angle}\,\left(K_{01}(\phi,\alpha)+K_{11}(\phi)y+K_{02}(\phi){\mathtt{w}}+\nabla_{{\mathtt{w}}}K_{\geq
3}(\phi,y,{\mathtt{w}})\right)\,,\end{cases}$ (6.7)
where $\partial_{\phi}K_{10}^{\top}$ is the $\nu\times\nu$ transposed matrix
and
$\partial_{\phi}K_{01}^{\top},K_{11}^{\top}:\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\rightarrow{\mathbb{R}}^{\nu}$
are defined by the duality relation
$(\partial_{\phi}K_{01}[{\widehat{\phi}}],{\mathtt{w}})_{L^{2}}={\widehat{\phi}}\cdot[\partial_{\phi}K_{01}]^{\top}{\mathtt{w}}$
for any ${\widehat{\phi}}\in{\mathbb{R}}^{\nu}$,
${\mathtt{w}}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.
The terms $K_{00},K_{01}$, $K_{10}-\omega$ in the Taylor expansion (6.6)
vanish at an exact solution: indeed, arguing as in Lemma 5.4 in [2], there is
$\sigma:=\sigma(\nu,\tau)>0$, such that, for all $s\geq s_{0}$,
$\left\|\partial_{\phi}K_{00}(\cdot,\alpha_{0})\right\|_{s}^{k_{0},\upsilon}+\left\|K_{10}(\cdot,\alpha_{0})-\omega\right\|_{s}^{k_{0},\upsilon}+\left\|K_{01}(\cdot,\alpha_{0})\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\left\|Z\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|Z\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\,.$
(6.8)
Under the linear change of variables
$DG_{\delta}({\varphi},0,0)\begin{pmatrix}{\widehat{\phi}}\\\ {\widehat{y}}\\\
{\widehat{{\mathtt{w}}}}\end{pmatrix}:=\begin{pmatrix}\partial_{\phi}\theta_{0}({\varphi})&0&0\\\
\partial_{\phi}I_{\delta}({\varphi})&[\partial_{\phi}\theta_{0}({\varphi})]^{-\top}&[(\partial_{\theta}{\widetilde{w}}_{0})(\theta_{0}({\varphi}))]^{\top}J_{\angle}^{-1}\\\
\partial_{\phi}w_{0}({\varphi})&0&{\rm
Id}\end{pmatrix}\begin{pmatrix}{\widehat{\phi}}\\\ {\widehat{y}}\\\
{\widehat{{\mathtt{w}}}}\end{pmatrix}\,,$
the linearized operator ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$ is
approximately transformed into the one obtained when one linearizes the
Hamiltonian system (6.7) at $(\phi,y,{\mathtt{w}})=({\varphi},0,0)$,
differentiating also in $\alpha$ at $\alpha_{0}$ and changing
$\partial_{t}\rightsquigarrow\omega\cdot\partial_{\varphi}$, namely
$\begin{pmatrix}\widehat{\phi}\\\ \widehat{y}\\\ \widehat{\mathtt{w}}\\\
\widehat{\alpha}\end{pmatrix}\mapsto\begin{pmatrix}\omega\cdot\partial_{\varphi}{\widehat{\phi}}-\partial_{\phi}K_{10}({\varphi})[{\widehat{\phi}}]-\partial_{\alpha}K_{10}({\varphi})[{\widehat{\alpha}}]-K_{20}({\varphi}){\widehat{y}}-[K_{11}({\varphi})]^{\top}{\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{y}}+\partial_{\phi\phi}K_{00}({\varphi})[{\widehat{\phi}}]+\partial_{\alpha}\partial_{\phi}K_{00}({\varphi})[{\widehat{\alpha}}]+[\partial_{\phi}K_{10}({\varphi})]^{\top}{\widehat{y}}+[\partial_{\phi}K_{01}({\varphi})]^{\top}{\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{{\mathtt{w}}}}-J_{\angle}\,\big{(}\partial_{\phi}K_{01}({\varphi})[{\widehat{\phi}}]+\partial_{\alpha}K_{01}({\varphi})[{\widehat{\alpha}}]+K_{11}({\varphi}){\widehat{y}}+K_{02}({\varphi}){\widehat{{\mathtt{w}}}}\big{)}\end{pmatrix}.$
(6.9)
In order to construct an “almost approximate" inverse of (6.9), we need that
${\mathcal{L}}_{\omega}:=\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}\left(\omega\cdot\partial_{\varphi}-JK_{02}({\varphi})\right)|_{\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}}$
(6.10)
is "almost invertible" (on traveling waves) up to remainders of size
$O(N_{{\mathtt{n}}-1}^{-{{\mathtt{a}}}})$, where, for
${\mathtt{n}}\in{\mathbb{N}}_{0}$
$N_{\mathtt{n}}:=K_{\mathtt{n}}^{p}\,,\quad
K_{\mathtt{n}}:=K_{0}^{\chi^{\mathtt{n}}}\,,\quad\chi=3/2\,.$ (6.11)
The $(K_{\mathtt{n}})_{{\mathtt{n}}\geq 0}$ is the scale used in the nonlinear
Nash-Moser iteration of Section 9 and $(N_{\mathtt{n}})_{{\mathtt{n}}\geq 0}$
is the one in Lemma 7.7 and Theorem 8.2. Let
$H_{\angle}^{s}({\mathbb{T}}^{\nu+1}):=H^{s}({\mathbb{T}}^{\nu+1})\cap\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$.
* (AI)
Almost invertibility of ${\mathcal{L}}_{\omega}$: There exist positive real
numbers $\sigma$, $\mu({\mathtt{b}})$, ${\mathtt{a}}$, $p$, $K_{0}$ and a
subset
${\mathtt{\Lambda}}_{o}\subset{\mathtt{D}}{\mathtt{C}}(\upsilon,\tau)\times[\gamma_{1},\gamma_{2}]$
such that, for all $(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, the operator
${\mathcal{L}}_{\omega}$ may be decomposed as
${\mathcal{L}}_{\omega}={\mathcal{L}}_{\omega}^{<}+{\mathcal{R}}_{\omega}+{\mathcal{R}}_{\omega}^{\perp}\,,$
(6.12)
where, for any traveling wave function $g\in
H_{\angle}^{s+\sigma}({\mathbb{T}}^{\nu+1},{\mathbb{R}}^{2})$ and for any
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, there is a traveling wave solution
$h\in H_{\angle}^{s}({\mathbb{T}}^{\nu+1},{\mathbb{R}}^{2})$ of
${\mathcal{L}}_{\omega}^{<}h=g$ satisfying, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\sigma$,
$\left\|({\mathcal{L}}_{\omega}^{<})^{-1}g\right\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\left\|g\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|g\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({{\mathtt{b}}})+\sigma}^{k_{0},\upsilon}\big{)}\,.$
(6.13)
In addition, if $g$ is anti-reversible, then $h$ is reversible. Moreover, for
any $s_{0}\leq s\leq S-\mu({\mathtt{b}})-\sigma$, for any traveling wave
$h\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, the operators
${\mathcal{R}}_{\omega},{\mathcal{R}}_{\omega}^{\perp}$ satisfy the estimates
$\displaystyle\left\|{\mathcal{R}}_{\omega}h\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\varepsilon\upsilon^{-3}N_{{\mathtt{n}}-1}^{-{\mathtt{a}}}\big{(}\left\|h\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({\mathtt{b}})+\sigma}^{k_{0},\upsilon}\big{)}\,,$
(6.14)
$\displaystyle\left\|{\mathcal{R}}_{\omega}^{\perp}h\right\|_{s_{0}}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}K_{\mathtt{n}}^{-{\rm
b}}\big{(}\left\|h\right\|_{s_{0}+{\rm
b}+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu({\mathtt{b}})+\sigma+{\rm
b}}^{k_{0},\upsilon}\big{)}\,,\ \forall\,{\rm b}>0\,,$ (6.15)
$\displaystyle\left\|{\mathcal{R}}_{\omega}^{\perp}h\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\left\|h\right\|_{s+\sigma}^{k_{0},\upsilon}+\left\|h\right\|_{s_{0}+\sigma}^{k_{0},\upsilon}\left\|{\mathfrak{I}}_{0}\right\|_{s+\mu({\mathtt{b}})+\sigma}^{k_{0},\upsilon}\,.$
(6.16)
This assumption shall be verified by Theorem 8.9 at each step of the Nash-
Moser iteration.
In order to find an almost approximate inverse of the linear operator in (6.9)
(and so of ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{\delta})$), it is sufficient to
invert the operator
${\mathbb{D}}\big{[}{\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}}\big{]}:=\begin{pmatrix}\omega\cdot\partial_{\varphi}{\widehat{\phi}}-\partial_{\alpha}K_{10}({\varphi})[{\widehat{\alpha}}]-K_{20}({\varphi}){\widehat{y}}-K_{11}^{\top}({\varphi}){\widehat{{\mathtt{w}}}}\\\
\omega\cdot\partial_{\varphi}{\widehat{y}}+\partial_{\alpha}\partial_{\phi}K_{00}({\varphi})[{\widehat{\alpha}}]\\\
{\mathcal{L}}_{\omega}^{<}{\widehat{{\mathtt{w}}}}-J_{\angle}\left(\partial_{\alpha}K_{01}({\varphi})[{\widehat{\alpha}}]+K_{11}({\varphi}){\widehat{y}}\right)\end{pmatrix}$
(6.17)
obtained neglecting in (6.9) the terms $\partial_{\phi}K_{10}$,
$\partial_{\phi\phi}K_{00}$, $\partial_{\phi}K_{00}$, $\partial_{\phi}K_{01}$
(they vanish at an exact solution by (6.8)) and the small remainders
${\mathcal{R}}_{\omega}$, ${\mathcal{R}}_{\omega}^{\perp}$ appearing in
(6.12).
As in section 6 of [7] we have the following result, where we denote
$\|(\phi,y,{\mathtt{w}},\alpha)\|_{s}^{k_{0},\upsilon}:=\max\big{\\{}\|(\phi,y,{\mathtt{w}})\|_{s}^{k_{0},\upsilon},\left|\alpha\right|^{k_{0},\upsilon}\big{\\}}$
(see [7, Proposition 6.5]):
###### Proposition 6.3.
Assume (6.1) (with $\mu=\mu({{\mathtt{b}}})+\sigma$) and (AI). Then, for all
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$, for any anti-reversible traveling
wave variation $g=(g_{1},g_{2},g_{3})$, there exists a unique solution
${\mathbb{D}}^{-1}g:=({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}})$
of
${\mathbb{D}}({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}},{\widehat{\alpha}})=g$
where $({\widehat{\phi}},{\widehat{y}},{\widehat{{\mathtt{w}}}})$ is a
reversible traveling wave variation. Moreover, for any $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\sigma$,
$\|{\mathbb{D}}^{-1}g\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\sigma}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({{\mathtt{b}}})+\sigma}^{k_{0},\upsilon}\|g\|_{s_{0}+\sigma}^{k_{0},\upsilon}\big{)}$.
Finally we conclude that the operator
${\bf T}_{0}:={\bf
T}_{0}(i_{0}):=(D{\widetilde{G}}_{\delta})({\varphi},0,0)\circ{\mathbb{D}}^{-1}\circ(DG_{\delta})({\varphi},0,0)^{-1}$
(6.18)
is an almost approximate right inverse for ${\rm
d}_{i,\alpha}{\mathcal{F}}(i_{0})$, where
${\widetilde{G}}_{\delta}(\phi,y,{\mathtt{w}},\alpha):=\left(G_{\delta}(\phi,y,{\mathtt{w}}),\alpha\right)$
is the identity on the $\alpha$-component. Arguing exactly as in Theorem 6.6
in [7] we deduce the following.
###### Theorem 6.4.
(Almost approximate inverse) Assume (AI). Then there is
$\overline{\sigma}:=\overline{\sigma}(\tau,\nu,k_{0})>0$ such that, if (6.1)
holds with $\mu=\mu({\mathtt{b}})+\overline{\sigma}$, then, for all
$(\omega,\gamma)\in{\mathtt{\Lambda}}_{o}$ and for any anti-reversible
traveling wave variation $g:=(g_{1},g_{2},g_{3})$, the operator ${\bf T}_{0}$
defined in (6.18) satisfies, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\overline{\sigma}$,
$\|{\bf
T}_{0}g\|_{s}^{k_{0},\upsilon}\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,.$
(6.19)
Moreover, the first three components of ${\bf T}_{0}g$ form a reversible
traveling wave variation. Finally, ${\bf T}_{0}$ is an almost approximate
right inverse of ${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0})$, namely
${\rm d}_{i,\alpha}{\mathcal{F}}(i_{0})\circ{\bf T}_{0}-{\rm
Id}={\mathcal{P}}(i_{0})+{\mathcal{P}}_{\omega}(i_{0})+{\mathcal{P}}_{\omega}^{\perp}(i_{0})\,,$
where, for any traveling wave variation $g$, for all $s_{0}\leq s\leq
S-\mu({\mathtt{b}})-\overline{\sigma}$,
$\displaystyle\|{\mathcal{P}}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\upsilon^{-1}\Big{(}\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}$
(6.20)
$\displaystyle\qquad+\,\big{(}\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathcal{F}}(i_{0},\alpha_{0})\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\Big{)}\,,$
(6.21) $\displaystyle\|{\mathcal{P}}_{\omega}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\varepsilon\upsilon^{-4}N_{{\mathtt{n}}-1}^{-{\mathtt{a}}}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,,$
(6.22)
$\displaystyle\|{\mathcal{P}}_{\omega}^{\perp}g\|_{s_{0}}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S,b}\upsilon^{-1}K_{\mathtt{n}}^{-b}\left(\|g\|_{s_{0}+\overline{\sigma}+b}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s_{0}+\mu({\mathtt{b}})+b+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\right)\,,\quad\forall\,b>0\,,$
(6.23) $\displaystyle\|{\mathcal{P}}_{\omega}^{\perp}g\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{S}\upsilon^{-1}\big{(}\|g\|_{s+\overline{\sigma}}^{k_{0},\upsilon}+\|{\mathfrak{I}}_{0}\|_{s+\mu({\mathtt{b}})+\overline{\sigma}}^{k_{0},\upsilon}\|g\|_{s_{0}+\overline{\sigma}}^{k_{0},\upsilon}\big{)}\,.$
(6.24)
## 7 The linearized operator in the normal subspace
We now write an explicit expression of the linear operator
${\mathcal{L}}_{\omega}$ defined in (6.10). As in Lemma 7.1 in [7], since the
diffeomorphism $G_{\delta}$ in (6.5) is just a translation along the infinite
dimensional normal variable $w$, we have the following structural result.
###### Lemma 7.1.
The Hamiltonian operator ${\mathcal{L}}_{\omega}$ defined in (6.10), acting on
the normal subspace $\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$, has the
form
${\mathcal{L}}_{\omega}=\Pi_{{\mathbb{S}}^{+},\Sigma}^{\angle}({\mathcal{L}}-\varepsilon
JR)|_{\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}}\,,$ (7.1)
where:
1\. ${\mathcal{L}}$ is the Hamiltonian operator
${\mathcal{L}}:=\omega\cdot\partial_{\varphi}-J\partial_{u}\nabla_{u}{\mathcal{H}}(T_{\delta}(\varphi))\,,$
(7.2)
where ${\mathcal{H}}$ is the water waves Hamiltonian in the Wahlén variables
defined in (2.9), evaluated at the reversible traveling wave
$T_{\delta}(\phi):=\varepsilon A(i_{\delta}(\phi))=\varepsilon
A\left(\theta_{0}(\phi),I_{\delta}(\phi),w_{0}(\phi)\right)=\varepsilon
v^{\intercal}\left(\theta_{0}(\phi),I_{\delta}(\phi)\right)+\varepsilon
w_{0}(\phi)\,,$ (7.3)
the torus
$i_{\delta}({\varphi}):=(\theta_{0}({\varphi}),I_{\delta}({\varphi}),w_{0}({\varphi}))$
is defined in Lemma 6.2 and $A(\theta,I,w)$, $v^{\intercal}(\theta,I)$ in
(2.2); 2\. $R(\phi)$ has the ‘finite rank" form
$R(\phi)[h]=\sum_{j=1}^{\nu}\left(h,g_{j}\right)_{L^{2}}\chi_{j}\,,\quad\forall\,h\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}\,,$
(7.4)
for functions
$g_{j},\chi_{j}\in\mathfrak{H}_{{\mathbb{S}}^{+},\Sigma}^{\angle}$ which
satisfy, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$, for all
$j=1,\ldots,\nu$, for all $s\geq s_{0}$,
$\displaystyle\left\|g_{j}\right\|_{s}^{k_{0},\upsilon}+\left\|\chi_{j}\right\|_{s}^{k_{0},\upsilon}$
$\displaystyle\lesssim_{s}1+\left\|{\mathfrak{I}}_{\delta}\right\|_{s+\sigma}^{k_{0},\upsilon}\,,$
(7.5) $\displaystyle\left\|{\rm
d}_{i}g_{j}[\widehat{\imath}]\right\|_{s}+\left\|{\rm
d}_{i}\chi_{j}[\widehat{\imath}]\right\|_{s}$
$\displaystyle\lesssim_{s}\left\|\widehat{\imath}\right\|_{s+\sigma}+\left\|\widehat{\imath}\right\|_{s_{0}+\sigma}\left\|{\mathfrak{I}}_{\delta}\right\|_{s+\sigma}\,.$
The operator ${\mathcal{L}}_{\omega}$ is reversible and momentum preserving.
In order to compute $dX$ we use the "shape derivative" formula, see e.g. [25],
$G^{\prime}(\eta)[{\widehat{\eta}}]\psi:=\lim_{\epsilon\rightarrow
0}\tfrac{1}{\epsilon}\big{(}G(\eta+\epsilon{\widehat{\eta}})\psi-G(\eta)\psi\big{)}=-G(\eta)(B{\widehat{\eta}})-\partial_{x}(V{\widehat{\eta}})\,,$
(7.6)
where
$B(\eta,\psi):=\frac{G(\eta)\psi+\eta_{x}\psi_{x}}{1+\eta_{x}^{2}}\,,\quad
V(\eta,\psi):=\psi_{x}-B(\eta,\psi)\eta_{x}\,.$ (7.7)
Then, recalling (2.9), (2.7), (1.6) and (7.6) the operator ${\mathcal{L}}$ in
(7.2) is given by
$\displaystyle{\mathcal{L}}=\omega\cdot\partial_{\varphi}$
$\displaystyle+\begin{pmatrix}\partial_{x}{\widetilde{V}}+G(\eta)B&-G(\eta)\\\
g+B{\widetilde{V}}_{x}+BG(\eta)B&{\widetilde{V}}\partial_{x}-BG(\eta)\end{pmatrix}$
(7.8)
$\displaystyle+\frac{\gamma}{2}\begin{pmatrix}-G(\eta)\partial_{x}^{-1}&0\\\
\partial_{x}^{-1}G(\eta)B-BG(\eta)\partial_{x}^{-1}-\frac{\gamma}{2}\partial_{x}^{-1}G(\eta)\partial_{x}^{-1}&-\partial_{x}^{-1}G(\eta)\end{pmatrix}\,,$
where
${\widetilde{V}}:=V-\gamma\eta\,,$ (7.9)
and the functions $B:=B(\eta,\psi)$, $V:=V(\eta,\psi)$ in (7.8)-(7.9) are
evaluated at the reversible traveling wave
$(\eta,\psi):=WT_{\delta}({\varphi})$ where $T_{\delta}({\varphi})$ is defined
in (7.3).
Notation. In (7.8) and hereafter the function $B$ is identified with the
corresponding multiplication operators $h\mapsto Bh$, and, where there is no
parenthesis, composition of operators is understood. For example $BG(\eta)B$
means $B\circ G(\eta)\circ B$.
###### Remark 7.2.
We consider the operator ${\mathcal{L}}$ in (7.8) acting on (a dense subspace
of) the whole $L^{2}({\mathbb{T}})\times L^{2}({\mathbb{T}})$. In particular
we extend the operator $\partial_{x}^{-1}$ to act on the whole
$L^{2}({\mathbb{T}})$ as in (3.22).
The following algebraic properties are a direct consequence of the reversible
and space-invariance properties of the water waves equations explained in
Section 2 and the fact that the approximate solution
$(\eta,\zeta)=T_{\delta}({\varphi})$ is a reversible traveling wave (cfr.
Lemma 7.3 in [7]).
###### Lemma 7.3.
The functions $(\eta,\zeta)=T_{\delta}({\varphi})$ and $B,{\widetilde{V}}$
defined in (7.7), (7.9) are quasi-periodic traveling waves. The functions
$(\eta,\zeta)=T_{\delta}({\varphi})$ are $({\rm even}({\varphi},x),{\rm
odd}({\varphi},x))$, $B$ is ${\rm odd}({\varphi},x)$ and ${\widetilde{V}}$ is
${\rm even}({\varphi},x)$. The Hamiltonian operator ${\mathcal{L}}$ is
reversible and momentum preserving.
For the sequel we will always assume the following ansatz (satisfied by the
approximate solutions obtained along the nonlinear Nash-Moser iteration of
Section 9): for some constants $\mu_{0}:=\mu_{0}(\tau,\nu)>0$,
$\upsilon\in(0,1)$, (cfr. Lemma 6.2)
$\left\|{\mathfrak{I}}_{0}\right\|_{s_{0}+\mu_{0}}^{k_{0},\upsilon}\,,\
\left\|{\mathfrak{I}}_{\delta}\right\|_{s_{0}+\mu_{0}}^{k_{0},\upsilon}\leq
1\,.$ (7.10)
In order to estimate the variation of the eigenvalues with respect to the
approximate invariant torus, we need also to estimate the variation with
respect to the torus $i({\varphi})$ in another low norm $\left\|\
\right\|_{s_{1}}$ for all Sobolev indexes $s_{1}$ such that
$s_{1}+\sigma_{0}\leq s_{0}+\mu_{0}\,,\quad\text{ for some }\
\sigma_{0}:=\sigma_{0}(\tau,\nu)>0\,.$ (7.11)
Thus, by (7.10), we have
$\left\|{\mathfrak{I}}_{0}\right\|_{s_{1}+\sigma_{0}}^{k_{0},\upsilon}$,
$\left\|{\mathfrak{I}}_{\delta}\right\|_{s_{1}+\sigma_{0}}^{k_{0},\upsilon}\leq
1$. The constants $\mu_{0}$ and $\sigma_{0}$ represent the _loss of
derivatives_ accumulated along the reduction procedure of the next sections.
What is important is that they are independent of the Sobolev index $s$. In
the following sections we shall denote by $\sigma:=\sigma(\tau,\nu,k_{0})>0$,
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},\tau,\nu,k_{0})$,
$\sigma_{M}:=\sigma_{M}(k_{0},\tau,\nu)>0$, $\aleph_{M}(\alpha)$ constants
(which possibly increase from lemma to lemma) representing losses of
derivatives along the finitely many steps of the reduction procedure.
###### Remark 7.4.
In the next sections $\mu_{0}:=\mu_{0}(\tau,\nu,M,\alpha)>0$ will depend also
on indexes $M,\alpha$, whose maximal values will be fixed depending only on
$\tau$ and $\nu$ (and $k_{0}$ which is however considered an absolute constant
along the paper). In particular $M$ is fixed in (8.5), whereas the maximal
value of $\alpha$ depends on $M$, as explained in Remark 7.16.
###### Remark 7.5.
Starting from Section 7.2, we introduce in the estimates upper bounds on the
regularity $s\geq s_{0}$. We shall control the terms in Sobolev spaces $H^{s}$
with $s_{0}\leq s\leq S-\sigma$, where $\sigma$ denotes a loss of derivatives
of the finitely many steps of the reduction (possibly increasing along the
steps), whereas $S>s_{0}+k_{0}$ is any finite Sobolev index. The index $S$ has
to be taken finite in view of Lemma 7.7 (see also Appendix A). The largest
regularity index $S$ will be fixed in (9.3). In particular, it is compatible
with the condition (7.11), namely $s_{1}+\sigma_{0}\leq s_{0}+\mu_{0}<S$.
As a consequence of Moser composition Lemma 3.2 and (6.3), the Sobolev norm of
the function $u=T_{\delta}({\varphi})$ defined in (7.3) satisfies for all
$s\geq s_{0}$
$\left\|u\right\|_{s}^{k_{0},\upsilon}=\left\|\eta\right\|_{s}^{k_{0},\upsilon}+\left\|\zeta\right\|_{s}^{k_{0},\upsilon}\leq\varepsilon
C(s)\big{(}1+\left\|{\mathfrak{I}}_{0}\right\|_{s}^{k_{0},\upsilon}\big{)}$
(7.12)
(the map $A$ defined in (2.2) is smooth). Similarly, using (6.4),
$\left\|\Delta_{12}u\right\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{2}-i_{1}\right\|_{s_{1}}\,,\quad\text{
where }\quad\Delta_{12}u:=u(i_{2})-u(i_{1})\,.$
We finally recall that ${\mathfrak{I}}_{0}={\mathfrak{I}}_{0}(\omega,\gamma)$
is defined for all
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and that
the functions $B,{\widetilde{V}}$ and $c$ appearing in ${\mathcal{L}}$ in
(7.8) are ${\mathcal{C}}^{\infty}$ in $({\varphi},x)$, as
$u=(\eta,\zeta)=T_{\delta}({\varphi})$ is.
In Sections 7.1-7.6 we are going to make several transformations, whose aim is
to conjugate the operator ${\cal L}$ in (7.8) to a constant coefficients
Fourier multiplier, up to a pseudo-differential operator of order $-1/2$ plus
a remainder that satisfies tame estimates, see ${\cal L}_{8}$ in (7.138).
Finally, in Section 7.7 we shall conjugate the restricted operator ${\cal
L}_{\omega}$ in (7.1).
### 7.1 Linearized good unknown of Alinhac
The first step is to conjugate the linear operator ${\mathcal{L}}$ in (7.8) by
the symplectic (Definition 3.18) multiplication matrix operator
${\mathcal{Z}}:=\left(\begin{matrix}{\rm Id}&0\\\ B&{\rm
Id}\end{matrix}\right)\ ,\qquad{\mathcal{Z}}^{-1}=\left(\begin{matrix}{\rm
Id}&0\\\ -B&{\rm Id}\end{matrix}\right)\,,$
obtaining
$\displaystyle{\mathcal{L}}_{1}$
$\displaystyle:={\mathcal{Z}}^{-1}{\mathcal{L}}{\mathcal{Z}}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&-G(\eta)\\\
a&{\widetilde{V}}\partial_{x}\end{pmatrix}-\frac{\gamma}{2}\begin{pmatrix}G(\eta)\partial_{x}^{-1}&0\\\
\frac{\gamma}{2}\partial_{x}^{-1}G(\eta)\partial_{x}^{-1}&\partial_{x}^{-1}G(\eta)\end{pmatrix}\,,$
(7.13)
where $a$ is the function
$a:=g+{\widetilde{V}}B_{x}+\omega\cdot\partial_{\varphi}B\,.$ (7.14)
The matrix ${\mathcal{Z}}$ amounts to introduce, as in [25] and [9, 2], a
linearized version of the “good unknown of Alinhac".
###### Lemma 7.6.
The maps ${\mathcal{Z}}^{\pm 1}-{\rm Id}$ are ${\mathcal{D}}^{k_{0}}$-tame
with tame constants satisfying, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$,
for all $s\geq s_{0}$,
${\mathfrak{M}}_{{\mathcal{Z}}^{\pm 1}-{\rm Id}}(s)\,,\
{\mathfrak{M}}_{({\mathcal{Z}}^{\pm 1}-{\rm
Id})^{*}}(s)\lesssim_{s}\varepsilon\big{(}1+\left\|{\mathfrak{I}}_{0}\right\|_{s+\sigma}^{k_{0},\upsilon}\big{)}\,.$
The function $a$ in (7.14) is a quasi-periodic traveling wave ${\rm
even}({\varphi},x)$. There is $\sigma:=\sigma(\tau,\nu,k_{0})>0$ such that,
for all $s\geq s_{0}$,
$\left\|a-g\right\|_{s}^{k_{0},\upsilon}+\|{\widetilde{V}}\|_{s}^{k_{0},\upsilon}+\left\|B\right\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\big{)}\,.$
(7.15)
Moreover, for any $s_{1}$ as in (7.11),
$\displaystyle\left\|\Delta_{12}a\right\|_{s_{1}}+\|\Delta_{12}{\widetilde{V}}\|_{s_{1}}+\left\|\Delta_{12}B\right\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.16) $\displaystyle\|\Delta_{12}({\mathcal{Z}}^{\pm
1})h\|_{s_{1}},\|\Delta_{12}({\mathcal{Z}}^{\pm
1})^{*}h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\left\|h\right\|_{s_{1}}\,.$
(7.17)
The operator ${\mathcal{L}}_{1}$ is Hamiltonian, reversible and momentum
preserving.
###### Proof.
By the expressions of $B,\widetilde{V},a$ in (7.7), (7.9), (7.14) the
composition estimates of Lemma 3.2, (3.7) and the bounds for the Dirichlet-
Neumann operator in Lemma 3.10. Since $B$ is an ${\rm odd}(\varphi,x)$ quasi-
periodic traveling wave, the matrix operator ${\mathcal{Z}}$ is reversibility
and momentum preserving (Definitions 3.19 and 3.22). ∎
### 7.2 Almost-straightening of the first order transport operator
We now write the operator ${\mathcal{L}}_{1}$ in (7.13) as
${\mathcal{L}}_{1}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&0\\\
0&{\widetilde{V}}\partial_{x}\end{pmatrix}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-G(0)\\\
a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}+{\bf
R}_{1}\,,$ (7.18)
where, using the decomposition (3.32) of the Dirichlet-Neumann operator,
${\bf
R}_{1}:=-\begin{pmatrix}\frac{\gamma}{2}{\mathcal{R}}_{G}(\eta)\partial_{x}^{-1}&{\mathcal{R}}_{G}(\eta)\\\
\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}{\mathcal{R}}_{G}(\eta)\partial_{x}^{-1}&\frac{\gamma}{2}\partial_{x}^{-1}{\mathcal{R}}_{G}(\eta)\end{pmatrix}$
(7.19)
is a small remainder in ${\rm OP}S^{-\infty}$. The aim of this section is to
conjugate the variable coefficients quasi-periodic transport operator
${\mathcal{L}}_{\rm
TR}:=\omega\cdot\partial_{\varphi}+\begin{pmatrix}\partial_{x}{\widetilde{V}}&0\\\
0&{\widetilde{V}}\partial_{x}\end{pmatrix}$ (7.20)
to a constant coefficients transport operator
$\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\,\partial_{y}$,
up to an exponentially small remainder, see (7.28)-(7.29), where
${\mathtt{n}}\in{\mathbb{N}}_{0}$ and the scale
$(N_{{\mathtt{n}}})_{{\mathtt{n}}\in{\mathbb{N}}_{0}}$ is defined, for
$N_{0}>1$, by
$N_{{\mathtt{n}}}:=N_{0}^{\chi^{{\mathtt{n}}}}\,,\quad\chi=3/2\,,\quad
N_{-1}:=1\,.$ (7.21)
Such small remainder is left because we assume only finitely many non-
resonance conditions, see (7.26). This enables to deduce Lemma 7.9, and then
to formulate the non-resonance condition (5.15), stated in terms of the
“final" function ${\mathtt{m}}_{1}^{\infty}(\omega,\gamma)$, which implies
(7.26) at any step of the nonlinear Nash-Moser iteration of Section 9.
In the next lemma we conjugate ${\mathcal{L}}_{\rm TR}$ by a _symplectic_
(Definition 3.18) transformation
${\mathcal{E}}:=\begin{pmatrix}(1+\beta_{x}({\varphi},x))\circ{\mathcal{B}}&0\\\
0&{\mathcal{B}}\end{pmatrix}\,,\quad{\mathcal{E}}^{-1}:=\begin{pmatrix}{\mathcal{B}}^{-1}\circ(1+\beta_{x}({\varphi},x))^{-1}&0\\\
0&{\mathcal{B}}^{-1}\end{pmatrix}$ (7.22)
where the composition operator
$({\mathcal{B}}u)({\varphi},x):=u\left({\varphi},x+\beta({\varphi},x)\right)$
(7.23)
is induced by a ${\varphi}$-dependent diffeomorphism $y=x+\beta({\varphi},x)$
of the torus ${\mathbb{T}}_{x}$, for some small quasi-periodic traveling wave
$\beta:{\mathbb{T}}_{\varphi}^{\nu}\times{\mathbb{T}}_{x}\to{\mathbb{R}}$,
${\rm odd}({\varphi},x)$. Let
${\mathtt{b}}:=[{\mathtt{a}}]+2\in{\mathbb{N}}\,,\quad{\mathtt{a}}:=3(\tau_{1}+1)\geq
1\,,\quad\tau_{1}:=k_{0}+(k_{0}+1)\tau\,.$ (7.24)
###### Lemma 7.7.
(Almost-Straightening of the transport operator) There exists
$\tau_{2}(\tau,\nu)>\tau_{1}(\tau,\nu)+1+{\mathtt{a}}$ such that, for all
$S>s_{0}+k_{0}$, there are $N_{0}:=N_{0}(S,{\mathtt{b}})\in{\mathbb{N}}$ and
$\updelta:=\updelta(S,{\mathtt{b}})\in(0,1)$ such that, if
$N_{0}^{\tau_{2}}\varepsilon\upsilon^{-1}<\updelta$ the following holds true.
For any $\overline{\mathtt{n}}\in{\mathbb{N}}_{0}$:
1\. There exist a constant
${\mathtt{m}}_{1,\overline{\mathtt{n}}}:={\mathtt{m}}_{1,\overline{\mathtt{n}}}(\omega,\gamma)\in{\mathbb{R}}$,
where ${\mathtt{m}}_{1,0}=0$, defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$, and a
quasi-periodic traveling wave
$\beta({\varphi},x):=\beta_{\overline{\mathtt{n}}}({\varphi},x)$, ${\rm
odd}({\varphi},x)$, satisfying, for some $\sigma=\sigma(\tau,\nu,k_{0})>0$,
the estimates
$|{\mathtt{m}}_{1,\overline{\mathtt{n}}}|^{k_{0},\upsilon}\lesssim\varepsilon\,,\quad\|\beta\|_{s}^{k_{0},\upsilon}\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,,\quad\forall\,s_{0}\leq
s\leq S\,,$ (7.25)
independently of $\overline{\mathtt{n}}$;
2\. For any $(\omega,\gamma)$ in
$\displaystyle{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
$\displaystyle:={\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}({\mathtt{m}}_{1,\overline{\mathtt{n}}},2\upsilon,\tau)$
(7.26)
$\displaystyle:=\Big{\\{}(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]\,:\,|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\vec{\jmath})\cdot\ell|\geq
2\upsilon\braket{\ell}^{-\tau}\,\ \forall\,0<|\ell|\leq
N_{\overline{\mathtt{n}}}\Big{\\}}$
the operator ${\mathcal{L}}_{\rm TR}$ in (7.20) is conjugated to
${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm
TR}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\,\partial_{y}+{\bf
P}_{2}^{\perp}\,,$ (7.27)
where
${\bf
P}_{2}^{\perp}:=\begin{pmatrix}\partial_{y}p_{\overline{\mathtt{n}}}&0\\\
0&p_{\overline{\mathtt{n}}}\partial_{y}\end{pmatrix}\,,$ (7.28)
and the real, quasi-periodic traveling wave function
$p_{\overline{\mathtt{n}}}({\varphi},y)$, ${\rm even}({\varphi},y)$,
satisfies, for some $\sigma=\sigma(\tau,\nu,k_{0})>0$ and for any $s_{0}\leq
s\leq S$,
$\|p_{\overline{\mathtt{n}}}\|_{s}^{k_{0},\upsilon}\lesssim_{s,{\mathtt{b}}}\varepsilon\,N_{\overline{\mathtt{n}}-1}^{-{\mathtt{a}}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,;$
(7.29)
3\. The operators ${\mathcal{E}}^{\pm}$ are
${\mathcal{D}}^{k_{0}}$-$(k_{0}+1)$-tame, the operators ${\mathcal{E}}^{\pm
1}-{\rm Id}$, $({\mathcal{E}}^{\pm 1}-{\rm Id})^{*}$ are
${\mathcal{D}}^{k_{0}}$-$(k_{0}+2)$-tame with tame constants satisfying, for
some $\sigma:=\sigma(\tau,\nu,k_{0})>0$ and for all $s_{0}\leq s\leq
S-\sigma$,
$\displaystyle{\mathfrak{M}}_{{\mathcal{E}}^{\pm
1}}(s)\lesssim_{S}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\,,\
{\mathfrak{M}}_{{\mathcal{E}}^{\pm 1}-{\rm
Id}}(s)+{\mathfrak{M}}_{\left({\mathcal{E}}^{\pm 1}-{\rm
Id}\right)^{*}}(s)\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma+{\mathtt{b}}}^{k_{0},\upsilon})\,.$
(7.30)
4\. Furthermore, for any $s_{1}$ as in (7.11),
$\displaystyle|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}|\lesssim\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\quad\|\Delta_{12}\beta\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\upsilon^{-1}\|i_{1}-i_{2}\|_{s_{1}+\sigma+{\mathtt{b}}}\,,$
(7.31)
$\displaystyle\|\Delta_{12}({\mathcal{A}})h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma+{\mathtt{b}}}\left\|h\right\|_{s_{1}+\sigma+{\mathtt{b}}}\,,\quad{\mathcal{A}}\in\\{{\mathcal{E}}^{\pm
1},({\mathcal{E}}^{\pm 1})^{*}\\}\,.$ (7.32)
###### Proof.
We apply Theorem A.2 and Corollary A.4 to the transport operator
$X_{0}=\omega\cdot\partial_{\varphi}+\widetilde{V}\partial_{x}$, which has the
form (A.1) with $p_{0}=\widetilde{V}$. By (7.15) and (7.10), the smallness
conditions (A.3) and (A.10) hold for
$N_{0}^{\tau_{2}}\varepsilon\upsilon^{-1}$ sufficiently small. Therefore there
exist a constant ${\mathtt{m}}_{1,\overline{\mathtt{n}}}\in{\mathbb{R}}$ and a
quasi-periodic traveling wave
$\beta({\varphi},x):=\beta_{\overline{\mathtt{n}}}({\varphi},x)$, ${\rm
odd}({\varphi},x)$, such that, for any $(\omega,\gamma)$ in
${\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)\subseteq{\mathtt{\Lambda}}_{\overline{\mathtt{n}}+1}^{\upsilon,\rm
T}\subseteq{\mathtt{\Lambda}}_{\overline{\mathtt{n}}}^{\upsilon,\rm T}$ (see
Corollary A.3) we have
${\mathcal{B}}_{\overline{\mathtt{n}}}^{-1}(\omega\cdot\partial_{\varphi}+\widetilde{V}\partial_{x}){\mathcal{B}}_{\overline{\mathtt{n}}}=\omega\cdot\partial_{\varphi}+({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}}({\varphi},y))\partial_{y}$
where the function $p_{\overline{\mathtt{n}}}$ satisfies (7.29) by (A.5) and
(7.15). The estimates (A.6), (A.15), (7.15) imply (7.25) and (7.30). The
conjugated operator of ${\mathcal{L}}_{\rm TR}$ in (7.20) is
${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm
TR}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}A_{1}&0\\\
0&({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}})\partial_{y}\end{pmatrix}$
where
$\omega\cdot\partial_{\varphi}+A_{1}={\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\big{(}\omega\cdot\partial_{\varphi}+\partial_{x}{\widetilde{V}}\big{)}(1+\beta_{x}){\mathcal{B}}$.
Since ${\mathcal{L}}_{\rm TR}$ is Hamiltonian (Definition 3.18), and the map
${\mathcal{E}}$ is symplectic, we have that
${\mathcal{E}}^{-1}{\mathcal{L}}_{\rm TR}{\mathcal{E}}$ is Hamiltonian as
well. In particular
$A_{1}=-(({\mathtt{m}}_{1,\overline{\mathtt{n}}}+p_{\overline{\mathtt{n}}})\partial_{y})^{*}={\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{y}+\partial_{y}p_{\overline{\mathtt{n}}}$.
This proves (7.27)-(7.28). The estimates (7.31)-(7.32) follow by
(A.11)-(A.12), the bound for
$\|\Delta_{12}\beta_{\overline{\mathtt{n}}}\|_{s_{1}}$ in Corollary A.4 and
(7.16)-(7.17). ∎
###### Remark 7.8.
Actually, for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
in (7.26), Theorem A.2 and Corollary A.3 would imply also the conjugation of
${\mathcal{L}}_{\rm TR}$ to the operator
$\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}+1}\partial_{y}$
for some ${\mathtt{m}}_{1,\overline{\mathtt{n}}+1}\in{\mathbb{R}}$, up to a
remainder ${\bf P}_{2}^{\perp}=O(\varepsilon
N_{\overline{\mathtt{n}}}^{-{\mathtt{a}}})$. For simplicity we stated only the
conjugation in (7.27). We shall use the non-resonance condition in (7.26) also
later in Sections 7.5, 7.6 .
The next lemma is needed in order to prove the inclusion of the Cantor sets
associated to two nearby approximate solutions.
###### Lemma 7.9.
Let $i_{1},i_{2}$ be close enough and $0<2\upsilon-\rho<2\upsilon<1$. Then
$\varepsilon
C(s_{1})N_{\overline{\mathtt{n}}}^{\tau+1}\|i_{1}-i_{2}\|_{s_{1}+\sigma}\leq\rho\quad\Rightarrow\quad{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)(i_{1})\subseteq{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon-\rho,\tau)(i_{2})\,.$
###### Proof.
For any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)(i_{1})$,
using also (7.31), we have, for any
$\ell\in{\mathbb{Z}}^{\nu}\setminus\\{0\\}$, $|\ell|\leq
N_{\overline{\mathtt{n}}}$,
$\displaystyle|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}(i_{2})\vec{\jmath})\cdot\ell|$
$\displaystyle\geq|(\omega-{\mathtt{m}}_{1,\overline{\mathtt{n}}}(i_{1})\vec{\jmath})\cdot\ell|-C|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}||\ell|$
$\displaystyle\geq\frac{2\upsilon}{\braket{\ell}^{\tau}}-C(s_{1})\varepsilon
N_{\overline{\mathtt{n}}}\|i_{1}-i_{2}\|_{s_{1}+\sigma}\geq\frac{2\upsilon-\rho}{\braket{\ell}^{\tau}}\,.$
We conclude that
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon-\rho,\tau)(i_{2})$.
∎
We now conjugate the whole operator ${\mathcal{L}}_{1}$ in (7.18)-(7.19) by
the operator ${\mathcal{E}}$ in (7.22).
We first compute the conjugation of the matrix
$\displaystyle{\mathcal{E}}^{-1}$
$\displaystyle\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-G(0)\\\
a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}{\mathcal{E}}$
$\displaystyle=\begin{pmatrix}-\frac{\gamma}{2}{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}&-{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0){\mathcal{B}}\\\
{\mathcal{B}}^{-1}\big{(}a-\left(\frac{\gamma}{2}\right)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}(1+\beta_{x}){\mathcal{B}}&-\frac{\gamma}{2}{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}}\end{pmatrix}\,.$
The multiplication operator for $a({\varphi},x)$ is transformed into the
multiplication operator for the function
${\mathcal{B}}^{-1}a(1+\beta_{x}){\mathcal{B}}={\mathcal{B}}^{-1}\big{(}a(1+\beta_{x})\big{)}\,.$
(7.33)
We write the Dirichlet-Neumann operator $G(0)$ in (1.9) as
$G(0)=G(0,{\mathtt{h}})=\partial_{x}{\mathcal{H}}T({\mathtt{h}})\,,$ (7.34)
where ${\mathcal{H}}$ is the Hilbert transform defined in (3.21) and
$T({\mathtt{h}}):=\begin{cases}\tanh({\mathtt{h}}|D|)={\rm Id}+{\rm
Op}(r_{\mathtt{h}})&\text{ if }{\mathtt{h}}<+\infty\,,\qquad
r_{{\mathtt{h}}}(\xi):=-\frac{2}{1+e^{2{\mathtt{h}}|\xi|\chi(\xi)}}\in
S^{-\infty}\,,\\\ {\rm Id}&\text{ if }{\mathtt{h}}=\infty\,.\end{cases}$
(7.35)
We have the conjugation formula (see formula (7.42) in [2])
${\mathcal{B}}^{-1}G(0){\mathcal{B}}=\left\\{{\mathcal{B}}^{-1}(1+\beta_{x})\right\\}G(0)+{\mathcal{R}}_{1}\,,$
(7.36)
where
${\mathcal{R}}_{1}:=\left\\{{\mathcal{B}}^{-1}(1+\beta_{x})\right\\}\partial_{y}\big{(}\l{\mathcal{H}}\left({\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})\right)+\left({\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}\right)({\mathcal{B}}^{-1}T({\mathtt{h}}){\mathcal{B}})\big{)}\,.$
The operator ${\mathcal{R}}_{1}$ is in ${\rm OP}S^{-\infty}$ because both
${\mathcal{B}}^{-1}{\rm Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})$ and
${\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}$ are in ${\rm
OP}S^{-\infty}$ and there is ${\sigma}>0$ such that, for any
$m\in{\mathbb{N}}$, $\alpha\in{\mathbb{N}}_{0}$ and $s\geq s_{0}$,
$\displaystyle\|{\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}}\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|\beta\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,,$
(7.37) $\displaystyle\|{\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm
Op}(r_{\mathtt{h}})\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|\beta\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,.$
The first estimate is given in Lemmata 2.36 and 2.32 in [9], whereas the
second one follows by that fact that $r_{\mathtt{h}}\in S^{-\infty}$ (see
(7.35)), Lemma 2.18 in [2] and Lemmata 2.34 and 2.32 in [9]. Therefore by
(7.36) we obtain
${\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0){\mathcal{B}}=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}{\mathcal{B}}^{-1}G(0){\mathcal{B}}=G(0)+{\mathcal{R}}_{B}\,,$
(7.38)
where
${\mathcal{R}}_{B}:=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}\,{\mathcal{R}}_{1}\,.$
(7.39)
Next we transform $G(0)\partial_{x}^{-1}$. By (7.34) and using the identities
${\mathcal{H}}\partial_{x}\partial_{x}^{-1}={\mathcal{H}}$ and
${\mathcal{H}}T({\mathtt{h}})=\partial_{y}^{-1}G(0)$ on the periodic
functions, we have that
$\displaystyle{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}=G(0)\partial_{y}^{-1}+{\mathcal{R}}_{A}$
(7.40)
$\displaystyle{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}}=\partial_{y}^{-1}G(0)+{\mathcal{R}}_{D}\,,$
where
$\displaystyle{\mathcal{R}}_{D}$
$\displaystyle=({\mathcal{B}}^{-1}{\mathcal{H}}{\mathcal{B}}-{\mathcal{H}})({\mathcal{B}}^{-1}T({\mathtt{h}}){\mathcal{B}})+{\mathcal{H}}\big{(}{\mathcal{B}}^{-1}{\rm
Op}(r_{\mathtt{h}}){\mathcal{B}}-{\rm Op}(r_{\mathtt{h}})\big{)}\,,$ (7.41)
$\displaystyle{\mathcal{R}}_{A}$
$\displaystyle=\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}\big{[}{\mathcal{H}}T({\mathtt{h}}),\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}-1\big{]}$
$\displaystyle\ \
+\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\}{\mathcal{R}}_{D}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}\,.$
The operator ${\mathcal{R}}_{D}$ is in ${\rm OP}S^{-\infty}$ by (7.37),
(7.35). Also ${\mathcal{R}}_{A}$ is in ${\rm OP}S^{-\infty}$ using that, by
Lemma 2.35 of [9] and (7.35), there is ${\sigma}>0$ such that, for any
$m\in{\mathbb{N}}$, $s\geq s_{0}$, and $\alpha\in{\mathbb{N}}_{0}$,
$\|[{\mathcal{H}}T({\mathtt{h}}),{\widetilde{a}}]\|_{-m,s,\alpha}^{k_{0},\upsilon}\lesssim_{m,s,\alpha,k_{0}}\|{\widetilde{a}}\|_{s+m+\alpha+\sigma}^{k_{0},\upsilon}\,.$
(7.42)
Finally we conjugate $\partial_{x}^{-1}G(0)\partial_{x}^{-1}$. By the Egorov
Proposition 3.9 applied to $\partial_{x}^{-1}$, we have that, for any
$N\in{\mathbb{N}}$,
${\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}={\mathcal{B}}^{-1}\partial_{x}^{-1}{\mathcal{B}}\,\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}=\partial_{y}^{-1}+P^{(1)}_{-2,N}(\varphi,x,D)+{\mathtt{R}}_{N}\,,$
(7.43)
where $P^{(1)}_{-2,N}(\varphi,x,D)\in{\rm OP}S^{-2}$ is given by
$P^{(1)}_{-2,N}(\varphi,x,D):=\big{[}\\{{\mathcal{B}}^{-1}(1+\beta_{x})^{-1}\\},\partial_{y}^{-1}\big{]}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}+\sum_{j=1}^{N}p_{-1-j}\partial_{y}^{-1-j}\\{{\mathcal{B}}^{-1}(1+\beta_{x})\\}$
with functions $p_{-1-j}(\lambda;\varphi,y)$, $j=0,\ldots,N$, satisfying
(3.30) and ${\mathtt{R}}_{N}$ is a regularizing operator satisfying the
estimate (3.31). So, using (7.40) and (7.43), we obtain
$\displaystyle{\mathcal{B}}^{-1}\partial_{x}^{-1}G(0)\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}}$
$\displaystyle=({\mathcal{B}}^{-1}\partial_{x}^{-1}G(0){\mathcal{B}})({\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}})$
(7.44)
$\displaystyle=\partial_{y}^{-1}G(0)\partial_{y}^{-1}+P_{-2,N}^{(2)}+{\mathtt{R}}_{2,N}$
where
$\displaystyle P_{-2,N}^{(2)}$
$\displaystyle:=\partial_{y}^{-1}G(0)P^{(1)}_{-2,N}(\varphi,x,D)\in{\rm
OP}S^{-2}$ (7.45)
and ${\mathtt{R}}_{2,N}$ is the regularizing operator
${\mathtt{R}}_{2,N}:={\mathcal{R}}_{D}({\mathcal{B}}^{-1}\partial_{x}^{-1}(1+\beta_{x}){\mathcal{B}})+G(0)\partial_{y}^{-1}{\mathtt{R}}_{N}\,.$
(7.46)
In conclusion, by Lemma 7.7, (7.33), (7.38), (7.40) and (7.44) we obtain the
following lemma, which summarizes the main result of this section.
###### Lemma 7.10.
Let $N\in{\mathbb{N}}$. For any $\overline{\mathtt{n}}\in{\mathbb{N}}_{0}$ and
for all
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$,
the operator ${\mathcal{L}}_{1}$ in (7.18) is conjugated to the real,
Hamiltonian, reversible and momentum preserving operator
$\displaystyle{\mathcal{L}}_{2}:={\mathcal{E}}^{-1}{\mathcal{L}}_{1}{\mathcal{E}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{y}\,+$
$\displaystyle\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{y}^{-1}&-G(0)\\\
a_{1}-\left(\frac{\gamma}{2}\right)^{2}\partial_{y}^{-1}G(0)\partial_{y}^{-1}&-\frac{\gamma}{2}\partial_{y}^{-1}G(0)\end{pmatrix}$
(7.47) $\displaystyle+\begin{pmatrix}0&0\\\
-\left(\frac{\gamma}{2}\right)^{2}P_{-2,N}^{(2)}&0\end{pmatrix}+{\bf
R}_{2}^{\Psi}+{\bf T}_{2,N}+{\bf P}_{2}^{\perp}\,,$
defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$, where:
1\. The constant
${\mathtt{m}}_{1,\overline{\mathtt{n}}}={\mathtt{m}}_{1,\overline{\mathtt{n}}}(\omega,\gamma)\in{\mathbb{R}}$
satisfies
$|{\mathtt{m}}_{1,\overline{\mathtt{n}}}|^{k_{0},\upsilon}\lesssim\varepsilon$,
independently on $\overline{\mathtt{n}}$;
2\. The real quasi-periodic traveling wave
$a_{1}:={\mathcal{B}}^{-1}\big{(}a(1+\beta_{x})\big{)}$, ${\rm
even}({\varphi},x)$, satisfies, for some $\sigma:=\sigma(k_{0},\tau,\nu)>0$
and for all $s_{0}\leq s\leq S-\sigma$,
$\|a_{1}-g\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.48)
3\. The operator $P_{-2,N}^{(2)}$ is a pseudodifferential operator in ${\rm
OP}S^{-2}$, reversibility and momentum preserving, and satisfies, for some
$\sigma_{N}:=\sigma_{N}(\tau,\nu,N)>0$, for finitely many
$0\leq\alpha\leq\alpha(M)$ (fixed in Remark 7.16) and for all $s_{0}\leq s\leq
S-\sigma_{N}-\alpha$,
$\|P_{-2,N}^{(2)}\|_{-2,s,\alpha}^{k_{0},\upsilon}\lesssim_{s,N,\alpha}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\alpha}^{k_{0},\upsilon})\,;$
(7.49)
4\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+2$, the operator $\langle
D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf T}_{2,N}({\varphi}))\langle D\rangle^{n_{2}}$ is
${\mathcal{D}}^{k_{0}}$-tame with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},k_{0},\tau,\nu)>0$
and for any $s_{0}\leq s\leq S-\sigma_{N}({\mathtt{q}}_{0})$,
${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf T}_{2,N}({\varphi}))\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})}^{k_{0},\upsilon}\big{)}\,;$
(7.50)
5\. The operator ${\bf P}_{2}^{\perp}$ is defined in (7.28) and the function
$p_{\overline{\mathtt{n}}}$ satisfies (7.29);
6\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha(M)$, ${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq N-{\mathtt{q}}_{0}+1$,
$\displaystyle|\Delta_{12}{\mathtt{m}}_{1,\overline{\mathtt{n}}}|\lesssim_{s_{1}}\varepsilon\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\
\|\Delta_{12}a_{1}\|_{s_{1}}\lesssim\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.51)
$\displaystyle\|\Delta_{12}P_{-2,N}^{(2)}\|_{-2,s_{1},\alpha}\lesssim_{s_{1},N,\alpha}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\alpha}\,,$
(7.52)
$\displaystyle\left\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}({\bf
R}_{2}^{\Psi}(\varphi)+{\bf
T}_{2,N}({\varphi}))\braket{D}^{n_{2}}\right\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})}\,.$
(7.53)
###### Proof.
Item 1 follows by Lemma 7.7. The function $a_{1}$ satisfies (7.48) by (7.14),
(3.7), (7.15), (7.30), (7.25). The estimate (7.49) follows by (7.45),
Proposition 3.9 and Lemmata 3.5, 3.6, 3.8, 7.7. The operators ${\bf
R}_{2}^{\Psi}$ and ${\bf T}_{2,N}$ in (7.47) are
${\bf
R}_{2}^{\Psi}:=-\begin{pmatrix}\frac{\gamma}{2}{\mathcal{R}}_{A}&{\mathcal{R}}_{B}\\\
0&\frac{\gamma}{2}{\mathcal{R}}_{D}\end{pmatrix}+{\mathcal{E}}^{-1}{\bf
R}_{1}{\mathcal{E}}\,,\qquad{\bf
T}_{2,N}:=-\left(\frac{\gamma}{2}\right)^{2}\begin{pmatrix}0&0\\\
{\mathtt{R}}_{2,N}&0\end{pmatrix}\,,$
where ${\mathcal{R}}_{B}$, ${\mathcal{R}}_{A}$, ${\mathcal{R}}_{D}$, are
defined in (7.39), (7.41), and ${\bf R}_{1}$, ${\mathtt{R}}_{2,N}$ in (7.19),
(7.46). Thus the estimate (7.50) holds by Lemmata 3.12, 3.13, 7.7, (7.37),
(7.42), Proposition 3.9, Lemma 3.10, (7.25) and Lemmata 2.34, 2.32 in [9]. The
estimates (7.51)-(7.53) are proved similarly. ∎
### 7.3 Symmetrization of the order $1/2$
The goal of this section is to symmetrize the order $1/2$ of the quasi-
periodic Hamiltonian operator ${\mathcal{L}}_{2}$ in (7.47). From now on, we
neglect the contribution of the operator ${\bf P}_{2}^{\perp}$, which will be
conjugated in Section 7.7. For simplicity of notation we denote such operator
${\mathcal{L}}_{2}$ as well.
Step 1: We first conjugate the operator ${\mathcal{L}}_{2}$ in (7.47), where
we relabel the space variable $y\rightsquigarrow x$, by the real, symplectic,
reversibility preserving and momentum preserving transformation
${\widetilde{{\mathcal{M}}}}:=\begin{pmatrix}\Lambda&0\\\
0&\Lambda^{-1}\end{pmatrix}\,,\quad{\widetilde{{\mathcal{M}}}}^{-1}:=\begin{pmatrix}\Lambda^{-1}&0\\\
0&\Lambda\end{pmatrix}\,,$ (7.54)
where $\Lambda\in{\rm OP}S^{\frac{1}{4}}$ is the Fourier multiplier
$\Lambda:=\tfrac{1}{\sqrt{g}}\pi_{0}+M(D)\,,\quad\text{with
inverse}\quad\Lambda^{-1}:=\sqrt{g}\pi_{0}+M(D)^{-1}\in{\rm
OP}S^{-\frac{1}{4}}\,,$ (7.55)
with $\pi_{0}$ defined in (3.23) and (cfr. (2.16))
$M(D):=G(0)^{\frac{1}{4}}\big{(}g-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}^{-\frac{1}{4}}\in{\rm
OP}S^{\frac{1}{4}}\,.$ (7.56)
We have the identities $\Lambda^{-1}G(0)\Lambda^{-1}=\omega(\gamma,D)$ and
$\Lambda\big{(}g-\big{(}\tfrac{\gamma}{2}\big{)}^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda=\Lambda^{-1}G(0)\Lambda^{-1}+\pi_{0}=\omega(\gamma,D)+\pi_{0}\,,$
(7.57)
where $\omega(\gamma,D)\in{\rm OP}S^{\frac{1}{2}}$ is defined in (2.18).
By (7.47) we compute
$\displaystyle{\mathcal{L}}_{3}:={\widetilde{{\mathcal{M}}}}^{-1}{\mathcal{L}}_{2}{\widetilde{{\mathcal{M}}}}=$
$\displaystyle\
\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-\Lambda^{-1}G(0)\Lambda^{-1}\\\
\Lambda\big{(}a_{1}-(\frac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda&-\frac{\gamma}{2}G(0)\partial_{x}^{-1}\end{pmatrix}$
(7.58) $\displaystyle+\begin{pmatrix}0&0\\\ -(\frac{\gamma}{2})^{2}\Lambda
P_{-2,N}^{(2)}\Lambda&0\end{pmatrix}+{\widetilde{{\mathcal{M}}}}^{-1}{\bf
R}_{2}^{\Psi}{\widetilde{{\mathcal{M}}}}+{\widetilde{{\mathcal{M}}}}^{-1}{\bf
T}_{2,N}{\widetilde{{\mathcal{M}}}}\,.$
By (7.57), (7.55) and (7.56), we get
$\displaystyle\Lambda\big{(}a_{1}-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda=\Lambda\big{(}g-(\tfrac{\gamma}{2})^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}\big{)}\Lambda+\Lambda(a_{1}-g)\Lambda$
(7.59) $\displaystyle\ \ \ \ \ \ \
=\omega(\gamma,D)+(a_{1}-g)\Lambda^{2}+[\Lambda,a_{1}]\Lambda+\pi_{0}$
$\displaystyle\ \ \ \ \ \ \
=\big{(}1+\tfrac{a_{1}-g}{g}\big{)}\omega(\gamma,D)+\tfrac{a_{1}-g}{g}\big{(}g\Lambda^{2}-\omega(\gamma,D)\big{)}+[\Lambda,a_{1}]\Lambda+\pi_{0}$
$\displaystyle\ \ \ \ \ \ \
=a_{2}^{2}\omega(\gamma,D)+\tfrac{a_{1}-g}{g}(\tfrac{\gamma}{2})^{2}M(D)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}+[\Lambda,a_{1}]\Lambda+\pi_{0}+\tfrac{a_{1}-g}{g}\pi_{0}$
where $a_{2}$ is the real quasi-periodic traveling wave function (with $a_{1}$
defined in Lemma 7.10)
$a_{2}:=\sqrt{\tfrac{a_{1}}{g}}=\sqrt{1+\tfrac{a_{1}-g}{g}}\,,\quad{\rm
even}({\varphi},x)\,\,.$ (7.60)
Therefore, by (7.58), (7.57), (7.59) we obtain
$\displaystyle{\mathcal{L}}_{3}$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-\omega(\gamma,D)\\\
a_{2}\omega(\gamma,D)a_{2}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}+\begin{pmatrix}0&0\\\
\pi_{0}&0\end{pmatrix}$ (7.61) $\displaystyle+\begin{pmatrix}0&0\\\
C_{3}&0\end{pmatrix}+{\bf R}_{3}^{\Psi}+{\bf T}_{3,N}\,,$
where
$C_{3}:=a_{2}[a_{2},\omega(\gamma,D)]+\tfrac{a_{1}-g}{g}(\tfrac{\gamma}{2})^{2}M(D)^{2}\partial_{x}^{-1}G(0)\partial_{x}^{-1}+[\Lambda,a_{1}]\Lambda-(\tfrac{\gamma}{2})^{2}\Lambda
P_{-2,N}^{(2)}\Lambda$ (7.62)
is in ${\rm OP}S^{-\frac{1}{2}}$ and
${\bf R}_{3}^{\Psi}:={\widetilde{{\mathcal{M}}}}^{-1}{\bf
R}_{2}^{\Psi}{\widetilde{{\mathcal{M}}}}+\begin{pmatrix}0&0\\\
(\tfrac{a_{1}}{g}-1)\pi_{0}&0\end{pmatrix}\,,\quad{\bf
T}_{3,N}:={\widetilde{{\mathcal{M}}}}^{-1}{\bf
T}_{2,N}{\widetilde{{\mathcal{M}}}}\,.$ (7.63)
The operator ${\mathcal{L}}_{3}$ in (7.61) is Hamiltonian, reversible and
momentum preserving.
Step 2: We now conjugate the operator ${\mathcal{L}}_{3}$ in (7.61) with the
symplectic matrix of multiplication operators
${\mathcal{Q}}:=\begin{pmatrix}q&0\\\ 0&q^{-1}\end{pmatrix}\
,\qquad{\mathcal{Q}}^{-1}:=\begin{pmatrix}q^{-1}&0\\\ 0&q\end{pmatrix}\,,$
where $q$ is a real function, close to $1$, to be determined, see (7.69). We
have that
$\displaystyle{\mathcal{L}}_{4}:={\mathcal{Q}}^{-1}{\mathcal{L}}_{3}{\mathcal{Q}}=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}A&B\\\
C&D\end{pmatrix}+{\mathcal{Q}}^{-1}({\bf R}_{3}^{\Psi}+{\bf
T}_{3,N}){\mathcal{Q}}\,,$ (7.64)
where (actually $D=-A^{*}$, see Definition 3.18)
$\displaystyle
A:=-\tfrac{\gamma}{2}q^{-1}G(0)\partial_{x}^{-1}q+{\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}+q^{-1}(\omega\cdot\partial_{\varphi}q)\,,$
(7.65) $\displaystyle B:=-q^{-1}\omega(\gamma,D)q^{-1}\,,$ (7.66)
$\displaystyle C:=qa_{2}\omega(\gamma,D)a_{2}q+q\pi_{0}q+qC_{3}q\,,$ (7.67)
$\displaystyle
D:=-\tfrac{\gamma}{2}q\partial_{x}^{-1}G(0)q^{-1}-{\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}-q^{-1}(\omega\cdot\partial_{\varphi}q)\,.$
(7.68)
We choose the function $q$ so that the coefficients of the highest order terms
of the off-diagonal self-adjoint operators $B$ and $C$ satisfy
$q^{-1}=qa_{2}$, namely as the real quasi-periodic traveling wave, ${\rm
even}({\varphi},x)$
$q({\varphi},x):=a_{2}({\varphi},x)^{-\frac{1}{2}}\,.$ (7.69)
Thus ${\mathcal{Q}}$ is reversibility and momentum preserving.
In view of (7.65)-(7.68) and (7.69) the operator ${\mathcal{L}}_{4}$ in (7.64)
becomes
$\displaystyle{\mathcal{L}}_{4}$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+\begin{pmatrix}-\frac{\gamma}{2}G(0)\partial_{x}^{-1}&-a_{2}^{\frac{1}{2}}\omega(\gamma,D)a_{2}^{\frac{1}{2}}\\\
a_{2}^{\frac{1}{2}}\omega(\gamma,D)a_{2}^{\frac{1}{2}}&-\frac{\gamma}{2}\partial_{x}^{-1}G(0)\end{pmatrix}$
(7.70) $\displaystyle\ \ \ +\begin{pmatrix}0&0\\\
\pi_{0}&0\end{pmatrix}+\begin{pmatrix}a_{3}&0\\\
C_{4}&-a_{3}\end{pmatrix}+{\bf R}_{4}^{\Psi}+{\bf T}_{4,N}\,,$
where $a_{3}$ is the real quasi-periodic traveling wave function, ${\rm
odd}({\varphi},x)$,
$\displaystyle a_{3}$
$\displaystyle:={\mathtt{m}}_{1,\overline{\mathtt{n}}}q^{-1}q_{x}+q^{-1}(\omega\cdot\partial_{\varphi}q)\,,\quad
C_{4}:=qC_{3}q\in{\rm OP}S^{-\frac{1}{2}}\,,$ (7.71)
and ${\bf R}_{4}^{\Psi},{\bf T}_{4,N}$ are the smoothing remainders (recall
that $G(0)\partial_{x}^{-1}={\mathcal{H}}T({\mathtt{h}})$)
$\displaystyle{\bf
R}_{4}^{\Psi}:=\begin{pmatrix}-\frac{\gamma}{2}q^{-1}[{\mathcal{H}}T({\mathtt{h}}),q-1]&0\\\
q\pi_{0}q-\pi_{0}&-\frac{\gamma}{2}[q-1,{\mathcal{H}}T({\mathtt{h}})]q^{-1}\end{pmatrix}+{\mathcal{Q}}^{-1}{\bf
R}_{3}^{\Psi}{\mathcal{Q}}\in{\rm OP}S^{-\infty}\,,$ (7.72) $\displaystyle{\bf
T}_{4,N}:={\mathcal{Q}}^{-1}{\bf T}_{3,N}{\mathcal{Q}}\,.$
The operator ${\mathcal{L}}_{4}$ in (7.70) is Hamiltonian, reversible and
momentum preserving.
Step 3: We finally move in complex coordinates, conjugating the operator
${\mathcal{L}}_{4}$ in (7.70) via the transformation ${\mathcal{C}}$ defined
in (2.19). The main result of this section is the following lemma.
###### Lemma 7.11.
Let $N\in{\mathbb{N}}$, ${\mathtt{q}}_{0}\in{\mathbb{N}}_{0}$. We have that
$\displaystyle{\mathcal{L}}_{5}:=$
$\displaystyle\,({\widetilde{{\mathcal{M}}}}{\mathcal{Q}}{\mathcal{C}})^{-1}{\mathcal{L}}_{2}{\widetilde{{\mathcal{M}}}}{\mathcal{Q}}{\mathcal{C}}$
(7.73) $\displaystyle=$
$\displaystyle\,\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}({\varphi},x){\bf{\Omega}}(\gamma,D)+{\rm
i}\,{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf R}_{5}^{(-\frac{1}{2},d)}+{\bf
R}_{5}^{(0,o)}+{\bf T}_{5,N}\,,$
where:
1\. The real quasi-periodic traveling wave $a_{2}({\varphi},x)$ defined in
(7.60), ${\rm even}({\varphi},x)$, satisfies, for some
$\sigma=\sigma(k_{0},\tau,\nu)>$ and for any $s_{0}\leq s\leq S-\sigma$,
$\|a_{2}-1\|_{s}^{k_{0},\upsilon}\lesssim\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.74)
2\. ${\bf{\Omega}}(\gamma,D)$ is the matrix of Fourier multipliers (see
(2.20), (2.21))
${\bf{\Omega}}(\gamma,D)=\begin{pmatrix}\Omega(\gamma,D)&0\\\
0&-\overline{\Omega(\gamma,D)}\end{pmatrix},\quad\Omega(\gamma,D)=\omega(\gamma,D)+{\rm
i}\,\frac{\gamma}{2}\partial_{x}^{-1}G(0)\,;$ (7.75)
3\. The operator
${\bf{\Pi}}_{0}:=\frac{1}{2}\begin{pmatrix}\pi_{0}&\pi_{0}\\\
-\pi_{0}&-\pi_{0}\end{pmatrix}\,.$
4\. The real quasi-periodic traveling wave
$a_{4}({\varphi},x):=\tfrac{\gamma}{2}(a_{2}({\varphi},x)-1)$, ${\rm
even}({\varphi},x)$, satisfies, for some $\sigma:=$ $\sigma(k_{0},\tau,\nu)>0$
and for all $s_{0}\leq s\leq S-\sigma$,
$\|a_{4}\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,;$
(7.76)
5\. ${\bf R}_{5}^{(-\frac{1}{2},d)}\in{\rm OP}S^{-\frac{1}{2}}$ and ${\bf
R}_{5}^{(0,o)}\in{\rm OP}S^{0}$ are pseudodifferential operators of the form
$\displaystyle\footnotesize{\bf
R}_{5}^{(-\frac{1}{2},d)}:=\begin{pmatrix}r_{5}^{(d)}({\varphi},x,D)&0\\\
0&\overline{r_{5}^{(d)}({\varphi},x,D)}\end{pmatrix},\quad{\bf
R}_{5}^{(0,o)}:=\begin{pmatrix}0&r_{5}^{(o)}({\varphi},x,D)\\\
\overline{r_{5}^{(o)}({\varphi},x,D)}&0\end{pmatrix}\,,$
reversibility and momentum preserving, satisfying, for some
$\sigma_{N}:=\sigma(\tau,\nu,N)>0$, for finitely many
$0\leq\alpha\leq\alpha(M)$ (fixed in Remark 7.16), and for all $s_{0}\leq
s\leq S-\sigma_{N}-3\alpha$,
$\|{\bf
R}_{5}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}+\|{\bf
R}_{5}^{(0,o)}\|_{0,s,\alpha}^{k_{0},\upsilon}\lesssim_{s,N,\alpha}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+3\alpha}^{k_{0},\upsilon})\,;$
(7.77)
6\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+\frac{3}{2}$, the operator
$\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{5,N}(\varphi)\langle D\rangle^{n_{2}}$ is ${\mathcal{D}}^{k_{0}}$-tame
with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}({\mathtt{q}}_{0},k_{0},\tau,\nu)>0$
and for any $s_{0}\leq s\leq S-\sigma_{N}({\mathtt{q}}_{0})$,
${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{5,N}(\varphi)\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,N,{\mathtt{q}}_{0}}\varepsilon\upsilon^{-1}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})}^{k_{0},\upsilon}\big{)}\,;$
(7.78)
7\. The operators ${\mathcal{Q}}^{\pm 1}$, ${\mathcal{Q}}^{\pm 1}-{\rm Id}$,
$({\mathcal{Q}}^{\pm 1}-{\rm Id})^{*}$ are ${\mathcal{D}}^{k_{0}}$-tame with
tame constants satisfying, for some $\sigma:=\sigma(\tau,\nu,k_{0})>0$ and for
all $s_{0}\leq s\leq S-\sigma$,
$\displaystyle{\mathfrak{M}}_{{\mathcal{Q}}^{\pm
1}}(s)\lesssim_{S}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon}\,,\ \
{\mathfrak{M}}_{{\mathcal{Q}}^{\pm 1}-{\rm
Id}}(s)+{\mathfrak{M}}_{\left({\mathcal{Q}}^{\pm 1}-{\rm
Id}\right)^{*}}(s)\lesssim_{S}\varepsilon\upsilon^{-1}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,.$
(7.79)
8\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha(M)$, ${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq
N-{\mathtt{q}}_{0}+\frac{1}{2}$,
$\displaystyle\|\Delta_{12}({\mathcal{A}})h\|_{s_{1}}\lesssim_{s_{1}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\left\|h\right\|_{s_{1}+\sigma}\,,\quad{\mathcal{A}}\in\\{{\mathcal{Q}}^{\pm
1}=({\mathcal{Q}}^{\pm 1})^{*}\\}\,,$ (7.80)
$\displaystyle\|\Delta_{12}a_{2}\|_{s_{1}}\lesssim_{s_{1}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,\
\|\Delta_{12}a_{4}\|_{s_{1}}\lesssim\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma}\,,$
(7.81) $\displaystyle\|\Delta_{12}{\bf
R}_{5}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s_{1},\alpha}+\|\Delta_{12}{\bf
R}_{5}^{(0,o)}\|_{0,s_{1},\alpha}\lesssim_{s_{1},N,\alpha}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+2\alpha}\,,$
(7.82)
$\displaystyle\left\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}{\bf
T}_{5,N}({\varphi})\braket{D}^{n_{2}}\right\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})}\,.$
(7.83)
The real operator ${\mathcal{L}}_{5}$ is Hamiltonian, reversible and momentum
preserving.
###### Proof.
By the expression of ${\mathcal{L}}_{4}$ in (7.70) and (3.17) we obtain that
${\cal L}_{5}$ has the form (7.73) with
$\displaystyle r_{5}^{(d)}$
$\displaystyle:=\tfrac{\gamma}{2}(a_{2}-1){\mathcal{H}}(T({\mathtt{h}})-1)+{\rm
i}\big{(}\tfrac{1}{2}C_{4}+a_{2}^{\frac{1}{2}}[\omega(\gamma,D),a_{2}^{\frac{1}{2}}]\big{)}\in{\rm
OP}S^{-\frac{1}{2}}\,,$ (7.84) $\displaystyle r_{5}^{(o)}$
$\displaystyle:=a_{3}+\tfrac{{\rm i}}{2}C_{4}\in{\rm OP}S^{0}$
(with $C_{4}$ given in (7.71)) and ${\bf T}_{5,N}:={\mathcal{C}}^{-1}({\bf
R}_{4}^{\Psi}+{\bf T}_{4,N}){\mathcal{C}}$. The function $q$ defined in
(7.69), with $a_{2}$ in (7.60), satisfies, by (7.48) and Lemma 3.2, for all
$s_{0}\leq s\leq S-\sigma$,
$\|q^{\pm
1}-1\|_{s}^{k_{0},\upsilon}\lesssim_{s}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma}^{k_{0},\upsilon})\,.$
(7.85)
The estimates (7.74) and (7.76) follows by (7.60) and (7.85). The estimate
(7.77) follows by (7.84), (7.74), (7.69), (7.62), (7.60), (7.48), (7.49),
(7.71), (7.55), (2.16), Lemma 7.10. The estimate (7.78) follows by (7.72),
(7.63), (7.42), (7.50), (7.48) Lemmata 3.12, 3.13, (7.85). The estimates
(7.79) follow by Lemmata 3.13 and (7.85). The estimates (7.80)- (7.83) are
proved similarly. ∎
### 7.4 Symmetrization up to smoothing remainders
The goal of this section is to transform the operator ${\mathcal{L}}_{5}$ in
(7.73) into the operator ${\mathcal{L}}_{6}$ in (7.88) which is block diagonal
up to a regularizing remainder. From this step we do not preserve any further
the Hamiltonian structure, but only the reversible and momentum preserving one
(it is sufficient for proving Theorem 5.1).
###### Lemma 7.12.
Fix ${\mathfrak{m}},N\in{\mathbb{N}}$, ${\mathtt{q}}_{0}\in{\mathbb{N}}_{0}$.
There exist real, reversibility and momentum preserving operator matrices
$\\{{\bf X}_{k}\\}_{k=1}^{{\mathfrak{m}}}$ of the form
${\bf X}_{k}:=\begin{pmatrix}0&\chi_{k}({\varphi},x,D)\\\
\overline{\chi_{k}({\varphi},x,D)}&0\end{pmatrix},\qquad\chi_{k}({\varphi},x,\xi)\in
S^{-\frac{k}{2}}\,,$ (7.86)
such that, conjugating the operator ${\mathcal{L}}_{5}$ in (7.73) via the map
${\bf{\Phi}}_{{\mathfrak{m}}}:=e^{{\bf X}_{1}}\circ\cdots\circ e^{{\bf
X}_{{\mathfrak{m}}}}\,,$ (7.87)
we obtain the real, reversible and momentum preserving operator
$\displaystyle{\mathcal{L}}_{6}$
$\displaystyle:={\mathcal{L}}_{6}^{({\mathfrak{m}})}:={\bf{\Phi}}_{{\mathfrak{m}}}^{-1}\,{\mathcal{L}}_{5}\,{\bf{\Phi}}_{{\mathfrak{m}}}$
(7.88)
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)+{\rm i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6}^{(-\frac{1}{2},d)}+{\bf R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}+{\bf
T}_{6,N}\,,$
where:
1\. ${\bf R}_{6}^{(-\frac{1}{2},d)}$ is a block-diagonal operator
$\displaystyle{\bf R}_{6}^{(-\frac{1}{2},d)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}$
$\displaystyle:=\begin{pmatrix}r_{6}^{(d)}({\varphi},x,D)&0\\\
0&\overline{r_{6}^{(d)}({\varphi},x,D)}\end{pmatrix}\in{\rm
OP}S^{-\frac{1}{2}}\,,$
${\bf R}_{6}^{(-\frac{{\mathtt{m}}}{2},o)}$ is a smoothing off diagonal
remainder
$\displaystyle{\bf R}_{6}^{(-\frac{{\mathtt{m}}}{2},o)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}$
$\displaystyle:=\begin{pmatrix}0&r_{6}^{(o)}({\varphi},x,D)\\\
\overline{r_{6}^{(o)}({\varphi},x,D)}&0\end{pmatrix}\in{\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}}\,,$ (7.89)
satisfying for finitely many $0\leq\alpha\leq\alpha({\mathfrak{m}})$ (fixed in
Remark 7.16), for some $\sigma_{N}:=\sigma_{N}(k_{0},\tau,\nu,N)>0$,
$\aleph_{{\mathfrak{m}}}(\alpha)>0$ and for all $s_{0}\leq s\leq
S-\sigma_{N}-\aleph_{{\mathfrak{m}}}(\alpha)$,
$\displaystyle\|{\bf
R}_{6}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}+\|{\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}\|_{-\frac{{\mathfrak{m}}}{2},s,\alpha}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},N,\alpha}\varepsilon{\upsilon^{-1}}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}}(\alpha)}^{k_{0},\upsilon}\big{)}\,.$
(7.90)
Both ${\bf R}_{6}^{(-\frac{1}{2},d)}$ and ${\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ are reversible and momentum
preserving;
2\. For any ${\mathtt{q}}\in{\mathbb{N}}^{\nu}_{0}$ with
$|{\mathtt{q}}|\leq{\mathtt{q}}_{0}$, $n_{1},n_{2}\in{\mathbb{N}}_{0}$ with
$n_{1}+n_{2}\leq N-(k_{0}+{\mathtt{q}}_{0})+\frac{3}{2}$, the operator
$\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{6,N}(\varphi)\langle D\rangle^{n_{2}}$ is ${\mathcal{D}}^{k_{0}}$-tame
with a tame constant satisfying, for some
$\sigma_{N}({\mathtt{q}}_{0}):=\sigma_{N}(k_{0},\tau,\nu,{\mathtt{q}}_{0})$,
for any $s_{0}\leq s\leq
S-\sigma_{N}({\mathtt{q}}_{0})-\aleph_{{\mathfrak{m}}}(0)$,
${\mathfrak{M}}_{\langle D\rangle^{n_{1}}\partial_{\varphi}^{\mathtt{q}}{\bf
T}_{6,N}(\varphi)\langle
D\rangle^{n_{2}}}(s)\lesssim_{S,{\mathfrak{m}},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}({\mathtt{q}}_{0})+\aleph_{{\mathfrak{m}}}(0)}^{k_{0},\upsilon})\,.$
(7.91)
3\. The conjugation map ${\bf{\Phi}}_{{\mathfrak{m}}}$ in (7.87) satisfies,
for all $s_{0}\leq s\leq S-\sigma_{N}-\aleph_{{\mathfrak{m}}}(0)$,
$\|{\bf{\Phi}}_{{\mathfrak{m}}}^{\pm 1}-{\rm
Id}\|_{0,s,0}^{k_{0},\upsilon}+\|\left({\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1}-{\rm
Id}\right)^{*}\|_{0,s,0}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},N}\varepsilon{\upsilon^{-1}}(1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}}(0)}^{k_{0},\upsilon})\,.$
(7.92)
4\. Furthermore, for any $s_{1}$ as in (7.11), finitely many
$0\leq\alpha\leq\alpha({\mathfrak{m}})$,
${\mathtt{q}}\in{\mathbb{N}}_{0}^{\nu}$, with
$\left|{\mathtt{q}}\right|\leq{\mathtt{q}}_{0}$, and
$n_{1},n_{2}\in{\mathbb{N}}_{0}$, with $n_{1}+n_{2}\leq
N-{\mathtt{q}}_{0}+\frac{1}{2}$, we have
$\displaystyle\|\Delta_{12}{\bf
R}_{6}^{(-\frac{1}{2},d)}\|_{-\frac{1}{2},s_{1},\alpha}+\|\Delta_{12}{\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}\|_{-\frac{{\mathfrak{m}}}{2},s_{1},\alpha}\lesssim_{s_{1},{\mathfrak{m}},N,\alpha}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\aleph_{{\mathfrak{m}}}(\alpha)}\,,$
(7.93)
$\displaystyle\|\braket{D}^{n_{1}}\partial_{\varphi}^{\mathtt{q}}\Delta_{12}{\bf
T}_{6,N}\braket{D}^{n_{2}}\|_{{\mathcal{L}}(H^{s_{1}})}\lesssim_{s_{1},{\mathfrak{m}},N,{\mathtt{q}}_{0}}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}({\mathtt{q}}_{0})+\aleph_{{\mathfrak{m}}}(0)}\,,$
(7.94) $\displaystyle\|\Delta_{12}{\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1}\|_{0,s_{1},0}+\|\Delta_{12}({\bf{\Phi}}_{{\mathfrak{m}}}^{\pm
1})^{*}\|_{0,s_{1},0}\lesssim_{s_{1},{\mathfrak{m}},N}\varepsilon{\upsilon^{-1}}\left\|i_{1}-i_{2}\right\|_{s_{1}+\sigma_{N}+\aleph_{{\mathfrak{m}}}(0)}\,.$
(7.95)
###### Proof.
The proof is inductive. The operator
${\mathcal{L}}_{6}^{(0)}:={\mathcal{L}}_{5}$ satisfies (7.90)-(7.91) with
$\aleph_{0}(\alpha):=3\alpha$, by (7.77)-(7.78). Suppose we have done already
${\mathfrak{m}}$ steps obtaining an operator
${\mathcal{L}}_{6}^{({\mathfrak{m}})}$ as in (7.88) with ${\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}:={\bf R}_{6}^{(-\frac{1}{2},d)}$ and
${\bf R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},o)}:={\bf
R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ and the remainder
${\bf\Phi}_{{\mathfrak{m}}}^{-1}{\bf T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}}$,
instead of ${\bf T}_{6,N}$. We now show how to define
${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$. Let
$\chi_{{\mathfrak{m}}+1}({\varphi},x,\xi):=-\big{(}2{\rm
i}\,a_{2}({\varphi},x)\omega(\gamma,\xi)\big{)}^{-1}r_{6,{\mathfrak{m}}}^{(o)}({\varphi},x,\xi)\chi(\xi)\in
S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}\,,$ (7.96)
where $\chi$ is the cut-off function defined in (3.11) and
$\omega(\gamma,\xi)$ is the symbol (cfr. (2.18))
$\omega(\gamma,\xi):=\sqrt{G(0;\xi)\Big{(}g+\frac{\gamma^{2}}{4}\frac{G(0;\xi)}{\xi^{2}}\Big{)}}\in
S^{\frac{1}{2}}\,,\ \
G(0;\xi):=\begin{cases}\chi(\xi)|\xi|\tanh({\mathtt{h}}|\xi|)\,,\
{\mathtt{h}}<+\infty\cr\chi(\xi)|\xi|\,,\qquad\qquad\ \,\
{\mathtt{h}}=+\infty\,.\end{cases}$
Note that $\chi_{{\mathfrak{m}}+1}$ in (7.96) is well defined because
$\omega(\gamma,\xi)$ is positive on the support of $\chi(\xi)$ and
$a_{2}({\varphi},x)$ is close to 1. We conjugate the operator
${\mathcal{L}}_{6}^{({\mathfrak{m}})}$ in (7.88) by the flow generated by
${\bf X}_{{\mathfrak{m}}+1}$ of the form (7.86) with
$\chi_{{\mathfrak{m}}+1}(\varphi,x,\xi)$ defined in (7.96). By (7.90) and
(7.75), for suitable constants
$\aleph_{{\mathfrak{m}}+1}(\alpha)>\aleph_{{\mathfrak{m}}}(\alpha)$, for
finitely many $\alpha\in{\mathbb{N}}_{0}$ and for any $s_{0}\leq s\leq
S-\sigma_{N}-\aleph_{{\mathfrak{m}}+1}(\alpha)$,
$\|{\bf
X}_{{\mathfrak{m}}+1}\|_{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2},s,\alpha}^{k_{0},\upsilon}\lesssim_{s,{\mathfrak{m}},\alpha}\varepsilon{\upsilon^{-1}}\big{(}1+\|{\mathfrak{I}}_{0}\|_{s+\sigma_{N}+\aleph_{{\mathfrak{m}}+1}(\alpha)}^{k_{0},\upsilon}\big{)}\,.$
(7.97)
Therefore, by Lemmata 3.7, 3.5 and the induction assumption (7.92) for
${\bf{\Phi}}_{{\mathfrak{m}}}$, the conjugation map
${\bf{\Phi}}_{{\mathfrak{m}}+1}:={\bf{\Phi}}_{{\mathfrak{m}}}e^{{\bf
X}_{{\mathfrak{m}}+1}}$ is well defined and satisfies estimate (7.92) with
${\mathfrak{m}}+1$. By the Lie expansion (3.18) we have
$\displaystyle{\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ $\displaystyle:=e^{-{\bf
X}_{{\mathfrak{m}}+1}}\,{\mathcal{L}}_{6}^{({\mathfrak{m}})}\,e^{{\bf
X}_{{\mathfrak{m}}+1}}$ (7.98)
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}a_{2}{\bf{\Omega}}(\gamma,D)+{\rm i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}$ $\displaystyle-\big{[}{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\big{]}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}+{\bf\Phi}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}+1}$ $\displaystyle-\int_{0}^{1}e^{-\tau{\bf
X}_{{\mathfrak{m}}+1}}\big{[}{\bf
X}_{{\mathfrak{m}}+1}\,,\,\omega\cdot\partial_{\varphi}+{\rm
i}{\bf{\Pi}}_{0}+a_{4}{\mathcal{H}}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}\big{]}e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}$ (7.99)
$\displaystyle-\int_{0}^{1}e^{-\tau{\bf X}_{{\mathfrak{m}}+1}}\left[{\bf
X}_{{\mathfrak{m}}+1},{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}\right]e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}$ (7.100)
$\displaystyle+\int_{0}^{1}(1-\tau)e^{-\tau{\bf
X}_{{\mathfrak{m}}+1}}\left[{\bf X}_{{\mathfrak{m}}+1},\left[{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\right]\right]e^{\tau{\bf
X}_{{\mathfrak{m}}+1}}\,{\rm d}{\tau}\,.$ (7.101)
In view of (7.86), (7.75) and (7.89), we have that
$-\big{[}{\bf
X}_{{\mathfrak{m}}+1},{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,a_{2}{\bf{\Omega}}(\gamma,D)\big{]}+{\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}=\begin{pmatrix}0&Z_{{\mathfrak{m}}+1}\\\
\overline{Z_{{\mathfrak{m}}+1}}&0\end{pmatrix}=:{\bf Z}_{{\mathfrak{m}}+1}\,,$
where, denoting for brevity
$\chi_{{\mathfrak{m}}+1}:=\chi_{{\mathfrak{m}}+1}({\varphi},x,\xi)$, it
results
$\displaystyle Z_{{\mathfrak{m}}+1}$ $\displaystyle={\rm i}\left({\rm
Op}(\chi_{{\mathfrak{m}}+1})a_{2}\,\omega(\gamma,D)+a_{2}\,\omega(\gamma,D){\rm
Op}(\chi_{{\mathfrak{m}}+1})\right)$ $\displaystyle\quad+\left[{\rm
Op}(\chi_{{\mathfrak{m}}+1}),-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+a_{2}\,\tfrac{\gamma}{2}\partial_{x}^{-1}G(0)\right]+{\rm
Op}(r_{6,{\mathfrak{m}}}^{(o)})\,.$
By (3.24), (3.26) and since $\chi_{{\mathfrak{m}}+1}\in
S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}$ by (7.96), we have that
${\rm
Op}(\chi_{{\mathfrak{m}}+1})a_{2}\omega(\gamma,D)+a_{2}\omega(\gamma,D){\rm
Op}(\chi_{{\mathfrak{m}}+1})={\rm
Op}\big{(}2a_{2}\omega(\gamma,\xi)\chi_{{\mathfrak{m}}+1}\big{)}+{\mathtt{r}}_{{\mathfrak{m}}+1}\,,$
where ${\mathtt{r}}_{{\mathfrak{m}}+1}$ is in ${\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}-1}$. By (7.96) and (7.4)
$Z_{{\mathfrak{m}}+1}={\rm i}{\mathtt{r}}_{{\mathfrak{m}}+1}+\left[{\rm
Op}(\chi_{{\mathfrak{m}}+1}),-{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+a_{2}\tfrac{\gamma}{2}\partial_{x}^{-1}G(0)\right]+{\rm
Op}(r_{6,{\mathfrak{m}}}^{(o)}(1-\chi(\xi)))\in{\rm
OP}S^{-\frac{{\mathfrak{m}}}{2}-\frac{1}{2}}\,.$
The remaining pseudodifferential operators in (7.99)-(7.101) have order ${\rm
OP}S^{-\frac{{\mathfrak{m}}+1}{2}}$. Therefore the operator
${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ in (7.98) has the form (7.88) at
${\mathfrak{m}}+1$ with
${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{1}{2},d)}+{\bf
R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}:={\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{1}{2},d)}+{\bf
Z}_{{\mathfrak{m}}+1}+\eqref{geno12}+\eqref{geno13}+\eqref{geno14}$ (7.102)
and a smoothing remainder ${\bf\Phi}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf\Phi}_{{\mathfrak{m}}+1}$. By Lemmata 3.5, 3.6, (7.90), (7.97),
(7.76), we conclude that ${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{1}{2},d)}$ and
${\bf R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}$ satisfy
(7.90) at order ${\mathfrak{m}}+1$ for suitable constants
$\aleph_{{\mathfrak{m}}+1}(\alpha)>\aleph_{{\mathfrak{m}}}(\alpha)$. Moreover
the operator ${\bf{\Phi}}_{{\mathfrak{m}}+1}^{-1}{\bf
T}_{5,N}{\bf{\Phi}}_{{\mathfrak{m}}+1}$ satisfies (7.91) at order
${\mathfrak{m}}+1$ by Lemmata 3.12, 3.13 and (7.78), (7.92). Estimates
(7.93)-(7.95) follow similarly. By (7.96), Lemmata 3.20, 3.24, and the
induction assumption that ${\bf
R}_{6,{\mathfrak{m}}}^{(-\frac{{\mathfrak{m}}}{2},o)}$ is reversible and
momentum preserving, we get that ${\bf X}_{{\mathfrak{m}}+1}$ is reversibility
and momentum preserving, and so are $e^{\pm{\bf X}_{{\mathfrak{m}}+1}}$. We
deduce that ${\mathcal{L}}_{6}^{({\mathfrak{m}}+1)}$ is reversible and
momentum preserving, in particular ${\bf
R}_{6,{\mathfrak{m}}+1}^{(-\frac{{\mathfrak{m}}+1}{2},o)}$ in (7.102). ∎
###### Remark 7.13.
The number of regularizing iterations ${\mathfrak{m}}\in{\mathbb{N}}$ will be
fixed by the KAM reduction scheme in Section 8, more precisely we take
${\mathfrak{m}}=2M$ with $M$ in (8.5). Note that it is independent of the
Sobolev index $s$.
So far the operator ${\mathcal{L}}_{6}$ of Lemma 7.12 depends on two indexes
${\mathfrak{m}},N$ which provide respectively the order of the regularizing
off-diagonal remainder ${\bf R}_{6}^{(-\frac{{\mathfrak{m}}}{2},o)}$ and of
the smoothing tame operator ${\bf T}_{6,N}$. From now on we fix
${\mathfrak{m}}:=2M\,,\ M\in{\mathbb{N}}\,,\quad N=M\,.$ (7.103)
### 7.5 Reduction of the order 1/2
The goal of this section is to transform the operator ${\mathcal{L}}_{6}$ in
(7.88) with ${\mathfrak{m}}:=2M$, $N=M$ (cfr. (7.103)), into the operator
${\mathcal{L}}_{7}$ in (7.117) whose coefficient in front of
${\bf{\Omega}}(\gamma,D)$ is constant. First we rewrite
${\mathcal{L}}_{6}=\omega\cdot\partial_{\varphi}+\begin{pmatrix}P_{6}&0\\\
0&\overline{P_{6}}\end{pmatrix}+{\rm i}{\bf{\Pi}}_{0}+{\bf
R}_{6}^{(-M,o)}+{\bf T}_{6,M}\,,$
having denoted
$P_{6}:=P_{6}({\varphi},x,D):={\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}a_{2}({\varphi},x)\Omega(\gamma,D)+a_{4}{\mathcal{H}}+r_{6}^{(d)}({\varphi},x,D)\,.$
(7.104)
We conjugate ${\mathcal{L}}_{6}$ through the real operator
${\bf{\Phi}}({\varphi}):=\begin{pmatrix}\Phi({\varphi})&0\\\
0&\overline{\Phi}({\varphi})\end{pmatrix}$ (7.105)
where $\Phi({\varphi}):=\Phi^{\tau}({\varphi})|_{\tau=1}$ is the time $1$-flow
of the PDE
$\begin{cases}\partial_{\tau}\Phi^{\tau}({\varphi})={\rm
i}A({\varphi})\Phi^{\tau}({\varphi})\,,\\\ \Phi^{0}({\varphi})={\rm
Id}\,,\end{cases}\qquad A({\varphi}):=b({\varphi},x)|D|^{\frac{1}{2}}\,,$
(7.106)
and $b({\varphi},x)$ is a real quasi-periodic traveling wave, ${\rm
odd}({\varphi},x)$, chosen later, see (7.114). Thus ${\rm
i}b({\varphi},x)|D|^{\frac{1}{2}}$ is reversibility and momentum preserving as
well as ${\bf{\Phi}}({\varphi})$. Moreover
$\Phi\pi_{0}=\pi_{0}=\Phi^{-1}\pi_{0}$, which implies
${\bf{\Phi}}^{-1}{\bf{\Pi}}_{0}{\bf{\Phi}}={\bf{\Pi}}_{0}{\bf{\Phi}}\,.$
(7.107)
By the Lie expansion (3.18) we have
$\displaystyle\Phi^{-1}P_{6}\Phi$ $\displaystyle=P_{6}-{\rm
i}[A,P_{6}]-\frac{1}{2}[A,[A,P_{6}]]+\sum_{n=3}^{2M+1}\frac{(-{\rm
i})^{n}}{n!}{\rm ad}_{A({\varphi})}^{n}(P_{6})+T_{M}\,,$ (7.108)
$\displaystyle T_{M}$ $\displaystyle:=\frac{(-{\rm
i})^{2M+2}}{(2M+1)!}\int_{0}^{1}(1-\tau)^{2M+1}\Phi^{-\tau}({\varphi})\,{\rm
ad}_{A({\varphi})}^{2M+2}(P_{6})\,\Phi^{\tau}({\varphi}){\rm d}\tau\,,$
and, by (3.19),
$\displaystyle\Phi^{-1}\circ\omega\cdot\partial_{\varphi}\circ\Phi$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\rm
i}(\omega\cdot\partial_{\varphi}A)+\frac{1}{2}[A,\omega\cdot\partial_{\varphi}A]-\sum_{n=3}^{2M+1}\frac{(-{\rm
i})^{n}}{n!}{\rm
ad}_{A({\varphi})}^{n-1}(\omega\cdot\partial_{\varphi}A({\varphi}))+T_{M}^{\prime}\,,$
$\displaystyle T_{M}^{\prime}$ $\displaystyle:=-\frac{(-{\rm
i})^{2M+2}}{(2M+1)!}\int_{0}^{1}(1-\tau)^{2M+1}\Phi^{-\tau}({\varphi})\,{\rm
ad}_{A({\varphi})}^{2M+1}(\omega\cdot\partial_{\varphi}A({\varphi}))\,\Phi^{\tau}({\varphi}){\rm
d}\tau\,.$ (7.109)
Note that ${\rm ad}_{A({\varphi})}^{2M+2}(P_{6})$ and ${\rm
ad}_{A({\varphi})}^{2M+1}(\omega\cdot\partial_{\varphi}A({\varphi}))$ are in
${\rm OP}S^{-M}$. We now determine the pseudo-differential term of order $1/2$
in (7.108)-(7.109). We use the expansion of the linear dispersion operator
$\Omega(\gamma,D)$, defined by (4.1), (1.10), and, since $j\to
c_{j}(\gamma)\in S^{0}$ (see (4.14)),
$\Omega(\gamma,D)=\sqrt{g}|D|^{\frac{1}{2}}+{\rm
i}\,\tfrac{\gamma}{2}{\mathcal{H}}+r_{-\frac{1}{2}}(\gamma,D)\,,\quad
r_{-\frac{1}{2}}(\gamma,D)\in{\rm OP}S^{-\frac{1}{2}}\,,$ (7.110)
where ${\mathcal{H}}$ is the Hilbert transform in (3.21). By (7.104), that
$A=b|D|^{\frac{1}{2}}$, (3.27), (7.110) we get
$\displaystyle[A,P_{6}]$
$\displaystyle=\big{[}b|D|^{\frac{1}{2}},{\mathtt{m}}_{1,{\mathtt{n}}}\partial_{x}+{\rm
i}\,\sqrt{g}a_{2}|D|^{\frac{1}{2}}+(a_{4}-\tfrac{\gamma}{2}a_{2}){\mathcal{H}}+r_{6}^{(d)}(x,D)+{\rm
i}\,a_{2}r_{-\frac{1}{2}}(\gamma,D)\big{]}$
$\displaystyle=-{\mathtt{m}}_{1,\overline{\mathtt{n}}}b_{x}|D|^{\frac{1}{2}}-{\rm
i}\tfrac{\sqrt{g}}{2}(b_{x}a_{2}-(a_{2})_{x}b){\mathcal{H}}+{\rm
Op}(r_{b,-\frac{1}{2}})\,,$ (7.111)
where $r_{b,-\frac{1}{2}}\in S^{-\frac{1}{2}}$ is small with $b$. As a
consequence, the contribution at order $\frac{1}{2}$ of the operator ${\rm
i}\,\omega\cdot\partial_{\varphi}A+P_{6}-{\rm i}[A,P_{6}]$ is ${\rm
i}\big{(}\omega\cdot\partial_{\varphi}b+{\mathtt{m}}_{1,\overline{\mathtt{n}}}b_{x}+\sqrt{g}\,a_{2})|D|^{\frac{1}{2}}$.
We choose $b({\varphi},x)$ as the solution of
$(\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x})b+\sqrt{g}\,\Pi_{N_{\overline{\mathtt{n}}}}\,a_{2}=\sqrt{g}\,{\mathtt{m}}_{\frac{1}{2}}$
(7.112)
where ${\mathtt{m}}_{\frac{1}{2}}$ is the average (see (3.6))
${\mathtt{m}}_{\frac{1}{2}}:=\braket{a_{2}}_{{\varphi},x}\,.$ (7.113)
We define $b({\varphi},x)$ to be the real, ${\rm odd}({\varphi},x)$, quasi-
periodic traveling wave
$b({\varphi},x):=-\sqrt{g}(\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x})_{\rm
ext}^{-1}\big{(}\Pi_{N_{\overline{\mathtt{n}}}}a_{2}({\varphi},x)-{\mathtt{m}}_{\frac{1}{2}}\big{)}$
(7.114)
recall (3.10). Note that $b({\varphi},x)$ and ${\mathtt{m}}_{\frac{1}{2}}$ are
defined for any
$(\omega,\gamma)\in{\mathbb{R}}^{\nu}\times[\gamma_{1},\gamma_{2}]$ and that,
for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$
defined in (7.26), it solves (7.112).
We deduce by (7.108), (7.109), (7.104), (7.5)-(7.114), that, for any
$(\omega,\gamma)\in{\mathtt{T}}{\mathtt{C}}_{\overline{\mathtt{n}}+1}(2\upsilon,\tau)$,
$\displaystyle L_{7}$
$\displaystyle:=\Phi^{-1}({\varphi})\left(\omega\cdot\partial_{\varphi}+P_{6}\right)\Phi({\varphi})$
$\displaystyle=\omega\cdot\partial_{\varphi}+{\mathtt{m}}_{1,\overline{\mathtt{n}}}\partial_{x}+{\rm
i}\,{\mathtt{m}}_{\frac{1}{2}}\Omega(\gamma,D)+a_{5}{\mathcal{H}}+{\rm
Op}(r_{7}^{(d)})+T_{M}+T_{M}^{\prime}+{\rm
i}\sqrt{g}(\Pi_{N_{\overline{\mathtt{n}}}}^{\perp}a_{2})|D|^{\frac{1}{2}}\,,$
where $a_{5}({\varphi},x)$ is the real function (using that
$a_{4}=\frac{\gamma}{2}(a_{2}-1)$)
$\displaystyle a_{5}:=$
$\displaystyle\,\tfrac{\gamma}{2}({\mathtt{m}}_{\frac{1}{2}}-1)-\tfrac{\sqrt{g}}{2}(b_{x}a_{2}-(a_{2})_{x}b)$
(7.115)
|
11institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
11email<EMAIL_ADDRESS>22institutetext: Departamento de
Física-Icex-UFMG Antônio Carlos, 6627, 31270-901. Belo Horizonte, MG, Brazil
33institutetext: Univ. de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400
Toulouse, France 44institutetext: Infrared Science Archive (IRSA), IPAC,
California Institute of Technology, 1200 E. California Blvd., Pasadena, CA,
91125, USA 55institutetext: Université de Montréal, Département de Physique,
IREX, Montréal, QC H3C 3J7, Canada
# New insights on the near-infrared veiling of young stars using CFHT/SPIRou
data††thanks: Based on observations obtained at the Canada-France-Hawaii Tele-
scope (CFHT) which is operated by the National Research Council (NRC) of
Canada, the Institut National des Sciences de l’Univers of the Centre National
de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.
Based on observations obtained with SPIRou, an international project led by
Institut de Recherche en Astrophysique et Planétologie, Toulouse, France.
A. P. Sousa 11 J. Bouvier 11 S. H. P. Alencar 22 J.-F. Donati 33 C. Dougados
11 E. Alecian 11 A. Carmona 11 L. Rebull 44 N. Cook 55 E. Artigau 55 P. Fouqué
33 R. Doyon 55 the SLS consortium
(Received September 15, 1996; accepted March 16, 1997)
###### Abstract
Context. Veiling is ubiquitous at different wavelength ranges in classical T
Tauri stars. However, the origin of the veiling in the infrared (IR) domain is
not well understood at present. The accretion spot alone is not enough to
explain the shallow photospheric IR lines in accreting systems, suggesting
that another source is contributing to the veiling in the near-infrared (NIR).
The inner disk is often quoted as the additional emitting source meant to
explain the IR veiling.
Aims. In this work, we aim to measure and discuss the NIR veiling to
understand its origins and variability timescale.
Methods. We used a sample of 14 accreting stars observed with the CFHT/SPIRou
spectrograph, within the framework of the SPIRou Legacy Survey, to measure the
NIR veiling along the $YJHK$ bands. We compared the veiling measurements with
accretion and inner disk diagnostics. We also analyzed circumstellar emission
lines and photometric observations from the literature.
Results. The measured veiling grows from the $Y$ to the $K$ band for most of
the targets in our sample. The IR veiling agrees with NIR emission excess
obtained using photometric data. However, we also find a linear correlation
between the veiling and the accretion properties of the system, showing that
accretion contributes to the inner disk heating and, consequently, to the
inner disk emission excess. We also show a connection between the NIR veiling
and the system’s inclination with respect to our line of sight. This is
probably due to the reduction of the visible part of the inner disk edge,
where the NIR emission excess is expected to arise, as the inclination of the
system increases. Our search for periods on the veiling variability showed
that the IR veiling is not clearly periodic in the typical timescale of
stellar rotation – which, again, is broadly consistent with the idea that the
veiling comes from the inner disk region. The NIR veiling appears variable on
a timescale of a day, showing the night-by-night dynamics of the optical
veiling variability. In the long term, the mean NIR veiling seems to be stable
for most of the targets on timescales of a month to a few years. However,
during occasional episodes of high accretion in classical T Tauri stars, which
affect the system’s dynamic, the veiling also seems to be much more prominent
at such times, as we found in the case of the target RU Lup.
Conclusions. We provide further evidence that for most targets in our sample,
the veiling that mainly occurs in the $JHK$ bands arises from dust in the
inner disk.
###### Key Words.:
Stars: pre-main sequence – Accretion, accretion disks – Stars: variables: T
Tauri
## 1 Introduction
The photospheric lines of young low-mass accreting systems, commonly referred
to as Classical T Tauri stars (CTTS), are shallower and present smaller
equivalent widths than those of non-accreting stars with a similar spectral
type. This phenomenon is known as the veiling of the photospheric lines (e.g.,
Hartigan et al., 1991; Valenti et al., 1993; Folha & Emerson, 1999; Fischer et
al., 2011). The presence of veiling suggests an additional emitting source,
beyond the stellar photosphere contributing to the spectra of the targets,
that is responsible for filling in the photospheric lines (e.g., Hartmann &
Kenyon, 1990; Gullbring et al., 1998; Calvet & Gullbring, 1998; Johns-Krull &
Valenti, 2001).
The veiling in the optical and in the IR wavelengths has been studied using
different approaches in the aim to understand its origins and variability
(e.g., Basri & Batalha, 1990; Edwards et al., 2006; Fischer et al., 2011;
Antoniucci et al., 2017; Gully-Santiago et al., 2017; Ingleby et al., 2013;
McClure et al., 2013; Kidder et al., 2021). The veiling variability along the
stellar spectra depends on wavelength (e.g., Fischer et al., 2011; Faesi et
al., 2012; McClure et al., 2013; Rei et al., 2018), and optical veiling is
often associated with the accretion process, while the accretion shock is
thought to be at the origin of an additional continuum emitting source to the
stellar spectrum. (e.g., Gullbring et al., 1998). Usually, the accretion spot
presents an emission contribution maximum around the ultraviolet domain, which
decreases as the wavelength increases (e.g., Calvet & Gullbring, 1998).
Nevertheless, we do not expect a significant contribution of the accretion
spot continuum emission in the IR wavelength; therefore, the accretion spot
alone cannot explain the veiling in the IR domain.
In the IR region, the veiling increases with wavelength and in some cases, it
becomes greater than the veiling in the optical domain (e.g., McClure et al.,
2013). The central star illuminates the inner disk and this region absorbs
photons from the star, the accretion spot, and even the accretion funnel, then
re-emitting them in the infrared (IR) as the system rotates (e.g., Chiang &
Goldreich, 1997, 1999). Therefore, the inner disk is suggested as the origin
of the additional continuum emission that is essential to explaining the near-
infrared (NIR) veiling, although the measured veiling is often too great to be
explained as coming merely from the disk emission, based on model predictions
(e.g., Folha & Emerson, 1999; Johns-Krull & Valenti, 2001). Many authors have
used different techniques to connect the observed veiling with inner disk
emission, such as measuring the temperature of the region where the veiling
comes from and using a black body fit to the veiling. For most of these
systems, they found temperatures compatible with the dust temperature in the
inner disk (e.g., Fischer et al., 2011; Antoniucci et al., 2017; Alcalá et
al., 2021). For a few targets, the blackbody temperature measured using
veiling is too high for dust to survive in the inner disk; this would indicate
that the veiling should arise from the gas in the inner disk inside the star-
disk co-rotation radius (e.g., Antoniucci et al., 2017; Alcalá et al., 2021).
However, McClure et al. (2013) found no evidence of hot gas inside the inner
disk, which would be responsible for the NIR veiling. Instead, they explained
the IR veiling as the combined emission from the accretion shock on the
stellar surface and dust around the sublimation rim of the inner disk.
The veiling around $1$$\mu$m and the veiling in the $K$ band and beyond can
also have different origins. While the accretion spot emission contribution is
not very substantial around $1$$\mu$m, we do not expect a significant
contribution from the inner disk either. The origin of the veiling in this
spectral domain is poorly understood. However, significant veiling was
measured around $1$$\mu$m, primarily for high accretion rate systems (e.g.,
Edwards et al., 2006; Fischer et al., 2011; Ingleby et al., 2013). In the
literature, there are only a few plausible explanations given for that case of
veiling, such as a contribution from emission lines, filling in of the
photospheric lines, or origination from the accretion shock (e.g., Sicilia-
Aguilar et al., 2015; Dodin & Lamzin, 2013). Even if we do not detect these
extra emission lines directly, they can contribute to making the photospheric
lines of the stellar spectra shallower, which increases the veiling
measurement.
We cannot exclude other possible explanations to the IR veiling, such as an
envelope around a star that can also be a source of additional emission to the
photospheric continuum, with emission compatible with NIR veiling (e.g.,
Calvet et al., 1997). However, CTTSs are usually Class II stars and we would
not expect a significant contribution from a dusty envelope. Furthermore, the
veiling in the $K$ band is higher than the dust envelope emission can explain
(Folha & Emerson, 1999).
In this work, we study the veiling and its variability in the NIR, using a
sample of young accreting stars observed with the Canada-France-Hawaii
Telescope SPectropolarimètre InfraROUge (CFHT/SPIRou). Our sample comprises
stars with different properties, such as the mass accretion rate, spectral
type, and inclination with respect to our line of sight. We computed the
veiling for the $YJHK$ bands and compared the results with accretion and inner
disk diagnostics.
We organized the paper as follows. In Sect. 2, we present the sample of stars
that we used in this work, and we describe the data used. In Sect. 3 we show
the procedures to measure the veiling. We describe the results obtained from
the veiling measurements in Sect. 4. In Sect. 5, we discuss the possible
origin of the veiling, and we compare our results with those of previous
works. In Sect. 6, we present our conclusions.
## 2 Observations and targets selection
Table 1: Sample of stars and number of observations per observational period
Star | SpT | v$\sin{i}$ | Av | i∗aa$a$ The inclination (see references) between the stellar rotation axis and our line of sight obtained from v$\sin{i}$, period, and radius of each target, except for the inclinations of DO Tau and DG Tau that were obtained from the outer disk parameters. | 2018 | 2019a | 2019b | 2020a | 2020b | 2021a | 2021b | 2022a | Referencesbb$b$ References for the SpT, $v\sin{i}$, Av and inclination ($i$), respectively.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| | (km/s) | (mag) | (∘) | | | | | | | | |
Accreting systems |
CI Tau | K4 | 9.5$\pm$0.5 | 0.65 | 55${}^{+35}_{-10}$ | 2 | 6 | 26 | 5 | 39 | - | - | - | (1),(2),(2),(2)
DoAr 44 | K2-K3 | 17.0$\pm$1.1 | 2.0$\pm$0.2 | 30$\pm$5 | - | 8 | - | - | - | - | - | - | (3),(3),(3),(3)
GQ Lup | K7 | 5$\pm$1 | 0.7 | $\sim$30 | - | 8 | - | 18 | 6 | - | - | - | (4),(5),(4),(5)
TW Hya | K6 | 6.0$\pm$1.2 | 0.0 | 18$\pm$10 | - | 12 | - | 14 | - | 27 | - | - | (1),(1),(6),(7)
V2129 Oph | K5 | 14.5$\pm$0.3 | 0.6 | 60 | 9 | - | - | 17 | 8 | - | - | - | (1),(1),(8),(9)
BP Tau | K5 | 9.0$\pm$0.5 | 0.45 | $\sim$45 | - | - | 21 | - | - | - | 34 | - | (10),(11),(6),(11)
V347 Aur | M2-M3 | 11.7${}^{+0.16}_{-0.24}$ | 3.4 | 40 | - | - | 18 | - | 13 | 12 | 22 | - | (12),(13),(12),(14)
DG Tau | K6 | 24.7$\pm$0.7 | 1.60$\pm$0.15 | 38$\pm$2 | - | - | - | 2 | 29 | - | - | - | (10),(10),(6),(15)
RU Lup | K7 | 8.5$\pm$4.8 | 0.0 | 24 | - | - | - | 9 | - | 17 | 13 | 1 | (16),(17),(14),(18)
V2247 Oph | M0 | 20.5$\pm$0.5 | 0.98$\pm$0.02 | 45$\pm$10 | - | - | - | 9 | 7 | - | - | - | (19),(20),(19),(20)
DO Tau | M0 | 14.3$\pm$0.5 | 0.75 | 37.0$\pm$3.7 | - | - | - | 3 | 8 | - | - | - | (6),(21),(6),(15)
J1604∗ | K3 | 17.3$\pm$0.4 | 1.0 | ¿61 | - | - | - | 12 | - | - | - | - | (22),(22),(24),(23)
PDS 70 | K7 | 16.0$\pm$0.5 | 0.01$\pm$0.07 | 50$\pm$8 | - | - | - | 4 | - | - | - | 6 | (19),(25),(19),(25)
GM Aur | K4-K5 | 14.9$\pm$0.3 | 0.3$\pm$0.3 | $\geq$63 | - | - | - | - | 2 | - | 34 | - | (26),(26),(26),(26)
Non-accreting systems |
V819 Tau | K4 | 9.5 | | | - | - | - | 1 | - | - | - | - | (27),(27)
TWA 25 | M0.5 | 12.9$\pm$1.2 | | | - | - | 25 | 14 | - | - | - | - | (6),(1)
TWA 9A | K6 | 7$\pm$3 | | | - | - | - | 1 | - | - | - | - | (6),(1)
111 $*$$*$footnotetext: RX J1604.3-2130A, elsewhere in the text, we used
J1604 for brevity. (1) Torres et al. (2006); (2) Donati et al. (2020a); (3)
Bouvier et al. (2020); (4) Alcalá et al. (2017); (5) Donati et al. (2012); (6)
Herczeg & Hillenbrand (2014); (7) Alencar & Batalha (2002); (8) Donati et al.
(2007); (9) Alencar et al. (2012); (10) Nguyen et al. (2012); (11) Donati et
al. (2008); (12) Connelley & Greene (2010); (13) Flores et al. (2019); (14)
Alecian et al. (in prep.); (15) Simon et al. (2016); (16) Alcalá et al.
(2014); (17) Frasca et al. (2017); (18) Stempels et al. (2007); (19) Pecaut &
Mamajek (2016); (20) Donati et al. (2010); (21) Kounkel et al. (2019); (22)
Dahm et al. (2012); (23) Davies (2019); (24) Preibisch & Zinnecker (1999);
(25) Thanathibodee et al. (2020); (26) Bouvier et al. (in prep.); (27) Donati
et al. (2015).
The sample of stars used in this work is composed of well-known young stars
that are part of the SPIRou Legacy Survey-SLS science program: ”Magnetic PMS
star/planet survey” of some 50 class I, II, and III stars. The
SPectropolarimètre InfraROUge (SPIRou) is a high-resolution velocimeter and
polarimeter spectrograph ($R\sim 75\,000$) covering the NIR wavelength range
$\sim 0.98-2.35\,\mu$m, corresponding to the spectral domain of the $YJHK$
bands (Donati et al., 2018). The main science goals of CFHT/SPIRou Legacy
Survey are the search for and characterization of planets around low-mass
stars, and investigating the impact of the stellar magnetic field on planet
and star formation in young systems (Donati et al., 2020b).
We aim to investigate the veiling of accreting young stars. Therefore, we
selected, among the sample of stars observed by the SLS, 13 stars reported as
accreting systems in the literature and for which we had a reasonable number
of observations in time in comparison to the stellar rotation period. Most of
these targets are classified as Class II and CTTS systems, and only V347 Aur
is a Class I target. In addition, we added the T Tauri star J1604 (RX
J1604.3-2130A) to the sample, which is not part of the SLS program, however,
its CFHT/SPIRou observations are available.
In Table 1, we show the list of young stars analyzed in this work and the
number of observations that we have for each target and each observational
period. We also list three non-accreting T Tauri stars that cover our sample’s
spectral types and are also slow rotators ($v\sin{i}<15\,$km/s). We used these
stars as templates to compute veiling and the residual profiles, as described
in the following sections.
Each CFHT/SPIRou observation consists of four sub-exposures to measure Stokes
V, taken at different orientations of the polarimeter and used to compute the
non-polarized and the circularly polarized profiles. As the focus of this work
is to analyze only the non-polarized component of the spectrum, we averaged
the four sub-exposures to increase the signal-to-noise ratio (S/N) of the
spectra obtained at each night. For the non-accreting systems, we averaged all
the observations obtaining a mean spectra that were used as templates. The
CFHT/SPIRou data were reduced, while the telluric-corrected spectra were
obtained using the data reduction system APERO, versions 0.6.131, 0.6.132, and
0.7.232 (Cook et al., 2022). The spectra were corrected for the barycentric
velocity and locally normalized to the continuum level, using a polynomial
function to fit the continuum.
## 3 Procedures to measure the veiling
Figure 1: Examples of the four spectral regions used to measure the veiling of
CI Tau. In each panel, we show two different nights, representing a small and
a high veiling estimated for this target. We show the CI Tau spectrum in black
and the residual profile (in red) obtained after subtracting the veiled
template. The V819 Tau template spectra are also displayed before (orange) and
after (blue) applying the veiling correction. The photospheric lines were
removed in the residual profiles, showing that the veiling was accurately
determined for most nights.
We computed the IR veiling of the targets following the method described by
Hartigan et al. (1989), where we compare the spectra of the target with the
spectrum of a non-accreting T Tauri star of a similar spectral type. The
Zeeman broadening of the photospheric lines can affect the veiling
measurements. Therefore, we used WTTSs as templates that should present a
similar magnetic activity as the CTTS (e.g., Johns-Krull et al., 2000). The
WTTSs also present comparable physical properties to the CTTSs, such as
chromospheric activity and surface gravity, which make WTTSs stars suitable to
measure the veiling in accreting systems. We list the templates applied to
each star in Table 2.
Before comparing the target and template spectra, we shifted and broadened the
template spectra to match the target spectra, using the radial velocities of
the targets.222 For most targets, the radial velocities were computed using
the CCF profiles generated by the SPIRou pipeline, implementing a numerical
mask corresponding to the target spectral type. For TW Hya, we computed the
radial velocity cross-correlating the target spectra with a WTTS with a
similar spectral type. and the literature $v\sin{i}$ values of the targets.
Due to the IR veiling wavelength dependence (e.g., Alcalá et al., 2021), we
measured the veiling in four different spectral regions, 10710Å-10810Å,
12840Å-12910Å, 16120Å-16220Å, and 22600Å-22690Å, which we call $r_{Y}$,
$r_{J}$, $r_{H}$, and $r_{K}$, representing the $YJHK$ veilings, respectively.
The stars RU Lup, DO Tau and, DG Tau present many emission lines along the
spectra, probably originating from the accretion shock, which is a
characteristic of high-mass accretion rate systems, which prevented us from
using the same $Y$ and $J$ spectral regions for the veiling calculations.
Then, for these targets, we used the spectral regions 10760Å-10785Å and
12400Å-12550Å to measure the $r_{Y}$ and $r_{J}$ veilings, respectively. On
some nights, the 10710Å-10810Å region of the TW Hya and V2247 Oph spectra
presented features that made veiling measurements impossible and we had to use
the 10864Å-10920Å region to measure $r_{Y}$ veiling instead.
We determined the best veiling value for each target spectrum through a
$\chi^{2}$ minimization routine. The veiling was defined as the ratio of the
continuum excess flux to the stellar photospheric flux
($r_{\lambda}=F_{\lambda_{Excess}}/F_{\lambda_{Phot}}$). Then, zero veiling
means the system has no additional excess to the stellar photosphere. In the
spectral range of our sample (K2 to M3), the choice of template does not
interfere much in the computed veiling, as shown by Folha & Emerson (1999),
since the difference between templates of different spectral types is small
and produces an almost null relative veiling, within the uncertainties. We
also measured the systematic veiling between our templates, which corresponds
to the average veiling resulting from the computed veiling comparing each
template with the other. We found it to be $r_{Y}=0.098\pm 0.068$,
$r_{J}=0.02\pm 0.02$, $r_{H}=0.05\pm 0.02$ and, $r_{K}=0.04\pm 0.02$, which
should only affect the veiling determination of stars with very small or no
veiling.
In Fig. 1, we present the results for the four photospheric regions used to
measure the veiling for CI Tau from two nights, representing spectra with
smaller and higher veiling values. We show the target spectrum, and the
unveiled and veiled template spectra. We also computed the residual profile,
which is the target’s spectrum subtracted from the veiled template. Most of
the residual profiles show almost no features at the location of photospheric
lines, indicating that the veiling measurements were correctly determined for
most nights, and that the photospheric lines were correctly removed. However,
some of the spectra were quite noisy and this affected the veiling
determinations. The veiling error obtained for each night, written in the
plot, comes from a Chi-square minimization process, where we compare the
target with the template spectra. Besides taking into account the noise of the
target and template spectra333 We used as the spectral noise, the standard
deviation measured in the continuum of each night, in the spectral region used
to measure the veiling. to compute the veiling, there are other errors, such
as those associated with the normalization processes that we did not consider
in estimating the veiling error.
## 4 Results
We present in Fig. 2 the NIR average veiling over the observation nights,
measured for all the four regions, referred to as $r_{Y}$, $r_{J}$, $r_{H}$,
and $r_{K}$. For readability, we split the targets into two groups; systems
with $r_{K}$ higher and smaller than 1. We also show the averaged veiling
obtained for each target in Table 2. The veiling values increase from the $Y$
to the $K$ band for most of the targets, similar to the results found in
previous works (e.g., McClure et al., 2013; Sousa et al., 2021; Alcalá et al.,
2021). Besides this general result, the average veiling for some individual
targets remains the same from the $Y$ to $K$ band. For example, the veiling
measured for V2247 Oph. However, this target does not present significant
veiling in any band, probably due to the fact that it is a more evolved
system, where the accretion and the dust in the inner disk are too faint to be
detected. Furthermore, this M-type star makes detecting excess in the IR
difficult due to the low contrast between the stellar photosphere and the
inner disk emission (e.g., Ercolano et al., 2009). Another example is CI Tau,
where the average veiling decreases from the $Y$ to the $H$ band, followed by
an increase to the $K$ band. In that case, each band’s veiling variability is
high and the difference in the $Y$ to $H$ average veiling is smaller than the
standard deviation.
Figure 2: Average NIR veiling (left) and the veiling variability diagnostic
(right) measured in different wavelength regions. The top and bottom panels
show the targets with $r_{K}$ higher and lower than 1, respectively. The
veiling increases from the $Y$ to the $K$ band for most of the targets. The
error bars in the left panel represent the standard deviation of the average
veiling over all the observation nights. In this figure and the following
figures, the color and symbol codes in the panels identify each target.
Figure 3: Veiling variability diagnostic as a function of the average
veiling. The rms value refers to the root-mean-square of the veiling
variability and $\sigma$ is the average error on the veiling measurements.
Each panel shows the veiling $r_{\lambda}$ measured in a different band
($YJHK$). Systems with higher veiling also present higher veiling variability.
Due to the small number of photospheric lines and the lower S/N values in the
$Y$ region of the spectra, the veiling in this region was not as well
determined, compared to the other bands. For some nights or targets, the
veiling is even slightly negative, which has no physical meaning – in such
cases, we can assume that the system in this region has zero veiling.
We also checked the variability of the veiling measured in each band,
computing for each target the RMS of the $YJHK$ veilings, using all the
observed nights. To characterize the variability over the noise, we subtracted
the average error of the veiling ($\sigma$) from the RMS. Then, we computed
$S=\sqrt{rms^{2}-\sigma^{2}}$, as a veiling variability diagnostic, following
Cody et al. (2014). We show the results in Figs. 2 and 3 as a function of the
band and veiling measured, respectively. Most of the targets present
variability above the error level. The veiling in the $H$ band appears to be
less variable than the other bands, while the $K$ band presents the highest
variability, driven by the systems with the highest veilings, which are also
the most variable ones (see Fig. 3).
The average veiling measured in RU Lup is relatively high compared to the
other targets, mainly in the $K$ band. These high average veilings are
primarily due to the veiling measured in the 2021a period of observations (see
Sect. 4.3). In that period, RU Lup also presented an increase in the
photometric AAVSO $V$ band brightness (about $0.3$ mag smaller than the next
observational period, where the veiling starts to go down). Therefore, the
average veiling of RU Lup probably is overestimated and does not represent a
quiescent value.
### 4.1 Veiling compared to inner disk diagnostics
Figure 4: Average NIR veiling ($r_{\lambda}$) as a function of spectral
energy distribution slope between 2 and 8 $\mu$m ($\alpha_{2\\_8}$), which we
used as the NIR disk emission diagnostic. The solid line is the linear fit to
the data, and the correspondent slope is written in each panel. The NIR
veiling seems to scale with the SED slope.
The emission from the inner disk is claimed to contribute to the veiling. In
such a case, we would expect a correlation between the veiling and other inner
disk emission diagnostics from the photometric data.
The slope of the spectral energy distribution (SED) is often used as a disk
emission diagnostic (e.g., Lada et al., 2006; Muzerolle et al., 2010; Teixeira
et al., 2012). Using photometric data from different surveys, such as SDSS
(Gunn et al., 1998), Gaia DR2 (Gaia Collaboration et al., 2018), 2MASS, WISE
(Wright et al., 2010), Spitzer (Fazio et al., 2004; Rieke et al., 2004),
Herschel (PACS), Akari (IRC and FIS), we constructed the SED of the targets
and measured the slope of the SED between 2 and 8 $\mu$m ($\alpha_{2-8}$),
which is the spectral range that indicates significant inner disk emission.
The $\alpha_{2-8}$ slope is smaller (more negative) for systems with less or
no inner disk emission, and is higher (and even positive) for systems that
present inner disk emission excess (e.g., Lada et al., 2006; Muzerolle et al.,
2010). We list the $\alpha_{2-8}$ slope computed for the targets in Table 2.
We show in Fig. 4 the NIR veiling in the four spectral regions ($YJHK$),
averaged over all the observation nights, as a function of $\alpha_{2-8}$. The
veiling presents a clear linear correlation with the SED slope, mainly from
the $J$ to $K$ band. The $Y$ band veiling values are still correlated to the
SED slope but less than the other bands, probably due to the contribution from
the inner disk emission becoming more important at longer wavelength. The
$W_{1}-W_{2}$ ([3.4]-[4.6]) color index (not shown here) comes from the WISE
telescope (Wright et al., 2010) is also correlated with the NIR veiling,
presenting similar results as $\alpha_{2-8}$.
Figure 5: Comparison between the color excess computed using the average NIR
veiling and the 2MASS photometry. left: $(H-K_{\rm s})_{excess}$, right:
$(J-K_{\rm s})_{excess}$. See the text for the color excess definition. The
dashed line represents a slope equal to 1. The NIR color excesses computed
using the average veiling and the 2MASS magnitudes agree for most targets.
We could expect that the NIR excess computed using the veiling would scale
with the emission excess from NIR photometric data, which is often used as an
inner disk emission indicator (Hillenbrand et al., 1998; Rebull, 2001; Rebull
et al., 2002). We computed the $(H-K_{\rm s})_{excess}$, using the observed
$H-K_{\rm s}$ color from 2MASS, corrected for extinction, using the $A_{V}$
quoted in Table 1 and the SVO Filter (Rodrigo et al., 2012; Rodrigo & Solano,
2020) $A_{\lambda}/A_{V}$ relations to obtain the $A_{H}$ and $A_{K}$
extinctions. Then, we compared this dereddened color excess to the intrinsic
color $(H-K_{\rm s})_{o}$ expected for an object with the same spectral type
(Pecaut & Mamajek, 2013). The color excess is $(H-K_{\rm
s})_{excess}=(H-K_{\rm s})_{obs,dred}-(H-K_{\rm s})_{o}$. We also relate the
color excess and the excess flux, leading to $(H-K_{\rm
s})_{excess}=-2.5\log{[(1+r_{H})/(1+r_{K})]}$, where we used the veiling
definition as $r_{\lambda}=F_{\lambda_{excess}}/F_{\lambda_{Phot}}$. Then we
can directly compare the color excess computed using the veiling and using the
photometric measurements. We compare the two sides of this equation in Fig. 5,
which shows a linear tendency and similar values considering the error of the
measurements. Performing similar procedures, we computed the $(J-K_{\rm
s})_{excess}$, and the results are also presented in Fig. 5. The color excess
computed using the 2MASS photometry is dependent on the $A_{v}$ of the
systems, and the targets V347 Aur and CI Tau present discrepant $A_{v}$ values
in the literature. CI Tau presents $A_{v}=0.65\,$mag (Donati et al., 2020a)
and $A_{V}=1.90\,$mag (Herczeg & Hillenbrand, 2014), while the $(J-K_{\rm
s})_{excess}$ computed using the largest $A_{V}$ better agrees with the color
excess calculated using veiling, the $(H-K_{\rm s})_{excess}$, and the mass
accretion rate computed in the next section seem to be in better agreement
with $A_{V}=0.65\,$mag; thus, we used this extinction value in the paper. The
$A_{V}$ range of V347 Aur is even larger, with values from $2.2$ to $7.5\,$mag
(e.g., Dahm & Hillenbrand, 2020). The $A_{V}=3.4\,$mag computed using NIR
colors by Connelley & Greene (2010) seems to better represent the value
obtained from the veiling calculations.
The 2MASS photometric magnitudes used for this analysis were obtained years
before our observations. However, we do not expect a significant change in the
NIR magnitudes over the next few years, aside from the daily timescale
variation (e.g., Carpenter et al., 2001). It is likely that RU Lup will stand
as an exception, as the average veiling was measured in a non-quiescent
period. Then, the color excess computed using the veiling is higher than that
obtained using the 2MASS magnitudes.
All these relations between the NIR veiling and the inner disk emission
diagnostics show that a higher veiled system also presents higher inner disk
emission, which is expected if the veiling has a contribution from the inner
disk. However, to draw this conclusion, we assumed that these inner disk
indicators, such as the slope of the spectral energy distribution, are
suitable inner disk diagnostics. Kidder et al. (2021) showed that some of the
targets classified as Class III using these disk indicators still show some
inner disk emission based on the $K$ band excess. They checked the emission
excess of V819 Tau, which we used as a template to compute the veiling, but
the $H$ and $K$ excesses found were very small, similar to the systematic
veiling we obtained in Sect. 3. The relation between veiling and inner disk
emission is clear for the $K$ band and less for other bands, probably due to
the influence of another additional continuum source for the other spectral
regions and/or a smaller contribution from the inner disk to these shorter
wavelengths.
### 4.2 Veiling compared to accretion diagnostics
Figure 6: Accretion diagnostics as a function of average NIR veiling. From
left to right: Average equivalent width corrected from the veiling of
${\rm{He\textsc{I}}}$ (10830 Å), ${\rm Pa}\beta$ , and ${\rm Br}\gamma$ lines.
We show a connection between the system’s accretion diagnostic and the NIR
veiling.
We know that CTTSs are still accreting gas from the disk and accreting systems
typically present strong and variable emission lines that form in the
accretion funnel or in the disk wind (e.g., Muzerolle et al., 1998; White &
Basri, 2003; Edwards et al., 2003; Kwan & Fischer, 2011; Alencar et al.,
2012). The CFHT/SPIRou wavelength range includes some emission lines from
hydrogen and helium, such as ${\rm Pa}\beta$ and ${\rm Br}\gamma$ as well as
the ${\rm{He\textsc{I}}}$ (10830 Å) triplet; in particular, the latter is very
sensitive to accretion and ejection processes (e.g., Kwan et al., 2007; Sousa
et al., 2021). The dynamics of the circumstellar lines for this sample of
stars will be analyzed in an accompanying paper (Sousa et al. in prep.).
We measured the equivalent width of the circumstellar lines, and the average
over all observing nights is listed in Table 2. First, we used the equivalent
width as an accretion diagnostic (Alcalá et al., 2017), as systems that
present larger equivalent widths are supposed to present higher mass accretion
rates as well.
Table 2: Parameters of the targets derived in this work.
Star | Std | $\mathrm{v_{rad}}$ | $\mathrm{r_{Y}}$ | $\mathrm{r_{J}}$ | $\mathrm{r_{H}}$ | $\mathrm{r_{K}}$ | $\mathrm{EW}_{\mathrm{HeI}\ }$aa$a$We used the convention of positive equivalent width for emission lines, and negative values for absorption lines. | $\mathrm{EW}_{\mathrm{Pa\beta}}$aa$a$We used the convention of positive equivalent width for emission lines, and negative values for absorption lines. | $\mathrm{EW}_{\mathrm{Br\gamma}}$aa$a$We used the convention of positive equivalent width for emission lines, and negative values for absorption lines. | $\mathrm{\dot{M}_{\mathrm{Pa\beta}}}$ | $\mathrm{\dot{M}_{\mathrm{Br\gamma}}}$ | $\alpha_{2\\_8}$bb$b$ Slope of the spectral energy distribution measured between 2 and 8 $\mu$m.
---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | | | | | | $\times 10^{-8}$ | $\times 10^{-8}$ |
| | ($\mathrm{km}\ \mathrm{s}^{-1})$ | | | | | $(\mathring{\mathrm{A}})$ | $(\mathring{\mathrm{A}})$ | $(\mathring{\mathrm{A}})$ | $(M_{\odot}\mathrm{yr}^{-1})$ | $(M_{\odot}\mathrm{yr}^{-1})$ |
CI Tau | V819TAU | 16.3 $\pm$ 0.5 | 0.53 $\pm$ 0.28 | 0.49 $\pm$ 0.17 | 0.39 $\pm$ 0.09 | 0.92 $\pm$ 0.23 | 9.5 $\pm$ 3.2 | 14.0 $\pm$ 2.6 | 7.9 $\pm$ 1.8 | 2.2 | 4.2 | -0.84
DoAr 44 | V819TAU | -6.1 $\pm$ 0.7 | 0.20 $\pm$ 0.09 | 0.48 $\pm$ 0.10 | 0.60 $\pm$ 0.06 | 1.28 $\pm$ 0.05 | 5.4 $\pm$ 1.4 | 9.1 $\pm$ 0.8 | 3.8 $\pm$ 0.5 | 1.7 | 1.5 | -1.09
GQ Lup | TWA9A | -3.0 $\pm$ 0.3 | 0.19 $\pm$ 0.09 | 0.58 $\pm$ 0.11 | 0.78 $\pm$ 0.14 | 1.87 $\pm$ 0.26 | 0.9 $\pm$ 2.0 | 6.3 $\pm$ 1.6 | 2.0 $\pm$ 0.7 | 2.5 | 1.9 | -0.89
TW Hya | TWA25 | 12.4 $\pm$ 0.1 | 0.08 $\pm$ 0.07 | 0.09 $\pm$ 0.06 | 0.10 $\pm$ 0.05 | 0.23 $\pm$ 0.11 | 5.1 $\pm$ 4.9 | 15.9 $\pm$ 6.2 | 6.8 $\pm$ 2.6 | 0.6 | 0.3 | -1.92
V2129 Oph | V819TAU | -7.1 $\pm$ 0.6 | 0.05 $\pm$ 0.10 | 0.11 $\pm$ 0.06 | 0.25 $\pm$ 0.07 | 0.76 $\pm$ 0.16 | 0.9 $\pm$ 1.1 | 1.0 $\pm$ 0.7 | 0.5 $\pm$ 0.3 | 0.2 | 0.1 | -1.62
BP Tau | V819TAU | 15.2 $\pm$ 0.6 | 0.22 $\pm$ 0.09 | 0.36 $\pm$ 0.06 | 0.36 $\pm$ 0.09 | 0.78 $\pm$ 0.10 | 3.9 $\pm$ 1.4 | 7.7 $\pm$ 1.4 | 3.2 $\pm$ 0.8 | 1.3 | 1.1 | -1.69
DG Tau | TWA9A | 15.1 $\pm$ 3.4 | 0.29 $\pm$ 0.22 | 0.80 $\pm$ 0.20 | 1.09 $\pm$ 0.16 | 2.90 $\pm$ 0.51 | 11.7 $\pm$ 2.7 | 16.3 $\pm$ 2.1 | 6.8 $\pm$ 1.0 | 6.4 | 7.6 | -0.26
RU Lup | TWA9A | -1.3 $\pm$ 1.4 | 1.14 $\pm$ 0.51 | 1.70 $\pm$ 0.65 | 1.65 $\pm$ 0.44 | 6.27 $\pm$ 2.97 | 23.2 $\pm$ 4.0 | 30.1 $\pm$ 3.0 | 12.2 $\pm$ 1.5 | 7.8 | 11.9 | -0.21
V2247 Oph | TWA25 | -5.8 $\pm$ 0.5 | -0.02 $\pm$ 0.07 | -0.06 $\pm$ 0.01 | 0.01 $\pm$ 0.03 | -0.03 $\pm$ 0.02 | 0.0 $\pm$ 0.2 | 0.1 $\pm$ 0.1 | -0.1 $\pm$ 0.2 | 0.009 | - | -2.54
DO Tau | TWA25 | 16.1 $\pm$ 1.0 | 0.50 $\pm$ 0.40 | 0.78 $\pm$ 0.41 | 1.07 $\pm$ 0.32 | 2.33 $\pm$ 0.61 | 1.0 $\pm$ 1.8 | 12.7 $\pm$ 3.9 | 4.0 $\pm$ 1.9 | 2.3 | 3.4 | -0.54
V347 Aur | TWA25 | 8.1 $\pm$ 0.6 | 0.31 $\pm$ 0.19 | 0.46 $\pm$ 0.26 | 0.37 $\pm$ 0.15 | 1.12 $\pm$ 0.31 | -0.9 $\pm$ 1.1 | 6.7 $\pm$ 2.1 | 2.9 $\pm$ 1.0 | 7.1 | 7.3 | -0.62
J1604 | V819TAU | -5.8 $\pm$ 0.8 | -0.14 $\pm$ 0.11 | 0.20 $\pm$ 0.07 | 0.22 $\pm$ 0.13 | 0.81 $\pm$ 0.23 | -2.6 $\pm$ 1.8 | -0.1 $\pm$ 0.2 | 0.3 $\pm$ 0.1 | - | 0.013 | -2.81
PDS 70 | TWA9A | 5.0 $\pm$ 0.4 | 0.02 $\pm$ 0.11 | 0.18 $\pm$ 0.04 | 0.19 $\pm$ 0.07 | 0.34 $\pm$ 0.08 | -0.7 $\pm$ 0.9 | 0.2 $\pm$ 0.2 | 0.1 $\pm$ 0.2 | 0.007 | 0.003 | -
GM Aur | TWA9A | 14.5 $\pm$ 0.3 | -0.00 $\pm$ 0.06 | 0.13 $\pm$ 0.09 | 0.11 $\pm$ 0.06 | 0.31 $\pm$ 0.10 | 3.4 $\pm$ 2.8 | 10.2 $\pm$ 3.2 | 4.7 $\pm$ 1.9 | 1.3 | 1.0 | -
444
We corrected the equivalent width for the veiling as
$EW=EW_{measured}(r_{\lambda}+1)$, where the $r_{\lambda}$ represents the
veiling computed close to each emission line. In Fig. 6, we show the veiling
as a function of the veiling corrected equivalent width of the circumstellar
emission lines. We see a clear relationship between the NIR veiling and the
accretion diagnostics. It means that higher mass-accretion rate systems also
present a higher degree of veiling; this is a similar result to that found by
Folha & Emerson (1999), demonstrating that although the veiling shows a
contribution from the inner disk emission, it also suggests a connection with
the accretion process.
We do not have photometric data simultaneous with our spectra to accurately
compute the mass accretion rates using the equivalent width of the emission
lines. However, most of our systems’ NIR magnitudes are relatively long-term
stable. We used the 2MASS $J$ and $K$ magnitudes to estimate the continuum
flux and then calculate the mass accretion rate using the ${\rm Pa}\beta$ and
${\rm Br}\gamma$ lines, respectively. The star V347 Aur is known to present
long-term photometric variations (e.g., Dahm & Hillenbrand, 2020), and we did
not compute the mass accretion rate of this target.
We followed the procedures described by Gullbring et al. (1998) to compute the
line flux and luminosity. The stellar parameters used are listed in Table 3,
and we used the stellar distance from the Gaia collaboration (Gaia
Collaboration et al., 2021). We dereddened the 2MASS magnitudes using the same
method described in the previous section. To compute the accretion luminosity,
we used the fits proposed by Alcalá et al. (2017), which show the relation
between the line and accretion luminosities. Then, we determined the accretion
rate setting the system inner radius as $5R_{\ast}$ (Gullbring et al., 1998).
In Table 2, we show the individual mass accretion rates computed using the
${\rm Pa}\beta$ and ${\rm Br}\gamma$ lines. In Fig. 7, we show the average
mass accretion rate as a function of the $Y$ to $K$ band veiling. Once again,
we can connect the highest accreting system with the highest NIR veiling
computed in the four bands.
Figure 7: Average mass accretion rate as a function of average NIR veiling.
The average mass accretion was computed using the line fluxes of ${\rm
Pa}\beta$ and ${\rm Br}\gamma$. The error bar is the standard deviation
between the two measurements. We see an association between the veiling and
the mass accretion rate. Table 3: Stellar parameters
Star | M∗ | R∗ | ref
---|---|---|---
| (M☉) | (R☉) |
CI Tau | 0.90 | 2.0 | (1)
DoAr 44 | 1.20 | 2.0 | (2)
GQ Lup | 0.86 | 2.26 | (3)
TW Hya | 0.80 | 1.1 | (4)
V2129 Oph | 1.35 | 2.1 | (5)
BP Tau | 0.70 | 1.99 | (6)
DG Tau | 0.65 | 2.05 | (6)
RU Lup | 1.15 | 2.39 | (7)
V2247 Oph | 0.35 | 2.0 | (8)
DO Tau | 0.60 | 2.0 | (9)
J1604 | 1.24 | 1.4 | (10)
PDS 70 | 0.76 | 1.26 | (11)
GM Aur | 0.95 | 1.71 | (12)
(1) Donati et al. (2020a); (2) Bouvier et al. (2020); (3) Alcalá et al.
(2017); (4) Donati et al. (2011); (5) Alencar et al. (2012); (6) Johns-Krull
(2007); (7) Alcalá et al. (2014); (8) Donati et al. (2010); (9) Ricci et al.
(2010); (10) Sicilia-Aguilar et al. (2020); (11) Müller et al. (2018); (12)
Bouvier et. al. (in prep.)
### 4.3 Veiling night-to-night variability
In this study, we have access to observations obtained on different nights and
sometimes different observational periods for our sample of stars. This
allowed us to analyze the night-to-night veiling variation and a possible
long-term veiling variability on a timescale of two years. In Fig. 8, we show
the veiling measured as a function of the observation dates. We used the
modified Lomb-Scargle periodogram (Horne & Baliunas, 1986) to study a possible
periodicity of the veiling variations. We performed the periodogram analysis
of the veiling in two ways: using all the observed nights and computing one
periodogram per observational period. We searched for periods from 2 to 15
days or limited the search to the number of observed days, when the target was
observed for less than 15 days. However, we did not find a significant
periodic signal for most systems. We also phased the veiling using the known
stellar rotation period of the targets, and again, most of the systems do not
seem to vary in phase with the stellar rotation.
In Sect. 4, we used the veiling error as a threshold of the veiling
variability, using the RMS as a variability diagnostic, see Fig. 2. Then, any
point above this limit represents a true variability of the veiling; below
this threshold, the variability is at the same level as the errors associated
with the veiling. We can see that the veiling RMS presents values above the
threshold of veiling variability for most of the targets and veiling bands.
Only V2247 Oph shows most values below the threshold limit, while a few bands
of TW Hya and DoAr 44 present most of the RMS values below the threshold.
Then, for V2247 Oph, we cannot attribute a variability to the veiling, which
is consistent with this target presenting a small veiling and weak level of
accretion and inner disk emission diagnostics.
We know the veiling is generally variable and we can trust the detection of
veiling variability for most targets. Figure 8 shows that for all the veiling
variable targets, the veiling varies on a timescale of at least one day and
the veiling computed in the four spectral regions presents the same
variability timescale. These results show that whatever region the IR veiling
comes from, this region is dynamic and its flux changes on a timescale of
days.
Figure 8: Night-by-night veiling values measured in four different spectral
regions. We show each observational period per target in an individual panel.
A missing veiling point in a specific band means that the spectral regions
were subject to effects that prevented the veiling from being measured. In
addition to the variability in terms of veiling, it is also not clearly
periodic, at least on the timescale of stellar rotation. For more details, see
text.
We also investigated veiling variability on a timescale of months to a few
years; a change in the veiling in that timescale can have a different origin
from the day-scale veiling variability, discussed above. The latter can be
associated with the dynamic of the system’s rotation, while the possible long-
term veiling variability reflects a change in the system’s accretion and/or
inner disk conditions. In Figure 9, we present the averaged veiling measured
at each observational period. This plot displays nine systems that were
observed in more than one observational season. Most targets do not present a
significant difference in the veiling along the observational period. The $K$
band veiling of CI Tau and the $YJH$ veiling of DO Tau show a possible small
change in this timescale. Then, we conclude that the veiling variability on a
timescale of months-to-years is on the same order of magnitude as the day-to-
day variability. RU Lup is the unique target with an evident change in the
veiling along the observational periods in the four bands, but much more
pronounced in the $K$ band, along with a high standard deviation. We associate
this change in the veiling with an occasional high accretion episode that
occurred in 2021a, and despite the veiling still being high in 2021b, it seems
to start to diminish later on. In 2022a, it is even smaller, but we have only
one observation to serve as the basis for this assumption. The circumstellar
emission lines’ equivalent widths corroborate with this assumption, as they
increase in 2021a and start to decrease in the subsequent observation periods,
similarly to the veiling. In contrast, the average veiling is stable, at least
for most of the stars we analyzed, except for very high accreting systems,
such as RU Lup, which can present episodic high veiling. Furthermore, in the
same observational period, some targets show a few episodes of an increase in
veiling, such as V347 Aur (Fig. 8); however, the average veiling values are
still sustained.
Figure 9: Average of the NIR veiling computed at each observational period,
with the error bar as the standard deviation of the computed mean veiling. We
merged the measured veiling in 2020a and 2020b of the stars V2129 Oph and GQ
Lup, as they are successive observations. The mean NIR veiling seems to be
stable for most of the targets for a few months or years.
## 5 Discussion
The dependence of veiling on wavelength is ubiquitous from the UV to the NIR
range. The veiling in the optical domain decreases from the blue to the red
part of the spectra, which is an effect of the decrease of the accretion spot
continuum contribution (e.g., Calvet & Gullbring, 1998); also, the veiling
also does not vary for some wavelength ranges (e.g., Basri & Batalha, 1990).
On the other hand, in the IR range, the veiling increases with wavelength, as
seen in Figs. 2 and 8, which is in agreement with similar results in the
literature (e.g., Fischer et al., 2011; Alcalá et al., 2021)
Figure 10: $YJH$ veiling as a function of the $K$ band veiling. Each veiling
value corresponds to the mean of all the observed nights, and the error bar is
its standard deviation. The solid line is the linear fit to the data, and the
fitted line’s slope is written in each panel. The correlation between the $K$
band and the $YJH$ bands veilings increases from the $Y$ to the $H$ band.
The average veiling value for the entire sample of CTTS is
$\left<r_{Y}\right>=0.2\pm 0.3$, $\left<r_{J}\right>=0.4\pm 0.4$,
$\left<r_{H}\right>=0.5\pm 0.5$, and $\left<r_{K}\right>=1.4\pm 1.6$. We note
that in the $Y$ band, the average veiling is the lowest. Over these
wavelengths from $Y$ to $K$, the veiling can have contributions from different
sources. For example, we expect the veiling in the $K$ band to present more
contributions from the inner disk than in the $Y$ band. In Fig. 10, we show
the $YJH$ veiling as a function of the $K$ band veiling. We can see that the
$J$ and $H$ veilings seem to increase as the $K$ band veiling increases;
however, there is smaller correlation with the $Y$ band veiling. These results
are supported by the analysis of the correlation of the veiling samples and
the linear fit showed in the Fig. 10. We computed the linear correlation
coefficient (r) between two samples, where r=1 represents a perfect
correlation and when r=0 there is no correlation. The $Y$ and $K$ band
veilings present a correlation coefficient of 0.87, while the coefficient of
the $J$ and $H$ and the $K$ band are 0.98 and 0.96, respectively. Similar
results were found by Cieza et al. (2005), where they compare the excess in
the $J$ and $H$ bands with the $K$ band excess, showing that both presented a
linear correlation with the $K$ band excess, and this was explained as due to
the $JHK$ excess arising from the same region.
The NIR veiling should be the result of a combination of physical processes.
Alcalá et al. (2021) computed the NIR veiling at several wavelengths for a
sample of very high-mass accretion rate systems, including DG Tau (included in
our sample). These authors fitted the veiling as a function of wavelength
using a blackbody function and found temperatures compatible with the presence
of dust in the inner disk. However, the temperature was too high ($>2000\,K$)
for the dust to survive in a few of their cases. They also argued that the
veiling should have a contribution from the hot gas inside the disk
sublimation radius, and similar results were found by Fischer et al. (2011).
To investigate this proposition, we looked at the CO bandhead at 2.3$\mu$m of
the CFHT/SPIRou data. This band, when in emission, is expected to form in the
hot gas in the inner disk. Using the $K$ band veiling, we veiled the template
and removed the photospheric lines of the CO bandhead to obtain the residual
CO profiles. Most targets do not present clear signs of CO emission, showing
that this band is strictly photospheric. However, a few residual profiles of
V347 Aur, which is a Class I object, along with DO Tau and RU Lup, which are
strong accretors, present CO emission in some observations, indicating the
presence of hot gas in the inner disk. In particular, RU Lup presents these
hints of CO emission in the observational period when the veiling was high,
and the system probably ensued a high episodic accretion. A further analysis
of the CO bandhead is beyond the scope of this paper and it will be carried
out in a dedicated paper exploring the significance of these CO emissions.
In the previous section, we show that the NIR veiling, mainly $r_{K}$,
presents a good correlation with the inner disk emission diagnostics obtained
from the NIR photometric data and SED fit, demonstrating that the NIR veiling
has an important contribution from the inner disk. However, we also see a
correlation between veiling and accretion diagnostics. A high accretion-rate
system presents larger veiling values in the IR. This shows that high-mass
accretion rate systems should feature higher inner disk heating, thus higher
temperatures and stronger inner disk emission excess as a consequence.
Espaillat et al. (2022) fit most of the continuum spectra from NUV to NIR of
the accreting star CVSO 109A quite well, using a combination of emission from
the accretion shock (multiple funnel flow model) on the stellar surface and
emission from the irradiated inner edge of the dusty disk. However, the inner
disk and accretion shock model do not adequately reproduce the continuum
excess in the $Y$ to $J$ band.
Indeed, while the veiling in the $J$ to $K$ band seems to point to a
significant contribution on the part of the inner disk emission excess, the
$Y$ band veiling origin is still unknown. Dodin & Lamzin (2013) predicted
significant veiling in the $Y$ band from the accretion spot. They argued that
the accretion spot (continuum emission and emission lines formed in the
accretion shock) could account for optical and near IR veiling up to the $J$
band. Unfortunately, our $Y$ band veiling was not as well computed as for the
other bands due to several issues in this spectral region and given the
photospheric lines are not quite so prominent. Despite these obstacles, the Y
band veiling agrees on a smaller scale with both accretion and inner disk
diagnostics.
Figure 11: Average NIR veiling as a function of the system’s inclination with
respect to our line of sight. The solid line is the data’s linear fit, and the
fitted line’s slope is given in each panel. We do not consider the discrepant
targets (TW Hya and V2247 Oph) to fit the data. The NIR veilings tend to be
anti-correlated with the system’s inclination, see text.
We checked if the inclination of the system has any impact on the measured
veiling values. In Fig. 11, we show the NIR veiling as a function of the
inclination of the system with respect to our line of sight, listed in Table
1. In addition, two discrepant systems (TW Hya and V2247 Oph), we can see an
anti-correlation between veiling and the system’s inclination. This anti-
correlation is not pronounced, as we can see in the linear fit slope, due to
the spread of points (the correlation coefficients between the inclination and
the $YJHK$ veilings are -0.64, -0.76, -0.80, and -0.70, respectively), but the
decreasing tendency of veiling with inclination is clear. We also advise that
the inclinations used for DG Tau and DO Tau are the outer disk inclination,
while the disk and the stellar inclination are not necessarily the same (Bohn
et al., 2022). If confirmed, this anti-correlation can be due to a geometric
effect: the more the system is inclined, the less we see (from the inner disk
edge) where the nIR veiling is supposed to arise. The two targets, TW Hya and
V2247 Oph, which do not seem to follow this tendency, do not have dust in the
inner disk any longer and they are known to have gaps or holes in their inner
disks (Calvet et al., 2002; Pontoppidan et al., 2008). In that case,
independently of the system’s inclination, we would not expect to detect IR
veiling, assuming that the IR veiling is due to dust emission in the inner
disk.
## 6 Conclusion
In this work, we analyze the NIR veiling computed using high-resolution data
from CFHT/SPIRou of a sample of 14 low-mass young stars. We found the veiling
to increase from the $Y$ to the $K$ band, as a result of the increase of the
emission contribution from the inner disk as a function of wavelength.
The veiling correlates with other photometric inner disk diagnostics, such as
color excess and the slope of the spectral energy distribution, mainly in the
$JHK$ band, providing further evidence that the NIR veiling arises from hot
dust in the inner disk. We also found a linear correlation between veiling and
the accretion properties of the system. This shows that accretion contributes
to inner disk heating and, consequently, to the inner disk emission excess.
This effect is enhanced in high-mass accretion rate systems that also present
a denser inner disk and higher inner disk emission (e.g., Sullivan & Kraus,
2022).
We analyzed the NIR veiling variability through the modified Lomb-Scargle
periodogram and we did not find any significant periodic signal in the four
bands in timescales typical of stellar rotation ($<15\,$days), which also
suggests the veiling comes from the dust emission in the inner disk. However,
we show that the veiling is variable for most targets on a timescale of at
least one day. Besides the night-by-night veiling variability, the mean NIR
veiling per season appears to be mostly stable, for most targets and on
timescales of several months to years.
###### Acknowledgements.
We thank the referee for the suggestions that helped to clarify this paper. We
want to thank Claire Moutou, Sylvie Cabrit, Nicolas Grosso, and Konstantin
Grankin for carefully reading the manuscript and giving suggestions to improve
the paper. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 742095; SPIDI: Star-Planets-Inner Disk-
Interactions; http://www.spidi-eu.org and grant agreement No. 740651
NewWorlds). We acknowledge financial support from CNPq, CAPES and Fapemig. We
acknowledge funding from the French National Research Agency (ANR) under
contract number ANR-18-CE31-0019 (SPlaSH). This research has made use of the
SVO Filter Profile Service (http://svo2.cab.inta-csic.es/theory/fps/)
supported from the Spanish MINECO through grant AYA2017-84089. The authors
wish to recognize and acknowledge the very significant cultural role and
reverence that the summit of MaunaKea has always had within the indigenous
Hawaiian community. We are most fortunate to have the opportunity to conduct
observations from this mountain. We acknowledge with thanks the variable star
observations from the AAVSO International Database contributed by observers
worldwide and used in this research.
## References
* Alcalá et al. (2021) Alcalá, J. M., Gangi, M., Biazzo, K., et al. 2021, A&A, 652, A72
* Alcalá et al. (2017) Alcalá, J. M., Manara, C. F., Natta, A., et al. 2017, A&A, 600, A20
* Alcalá et al. (2014) Alcalá, J. M., Natta, A., Manara, C. F., et al. 2014, A&A, 561, A2
* Alencar & Batalha (2002) Alencar, S. H. P. & Batalha, C. 2002, ApJ, 571, 378
* Alencar et al. (2012) Alencar, S. H. P., Bouvier, J., Walter, F. M., et al. 2012, A&A, 541, A116
* Antoniucci et al. (2017) Antoniucci, S., Nisini, B., Biazzo, K., et al. 2017, A&A, 606, A48
* Basri & Batalha (1990) Basri, G. & Batalha, C. 1990, ApJ, 363, 654
* Bohn et al. (2022) Bohn, A. J., Benisty, M., Perraut, K., et al. 2022, A&A, 658, A183
* Bouvier et al. (2020) Bouvier, J., Alecian, E., Alencar, S. H. P., et al. 2020, A&A, 643, A99
* Calvet et al. (2002) Calvet, N., D’Alessio, P., Hartmann, L., et al. 2002, ApJ, 568, 1008
* Calvet & Gullbring (1998) Calvet, N. & Gullbring, E. 1998, ApJ, 509, 802
* Calvet et al. (1997) Calvet, N., Hartmann, L., & Strom, S. E. 1997, ApJ, 481, 912
* Carpenter et al. (2001) Carpenter, J. M., Hillenbrand, L. A., & Skrutskie, M. F. 2001, AJ, 121, 3160
* Chiang & Goldreich (1997) Chiang, E. I. & Goldreich, P. 1997, ApJ, 490, 368
* Chiang & Goldreich (1999) Chiang, E. I. & Goldreich, P. 1999, ApJ, 519, 279
* Cieza et al. (2005) Cieza, L. A., Kessler-Silacci, J. E., Jaffe, D. T., Harvey, P. M., & Evans, Neal J., I. 2005, ApJ, 635, 422
* Cody et al. (2014) Cody, A. M., Stauffer, J., Baglin, A., et al. 2014, AJ, 147, 82
* Connelley & Greene (2010) Connelley, M. S. & Greene, T. P. 2010, AJ, 140, 1214
* Cook et al. (2022) Cook, N. J., Artigau, É., Doyon, R., et al. 2022, arXiv e-prints, arXiv:2211.01358
* Dahm & Hillenbrand (2020) Dahm, S. E. & Hillenbrand, L. A. 2020, AJ, 160, 278
* Dahm et al. (2012) Dahm, S. E., Slesnick, C. L., & White, R. J. 2012, ApJ, 745, 56
* Davies (2019) Davies, C. L. 2019, MNRAS, 484, 1926
* Dodin & Lamzin (2013) Dodin, A. V. & Lamzin, S. A. 2013, Astronomy Letters, 39, 389
* Donati et al. (2007) Donati, J., Jardine, M. M., Gregory, S. G., et al. 2007, MNRAS, 380, 1297
* Donati et al. (2020a) Donati, J. F., Bouvier, J., Alencar, S. H., et al. 2020a, MNRAS, 491, 5660
* Donati et al. (2011) Donati, J. F., Gregory, S. G., Alencar, S. H. P., et al. 2011, MNRAS, 417, 472
* Donati et al. (2012) Donati, J. F., Gregory, S. G., Alencar, S. H. P., et al. 2012, MNRAS, 425, 2948
* Donati et al. (2015) Donati, J. F., Hébrard, E., Hussain, G. A. J., et al. 2015, MNRAS, 453, 3706
* Donati et al. (2008) Donati, J. F., Jardine, M. M., Gregory, S. G., et al. 2008, MNRAS, 386, 1234
* Donati et al. (2018) Donati, J.-F., Kouach, D., Lacombe, M., et al. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte, 107
* Donati et al. (2020b) Donati, J. F., Kouach, D., Moutou, C., et al. 2020b, MNRAS, 498, 5684
* Donati et al. (2010) Donati, J. F., Skelly, M. B., Bouvier, J., et al. 2010, MNRAS, 402, 1426
* Edwards et al. (2006) Edwards, S., Fischer, W., Hillenbrand, L., & Kwan, J. 2006, ApJ, 646, 319
* Edwards et al. (2003) Edwards, S., Fischer, W., Kwan, J., Hillenbrand, L., & Dupree, A. K. 2003, ApJL, 599, L41
* Ercolano et al. (2009) Ercolano, B., Clarke, C. J., & Robitaille, T. P. 2009, MNRAS, 394, L141
* Espaillat et al. (2022) Espaillat, C. C., Herczeg, G. J., Thanathibodee, T., et al. 2022, AJ, 163, 114
* Faesi et al. (2012) Faesi, C. M., Covey, K. R., Gutermuth, R., et al. 2012, PASP, 124, 1137
* Fazio et al. (2004) Fazio, G. G., Hora, J. L., Allen, L. E., et al. 2004, ApJS, 154, 10
* Fischer et al. (2011) Fischer, W., Edwards, S., Hillenbrand, L., & Kwan, J. 2011, ApJ, 730, 73
* Flores et al. (2019) Flores, C., Connelley, M. S., Reipurth, B., & Boogert, A. 2019, ApJ, 882, 75
* Folha & Emerson (1999) Folha, D. F. M. & Emerson, J. P. 1999, A&A, 352, 517
* Frasca et al. (2017) Frasca, A., Biazzo, K., Alcalá, J. M., et al. 2017, A&A, 602, A33
* Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
* Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
* Gullbring et al. (1998) Gullbring, E., Hartmann, L., Briceño, C., & Calvet, N. 1998, APJ, 492, 323
* Gully-Santiago et al. (2017) Gully-Santiago, M. A., Herczeg, G. J., Czekala, I., et al. 2017, ApJ, 836, 200
* Gunn et al. (1998) Gunn, J. E., Carr, M., Rockosi, C., et al. 1998, AJ, 116, 3040
* Hartigan et al. (1989) Hartigan, P., Hartmann, L., Kenyon, S., Hewett, R., & Stauffer, J. 1989, ApJS, 70, 899
* Hartigan et al. (1991) Hartigan, P., Kenyon, S. J., Hartmann, L., et al. 1991, ApJ, 382, 617
* Hartmann & Kenyon (1990) Hartmann, L. W. & Kenyon, S. J. 1990, ApJ, 349, 190
* Herczeg & Hillenbrand (2014) Herczeg, G. J. & Hillenbrand, L. A. 2014, ApJ, 786, 97
* Hillenbrand et al. (1998) Hillenbrand, L. A., Strom, S. E., Calvet, N., et al. 1998, AJ, 116, 1816
* Horne & Baliunas (1986) Horne, J. H. & Baliunas, S. L. 1986, APJ, 302, 757
* Ingleby et al. (2013) Ingleby, L., Calvet, N., Herczeg, G., et al. 2013, ApJ, 767, 112
* Johns-Krull (2007) Johns-Krull, C. M. 2007, ApJ, 664, 975
* Johns-Krull & Valenti (2001) Johns-Krull, C. M. & Valenti, J. A. 2001, ApJ, 561, 1060
* Johns-Krull et al. (2000) Johns-Krull, C. M., Valenti, J. A., & Linsky, J. L. 2000, ApJ, 539, 815
* Kidder et al. (2021) Kidder, B., Mace, G., López-Valdivia, R., et al. 2021, ApJ, 922, 27
* Kounkel et al. (2019) Kounkel, M., Covey, K., Moe, M., et al. 2019, AJ, 157, 196
* Kwan et al. (2007) Kwan, J., Edwards, S., & Fischer, W. 2007, ApJ, 657, 897
* Kwan & Fischer (2011) Kwan, J. & Fischer, W. 2011, MNRAS, 411, 2383
* Lada et al. (2006) Lada, C. J., Muench, A. A., Luhman, K. L., et al. 2006, ApJ, 131, 1574
* McClure et al. (2013) McClure, M. K., Calvet, N., Espaillat, C., et al. 2013, ApJ, 769, 73
* Müller et al. (2018) Müller, A., Keppler, M., Henning, T., et al. 2018, A&A, 617, L2
* Muzerolle et al. (2010) Muzerolle, J., Allen, L. E., Megeath, S. T., Hernández, J., & Gutermuth, R. A. 2010, ApJ, 708, 1107
* Muzerolle et al. (1998) Muzerolle, J., Hartmann, L., & Calvet, N. 1998, AJ, 116, 455
* Nguyen et al. (2012) Nguyen, D. C., Brandeker, A., van Kerkwijk, M. H., & Jayawardhana, R. 2012, ApJ, 745, 119
* Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9
* Pecaut & Mamajek (2016) Pecaut, M. J. & Mamajek, E. E. 2016, MNRAS, 461, 794
* Pontoppidan et al. (2008) Pontoppidan, K. M., Blake, G. A., van Dishoeck, E. F., et al. 2008, ApJ, 684, 1323
* Preibisch & Zinnecker (1999) Preibisch, T. & Zinnecker, H. 1999, AJ, 117, 2381
* Rebull (2001) Rebull, L. M. 2001, AJ, 121, 1676
* Rebull et al. (2002) Rebull, L. M., Makidon, R. B., Strom, S. E., et al. 2002, AJ, 123, 1528
* Rei et al. (2018) Rei, A. C. S., Petrov, P. P., & Gameiro, J. F. 2018, A&A, 610, A40
* Ricci et al. (2010) Ricci, L., Testi, L., Natta, A., et al. 2010, A&A, 512, A15
* Rieke et al. (2004) Rieke, G. H., Young, E. T., Engelbracht, C. W., et al. 2004, ApJS, 154, 25
* Rodrigo & Solano (2020) Rodrigo, C. & Solano, E. 2020, in XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society, 182
* Rodrigo et al. (2012) Rodrigo, C., Solano, E., & Bayo, A. 2012, SVO Filter Profile Service Version 1.0, IVOA Working Draft 15 October 2012
* Sicilia-Aguilar et al. (2015) Sicilia-Aguilar, A., Fang, M., Roccatagliata, V., et al. 2015, A&A, 580, A82
* Sicilia-Aguilar et al. (2020) Sicilia-Aguilar, A., Manara, C. F., de Boer, J., et al. 2020, A&A, 633, A37
* Simon et al. (2016) Simon, M. N., Pascucci, I., Edwards, S., et al. 2016, ApJ, 831, 169
* Sousa et al. (2021) Sousa, A. P., Bouvier, J., Alencar, S. H. P., et al. 2021, A&A, 649, A68
* Stempels et al. (2007) Stempels, H. C., Gahm, G. F., & Petrov, P. P. 2007, A&A, 461, 253
* Sullivan & Kraus (2022) Sullivan, K. & Kraus, A. L. 2022, ApJ, 928, 134
* Teixeira et al. (2012) Teixeira, P. S., Lada, C. J., Marengo, M., & Lada, E. A. 2012, A&A, 540, A83
* Thanathibodee et al. (2020) Thanathibodee, T., Molina, B., Calvet, N., et al. 2020, ApJ, 892, 81
* Torres et al. (2006) Torres, C. A. O., Quast, G. R., da Silva, L., et al. 2006, A&A, 460, 695
* Valenti et al. (1993) Valenti, J. A., Basri, G., & Johns, C. M. 1993, AJ, 106, 2024
* White & Basri (2003) White, R. J. & Basri, G. 2003, ApJ, 582, 1109
* Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
|
# Improving Factual Accuracy of Neural Table-to-Text Output by Addressing
Input Problems in ToTTo
Barkavi Sundararajan Somayajulu Sripada Ehud Reiter
Department of Computing Science, University of Aberdeen
{b.sundararajan.21, yaji.sripada<EMAIL_ADDRESS>
###### Abstract
Neural Table-to-Text models tend to hallucinate, producing texts that contain
factual errors. We investigate whether such errors in the output can be traced
back to problems with the input. We manually annotated 1,837 texts generated
by multiple models in the _politics_ domain of the ToTTo dataset. We identify
the input problems that are responsible for many output errors and show that
fixing these inputs reduces factual errors by between 52% and 76% (depending
on the model). In addition, we observe that models struggle in processing
tabular inputs that are structured in a non-standard way, particularly when
the input lacks distinct row and column values or when the column headers are
not correctly mapped to corresponding values.
Improving Factual Accuracy of Neural Table-to-Text Output by Addressing Input
Problems in ToTTo
Barkavi Sundararajan and Somayajulu Sripada and Ehud Reiter Department of
Computing Science, University of Aberdeen {b.sundararajan.21, yaji.sripada,
<EMAIL_ADDRESS>
## 1 Introduction
Table-to-Text generation refers to the task of generating natural language
descriptions from tabular data (Parikh et al., 2020; Chen et al., 2020a, b)
and is widely used in several application domains such as medical diagnosis
(Pauws et al., 2018), financial (Zhu et al., 2023) and weather reporting
(Sripada et al., 2002; Gkatzia et al., 2017; Upadhyay and Massie, 2022) and
sports summaries (Thomson et al., 2020). Neural language models are known to
generate fluent texts (Ji et al., 2023) but may generate outputs that are
factually incorrect or unrelated to the provided input data. Such undesirable
generation is called ‘hallucination’ (Wang and Sennrich, 2020; Raunak et al.,
2021; Ji et al., 2023).
Previous studies on Table-to-Text tasks adopt traditional seq2seq methods to
generate table descriptions (Wiseman et al., 2017; Puduppully et al., 2019;
Rebuffel et al., 2019). Recently, Transformer based models (Devlin et al.,
2019; Raffel et al., 2020; OpenAI, 2023) have shown remarkable progress in
language generation from textual input (Badaro et al., 2023), however tabular
data still needs more improvement to control hallucinations (Rebuffel et al.,
2021). Neural models struggle with tabular data, especially when the inputs do
not have distinct cell values from rows and columns mapped along with their
respective headers. These input problems lead the model to generate more
factual errors (Kasner and Dušek, 2024).
Using the ToTTo tabular dataset (Parikh et al., 2020), we identify and address
input problems that are responsible for factual errors. Some common tabular
input problems in the ToTTo are i. ‘non-atomic’ cell values, where a column
contains multiple values such as leader name, party name and % of votes in one
cell rather than a single indivisible value, ii. missing important cell values
in the input (see Table 1(b)) and iii. nested column headers and row headers
in the Wikipedia
tables111https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Accessibility/Data_tables_tutorial#Column_headers:_bad_example
that lead to incorrect mapping of the cell values.
(a) Original cells highlighted in Yellow as Input
1996 United States House of Representatives election
---
Party | Candidate | Votes | %
Democratic | Eleanor Holmes Norton (inc.) | 134,996 | 90.00
Republican | Sprague Simonds | 11,306 | 7.54
Llama 2-13B Output for tabular input a:
---
Eleanor Holmes Norton (inc.) won with 7.54%U of the vote. Sprague Simonds was
the Republican candidate and received 22.38%U of the vote.
(b) Corrected Tabular data by including relevant cells
1996 United States House of Representatives election
---
Party | Candidate | Votes | %
Democratic | Eleanor Holmes Norton (inc.) | 134,996 | 90.00
Republican | Sprague Simonds | 11,306 | 7.54
Llama 2-13B Output for tabular input b:
---
Democratic Party candidate Eleanor Holmes Norton won with 90% of the vote.
Republican Party candidate Sprague Simonds received 7.54%.
Table 1: ToTTo example: Highlighted cells in yellow are passed as input to the
model. Passing the appropriate cells (i.e., % votes and party name) as input,
as shown in Table 1(b) fixes the factual errors. Compare the Table 1(b) output
(with no errors) to the Table 1(a) output (with NUMBER errors denoted by a
superscript U). Table 2: Linearized Input from (a) and (b) is passed as input
to Llama 2-13B model. For example, we present the Linearized Input for Table
1(a) here. The corresponding input for Table 1(b) is not shown, but it will
include all related cells highlighted in yellow in (b).
<page_title> 1996 United States House of Representatives election
</page_title> <table> <cell> Eleanor Holmes Norton (inc.) <col_header>
Candidate </col_header> </cell> <cell> Republican <col_header> Party
</col_header> </cell> <cell> Sprague Simonds <col_header> Candidate
</col_header> </cell> <cell> 7.54 <col_header> % </col_header> </cell>
</table>
Table 1(b) presents a sample from ToTTo. Only the highlighted cells from Table
1(a) are passed to the model (as shown in Table 1(b)). Passing Norton’s % of
votes and her party name (compare Table 1(b) to Table 1(a)) eliminates the
hallucinated % of votes (see output for Table 1(b)); this correction is based
on the input problem described in Sec. 5.2.
In this paper, we score the quality of output texts by manually annotating
output errors instead of using automatic evaluation metric scores such as BLEU
(Papineni et al., 2002), ROUGE (Lin, 2004), PARENT (Dhingra et al., 2019) and
BLEURT (Sellam et al., 2020). We conducted a pilot study, where we fine-tuned
T5-base and T5-large models (Raffel et al., 2020), analysing 1,677 _politics_
domain texts from the ToTTo dataset through manual error analysis adopted from
Thomson and Reiter (2020). These manual error annotations allowed us to
identify patterns of errors in the generated text which were then traced back
to input problems.
Our approach is summarised as follows:
1. I.
We systematically correct the tabular inputs for the _politics_ domain in
ToTTo to adhere to a standard form to ensure the generation of factual texts
from neural models. The correction procedure is elaborated in Sec. 5.2 and is
supplemented by pseudocode in App. D.
1. (a)
We apply this correction to a larger subset of 210 samples, resulting in a 62%
decrease in factual errors for T5-base and a 57% decrease for T5-large in the
generated text (Sec. 6.1).
2. (b)
We conduct experiments on Llama 2-7B and Llama 2-13B models (Touvron et al.,
2023) with 40 challenging samples selected from the previous 210 samples.
Tailoring zero-shot prompts for specific input and error annotation on 160
texts showed that correcting input reduces factual errors by 52% in Llama 2-7B
and 76% in Llama 2-13B (Sec. 6.2).
2. II.
The manual error annotation methodology adopted from Thomson and Reiter (2020)
is detailed in App. B; this builds on the work of Sundararajan et al. (2022)
for ToTTo _politics_ domain outputs222Error annotation guidelines and sample
annotations from our human evaluation is available at
https://github.com/BarkaviSJ/totto_politics_human_annotations. The inter-
annotation agreement on the error annotation was good, with a Fleiss’ Kappa of
0.622 (Sec. 7).
### 1.1 Table to Text dataset, ToTTo
ToTTo is an open-domain English language dataset (Parikh et al., 2020), where
the input $X$ is taken from Wikipedia table $T$, which includes the table’s
metadata (such as title) and a set of highlighted cells, along with their
headers. This structured information is flattened to create a linearized text
representation of the table, as mentioned in Table 1(b). This crowdsourced
dataset is paired with a natural language description, denoted as output $Y$,
comprising a sequence of tokens $y_{1},y_{2},\ldots,y_{n}$, which provides a
coherent summary of the content present in the input table $X$.
These input-output pairs from the ToTTo dataset can be used for fine-tuning or
prompting the neural language models. As shown in Table 1(a), the Input X is
often observed to be problematic and fixing these problems is the main focus
of this paper.
## 2 Related Work
Prior work on ToTTo: Wang et al. (2022) proposed a framework, LATTICE, that
preserves the structural information of the cell values (tokens) within the
same rows or columns, and by removing the attention mechanism flow for other
unrelated tokens. Hu et al. (2023) incorporates content planning and
generation from Su et al. (2021) and synthetically added noisy cells in their
fine-tuning regime. While these approaches are model agnostic and improved
automatic metric scores such as BLEU (Papineni et al., 2002), PARENT (Dhingra
et al., 2019) and BLEURT (Sellam et al., 2020) by few points in the
leaderboard 333https://github.com/google-research-datasets/ToTTo, the
fundamental problem with the tabular input remains.
Chen et al. (2022) acknowledged that the target cells in the ToTTo tabular
input are not always highlighted in their work titled ‘Table Structure
Understanding and Text Deliberation Approach’. They used a template to extract
all facts from the raw table for 1.2K training samples (only for the inputs
with rows and columns fewer than 8) and employed hierarchical multi-head
attention to capture the structural information in their fine-tuning process.
Though this approach promises to retain the facts from raw tables, it only
addresses simpler tables with fewer than 8 rows and columns, still has
limitations for longer tables, complex tabular structures and non-atomic
tabular cells.
Our focus on correcting inputs aiming to achieve factually correct outputs
aligns with the work of Dušek et al. (2019) on the E2E dataset (Novikova et
al., 2017). Their study also demonstrated that improving inputs in an NLG
dataset helps in improving model outputs.
Error Analysis: While the automatic metric scores such as BLEU, PARENT and
BLEURT help evaluate the model’s performance at a high level, relying solely
on these metrics will not address specific weaknesses i.e., lower metric
scores do not provide insights into specific error types in the output. We
follow guidelines from van Miltenburg et al. (2021) to perform error analysis
in NLG systems and investigate errors in the output at a more granular level
by adopting the manual error annotation approach from Thomson and Reiter
(2020); Sundararajan et al. (2022); Thomson et al. (2023).
Maynez et al. (2020) also emphasized automatic metrics are not sufficient to
study the hallucination problem and provided a detailed study on intrinsic and
extrinsic hallucination in abstractive summarization. We studied hallucination
in our evaluation scheme by annotating different categories of errors in the
output tokens (single token or group of tokens). Mapping our adopted
methodology to Maynez et al. (2020)’s work, intrinsic is the main error
category occurring when generated outputs are not faithful to the given input.
It includes WORDW, NAMEN, DATE_DIMENSIOND, NUMBERU, CONTEXTC and OTHERO)
from our error categories. Extrinsic refers to our ADDITIONA category (see
Sec. B.3).
LLM prompts: Empirical evaluation of prompting strategies on the three large
language models (LLMs) by Sivarajkumar et al. (2023) in clinical NLP found
that tailoring task-specific prompt is crucial for achieving accuracy. In a
study on Text-to-SQL, Chang and Fosler-Lussier (2023) investigated zero-shot
prompting strategies, highlighting the significance of table representation.
Their findings indicated that normalized database prompts outperformed
unnormalized ones; this motivates our initial step of correcting tabular
inputs to a standard form. In our work, we leveraged a recent LLM, Llama 2
(Touvron et al., 2023) and tailored our zero-shot prompt (Kojima et al., 2022)
specific to the content of each table.
## 3 Pilot Study
### 3.1 Methodology
We only look at the _politics_ domain on the ToTTo validation set. We build
upon the work of Sundararajan et al. (2022) to identify the causes of output
errors in T5 models. In this paper, we go beyond error annotations to fix
these errors in our main study, both in T5 and Llama 2 models (detailed in
Sec. 5).
The error categories mentioned in Sundararajan et al. (2022) are: WORDW,
NAMEN, DATE_DIMENSIOND, NUMBERU, OTHERO, CONTEXTC, NOT_CHECKABLENC and
NON_ENGLISHNE. In this work, we excluded NOT_CHECKABLENC and introduced a new
error category, ADDITIONA, which is used when the generated text has added
words or phrases that diverge from the input. Definitions for all error
categories are provided in Sec. B.3.
### 3.2 Insights from our Pilot Study
For the pilot study, we fine-tuned both the T5-base (T5-b) and the T5-large
(T5-l) models on the ToTTo dataset by following the baseline approach (Kale
and Rastogi, 2020). The hyperparameters and fine-tuning details for these two
models are shown in App. A.
Category | T5-base (T5-b) | T5-large (T5-l)
---|---|---
| Count | % | Count | %
No Error | 358 | 47 | 450 | 60
Omissions | 272 | 36 | 218 | 29
Errors | 124 | 17 | 86 | 11
Total Count | 754 | 754
Table 3: Pilot Study analysis: T5-b and T5-l Models in ToTTo Politics Domain.
‘Errors’ category is the main focus of our work; ‘omissions’ are excluded.
Our analysis (presented in Table 3) shows that:
No Error: 47% of the samples from T5-b and 60% of the samples from T5-l are
error-free.
Omissions: Omissions occur when the generated text fails to mention some
information from the input (González Corbelle et al., 2022) without making any
factual errors. If the output has errors and omissions, we classify it as an
error. The T5-b had 36% omissions and T5-l had 29%.
Errors: Our analysis revealed that T5-b made factual errors in 17% and T5-l
made factual errors in 11% of the total samples.
Hypothesis: Based on the insights from this analysis, we hypothesize that when
tabular input data is structured in non-standard ways, models struggle to
interpret these ambiguous inputs leading to generate factually inaccurate
output. We test this hypothesis by addressing input problems related to non-
standard tabular structures.
## 4 Input Problems
Due to the practical challenges involved in improving the tabular input for
the entire dataset, which includes unique headers and tabular structures for
each input, our focus is on analyzing a subset of samples containing errors.
We examined 124 error samples from the T5-b and 86 error samples from T5-l
models within the ToTTo _politics_ domain, as identified in Table 3. We aim to
segregate the errors originating from non-standard or illogical nested table
structures. We categorize these input problems into two broad categories, as
briefly elaborated upon in Sec. 4.1 and Sec. 4.2.
### 4.1 Generic Input Problems
Non-atomic tabular cell values: When a table cell contains multiple atomic
values (see Table 4(c)). Examples of such non-atomic forms include multiple
leaders’ names, votes, term dates, or election years all in a single cell. We
further categorize these problems into ‘single record lacking atomicity’ and
‘multiple records lacking atomicity’ (shown in Fig. 2 in App. F) to
demonstrate how the models struggle when records of multiple leaders lack
atomicity.
Complex table type: When a table contains election results in sentence form,
models struggle to interpret and generate meaningful texts because the
sentence form data lacks the needed context (see Table 11(d) in App. F).
Insufficient input: In some cases, the necessary cells are not highlighted in
the tabular input, resulting in incorrect outputs. Our analyses in Table 1(b)
and Table 5(c) demonstrate that outputs become factually correct when relevant
cells are included.
Longer table input: Models often struggle to generate accurate texts for
lengthy table inputs, especially when the data is not in a standard form and
lacks clear cell relationships.
### 4.2 Politics Domain-Specific Input Problems
Politics specific headers: In ToTTo, the use of symbols, for example, ‘+’ or
‘-’ instead of having clear semantic terms like _‘swing percentage’ or ‘%
change compared to previous election’_ as column headers caused 5% of errors.
This lack of semantic guidance in the input made it difficult for models to
accurately generate the correct output text (see Table 10(c) in App. F).
List of leader names in the input: In the _politics_ domain, we observed a
specific issue when input data contains a list of leader names. Models tend to
favour the leader whose name appears first in the list (for example, either
from the table title or as the first leader in highlighted cells), even if
they have lost the election (see Table 13(c), Table 14, Table 15(c) in App.
F). This becomes even more challenging when leader names are associated either
with missing values (opponent leader’s name or vote count) in other columns in
the input or when the tabular cell values are non-atomic.
The manual fixes we applied to each of these input problems with examples are
detailed in Sec. 5.2.
## 5 Main Study
(a) Input Table with Original Cells highlighted in Yellow
Page Title: 2014 United States Senate elections
---
Section Title: Elections leading to the next Congress
State | Candidates
Alaska | Dan Sullivan (Republican) 48.0% Mark Begich (Democratic) 45.8% Mark Fish (Libertarian) 3.7% Ted Gianoutsos (Independent) 2.0%
Virginia | Mark Warner (Democratic) 49.1% Ed Gillespie (Republican) 48.3% Robert Sarvis (Libertarian) 2.4%
(b) Input Table with Corrected Cells highlighted in Yellow
State | Candidates | Party | % votes
---|---|---|---
Alaska | Dan Sullivan | Republican | 48.0%
Alaska | Mark Begich | Democratic | 45.8%
Virginia | Mark Warner | Democratic | 49.1%
Virginia | Ed Gillespie | Republican | 48.3%
(c) Outputs Before and After Input Data Correction T5-base before fix: Alaska:
Dan Sullivan (R) defeated Mark Begich (D) 45.8% toW 3.7%U and Ted Gianoutsos
(Independent) 2.0%, and Virginia: Mark Warner won 49.1% toW 2.4%U.
---
T5-base after fix: Alaska was won by Dan Sullivan (R) with 48.0% of the vote,
and Virginia was won by Mark Warner (D) with 49.1% of the vote.
Llama 2-7B before fix: 2014 United States Senate elections were held on
November 4A, 2014. The Republican Party gained control of the Senate for the
first time since 2006. The Democrats lost their majority in the Senate and
Republicans won a net gain of eight seats to take control of the chamber with
a 54-46 majority. The Republicans also took control of the House of
Representatives, giving them full control of Congress for the first time since
2010.A
Llama 2-7B after fix: In the United States Senate elections of 2014, Dan
Sullivan (Republican) won in Alaska with 48.0% of votes against Mark Begich
(Democratic) who got 45.8%. In Virginia, Mark Warner (Democratic) won with
49.1% of votes against Ed Gillespie (Republican) who got 48.3%.
Table 4: Fixing Multiple leaders’ votes that are not atomic, as detailed in
Sec. 5.2.1. ‘Before fix’ shows the errors produced by models with Table 4(b)
as input and ‘After fix’ shows both models did not produce any factual error
with Table 4(b) as input. The error tokens in outputs are annotated with
respective error code in superscript (W for WORD, U for NUMBER and A for
ADDITION errors).
### 5.1 Llama 2 Models
In our main study, in addition to T5 models, we included Llama 2-7B (L-7B) and
Llama 2-13B (L-13B) models to test our hypothesis on non-standard tabular
inputs. We received official model weights from Meta444We downloaded weights
from https://ai.meta.com/resources/models-and-libraries/llama-downloads/ and
quantized our models by following the installation instructions from
https://github.com/ggerganov/llama.cpp. and then quantized our Llama 2 models.
We ran experiments using the full model weights and the model with 4-bit
integer quantization for the ToTTo dataset (_politics_ domain). Both produced
comparable output quality with no difference in errors for the quantized
models. Therefore, we finalized our experiments with 4-bit quantized models to
save computational resources and used them for local inference on a MacOS CPU.
We set the temperature to 0.0 as it produced fewer errors. Further details on
data corrections for these models are discussed in Sec. 5.4.
(a) Input Table with Original Cells highlighted in Yellow
Page Title: Minnesota Amendment 2
---
Section Title: Results
Choice | Votes | % Votes
Referendum failed No | 1,588,752 | 53.84
Yes | 1,362,030 | 46.16
(b) Input Table with Corrected Cells highlighted in Yellow
Choice | Votes | % Votes
---|---|---
Referendum failed No | 1,588,752 | 53.84
Yes | 1,362,030 | 46.16
(c) Outputs Before and After Input Data Correction T5-base before fix The
Minnesota Amendment 2 was approvedW by 53.84 percent of voters, with a
turnout ofC 46.16 percent.
---
T5-base after fix: Minnesota Amendment 2 was defeated by 53.84 percent of
voters, while 46.16 percent voted in favor of the measure.
Llama 2-7B before fix: 53.84% of Minnesota voters voted forW Amendment 2.
Llama 2-7B after fix: 53.84% of the voters voted no on Amendment 2, while
46.16% voted yes.%.
Table 5: Fixing Insufficient input problems, as detailed in Sec. 5.2.3.
‘Before’ shows WORD and CONTEXT errors due to missing choices of votes.
‘After’ shows these errors are resolved when these cell values are provided.
### 5.2 Manual Fixes for Input Problems
We followed a systematic procedure to apply manual fixes to specific input
problems, as discussed from Sec. 5.2.1 to Sec. 5.2.6. A supplementary
pseudocode for these fixes is also provided in App. D. The algorithm 1 takes
tabular data and title (metadata) from ToTTo as input parameters to execute
six functions to gather insights and return with corrected tabular data. The
first two functions, DataSizeManageableForLongTable and
IdentifyLeaderNameOrder do not correct the tabular input but provide insights
on the input problems leading to factual errors in the outputs. The other four
functions correct the input problems as shown in algorithm 4, algorithm 5,
algorithm 6 and algorithm 7.
#### 5.2.1 Non-Atomic Cell Values
Correction: We corrected the input to store individual leader data, including
votes, party, and election details, as a separate column variable. In Table
4(b), when the ‘candidates’ column is not atomic, all models made errors for
this tabular data. The corrections involved separating each leader’s party and
votes into separate columns (see Table 4(b)). We passed the leading two
leaders’ data, excluding the remaining leaders, in separate rows for the
respective states (Alaska and Virginia). After input correction, T5-b and L-7B
outputs in Table 4(c) were error-free. While some models omitted other
candidate and election details, all models corrected the previous factual
errors after the correction.
#### 5.2.2 Complex Table Type
Correction: Complex tabular structures were generally difficult to correct
because the election results are in sentence form e.g., see ‘Results’ column
in Table 11(d) in App. F. We could only make corrections to the ‘non-atomic’
cell values in ‘candidates’ column. Three models generated ‘the incumbent
senator Coe I. Crawford lost renomination to Edwin S. Johnson, a Democrat
candidateC.’ Losing renomination is when the current senator did not lose the
seat to the opponent, but rather failed to be nominated in the primary to
stand for reelection. This resulted in CONTEXTC error, specifically when the
models made unsupported assumptions about the given input data, as defined in
App. B. Even after correction, all models except L-13B made this CONTEXTC
error (see Table 11(d)).
#### 5.2.3 Insufficient Input
Correction: This is a data annotation problem because it is related to how the
ToTTo dataset was created by pairing the sentence description and highlighting
only a subset of cells from the Wikipedia table as Input (X). This problem is
straightforward to fix by including the missing cells or headers from the
tabular data.
In Table 5(c), when the choices for the Minnesota Amendment (i.e., ‘Referendum
failed No’ and ‘Yes’ cells) were not included, all models incorrectly
generated the favour of votes (‘approved’, ‘for’) and made an unsupported
assumption regarding the turnout percentage. As shown in Table 5(c), all
models corrected the errors after including the vote choices.
#### 5.2.4 Longer Table Input
Correction: When dealing with straightforward tabular data, it was easier to
correct the input either by i. correcting the cell values to ensure atomicity
and/or ii. including the missing cells. However, this longer tabular input not
only had over 100 rows but also included complex structures with nested column
headers and row headers which posed difficulties in correcting the input.
Large models such as L-13B and T5-l could process tables with fewer than 20
rows for straightforward inputs. However, T5-b and L-7B struggled to produce
factual information even for 20 rows. In our correction procedure, we chose an
upper limit of 20 rows and 10 columns to simplify the longer tabular input. A
simplified example of this problem with fewer rows is shown in Table 16 in
App. F.
#### 5.2.5 ToTTo: Politics Specific
Correction: We included appropriate semantic cues in the headers to help the
model differentiate between the generic percentage of votes and the swing
percentage of votes. For other errors, we expanded the abbreviation of
coalition party names to avoid errors. In Table 10(c) from App. F, the
original input only had $\pm$ in the header, resulting in errors from all
models including L-13B. After explicitly including ‘$\pm$ seats compared to
the previous election’, both Llama 2 models corrected the previous error.
However, both fine-tuned T5 models committed mistakes even after this
correction.
Specific Semantic Generation Issue: In some cases, the model cannot determine
whether a minister was ‘shortest-lived’ or ‘longest-lived’ only based on their
lifespan, birth, and death years from the input. It may require additional
context to produce accurate text. One such example is shown in Table 12. All
models with the original data produced incorrect WORDW. After we included the
‘Total time in office’ cell and customized the prompt for Llama 2, both Llama
2 models corrected the WORDW error. However, both T5 models did not show any
improvement after including this detail. This could be due to the strong
influence of the patterns learned from the fine-tuning process for similar
tabular structures.
#### 5.2.6 List of Leader Names
Correction: We did not change the order of the leader’s name in the title, but
we addressed missing vote counts and leader names to map the relations for
each record as shown in Table 13(c), Table 14, and Table 15(c) in the App. F.
In Table 14, the title leader ‘Chuck DeVore’ lost the election among four
other party nominees. Both T5 models made errors by focusing on ‘Chuck DeVore’
as the main candidate, resulting in WORDW and CONTEXTC errors. NUMBERU
errors are present in all models due to missing votes. After including the
votes and party details for all candidates, both Llama 2 models corrected all
errors. However, both T5 models still made errors stating the title leader
either defeated or being defeated by all candidates, possibly due to the
learned patterns during fine-tuning on the ToTTo dataset.
In Table 15(c), the title leader, ‘Joseph Haslet’ ran as a candidate for
Governor office in 1804 and 1807 but lost both elections. All models
incorrectly generated Haslet won in both years. The correction is only limited
to passing the party name and modifying the header from ‘Subject’ to
‘Candidate’. Despite this change, all models continued to generate errors.
Both T5 models struggled to generate the right winning candidate in 5 out of
11 samples (Fig. 2(a)) for this input with the same set of headers.
### 5.3 T5 Models on Corrected Data
Based on the insights gathered from different input problems in Sec. 4, we
followed the procedure and manually corrected the original tabular input for
210 samples (taken from 124 and 86 ‘errors’ category samples in Table 3). We
ran both T5-b and T5-l models on the corrected data.
### 5.4 Llama 2 Models on Corrected Data
From the corrected data, as described in Sec. 5.3, we selected 40 challenging
samples from the previous 210 samples as inputs to the Llama 2-7B (L-7B) and
Llama 2-13B (L-13B) models. These samples were chosen to cover each of the
input problem types described in Sec. 4, guaranteeing a thorough analysis.
Within these 40 samples, 21 are associated with general input problems and the
remaining 19 are specific to the _politics_ input problems, as shown in Fig.
2(c) and Fig. 2(d) (in App. F).
We studied the _zero-shot_ capabilities of L-7B and L-13B models for the 40
challenging samples. For each sample, we employed different prompts tailoring
to the content of the tabular input (Kojima et al., 2022). Two common prompts
we used are shown in App. C. For each sample and each of L-7B, and L-13B, we
examined the outputs before and after correcting the tabular input.
## 6 Results and Discussion
We manually annotated the outputs from all models working on corrected input
data, following the same procedure outlined in Sec. 3.1 and defined in App. B.
In this section, we summarize the results of the total input data corrections
and discuss them. The previous section also described and discussed error
analysis locally while presenting individual corrections.
### 6.1 Error Reductions in T5 Models
The input corrections significantly reduced factual errors, as evident in
Table 6, which provides a comparison of error counts ‘before’ and ‘after’
input data fixes for each error category. It should be noted that _no prompt
or instruction_ was provided to these two fine-tuned T5 models.
Category | T5-base (T5-b) | T5-large (T5-l)
---|---|---
Before | After | Before | After
WORD | 62 | 20 | 31 | 12
NAME | 13 | 3 | 12 | 2
DATE_DIMENSION | 7 | 1 | 6 | 1
NUMBER | 12 | 1 | 5 | 0
OTHER | 5 | 2 | 2 | 0
CONTEXT | 13 | 3 | 16 | 4
ADDITION | 2 | 1 | 2 | 1
NON-ENGLISH | 20 | 20 | 21 | 21
Total errors | 134 | 51 | 95 | 41
Error reduction | 62% | 57%
Table 6: Count of individual error annotations for 210 samples. The table
compares the error counts between ‘before’ and ‘after’ applying input
correction. Each sample can contain multiple errors.
Category T5-base (T5-b) baseline T5-large (T5-l) baseline Llama 2-7B (L-7B)
Llama 2-13B (L-13B) Before After Before After Before After Before After WORD
27 18 21 12 23 15 22 5 NAME 7 3 6 2 2 1 3 0 DATE_DIM 2 1 1 1 2 0 1 0 CONTEXT 8
3 7 2 6 2 4 0 NUMBER 4 1 1 0 7 1 6 1 OTHER 0 0 0 0 5 0 1 0 ADDITION 0 0 0 0 5
5 7 5 Total errors 48 26 36 17 50 24 45 11 Error
reduction(%) 46% 53% 52% 76%
Table 7: Individual error count for the 40 challenging samples selected from
the previous 210 in Table 6. It shows the comparison of the errors annotated
for original data versus corrected data in T5 and Llama 2 models.
Category T5-base (T5-b) baseline T5-large (T5-l) baseline Llama 2-7B (L-7B)
Llama 2-13B (L-13B) Before After Before After Before After Before After No
error
(Higher is better) 0 12 3 16 2 17 5 23 Omissions 0 6 3 8 2 4 4 8
Table 8: Comparison of ‘No error’ and ‘Omissions’ unique count for the same 40
samples. i. Increase in ‘No error’ count indicates error-free outputs after
addressing input problems. ii. Some outputs stopped making factual errors
after correction but instead omitted part of the input content, resulting in a
higher omission count post-fix.
### 6.2 Error Reductions in Llama 2 Models
In Table 8, we present the error analysis of 40 difficult samples, validated
using L-7B and L-13B models with tailored prompts for each input. Outputs were
manually error annotated, and the table compares the error counts before and
after input correction. The _L-7B and L-13B_ models showed reductions of 52%
and 76% in errors, after addressing tabular input issues. The _T5-b and T5-l
models_ demonstrated error reductions of 46% and 53% respectively for the 40
difficult samples considered, which was shown for comparison purposes. The
insights gathered from the individual error categories after input correction
are mentioned below.
* •
WORDW errors, predominantly incorrect verbs and prepositions, were most common
among all models. Post-correction, the L-13B model reduced this error category
by 77%, while the T5-b, T5-l, and L-7B models achieved reductions of 33%, 42%,
and 34%, respectively.
* •
Input correction led to a reduction in NAMEN and CONTEXTC errors across all
models.
* •
The original input data exhibited more NUMBERU errors in Llama 2 models
compared to T5, which were significantly reduced after correction. Both Llama
2 models completely resolved DATE_DIMENSIOND and OTHERO errors.
* •
ADDITIONA errors are more common in Llama 2 models than in T5. Despite
including the prompt ‘Use only the information mentioned in the input table
data’ and correcting the tabular data, both L-7B and L-13B models still
produced five ADDITION errors each.
Table 8 shows two rows of data from our analysis of the 40 challenging
samples. The first row shows that corrected data leads to an increased number
of samples with ‘no error’ (Higher is better). However, the second row shows
that omissions have increased after input corrections. We observed that the
models omitted part of the information either when the corrected tabular data
had multiple column variables for more than two records or when the tabular
structure was complex. In the case of fine-tuned models, it learned to omit
some information during the fine-tuning process. In our future work, we plan
to study the reasons why this is happening.
ERROR ALL AGREE WORD NAME DATE_DIM NUMBER OTHER CONTEXT ADDITION NO ERROR
TOTAL WORD 17 0 0 0 2 0 9 3 0 31 NAME 3 0 0 0 0 0 2 0 0 5 DATE_DIM 1 0 0 0 0 0
2 0 0 3 NUMBER 2 2 0 0 0 0 0 0 0 4 OTHER 2 0 0 0 0 0 0 0 1 3 CONTEXT 7 9 2 2 0
0 0 6 0 26 ADDITION 12 1 0 0 0 0 5 0 0 18 NO ERROR 15 0 0 0 0 1 0 0 0 16
Fleiss’ kappa overall agreement for three annotators, $k={(pa-pe)}/{(1-pe)}$ =
0.622
---
Table 9: Fleiss‘ Kappa coefficient: overall agreement on 60 samples among
three annotators. ‘All agree’ column signifies unanimous agreement on error
types, while other columns show unique error selections.
### 6.3 Model-Specific Results for Different Input Problems
In App. E, Fig. 2 presents how the four models are performing before (left
bars) and after input corrections (right bars) for each input problem type.
Non-atomic cell values: T5-l and L-13B models corrected over 90% of the errors
for this problem type, single record and multiple records lacking atomicity
(see red and green bars in Fig. 2, App. F). T5-b and L-7B models corrected
over 60% of the errors but still struggled with a few samples even after
correction. For example, Fig. 2(b) shows that T5-large was making more errors
for the input problem type labelled ‘Multiple records lacking atomicity’
before correction (in red) and made significant reductions after correction
(in green).
Complex table: Due to the limitations in some tabular inputs, which require
additional context, T5-b could not fix the errors. T5-l omitted the error for
one sample, while L-7B and L-13B models partially fixed this input problem
(see Fig. 2).
Insufficient input: This data annotation problem fixed all factual errors in
T5-b, T5-l and L-13B after correction except for L-7B which added additional
information for one sample.
Longer table input: Large models such as L-13B, L-7B and T5-l corrected
factual errors for straightforward inputs, especially for tables with fewer
than 20 rows. However, T5-b struggled the most to produce factual information.
Politics specific semantic issue: For the corrected input, L-13B fixed the
factual mistakes for 5 out of 8 samples and L-7B fixed factual errors for 4
out of 8 samples. Both T5 models corrected 3 out of 8 samples (see Fig. 2).
List of leader names: T5 models struggled the most to correct factual errors
for this problem. After correction, the L-7B model corrected three samples but
produced factual errors for the remaining eight samples. L-13B made factual
errors only for two samples (see Fig. 2).
While some models struggle with specific inputs, particularly regarding leader
name order, tables requiring additional context for complex tabular
structures, and semantic issues with symbols, our overall results indicate
that correcting tabular inputs improves model outputs.
## 7 Inter-Annotation Experiment
One of the authors manually annotated errors in 1,508 outputs before input
correction and 169 problematic outputs after correction (a total of 1,677 from
both T5 models). Similarly, the annotation for both Llama 2 models covered 160
outputs before and after input correction.
Two additional annotators annotated 60 outputs each, generated by four
different models both before and after input correction. We provided them with
a detailed document that included definitions of error categories, guidelines,
tabular inputs and output texts for error annotation555We release our error
annotations from our human evaluation on
https://github.com/BarkaviSJ/totto_politics_human_annotations.. Annotators
followed the guidelines and marked the errors. Each annotator spent
approximately 3 hours to complete this experiment.
The annotated errors are shown in a confusion matrix in Table 9, where the
‘all agree’ column shows all annotators agreed on the same error type and
other columns show how often each annotator selected a different error type.
Although the correct token was usually chosen, disagreements primarily
occurred in choosing CONTEXT, ADDITION and WORD error types, as shown in red
in Table 9. This might be due to the potential similarities in the definitions
of CONTEXT errors and ADDITION errors (see Sec. B.3). While WORD error is
comparatively straightforward, one annotator chose CONTEXT errors instead of
WORD errors for a few outputs. Cases where an annotator did not mark any
errors were in the minority. The inter-annotation agreement on error category
classification for 60 outputs, as shown by a Fleiss’ kappa of 0.622, indicates
substantial agreement between the three annotators.
## 8 Conclusion
This paper presented a study that quantitatively demonstrates that fixing
input problems such as insufficient data and data records containing non-
atomic content improves the factual accuracy of output text by as high as 76%
for one of the study models. Correcting inputs also seems to improve the
number of entirely error-free output texts. However, we still need to
investigate why errors categorised as ‘omissions’ increase after input
corrections. In our future work, we aim to explore other tabular datasets for
problems with input data and study the generalization of the fixes explored in
the current work.
## Limitations
We acknowledge some limitations in this work. First, we only looked at ToTTo
and our scope of corrections to tabular input is limited to errors identified
within the _politics_ domain of the ToTTo validation set. Second, we did not
extend the correction of tabular input for the _politics_ domain to the
training split of the ToTTo dataset due to the time-consuming process of
handling different table headers and metadata. Third, our experiment results
are restricted to two specific models (T5 and Llama 2) and may not generalize
to other models.
In our future work, we aim to simplify the definitions of the ‘context’ and
‘addition’ error categories, as the annotation experiment revealed
disagreement in choosing these error types for some samples, despite
annotators marking the same error token.
## Ethics Statement
This work seeks to address input problems in non-standard tabular structures
to reduce factual errors in the output text. We utilized the open-source
dataset, ToTTo and maintained the same ground-truth generation as the original
dataset. Our input correction did not introduce any further social bias to
this dataset. We adopted an error annotation methodology to annotate factual
errors and one of the authors performed manual error analysis for this
complete study. We sought consent from two additional annotators, the
annotators volunteered to participate and annotated errors for 60 output texts
each. They had the right to withdraw from the study at any point without
facing any consequences. We provided necessary guidelines, instructions and
examples for them to annotate errors.
## Acknowledgements
We thank Craig Thomson and Adarsa Sivaprasad for their hard work in helping
with the annotations in this paper. We thank the anonymous reviewers for their
detailed feedback and suggestions which have significantly improved this work.
We also thank the NLG (CLAN) reading group at the University of Aberdeen for
their invaluable feedback.
## References
* Badaro et al. (2023) Gilbert Badaro, Mohammed Saeed, and Paolo Papotti. 2023. Transformers for tabular data representation: A survey of models and applications. _Transactions of the Association for Computational Linguistics_ , 11:227–249.
* Chang and Fosler-Lussier (2023) Shuaichen Chang and Eric Fosler-Lussier. 2023. How to prompt llms for text-to-sql: A study in zero-shot, single-domain, and cross-domain settings.
* Chen et al. (2022) Miao Chen, Xinjiang Lu, Tong Xu, Yanyan Li, Zhou Jingbo, Dejing Dou, and Hui Xiong. 2022. Towards table-to-text generation with pretrained language model: A table structure understanding and text deliberating approach. In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 8199–8210, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
* Chen et al. (2020a) Wenhu Chen, Jianshu Chen, Yu Su, Zhiyu Chen, and William Yang Wang. 2020a. Logical natural language generation from open-domain tables. _CoRR_ , abs/2004.10404.
* Chen et al. (2020b) Zhiyu Chen, Wenhu Chen, Hanwen Zha, Xiyou Zhou, Yunkai Zhang, Sairam Sundaresan, and William Yang Wang. 2020b. Logic2Text: High-fidelity natural language generation from logical forms. In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 2096–2111, Online. Association for Computational Linguistics.
* Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. Bert: Pre-training of deep bidirectional transformers for language understanding. In _North American Chapter of the Association for Computational Linguistics_.
* Dhingra et al. (2019) Bhuwan Dhingra, Manaal Faruqui, Ankur Parikh, Ming-Wei Chang, Dipanjan Das, and William Cohen. 2019. Handling divergent reference texts when evaluating table-to-text generation. In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 4884–4895, Florence, Italy. Association for Computational Linguistics.
* Dušek et al. (2019) Ondřej Dušek, David M. Howcroft, and Verena Rieser. 2019. Semantic noise matters for neural natural language generation. In _Proceedings of the 12th International Conference on Natural Language Generation_ , pages 421–426, Tokyo, Japan. Association for Computational Linguistics.
* Gkatzia et al. (2017) Dimitra Gkatzia, Oliver Lemon, and Verena Rieser. 2017. Data-to-text generation improves decision-making under uncertainty. _IEEE Computational Intelligence Magazine_ , 12(3):10–17.
* González Corbelle et al. (2022) Javier González Corbelle, Alberto Bugarín-Diz, Jose Alonso-Moral, and Juan Taboada. 2022. Dealing with hallucination and omission in neural natural language generation: A use case on meteorology. In _Proceedings of the 15th International Conference on Natural Language Generation_ , pages 121–130, Waterville, Maine, USA and virtual meeting. Association for Computational Linguistics.
* Hu et al. (2023) Hanxu Hu, Yunqing Liu, Zhongyi Yu, and Laura Perez-Beltrachini. 2023. Improving user controlled table-to-text generation robustness. In _Findings of the Association for Computational Linguistics: EACL 2023_ , pages 2317–2324, Dubrovnik, Croatia. Association for Computational Linguistics.
* Ji et al. (2023) Ziwei Ji, Nayeon Lee, Rita Frieske, Tiezheng Yu, Dan Su, Yan Xu, Etsuko Ishii, Ye Jin Bang, Andrea Madotto, and Pascale Fung. 2023. Survey of hallucination in natural language generation. _ACM Computing Surveys_ , 55(12):1–38.
* Kale and Rastogi (2020) Mihir Kale and Abhinav Rastogi. 2020. Text-to-text pre-training for data-to-text tasks. In _Proceedings of the 13th International Conference on Natural Language Generation_ , pages 97–102, Dublin, Ireland. Association for Computational Linguistics.
* Kasner and Dušek (2024) Zdeněk Kasner and Ondřej Dušek. 2024. Beyond reference-based metrics: Analyzing behaviors of open llms on data-to-text generation. _arXiv preprint arXiv:2401.10186_.
* Kojima et al. (2022) Takeshi Kojima, Shixiang (Shane) Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. 2022. Large language models are zero-shot reasoners. In _Advances in Neural Information Processing Systems_ , volume 35, pages 22199–22213.
* Lin (2004) Chin-Yew Lin. 2004. ROUGE: A package for automatic evaluation of summaries. In _Text Summarization Branches Out_ , pages 74–81, Barcelona, Spain. Association for Computational Linguistics.
* Maynez et al. (2020) Joshua Maynez, Shashi Narayan, Bernd Bohnet, and Ryan McDonald. 2020. On faithfulness and factuality in abstractive summarization. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 1906–1919, Online. Association for Computational Linguistics.
* Novikova et al. (2017) Jekaterina Novikova, Ondřej Dušek, and Verena Rieser. 2017. The E2E dataset: New challenges for end-to-end generation. In _Proceedings of the 18th Annual SIGdial Meeting on Discourse and Dialogue_ , pages 201–206, Saarbrücken, Germany. Association for Computational Linguistics.
* OpenAI (2023) OpenAI. 2023. Gpt-4 technical report. _ArXiv_ , abs/2303.08774.
* Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002. Bleu: a method for automatic evaluation of machine translation. In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics_ , pages 311–318, Philadelphia, Pennsylvania, USA. Association for Computational Linguistics.
* Parikh et al. (2020) Ankur Parikh, Xuezhi Wang, Sebastian Gehrmann, Manaal Faruqui, Bhuwan Dhingra, Diyi Yang, and Dipanjan Das. 2020. ToTTo: A controlled table-to-text generation dataset. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1173–1186, Online. Association for Computational Linguistics.
* Pauws et al. (2018) Steffen C. Pauws, Albert Gatt, Emiel J. Krahmer, and Ehud Reiter. 2018. Making effective use of healthcare data using data-to-text technology. In _Data Science for Healthcare_.
* Puduppully et al. (2019) Ratish Puduppully, Li Dong, and Mirella Lapata. 2019. Data-to-text generation with content selection and planning. _Proceedings of the AAAI Conference on Artificial Intelligence_ , 33(01):6908–6915.
* Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020. Exploring the limits of transfer learning with a unified text-to-text transformer. _Journal of Machine Learning Research_ , 21(140):1–67.
* Raunak et al. (2021) Vikas Raunak, Arul Menezes, and Marcin Junczys-Dowmunt. 2021. The curious case of hallucinations in neural machine translation. In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1172–1183, Online. Association for Computational Linguistics.
* Rebuffel et al. (2021) Clément Rebuffel, Marco Roberti, Laure Soulier, Geoffrey Scoutheeten, Rossella Cancelliere, and Patrick Gallinari. 2021. Controlling hallucinations at word level in data-to-text generation. _Data Mining and Knowledge Discovery_ , 36:318 – 354.
* Rebuffel et al. (2019) Clément Rebuffel, Laure Soulier, Geoffrey Scoutheeten, and Patrick Gallinari. 2019. A hierarchical model for data-to-text generation. _Advances in Information Retrieval_ , 12035:65 – 80.
* Sellam et al. (2020) Thibault Sellam, Dipanjan Das, and Ankur Parikh. 2020. BLEURT: Learning robust metrics for text generation. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7881–7892, Online. Association for Computational Linguistics.
* Sivarajkumar et al. (2023) Sonish Sivarajkumar, Mark Kelley, Alyssa Samolyk-Mazzanti, Shyam Visweswaran, and Yanshan Wang. 2023. An empirical evaluation of prompting strategies for large language models in zero-shot clinical natural language processing. _arXiv preprint arXiv:2309.08008_.
* Sripada et al. (2002) Somayajulu Sripada, Ehud Reiter, Jim Hunter, and Jin Yu. 2002. Sumtime-meteo: Parallel corpus of naturally occurring forecast texts and weather data. _Computing Science Department, University of Aberdeen, Aberdeen, Scotland, Tech. Rep. AUCS/TR0201_.
* Su et al. (2021) Yixuan Su, David Vandyke, Sihui Wang, Yimai Fang, and Nigel Collier. 2021. Plan-then-generate: Controlled data-to-text generation via planning. In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 895–909, Punta Cana, Dominican Republic. Association for Computational Linguistics.
* Sundararajan et al. (2022) Barkavi Sundararajan, Somayajulu Sripada, and Ehud Reiter. 2022. Error analysis of ToTTo table-to-text neural NLG models. In _Proceedings of the 2nd Workshop on Natural Language Generation, Evaluation, and Metrics (GEM)_ , pages 456–470, Abu Dhabi, United Arab Emirates (Hybrid). Association for Computational Linguistics.
* Thomson and Reiter (2020) Craig Thomson and Ehud Reiter. 2020. A gold standard methodology for evaluating accuracy in data-to-text systems. In _Proceedings of the 13th International Conference on Natural Language Generation_ , pages 158–168, Dublin, Ireland. Association for Computational Linguistics.
* Thomson et al. (2020) Craig Thomson, Ehud Reiter, and Somayajulu Sripada. 2020. SportSett:basketball - a robust and maintainable data-set for natural language generation. In _Proceedings of the Workshop on Intelligent Information Processing and Natural Language Generation_ , pages 32–40, Santiago de Compostela, Spain. Association for Computational Lingustics.
* Thomson et al. (2023) Craig Thomson, Ehud Reiter, and Barkavi Sundararajan. 2023. Evaluating factual accuracy in complex data-to-text. _Computer Speech & Language_, 80:101482.
* Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. 2023. Llama 2: Open foundation and fine-tuned chat models. _arXiv preprint arXiv:2307.09288_.
* Upadhyay and Massie (2022) Ashish Upadhyay and Stewart Massie. 2022. Content type profiling of data-to-text generation datasets. In _Proceedings of the 29th International Conference on Computational Linguistics_ , pages 5770–5782, Gyeongju, Republic of Korea. International Committee on Computational Linguistics.
* van Miltenburg et al. (2021) Emiel van Miltenburg, Miruna Clinciu, Ondřej Dušek, Dimitra Gkatzia, Stephanie Inglis, Leo Leppänen, Saad Mahamood, Emma Manning, Stephanie Schoch, Craig Thomson, and Luou Wen. 2021. Underreporting of errors in NLG output, and what to do about it. In _Proceedings of the 14th International Conference on Natural Language Generation_ , pages 140–153, Aberdeen, Scotland, UK. Association for Computational Linguistics.
* Wang and Sennrich (2020) Chaojun Wang and Rico Sennrich. 2020. On exposure bias, hallucination and domain shift in neural machine translation. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 3544–3552, Online. Association for Computational Linguistics.
* Wang et al. (2022) Fei Wang, Zhewei Xu, Pedro Szekely, and Muhao Chen. 2022. Robust (controlled) table-to-text generation with structure-aware equivariance learning. In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5037–5048, Seattle, United States. Association for Computational Linguistics.
* Wiseman et al. (2017) Sam Wiseman, Stuart Shieber, and Alexander Rush. 2017. Challenges in data-to-document generation. In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 2253–2263, Copenhagen, Denmark. Association for Computational Linguistics.
* Zhu et al. (2023) Fengbin Zhu, Moxin Li, Junbin Xiao, Fuli Feng, Chao Wang, and Tat Seng Chua. 2023. Soargraph: Numerical reasoning over financial table-text data via semantic-oriented hierarchical graphs. In _Companion Proceedings of the ACM Web Conference 2023_ , WWW ’23 Companion, page 1236–1244, New York, NY, USA. Association for Computing Machinery.
## Appendix A Model fine-tuning specifications
The first model, T5-base (T5-b), was fine-tuned on the full ToTTo training set
of 120,761 samples on a commodity server with a GeForce RTX 2080 GPU. The
training took around seven days. The second model, T5-large (T5-l), was fine-
tuned on a subset of 50,000 ToTTo samples on a secure cloud instance with an
NVIDIA A100 GPU, completing in around 48 hours. Both models were fine-tuned
using a constant learning rate of 0.0001, with the encoder’s maximum length
set to 512 tokens and the decoder’s maximum length set to 128 tokens for
ToTTo’s generation task (Kale and Rastogi, 2020). Single-precision floating-
point format (FP32) was employed for training on their respective GPU servers.
The batch size, beam size and training steps for each model are shown in Fig.
1.
Models | Batch size | Beam Size | Training steps
---|---|---|---
T5-base | 2 | 10 | 180,000
T5-large | 4 | 5 | 9,000
Figure 1: Model Specifications
## Appendix B Inter-Annotation Procedure for Participants
### B.1 Overview
The Input Data from the ToTTo dataset, includes the Page title, Section Title
and a Table with highlighted cells in yellow. These key parameters are
conditionally used for training the neural models to summarize a meaningful
and factual Text (as Output) focusing on: (i). highlighted cells in the table
(ii). their corresponding header values (iii) The main Title and (iv) The
Section Title.
For each of these tabular input data, we provided outputs generated by
different neural language models to annotators. Our goal is to evaluate
whether the neural outputs remain faithful and produce factually accurate
information based on the four parameters from the tabular input. The complete
table, including the non-highlighted cells, is provided to offer a clearer
understanding of the error annotation task.
### B.2 Domain
The inputs provided are specific to the domain of _politics_ , sourced from
Wikipedia tables (as part of the open-domain ToTTo dataset). The political
data within these tables is not limited to a single demography. Instead, it
encompasses various details from the election processes across multiple
countries, including:
* •
Election specifics such as Presidential, state, by-elections, council,
district, Legislative Assembly, and other elections unique to particular
countries.
* •
Information about Governors, Mayors, Ministers, and Ambassadors (about Foreign
Affairs).
* •
Details regarding the Speaker of the Assembly.
The first item related to election details is predominantly used in this error
annotation task.
### B.3 Error Annotation guidelines
We are only interested if the highlighted cell values from the table were used
to produce a factually correct sentence. Please also pay attention to the non-
highlighted cells in the same row as the highlighted cells, as this might be
required in some inputs to generate a meaningful sentence. Other non-
highlighted values in the table are not expected to be used for your
evaluation. Please read through the output texts and annotate cases for the
error categories as mentioned below. Each error is denoted with a superscript
for better readability.
##### NAMEN
* •
When names of the Party, Leader, place (Electorate), Ambassador etc., are
wrong (mostly nouns).
* •
Annotation includes both single tokens or multiple tokens to include the
complete names.
* •
Example Output text: Urban AhlinN is the Deputy Speaker of the Riksdag.
Remarks: NAME error because the correct deputy speaker was Tobias Billström as
per the tabular data.
* •
Example Output text: Kansas was won by Mitt Romney, Paul Ryan, Barack ObamaN,
and Joe BidenN, with Romney winning 59.66% of the popular vote, six electoral
votes and 38.05 percent. In this example, Barack Obama and Joe Biden are two
NAME errors because they did not win the election.
* •
In general, WednesdayN instead of Tuesday is a NAME error but MayD instead
of April is a DATE_DIMENSION error.
##### NUMBERU
* •
When the number of seats and/or the number of votes and/or % of votes are
incorrect. A single token is marked as an error.
* •
When the A-party won with a majority of 5.5%. But the correct one is 4.4%.
5.5%U is a NUMBER error.
* •
Output: The voter turnout was 8,90%, with 10,052 votes. Remarks: The actual
turnout was 81.90%U. Please note: the error here is NOT with the comma used
as decimal (as it is an acceptable decimal operator for international use);
Error because the number 81.90 was incorrect.
##### DATE_DIMENSIOND
* •
When the Date and/or Month and/or Year are wrong in the generated text, it is
annotated as one error.
* •
Example Output text: Cletus Avoka was the Minister for the Interior in the
Mills government from 2009 to 2012D. Remarks: 2010 is the right end term of
the year.
* •
As a general note, if the Output text did not capture Month and/or Date, but
has the correct year, then this is NOT an ERROR. It could go to OMISSION with
remarks, Omission of Date and Month.
##### WORDW
* •
When incorrect words such as verbs, prepositions, adjectives, adverbs and
conjunctions are found in the output.
* •
Single token is marked as a WORD error in most cases. Multiple tokens are
annotated when the auxiliary verb (was), an extension of a prepositional
phrase (along with) and others are incorrect.
* •
Example Output text: Carly Fiorina defeated Republican Tom Campbell with 56.4%
of the vote to DeVore’s 19.3%, along withW Al Ramirez and Tim Kalemkarian.
Remarks: Fiorina independently defeated all the leaders, so ‘along with’ is
wrong.
* •
Example Output text: Ling wonW the 2016 senate district against Democrat Josh
Newman, with 49.6 percent of the vote. Remarks: Ling lost the election as per
the tabular data.
* •
Some of the common WORD errors found in this data are _won, defeated, lost,
succeeded_ , adjectives such as _current_ governor other prepositions (_since,
in_ and so on).
##### CONTEXTC
* •
When the model’s output presents information that contradicts or makes
unsupported assumptions about the given input data. Group of tokens/span of
text are annotated as CONTEXT error.
* •
It can sometimes be tricky to check for this type of error. Please follow the
below sequence before marking this error.
* –
In case of simple misrepresentation based on the information in the input
data, it would be easier to mark the token as NAME, NUMBER, DATE_DIMENSION or
WORD error.
* –
In the case of a complex table structure, the outputs are likely to mess up
completely with the overall information in the provided input data. In this
case, it is hard to mark individual errors. Please go ahead and mark the group
of tokens/ span of text and annotate it as a CONTEXT error.
* –
For example, the output is: In the 2006 election for mayor of Florence
PendletonC, Michael D. Brown received 62,415 votes while Philip Pannell
received 21,552 votes and write-in candidatesC received 1,363 votes.
Annotation remarks:
* *
for mayor of Florence PendletonC \- the name of a person is misrepresented as
electorate (jurisdiction) in the output.
* *
write-in candidatesC \- this implies there was more than one write-in
candidate.
##### ADDITIONA
* •
When the model’s outputs have added words, phrases, or details that either
diverge from the input’s main topic or are unsupported by the given context.
* •
Single or group of tokens are annotated as ADDITION error.
* •
Example Output text: 2007 Algerian legislative election was held on May 17,
2007A. The results were as follows: 24 political parties won a total of 389
seats in the National Assembly. Remarks: The date and month are additional
information, that are not provided in the input. This is marked as an ADDITION
error. Other details in the output are correct.
##### OTHERO
* •
When the output repeats the same input multiple times producing garbage data.
In some cases, when the table data has the political party name in the
abbreviation, it produces garbage output. For example, when the tabular data
has a party name in abbreviation, it tries to produce a strange output. Output
text: GSSSDULSVDHSSO gained 5.31% of the vote
* •
When the output is incomplete for longer table input or when the output
repeats the same input multiple times without producing a complete text.
* •
When punctuation symbols are placed in inappropriate places, for example, an
apostrophe is missed for the Name of the Leader or Place.
##### NON-ENGLISHNE
* •
when the Unicode characters in non-English names are either replaced with
special characters or when these Unicode characters are omitted.
* •
For example, Pawe GraNE is a member of Sejm. Remarks: Paweł Graś is the
correct name here.
## Appendix C Llama 2 prompts
The below prompt is for the corrected tabular data when the input table has
party name, candidate and votes.
> ‘Given the input table data, the task is to:
> (i). Identify the party name, candidate name, and the number of votes
> received by each candidate.
> (ii). Determine the winner based on the highest number of votes. Then, put
> together the gathered information from (i) and (ii) into a single coherent
> sentence. Input table data: <Linearized table data>’
Below is one of the prompts used for the original table data without
mentioning any specific fields.
> ‘The task is to summarize the information from the given input table data
> into a single coherent sentence. Use only the information mentioned in the
> input table data. Input table data is: <Linearized table data>’
## Appendix D Steps for Correcting ToTTo Tabular Input
Algorithm 1 Manual Correction Procedure for ToTTo Tabular Input (elaborated in
Sec. 5.2). This main function takes tabular data and title details as input
parameters and returns the corrected tabular data. In all functions, we
excluded row and column indices for simplicity and readability.
function MainCorrectionProcedure($tabularData,title$)
if not DataSizeManageableForLongTable($tabularData$) then
return "Please simplify the tabular data with fewer records", null
end if
$leaderName,recordedData\leftarrow\textsc{IdentifyLeaderNameOrder}(tabularData,title)$
$correctionsMade\leftarrow\textbf{false}$
$correctionsMade\leftarrow\textsc{CorrectNonAtomicCells}(tabularData)\textbf{
or }correctionsMade$
$correctionsMade\leftarrow\textsc{UpdateHeaders}(tabularData)\textbf{ or
}correctionsMade$
$correctionsMade\leftarrow\textsc{AddressMissingValues}(tabularData)\textbf{
or }correctionsMade$
$correctionsMade\leftarrow\textsc{ReplaceSymbols}(tabularData)\textbf{ or
}correctionsMade$
if $correctionsMade$ then
return $(tabularData$, "Leader Data: " + $recordedData)$ $\triangleright$
Returns corrected input and leader data
else
return $(tabularData$, "Leader Data: " + $recordedData)$ $\triangleright$
Returns original input (if no issues) and leader data
end if
end function
Algorithm 2 Verify the number of rows and columns in Longer Tabular Input
(discussed in Sec. 5.2.4). This function validates the maximum allowable
number of rows and columns for the tabular data and returns true or false to
the main function.
function DataSizeManageableForLongTable($tabularData$)
$maxRows\leftarrow 20$
$maxCols\leftarrow 10$
return $(\text{length}(\textit{tabularData})\leq maxRows)\textbf{ and
}(\text{length}(\textit{tabularData}[0])\leq maxCols)$
end function
Algorithm 3 Identify Leader Name Order in Title and Table Rows (discussed in
Sec. 5.2.6). This function verifies three main scenarios for the order of
leader names in the input. It returns a tuple with title information for
leader name and a list of leader names from tabular input depending on the
scenario. Description for each scenario is briefly commented.
function IdentifyLeaderNameOrder($\textit{tabularData},\textit{title}$)
$leaderNameFromTitle\leftarrow\text{ExtractLeaderNameFromTitle}(\textit{title})$
$leaderNamesFromRows\leftarrow[\,]$
for each row in tabularData do
for each cell in current row do
if leader_name is found in cell then
Add leader_name to $leaderNamesFromRows$ $\triangleright$ Record leader’s
names from rows
end if
end for
end for
$\triangleright$ Scenario 1: Leader’s name found in title
if leaderNameFromTitle is not None then
$recordedData\leftarrow[\,]$ $\triangleright$ To store sequential order of
leader names
Add $leaderNameFromTitle$ to $recordedData$
for each leader_name in leaderNamesFromRows do
Add leader_name to $recordedData$
end for
return ($leaderNameFromTitle,recordedData$)
$\triangleright$ for e.g., this could return ("b", ["a", "b", "c"])
else
$\triangleright$ Scenario 2: Leader’s name not found in title
if $\text{length of }leaderNamesFromRows>0$ then $\triangleright$ if leader’s
name is in at least one row
$recordedData\leftarrow\text{"Leader name not in title"}$
for each leader_name in leaderNamesFromRows do
Add leader_name to $recordedData$
end for
return ($leaderNameFromTitle,recordedData$)
$\triangleright$ for e.g., this could return (None, ["Leader name not in
title", "a", "b", "c"])
else
$\triangleright$ Scenario 3: Leader name neither in title nor in rows
return ($leaderNameFromTitle,\text{"Leader name not in table"}$)
$\triangleright$ for e.g., this could return (None, ["Leader name not in
table"])
end if
end if
end function
Algorithm 4 Correct Non-Atomic Cells (discussed in Sec. 5.2.1). If a non-
atomic cell is found, the CorrectCell function separates multiple atomic
values into individual columns. We follow our manual correction procedure in
CellIsNonAtomic and CorrectCell. The function returns the corrected tabular
data along with a flag indicating if any corrections were made.
function CorrectNonAtomicCells($tabularData$)
$correctionsMade\leftarrow\textbf{false}$
for all $row\textbf{ in }tabularData$ do
for all $cell\textbf{ in }row$ do
if $\text{CellIsNonAtomic}(cell)$ then
$correctedCell\leftarrow\text{CorrectCell}(cell)$ $\triangleright$ Separate
multiple values into individual columns
$tabularData[cell]\leftarrow correctedCell$ $\triangleright$ Update the
corrected cell value
$correctionsMade\leftarrow\textbf{true}$
end if
end for
end for
return ($tabularData,correctionsMade$)
end function
Algorithm 5 Update Column and Row Headers to Atomic cells (discussed in Sec.
5.2.4). For each cell, the tabular input has header_value as true or false.
When the header value is true, we manually verify the function
HeaderIsIncorrect and correct the logic in UpdateHeader(cell). The function
returns the corrected tabular data with corrections made flag.
function CorrectHeaders($tabularData$)
$correctionsMade\leftarrow\textbf{false}$
for all $row\textbf{ in }tabularData$ do
for all $cell\textbf{ in }row$ do
if $\text{cell.header\\_value}=\textbf{true}$ then
if $\text{HeaderIsIncorrect}(cell)$ then
$correctedHeader\leftarrow\text{UpdateHeader}(cell)$ $\triangleright$ Update
column and row headers
$tabularData[cell]\leftarrow correctedHeader$ $\triangleright$ Update the
corrected header
$correctionsMade\leftarrow\textbf{true}$
end if
end if
end for
end for
return $(tabularData,correctionsMade)$
end function
Algorithm 6 Address Missing cell Values (discussed in Sec. 5.2.3). For each
row, we verify the missing cell values in RowHasMissingValues and pass the
right cell values in FillCellValues through a manual process. The function
returns the corrected tabular data with corrections made flag.
function AddressMissingValues($tabularData$)
$correctionsMade\leftarrow\textbf{false}$
for all $row\textbf{ in }tabularData$ do
if $\text{RowHasMissingValues}(row)$ then
$correctedRow\leftarrow\text{FillCellValues}(row)$ $\triangleright$ Pass
missing cells in the row
${tabularData}[row]\leftarrow correctedRow$
$correctionsMade\leftarrow\textbf{true}$
end if
end for
return $(tabularData,correctionsMade)$
end function
Algorithm 7 Replace Politics Domain-Specific Symbols and Abbreviate Party
Names with Correct Semantics/words (discussed in Sec. 5.2.5). For all cells in
the table, we verify containsDomainSpecificSymbols and
containsPartyAbbreviations functions and correct the values using
substituteSymbolWithEquivalent and containsPartyAbbreviations through a manual
process. The function returns the corrected tabular data with corrections made
flag.
function replaceSymbols(tabularData)
$correctionsMade\leftarrow\textbf{false}$
for all $cell\textbf{ in }\textit{tabularData}$ do
if $\text{containsDomainSpecificSymbols}(cell)$ then
$correctedValue\leftarrow\text{substituteSymbolWithEquivalent}(cell)$
$\triangleright$ Replace symbols with words
$tabularData[cell]\leftarrow correctedValue$ $\triangleright$ Update the
corrected cell value
$correctionsMade\leftarrow\textbf{true}$
else if $\text{containsPartyAbbreviations}(cell)$ then
$correctedValue\leftarrow\text{abbreviatePartyNames}(cell)$ $\triangleright$
Abbreviate party names
$\textit{tabularData}[cell]\leftarrow correctedvalue$ $\triangleright$ Update
the corrected cell value
$correctionsMade\leftarrow\textbf{true}$
end if
end for
return $(tabularData,correctionsMade)$
end function
In addition to the correction procedure detailed in Sec. 5.2, which provides
examples of the corrections applied to each tabular input problems, we present
a supplementary pseudocode in this section. In algorithm 1, we have a generic
function that takes tabular data and title (metadata) from ToTTo as input
parameters. This main algorithm executes six different functions.
The first two functions, DataSizeManageableForLongTable and
IdentifyLeaderNameOrder provide insights on the input problems leading to
factual errors in the outputs but do not correct the tabular input. The other
four functions, namely CorrectNonAtomicCells, UpdateHeaders,
AddressMissingValues, and ReplaceSymbols correct the input problems as
presented in algorithm 4, algorithm 5, algorithm 6, and algorithm 7. We
provide brief descriptions for each algorithm in the caption and comments.
## Appendix E Input Problem Types for four models
Fig. 2 shows how each of the four models is performing before and after data
corrections for each of the problem types. The left bars for each input
problem type represent output scores before data correction, while the right
bars represent scores after data correction. Colour coding of output scores
helps to identify and understand errors: green indicates no error, red
indicates an error, and yellow indicates an omission.
The number of samples for each model differs because we focus on outputs only
when the model had ‘errors’ or ‘omissions’ before correction. This approach
emphasizes the actual improvement in corrections. For example, T5-large had
errors in 37 samples before correction (left bar), the after-correction (right
bar) also shows improvements for the same 37 samples.
(a) T5-base: Error Analysis of 40 samples.Single record lacking
atomicityMultiple records lacking atomicityComplex table typeInsufficient
inputLonger inputToTTo specificList of Leader names$2$$4$$6$$8$$10$ Input
Problem Types: before data fix (left bar) versus after data fix (right bar)
Samples countNo errorOmissionsErrors
(b) T5-large model: Error Analysis of 37 samples.Single record lacking
atomicityMultiple records lacking atomicityComplex table typeInsufficient
inputLonger inputToTTo specificList of Leader names$2$$4$$6$$8$$10$ Input
Problem Types: before data fix (left bar) versus after data fix (right bar)
Samples countNo ErrorOmissionsErrors
(c) Llama 2-7b: Error Analysis of 38 samples.Single record lacking
atomicityMultiple records lacking atomicityComplex table typeInsufficient
inputLonger inputToTTo specificList of Leader names$2$$4$$6$$8$$10$ Input
Problem Types: before data fix (left bar) versus after data fix (right bar)
Samples countNo ErrorOmissionsErrors
(d) Llama 2-13b: Error Analysis of 35 samples.Single record lacking
atomicityMultiple records lacking atomicityComplex table typeInsufficient
inputLonger inputToTTo specificList of Leader names$2$$4$$6$$8$$10$ Input
Problem Types: before data fix (left bar) versus after data fix (right bar)
Samples countNo ErrorOmissionsErrors
Figure 2: Comparison of different input problem types before (original data)
and after data correction across four models. It shows the count of unique
samples for three categories: (i) errors, (ii) omissions, and (iii) no errors.
## Appendix F Input Data Corrections and Output Text
In this section, the input tabular data with outputs supplements the examples
elaborated in Sec. 5.2 for each of the input problems as presented from Table
10(c) to Table 16.
(a) Input Table with Original Cells highlighted in Yellow Page Title: 2012
Cardiff South and Penarth by-election
---
Section Title: By-election result
Party | % | $\pm$
Conservative | 19.9 | 8.4
Liberal Democrat | 10.8 | -11.5
(b) Input Table with Corrected Cells highlighted in Yellow Page Title: 2012
Cardiff South and Penarth by-election
---
Section Title: By-election result
Party | % | $\pm$ % difference with previous election
Conservative | 19.9 | 8.4
Liberal Democrat | 10.8 | -11.5
(c) Outputs Before and After Input Data Correction T5-large before fix: In the
2012 Cardiff South and Penarth by-election, the Liberal Democrats wonW 11.5%
of the vote.
---
T5-large after fix: In the 2012 Cardiff South and Penarth by-election, the
Liberal Democrats lost byW 11.5%.
Llama 2-13B before fix: 2012 Cardiff South and Penarth by-election was wonW
by the Liberal Democrats with a marginW of -11.5%.
Llama 2-13B after fix: 2012 Cardiff South and Penarth by-election saw Liberal
Democrat candidate lose by 11.5% of the vote compared to the previous
election.
Table 10: ToTTo specific: Swing percentage difference with previous election,
as detailed in Sec. 5.2.5. Before fix represents the output generated from
input Table 10(a) and after fix represents the output generated after
explicitly changing the header, as shown in input Table 10(b).
(a) Input Table with Original Cells highlighted in Yellow Page Title: 1914
United States Senate elections
---
Section Title: Races leading to the 64th Congress
State | Incumbent | Results | Candidates
| Senator | Party | |
South Dakota | Coe I.Crawford | Republican | Incumbent lost renomination. New senator elected. Democratic gain. | Edwin S. Johnson (Democratic) 48.32% Charles H. Burke (Republican) 44.47%
(b) Outputs with Original Input before Input Data Correction T5-base: In South
Dakota, Republican Coe I. Crawford lost renomination to Democrat Edwin S.
JohnsonC.
---
T5-large: South Dakota: Republican incumbent Coe I. Crawford lost his bid for
the seat to Democratic challenger Edwin S. JohnsonC.
Llama 2-7B: 1914 United States Senate elections in South Dakota were held on
November 3A, 1914. Incumbent Republican Senator Coe I. Crawford lost
renomination to Democratic candidate Edwin S. JohnsonC. Johnson defeated
Charles H. DunniganN by a margin ofW 48.32% to 47.56%U. This was a
Democratic gain from the previous election.
Llama 2-13B: 1914 United States Senate elections were held on November 3A,
1914 to elect members of the United States Senate in various states. The
Republican Party lost one seat and the Democratic Party gained one seat.
(c) Input Table with Corrected Cells highlighted in Yellow Page Title: 1914 United States Senate elections |
---|---
Section Title: Races leading to the 64th Congress |
State | Incumbent Senator | Incumbent Party | Results | Candidate Senator | Candidate Party | Candidate votes %
South Dakota | Coe I.Crawford | Republican | Incumbent lost renomination. New senator elected. Democratic gain. | Edwin S. Johnson | Democratic | 48.32%
South Dakota | Coe I.Crawford | Republican | Incumbent lost renomination. New senator elected. Democratic gain. | Charles H. Burke | Republican | 44.47%
(d) Outputs After Input Data Correction T5-base: In South Dakota, Republican
Coe I. Crawford lost renomination to Democrat Edwin S. JohnsonC.
---
T5-large: South Dakota: Republican incumbent Coe I. Crawford lost his bid for
the seat to Democratic challenger Edwin S. JohnsonC.
Llama 2-7B: 1914 United States Senate elections were held on November 3,A
1914\. In South Dakota, Coe I. Crawford, the Republican incumbent Senator lost
renomination to Edwin S. JohnsonC, a Democratic candidate. Charles H., the
Republican candidate also ran for the seat but he was defeated by Johnson.
Therefore, it is a Democratic gain in this election.
Llama 2-13B: 1914 United States Senate elections were held on November 3A,
1914 to elect members of the United States Senate in various states. Races
leading to the 64th Congress included South Dakota where Coe I. Crawford
(Republican) lost renomination and Edwin S. Johnson (Democratic) was elected.
Table 11: Complex Table input problems, as detailed in Sec. 5.2.2. Difficult
to fix this complex input type, yet Llama 2-13B produced most of the facts
right except for ADDITION error. Lost renomination ‘to Democrat candidate’ is
a CONTEXT error, because the model made unsupported assumptions about the
given input. Losing renomination is when the current senator failed to be
nominated in the primary to stand for reelection.
(a) Input Table with Original Cells highlighted in Yellow Page Title: List of
Japanese prime ministers by longevity
---
Section Title: Longevity of Japanese prime ministers
Prime Minister | Total time in Office | Date of Death | Lifespan
Sanetomi Sanjō | 60 days | 28 Feb 1891 | 53 years, 352 days
(b) Input Table with Corrected Cells highlighted in Yellow Page Title: List of
Japanese prime ministers by longevity
---
Section Title: Longevity of Japanese prime ministers
Prime Minister | Total time in Office | Date of Death | Lifespan
Sanetomi Sanjō | 60 days | 28 Feb 1891 | 53 years, 352 days
(c) Outputs Before and After Input Data Correction T5-base and T5-large before
and after fix (no improvement): Sanj SanetomiNE was the longest-livedW prime
minister in Japan at 53 years, 352 days.
---
Llama 2-7B before fix: The longest-servingW Japanese prime minister was Sanjō
Sanetomi who served for 53 years and 352 days.
Llama 2-7B after data fix: Sanjō Sanetomi was the Japanese prime minister and
served for a total of 60 days in office. He died at age 53 years and 352 days.
Llama 2-13B before data fix: The longest-livedW Japanese prime minister was
Sanjō Sanetomi, who served for 53 years and 352 days.
Llama 2-13B after data fix: Sanjō Sanetomi was the prime minister of Japan for
a total of 60 days. He lived for 53 years and 352 days.
Table 12: ToTTo specific - WORD error made by all models before fix, as
detailed in Sec. 5.2.5. After including ‘Total time in Office’, Llama 2 models
fixed the ‘longest-lived’ word error and introduced other addition errors. We
then revised the prompt and fixed the addition errors. T5 models had no
improvements after including information.
(a) Input Table with Original Cells highlighted in Yellow Page Title: Ling
Ling Chang
---
Section Title: 2016 Senate run
California’s 29th State Senate district election
Party | Candidate | % Votes
Democratic | Josh Newman | 50.4
Republican | Ling Ling Chang | 49.6
(b) Input Table with Corrected Cells highlighted in Yellow Page Title: Ling
Ling Chang
---
Section Title: 2016 Senate run
California’s 29th State Senate district election
Party | Candidate | % Votes
Democratic | Josh Newman | 50.4
Republican | Ling Ling Chang | 49.6
(c) Outputs Before and After Input Data Correction T5-base and T5-large before
fix: Chang defeatedW Democrat Josh Newman with 49.6% of the vote.
---
T5-base and T5-large after fix: Ling Ling Chang was defeated by Josh Newman, a
Democrat, with 50.4% of the vote to Chang’s 49.6%.
Llama 2-13B before fix: Ling Ling Chang wonW the 2016 Senate run with 49.6%
of the vote.
Llama 2-13B after fix: Ling Ling Chang, who was running for the Senate seat in
California, received 49.6% of votes while Josh Newman, her opponent, received
50.4%.
Table 13: Fixing ToTTo specific input problems: List of leader names for two
leaders. ‘Before’ shows WORD errors due to missing vote percentages and party
names for a leader. ‘After’ shows these errors are resolved by all models when
the missing values are provided. Key point is T5 model did not produce error
even though the title leader ‘Ling Ling Chang’ lost the election.
(a) Input Table with Original Cells highlighted in Yellow Page Title: Chuck
DeVore
---
Section Title: As U.S. Senate Candidate
Party | Candidate | Votes | %
Republican | Carly Fiorina | 1,315,429 | 56.4
Republican | Tom Campbell | 504,289 | 21.7
Republican | Chuck DeVore | 452,577 | 19.3
Republican | Al Ramirez | 42,149 | 1.8
Republican | Tim Kalemkarian | 19,598 | 0.8
(b) Input Table with Corrected Cells highlighted in Yellow Page Title: Chuck
DeVore
---
Section Title: As U.S. Senate Candidate
Party | Candidate | Votes | %
Republican | Carly Fiorina | 1,315,429 | 56.4
Republican | Tom Campbell | 504,289 | 21.7
Republican | Chuck DeVore | 452,577 | 19.3
Republican | Al Ramirez | 42,149 | 1.8
Republican | Tim Kalemkarian | 19,598 | 0.8
(c) Outputs Before and After Input Data Correction T5-base before fix: DeVore
defeatedW Republican Tom Campbell with 56.4%U of the vote, Al Ramirez with
19.3%, and Tim Kalemkarian with 19.3%C.
---
T5-base after fix: DeVore defeatedW Republican nominees Carly Fiorina (56.4%)
, Tom Campbell (21.7%), Al Ramirez (1.8%) and Tim Kalemkarian (0.8%)C.
T5-large before fix: DeVore was defeated by Republican Tom Campbell (56.4%)U,
Al Ramirez (19.3%), and Tim Kalemkarian (21.7%)C.
T5-large after fix: DeVore was defeated in the U.S. Senate race by Republican
nominees Carly Fiorina (56.4%), Tom Campbell (21.7%) , Al Ramirez (1.8%) and
Tim Kalemkarian (0.8%).C
Llama 2-7B before fix: Chuck DeVore is a Republican candidate for the U.S.
Senate in California. He received 19.3% of the vote in the primary election,
which was held on June 8th, 2010. The other candidates were Carly Fiorina
(56.4%), Tom Campbell (24.1%)U and Al Ramirez (10.2%)U. Tim Kalemkarian did
not run for this officeC.
Llama 2-7B after fix: The number of votes for each leader is as follows: Carly
Fiorina - 56.4% Tom Campbell - 21.7% Chuck DeVore - 19.3% Al Ramirez - 1.8%
Tim Kalemkarian - 0.8%. Carly Fiorina won the Republican primary for U.S.
Senate in California with 56.4 percent of the vote, followed by Tom Campbell
at 21.7 percent and Chuck DeVore at 19.3 percent.
Llama 2-13B before fix: Carly Fiorina is the leading candidate for U.S. Senate
with 56.4% of the vote, followed by Tom Campbell with 20.3%U, Chuck DeVore
with 19.3%, and Al Ramirez with 4.0%U.
Llama 2-13B after fix: The Republican Party candidate who won the most votes
in the U.S. Senate election was Carly Fiorina with 56.4% of the total votes.
Table 14: ToTTo specific input problems: List of Leader names for five
leaders, as detailed in Sec. 5.2.6. All models swapped the defeated leader
names and/or % of votes, resulting in WORD, NUMBER and CONTEXT errors. After
including the party names and % of votes for all leaders, Llama 2 models
corrected the errors. Fine-tuned T5 models still made errors. This could be
because of the pattern learned from the fine-tuned training data of ToTTo.
(a) Input Table with Original Cells highlighted in Yellow Page Title: Joseph
Haslet
---
Section Title: Almanac
Year | Office | Subject | Party | % Votes | Opponent | Party | % Votes
1804 | Governor | Joseph Haslet | Republican | 48% | Nathaniel Mitchell | Federalist | 52%
1807 | Governor | Joseph Haslet | Republican | 48% | George Truitt | Federalist | 52%
(b) Input Table with Corrected Cells highlighted in Yellow Page Title: Joseph
Haslet
---
Section Title: Almanac
Year | Office | Candidate | Party | % Votes | Opponent | Party | % Votes
1804 | Governor | Joseph Haslet | Republican | 48% | Nathaniel Mitchell | Federalist | 52%
1807 | Governor | Joseph Haslet | Republican | 48% | George Truitt | Federalist | 52%
(c) Outputs Before and After Input Data Correction T5-base before fix: Joseph
Haslet was electedW Governor in 1804, defeatingW Federalist Party candidate
Nathaniel MitchellN in the election of 1807.
---
T5-base after fix: Joseph Haslet was electedW Governor ofW the Republican
Party in 1804 and 1807, defeatingW the Federalist Party’s Nathaniel Mitchell
and George Truitt.
T5-large before fix: Joseph Haslet was electedW Governor in 1804,
defeatingW Federalist Party candidate Nathaniel Mitchell in the election of
1807.
T5-large after fix: In 1804 and 1807, Republican Joseph Haslet defeatedW
Federalist Nathaniel Mitchell and Federalist George Truitt.
Llama 2-13B before fix: 1804, Joseph Haslet was electedW governor with 48% of
the vote. In 1807, he was re-electedW with 48% of the vote.
Llama 2-13B after fix: 1804: Joseph Haslet (Republican) wonW with 48% of the
votes. 1807: George Truitt (Federalist) won with 52% of the votes.
Table 15: ToTTo specific input problems: List of Leader names, as detailed in
Sec. 5.2.6. Correction for this input is including the party name and
modifying the header ‘Subject’ to ‘Candidate’. All models generated Joseph
Haslet to be the winning candidate including the Llama 2 models that had
specific instructions.
(a) Input Table with Original Cells highlighted in Yellow Page Title: List of
members of the United States House of Representatives in the 67th Congress by
seniority
---
Section Title: List of Representatives by seniority
Rank | Representative | Party | District | Seniority date
Twenty-three non-consecutive terms
1 | Joseph G. Cannon | R | IL-18 | March 4, 1915
Fifteen consecutive terms
2 | Frederick H. Gillett | R | MA-2 | March 4, 1893
Fourteen non-consecutive terms
3 | Henry A. Cooper | R | WI-1 | March 4, 1921
One term
4 | … | … | … | …
5 | … | … | … | …
433 | … | … | … | …
434 | … | … | … | …
435 | Adam M. Wyant | PA-22 | WI-1 | March 4, 1921
Members joining the House, after the start of the Congress
… | … | … | … | …
(b) Outputs with Original Input before Input Data Correction T5-large before
fix: The 435th member of the United States House of Representatives in the
65th Congress was a member who joined the House after the start of CongressC.
---
Llama2-13B before fix: List of Representatives by seniority The table below
lists the 100.O
(c) Input Table with Corrected Cells highlighted in Yellow Page Title: List of
members of the United States House of Representatives in the 67th Congress by
seniority
---
Section Title: List of Representatives by seniority
Rank | Representative | Party | District | Seniority date | Term details
1 | Joseph G. Cannon | R | IL-18 | March 4, 1915 | Twenty-three non-consecutive terms
2 | Frederick H. Gillett | R | MA-2 | March 4, 1893 | Fifteen consecutive terms
3 | Henry A. Cooper | R | WI-1 | March 4, 1921 | Fourteen non-consecutive terms
4 | … | … | … | … | One term
5 | … | … | … | … | One term
433 | … | … | … | … | One term
434 | … | … | … | … | One term
435 | Adam M. Wyant | PA-22 | WI-1 | March 4, 1921 | One term
(d) Outputs After Input Data Correction T5-large after fix: There are 435
members of the United States House of Representatives in the 65th Congress.
---
Llama2-13B after fix: The table shows that there are 435 representatives in
the 65th Congress, and they are ranked according to their seniority.
Table 16: Longer Table input problems, as detailed in Sec. 5.2.4.
(a). Original tabular input had 435 rows with term details and other details
from nested headers (irrelevant details).
(b). The fine-tuned T5-large model hallucinated the output text with the
irrelevant header information (nested header issue). Llama 2-13B also
struggled to produce the right text and generated incomplete output. Llama
2-7B produced longer garbage output.
(c). First, we fixed the nested header by creating a separate column. Then, we
only passed 20 records (first 10 rank and 426 to 435 rank) along with title
information. We did not pass the irrelevant term details to the corrected
data.
(d). This simplified input records with relevant header details (Rank) fixed
the errors in both models.
|
# Determination of the asymptotic limits of adaptive photon counting
measurements for coherent-state optical phase estimation
M. A. Rodríguez-García Instituto de Investigaciones en Matemáticas Aplicadas
y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria,
Ciudad de. México 04510, Mexico M. T. DiMario Center for Quantum Information
and Control, Department of Physics and Astronomy, University of New Mexico,
Albuquerque, New Mexico 87131 Joint Quantum Institute, National Institute of
Standards and Technology and the University of Maryland, College Park,
Maryland 20742 P. Barberis-Blostein Instituto de Investigaciones en
Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México,
Ciudad Universitaria, Ciudad de. México 04510, Mexico F. E. Becerra
<EMAIL_ADDRESS>Center for Quantum Information and Control, Department of
Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131
###### Abstract
Physical realizations of the canonical phase measurement for the optical phase
are unknown. Single-shot phase estimation, which aims to determine the phase
of an optical field in a single shot, is critical in quantum information
processing and metrology. Here we present a family of strategies for single-
shot phase estimation of coherent states based on adaptive non-Gaussian,
photon counting, measurements with coherent displacements that maximize
information gain as the measurement progresses, which have higher
sensitivities over the best known adaptive Gaussian strategies. To gain
understanding about their fundamental characteristics and demonstrate their
superior performance, we develop a comprehensive statistical analysis based on
Bayesian optimal design of experiments, which provides a natural description
of these non-Gaussian strategies. This mathematical framework, together with
numerical analysis and Monte Carlo methods, allows us to determine the
asymptotic limits in sensitivity of strategies based on photon counting
designed to maximize information gain, which up to now had been a challenging
problem. Moreover, we show that these non-Gaussian phase estimation strategies
have the same functional form as the canonical phase measurement in the
asymptotic limit differing only by a scaling factor, thus providing the
highest sensitivity among physically-realizable measurements for single-shot
phase estimation of coherent states known to date. This work shines light into
the potential of optimized non-Gaussian measurements based on photon counting
for optical quantum metrology and phase estimation.
## I Introduction
Optical phase estimation is ubiquitous in many fundamental and practical
problems ranging from quantum state preparation [1, 2, 3], sensing [4],
communications [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], and information processing
[15]. In a photonic metrology problem [16, 17, 18, 19], an optical probe state
interacts with a physical system to interrogate its properties. This
interaction maps parameters of the system to the state of the optical probe,
where an optimal readout can be performed [17, 18, 15, 19]. When the physical
property of the system is mapped onto the phase of the optical probe, the
optimal quantum measurement is the canonical phase measurement [20], which
consists of projections onto phase eigenstates [21]. However, while
theoretically the canonical phase measurement exists, the physical realization
of projections onto phase eigenstates are not physically known [22]. Thus in
practical estimation problems in quantum metrology one seeks to determine the
limits in precision of physically realizable measurements, and the degree with
which they approach to the fundamental quantum limit in sensitivity [23, 24,
25, 26, 27, 17].
Physically realizable measurements of the optical phase have been widely
investigated [20, 28, 21] for diverse metrological problems with quantum and
classical fields [29, 30, 31, 32, 33] including sensing small deviations from
a known phase [29, 30, 31, 34, 32, 33] and estimation with repeated sampling
[35, 36] and measurements [30, 37, 38, 39, 40, 41]. Beyond these specific
estimation problems, measurements of the phase of a single optical mode in a
single-shot are central for quantum state preparation and detection [42, 43],
waveform estimation and sensing [44, 45, 46, 47], and quantum information
processing [48, 49, 50]. The standard measurement for optical phase estimation
is the heterodyne measurement [21], which samples both quadratures of the
electromagnetic field simultaneously from which the phase can be estimated.
However, the achievable sensitivity of the heterodyne measurement [21] is far
below the ultimate measurement sensitivity allowed by physics, given by the
canonical phase measurement [51, 21]. Adaptive measurement techniques based on
homodyne detection, a Gaussian measurement, can be used to align the phase
quadrature of the optical field with the measurement setting where the
homodyne measurement provides maximum sensitivity [32]. Adaptive homodyne has
been theoretically shown to surpass the heterodyne limit and get closer to the
canonical phase measurement for optical phase estimation of coherent states
[20, 21], providing the most sensitive Gaussian measurement of the optical
phase so far [21].
In a complementary measurement paradigm, quantum measurements of coherent
states based on photon counting, displacement operations, and feedback [8, 2,
9, 10, 11, 12, 13, 6, 52, 14] have enabled state discrimination below the
Gaussian sensitivity limits and approaching the Helstrom bound [5]. Recently,
some of the authors proposed and demonstrated a non-Gaussian measurement
strategy for single-shot phase estimation of coherent states, able to surpass
the heterodyne limit and approach the sensitivity limit of a canonical phase
measurement in the presence of loss and noise of real systems [2]. These
measurement strategies are based on realizing coherent displacements of the
input field and monitoring the output field with photon number resolving
detection. The information from the detection outcomes is then used to
implement real time feedback of displacement operations optimized to maximize
the measurement sensitivity of the phase of the input state. This estimation
strategy is the most sensitive single-shot non-Gaussian measurement of a
completely unknown phase encoded in optical coherent states so far [2]. In
this strategy the displacement operation optimization is realized by
maximizing either the information gain in subsequent adaptive steps of the
measurement or the sharpness of the posterior phase distribution. While these
cost functions are functionally different, both perform similarly and get
close to the ultimate sensitivity allowed by physics, the Crámer-Rao lower
bound (CRLB), in the limit of many photons and many adaptive measurements.
While the work in Ref. [2] demonstrated the potential of non-Gaussian
measurements for single-shot phase estimation, the superiority over adaptive
homodyne detection was not proven. A deeper understanding of the properties of
convergence and ultimate limits of the estimators produced by non-Gaussian
measurements is still missing. This is an open problem due to the complexity
associated with the analysis of these adaptive non-Gaussian strategies.
In this article we investigate a family of adaptive non-Gaussian strategies
based on photon counting for single-shot optical phase estimation, and assess
their performance compared to Gaussian measurements and the canonical phase
measurement. To analyze these non-Gaussian strategies, we use the mathematical
framework of Bayesian optimal design of experiments, which provides a natural
description of non-Gaussian adaptive strategies, allowing us to investigate
their fundamental characteristics and determine the limits in sensitivity,
which up to now has been a challenging problem. Our work provides a
comprehensive statistical analysis of adaptive non-Gaussian measurements for
parameter estimation, and the requirements to approach optimal bounds in the
asymptotic limit. We show that strategies based on photon counting and
feedback for single-shot phase estimation of coherent states provide superior
sensitivity over the best known adaptive Gaussian strategies, having the same
functional form as the canonical phase measurement in the asymptotic limit,
differing only by a scaling factor. This work provides a deep insight into the
potential of optimized non-Gaussian measurements for quantum communication,
metrology, sensing, and information processing.
## II Results
### II.1 Holevo Variance of Non-Gaussian Estimation Strategies
The non-Gaussian phase estimation strategies investigated here are based on
photon counting, displacement operations, and feedback, and are optimized by
maximizing a specific cost function. These strategies maximize either the
estimation precision (by minimizing the Holevo variance [51]), or the
information gain about the unknown parameter based on entropy measures
including mutual information, the Kullback-Lieber divergence, and the
conditional entropy [53, 54]. We note that these cost functions produce non-
identifiable likelihood functions that do not allow to correctly estimate a
cyclic parameter, such as the phase [55]. To address this problem, these non-
Gaussian strategies use the Fisher information to optimize the displacement
operations, which are the dynamical control variable in the strategy, to
guarantee that these cost functions provide identifiable likelihood functions,
and to enable optical phase estimation with near optimal performance.
In the problem of single-shot phase estimation with coherent states, an
electromagnetic field in a coherent state
$\rho_{0}=\left|\alpha\right\rangle\left\langle\alpha\right|$ interacts with a
physical system and experiences a unitary transformation
$e^{\mathrm{i}\phi\hat{n}}$, where $\hat{n}$ is the number operator. The phase
$\phi$ induced in the coherent state carries information about the system,
which can be extracted by a measurement of the output state
$\rho(\phi)=e^{\mathrm{i}\phi\hat{n}}\rho_{0}e^{-\mathrm{i}\phi\hat{n}}=\left|e^{\mathrm{i}\phi}\alpha\right\rangle\left\langle
e^{-\mathrm{i}\phi}\alpha\right|\,$.
Measurements onto $\rho(\phi)$, together with an estimator $\widehat{\phi}$ on
the measurement outcomes, provide an estimate of $\phi$, and a measurement
strategy aims to obtain the best estimation of the physical parameter. The
efficiency of such a strategy is characterized by the estimation variance as a
function of the number of photons in the coherent state. The most efficient
strategy provides a variance at the highest convergence rate towards zero as
the number of photons increase.
The standard measurement paradigm for phase estimation of Gaussian states is
the heterodyne measurement (a Gaussian measurement), with an estimator
variance of
$\text{Var}\left[\widehat{\phi}_{\text{Het}}\right]=1/2\lvert\alpha\rvert^{2}$.
Within the paradigm of Gaussian measurements, adaptive homodyne strategies
optimized to minimize the Holevo variance have achieved the best performance
among Gaussian measurements for single-shot phase estimation of coherent
states [20]. The best adaptive Gaussian measurement reported to date, termed
the Adaptive Mark-II (MKII), achieves a Holevo variance in the limit of large
number of photons of:
$\text{Var}\left[\widehat{\phi}_{\text{MKII}}\right]\approx\frac{1}{4\lvert\alpha\rvert^{2}}+\frac{1}{8\lvert\alpha\rvert^{3}}\,.$
(1)
While this optimized Gaussian strategy surpasses the heterodyne limit, it has
an error of order $1/|\alpha|^{3}$ above the Cramér-Rao Lower Bound
($1/4\lvert\alpha\rvert^{2}$), and does not reach the performance of the
canonical phase measurement [21]:
$\text{Var}\left[\widehat{\phi}_{\text{CPM}}\right]\approx\frac{1}{4\lvert\alpha\rvert^{2}}+\frac{5}{32\lvert\alpha\rvert^{4}}\,.$
(2)
In this work, we numerically show that non-Gaussian strategies for single-shot
phase estimation based on photon counting, optimized displacement operations,
and real-time feedback achieve an estimator variance smaller than Gaussian
strategies with an asymptotic scaling in the limit of high mean photon numbers
of:
$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]\approx\frac{0.250\pm
0.001}{\lvert\alpha\rvert^{2}}+\frac{0.520\pm
0.010}{\lvert\alpha\rvert^{4}}\,.$ (3)
Figure 1A summarizes the main result comparing the three asymptotic variances
for the canonical phase measurement (solid blue), MKII (solid red), and non-
Gaussian strategies (solid green and points). Figure 1B, shows the excess
Holevo variance compared to the canonical phase measurement
($\text{Var}[\cdot]-\text{Var}[\hat{\phi}_{\text{CPM}}]$) for Heterodyne
(purple), MKII (red), and non-Gaussian strategies (green).
Fig. 1: Asymptotic variances for different phase measurements. a: Holevo
variance in the limit of large mean photon (MPN) $\lvert{\alpha}\rvert^{2}$
for the canonical phase measurement in Eq. (2), the most sensitive Gaussian
measurement known to date (MKII) [21] in Eq. (1), and the non-Gaussian
strategies in Eq. (3). The points show numerically values for the non-Gaussian
strategies in a region where the analytical expression is not valid. Error
bars represent a 1-$\sigma$ standard deviation. b: Excess phase variance
($\text{Var}[\cdot]$ \- $\text{Var}[\widehat{\phi}_{\text{CPM}}]$) for
Heterodyne, MKII, and non-Gaussian strategies.
These non-Gaussian strategies implement a series of adaptive steps with
displacement operations optimized to maximize information gain, while ensuring
efficient phase estimators in the asymptotic limit. These strategies surpass
the best known Gaussian strategy in Eq. (1), and have the same functional form
as the canonical phase measurement in the asymptotic limit, differing only by
a scaling factor, thus providing the highest sensitivity among physically-
realizable measurements for single-shot phase estimation of coherent states
known to date.
Achieving this performance with non-Gaussian estimation strategies, however,
requires a deep understanding of the measurement process. To gain this
understanding, we use the mathematical framework of Bayesian optimal
experimental design, which provides a natural description of adaptive non-
Gaussian measurements. This allows us to optimize these strategies for single-
shot phase estimation with a Holevo variance given by Eq. (3).
### II.2 Bayesian optimal design of experiments
Phase estimation of coherent states based on photon counting with adaptive
coherent displacement operations can be defined in the context of Bayesian
optimal design of experiments. Optimal design of experiments allows for
improving statistical inferences about quantities of interest by appropriately
selecting the values of control variables known as designs [54, 56, 57]. In
this framework, it is assumed that the experimental data $y$ (the measurement
outcomes) can be modeled as an element of the set
$\mathcal{P}_{\Phi}=\left\\{p(y\mid\mathbf{d},\mathbf{\phi}),\,\,\,\mathbf{d}\in\mathcal{D}\,\,\,\mathbf{\phi}\in\Phi\right\\},$
(4)
where $\mathbf{d}$ is a parameter called design chosen from some set
$\mathcal{D}$ called design space, $\bm{\phi}\in\Phi$ is an unknown parameter
to be estimated, and the data $y$ comes from a sample space
$\mathcal{Y}\subseteq\mathbb{R}$. In this paradigm, the experimenter has full
control over the designs $\mathbf{d}$ and the ability to adjust them prior to
making a measurement. This allows for optimizing such measurement for
estimating the unknown parameter $\bm{\phi}$. Bayesian optimal design of
experiments goes beyond standard methods for parameter estimation based on
Bayesian statistical inference [58, 59, 60, 61, 62], by providing the suitable
mathematical framework to ensure optimal designs to find efficient estimators
for a general parameter space [63, 64, 65, 66].
In the Bayesian approach of optimal design [54, 56], the initial lack of
knowledge about $\bm{\phi}$ is modeled as a prior probability distribution
$p(\bm{\phi})$. The measurement aims to reduce the uncertainty of $\bm{\phi}$
as much as possible by the use of Bayes’ theorem over the prior distribution.
In an estimation problem, the optimal choice for the designs
$\mathbf{d}\in\mathcal{D}$ maximize the expected value of a cost function
$U(\bm{d},\bm{\phi},y)$ with respect to the possible outcomes of $y$ and
$\bm{\phi}$:
$\begin{split}\mathbf{d}^{\text{opt}}&=\arg\max_{\mathbf{d}\in\mathcal{D}}\text{E}\left[U(\mathbf{d},\bm{\phi},y)\right]\\\
&=\arg\max_{\mathbf{d}\in\mathcal{D}}\int_{\mathcal{Y}}\int_{\Phi}U(\mathbf{d},\bm{\phi},y)p(\bm{\phi}\mid\mathbf{d},y)d\bm{\phi}p(y\mid\mathbf{d})dy\\\
&=\arg\max_{\mathbf{d}\in\mathcal{D}}\int_{\mathcal{Y}}\int_{\Phi}U(\mathbf{d},\bm{\phi},y)p\left(\bm{\phi},y\mid\mathbf{d}\right)d\bm{\phi}dy\,.\end{split}$
(5)
A standard approach in optimal design of experiments considers choosing
$\mathbf{d}^{\text{opt}}$ at the beginning of the experiment an then sample
data from $p(y\mid\mathbf{d}^{\text{opt}},\bm{\phi})$ for all subsequent
trials. An alternative approach considers dynamically updating
$\mathbf{d}^{\text{opt}}$ on each trial, as more data is collected. The
advantage of this approach is that adaptive estimation strategies are never
less efficient than the non-adaptive ones [67].
The implementation of Bayesian optimal design of experiments requires a cost
function, such as the Kullback-Lieber divergence between the prior and
posterior distributions [68]:
$\begin{split}U(\mathbf{d},y)&=D_{\text{KL}}\left[p(\bm{\phi}\mid\mathbf{d},y)\mid\mid
p(\bm{\phi})\right]\\\
&=\int_{\Phi}p(\bm{\phi}\mid\mathbf{d},y)\log\left[\frac{p(\bm{\phi}\mid\mathbf{d},y)}{p(\bm{\phi})}\right]d\bm{\phi}.\end{split}$
(6)
The Kullback-Lieber divergence provides a distance between probability
distribution $p(\bm{\phi}\mid\mathbf{d},y)$ and $p(\bm{\phi})$ [53]. If
$p(\bm{\phi}\mid\mathbf{d},y)$ is equal to $p(\bm{\phi})$ then
$U(\mathbf{d},y)=0$ and there is not any gain of information about $\bm{\phi}$
by measuring with design $\mathbf{d}$ and outcome $y$.
Another cost function considered in optimal design is the conditional entropy
between the plausible values of $\bm{\phi}$ and $y$
$\begin{split}U(\mathbf{d})&=-H(\bm{\phi}\mid Y)\\\
&=-\sum_{y\in\mathcal{Y}}p(y)\int_{\Phi}p(\bm{\phi}\mid\mathbf{d},y)\log\left[p(\bm{\phi}\mid\mathbf{d},y)\right]d\bm{\phi},\end{split}$
(7)
which is a measure of how much information is needed to describe the outcomes
of the random variable $\bm{\phi}$ given that the value of another random
variable $Y$ is known [53]. However, we note that cost functions Eq. (6) and
Eq. (7) can be related to the mutual information [53]:
$\begin{split}U(\mathbf{d})=I(\bm{\phi}\mid
Y)&=\text{E}_{y}\left[D_{\text{KL}}\left[p(\bm{\phi}\mid\mathbf{d},y)\mid\mid
p(\bm{\phi})\right]\right]\\\ &=H(p(\bm{\phi}))-H(\bm{\phi}\mid
Y).\end{split}$ (8)
As a result, designs $d$ maximizing any of these cost functions in Eq. (6),
Eq. (7), or Eq. (8) are equivalent [54, 69, 56]. Moreover, in the asymptotic
limit, maximizing these cost functions is equivalent to minimizing the
determinant of the covariance matrix (D-optimality design criteria), that is,
in the asymptotic limit, optimizing these cost functions is equivalent to
optimizing any member within the family of D-optimal designs [54, 68].
Our goal is to apply the theory of Bayesian optimal design of experiments to
the problem of phase estimation of coherent states with photon counting and
adaptive coherent displacement operations. The adaptive non-Gaussian
estimation strategy consists of several parts: i) in the first adaptive step,
it uses an specific cost function and the prior information to choose the
design; ii) then, it performs a measurement; iii) based on the measurement
outcome, it uses Bayes’ theorem to update the probability distribution; iv)
and lastly, it uses a recursive approach, where this posterior probability
distribution becomes the prior of the subsequent measurement step i). The
estimation of $\bm{\phi}$ at each adaptive step is obtained from the maximum
posterior estimator (MAP) of the posterior probability distribution. This
approach requires that the MAP converge to the true value of the parameter
when the number of measurements increases.
In the adaptive mathematical framework of optimal experimental design,
Paninski [67] proved that under a set of regular modelling conditions and in
the case when $\phi\in\mathbb{R}$, cost functions based on mutual information
can allow for designs that lead to asymptotically consistent and efficient MAP
estimators with variance
$\sigma^{2}_{\text{INFO}}=\left(\underset{C\in
co\left(F(\phi;\mathbf{d})\right)}{\operatorname{arg}\,\operatorname{max}}\;\left\lvert
C\right\rvert\right)^{-1}.$ (9)
Here $co\left(F(\phi;\mathbf{d})\right)$ denotes the convex closure of the set
of Fisher information functions $F(\phi;\mathbf{d})$. In other words, the
estimations produced by $\widehat{\phi}$ converge to $\phi$ (consistency), and
the distribution of $\widehat{\phi}$ tends to a normal distribution with mean
$\phi$ and variance given by Eq. (9) when the number of adaptive steps tends
to infinity (efficiency).
Formally, the regularity conditions introduced in [67] can be stated as
follows:
1. 1.
The parameter space $\Phi$ is a compact metric space.
2. 2.
The log likelihood, $\log\left(p(y\mid\phi,\mathbf{d})\right)$ is uniformly
Lipschitz in $\phi$ with respect to some dominating measure on $\mathcal{Y}$.
3. 3.
The likelihood function is identifiable for $\phi$, that is, the likelihood
function has a unique global maximum.
4. 4.
The prior distribution, assigns a positive probability to any neighborhood of
the real value of $\phi$.
5. 5.
The Fisher information functions $F(\phi;\mathbf{d})$ are well defined for any
$\phi\in\Phi$ and $\mathbf{d}\in\mathcal{D}$.
6. 6.
The maximum of the convex closure of the set of Fisher information functions
$\left\\{F(\phi,\mathbf{d})\mid\mathbf{d}\in\mathcal{D},\,\phi\in\Phi\right\\}$
must be positive-definite, i.e., $\max_{C\in
co\left(F(\phi;\mathbf{d})\right)}\left\lvert C\right\rvert>0$.
We note that in the case of estimation of a scalar parameter, the maximization
of mutual information is equivalent to the minimization of the mean square
error (MSE) [67]:
$\begin{split}\rm{MSE}(\widehat{\phi})&=\rm{E}\left[\right(\widehat{\phi}-\phi\left){}^{2}\right]\\\
&=\text{Var}\left(\widehat{\phi}\right)+\left(\rm{E}\left[\widehat{\phi}\right]-\phi\right)^{2}.\end{split}$
(10)
This shows that the MSE is a trade-off between the estimator’s variance and
its bias. As a result, since the phase is a scalar quantity, in the asymptotic
limit both the mutual information Eq. (8) and the mean square error Eq. (10)
are appropriate cost functions to find optimal estimation strategies. Even
more, for unbiased estimators (such as the MAP estimator on the asymptotic
limit), the MSE corresponds to the estimator variance, then the maximization
of mutual information is equivalent to the minimization of estimator variance.
In practice, however, an estimation strategy would use a cost function that
can be calculated efficiently and with a high rate of convergence.
Phase estimation in coherent states.
Optimal phase estimation of coherent states of light aims to obtain the best
estimate, from the outcomes of a physical measurement, of an unknown phase
$\phi\in[0,2\pi)$ encoded in a coherent state
$\rho(\phi)=\left|e^{i\phi}\alpha\right\rangle\left\langle
e^{-i\phi}\alpha\right|\,$. The most general description of a physical
measurement is given by a POVM, a positive operator-valued measure. A
measurement $M$ with a discrete set of outcomes $\left\\{m\,|\,m\in
S\subseteq\mathbb{Z}\right\\}$, can be represented as a POVM
$M=\left\\{M(m)\,|\,m\in S\right\\}$, where the operators $M(m)$ are positive
bounded $M(m)>0$ and resolve the identity $\sum_{m}M(m)=I$, ${\forall m\in S}$
[70]. By the Born rule, the probability for $m$ conditioned to $\phi$ is
$\text{ Tr}\left[M(m)\rho(\phi)\right]=p\left(m\mid\phi\right)\,.$ (11)
According to information theory, if an estimator $\widehat{\phi}$ of ${\phi}$
is constructed using a sample from a POVM $M$, the limit in the accuracy of
$\widehat{\phi}$ is given by the Crámer-Rao Bound [71, 72]
$\text{Var}_{\phi}\left[\widehat{\phi}\right]\geq\frac{1}{F_{M}(\phi)}.$ (12)
Here $F_{M}(\phi)$ is the Fisher information of $M$ about $\phi$, which
quantifies how much information about $\phi$ is carried in a sample from $M$:
$F_{M}(\phi)=\text{E}_{\phi}\left[\left(\frac{\partial}{\partial\phi}\left[\log\left(p\left(m\mid\phi\right)\right)\right]\right)^{2}\right].$
(13)
Since the Fisher information in Eq. (13) depends on the POVM $M$, the
maximization of the Fisher information over all POVMs provides the lowest
possible Cramér-Rao bound. This maximum Fisher information over all POVMs is
known as the quantum Fisher information $F_{\text{Q}}$ (QFI), and the lowest
possible bound in the accuracy of $\widehat{\phi}$ is known as the quantum
Cramér-Rao bound (QCRB) [5, 73, 72]. In the case of phase estimation of
coherent states $F_{\text{Q}}=4\lvert\alpha\rvert^{2}$, and the QCRB is:
$\text{Var}_{\phi}\left[\widehat{\phi}\right]\geq\frac{1}{4\lvert\alpha\rvert^{2}}.$
(14)
In the limit of large photon number $\lvert\alpha\rvert\gg 1$, the QCRB is
saturated by the canonical phase measurement (the optimal phase measurement),
which is described by the POVM [51]:
$M(\widehat{\phi})=\frac{1}{2\pi}\sum_{n,m=0}^{\infty}\left|n\right\rangle\left\langle
m\right|e^{\mathrm{i}\left(n-m\right)\widehat{\phi}},$ (15)
where $\left|n\right\rangle$ is an eigenstate of the number operator
$\hat{n}$. The operator $M(\widehat{\phi})$ is an element of the canonical
phase measurement whose outcome is a number
$\widehat{\phi}\in\left[0,2\pi\right)$, which can be used as an estimation for
$\phi$.
The optimality of the canonical phase measurement indicates that its Holevo
variance
$V_{\text{CPM}}=\left\lvert
e^{-\alpha^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{n!\sqrt{n+1}}\right\rvert^{2}-1\,,$
(16)
is the fundamental bound of estimation precision [51, 20]. Although there are
proposals that attempt to implement this POVM, they have not been able to
reach the fundamental bound Eq. (16), or Eq. (2). For instance, in [74] it was
possible to obtain the canonical measurement distribution as the marginal of a
joint measurement in phase space producing a worse performance in the context
of phase estimation. Moreover, the canonical phase measurement was implemented
for the case of one-photon wave packet using quantum feedback [22]. However,
for the case of higher dimensional states, such as coherent states, this
problem remains open.
While there is not a satisfactory known method to implement the canonical
phase measurement, Gaussian strategies serve as a standard of physically
realizable measurement techniques for phase estimation of coherent states. The
natural benchmark in the Gaussian strategies is the heterodyne detection,
whose variance is lower bounded by
$\text{Var}\left[\widehat{\phi}_{\text{Het}}\right]=1/2\lvert\alpha\rvert^{2}$
[21]. Several adaptive Gaussian schemes have been shown to exceed the lower
bound for heterodyne detection. The most efficient Gaussian measurement
reported to date, termed the Adaptive Mark-II (MkII) strategy [20], has a
variance in the limit of $\lvert\alpha\rvert\gg 1$ given by Eq. (1).
Nonetheless, these adaptive Gaussian strategies cannot reach the performance
for the canonical phase measurement in Eq. (2) [21].
The proposed non-Gaussian strategies for single-shot phase estimation of
coherent states are based on optimized adaptive measurements with photon
number resolving detection. These non-Gaussian strategies are able to
outperform the best known Gaussian strategies and closely follow the
performance of the canonical phase measurement in the limit of large photon
number.
### II.3 Adaptive non-Gaussian phase estimation
The proposed non-Gaussian adaptive estimation strategies based on adaptive
photon counting [2] aim to estimate the phase $\phi\in\Phi=[0,2\pi)$ of a
coherent state
$\rho(\phi)=\left|e^{\mathrm{i}\phi}\alpha\right\rangle\left\langle
e^{-\mathrm{i}\phi}\alpha\right|\,$ with mean photon number
$\text{E}\left[\hat{n}\right]=\lvert{\alpha}\rvert^{2}$ using a finite number
of adaptive measurement steps, and based on the prior information $p(\phi)$
about $\phi$. In every adaptive step $l=1,2,\cdots,L$, the input coherent
state with energy $\lvert{\alpha}\rvert^{2}/L$ interferes with a local
oscillator field, which implements a displacement operation
$\hat{D}\left(\beta\right)\left|\alpha\right\rangle=\left|\alpha+\beta\right\rangle,\,\beta\in\mathbb{C},$
with phase and amplitude chosen by some rule, in general depending on previous
measurement outcomes. This is followed by a photon number detection
measurement with a given photon number resolution (PNR) $m$ of the detectors
[6], $m\in\mathbb{N}$. In practice, since the energy in each adaptive step is
$\lvert{\alpha}\rvert^{2}/L$, the strategy will only require moderate PNR
resolution ($m<10$) as $L$ increases.
In the first adaptive measurement $l=1$ [2], the strategy makes a random guess
hypothesis $\beta_{0}\in\mathbb{C}$ about the optimal $\beta$, and applies the
POVM
$\displaystyle\left\\{\hat{D}(\beta_{0})\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta_{0})\right\\}_{n=0}^{m-1}\cup$
$\displaystyle\left\\{\mathbb{I}-\sum_{n=0}^{m-1}\hat{D}(\beta_{0})\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta_{0})\right\\}$ (17)
over the state $\left|e^{\mathrm{i}\phi}\alpha/\sqrt{L}\right\rangle$. In the
POVM in Eq. (II.3), the sum over Fock states $\left|n\right\rangle\left\langle
n\right|$ models the photon detection on the displaced state with a detector
with PNR($m$) [6]. The corresponding Wigner function describing a photon-
number resolving detector shows non-Gaussian features with negative values.
For this reason, these adaptive estimation techniques are called “non-
Gaussian”, despite that the estimator produced is asymptotically normal (this
result will be proved in the remainder of this section).
Given the detection outcome $n_{1}$ in $l=1$, the posterior probability
distribution becomes
$p\left(\phi\mid n_{1};\beta_{0}\right)\propto\mathcal{L}(\phi\mid
n_{1};\beta_{0})p(\phi),$ (18)
where $\mathcal{L}\left(\phi\mid n_{1};\beta_{0}\right)$ is the likelihood
function given by
$\begin{split}\mathcal{L}\left(\phi\mid
n_{1};\beta_{0}\right)&=p\left(n_{1}\mid\phi;\beta_{0}\right)\\\
&=\operatorname{Tr}\left[\hat{D}(\beta_{0})\left|n_{1}\right\rangle\left\langle
n_{1}\right|\hat{D}^{\dagger}(\beta_{0})\rho(\phi)\right]\,.\end{split}$ (19)
The phase estimate $\phi_{1}$ in this adaptive step corresponds to the MAP
estimator $\widehat{\phi}_{\text{MAP}}$,
$\phi_{1}=\widehat{\phi}_{\text{MAP}}(n_{1})$, with the posterior probability
distribution in Eq. (18). Using the posterior phase distribution in Eq. (18)
as the prior for the next adaptive step $l=2$, the strategy optimizes a cost
function $U(\beta)$ to obtain the next value of $\beta$ denoted as
$\beta_{1}$, and implements the POVM in Eq. (II.3) with $\beta_{1}$. The
Bayesian updating procedure is repeated at each step $l\geq 2$. After $l$
adaptive measurements the posterior probability distribution becomes
$\begin{split}p(\phi\mid\mathbf{n},\bm{\beta})&=p(\phi\mid
n_{l},...,n_{1},\beta_{l-1},...,\beta_{0})\\\
&\propto\prod_{i=1}^{l}p(n_{i}\mid\phi,\beta_{i-1})p(\phi).\end{split}$ (20)
Here $n_{i}$ is the observed photon detection at step $i$. Using the MAP on
this phase distribution, we obtain the $lth$ estimation $\widehat{\phi}_{l}$.
The procedure is repeated until the last measurement step $L$.
This parameter estimation strategy is a particular case of Bayesian optimal
design of experiments, where the parameters $\beta\in\mathbb{C}$ are the
designs, and which are optimized to estimate a phase $\phi\in[0,2\pi)$. Since
the optimal value for $\beta$ on each adaptive step depends on all previous
detection results, the cost function to be optimized is a function of the
posterior distribution in Eq. (20). A suitable choice of cost function, such
as the mutual information or the estimator variance, can provide a sequence of
estimations $\widehat{\phi}_{n}$ that approaches the true value of $\phi$ [2].
In the case of estimation of cyclic parameters, such as phase estimation, the
posterior distribution in Eq. (20) is $2\pi$ periodic, and the moments of
$\widehat{\phi}$ cannot be calculated as in the linear case [51]. In such
situations, the first moment of the cyclic random variable $X$ is defined as
$\rm{E}\left[e^{\mathrm{i}X}\right]$, and the dispersion of an estimator
$\widehat{\phi}$ is calculated using the Holevo Variance [51]:
$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]=\frac{1}{\left\lvert\rm{E}\left[e^{\mathrm{i}\widehat{\phi}}\right]\right\rvert^{2}}-1\,,$
(21)
which is the analogous to the mean square error. The minimization of the
uncertainty about $\phi$ (positive square root of Eq. (21)), requires
maximization of
$S(\beta,m)=\left\lvert\rm{E}\left[e^{\mathrm{i}\widehat{\phi}}\right]\right\rvert$,
known as the sharpness of the distribution. Then, the suitable cost function
for the adaptive protocol is the average sharpness:
$\bar{S}(\beta,m)=\sum_{n=0}^{m}p(n)\left\lvert\int_{\Phi}e^{\mathrm{i}\phi}p\left(\phi\mid
n,\beta\right)d\phi\right\rvert.$ (22)
Identifiability of Likelihood.
To guarantee a consistent asymptotic estimator the optimized estimation
strategies require to satisfy the regularity conditions 1-6 described in Sec.
II.2. For phase estimation, the regularity conditions $1\text{ and }2$ are
satisfied given that $\phi$ is an interior point of $\Phi=[0,2\pi)$. Moreover,
given that the probability
$\begin{split}&p(n\mid\phi;\beta)=\text{Tr}\left[\left|\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}\right\rangle\left\langle\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}\right|\hat{D}(\beta)\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta)\right]\\\
&=\frac{\exp\left(-\lvert\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}-\beta\rvert^{2}\right)\lvert\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}-\beta\rvert^{2n}}{n!},\,\alpha\in\mathbb{R}_{+},\,\beta\in\mathbb{C}\end{split}$
(23)
is well defined and two times differentiable, the conditions $4$, $5$, and $6$
are directly satisfied. However, the condition $3$ is not satisfied in
general. If one chooses $\beta$ as the value that maximizes the mutual
information (8) or the average sharpness (22), the resulting likelihood
function can have in general two maxima, that is, a non-identifiable
likelihood function [30]. In that case it is not possible to guarantee the
existence of a consistent estimator.
To address the challenge of working with non-identifiable likelihood
functions, we use designs with a fixed relation between the phase of $\beta$
and the amplitude $|\beta|$, given by $\beta=f(\theta)e^{\mathrm{i}\theta}$,
with $\theta\in[0,2\pi)$ and $f(\theta)$ a real function. These experimental
designs result in a cost function $U$ that is a function of $\theta$.
To see how this method solves the problem of non-identifiability, we observe
that the probability $p(n\mid\phi;\beta)$ in Eq. (23) is Poisson distributed,
$\begin{split}p(n\mid\phi;\beta=\lvert\beta\rvert
e^{\mathrm{i}\theta})&=\frac{e^{-\lambda}\cdot\lambda^{n}}{n!},\end{split}$
(24)
with
$\lambda=\lvert\alpha\rvert^{2}/L+\lvert\beta\rvert^{2}-2\lvert\alpha\rvert\lvert\beta\rvert\cos\left(\phi-\theta\right)/\sqrt{L}$.
Then, for $L$ adaptive steps with results $\mathbf{n}=(n_{1},...,n_{L})$, the
likelihood function is the product of $L$ probability functions of the form of
Eq. (24):
$\begin{split}\mathcal{L}(\mathbf{n},\phi)=\prod_{i=1}^{L}p(n_{i}\mid\phi;\beta_{i}=\lvert\beta\rvert
e^{\mathrm{i}\theta_{i}})&=\prod_{i=1}^{L}\frac{e^{-\lambda_{i}}\cdot\lambda_{i}^{n_{i}}}{n_{i}!}\,.\end{split}$
(25)
Here, each $\theta_{i}$ depends on the cost function and previous POVM
outcomes. The choice of the experimental designs with $|\beta|=f(\theta)$ can
force the adaptive strategy to change $\theta_{i}$ in each step. In this case,
the likelihood function in Eq. (25) becomes identifiable because the product
of probability functions with different $\theta_{i}$ produces a likelihood
with a global maximum. To see this, note that if $\theta_{i}=\theta$ is fixed,
even if the parameter $|\beta|$ is different in each adaptive step of the
protocol, the likelihood function from a sequence of independent random
variables with probability function given by Eq. (24) has two maxima over
$\Phi$. One of the maxima will always be around $\phi$ and the other will
depend on the value of $\theta$. On the other hand, if the strategy allows
$\theta$ to change at each adaptive step, the second maximum is suppressed,
since the functions whose product constitutes the likelihood Eq. (25) will
each have a second maxima at different positions
$\theta_{i}\neq\theta_{j}\,(i\neq j)$. As a result, with the experimental
designs $\beta=f(\theta)\exp(\mathrm{i}\theta)$, the likelihood functions
satisfy all the regularity conditions described in Sec. II.2.
Bayesian optimal design of $|\beta|$.
Any viable estimation strategy should aim to achieve the QCRB (14). Therefore,
natural choice for $\lvert{\beta}\rvert$ is the one that maximizes the Fisher
information. For a discrete random variable, the Fisher information is given
by [72, 75]:
$F(\phi)=\sum_{n=0}^{\infty}p(n\mid\phi)\left(\frac{\partial}{\partial\phi}\log\left(p(n\mid\phi)\right)\right)^{2}.$
(26)
The Fisher information for a particular design $\beta=\lvert{\beta}\rvert
e^{\mathrm{i}\theta}$ for Poisson distributions Eq. (24) results in
$F(\phi;\beta)=\frac{4\lvert\alpha\rvert^{2}\lvert\beta\sin^{2}(\phi-\theta)\rvert^{2}}{\lvert\alpha\rvert^{2}+L\lvert\beta\rvert^{2}-2\lvert\alpha\rvert\lvert\beta\rvert\cos(\phi-\theta)\sqrt{L}}.$
(27)
Optimizing over $|\beta|$, the Fisher information becomes:
$F(\phi,\beta_{\text{opt}})=4\lvert\alpha\rvert^{2}/L,$ (28)
where
$\beta_{\text{opt}}=\frac{\lvert\alpha\rvert}{\sqrt{L}\cos(\phi-\theta)}$ (29)
is the value of $|\beta|$ that maximizes the Fisher information.
Unfortunately, since $\beta_{\text{opt}}$ depends on $\phi$, its
implementation is not practical, because it would be required to already know
a priori the very same parameter that one wants to estimate. To address this
problem, the optimal Bayesian design $\beta_{\text{opt}}$ can be estimated as:
$\widehat{\beta}_{\text{opt}}=\frac{\lvert\alpha\rvert}{\sqrt{L}\cos(\widehat{\phi}-\theta)},$
(30)
where $\widehat{\phi}$ is the MAP estimator. As a result, the non-Gaussian
estimation strategy has now only one design, the phase $\theta$. With
$\beta=\widehat{\beta}_{\text{opt}}$, the Fisher information becomes:
$F(\Delta,\widehat{\beta}_{\text{opt}})=\frac{4\,\sin^{2}(\Delta)\,\alpha^{2}}{L(\cos^{2}\left(\delta+\Delta\right)-2\,\cos(\Delta)\,\cos\left(\delta+\Delta\right)+1)}\,,$
(31)
where $\delta=\widehat{\phi}-\phi$ and $\Delta=\phi-\theta$.
Note that $F(\Delta,\widehat{\beta}_{\text{opt}})$ becomes a random variable
with outcomes depending on $\widehat{\phi}$ through
$\widehat{\beta}_{\text{opt}}$. Moreover, for a random initial design
$\theta_{1}$, a cost function given by the expected sharpness or the mutual
information with $\widehat{\beta}_{\text{opt}}$ in Eq. (30), the likelihood
function in Eq. (25) becomes identifiable. As a result, the non-Gaussian
strategy then leads to consistent and efficient MAP estimators [67].
Asymptotic behavior.
In general $\delta=\widehat{\phi}-\phi\neq 0$, and the Fisher information has
a strong dependence on $\theta$. For example, if $\theta\to\phi$ then
$F(\Delta=0,\widehat{\beta}_{\text{opt}})\to 0$, and negligible information is
gained when the system is measured. Therefore, the optimal value of $\theta$
should result in the value of $\Delta$ that maximizes the expected value of
$F(\Delta,\widehat{\beta}_{\text{opt}})$. By observing that the expected value
$\rm{E}[\delta^{2n+1}]=0$, with $n\in\mathbb{N}$, we see that the optimal
value of $\Delta$ is $\pi/2$. As a result, an efficient adaptive strategy
would make $\Delta_{L}=\phi-\theta_{L}$ tend to $\pi/2$ as $L$ increases.
However, in the limit of $\Delta\to\pi/2$ and $\delta\to 0$,
$|\widehat{\beta}_{\text{opt}}|\to\infty$ (see Eq. (30)), and we expect that
when the strategy is implemented $\Delta<\pi/2$. These findings are consistent
with our numerical simulations of the non-Gaussian adaptive strategy, where we
observe that for $L\gg 1$, $\Delta\to\pi/2-\epsilon$, with $\epsilon$ a small
positive real number, and $|\beta|$ stays finite around a value that the PNR
detector in the strategy can resolve. Note that $\hat{\beta}_{\text{opt}}$,
which maximizes the Fisher information given $\theta$, does not necessarily
maximizes the cost function $U(\beta)$ for $\beta\in\mathbb{C}$. However
restricting $\beta$ to the set that maximizes the Fisher information makes the
phase of $\beta$ change in each step forcing the likelihood to be
identifiable. Moreover, when $\hat{\phi}\to\phi$, $\hat{\beta}_{\text{opt}}$
tends to the value that maximizes $U$ [2].
Given that the Fisher Information is additive, for any $L$ measurements made
using the optimal design, $\beta_{\text{opt}}$ in Eq. (29), the total Fisher
information for this design corresponds to the quantum Fisher information
$4|\alpha|^{2}$. However, since $\beta_{\text{opt}}$ is not known, it is not
possible to choose a design $\beta$ for which its Fisher information equals
the quantum Fisher information. This is already implied by the fact that the
canonical phase measurement does not saturate the Cramér-Rao bound. To show
this, we observe that for and estimator very close to the true value
$\delta=\widehat{\phi}-\phi\ll 1$ for $L$ adaptive measurements, the Fisher
information (Eq. (31)) is
$F(\phi,\widehat{\beta}_{\text{opt}})\approx 4\,|\alpha|^{2}(1-\delta^{2})\,.$
(32)
On the other hand, the best possible estimator of $\delta$ in each step
satisfies $E\left[\delta^{2}\right]\geq 1/4|\alpha|^{2}$, so that
$F(\phi,\widehat{\beta}_{\text{opt}})\lesssim 4\,|\alpha|^{2}-1\,,$ (33)
independently of $L$. We conclude that the adaptive non-Gaussian strategies do
not saturate the Cramér-Rao bound for finite $|\alpha|$. Nevertheless, these
adaptive estimation schemes outperform the most sensitive Gaussian strategy
known to date, and show a similar asymptotic scaling as the canonical phase
measurement.
### II.4 Performance of Non-Gaussian Adaptive Strategies
We numerically investigate the performance of the estimator produced by non-
Gaussian adaptive strategies for phase estimation for different number of
adaptive steps $L$, photon number resolution PNR($m$), and average photon
number $|\alpha|^{2}$. To assess the performance of these strategies, we
compare our results with the best known Gaussian measurement for phase
estimation, termed Mark II (MKII) strategy [21], and with the canonical phase
measurement. As discussed in Sec. II.2, the performance of non-Gaussian
adaptive strategies using cost functions including the Kullback-Lieber
divergence Eq. (6), conditional entropy Eq. (7), mutual information Eq. (8),
and expected sharpness Eq. (22) are equivalent in the asymptotic limit. In our
numerical simulations, however, we use the expected sharpness in Eq. (22) as
the cost function for the optimization of the strategy, because it
substantially reduces the number of operations for this optimization and the
computational overhead.
In our numerical analysis, we use Monte Carlo simulations and generate
sufficient numerical data samples to reduce statistical uncertainties. Fig. 2A
shows the Holevo variance for the non-Gaussian adaptive strategy, as a
function of the number of adaptive steps $L$ for $|\alpha|^{2}=1$, for
different PNR: $m=1,3,6$. We observe that the adaptive non-Gaussian scheme
with PNR($1$) and $L\geq 100$ (green dots with error bars) outperforms the
most sensitive Gaussian strategy known to date, the MKII, (red dashed line).
Moreover, strategies with PNR($3$) (yellow dots with error bars) and PNR($6$)
(light blue dots with error bars) outperform the MKII strategy with only
$L\approx 30$ and $L\approx 20$, respectively, achieving a smaller Holevo
variance with fewer adaptive measurements compared to PNR(1). We have observed
similar behavior for non-Gaussian strategies optimizing different cost
functions, such as the mutual information.
To investigate the asymptotic behavior of the adaptive non-Gaussian strategy,
we assume an exponential dependence for the Holevo variance with $L$ (solid
lines in Fig. 2A):
$y(L,\alpha)=\text{A}e^{-\text{B}\cdot L}+\text{C}.$ (34)
The exponential model is a few parameter model that allows us to quantify
asymptotic trends when the datasets have rapidly decaying tails. This is a
widely used model for studying the asymptotic scaling of estimators as a
function of resources in diverse metrological problems [64, 76, 77].
We fit the numerical data from Monte Carlo simulations to Eq. (34) to estimate
the constants $A$, $B$, and $C$. Our results show that the asymptotic constant
$C=0.751\pm 0.002$ for PNR(1), $C=0.719\pm 0.004$ for PNR(3), and $C=0.714\pm
0.003$ for PNR(6). We observe that these values are smaller than the asymptote
of the MKII Gaussian strategy, but larger than the one for the canonical phase
measurement ($0.673$, blue dashed line). We note that while $C$ for PNR(3) is
larger than for PNR(6), they are statistically equal due to their
uncertainties. This prevents us from drawing any conclusions for larger values
of $m$.
Fig. 2: Asymptotic limit for the estimator of Holevo variance and optimal
design. a: Holevo variance for $\alpha=1$ and PNR $m=1,3,6$ as a function of
adaptive steps $L$. Note that the non-Gaussian strategy surpasses the MKII
strategy (red dashed line at $y=0.767$) with $L>100$, $L>30$ and $L>20$ for
PNR($1$), PNR($3$) and PNR($6$), respectively. The lines are obtained fitting
the exponential model Eq. (34). Estimates for these strategies result in
$(A,B,C,\mathrm{RSE})=(0.11\pm 0.02,0.059\pm 0.014,0.7145\pm 0.003,0.00058)$
for PNR($6$), $(A,B,C,\mathrm{RSE})=(0.22\pm 0.04,0.082\pm 0.0157,0.719\pm
0.004,0.00076)$ for PNR($3$), $(A,B,C,\mathrm{RSE})=(0.41\pm 0.05,B=0.117\pm
0.013,0.7517\pm 0.0027,0.00052)$ for PNR($1$). The shaded region represents
one standard deviation. b: Optimal design as a function of $L$. As $L$
increases the optimal design tends to $\pi/2$. Numerical data are obtained
with $10000$ Monte Carlo sequences. Error bars represent a 1-$\sigma$ standard
deviation.
Figure 2b shows the design $\Delta=\widehat{\phi}-\theta$, the phase of the
displacement field, as a function of $L$. We observe that for non-Gaussian
adaptive strategies with increasing PNR, $\Delta$ tends to $\pi/2$ for large
$L$ ($L=200$). This observation is consistent with the theoretical framework
of optimal design of experiments (Sec. II.3), which states that
$\Delta_{optimal}=\pi/2$ for $L\to\infty$. Moreover, as PNR increases, the
strategies show a faster convergence to the asymptotic value of the Holevo
variance, which translates in a smaller error in the estimation (see Fig. 2a.)
These observations further support our theoretical model of non-Gaussian
strategies for phase estimation.
Fig. 3: Estimator variance produced by the non-Gaussian strategy. Holevo
variance of adaptive Non-Gaussian strategies as a function of
$\lvert\alpha\rvert^{2}$, normalized to QCRB, for different adaptive
measurement steps (solid lines), together with the canonical phase measurement
(blue dashed line). For $L=100$ the Holevo variance differs from the QCRB by
$3\%$ exemplifying the tendency to the ultimate precision limit in the regime
of large number of photons in the asymptotic limit. Simulations consist of
$10000$ Monte Carlo simulation sequences with PNR $m=3$.
We investigate the asymptotic performance of the estimator variance produced
by the non-Gaussian strategy for large $|\alpha|^{2}$. Figure 3 shows the
Holevo variance for the non-Gaussian adaptive strategy as a function of
$|\alpha|^{2}$ for different $L$ normalized to the QCRB. We observe that for
large $|\alpha|^{2}$ (with $L=100$), the adaptive non-Gaussian strategy tends
to the CRLB.
To build a model for the performance of this strategy for large
$|\alpha|^{2}$, we propose three candidate models for the Holevo variance for
$L\gg 1$:
$\displaystyle\widehat{y}_{1}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{2}}{\lvert\alpha\rvert^{3}}+\frac{A_{3}}{\lvert\alpha\rvert^{4}},$
(35a) $\displaystyle\widehat{y}_{2}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{2}}{\lvert\alpha\rvert^{3}},$
(35b) $\displaystyle\widehat{y}_{3}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{3}}{\lvert\alpha\rvert^{4}}\,,$
(35c)
to describe our numerical observations in Fig. 3 based on the Monte Carlo
simulations. The model that best describes our observations allows us to
determine, with a certain degree of confidence, the performance of non-
Gaussian adaptive strategies, and compare them with the best Gaussian
strategy, Eq. (1), and the canonical phase measurement, Eq. (2).
We discriminate among plausible candidate models using the technique of
backward elimination [78]. We start with the candidate $\widehat{y}_{1}$, Eq.
(35a), and test the deletion of each variable $A_{i}$ using the $p$-value of a
hypothesis testing procedure ($H_{0}:\,A_{i}=0$, $H_{A}:\,A_{i}\neq 0$). Given
that $p>0.1$ for $A_{2}$ and $p<0.001$ for $A_{1}$ and $A_{3}$, we conclude,
with confidence larger than $99\%$ , that the model reflecting the behavior of
the data presented in Fig. 3 is $\widehat{y}_{3}$, Eq. (35c).
To find $A_{1}$ and $A_{3}$ in the limit $L\to\infty$ in model
$\widehat{y}_{3}$, we fit the Holevo variance as a function of $|\alpha|^{2}$
for $\lvert{\alpha}\rvert^{2}>5$ to the model $\widehat{y}_{3}$ for different
values of $L$. Given a number of adaptive steps $L$, each fitting provides a
set of coefficients $\left\\{A_{1},A_{3}\right\\}$. Hence, to obtain the trend
of each coefficient $A_{1}$ and $A_{2}$ as $L$ increases, we fit them to an
exponential model of the form $A_{i}=D_{i}\exp\left(-E_{i}L\right)+F_{i}$.
Fig. 4 shows the coefficients $A_{1}$ and $A_{3}$ as a function of $L$.
Adjusting this exponential model and observing that in the limit $L\gg 1$
$A_{i}\rightarrow F_{i}$, we obtain $A_{1}=0.250\pm 0.001$ and $A_{3}=0.52\pm
0.01$. Therefore, we conclude with a $99\%$ confidence level that the Holevo
variance for the adaptive non-Gaussian strategy in the limit $L\to\infty$ for
large $|\alpha|$ is:
$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]\approx\frac{0.250\pm
0.001}{\lvert\alpha\rvert^{2}}+\frac{0.520\pm
0.010}{\lvert\alpha\rvert^{4}}\,.$ (36)
This equation is the main result of this work, also shown in Eq. (3). We
observe that the Holevo variance for the adaptive non-Gaussian strategy has a
similar dependance with $|\alpha|$ as the canonical phase measurement,
differing only in the scaling of the correction term of order
$1/|\alpha|^{4}$, see Eq. (2). Moreover, we note that the best Gaussian
strategy known to date, the MKII adaptive homodyne, has a correction term of
order $1/|\alpha|^{3}$ [20], see Eq. (1). Then, for large $|\alpha|$, the non-
Gaussian estimation strategies show a much better scaling in the Holevo
variance than the best known Gaussian strategy, and closely follows the
canonical phase measurement. Furthermore, these non-Gaussian phase estimation
strategies can be implemented with current technologies [2], and our work
demonstrates their superior performance over all the physically realizable
strategies for single-shot phase estimation of coherent states reported to
date.
Fig. 4: Coefficients of model $\widehat{y}_{3}$ in Eq. (35c) for the Holevo
Variance $\text{Var}_{\text{H}}[\widehat{\phi}]$, as a function of $L$. Note
that in the limit of $L\to\infty$ the coefficient $A_{1}$ for the term
$1/\lvert\alpha\rvert^{2}$ tends to $0.25$, and $A_{3}$ of term
$1/\lvert\alpha\rvert^{4}$ tends to $0.52$. Curves are fits to an exponential
model $A_{i}=D_{i}\exp(-E_{i}L)+F_{i}$ with coefficients
$(D,E,F,\mathrm{RSE})=(0.065\pm 0.013,0.060\pm 0.010,0.250\pm 0.001,1.87\times
10^{-7})$ for the left panel, and $(D,E,F,\mathrm{RSE})=(0.095\pm
0.015,0.279\pm 0.119,0.522\pm 0.010,1.82\times 10^{-5})$ for the right panel.
The shaded regions represent a 1-$\sigma$ standard deviation.
The optimized non-Gaussian adaptive strategies based on photon counting
analyzed in this work are not the only possible strategies for single-shot
phase estimation, and there could be other strategies based on photon counting
with better performances. In this work, we studied estimation strategies using
cost functions that are consistent with D-optimal designs. However, there may
be other cost functions that could provide a further improvements to non-
Gaussian strategies. Moreover, we note that the relation in Eq. (30) between
the phase and the magnitude of the displacement field was used to obtain
identifiable likelihood functions and ensure efficient estimators in the
asymptotic limit. While we chose the relation in Eq. (30) because it maximizes
the Fisher information, there is no mathematical proof that this choice is
optimal, or that other choices for this relation would not provide higher
sensitivities. Finally, in the presented adaptive non-Gaussian strategies for
phase estimation, the displacement operations were optimized in every adaptive
step at a time, i. e. using local optimizations in the adaptive steps. We
note, however, that local optimal strategies do not necessarily ensure global
optimality [13]. Strategies with global optimizations, where all adaptive
steps are optimized simultaneously, could probably lead to better
performances. However, the computational overhead required for performing
global optimizations beyond $L=10$ adaptive steps prevents us from being able
to investigate global optimized strategies. To overcome these limitations,
future investigations could make use of machine learning methods, such as
neural networks and reinforcement learning [79], to lower the complexity of
these calculations.
## III Discussion
Non-Gaussian measurement strategies for phase estimation approaching the
quantum limits in sensitivity, as set by the canonical phase measurement, can
become a tool for enhancing the performance of diverse protocols in quantum
sensing, communications, and information processing. Optical phase estimation
approaching the quantum limit can be used to prepare highly squeezed atomic
spin states based on measurement backaction [42] for quantum sensing and
metrology [80]; enhance phase contrast imaging of biological samples with
optical probes at the few photon level to avoid photodamage and ensure
integrity of the sample; and to enhance fidelities in quantum communications
with phase coherent states that require phase tracking close to the quantum
level between receiver and transmitter with few-photon pulses [81, 82], while
allowing for quantum receivers to decode information encoded in coherent
states below the quantum noise limit [83, 13, 52].
As a direct application for quantum information processing, non-Gaussian
measurements based on photon counting and displacement operators can be used
for full reconstruction of quantum states with on-off detectors [84] and PNR
detectors [85] in a multi-copy state setting by sampling phase space with non-
Gaussian projections. The theoretical methods to assess the performance of
adaptive non-Gaussian measures for phase estimation described in this work
could be used to study methods for adaptive quantum tomography [86, 87] based
on photon counting for high dimensional quantum states, and investigate their
asymptotic advantages over adaptive homodyne tomography [88].
From the practical point of view, our work provides insight into the design of
practical, highly efficient measurement strategies for phase estimation based
on photon counting. It shows that non-Gaussian strategies optimizing any cost
function within the family of D-optimal designs are equally efficient, and
demonstrates the advantages of higher photon number resolution in the
strategies to reduce estimation errors. This understanding allows the
experimenter to chose cost functions that can be efficiently calculated and
optimized to achieve higher convergence rates, while selecting a PNR given the
desired/target error budget in the estimation problem for specific
applications. This knowledge will be critical for the design and development
of future sensors based on photon counting for diverse applications in
communication, phase-contrast imaging, metrology, and information processing.
In conclusion, we investigate a family of non-Gaussian strategies for single-
shot phase estimation of optical coherent states. These strategies are based
on adaptive photon counting measurements with a finite number of adaptive
steps implementing coherent displacement operations, optimized to maximize
information gain as the measurement progresses [2]. We develop a comprehensive
statistical analysis based on Bayesian optimal design of experiments that
provides a natural description of adaptive non-Gaussian strategies. This
theoretical framework gives a fundamental understanding on how to optimize
these strategies to enable efficient estimators with a high degree of
convergence towards the ultimate limit, the Cramér Rao lower bound.
We use numerical simulations to show that optimized adaptive non-Gaussian
strategies producing an asymptotically efficient normal estimator achieve a
much higher sensitivity than the best Gaussian strategy known to date, which
is based on adaptive homodyne [20]. Moreover, we show that the Holevo variance
of the estimator for the adaptive non-Gaussian strategy has a similar
dependence as the canonical phase measurement in the asymptotic limit of large
photon numbers, differing by a scaling factor in the second-order correction
term. Our work complements the work in single-shot phase measurements for
single-photon wave packets in two dimensions [22] using quantum feedback with
Gaussian measurements, and paves the way for the realization of optimized
feedback measurements approaching the canonical phase measurement for higher
dimensional states based on non-Gaussian operations.
DATA AVAILABILITY
The data that support the findings of this study are available from the
authors upon request.
CODE AVAILABILITY
Our numerical results have been implemented using MATLAB and Julia language. A
functions to reproduce the numerical results can be publicly found in the
repository [89]. To obtain a set of estimations produced by the non Gaussian
strategy, use the program: ` Bootstrap_phase_estimation.jl`, setting the
desired values of $L$, Bootstrap size, MPN, in the `Adapt_Steps`, `Boot_Reps`,
and `MPNS` variables.
ACKNOWLEDGMENTS
We thank Laboratorio Universitario de Cómputo de Alto Rendimiento (LUCAR) of
IIMAS-UNAM for their service on information processing. This research was
supported by the Grant No. UNAM-DGAPA-PAPIIT IG101421 and the National Science
Foundation (NSF) grans # PHY-1653670 and PHY-2210447.
AUTHOR CONTRIBUTIONS
F.E.B. and P.B. conceived the idea and supervised the work. M.A.R. and P.B.
conducted the theoretical and statistical study. M.T.D. and M.A.R. contributed
in the code for simulations. All authors contributed to the analysis of the
theoretical and numerical results and contributed to writing the manuscript.
COMPETING INTERESTS
The authors declare that there are no competing interests.
## References
* [1] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: Beating the standard quantum limit. _Science_ 306, 1330–1336 (2004). URL https://science.sciencemag.org/content/306/5700/1330.
* [2] DiMario, M. T. & Becerra, F. E. Single-shot non-gaussian measurements for optical phase estimation. _Phys. Rev. Lett._ 125, 120505 (2020). URL https://link.aps.org/doi/10.1103/PhysRevLett.125.120505.
* [3] Demkowicz-Dobrzański, R., Kołodyński, J. & Guţă, M. The elusive heisenberg limit in quantum-enhanced metrology. _Nat. Commun._ 3, 1–8 (2012).
* [4] Abbott, B. P. e. a. Observation of gravitational waves from a binary black hole merger. _Phys. Rev. Lett._ 116, 061102 (2016). URL https://link.aps.org/doi/10.1103/PhysRevLett.116.061102.
* [5] Helstrom, C. _Quantum Detection and Estimation Theory_. Mathematics in Science and Engineering: a series of monographs and textbooks (Academic Press, 1976). URL https://books.google.com.mx/books?id=fv9SAAAAMAAJ.
* [6] Becerra, F. E., Fan, J. & Migdall, A. Photon number resolution enables quantum receiver for realistic coherent optical communications. _Nat. Photonics_ 9, 48–53 (2015).
* [7] Dolinar, S. J. An optimum receiver for the binary coherent state quantum channel. Research Laboratory of Electronics, MIT, Quarterly Progress Report No. 111 (1973), p. 115.
* [8] DiMario, M. T. & Becerra, F. E. Channel-noise tracking for sub-shot-noise-limited receivers with neural networks. _Phys. Rev. Res._ 3, 013200 (2021). URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.013200.
* [9] DiMario, M. T. & Becerra, F. E. Phase tracking for sub-shot-noise-limited receivers. _Phys. Rev. Res._ 2, 023384 (2020). URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.023384.
* [10] DiMario, M., Kunz, L., Banaszek, K. & Becerra, F. Optimized communication strategies with binary coherent states over phase noise channels. _npj Quantum Inf._ 5, 1–7 (2019).
* [11] DiMario, M. T. & Becerra, F. E. Robust measurement for the discrimination of binary coherent states. _Phys. Rev. Lett._ 121, 023603 (2018). URL https://link.aps.org/doi/10.1103/PhysRevLett.121.023603.
* [12] DiMario, M. T., Carrasco, E., Jackson, R. A. & Becerra, F. E. Implementation of a single-shot receiver for quaternary phase-shift keyed coherent states. _J. Opt. Soc. Am. B_ 35, 568–574 (2018). URL http://josab.osa.org/abstract.cfm?URI=josab-35-3-568.
* [13] Ferdinand, A., DiMario, M. & Becerra, F. Multi-state discrimination below the quantum noise limit at the single-photon level. _npj Quantum Inf._ 3, 1–7 (2017).
* [14] Becerra, F. _et al._ Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination. _Nat. Photonics_ 7, 147–152 (2013).
* [15] Slussarenko, S. & Pryde, G. J. Photonic quantum information processing: A concise review. _Appl. Phys. Rev._ 6, 041303 (2019).
* [16] Plenio, M. B. Logarithmic negativity: A full entanglement monotone that is not convex. _Phys. Rev. Lett._ 95, 090503 (2005). URL https://link.aps.org/doi/10.1103/PhysRevLett.95.090503.
* [17] Polino, E., Valeri, M., Spagnolo, N. & Sciarrino, F. Photonic quantum metrology. _AVS Quantum Science_ 2, 024703 (2020).
* [18] Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. _Nat. Photonics_ 5, 222–229 (2011).
* [19] Flamini, F., Spagnolo, N. & Sciarrino, F. Photonic quantum information processing: a review. _Rep. Prog. Phys._ 82, 016001 (2018). URL https://doi.org/10.1088/1361-6633/aad5b2.
* [20] Wiseman, H. M. Adaptive phase measurements of optical modes: Going beyond the marginal $q$ distribution. _Phys. Rev. Lett._ 75, 4587–4590 (1995). URL https://link.aps.org/doi/10.1103/PhysRevLett.75.4587.
* [21] Wiseman, H. M. & Killip, R. B. Adaptive single-shot phase measurements: The full quantum theory. _Phys. Rev. A_ 57, 2169–2185 (1998).
* [22] Martin, L. S., Livingston, W. P., Hacohen-Gourgy, S., Wiseman, H. M. & Siddiqi, I. Implementation of a canonical phase measurement with quantum feedback. _Nat. Phys._ 16, 1046–1049 (2020). URL https://doi.org/10.1038/s41567-020-0939-0.
* [23] Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of quantum states. _Phys. Rev. Lett._ 72, 3439–3443 (1994).
* [24] Demkowicz-Dobrzański, R., Jarzyna, M. & Kołodyński, J. Quantum limits in optical interferometry. _Prog. Opt._ 60, 345–435 (2015).
* [25] Genoni, M. G., Olivares, S. & Paris, M. G. A. Optical phase estimation in the presence of phase diffusion. _Phys. Rev. Lett._ 106, 153603 (2011).
* [26] Lee, C., Oh, C., Jeong, H., Rockstuhl, C. & Lee, S.-Y. Using states with a large photon number variance to increase quantum fisher information in single-mode phase estimation. _J. Phys. Commun._ 3, 115008 (2019). URL https://doi.org/10.1088/2399-6528/ab524a.
* [27] Bradshaw, M., Lam, P. K. & Assad, S. M. Ultimate precision of joint quadrature parameter estimation with a gaussian probe. _Phys. Rev. A_ 97, 012106 (2018).
* [28] Wiseman, H. M. & Killip, R. B. Adaptive single-shot phase measurements: A semiclassical approach. _Phys. Rev. A_ 56, 944–957 (1997).
* [29] Anisimov, P. M. _et al._ Quantum metrology with two-mode squeezed vacuum: Parity detection beats the heisenberg limit. _Phys. Rev. Lett._ 104, 103602 (2010).
* [30] Huang, Z., Motes, K. R., Anisimov, P. M., Dowling, J. P. & Berry, D. W. Adaptive phase estimation with two-mode squeezed vacuum and parity measurement. _Phys. Rev. A_ 95, 053837 (2017). URL https://link.aps.org/doi/10.1103/PhysRevA.95.053837.
* [31] Anderson, B. E. _et al._ Phase sensing beyond the standard quantum limit with a variation on the su(1,1) interferometer. _Optica_ 4, 752–756 (2017).
* [32] Izumi, S. _et al._ Optical phase estimation via the coherent state and displaced-photon counting. _Phys. Rev. A_ 94, 033842 (2016).
* [33] Slussarenko, S. _et al._ Unconditional violation of the shot noise limit in photonic quantum metrology. _Nat. Photonics_ 11, 700–703 (2017).
* [34] Anderson, B. E., Schmittberger, B. L., Gupta, P., Jones, K. M. & Lett, P. D. Optimal phase measurements with bright- and vacuum-seeded su(1,1) interferometers. _Phys. Rev. A_ 95, 063843 (2017).
* [35] Daryanoosh, S., Slussarenko, S., Berry, D. W., Wiseman, H. M. & Pryde, G. J. Experimental optical phase measurement approaching the exact heisenberg limit. _Nat. Commun._ 9, 4606 (2018).
* [36] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. _Nature_ 450, 393–396 (2007).
* [37] Hentschel, A. & Sanders, B. C. Machine learning for precise quantum measurement. _Phys. Rev. Lett._ 104, 063603 (2010).
* [38] Hou, Z. _et al._ Control-enhanced sequential scheme for general quantum parameter estimation at the heisenberg limit. _Phys. Rev. Lett._ 123, 040501 (2019).
* [39] Larson, W. & Saleh, B. E. A. Supersensitive ancilla-based adaptive quantum phase estimation. _Phys. Rev. A_ 96, 042110 (2017).
* [40] Lumino, A. _et al._ Experimental phase estimation enhanced by machine learning. _Phys. Rev. Applied_ 10, 044033 (2018).
* [41] Zheng, K., Xu, H., Zhang, A., Ning, X. & Zhang, L. Ab initio phase estimation at the shot noise limit with on–off measurement. _Quantum Inf. Process._ 18, 329 (2019).
* [42] Deutsch, I. H. & Jessen, P. S. Quantum control and measurement of atomic spins in polarization spectroscopy. _Opt. Commun._ 283, 681–694 (2010). URL https://www.sciencedirect.com/science/article/pii/S0030401809010517. Quo vadis Quantum Optics?
* [43] Bouchoule, I. & Mølmer, K. Preparation of spin-squeezed atomic states by optical-phase-shift measurement. _Phys. Rev. A_ 66, 043811 (2002).
* [44] Iwasawa, K. _et al._ Quantum-limited mirror-motion estimation. _Phys. Rev. Lett._ 111, 163602 (2013).
* [45] Tsang, M. Quantum metrology with open dynamical systems. _New Journ. of Phys._ 15, 073005 (2013).
* [46] Aasi, A. J. A. B. e. a., J. Enhanced sensitivity of the ligo gravitational wave detector by using squeezed states of light. _Nat. Photonics_ 7, 613–619 (2013).
* [47] Nair, R., Yen, B. J., Guha, S., Shapiro, J. H. & Pirandola, S. Symmetric $m$-ary phase discrimination using quantum-optical probe states. _Phys. Rev. A_ 86, 022306 (2012).
* [48] van Loock, P., Lütkenhaus, N., Munro, W. J. & Nemoto, K. Quantum repeaters using coherent-state communication. _Phys. Rev. A_ 78, 062319 (2008).
* [49] Munro, W. J., Nemoto, K. & Spiller, T. P. Weak nonlinearities: a new route to optical quantum computation. _New J. of Phys._ 7, 137 (2005).
* [50] Nemoto, K. & Munro, W. J. Nearly deterministic linear optical controlled-not gate. _Phys. Rev. Lett._ 93, 250502 (2004).
* [51] Holevo, A. S. _Probabilistic and Statistical Aspects of Quantum Theory; 2nd ed._ Publications of the Scuola Normale Superiore. Monographs (Springer, Dordrecht, 2011). URL https://cds.cern.ch/record/1414149.
* [52] Becerra, F., Fan, J. & Migdall, A. Photon number resolution enables quantum receiver for realistic coherent optical communications. _Nat. Photonics_ 9, 48–53 (2015).
* [53] Cover, T. M. & Thomas, J. A. _Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)_ (Wiley-Interscience, USA, 2006).
* [54] Chaloner, K. & Verdinelli, I. Bayesian Experimental Design: A Review. _Stat. Sci._ 10, 273 – 304 (1995). URL https://doi.org/10.1214/ss/1177009939.
* [55] Rodríguez-García, M. A., Castillo, I. P. & Barberis-Blostein, P. Efficient qubit phase estimation using adaptive measurements. _Quantum_ 5, 467 (2021). URL https://doi.org/10.22331/q-2021-06-04-467.
* [56] Ryan, E., Drovandi, C., McGree, J. & Pettitt, T. A review of modern computational algorithms for bayesian optimal design. _Int. Stat. Rev._ 84, 128–154 (2016). URL https://eprints.qut.edu.au/75000/.
* [57] Suzuki, J. Quantum-state estimation problem via optimal design of experiments. _Int. J. Quantum Inf._ 19, 2040007 (2021). URL https://doi.org/10.1142/S0219749920400079. eprint https://doi.org/10.1142/S0219749920400079.
* [58] Morelli, S., Usui, A., Agudelo, E. & Friis, N. Bayesian parameter estimation using gaussian states and measurements. _Quantum Sci. Technol._ 6, 025018 (2021). URL https://doi.org/10.1088/2058-9565/abd83d.
* [59] Martínez-García, F., Vodola, D. & Müller, M. Adaptive bayesian phase estimation for quantum error correcting codes. _New J. Phys._ 21, 123027 (2019). URL https://doi.org/10.1088/1367-2630/ab5c51.
* [60] Berni, A. A. _et al._ Ab initio quantum-enhanced optical phase estimation using real-time feedback control. _Nat. Photonics_ 9, 577–581 (2015). URL https://doi.org/10.1038/nphoton.2015.139.
* [61] Genoni, M. G. _et al._ Optical interferometry in the presence of large phase diffusion. _Phys. Rev. A_ 85, 043817 (2012). URL https://link.aps.org/doi/10.1103/PhysRevA.85.043817.
* [62] Oh, C. & Son, W. Sub shot-noise frequency estimation with bounded a priori knowledge. _J. Phys. A Math. Theor._ 48, 045304 (2014). URL https://doi.org/10.1088/1751-8113/48/4/045304.
* [63] Macieszczak, K., Fraas, M. & Demkowicz-Dobrzański, R. Bayesian quantum frequency estimation in presence of collective dephasing. _New J. Phys._ 16, 113002 (2014). URL https://doi.org/10.1088/1367-2630/16/11/113002.
* [64] Glatthard, J. _et al._ Optimal cold atom thermometry using adaptive bayesian strategies (2022). URL https://arxiv.org/abs/2204.11816.
* [65] McMichael, R. D., Dushenko, S. & Blakley, S. M. Sequential bayesian experiment design for adaptive ramsey sequence measurements. _J. Appl. Phys._ 130, 144401 (2021). URL https://doi.org/10.1063/5.0055630. eprint https://doi.org/10.1063/5.0055630.
* [66] Kleinegesse, S. & Gutmann, M. U. Efficient bayesian experimental design for implicit models. In _AISTATS_ (2019).
* [67] Paninski, L. Asymptotic theory of information-theoretic experimental design. _Neural Comput._ 17, 1480–1507 (2005). URL https://doi.org/10.1162/0899766053723032.
* [68] Verdinelli, I. A note on bayesian design for the normal linear model with unknown error variance. _Biometrika_ 87, 222–227 (2000). URL http://www.jstor.org/stable/2673577.
* [69] Ryan, E., Drovandi, C., Thompson, H. & Pettitt, T. Towards bayesian experimental design for nonlinear models that require a large number of sampling times 70, 45–60 (2014). URL https://eprints.qut.edu.au/56522/.
* [70] Nielsen, M. A. & Chuang, I. L. _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
* [71] Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of quantum states. _Phys. Rev. Lett._ 72, 3439–3443 (1994). URL https://link.aps.org/doi/10.1103/PhysRevLett.72.3439.
* [72] Paris, M. G. Quantum estimation for quantum technology. _Int. J. Quantum Inf._ 7, 125–137 (2009).
* [73] Holevo, A. Statistical decision theory for quantum systems. _J. Multivar. Anal._ 3, 337 – 394 (1973). URL http://www.sciencedirect.com/science/article/pii/0047259X73900286.
* [74] Pellonpää, J.-P. & Schultz, J. Measuring the canonical phase with phase-space measurements. _Phys. Rev. A_ 88, 012121 (2013). URL https://link.aps.org/doi/10.1103/PhysRevA.88.012121.
* [75] Escher, B. M., de Matos Filho, R. L. & Davidovich, L. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. _Nat. Phys._ 7, 406–411 (2011). URL https://doi.org/10.1038/nphys1958.
* [76] Wang, P. _et al._ Single ion qubit with estimated coherence time exceeding one hour. _Nat. Commun._ 12, 233 (2021). URL https://doi.org/10.1038/s41467-020-20330-w.
* [77] Hall, M. J. W. & Wiseman, H. M. Does nonlinear metrology offer improved resolution? answers from quantum information theory. _Phys. Rev. X_ 2, 041006 (2012). URL https://link.aps.org/doi/10.1103/PhysRevX.2.041006.
* [78] Young, D. S. _Handbook of regression methods_. A Chapman and Hall Book (2017).
* [79] DiMario, M. T. & Becerra, F. E. Channel-noise tracking for sub-shot-noise-limited receivers with neural networks. _Phys. Rev. Res._ 3, 013200 (2021). URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.013200.
* [80] Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. _Rev. Mod. Phys._ 90, 035005 (2018). URL https://link.aps.org/doi/10.1103/RevModPhys.90.035005.
* [81] Qi, B., Lougovski, P., Pooser, R., Grice, W. & Bobrek, M. Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection. _Phys. Rev. X_ 5, 041009 (2015). URL https://link.aps.org/doi/10.1103/PhysRevX.5.041009.
* [82] Soh, D. B. S. _et al._ Self-referenced continuous-variable quantum key distribution protocol. _Phys. Rev. X_ 5, 041010 (2015). URL https://link.aps.org/doi/10.1103/PhysRevX.5.041010.
* [83] DiMario, M. T. & Becerra, F. E. Robust measurement for the discrimination of binary coherent states. _Phys. Rev. Lett._ 121, 023603 (2018).
* [84] Allevi, A. _et al._ State reconstruction by on/off measurements. _Phys. Rev. A_ 80, 022114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.022114.
* [85] Nehra, R. _et al._ State-independent quantum state tomography by photon-number-resolving measurements. _Optica_ 6, 1356–1360 (2019). URL http://opg.optica.org/optica/abstract.cfm?URI=optica-6-10-1356.
* [86] Huszár, F. & Houlsby, N. M. T. Adaptive bayesian quantum tomography. _Phys. Rev. A_ 85, 052120 (2012). URL https://link.aps.org/doi/10.1103/PhysRevA.85.052120.
* [87] Granade, C., Ferrie, C. & Flammia, S. T. Practical adaptive quantum tomography. _New J. Phys._ 19, 113017 (2017). URL https://doi.org/10.1088/1367-2630/aa8fe6.
* [88] D’Ariano, G. M. & Paris, M. G. A. Adaptive quantum homodyne tomography. _Phys. Rev. A_ 60, 518–528 (1999). URL https://link.aps.org/doi/10.1103/PhysRevA.60.518.
* [89] Rodríguez-García, M. A. adaptive_photon_counting_for_coherent-state (2022). URL https://github.com/Gateishion/adaptive_photon_counting_for_coherent-state.
|
# Energy-efficient spin injector into semiconductors driven by elastic waves
Andrei V. Azovtsev∗ Andrei I. Nikitchenko Nikolay A. Pertsev Ioffe Institute
194021 St. Petersburg Russia
<EMAIL_ADDRESS>
###### Abstract
Generation of significant spin imbalance in nonmagnetic semiconductors is
crucial for the functioning of many spintronic devices, such as magnetic
diodes and transistors, spin-based logic gates, and spin-polarized lasers. An
attractive design of spin injectors into semiconductors is based on a spin
pumping from a precessing ferromagnet, but the classical excitation of
magnetization precession by a microwave magnetic field leads to the high power
consumption of the device. Here we describe theoretically a spin injector with
greatly reduced energy losses, in which the magnetic dynamics is excited by an
elastic wave generated in a ferromagnet-semiconductor heterostructure by an
attached piezoelectric transducer. To demonstrate the efficient functioning of
such an injector, we first perform micromagnetoelastic simulations of the
coupled elastic and magnetic dynamics in $\mathrm{Ni}$ films and
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers traversed by plane longitudinal and
shear waves. For thick $\mathrm{Ni}$ films, it is shown that a monochromatic
acoustic wave generates a spin wave with the same frequency and wavelength,
which propagates together with the driving wave over distances of several
micrometers at the excitation frequencies $\nu\approx 10$ GHz close to the
frequency of ferromagnetic resonance. The simulations of
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers with $\mathrm{Ni}$ thicknesses
comparable to the wavelength of the injected acoustic wave demonstrate the
development of a steady-state magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface. The amplitude of such a precession has
a maximum at $\mathrm{Ni}$ thickness amounting to three quarters of the
wavelength of the elastic wave, which is explained by an analytical model.
Using simulation data obtained for the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, we evaluate the spin current pumped
into $\mathrm{GaAs}$ and calculate the spin accumulation in the semiconducting
layer by solving the spin diffusion equation. Then the electrical signals
resulting from the spin flow and the inverse spin Hall effect are determined
via the numerical solution of the Laplace’s equation. It is shown that
amplitudes of these ac signals near the interface are large enough for
experimental measurement, which indicates an efficient acoustically driven
spin pumping into $\mathrm{GaAs}$ and rather high spin accumulation in this
semiconductor.
## I Introduction
Semiconductors are attractive for the development of spintronic devices due to
their large spin diffusion lengths in comparison with transition metals Hägele
_et al._ (1998); Kikkawa and Awschalom (1999), long spin relaxation times Bhat
and Kumar (2014), and the possibility of manipulating the electrons’ spin by
polarized light Putikka and Joynt (2004); Kokurin _et al._ (2013). However,
the application of conventional nonmagnetic semiconductors in spintronics
requires the generation of an internal spin imbalance by an external stimulus
or via an attached magnetic material Hirohata _et al._ (2020). The simplest
method to create such an imbalance would be the direct injection of a spin-
polarized charge current from a metallic ferromagnet through Ohmic contact,
but the conductance mismatch at the semiconductor-metal interface makes this
method inefficient Schmidt _et al._ (2000). The presence of a thin insulating
interlayer acting as a tunnel barrier solves the mismatch problem Hanbicki and
Jonker (2002); Jiang _et al._ (2005); Dash _et al._ (2009); Kamerbeek _et
al._ (2014), but requires the fabrication of a high quality interlayer unless
the formation of a natural Schottky barrier with the appropriate parameters
occurs Hanbicki and Jonker (2002). Alternatively, the spin imbalance in the
semiconductor can be created by bringing it into a direct contact with a
precessing ferromagnet Brataas _et al._ (2002); Tserkovnyak _et al._ (2005).
The resulting spin pumping into the nonmagnetic semiconductor is due to the
modulation of the interface scattering matrix by the coherent precession of
the magnetization Brataas _et al._ (2002).
Typically, in spin pumping experiments magnetization dynamics is excited by an
external microwave magnetic field with the frequency matching that of the
ferromagnetic resonance. Efficient generation of spin currents in normal
metals by this technique has been demonstrated experimentally Heinrich _et
al._ (2003); Saitoh _et al._ (2006); Bell _et al._ (2008); Mosendz _et al._
(2010); Czeschka _et al._ (2011); Tashiro _et al._ (2015). The spin pumping
into semiconductors from metallic ferromagnets Ando _et al._ (2011); Shikoh
_et al._ (2013); Lee _et al._ (2014); Wang _et al._ (2017) and ferrimagnetic
insulators Mendes _et al._ (2018) subjected to microwave radiation has been
revealed as well. However, the power consumption associated with the
generation of microwave magnetic fields appears to be rather high, which
impedes applications of magnetically driven spin injectors in low-power
spintronics. For this reason, alternative spin pumping techniques have been
studied during the past decade, one of which is based on the excitation of
magnetization dynamics in ferromagnets by injected elastic waves Weiler _et
al._ (2011); Uchida _et al._ (2011a); Weiler _et al._ (2012); Kamra _et
al._ (2015); Polzikova _et al._ (2016); Azovtsev and Pertsev (2016, 2017);
Polzikova _et al._ (2018); Azovtsev and Pertsev (2019); Alekseev _et al._
(2020). Since such waves can be generated by a piezoelectric transducer
coupled to the ferromagnet and subjected to an ac electric field, the power
consumption of elastically driven spin injectors is expected to be
comparatively low Alekseev _et al._ (2020); Cherepov _et al._ (2014);
Bhaskar _et al._ (2020). The experimental and theoretical studies have
demonstrated an efficient generation of spin currents in normal metals by
surface and bulk acoustic waves, but the strain-driven spin pumping into
semiconductors was not investigated so far.
In this paper, we theoretically describe a spin injector into nonmagnetic
semiconductors, which employs the spin pumping generated by a dynamically
strained ferromagnetic film. The injector has the form of a ferromagnet-
semiconductor bilayer coupled to a piezoelectric transducer excited by a
microwave voltage. Such a transducer creates a bulk elastic wave propagating
across the bilayer, which induces a radio-frequency magnetization precession
providing efficient spin pumping into the semiconducting layer. To quantify
the elastically driven magnetic dynamics in the ferromagnetic film, we employ
the state-of-the-art numerical simulations allowing for the two-way coupling
between spins and strains (see Sec. II). The simulations are performed for the
$(001)$-oriented $\mathrm{Ni}$ films and $\mathrm{Ni/GaAs}$ bilayers traversed
by plane longitudinal and transverse acoustic waves. For thick $\mathrm{Ni}$
films, tightly coupled elastic and magnetic dynamics are described (Sec. III),
which involve the generation of a spin wave carried by the propagating elastic
wave. In Sec. IV, we report the results of numerical simulations performed for
$\mathrm{Ni/GaAs}$ bilayers with the $\mathrm{Ni}$ thickness comparable to the
wavelength of the propagating elastic wave and discuss the influence of the
thickness of the ferromagnetic layer and the excitation frequency on the
amplitude of the magnetization precession at the $\mathrm{Ni|GaAs}$ interface
(Sec. IV). Numerical results obtained for the steady-state magnetization
precession at the interface are then used to calculate the spin pumping into
the $\mathrm{GaAs}$ film and to determine the spin accumulation in the
semiconductor by solving the spin diffusion equation (Sec. V). It is shown
that the proposed injector has a high efficiency ensuring significant spin
flux in $\mathrm{GaAs}$, which can be detected experimentally via the inverse
spin Hall effect.
$y$$z$$x$m$\psi$transducer$t_{\text{F}}$$t_{\text{N}}$$w_{\text{N}}$$h_{\text{N}}$k$\mathrm{GaAs}$$\mathrm{Ni}$Fe$V_{\text{s}}$
Figure 1: Ferromagnet-semiconductor heterostructure comprising $\mathrm{Ni}$
and $\mathrm{GaAs}$ layers. An elastic wave (longitudinal or transverse) with
the wave vector k is injected into the $\mathrm{Ni}$ layer by the attached
piezoelectric transducer. The thicknesses of the $\mathrm{Ni}$ and
$\mathrm{GaAs}$ layers are denoted by $t_{\text{F}}$ and $t_{\text{N}}$,
respectively. Precessing magnetization in Ni creates a spin imbalance in GaAs,
which then produces a measurable voltage $V_{\text{s}}$ between an attached
iron probe and a nonmagnetic contact.
## II Modeling of magnetoelastic phenomena in ferromagnetic heterostructures
Owing to the magnetoelastic coupling between spins and strains, the excitation
of an elastic wave in a ferromagnetic material can induce a precessional
motion of the magnetization and the generation of a spin wave Weiler _et al._
(2011); Uchida _et al._ (2011b); Weiler _et al._ (2012); Thevenard _et al._
(2014); Janušonis _et al._ (2015); Gowtham _et al._ (2015); Casals _et al._
(2020). The backaction of the induced magnetization precession on strain state
of the ferromagnet may significantly affect the propagation of the driving
elastic wave and lead to the appearance of additional “secondary” waves
Azovtsev and Pertsev (2017, 2019). Therefore, the two-way interplay between
elastic and magnetic variables Akhiezer _et al._ (1958) should be fully taken
into account for an accurate modeling of the magnetoelastic phenomena in
ferromagnets. Such micromagnetoelastic modeling can be realized via the
numerical solution of the system of differential equations comprising the
elastodynamic equation for the mechanical displacement u and the Landau-
Lifshitz-Gilbert (LLG) equation for the magnetization M Chen _et al._ (2017);
Azovtsev and Pertsev (2017, 2019, 2020). The elastodynamic equation should
allow for the magnetoelastic contribution $\delta\sigma_{ij}^{\text{ME}}$ to
the mechanical stresses $\sigma_{ij}$ in the ferromagnet, which can be
calculated as $\delta\sigma_{ij}^{\text{ME}}=\partial
F_{\text{ME}}/\partial\varepsilon_{ij}$, where $F_{\text{ME}}$ is the
magnetoelastic energy density, and $\varepsilon_{ij}$ are the elastic strains
($i,j=x,y,z$). The influence of strains on the magnetization orientation can
be quantified by adding a magnetoelastic term
$\textbf{H}_{\text{ME}}=-(1/\mu_{0})\partial F_{\text{ME}}/\partial\textbf{M}$
to the effective magnetic field $\textbf{H}_{\text{eff}}$ involved in the LLG
equation ($\mu_{0}$ being the magnetic constant). For cubic ferromagnets such
as nickel, the magnetoelastic contribution to the total energy density F can
be written as
$\displaystyle F_{\text{ME}}$
$\displaystyle=B_{1}[(m_{x}^{2}-\frac{1}{3})\varepsilon_{xx}+(m_{y}^{2}-\frac{1}{3})\varepsilon_{yy}+(m_{z}^{2}-\frac{1}{3})\varepsilon_{zz}]$
(1)
$\displaystyle+2B_{2}\left[m_{x}m_{y}\varepsilon_{xy}+m_{x}m_{z}\varepsilon_{xz}+m_{y}m_{z}\varepsilon_{yz}\right],$
where $\textbf{m}=\textbf{M}/M_{s}$ is the unit vector in the magnetization
direction, $M_{s}$ is the saturation magnetization regarded as a strain-
independent quantity, and $B_{1}$, $B_{2}$ are the magnetoelastic coupling
constants Kittel (1949).
In this work, we performed micromagnetoelastic simulations of $\mathrm{Ni}$
films and $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers subjected to a periodic
displacement $\textbf{u}(x=0,t)=\textbf{u}_{0}(t)$ imposed at the
$\mathrm{Ni}$ surface $x=0$ (Fig. 1). Such a displacement models the action of
a piezoelectric transducer coupled to $\mathrm{Ni}$ film and generates a plane
elastic wave traversing the heterostructure Azovtsev and Pertsev (2017, 2019).
To excite a longitudinal wave characterized by the strain
$\varepsilon_{xx}(x,t)$, we introduced the surface displacement with the
components $u^{0}_{y}=u^{0}_{z}=0$ and $u^{0}_{x}=u_{\text{max}}\sin{(2\pi\nu
t)}$, while a transverse wave with the shear strain $\varepsilon_{xz}(x,t)$
was created by setting $u^{0}_{x}=u^{0}_{y}=0$ and
$u^{0}_{z}=u_{\text{max}}\sin{(2\pi\nu t)}$. The excitation frequency $\nu$
was varied in a wide range spanning the resonance frequency $\nu_{\text{res}}$
of the coherent magnetization precession in the unstrained $\mathrm{Ni}$ film,
which was determined by simulations of the magnetization relaxation to its
equilibrium orientation. To ensure the same maximal strain in the elastic wave
at any excitation frequency $\nu$, the displacement amplitude $u_{\text{max}}$
was taken to be inversely proportional to $\nu$ Azovtsev and Pertsev (2019).
Namely, we used the relations
$u_{\text{max}}=\varepsilon_{xx}^{\text{max}}/k_{L}$ and
$u_{\text{max}}=2\varepsilon_{xz}^{\text{max}}/k_{T}$ for longitudinal and
transverse waves, respectively, where $k_{L}=2\pi\nu/c_{L}$ and
$k_{T}=2\pi\nu/c_{T}$ are the wave numbers of these waves having velocities
$c_{L}$ and $c_{T}$.
The magnetization dynamics in the $\mathrm{Ni}$ film was quantified using the
LLG equation with the effective magnetic field $\textbf{H}_{\text{eff}}$
comprising contributions resulting from the exchange interaction, cubic
magnetocrystalline anisotropy, magnetoelastic coupling, Zeeman energy, and
dipolar interactions between oscillating spins Azovtsev and Pertsev (2016).
For numerical calculations, we characterized $\mathrm{Ni}$ by the saturation
magnetization $M_{s}=4.78\times 10^{5}$ A m-1 Niitsu (2020), exchange constant
$A_{\text{ex}}=0.85\times 10^{-11}$ J m-1 Niitsu (2020), magnetocrystalline
anisotropy constants $K_{1}=-5.7\times 10^{3}$ J m-3, $K_{2}=-2.3\times
10^{3}$ J m-3 Stearns (1986), magnetoelastic constants $B_{1}=9.2\times
10^{6}$ J m-3, $B_{2}=10.7\times 10^{6}$ J m-3 Stearns (1986), and Gilbert
damping parameter of 0.045 Walowski _et al._ (2008). The elastic stiffnesses
$c_{11}$, $c_{44}$ and mass densities $\rho$ of $\mathrm{Ni}$ and
$\mathrm{GaAs}$, which are involved in their elastodynamic equations of
motion, were taken from Ref. Haynes (2016) and listed in Table 1 together with
the velocities $c_{L}$ and $c_{T}$ of elastic waves in these materials. No
elastic damping was added to the elastodynamic equation in our simulations for
the following reasons. First, we are interested in investigating the purely
magnetic damping of elastic waves in $\mathrm{Ni}$, and the introduction of
the intrinsic elastic damping would obscure simulation results described in
Sec. III. Second, in Sec. IV we consider $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayers comprising $\mathrm{Ni}$ films much thinner than the decay lengths of
longitudinal and transverse elastic waves in $\mathrm{Ni}$, which are measured
to be 5.8 and 29 $\mu$m respectively at the relevant frequency of 9.4 GHz
Homer _et al._ (1987).
Micromagnetoelastic simulations were performed with the aid of a homemade
software operating with a finite ensemble of nanoscale computational cells.
Our software solves the elastodynamic equations of $\mathrm{Ni}$ and
$\mathrm{GaAs}$ films by a finite-difference technique with a midpoint
derivative approximation and numerically integrates the LLG equation by the
projective Runge-Kutta algorithm. We employed a fixed integration step $\delta
t=100$ fs and set the size of cubic computation cells to $2$ nm, which is
smaller than the exchange length
$l_{\text{ex}}=\sqrt{2A_{\text{ex}}/(\mu_{0}M_{s}^{2})}\approx 7.7$ nm of
$\mathrm{Ni}$. The system of partial differential equations was appended by
appropriate boundary conditions. At the free surface of the $\mathrm{GaAs}$
layer, the stresses $\sigma_{ix}$ were set to zero, and the “free-surface”
condition $\partial\textbf{m}/\partial x=0$ was imposed at both boundaries of
the $\mathrm{Ni}$ layer. Since a unified ensemble of computational cells
covering the whole $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer was employed in the
simulations, the continuity conditions at the $\mathrm{Ni}|\mathrm{GaAs}$
interface were satisfied automatically. The layers comprising the
heterostructure were considered infinite in $y-z$ plane and the dynamical
quantities were allowed to change only along the $x$ direction.
| $\mathrm{Ni}$ | $\mathrm{GaAs}$
---|---|---
$c_{11}$ ($10^{11}$ J m-3) | 2.481 | 1.188
$c_{12}$ ($10^{11}$ J m-3) | 1.549 | 0.537
$c_{44}$ ($10^{11}$ J m-3) | 1.242 | 0.594
$\rho$ (kg m-3) | 8910 | 5317
$c_{L}$ (m s-1) | 5277 | 4726
$c_{T}$ (m s-1) | 3734 | 3344
Table 1: Elastic stiffnesses and mass densities of $\mathrm{Ni}$ and
$\mathrm{GaAs}$ Haynes (2016) used in numerical calculations. Velocities
$c_{L}=\sqrt{c_{11}/\rho}$ and $c_{T}=\sqrt{c_{44}/\rho}$ of longitudinal and
transverse elastic waves in these materials are also given for information.
## III Magnetic dynamics excited by longitudinal and transverse elastic waves
in thick Ni films
An elastic wave perturbs the ferromagnetic state when it creates a nonzero
magnetoelastic torque
$\textbf{T}_{\text{ME}}=\textbf{M}\times\textbf{H}_{\text{ME}}$ acting on the
magnetization M. In the case of the longitudinal wave $\varepsilon_{xx}(x,t)$,
the effective field $\textbf{H}_{\text{ME}}$ has the only nonzero component
$H_{x}^{\text{ME}}=-2B_{1}\varepsilon_{xx}m_{x}/M_{s}$. Therefore, an external
magnetic field H creating the direction cosine $m_{x}$ in the in-plane
magnetized $\mathrm{Ni}$ film should be applied to generate the magnetization
dynamics. To additionally stabilize the single-domain state, we introduced the
field along the [111] crystallographic direction (easy axis), taking
$H_{x}=H_{y}=H_{z}=1000$ Oe. At such field, the magnetization in the
unstrained $\mathrm{Ni}$ film has an elevation angle $\psi\approx 46^{\circ}$
($m_{x}=0.137$) and equal projections on [100] and [010] directions
($m_{y}=m_{z}=0.70044$). The same magnetic field was used in the simulations
of magnetic dynamics induced by the shear acoustic wave
$\varepsilon_{xz}(x,t)$, which facilitates the comparison of results obtained
for two types of elastic perturbations. It should be noted that shear waves
impose nonzero magnetoelastic torque $\textbf{T}_{\text{ME}}$ even on the in-
plane magnetized ferromagnetic films. Indeed, the effective field
$\textbf{H}_{\text{ME}}$ created by the strain $\varepsilon_{xz}$ has two
components, $H_{z}^{\text{ME}}=-2B_{2}\varepsilon_{xz}m_{x}/M_{s}$ and
$H_{x}^{\text{ME}}=-2B_{2}\varepsilon_{xz}m_{z}/M_{s}$, and the latter differs
from zero even at $m_{x}=0$ when $m_{z}\neq 0$. However, simulations show that
the amplitude of the magnetization precession increases significantly when
both $H_{x}^{\text{ME}}$ and $H_{z}^{\text{ME}}$ differ from zero.
At the chosen magnetic field H, the resonance frequency $\nu_{\text{res}}$ of
the unstrained $\mathrm{Ni}$ film with in-plane dimensions much larger than
the film thickness $t_{\text{F}}$ was found to be $9.6$ GHz. Accordingly, the
excitation frequency was varied in a wide range around 10 GHz. By selecting
the appropriate amplitudes $u_{\text{max}}(\nu)$ of the surface displacements
$u_{x}^{0}(t)$ or $u_{z}^{0}(t)$, we created the maximal strains
$\varepsilon^{\text{max}}_{xx}=\varepsilon^{\text{max}}_{xz}=10^{-4}$ in the
excited elastic waves near the $\mathrm{Ni}$ surface $x=0$. Simulations were
first performed for $\mathrm{Ni}$ films with thicknesses $t_{\text{F}}$ much
larger than the wave lengths $\lambda_{L}=c_{L}/\nu$ and
$\lambda_{T}=c_{T}/\nu$ of the pure elastic waves, which amount to
$\lambda_{L}=550$ nm and $\lambda_{T}=389$ nm at the resonance excitation
$\nu=\nu_{\text{res}}$. This allows the observation of several wave periods
inside the ferromagnetic film. Since in this section we concentrate on the
propagation of the elastic waves in $\mathrm{Ni}$ and their interaction with
the magnetic subsystem, the simulation time was limited by the period needed
for the wave to reach the opposite boundary of the $\mathrm{Ni}$ film. The
effects of the wave reflection from the $\mathrm{Ni}|\mathrm{GaAs}$ interface
are discussed in Section IV.
Figure 2: Spatial distributions of strains in the driving longitudinal (a)
and transverse (b) elastic waves and strain-induced variations of the
magnetization direction cosines $m_{i}$ in the 2-$\mu$m-thick $\mathrm{Ni}$
film. Excitation frequency $\nu$ is equal to the resonance frequency
$\nu_{\text{res}}=9.6$ GHz, and snapshots are taken at 0.37 ns (a) and 0.52 ns
(b).
The simulations of the coupled elastic and magnetic dynamics in thick
$\mathrm{Ni}$ films confirmed the creation of periodic, almost sinusoidal
elastic waves at all studied excitation frequencies $\nu$ (see Fig. 2). The
wave emerges at the $\mathrm{Ni}$ surface and propagates away with the
velocity $c_{L}$ or $c_{T}$ characteristic of a pure elastic wave despite an
inhomogeneous magnetization precession excited by the magnetoelastic torque
$\textbf{T}_{\text{ME}}$ (see Fig. 2). However, the strain-induced precession
manifests itself in the generation of the additional elastic waves caused by
the magnetoelastic feedback (see Fig. 3). These secondary strain waves were
already revealed by the micromagnetoelastic simulations performed for Fe81Ga19
and CoFe2O4 films Azovtsev and Pertsev (2017, 2019), but were not detected in
the simulations of the propagation of longitudinal elastic waves in
$\mathrm{Ni}$ Chen _et al._ (2017). When the driving wave is a longitudinal
one, two transverse secondary waves with the strains $\varepsilon_{xy}(x,t)$
and $\varepsilon_{xz}(x,t)$ having amplitudes $\sim 10^{-7}$ appear in
$\mathrm{Ni}$. Their profiles depend on the position in the film, exhibiting a
peculiar behavior similar to that of the secondary waves arising in CoFe2O4
excited by longitudinal elastic waves Azovtsev and Pertsev (2019). This
behavior is caused by the interference of the two components of each secondary
wave, which have the form of a shear wave with the wavelength $\lambda_{T}$
and velocity $c_{T}$ freely propagating from the $\mathrm{Ni}$ surface and a
forced shear wave with the wavelength $\lambda_{L}$ and velocity $c_{L}$
generated in the whole driving longitudinal wave. When the driving wave is a
transverse one [$\varepsilon_{xz}(x,t)$], a longitudinal secondary wave
$\varepsilon_{xx}(x,t)$ and another shear wave $\varepsilon_{xy}(x,t)$ with
amplitudes $\sim 10^{-6}$ are generated by the magnetization precession.
Similarly to the aforementioned situation, the longitudinal wave appears to be
a superposition of a free wave with the wavelength $\lambda_{L}$ and velocity
$c_{L}$ and a forced wave with the parameters $\lambda_{T}$ and $c_{T}$. In
contrast, the secondary shear wave can be regarded as a single wave because
its wavelength $\lambda_{T}$ and velocity $c_{T}$ match those of the driving
wave.
Figure 3: Secondary elastic waves generated by the magnetization precession
induced by the primary longitudinal (a) and transverse (b) waves in the
2-$\mu$m-thick $\mathrm{Ni}$ film. Snapshots are taken at 0.37 ns (a) and 0.35
ns (b).
The magnetization precession induced by the primary elastic wave also affects
its propagation at long distances from the $\mathrm{Ni}$ surface. A careful
evaluation of the local strain amplitudes $\varepsilon_{xx}^{\text{max}}$ and
$\varepsilon_{xz}^{\text{max}}$ in the driving longitudinal and transverse
waves reveals that they decrease with the increasing distance $x$ from the
$\mathrm{Ni}$ surface. This decay is caused by the energy transfer to the
magnetic subsystem, where the strain-driven magnetization precession is
hindered by the Gilbert damping Azovtsev and Pertsev (2019). The analysis of
the simulation results shows that the dependences
$\varepsilon_{xx}^{\text{max}}(x)$ and $\varepsilon_{xz}^{\text{max}}(x)$ can
be fitted by an exponential function $e^{-x/L_{\text{dec}}}$, where the decay
length $L_{\text{dec}}$ depends on the wave frequency $\nu$. At the resonance
excitation $\nu=9.6$ GHz, $L_{\text{dec}}$ amounts to approximately 350 $\mu$m
for the longitudinal wave and about 19 $\mu$m for the shear wave. This finding
explains why no damping of magnetic origin was detected in simulations of the
propagation of longitudinal elastic waves in $\mathrm{Ni}$ through a short
distance of 300 nm Chen _et al._ (2017). At the same time, it was shown
experimentally that surface acoustic waves (SAWs) can propagate in a
$\mathrm{Ni}$ film over the distance of several millimeters Casals _et al._
(2020). This absence of significant damping observed for the studied SAWs with
frequencies not exceeding 500 MHz is very different from the results of Homer
et al. Homer _et al._ (1987), who reported the decay length
$L_{\text{dec}}\approx 5.8\leavevmode\nobreak\ \mu$m for the longitudinal wave
with the frequency of 9.4 GHz in $\mathrm{Ni}$. The reason for such a
difference most probably lies in a drastic reduction of damping, which should
happen when the frequency of the elastic wave changes from about 10 GHz to
several hundreds of MHz. As for the damping of transverse elastic waves in
$\mathrm{Ni}$, our results show that the magnetic damping of elastic waves
($L_{\text{dec}}\approx 19\leavevmode\nobreak\ \mu$m) could be stronger than
the damping of electronic origin (measured $L_{\text{dec}}\approx
29\leavevmode\nobreak\ \mu$m Homer _et al._ (1987)) at wave frequencies
around 10 GHz.
Most important of all, the magnetoelastic interaction leads to the formation
of a spin wave tightly coupled to the driving elastic wave. The spin waves
predicted by our simulations have sinusoidal time dependences under both types
of elastic excitation, which differs from a non-sinusoidal time dependence
reported in Ref. Chen _et al._ (2017) for the spin wave generated by the
longitudinal acoustic wave having near-resonance frequency of 10 GHz.
Regarding the spin wave amplitude, the transverse elastic wave appears to be
much more efficient for the generation of spin waves in $\mathrm{Ni}$ than the
longitudinal wave at the chosen equilibrium magnetization orientation [compare
panels (a) and (b) in Fig. 2]. Similarly to elastically generated spin waves
in Fe81Ga19 and CoFe2O4 films Azovtsev and Pertsev (2017, 2019), the spin wave
propagating in a thick $\mathrm{Ni}$ film has the same frequency and
wavelength as the driving strain wave. Since both waves (spin and elastic)
travel with the same velocity $c_{L}$ or $c_{T}$ and obey a purely elastic
dispersion relation $k_{L}=2\pi\nu/c_{L}$ or $k_{T}=2\pi\nu/c_{T}$, the
driving wave acts as a carrier of the spin wave having a forced character.
Furthermore, the decay length of the spin wave carried by the longitudinal
acoustic wave matches the decay length $L_{\text{dec}}\approx
350\leavevmode\nobreak\ \mu$m of the latter in our simulations, which agrees
with the behavior predicted for CoFe2O4 films Azovtsev and Pertsev (2019).
However, in the case of the excitation by a shear wave, the spin-wave decay
length is significantly smaller than that of the driving wave, being roughly 9
$\mu$m instead of 19 $\mu$m. Despite this peculiarity, the acoustically driven
spin waves with frequencies $\nu\approx 10$ GHz can still propagate in Ni over
long distances of several micrometers, which is important for magnon
spintronics.
## IV Magnetoelastic dynamics in Ni/GaAs bilayers
Now we turn our attention to $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers
Figure 4: Time dependence of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface excited by longitudinal (a) or
transverse (b) elastic waves with the frequency $\nu=\nu_{\text{res}}=9.6$
GHz. Thickness $t_{\text{F}}$ of $\mathrm{Ni}$ film in each case is equal to a
corresponding wavelength $\lambda_{L}$ or $\lambda_{T}$.
comprising relatively thin $\mathrm{Ni}$ layers with the thickness
$t_{\text{F}}$ comparable to the wavelength of the driving elastic wave with
the frequency $\nu\sim\nu_{\text{res}}$, which are most suitable for
applications in miniature spin injectors (see Sec. V). The magnetoelastic
dynamics in such bilayers is of a more complicated character due to
reflections of the elastic waves from the $\mathrm{Ni}|\mathrm{GaAs}$
interface and the $\mathrm{GaAs}$ free surface. Fortunately, owing to similar
acoustic impedances of $\mathrm{Ni}$ and $\mathrm{GaAs}$, the transmittance of
the driving longitudinal or transverse wave through the
$\mathrm{Ni}|\mathrm{GaAs}$ interface is close to unity (about 0.9 with
respect to energy). In contrast, the driving wave fully reflects from the
$\mathrm{GaAs}$ free surface, and the reflected wave strongly disturbs the
magnetization dynamics when it penetrates back into the $\mathrm{Ni}$ layer.
In order to avoid this complication, we imparted a strong artificial elastic
damping to $\mathrm{GaAs}$, which is sufficient to force any elastic wave to
vanish before it reaches the free surface, but does not change significantly
the strain dynamics at the $\mathrm{Ni}|\mathrm{GaAs}$ interface.
The simulations demonstrated that the elastically driven magnetization
dynamics in $\mathrm{Ni}$ layers with the thicknesses $t_{\text{F}}$ about
$\lambda_{L}$ or $\lambda_{T}$ remains to be highly inhomogeneous at the
resonance excitation $\nu=\nu_{\text{res}}$. Initially the magnetic dynamics
has the form of a spin wave, but it assumes a complex character after several
reflections of the driving elastic wave from the boundaries of the
$\mathrm{Ni}$ layer. However, near the interface the magnetization precesses
with a constant frequency and amplitude in a steady-state regime, which
settles in after a transition period of about 1 ns (Fig. 4). Performing a
series of simulations at different thicknesses of $\mathrm{Ni}$ layers, we
found that the amplitude of the magnetization precession at the interface has
local maxima at Ni thicknesses amounting to 0.25, 0.75, 1.25 and 1.75 of the
wavelength $\lambda_{L}$ or $\lambda_{T}$ (Fig. 5). This result differs from
that obtained for Fe81Ga19/Au and CoFe2O4/Pt bilayers in our previous works
Azovtsev and Pertsev (2017, 2019), where such amplitude maximizes at the
ferromagnet’s thickness equal to one wavelength of the driving elastic wave.
Figure 5: Amplitude of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface as a function of the $\mathrm{Ni}$
thickness $t_{\text{F}}$ normalized by the wavelength $\lambda$ of the driving
longitudinal or transverse elastic wave. The plots show the maximal change
$\Delta m_{x}$ of the out-of-plane direction cosine $m_{x}(x=t_{\text{F}},t)$
normalized by the largest value of $\Delta m_{x}$ in the studied thickness
range. The excitation frequency equals $\nu_{\text{res}}=9.6$ GHz.
To understand the revealed behavior of ferromagnetic-nonmagnetic (F/N)
bilayers, we investigated the dependence of the strain amplitude at the
interface on the thickness $t_{\text{F}}$ of the F layer. The analysis of the
results of simulations showed that in all studied bilayers the precession
amplitude in the steady-state regime becomes maximal whenever the strain
amplitude maximizes. Therefore, we considered a general elasticity problem of
finding the strain distribution in an elastic F/N bilayer subjected to a
periodic surface displacement
$u_{i}^{\text{F}}(x=0,t)=u_{\text{max}}e^{-i\omega t}$. Despite multiple
reflections of the elastic waves from the boundaries of the F layer, the
steady-state solution for the elastic displacement $u_{i}^{\text{F}}(x,t)$
inside this layer can be written as a superposition of two waves with the same
frequency $\omega=2\pi\nu$. Indeed, due to the principle of superposition in
linear elasticity any number of interfering sinusoidal waves with the same
frequency but different amplitudes and phases produce another sinusoidal wave
of the same frequency with its own amplitude and phase Sadd (2005). Hence, we
can write
$u_{i}^{\text{F}}(x,t)=A_{i}^{\text{F}}e^{i(k_{i}^{\text{F}}x-\omega
t)}+B_{i}^{\text{F}}e^{-i(k_{i}^{\text{F}}x+\omega t)},$ (2)
where the first term corresponds to the waves propagating towards the F$|$N
interface, while the second term describes the waves reflected from the F$|$N
interface; $A_{i}^{\text{F}}$ and $B_{i}^{\text{F}}$ are the unknown
amplitudes of these waves, and $k_{i}^{\text{F}}$ is the wavenumber of the
longitudinal ($i=x$) or transverse ($i=y$ or $z$) wave in the F layer.
Figure 6: Expected thickness dependences of the strain amplitudes at the
interface between $\mathrm{Ni}$ and $\mathrm{GaAs}$ layers and CoFe2O4 and Pt
layers. The amplitude of each strain is normalized by its maximal value. The
thickness $t_{\text{F}}$ of the ferromagnetic layer is normalized by the
wavelength $\lambda$ of the excited longitudinal or transverse elastic wave.
The excitation frequency equals 9.6 GHz ($\mathrm{Ni}$/$\mathrm{GaAs}$) and 11
GHz (CoFe2O4/Pt).
Since we neglect the reflections from the free surface of the N layer, only
the transmitted elastic wave exists in it, and the displacement
$u_{i}^{\text{N}}(x,t)$ has the form
$u_{i}^{\text{N}}(x,t)=A_{i}^{\text{N}}e^{i(k_{i}^{\text{N}}x-\omega t)},$ (3)
where $A_{i}^{\text{N}}$ and $k_{i}^{\text{N}}$ are the amplitude and
wavenumber of the transmitted wave. The mechanical boundary conditions at the
F$|$N interface $x=t_{\text{F}}$ yield the displacement continuity
$u_{i}^{\text{F}}(x=t_{\text{F}},t)=u_{i}^{\text{N}}(x=t_{\text{F}},t)$ and
the stress continuity
$\sigma_{ix}^{\text{F}}(x=t_{\text{F}},t)=\sigma_{ix}^{\text{N}}(x=t_{\text{F}},t)$.
In our model case, the stresses are given by the relations
$\sigma_{ix}^{\text{F}}(x,t)=c_{\alpha\alpha}^{\text{F}}(1/\sqrt{\alpha})\partial/\partial
x[u_{i}^{\text{F}}(x,t)]$ and
$\sigma_{ix}^{\text{N}}(x,t)=c_{\alpha\alpha}^{\text{N}}(1/\sqrt{\alpha})\partial/\partial
x[u_{i}^{\text{N}}(x,t)]$, where $c_{\alpha\alpha}^{\text{F}}$ and
$c_{\alpha\alpha}^{\text{N}}$ are the elastic stiffnesses of the F and N
layers, respectively ($\alpha=1$ at $i=x$ and $\alpha=4$ at $i=y$ or $z$).
Combining the boundary conditions at the F$|$N interface and the F surface
$x=0$ and using Eqs. (2) and (3), one can derive analytic relations for the
unknown amplitudes $A_{i}^{\text{F}}$, $B_{i}^{\text{F}}$, and
$A_{i}^{\text{N}}$. The substitution of these relations back to Eqs. (2) and
(3) yields the formulae for the displacements $u_{i}^{\text{F}}(x,t)$ and
$u_{i}^{\text{N}}(x,t)$, which render possible to calculate the strains
$\varepsilon_{ix}^{\text{F}}=(1/\sqrt{\alpha})\partial
u_{i}^{\text{F}}/\partial x$ and
$\varepsilon_{ix}^{\text{N}}=(1/\sqrt{\alpha})\partial
u_{i}^{\text{N}}/\partial x$ in the F and N layers. For the strains
$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)$ at the F$|$N interface after
some mathematical manipulations we obtain
$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)=\frac{2i\varepsilon_{ix}^{\text{max}}Z_{\alpha}e^{-i\omega
t}}{(1+Z_{\alpha})F^{-}+(1-Z_{\alpha})F^{+}},$ (4)
where
$Z_{\alpha}=\sqrt{\frac{c_{\alpha\alpha}^{\text{N}}\rho_{\text{N}}}{c_{\alpha\alpha}^{\text{F}}\rho_{\text{F}}}},F^{+}=e^{ik_{i}^{\text{F}}t_{\text{F}}},F^{-}=e^{-ik_{i}^{\text{F}}t_{\text{F}}}.$
Equation (4) shows that the amplitude of
$\varepsilon_{ix}^{\text{F}}(x=t_{\text{F}},t)$ depends on the input strain
$\varepsilon_{ix}^{\text{max}}=(1/\sqrt{\alpha})u_{\text{max}}k_{i}^{\text{F}}$,
the relative thickness $t_{\text{F}}/\lambda_{L}$ or
$t_{\text{F}}/\lambda_{T}$ of the F layer, and the dimensionless parameter
$Z_{\alpha}$ of the F/N bilayer, which is governed by the elastic stiffnesses
and densities of the involved materials. Using Eq. (4), we calculated the
dependences of the discussed strain amplitudes on the relative thickness of
the F layer for $\mathrm{Ni}$/$\mathrm{GaAs}$ and CoFe2O4/Pt bilayers
subjected to the resonance excitation $\nu=\nu_{\text{res}}$. The results
presented in Fig. 6 show that, for a given bilayer, the amplitudes of
$\varepsilon_{xx}^{\text{F}}(x=t_{\text{F}},t)$ and
$\varepsilon_{zx}^{\text{F}}(x=t_{\text{F}},t)$ normalized by their maximal
values follow similar curves (almost identical in case of
$\mathrm{Ni}$/$\mathrm{GaAs}$) when plotted as a function of
$t_{\text{F}}/\lambda_{L}$ and $t_{\text{F}}/\lambda_{T}$, respectively.
However, the maximal strain amplitude is reached at the thicknesses
$t_{\text{F}}=(0.25+0.5n)\lambda$ in $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers
and at $t_{\text{F}}=(0.5+0.5n)\lambda$ in CoFe2O4/Pt ones
($\lambda=\lambda_{L}$ or $\lambda_{T}$, $n=0,1,2,3$…). These conditions
explain dissimilar results of our micromagnetoelastic simulations performed
for $\mathrm{Ni}$/$\mathrm{GaAs}$ and CoFe2O4/Pt bilayers, which showed that
the precession amplitude at the interface has a maximum at
$t_{\text{F}}=0.75\lambda$ ($\mathrm{Ni}$/$\mathrm{GaAs}$) and
$t_{\text{F}}=\lambda$ (CoFe2O4/Pt). Furthermore, the analysis of Eq. (4)
reveals that the character of the strain-amplitude thickness dependence is
governed by the magnitude of the parameter $Z_{\alpha}$. Namely, when
$Z_{\alpha}<1$, the strain amplitude maximizes at
$t_{\text{F}}=(0.25+0.5n)\lambda$ as it happens in the
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers ($Z_{1}\approx Z_{4}\approx 0.53$),
whereas at $Z_{\alpha}>1$ the optimal thicknesses satisfy the condition
$t_{\text{F}}=(0.5+0.5n)\lambda$ holding for the CoFe2O4/Pt bilayers
($Z_{1}\approx 2.34$, $Z_{4}\approx 1.91$). The derived simple criteria open
the possibility to predict the optimal thickness of the ferromagnetic layer
that maximizes the strain and precession amplitudes at the interface for any
F/N bilayer.
Figure 7: Dependence of the amplitude of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface on the excitation frequency $\nu$. The
points show the maximal deviation $\Delta m_{x}(\nu)$ of the out-of-plane
magnetization direction cosine $m_{x}$ from its equilibrium value. The
thickness of the $\mathrm{Ni}$ layer equals three quarters of the wavelength
of the driving longitudinal or transverse elastic wave.
In conclusion of this section, we discuss the dependence of the amplitude of
the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface on
the excitation frequency $\nu$. For the optimal $\mathrm{Ni}$ thickness
$t_{\text{F}}=0.75\lambda$ and the driving waves with the initial strain
amplitudes
$\varepsilon_{xx}^{\text{max}}=\varepsilon_{xz}^{\text{max}}=10^{-4}$, the
simulations predict that the maximal deviation $\Delta m_{x}(\nu)$ of the
magnetization direction cosine $m_{x}$ from the equilibrium value varies with
the frequency as shown in Fig. 7. It can be seen that $\Delta m_{x}(\nu)$
reaches a peak at a frequency $\nu_{\text{max}}$ slightly higher than the
resonance frequency $\nu_{\text{res}}=9.6$ GHz. Namely, $\nu_{\text{max}}$
amounts to 9.9 GHz for the precession excited by the longitudinal elastic
waves and to 10 GHz for that induced by the transverse waves. In agreement
with the results described in Sec. III, the shear waves appear to be much more
efficient for the excitation of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface (see Fig. 7).
## V Spin pumping into GaAs layer
The magnetization precession occurring near the interface between the
ferromagnet and a nonmagnetic conductor generates the spin pumping into the
latter Tserkovnyak _et al._ (2005). Using the results obtained for the
magnetization dynamics $\mathbf{m}(x=t_{\mathrm{F}},t)$ induced by the elastic
waves at the $\mathrm{Ni}|\mathrm{GaAs}$ interface, we can calculate the spin
current flowing in the $\mathrm{GaAs}$ layer. The spin-current density
$\mathbf{J}_{s}(x,t)$ is a second-rank tensor characterizing the direction of
spin flow and the orientation and magnitude of the carried spin polarization
per unit volume Dyakonov and Perel (1971). In the vicinity of the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, the density
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$ of the spin current pumped into
$\mathrm{GaAs}$ can be evaluated via the approximate relation
$\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SP}}\simeq(\hbar/4\pi)\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\mathbf{m}\times\dot{\mathbf{m}}$,
where $\mathbf{e}_{n}$ is the unit vector normal to the interface and pointing
into $\mathrm{GaAs}$, $\hbar$ is the reduced Planck constant,
$g^{r}_{\uparrow\downarrow}$ is the reflection spin-mixing conductance per
unit area, and a small contribution caused by the imaginary part of
$g^{r}_{\uparrow\downarrow}$ is neglected Zwierzycki _et al._ (2005). Since
$\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}$ may be set equal to
$1.5\times 10^{17}$ m-2 for the $\mathrm{Ni}|\mathrm{GaAs}$ interface Ando
_et al._ (2011), the above relation and the simulation data on the temporal
variation of $\mathbf{m}(x=t_{\mathrm{F}},t)$ enable us to evaluate the spin-
current density $\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$.
Figure 8 shows time dependences of three nonzero components
$J^{\mathrm{SP}}_{xj}$ of the tensor $\mathbf{J}_{\mathrm{SP}}$, which settle
in the steady-state regime of the magnetization precession at the excitation
frequency $\nu=\nu_{\mathrm{max}}$. It can be seen that the transverse wave
creates much stronger spin pumping into $\mathrm{GaAs}$ than the longitudinal
one. For both types of elastic excitations, the amplitude of
$J^{\mathrm{SP}}_{xx}$ is about two times larger than almost equal amplitudes
of $J^{\mathrm{SP}}_{xy}$ and $J^{\mathrm{SP}}_{xz}$.
Figure 8: Time dependences of the spin-current densities
$J_{xj}^{\mathrm{SP}}$ pumped into $\mathrm{GaAs}$ by the longitudinal (a) and
transverse (b) elastic waves with the frequencies 9.9 GHz and 10 GHz,
respectively. The time period shown in the figure corresponds to the steady-
state regime of the magnetization precession at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface. The thickness of the $\mathrm{Ni}$
layer equals three quarters of the wavelength of the driving longitudinal or
transverse elastic wave.
The averaging over the period $1/\nu$ of (almost sinusoidal) spin-current
variations shows that $\langle J^{\mathrm{SP}}_{xx}\rangle_{t}$ is negligible,
whereas there are very small nonzero dc components $\langle
J^{\mathrm{SP}}_{xy}\rangle_{t}=\langle J^{\mathrm{SP}}_{xz}\rangle_{t}$ of
the pumped spin current. It should be noted that, owing to relatively small
reflection spin mixing conductance of the $\mathrm{Ni}|\mathrm{GaAs}$
interface, the spin pumping into $\mathrm{GaAs}$ does not significantly
increase the effective damping of the magnetization precession in
$\mathrm{Ni}$ Nikitchenko and Pertsev (2020).
The pumped spin current generates non-equilibrium spin accumulation
$\bm{\mu}_{s}(x,t)$, which gives rise to a spin backflow at the interface with
the density $\mathbf{J}_{\mathrm{SB}}(x=t_{\mathrm{F}},t)$ amounting to
$\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SB}}\approx-\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\bm{\mu}_{s}/4\pi$
Tserkovnyak _et al._ (2005). The overall spin-current density
$\mathbf{J}_{s}(x,t)$ decays inside the $\mathrm{GaAs}$ layer due to spin
relaxation and diffusion. The spatial distribution of the density
$\mathbf{J}_{s}$ depends on that of the spin accumulation $\bm{\mu_{s}}$
Tserkovnyak and Brataas (2002), being defined in our one-dimensional model by
the relation
$\mathbf{e}_{n}\cdot\mathbf{J}_{s}(x,t)=-[\sigma\hbar/(4e^{2})]\partial\bm{\mu}_{s}(x,t)/\partial
x$, where $e$ is the elementary positive charge, and $\sigma$ is the
electrical conductivity, which amounts to $3.68\times 10^{4}$ S m-1 for
n+-$\mathrm{GaAs}$ Kikkawa and Awschalom (1998); Nikitchenko and Pertsev
(2020). We find the spin accumulation $\bm{\mu}_{s}(x,t)$ by solving the
diffusion equation Tserkovnyak and Brataas (2002) appended by the boundary
conditions for the spin currents at the $\mathrm{Ni}|\mathrm{GaAs}$ interface
$x=t_{\mathrm{F}}$ and the $\mathrm{GaAs}$ free surface
$x=t_{\mathrm{F}}+t_{\mathrm{N}}$, which read
$\mathbf{J}_{s}(x=t_{\mathrm{F}})=\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}})+\mathbf{J}_{\mathrm{SB}}(x=t_{\mathrm{F}})$
and $\mathbf{J}_{s}(x=t_{\mathrm{F}}+t_{\mathrm{N}})=0$. The calculation
yields
$\bm{\mu}_{s}^{\omega}=\frac{4\pi
e^{2}\cosh{[\kappa(t_{\mathrm{N}}+t_{\mathrm{F}}-x)]}}{e^{2}\mathrm{Re}\big{[}g^{r}_{\uparrow\downarrow}\big{]}\cosh{(\kappa
t_{\mathrm{N}})}+\pi\sigma\hbar\kappa\sinh{(\kappa
t_{\mathrm{N}})}}\mathbf{e}_{n}\cdot\mathbf{J}_{\mathrm{SP}}^{\omega},$ (5)
where $\bm{\mu}_{s}^{\omega}$ and $\mathbf{J}_{\mathrm{SP}}^{\omega}$ denote
the complex amplitudes of the harmonics having the angular frequency $\omega$,
which represent the Fourier components of the spin accumulation
$\bm{\mu}_{s}(x,t)$ and spin pumping density
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$, and the parameter
$\kappa=\lambda_{\mathrm{sd}}^{-1}\sqrt{1+i\omega\tau_{\mathrm{sf}}}$ depends
on spin-diffusion length $\lambda_{\mathrm{sd}}$ and spin-flip relaxation time
$\tau_{\mathrm{sf}}$. Equation (5) differs from a similar relation derived in
Ref. Tserkovnyak and Brataas (2002) by the account of the spin backflow. In
the case of $\mathrm{GaAs}$, the spin backflow cannot be neglected because it
appears to be rather strong at $\lambda_{\mathrm{sd}}=2.32$ $\mu$m Kikkawa and
Awschalom (1998) and $\tau_{\mathrm{sf}}=0.9$ ns Bhat and Kumar (2014). Since
the elastically driven spin pumping is almost monochromatic in our setting,
Eq. (5) is valid for the sought relation between $\bm{\mu}_{s}(x,t)$ and
$\mathbf{J}_{\mathrm{SP}}(x=t_{\mathrm{F}},t)$ as well, which enables us to
calculate the overall spin-current density $\mathbf{J}_{s}(x,t)$.
Owing to the inverse spin Hall effect (ISHE), the spin current in the
$\mathrm{GaAs}$ layer generates a charge current with the density
$\mathbf{J}_{c}^{\mathrm{ISHE}}$ defined by the formula Mosendz _et al._
(2010)
$\mathbf{J}_{c}^{\mathrm{ISHE}}(x,t)=\alpha_{\mathrm{SH}}(2e/\hbar)\mathbf{e}_{n}\times[\mathbf{e}_{n}\cdot\mathbf{J}_{s}(x,t)],$
(6)
where $\alpha_{\mathrm{SH}}=0.007$ is the spin Hall angle of $\mathrm{GaAs}$
Ando _et al._ (2011). Under considered open-circuit electrical boundary
conditions, the transverse charge current $\mathbf{J}_{c}^{\mathrm{ISHE}}$
flowing along the interface should create a charge accumulation at the lateral
boundaries of the $\mathrm{GaAs}$ film. Such an accumulation induces an
electric field $\mathbf{E}$ in $\mathrm{GaAs}$, which causes a drift current
with the density $\mathbf{J}_{c}^{\mathrm{drift}}=\sigma\mathbf{E}$. To
calculate the spatial distribution of the electric potential $\varphi$ in the
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer and the total charge current density
$\mathbf{J}_{c}=\mathbf{J}_{c}^{\mathrm{ISHE}}+\mathbf{J}_{c}^{\mathrm{drift}}$,
we numerically solve the Laplace’s equation $\nabla^{2}\varphi=0$ with the
appropriate boundary conditions. The latter follow from the absence of charge
current across the outer surfaces of the bilayer, and the absence of
$\mathbf{J}_{c}^{\mathrm{ISHE}}$ inside $\mathrm{Ni}$. It should be noted that
the potential $\varphi$ should be regarded as a complex quantity since the
parameter $\kappa$ affecting the spin-current density $\mathbf{J}_{s}$
involved in Eq. (6) has a substantial complex part at
$\omega\tau_{\mathrm{sf}}>>1$.
In the numerical calculations, we consider only the component $J^{s}_{xy}$ of
the elastically generated spin current $\mathbf{J}_{s}$, because $J^{s}_{xx}$
does not create any charge flow, and the components $J^{s}_{xy}$ and
$J^{s}_{xz}$ have almost equal magnitudes and can be probed independently via
transverse voltages
$V_{z}^{\mathrm{ISHE}}=\varphi(z=w_{\mathrm{N}}/2)-\varphi(z=-w_{\mathrm{N}}/2)$
and $V_{y}^{\mathrm{ISHE}}=\varphi(y=0)-\varphi(y=-h_{\mathrm{N}})$,
respectively (Fig. 1). Figure 9 shows the amplitude $\delta
V_{z}^{\mathrm{ISHE}}(x)$ of the oscillating voltage
$V_{z}^{\mathrm{ISHE}}(x,t)$ calculated at the excitation frequency
$\nu=\nu_{\mathrm{max}}$ for the $\mathrm{GaAs}$ films with the thickness
$t_{\mathrm{N}}=5$ $\mu$m.
Figure 9: Amplitude $\delta V_{z}^{\mathrm{ISHE}}(x)$ of the ac voltage
between the lateral sides of the $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer excited
by longitudinal (a) and transverse (b) elastic waves with the frequencies 9.9
GHz and 10 GHz, respectively. The thickness of the $\mathrm{GaAs}$ layer
equals 5 $\mu$m, and its width $w_{\mathrm{N}}$ is indicated in the figure.
It can be seen that $\delta V_{z}^{\mathrm{ISHE}}$ varies nonmonotonically
with the distance $x-t_{\mathrm{F}}$ from the $\mathrm{Ni}|\mathrm{GaAs}$
interface, reaching its maximum inside the semiconductor at
$x-t_{\mathrm{F}}=100-200$ nm. The voltage amplitude $\delta
V_{z}^{\mathrm{ISHE}}(x)$ grows with the increasing width $w_{\mathrm{N}}$ of
the $\mathrm{GaAs}$ film, and the voltage peak becomes much higher at the
excitation of magnetization dynamics by the shear elastic wave (see Fig. 9).
Importantly, the transverse ac voltage $V_{z}^{\mathrm{ISHE}}$ characterizing
the spin pumping induced by either type of elastic waves is high enough for
the experimental measurement near the $\mathrm{Ni}|\mathrm{GaAs}$ interface.
Another method to evaluate the spin pumping into a normal metal or
semiconductor experimentally is known as a nonlocal spin detection scheme
Johnson and Silsbee (1985); Lou _et al._ (2007). This scheme measures a
voltage $V_{s}$ between a ferromagnetic probe and a nonmagnetic electrode
brought into contact with the semiconductor. Since the voltage $V_{s}$ is
directly proportional to the product
$\bm{\mu}_{s}\cdot\mathbf{M}_{\mathrm{probe}}$, where
$\mathbf{M}_{\mathrm{probe}}$ is the probe magnetization, it is possible to
detect all three components of the vector $\bm{\mu}_{s}$ by using differently
magnetized ferromagnetic contacts. As a representative example, we consider an
iron probe magnetized along the $x$ axis, which is placed on the lateral side
of the $\mathrm{GaAs}$ layer (Fig. 1), and a normal-metal electrode deposited
on the free surface $x=t_{\mathrm{F}}+t_{\mathrm{N}}$ of the 5-$\mu$m-thick
$\mathrm{GaAs}$ film. In this case, the spin voltage $V_{s}(x,t)$ is defined
by the relation
$V_{s}(x,t)=\eta_{\mathrm{IE}}p_{\mathrm{Fe}}\mu_{x}^{s}(x,t)/(2e)$, where
$\eta_{\mathrm{IE}}$ is the spin transmission efficiency of the
$\mathrm{GaAs}|\mathrm{Fe}$ interface, $p_{\mathrm{Fe}}$ is the spin
polarization of $\mathrm{Fe}$ at the Fermi level, and the spin accumulation
$\mu_{x}^{s}$ beneath the probe with nanoscale dimensions is assumed uniform.
Figure 10 shows the amplitude $\delta V_{s}(x)$ and phase $\phi_{s}(x)$ of the
ac spin voltage calculated using the parameters $\eta_{\mathrm{IE}}\approx
0.5$ and $p_{\mathrm{Fe}}\approx 0.42$ characteristic of the Schottky tunnel
barrier between Fe probe and n+-$\mathrm{GaAs}$ Lou _et al._ (2007).
Figure 10: Amplitude $\delta V_{s}$ and phase $\phi_{s}$ (inset) of the ac
spin voltage between the lateral Fe probe and a normal-metal contact at the
$\mathrm{GaAs}$ free surface plotted as a function of the distance
$x-t_{\mathrm{F}}$ from the $\mathrm{Ni}|\mathrm{GaAs}$ interface. The
$\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer is excited either by the longitudinal
wave with frequency 9.9 GHz or by the transverse wave with frequency 10 GHz.
The thickness of the $\mathrm{GaAs}$ layer equals 5 $\mu$m.
Importantly, the voltage amplitude $\delta V_{s}$ appears to be rather large
near the $\mathrm{Ni}|\mathrm{GaAs}$ interface, exceeding 650 nV under the
excitation by the transverse elastic wave and 80 nV in the bilayer excited by
the longitudinal wave (see Fig. 10). Although $\delta V_{s}(x)$ gradually
decreases with the increasing distance $x-t_{\mathrm{F}}$ from the
$\mathrm{Ni}|\mathrm{GaAs}$ interface, it remains to be measurable
experimentally even at the distances over 0.5 $\mu$m. The phase $\phi_{s}(x)$
of the spin voltage varies linearly inside the $\mathrm{GaAs}$ layer and
changes strongly already at $x-t_{\mathrm{F}}\sim 0.5$ $\mu$m (Fig. 10). This
result demonstrates that the phase difference between the spin accumulation
inside $\mathrm{GaAs}$ and the spin pumping at the $\mathrm{Ni}|\mathrm{GaAs}$
interface may be large owing to the condition $\omega\tau_{\mathrm{sf}}>>1$.
The input electric power $W$ necessary for the functioning of the proposed
spin injector can be estimated from the generated acoustic power using the
relation $W=\frac{1}{K^{2}}\frac{1}{2}A\rho c\omega^{2}u_{\text{max}}^{2}$,
where $K^{2}$ is the electromechanical transduction efficiency of the
piezoelectric transducer Bhaskar _et al._ (2020), $c$ is the velocity of the
generated longitudinal or transverse acoustic wave, and $A$ denotes the
dynamically strained area of the ferromagnetic film having the mass density
$\rho$. Expressing $u_{\text{max}}$ via the maximal strain
$\varepsilon_{\text{max}}$ in the acoustic wave, we obtain
$W=\alpha\frac{1}{K^{2}}\frac{1}{2}A\rho c^{3}\varepsilon_{\text{max}}^{2}$.
For the device with $A<25\leavevmode\nobreak\ \mu$m2 and $K^{2}=12\%$ Bhaskar
_et al._ (2020), which is driven by a transverse ($\alpha=4$) or longitudinal
($\alpha=1$) wave with $\varepsilon_{\text{max}}=10^{-4}$, the calculation
yields $W<2$ mW. This value is much smaller than the lowest power consumption
$W\approx 25$ mW of the spin injector driven by the microwave magnetic field
Ando _et al._ (2011).
## VI Conclusions
In this work, we theoretically studied the coupled elastic and spin dynamics
induced in $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayers by longitudinal and
transverse acoustic waves generated by the attached piezoelectric transducer
(Fig. 1). Using advanced micromagnetoelastic simulations, we first modeled the
elastically driven magnetization dynamics in thick $\mathrm{Ni}$ films at the
wave frequencies around the resonance frequency $\nu_{\mathrm{res}}$ of the
coherent magnetization precession in unstrained $\mathrm{Ni}$ film. The
simulations showed that this dynamics has the form of a forced spin wave
having the frequency and wavelength of the monochromatic driving wave.
Remarkably, the transverse elastic wave creates much stronger spin wave than
the longitudinal one at the considered external magnetic field (Fig. 2). The
backaction of travelling spin wave on the elastic dynamics manifests itself in
the generation of weak secondary elastic waves created by the magnetization
precession (see Fig. 3). These waves are characterized by oscillating strains
$\varepsilon_{ij}(x,t)$ different from the strain $\varepsilon_{xx}(x,t)$ or
$\varepsilon_{xz}(x,t)$ in the primary driving wave, and they were not
reported in the previous work on the modeling of magnetization dynamics
induced by longitudinal elastic waves in $\mathrm{Ni}$ Chen _et al._ (2017).
The magnetoelastic feedback also influences the driving elastic wave, leading
to a gradual reduction of its amplitude during the propagation in
$\mathrm{Ni}$, which adds to the “acoustic” decay caused by the wave
attenuation of electronic origin Homer _et al._ (1987). At the considered
wave frequencies $\nu\approx 10$ GHz, the decay resulting from the energy
transfer to the magnetic subsystem is stronger than the acoustic decay for the
transverse waves but small for longitudinal ones. Importantly, both types of
elastic waves are expected to carry spin signals over significant distances of
several micrometers in $\mathrm{Ni}$.
We also modeled the magnetoelastic dynamics of $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayers at the excitation frequencies $\nu\sim\nu_{\mathrm{res}}$, focusing
on the $\mathrm{Ni}$ thicknesses comparable to the wavelength of the injected
acoustic wave. The simulations allowed for the reflections of the elastic
waves from the boundaries of the $\mathrm{Ni}$ layer and demonstrated the
excitation of a nonhomogeneous magnetization dynamics in it. Importantly, a
steady-state magnetization precession with frequency equal to the excitation
frequency and constant amplitude was revealed at the
$\mathrm{Ni}|\mathrm{GaAs}$ interface after a short transition period of about
1 ns (Fig. 4). The simulations performed for $\mathrm{Ni}$ layers of different
thicknesses showed that the amplitude of stationary precession has a maximum
at $\mathrm{Ni}$ thickness amounting to three quarters of the driving elastic
wave wavelength. This finding, which differs from the results of simulations
carried out for $\mathrm{Fe}_{81}\mathrm{Ga}_{19}$/$\mathrm{Au}$ and
$\mathrm{CoFe}_{2}\mathrm{O}_{4}$/$\mathrm{Pt}$ bilayers Azovtsev and Pertsev
(2017, 2019), was explained by an analytical model giving simple criteria for
the optimal geometry of an elastic bilayer that maximizes the strain amplitude
at the interface.
Numerical results obtained for the steady-state magnetization precession at
the $\mathrm{Ni}|\mathrm{GaAs}$ interface were used to evaluate the spin-
current densities pumped into $\mathrm{GaAs}$ by the dynamically strained
$\mathrm{Ni}$ film (Fig. 8). The spin accumulation in the semiconductor was
then calculated by solving numerically the spin diffusion equation with the
account of the spin pumping into $\mathrm{GaAs}$ and the spin backflow into
$\mathrm{Ni}$. Since the spin current creates a charge current owing to the
ISHE, the spin generation in $\mathrm{GaAs}$ can be detected via electrical
measurements. Therefore, we also determined the distribution of the electric
potential in the $\mathrm{Ni}$/$\mathrm{GaAs}$ bilayer with open-circuit
electrical boundary conditions by numerically solving the Laplace’s equation.
This enabled us to evaluate the transverse voltage appearing between the
lateral sides of the dynamically strained $\mathrm{Ni}$/$\mathrm{GaAs}$
bilayer. It was shown that the amplitude of this ac voltage is large enough
for the experimental detection near the $\mathrm{Ni}|\mathrm{GaAs}$ interface
(Fig. 9). Furthermore, spin accumulation manifests itself in the voltage
between a ferromagnetic probe and a nonmagnetic electrode brought into contact
with the semiconductor (Fig. 1). Performing calculations of this ac spin
voltage, we found that it retains measurable amplitude even at the distances
over 0.5 $\mu$m from the $\mathrm{Ni}|\mathrm{GaAs}$ interface (Fig. 10).
Thus, our theoretical study of the $\mathrm{Ni}$/$\mathrm{GaAs}$
heterostructure demonstrated that the spin injector employing elastic waves is
promising for the spin generation in semiconductors. Since the proposed device
can be driven electrically via the strain-mediated magnetoelectric effect, it
has much lower power consumption than the spin injector excited by a microwave
magnetic field Ando _et al._ (2011).
## VII Acknowledgements
The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics ”BASIS”.
## References
* Hägele _et al._ (1998) D. Hägele, M. Oestreich, and W. W. Rühle, Appl. Phys. Lett. 73, 1580 (1998).
* Kikkawa and Awschalom (1999) J. M. Kikkawa and D. D. Awschalom, Physics Today 52, 33 (1999).
* Bhat and Kumar (2014) S. G. Bhat and P. S. A. Kumar, Sci. Rep. 4, 5588 (2014).
* Putikka and Joynt (2004) W. O. Putikka and R. Joynt, Phys. Rev. B 70, 113201 (2004).
* Kokurin _et al._ (2013) I. A. Kokurin, P. V. Petrov, and N. S. Averkiev, Semiconductors 47, 1232 (2013).
* Hirohata _et al._ (2020) A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Diény, P. Pirro, and B. Hillebrands, J. Magn. Magn. Mater. 509, 166711 (2020).
* Schmidt _et al._ (2000) G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B 62, R4790 (2000).
* Hanbicki and Jonker (2002) A. T. Hanbicki and B. T. Jonker, Appl. Phys. Lett. 80, 1240 (2002).
* Jiang _et al._ (2005) X. Jiang, R. Wang, R. M. Shelby, R. M. Macfarlane, S. R. Bank, J. S. Harris, and S. S. Parkin, Phys. Rev. Lett. 94, 056601 (2005).
* Dash _et al._ (2009) S. P. Dash, S. Sharma, R. S. Patel, M. P. de Jong, and R. Jansen, Nature 462, 491 (2009).
* Kamerbeek _et al._ (2014) A. M. Kamerbeek, E. K. de Vries, A. Dankert, S. P. Dash, B. J. van Wees, and T. Banerjee, Appl. Phys. Lett. 104, 212106 (2014).
* Brataas _et al._ (2002) A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404 (2002).
* Tserkovnyak _et al._ (2005) Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005).
* Heinrich _et al._ (2003) B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003).
* Saitoh _et al._ (2006) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006).
* Bell _et al._ (2008) C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys. Rev. Lett. 100, 047002 (2008).
* Mosendz _et al._ (2010) O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010).
* Czeschka _et al._ (2011) F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011).
* Tashiro _et al._ (2015) T. Tashiro, S. Matsuura, A. Nomura, S. Watanabe, K. Kang, H. Sirringhaus, and K. Ando, Sci. Rep. 5, 15158 (2015).
* Ando _et al._ (2011) K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C. H. W. Barnes, S. Maekawa, and E. Saitoh, Nat. Mat. 10, 655 (2011).
* Shikoh _et al._ (2013) E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110, 127201 (2013).
* Lee _et al._ (2014) J. Lee, L. Huang, D. Hung, T. Chiang, J. C. A. Huang, J. Liang, and S. Lee, Appl. Phys. Lett. 104, 052401 (2014).
* Wang _et al._ (2017) Y. Wang, R. Ramaswamy, M. Motapothula, K. Narayanapillai, D. Zhu, J. Yu, T. Venkatesan, and H. Yang, Nano Lett. 17, 7659 (2017).
* Mendes _et al._ (2018) J. B. S. Mendes, A. Aparecido-Ferreira, J. Holanda, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 112, 242407 (2018).
* Weiler _et al._ (2011) M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Phys. Rev. Lett. 106, 117601 (2011).
* Uchida _et al._ (2011a) K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, Nat. Mater. 10, 737 (2011a).
* Weiler _et al._ (2012) M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 176601 (2012).
* Kamra _et al._ (2015) A. Kamra, H. Keshtgar, P. Yan, and G. E. W. Bauer, Phys. Rev. B 91, 104409 (2015).
* Polzikova _et al._ (2016) N. I. Polzikova, S. G. Alekseev, I. I. Pyataikin, I. M. Kotelyanskii, V. A. Luzanov, and A. P. Orlov, AIP Advances 6, 056306 (2016).
* Azovtsev and Pertsev (2016) A. V. Azovtsev and N. A. Pertsev, Phys. Rev. B 94, 184401 (2016).
* Azovtsev and Pertsev (2017) A. V. Azovtsev and N. A. Pertsev, Appl. Phys. Lett. 111, 222403 (2017).
* Polzikova _et al._ (2018) N. I. Polzikova, S. G. Alekseev, V. A. Luzanov, and A. O. Raevskiy, Phys. Solid State 60, 2211 (2018).
* Azovtsev and Pertsev (2019) A. V. Azovtsev and N. A. Pertsev, Phys. Rev. B 100, 224405 (2019).
* Alekseev _et al._ (2020) S. G. Alekseev, S. E. Dizhur, N. I. Polzikova, V. A. Luzanov, A. O. Raevskiy, A. P. Orlov, V. A. Kotov, and S. A. Nikitov, Appl. Phys. Lett. 117, 072408 (2020).
* Cherepov _et al._ (2014) S. Cherepov, P. Khalili Amiri, J. G. Alzate, K. Wong, M. Lewis, P. Upadhyaya, J. Nath, M. Bao, A. Bur, T. Wu, G. P. Carman, A. Khitun, and K. L. Wang, Appl. Phys. Lett. 104, 082403 (2014).
* Bhaskar _et al._ (2020) U. K. Bhaskar, D. Tierno, G. Talmelli, F. Ciubotaru, C. Adelmann, and T. Devolder, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 67, 1284 (2020).
* Uchida _et al._ (2011b) K.-I. Uchida, T. An, Y. Kajiwara, M. Toda, and E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011b).
* Thevenard _et al._ (2014) L. Thevenard, C. Gourdon, J. Y. Prieur, H. J. von Bardeleben, S. Vincent, L. Becerra, L. Largeau, and J.-Y. Duquesne, Phys. Rev. B 90, 094401 (2014).
* Janušonis _et al._ (2015) J. Janušonis, C. L. Chang, P. H. M. van Loosdrecht, and R. I. Tobey, Appl. Phys. Lett. 106, 181601 (2015).
* Gowtham _et al._ (2015) P. G. Gowtham, T. Moriyama, D. C. Ralph, and R. A. Buhrman, J. Appl. Phys. 118, 233910 (2015).
* Casals _et al._ (2020) B. Casals, N. Statuto, M. Foerster, A. Hernández-Mínguez, R. Cichelero, P. Manshausen, A. Mandziak, L. Aballe, J. M. Hernàndez, and F. Macià, Phys. Rev. Lett. 124, 137202 (2020).
* Akhiezer _et al._ (1958) A. I. Akhiezer, V. G. Bar’iakhtar, and S. V. Peletminski, J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 228 (1958).
* Chen _et al._ (2017) C. Chen, A. Barra, A. Mal, G. Carman, and A. Sepulveda, Appl. Phys. Lett. 110, 072401 (2017).
* Azovtsev and Pertsev (2020) A. V. Azovtsev and N. A. Pertsev, Phys. Rev. Materials 4, 064418 (2020).
* Kittel (1949) C. Kittel, Rev. Mod. Phys. 21, 541 (1949).
* Niitsu (2020) K. Niitsu, J. Phys. D: Applied Physics 53, 39LT01 (2020).
* Stearns (1986) M. Stearns, in _Landolt-Börnstein - Group III Condensed Matter_ , Vol. 19a. Magnetic properties of 3d, 4d, and 5d elements, alloys and compounds, edited by H. Wijn (Springer-Verlag, Berlin‐Heidelberg‐New York‐Tokyo, 1986) Chap. 1.1.2. Fe, Co, Ni, pp. 24–51.
* Walowski _et al._ (2008) J. Walowski, M. D. Kaufmann, B. Lenk, C. Hamann, J. McCord, and M. Münzenberg, J. Phys. D: Applied Physics 41, 164016 (2008).
* Haynes (2016) W. M. Haynes, _CRC Handbook of Chemistry and Physics, 96th Edition (Internet Version 2016)_ (CRC Press/Taylor and Francis, Boca Raton, FL, 2016).
* Homer _et al._ (1987) R. Homer, G. C. Alexandrakis, and G. Dewar, J. Appl. Phys. 61, 4133 (1987).
* Sadd (2005) M. H. Sadd, _Elasticity. Theory, applications and numerics_ (Elsevier Butterworth-Heinemann, MA, USA, 2005).
* Dyakonov and Perel (1971) M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971).
* Zwierzycki _et al._ (2005) M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005).
* Nikitchenko and Pertsev (2020) A. I. Nikitchenko and N. A. Pertsev, Phys. Rev. Appl. 14, 034022 (2020).
* Tserkovnyak and Brataas (2002) Y. Tserkovnyak and A. Brataas, Phys. Rev. B 66, 224403 (2002).
* Kikkawa and Awschalom (1998) J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998).
* Johnson and Silsbee (1985) M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985).
* Lou _et al._ (2007) X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. Zhang, K. S. M. Reddy, S. D. Flexner, C. J. Palmstrøm, and P. A. Crowell, Nat. Phys. 3, 197 (2007).
|
††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.
# A heterogeneously integrated lithium niobate-on-silicon nitride photonic
platform
Mikhail Churaev Institute of Physics, Swiss Federal Institute of Technology
Lausanne (EPFL), CH-1015 Lausanne, Switzerland Rui Ning Wang Institute of
Physics, Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015
Lausanne, Switzerland Viacheslav Snigirev Institute of Physics, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Annina Riedhauser IBM Research - Europe, Zurich, Säumerstrasse 4, CH-8803
Rüschlikon, Switzerland Terence Blésin Institute of Physics, Swiss Federal
Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Charles Möhl IBM Research - Europe, Zurich, Säumerstrasse 4, CH-8803
Rüschlikon, Switzerland Miles A. Anderson Institute of Physics, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Anat Siddharth Institute of Physics, Swiss Federal Institute of Technology
Lausanne (EPFL), CH-1015 Lausanne, Switzerland Youri Popoff IBM Research -
Europe, Zurich, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland Integrated
Systems Laboratory, Swiss Federal Institute of Technology Zurich (ETH Zürich),
CH-8092 Zürich, Switzerland Ute Drechsler IBM Research - Europe, Zurich,
Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland Daniele Caimi IBM Research
- Europe, Zurich, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland Simon Hönl
IBM Research - Europe, Zurich, Säumerstrasse 4, CH-8803 Rüschlikon,
Switzerland Johann Riemensberger Institute of Physics, Swiss Federal
Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland Junqiu
Liu Institute of Physics, Swiss Federal Institute of Technology Lausanne
(EPFL), CH-1015 Lausanne, Switzerland Paul Seidler<EMAIL_ADDRESS>IBM
Research - Europe, Zurich, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland
Tobias J. Kippenberg<EMAIL_ADDRESS>Institute of Physics, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
The availability of thin-film lithium niobate on insulator (LNOI) and advances
in processing have led to the emergence of fully integrated LiNbO3 electro-
optic devices[1, 2, 3, 4], including low-voltage[5], high-speed modulators[6],
electro-optic frequency combs[7], and microwave-optical transducers [8, 9].
Yet to date, LiNbO3 photonic integrated circuits (PICs) have mostly been
fabricated using non-standard etching techniques that lack the reproducibility
routinely achieved in silicon photonics. Widespread future application of
thin-film LiNbO3 requires a reliable and scalable solution using standard
processing and precise lithographic control. Here we demonstrate a
heterogeneously integrated LiNbO3 photonic platform that overcomes the
abovementioned challenges by employing wafer-scale bonding of thin-film LiNbO3
to planarized low-loss silicon nitride (Si3N4) photonic integrated
circuits[10], a mature foundry-grade integrated photonic platform. The
resulting devices combine the substantial Pockels effect of LiNbO3 with the
scalability, high-yield, and complexity of the underlying Si3N4 PICs.
Importantly, the platform maintains the low propagation loss ($\mathbf{<0.1}$
dB/cm) and efficient fiber-to-chip coupling ($<$2.5 dB per facet) of the Si3N4
waveguides. We find that ten transitions between a mode confined in the Si3N4
PIC and the hybrid LiNbO3 mode produce less than 0.8 dB additional loss,
corresponding to a loss per transition not exceeding 0.1 dB. These nearly
lossless adiabatic transitions thus link the low-loss passive Si3N4 photonic
structures with electro-optic components. We demonstrate high-Q
microresonators, optical splitters, electrically tunable photonic dimers,
electro-optic frequency combs, and carrier-envelope phase detection of a
femtosecond laser on the same platform, thus providing a reliable and foundry-
ready solution to low-loss and complex LiNbO3 integrated photonic circuits.
Figure 1: Hybrid integrated Lithium Niobate photonics.(a) Conventional
approaches to lithium niobate photonics, consisting of traditional Ti or
proton exchange based waveguides, and the recently emerged integrated
photonics based on etching of thin-film LNOI (which is mostly based on ridge
waveguides) (b) The hybrid approach presented in this work, based on
heterogeneous integration of thin film lithium niobate with Si3N4. (c)
Schematics of our approach which is based on wafer bonding of 4” (100 mm) thin
film lithium niobate onto planarized ultra low loss $\mathrm{Si_{3}N_{4}}$
photonic integrated circuits. (d) Hybrid optical mode profile for a typical
waveguide used in this work (cf Methods). (e) AFM measurements of the Si3N4
wafer before bonding showing 400 pm RMS roughness over 5 $\mu$m by 5 $\mu$m
field of view. (f) Long-range profilometry scan of the wafer before bonding.
(g) False-coloured scanning electron micrograph showing the hybrid structure.
Modern society has an constantly increasing demand for optical communications
bandwidth, with aggregate data rates doubling every 18 months[11, 12], and
optical technology is coming ever closer to the central processing units[13].
Optical modulators play a crucial role in this context, providing the means to
transfer electronic signals to optical carriers. With the rise of commercial
integrated photonics [14], a wide variety of modulation platforms have been
demonstrated that are compatible with wafer-scale manufacturing, among which
silicon and indium phosphide are the most prominent [15, 16, 17, 18]. In the
last decade, alternative systems, including organic hybrids [19, 20],
plasmonic devices [21], and modulators based on two-dimensional materials [22,
23, 24, 25] have also been developed. Among all the materials used, lithium
niobate (LiNbO3) remains the most preferable because of its excellent physical
properties, and commercial availability [26]. Advances in wafer-scale transfer
of LiNbO3 thin-films via the SmartCut${}^{\textbf{TM}}$ technique, combined
with improvements in etching of LiNbO3, have enabled low-loss integrated
electro-optics [27, 28, 29]. This has led to several key demonstrations,
including ultra-high-$Q$ optical microresonators[28], efficient electro-optic
frequency comb generation[7], frequency converters [30], and non-reciprocal
devices [31, 32]. In addition, electro-optic modulation both at CMOS voltage
levels and at high speed (up to 100 GHz) has been achieved [5, 6], offering
routes toward compact integrated LiNbO3 modulators compatible with CMOS
microelectronics for applications ranging from classical communication for 5G
cellular networks and datacenter interconnects to quantum interfaces for
microwave-optical conversion [33, 34, 35], and topological photonics employing
synthetic dimensions [36, 37]. Besides the electro-optic applications,
integrated LiNbO3 PICs are also of high interest for nonlinear photonics, for
example, for efficient second-harmonic generation, optical squeezing, and
parametric amplification [38, 39, 40].
Despite the achievements to date, widespread adoption of LiNbO3 integrated
photonics is still impeded by several key issues. Current LNOI-based devices
are fabricated using specific non-conventional ion-beam etching (IBE) to
achieve smooth waveguide surfaces. Insufficient etch-mask selectivity leads to
the formation of shallow ridge waveguides that require more challenging
process control to achieve the desired geometries. This complicates the
establishment of a reliable process design kit (PDK) for integrated LiNbO3
platforms. Second, edge coupling between fibers and chips is challenging, as
the ridge waveguide structures demonstrated so far show significant coupling
loss, 5 to 10 dB per facet, [30] unless more complicated double-etching
techniques are used [41, 42]. Third, while record resonance quality factors (Q
$\approx$ 107, linear loss of 0.027 dB/cm) have been reported in LiNbO3
microresonators[28], this has only been demonstrated for selected optical
resonances and has not been achieved broadly in other recently reported works,
where losses are typically one order of magnitude higher (0.2 - 0.3 dB/cm, see
Supplementary Table 1 for comparison). For future applications, uniformly low
loss across a wafer using precise and mature lithographic processes, along
with efficient coupling, are necessary to develop a foundry-level technology
that includes PDKs with, e.g., splitters, arrayed-waveguide gratings, optical
filters or beamforming networks.
As an alternative to conventional bulk LiNbO3 and ridge-waveguide-based
photonic devices, hybrid platforms combining thin-film LiNbO3 with waveguides
made of Si, Si3N4, or Ta2O5 have been recently developed[43, 44, 45] (see Fig
1(a)-(c)). With proper geometry optimization, the heterogeneously integrated
LNOI devices can reach electro-optic performance comparable to that of the
all-LNOI platforms[46] ($\mathrm{V_{\pi}L}=2.3$ V$\cdot$cm). Heterogeneous
integration using the organic adhesive benzocyclobutene (BCB) to bond LNOI to
silicon and direct bonding of chiplets to silicon and silicon nitride PICs has
been demonstrated, leading to modulators operating at CMOS voltages [47, 44].
Even though the low-loss operation of hybrid Si3N4-LiNbO3 waveguides was
reported [48, 49], the previous works did not provide reliable studies of
wafer-scale uniformity, dispersion, insertion loss, or other essential aspects
of photonic integrated circuits. The approaches were aimed at specific device
applications and could not demonstrate all the benefits of heterogeneous
integration of LiNbO3 with a well-developed photonic platform. Moreover, the
wafer-level bonding required to achieve scalable PDKs was not shown. Here, we
demonstrate a high-yield, low-loss, integrated LiNbO3-Si3N4 photonic platform
that solves multiple issues of LNOI integrated photonics. The approach
circumvents the need for optimized IBE etching of LiNbO3 and opens up the
possibility of creating a wide range of low-loss integrated electro-optic
photonic circuits. This is achieved by wafer-scale heterogeneous integration
[50, 48] (i.e. direct wafer bonding [51]) of an LNOI wafer onto a patterned
and planarized ultra-low-loss Si3N4 substrate as depicted in Fig 1(c). Our
approach combines the maturity of Si3N4 integrated photonics with the large
Pockels effect of LiNbO3 and enables complex, hybrid PICs that incorporate
passive Si3N4 and electro-optic LiNbO3 and exhibit ultra-low propagation loss
(8.5 dB/m).
Figure 2: Optical loss measurements of the hybrid devices. (a) Photograph of
a fully bonded 2” lithium niobate wafer to a 4” photonic damascene wafer. (b)
Schematics of the chip facet with lithium niobate removed from the silicon
nitride inverse tapers. FDTD simulations of the mode transition at interface.
(c) Broadband transmission measurements of a 100 GHz FSR ring showing flat
coupling of resonances from 1260 nm to 1630 nm wavelength. (d) Extracted
integrated dispersion of the device. The inset shows a zoom-in of one of the
modes at 184 THz with amplitude modulation sidebands at 500 MHz and the
corresponding fitting curve. (e) Wafer map with averaged linewidth indicated
for the reference 21 GHz FSR rings. (f) Simulated optical mode confinement in
LiNbO3 as a function of optical frequency. Insets show typical mode profiles.
Grey-shadowed area represents the measurement bandwidth. (g) Loaded (red),
intrinsic (green), and coupling (blue) linewidth of the resonances presented
in (c). (h) Measured linewidth of 3 types of devices on the wafer accumulated
over 55 THz measurement bandwidth.
Fabrication process. The process flow for our hybrid PICs starts with the
fabrication of Si3N4 waveguide structures using the photonic Damascene process
[52, 10]. We use a 100 mm-diameter silicon wafer with 4 $\mu$m-thick wet
thermal SiO2, followed by deep-ultraviolet (DUV) stepper lithography, preform
dry etching, preform reflow, low-pressure chemical vapor deposition (LPCVD)
deposition of Si3N4, chemical-mechanical polishing (CMP), and SiO2 interlayer
deposition and annealing, as detailed in the Supplementary Information. The
Si3N4 photonic Damascene process is free of crack formation in the highly
tensile LPCVD Si3N4 film and provides high fabrication yield and ultra-low
propagation loss (1 dB/m). In addition, double-inverse nanotapers [53] are
incorporated for efficient edge coupling to lensed fibers. Previous devices
fabricated using this process have been the workhorse for numerous system-
level applications of soliton microcombs, ranging from coherent
telecommunication [54] to astrophysical spectrometer calibration [55] to
supercontinuum generation [56] and turnkey soliton generation [57].
One of the key advantages of the photonic Damascene process is the possibility
of obtaining an extremely flat wafer surface suitable for heterogeneous
integration. Specifically, we perform CMP on the SiO2 interlayer and bond the
fabricated Si3N4 Damascene substrate to a commercially available LNOI wafer
(NanoLN). The most critical constraints for achieving high bonding yield, the
surface roughness and topography, are measured prior to bonding. With CMP, the
long-range topography is reduced to a few nanometers over several hundred
microns, as shown in Fig. 1(f). Moreover, atomic force microscopy (AFM)
measurements over a range of a few microns reveal a root-mean-square (RMS)
roughness of 400 pm (Fig. 1(e)). This roughness level is sufficiently low for
direct wafer bonding. The donor and the acceptor wafer (the Si3N4 substrate
and the LNOI wafer, respectively) are cleaned, and both are coated by atomic
layer deposition (ALD) with a few nanometers of alumina (Al2O3). The wafers
are then bonded and annealed at 250∘C to enhance the bonding strength. See SI
for more technical information on fabrication. We have successfully bonded
several wafers, including whole 100-mm LNOI wafers, with a bonding yield close
to 100%, as evidenced by photoacoustic spectroscopy. Subsequent back-end
processing and electrode integration are described in the SI. A scanning
electron microscope image (Fig. 1(g)) of a cross-section of the layer
structure reveals clean bonding results. Finite-element-method simulations
(see Fig. 1(d)) indicate that, for the waveguide and wafer parameters used
here (cf. SI for waveguide geometry), the optical mode confinement factor for
LiNbO3 at the telecommunication wavelength of 1550 nm is
$\Gamma=\iint_{\mathrm{LiNbO_{3}}}|E|^{2}dS/\iint_{\mathrm{\Omega}}|E|^{2}dS=12\%$,
where $\Omega$ denotes the whole cross-section area. To demonstrate the
electro-optic capabilities of the heterogeneously integrated LiNbO3 photonic
circuits, we deposit tungsten ($\mathrm{W}$) electrodes on top of the LiNbO3
adjacent to the waveguides with an electrode-electrode gap of 6 $\mu$m.
To illustrate the versatility, lithographic precision, complexity, and yield
of the hybrid platform, we design a reticle with various devices. Figure 2(e)
shows the design layout of the Si3N4 photonic circuits for a 100-mm wafer; it
contains nine fields with 16 chips each – in total, more than 100 chips with
dimensions of 5 mm $\times$ 5 mm. The reticle includes chips with three
different types of devices: (1) microresonators with a free spectral range
(FSR) of either 100 GHz or 21 GHz, the former being used for electro-optic
comb generation; (2) photonic molecules consisting of a pair of coupled
microresonators each with a FSR of 50 GHz, as used for microwave-optical
conversion schemes; and (3) waveguides with a length of several centimeters
for supercontinuum generation (wafer D67b01).
Figure 3: Adiabatic mode transitions and hybrid optical splitters. (a) Colored
SEM image of a LiNbO3 taper fabricated to make a smooth optical mode
transition. (b) Optical microscope image of a chip used to estimate the
performance of interface tapers. Horizontal dotted lines mark waveguides with
two interface transitions (blue), four interface transitions (red), and six
interface transitions (green). The inset shows a zoom-in of a breakout region.
(c) Schematic of a tapered (adiabatic) interface transition. (d) Transmission
measurements of breakout waveguides described in (b) as well as a typical
straight (non-adiabatic) transition for comparison (orange line). (e)
Schematic of a straight (non-adiabatic) interface transition. (f) Transmission
measurements of waveguides with straight transition breakouts. (g) Optical
image, FDTD simulation, and schematic of a W-type 3-dB splitter. The geometry
is defined by the underlying Si3N4 layer. (h) Transmission measurements of two
ports of the splitter together with the transmission of a straight waveguide
as reference.
Optical loss characterization. Linear propagation loss is crucial in
determining the performance of integrated electro-optic devices, as it limits
the length of electro-optic modulators and the complexity of the associated
photonic circuit. To measure the linear optical loss, evanescent coupling
properties, and group velocity dispersion (GVD) of the hybrid structures, we
perform broadband frequency-comb-assisted spectroscopy [58] of multiple
microresonators across the entire wafer with three different external-cavity
diode lasers covering the wavelength ranges of 1260–1360 nm, 1355–1505 nm, and
1500–1630 nm. The intrinsic quality factors of individual 100 GHz microring
resonators reach up to $Q=3\times 10^{6}$, while the 50 GHz photonic dimers
(i.e., coupled microrings) and 21 GHz single rings exhibit even higher quality
factors up to $Q=4.5\times 10^{6}$. The latter corresponds to a linear
propagation loss of 8.5 dB/m. We observe an absorption peak at approximately
1420 nm (207 THz), which we associate with an overtone of OH-bond vibrations
in lithium niobate [59, 60]. As shown in Fig 2(g), optical losses rise with
increasing optical frequency. We associate this dependency with increased
whispering-gallery (radiation) loss as the mode shifts into the LiNbO3 thin
film at higher frequencies and becomes less confined (see Fig 2(f)). For the
same reason, we observe nearly uniform evanescent coupling of optical
microresonators over a span of 55 THz. The layout in Fig 2(e) is labeled with
the most probable linewidth measured for 21 GHz microresonators in various
regions, indicating the degree of variation across the wafer (see
Supplementary Information). These results not only demonstrate high yield and
wafer-scale fabrication but include some of the highest quality factors
achieved to date with integrated LiNbO3 devices. Notably, the $Q$s reported
here are not isolated values as in prior work on ridge resonators[28] but are
consistently high, i.e., we measure hundreds of resonances with $Q$ above
$4\times 10^{6}$ (linewidth below 50 MHz), as shown in Fig. 2(h).
Our hybrid platform also offers the possibility of precise dispersion
engineering; due to the interplay between material dispersion and the optical
mode distribution, the dispersion can be adjusted by varying the Si3N4
waveguide geometry or the LiNbO3 thickness. In this work, we designed the
structure to work in the near-zero GVD regime advantageous for broadband
optical frequency comb generation. As shown in Fig. 2(d) for a microresonator
with a FSR of 100 GHz, the measured integrated microresonator dispersion
($D_{\text{int}}$) only varies by 15 GHz over an optical bandwidth of 55 THz.
Interestingly, there is no constraint preventing the design of devices that
are uniformly coupled over a broad frequency range and, at the same time, have
extremely small dispersion.
Figure 4: Electro-optic frequency comb generation and tunable dimers using
heterogenously integrated $\mathrm{LiNbO_{3}}$ photonic circuits. (a)
Experimental setup for electro-optic frequency comb generation. MWG -
microwave generator, MWA - microwave amplifier, OSA - optical spectrum
analyser. (b) Mode coupling schematics and integrated dispersion of the
measured device.(c) Measured optical spectrum of the generated EO comb at 1552
nm central frequency. Dashed line corresponds to numerical simulations with
phase modulation amplitude $\beta$ = 0.14 $\pi$. (d) Examples of EO frequency
combs generated at 4 other pump wavelength. (e) Photonic dimer image and mode
hybridization illustration. (f) High-Q resonance splitting of a photonic dimer
without additional biasing. S - symmetric supermode, AS - antisymmetric. (g)
DC tuning of the photonic dimer mode hybridization, corresponding to linear
tuning of around 30 MHz/V for a single mode. (h) Echelle plot of the photonic
dimer transmission, showing mode hybridization over a broadband scanning
range.
Fiber-chip coupling and interface transitions. Efficient input coupling from
an optical fiber to the photonic chip is paramount for numerous applications.
For air-cladded LiNbO3 ridge waveguides, inverse tapers lead to fiber-chip
edge-coupling losses of around 10 dB per facet unless complicated multi-layer
etching is used[41, 61, 62]. This is due to the significant mode mismatch
between the lensed fiber mode (typically circular with about 2.5 $\mu$m
diameter) and the asymmetric mode of partially etched air-cladded ridge
waveguide structures. Some recent work on integrated LiNbO3 devices
demonstrated the possibility of using embedded silicon edge-couplers to
overcome this challenge[6]. In our case, the relative indices of refraction of
the materials would lead to significant coupling loss if the LiNbO3 layer
remained on top of underlying Si3N4 inverse tapers. Hence, we remove the
LiNbO3 from the coupling regions and rely on standard Damascene Si3N4 inverse
tapers[53]. While this provides an efficient input coupling, there remains the
challenge of the transition between the regions with and without LiNbO3. To
address this issue, we designed and implemented adiabatic tapers in the LiNbO3
layer, as shown in Fig.3(c). The tapers are 100 $\mathrm{\mu}$m long with a
tip width of 500 nm and a final width of 10 $\mathrm{\mu}$m. Both film removal
and etching of the tapers are done in a single fabrication step using argon
ion-beam etching with a photoresist etch mask (see the Supplementary
Information for details on taper fabrication). All the functional photonic
components do not depend on the etching of the chip interface, which is
employed only at mode transition regions and, due to the short taper length,
does not require low roughness. We thus keep the LiNbO3 layer unprocessed for
all the photonic components, where both roughness and precise alignment are
critical to achieving low optical losses.
To measure the efficiency of the adiabatic transitions and remove the
ambiguity associated with the fiber-chip coupling loss from the measurement,
we designed an experiment in which we introduce multiple breakouts on straight
waveguides, where the optical mode experiences transitions from Si3N4
waveguides into the hybridized mode and back as depicted in Fig. 3(b). We
fabricated waveguides with 2, 4, 6, and 10 transitions (input/output tapers
and 0, 1, 2, and 4 breakouts, respectively) to determine the increase in loss
due to each transition. Figure 3(d) shows the results of these measurements.
As a reference, we compare them with straight interfaces (schematic in
Fig.3(e)). As can be seen in Figure 3(f), each straight transition leads to
approximately 1 dB loss, whereas the tapered input/output behaves in this
measurement as a virtually lossless transition. Strikingly, for the case of
the tapered transitions, we observe hardly any difference between 2, 4 and 6
interfaces, as shown in Figure 3(d) and approximately only 0.8 dB additional
loss for ten interfaces. Considering the statistical uncertainty in the
measurements, we deduce a transition loss of $<$0.1 dB per taper. Calibration
of the transmission measurements is discussed in the Supplementary
Information.
The lithographic precision of the Si3N4 photonic circuit layer provides our
heterogeneous integration approach with versatiliy and robustness, as
confirmed by the implementation of a W-shaped 3-dB splitter/coupler[63] (see
Fig. 3(g)) that uses the hybrid Si3N4-LiNbO3 mode but is defined solely by
underlying Si3N4 inverse tapers. Splitters are important components for many
optical devices, such as electro-optic modulators, optical networks, and
lasers based on reflective semiconductor optical amplifiers. The elegance of
this type of splitter is in its simplicity of design. Due to the presence of
the LiNbO3 slab and the single-mode nature of our hybrid waveguides, the
optical mode is adiabatically transferred from the input arm to the output
arms. We make the tapered sections 100 $\mu$m long, ensuring a small footprint
for integrated components exploiting this design. Transmission measurements of
the device reveal a flat response, with power asymmetry between the two arms
not exceeding 1.7 dB and on-chip insertion loss not exceeding 1 dB in the
1500-1620 nm wavelength range (Fig.3(h)).
Electro-optic devices. To demonstrate the electro-optic performance achievable
with our high-quality factor, hybrid LiNbO3 microresonators, we generate
electro-optic frequency combs in the 21 GHz resonators pumped resonantly in
the telecommunications C-band. We apply a high-power microwave signal with a
frequency 20.97 GHz across the integrated electrodes such that microwave-
induced sidebands are resonantly enhanced (Fig 4(a)-(b)). Even though only 12%
of the optical mode is confined inside the lithium niobate, it is enough to
achieve strong modulation of the optical phase. We observe around 60 sidebands
within a 25 dB span for an injected RF power of 40 dBm, as depicted in Fig
4(c). In this experiment, the phase modulation amplitude corresponds to
approximately 0.14$\mathrm{\pi}$ (see Supplementary Information). The electro-
optic coupling is enhanced due to the device’s high quality factor and flat
dispersion. We also make use of the previously mentioned homogeneous coupling
of our hybrid devices at optical wavelengths ranging from 1260 nm to 1630 nm
to generate electro-optic comb at five different pump wavelengths (1290 nm,
1345 nm, 1500 nm, 1550 nm, and 1625 nm) on a single device (Fig 4(d)).
Moreover, according to the simulations, the geometry can be optimized for
maximum electro-optic efficiency with a characteristic $\mathrm{V_{\pi}\cdot
L}$ product comparable to (2x larger than) the performance of X-cut ridge-
waveguide platforms[5]. The reason is that, in ridge waveguides, most of the
electro-optic interaction occurs not in the ridge itself, but in the slab
layer, where the modulating electric field is a factor of magnitude stronger
(see Supplementary Information). Therefore, the ridge waveguides and hybrid
waveguides are conceptually similar in terms of electro-optic interactions
(i.e., the role of the slab in ridge waveguides is being taken over by the
bonded LiNbO3 layer in our hybrid structure). Similar studies on optimization
of electro-optic performance of heterogeneously integrated LiNbO3 devices can
also be found elsewhere[46]. As a further example of the electro-optic
capabilities, we fabricate photonic dimers, which are known building blocks
for quantum coherent transducers based on cavity electro-optics[64, 34, 65, 9,
66]. Figure 4(g) shows the mode hybridization in the system as a function of
the applied DC voltage. We observe frequency tuning of 30 MHz/V when a DC
voltage is applied to one of the rings. Moreover, the precise and mature
fabrication of the Si3N4 waveguides enables the creation of high-Q photonic
dimers exhibiting broadband normal mode splitting even at zero-bias (cf Figure
4(f)-(h)). The presence of avoided mode crossings for the symmetric supermode
(lower frequency) is common with the photonic dimer configuration[67]. In one
last experiment exploiting the $\chi^{(2)}$ nonlinearity of the LiNbO3, we
perform supercontinuum generation in the hybrid waveguides. We observe octave-
spanning supercontinuum generation mediated by the $\chi^{(3)}$ nonlinearity,
together with simultaneous second-harmonic generation due to the optical field
in the LiNbO3, allowing direct measurement of the carrier-envelope offset
frequency of the femtosecond pulse laser used as a pump. The details of this
experiment can be found in the Supplementary Information.
Summary. To conclude, we have demonstrated a hybrid Si3N4-LiNbO3 platform for
photonic integrated circuits using direct wafer-scale bonding that endows the
mature Si3N4 Damascene technology with the second-order nonlinearity
($\chi^{(2)}$ / Pockels effect) of LiNbO3. The heterogeneous integration
preserves the precise lithographic control, low propagation loss, and
efficient fiber-to-chip coupling of the underlying Si3N4 waveguides for use in
a variety of important photonic building blocks. We have also presented a
design for the transition from Si3N4 waveguides to hybrid Si3N4-LiNbO3
waveguides with a measured insertion loss not exceeding 0.1 dB per interface.
The ability to achieve low-loss transitions is essential for the realization
of complex devices, providing a bridge between passive silicon nitride
photonics and electro-optic devices. To the best of our knowledge, this is the
first time a heterogeneously integrated LiNbO3 photonic platform combines all
the beneficial features of Si3N4 PICs at wafer scale. A comparison of the
simultaneously achieved desirable features is given in Supplementary Table 1.
With further geometry optimization, the electro-optic performance can reach
levels comparable to that of ridge waveguide structures while keeping
propagation losses independent of the quality of the LiNbO3 etching. Possible
applications of our platform include photonic switching networks for
neuromorphic or quantum computing, devices for quantum state transduction from
microwave to optical photons, integrated electro-optic frequency comb sources,
on-chip generation of second-harmonic and squeezed light, as well as high-
speed electro-optic devices for optical communications or rapidly tunable,
low-noise lasers.
Acknowledgments: We thank the Operations Team of the Binnig and Rohrer
Nanotechnology Center (BRNC), and especially Diana Davila Pineda and Ronald
Grundbacher, for their help and support. Silicon nitride substrates were
fabricated in the EPFL center of MicroNanoTechnology (CMi). We also thank
Aleksandr Tusnin for his help in numerical simulations.
Funding Information: This work was supported by funding from the European
Union Horizon 2020 Research and Innovation Program under the Marie Skłodowska-
Curie grant agreement No. 722923 (OMT) and No. 812818 (MICROCOMB), as well as
under the FET-Proactive grant agreement No. 732894 (HOT). This work was also
supported by the Swiss National Science Foundation under grant agreement No.
176563 (BRIDGE) and 186364 (Sinergia).
Data Availability: The code and data used to produce the plots are available
from Zenodo.
## References
* Zhu _et al._ [2021] D. Zhu, L. Shao, M. Yu, R. Cheng, B. Desiatov, C. J. Xin, Y. Hu, J. Holzgrafe, S. Ghosh, A. Shams-Ansari, E. Puma, N. Sinclair, C. Reimer, M. Zhang, and M. Lončar, Adv. Opt. Photon. 13, 242 (2021).
* Wang _et al._ [2019a] C. Wang, M. Zhang, M. Yu, R. Zhu, H. Hu, and M. Loncar, Nature Communications 10, 978 (2019a).
* Desiatov _et al._ [2019] B. Desiatov, A. Shams-Ansari, M. Zhang, C. Wang, and M. Lončar, Optica 6, 380 (2019).
* Bahadori _et al._ [2020] M. Bahadori, Y. Yang, A. E. Hassanien, L. L. Goddard, and S. Gong, Opt. Express 28, 29644 (2020).
* Wang _et al._ [2018] C. Wang, M. , X. Chen, M. Bertrand, A. Shams-Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, Nature 562, 101 (2018).
* He _et al._ [2019a] M. He, M. Xu, Y. Ren, J. Jian, Z. Ruan, Y. Xu, S. Gao, S. Sun, X. Wen, L. Zhou, L. Liu, C. Guo, H. Chen, S. Yu, L. Liu, and X. Cai, Nature Photonics 13, 359 (2019a), arXiv:1807.10362 .
* Zhang _et al._ [2019a] M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Lončar, Nature 568, 373 (2019a), arXiv:1809.08636 .
* Holzgrafe _et al._ [2020] J. Holzgrafe, N. Sinclair, D. Zhu, A. Shams-Ansari, M. Colangelo, Y. Hu, M. Zhang, K. K. Berggren, and M. Lončar, Optica 7, 1714 (2020).
* McKenna _et al._ [2020] T. P. McKenna, J. D. Witmer, R. N. Patel, W. Jiang, R. V. Laer, P. Arrangoiz-Arriola, E. A. Wollack, J. F. Herrmann, and A. H. Safavi-Naeini, Optica 7, 1737 (2020).
* Liu _et al._ [2021] J. Liu, G. Huang, R. N. Wang, J. He, A. S. Raja, T. Liu, N. J. Engelsen, and T. J. Kippenberg, Nature Communications 12, 2236 (2021).
* Agrell _et al._ [2016] E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, Journal of Optics (United Kingdom) 18, 10.1088/2040-8978/18/6/063002 (2016).
* Kitayama _et al._ [2019] K. I. Kitayama, M. Notomi, M. Naruse, K. Inoue, S. Kawakami, and A. Uchida, APL Photonics 4, 10.1063/1.5108912 (2019).
* Caulfield and Dolev [2010] H. J. Caulfield and S. Dolev, Nature Photonics 4, 261 (2010).
* Thomson _et al._ [2016] D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, Journal of Optics 18, 073003 (2016).
* Xu _et al._ [2005] Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, Nature 435, 325 (2005).
* Maram _et al._ [2019] R. Maram, S. Kaushal, J. Azaña, and L. R. Chen, _Photonics_, Vol. 6 (2019).
* Ogiso _et al._ [2017] Y. Ogiso, J. Ozaki, Y. Ueda, N. Kashio, N. Kikuchi, E. Yamada, H. Tanobe, S. Kanazawa, H. Yamazaki, Y. Ohiso, T. Fujii, and M. Kohtoku, Journal of Lightwave Technology 35, 1450 (2017).
* Witzens [2018] J. Witzens, Proceedings of the IEEE 106, 2158 (2018).
* Lee _et al._ [2002] M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, Science 298, 1401 (2002).
* Alloatti _et al._ [2014] L. Alloatti, R. Palmer, S. Diebold, K. P. Pahl, B. Chen, R. Dinu, M. Fournier, J. M. Fedeli, T. Zwick, W. Freude, C. Koos, and J. Leuthold, Light: Science and Applications 3, 5 (2014).
* Haffner _et al._ [2015] C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, L. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, Nature Photonics 9, 525 (2015).
* Liu _et al._ [2011] M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
* Gruhler _et al._ [2013] N. Gruhler, C. Benz, H. Jang, J.-H. Ahn, R. Danneau, and W. H. P. Pernice, Opt. Express 21, 31678 (2013).
* Phare _et al._ [2015] C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, Nature Photonics 9, 511 (2015).
* Datta _et al._ [2020] I. Datta, S. H. Chae, G. R. Bhatt, M. A. Tadayon, B. Li, Y. Yu, C. Park, J. Park, L. Cao, D. N. Basov, J. Hone, and M. Lipson, Nature Photonics 14, 256 (2020).
* Poberaj _et al._ [2012] G. Poberaj, H. Hu, W. Sohler, and P. Günter, Laser & Photonics Reviews 6, 488 (2012), https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.201100035 .
* Krasnokutska _et al._ [2018] I. Krasnokutska, J.-L. J. Tambasco, X. Li, and A. Peruzzo, Optics Express 26, 897 (2018).
* Zhang _et al._ [2017] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, Optica 4, 1536 (2017).
* Luke _et al._ [2020] K. Luke, P. Kharel, C. Reimer, L. He, M. Loncar, and M. Zhang, Optics Express 28, 24452 (2020).
* Hu _et al._ [2020a] Y. Hu, M. Yu, D. Zhu, N. Sinclair, A. Shams-Ansari, L. Shao, J. Holzgrafe, E. Puma, M. Zhang, and M. Loncar, Arxiv (2020a), arXiv:2005.09621 .
* Shao _et al._ [2020] L. Shao, W. Mao, S. Maity, N. Sinclair, Y. Hu, L. Yang, and M. Lončar, Nature Electronics 3, 267 (2020).
* Sohn _et al._ [2021] D. Sohn, O. E. Örsel, and G. Bahl, arXiv preprint arXiv:2104.04803 (2021).
* Lambert _et al._ [2020] N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L. Schwefel, Advanced Quantum Technologies 3, 1900077 (2020), arXiv:1906.10255 .
* Javerzac-Galy _et al._ [2016] C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, Physical Review A 94, 1 (2016), arXiv:1512.06442 .
* Wang _et al._ [2019b] J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Nature Photonics 10.1038/s41566-019-0532-1 (2019b).
* Yuan _et al._ [2018] L. Yuan, Q. Lin, M. Xiao, and S. Fan, Optica 5, 1396 (2018), arXiv:1807.11468 .
* Hu _et al._ [2020b] Y. Hu, C. Reimer, A. Shams-Ansari, M. Zhang, and M. Loncar, Optica 7, 1189 (2020b).
* Yu _et al._ [2019] M. Yu, B. Desiatov, Y. Okawachi, A. L. Gaeta, and M. Lončar, Optics Letters 44, 1222 (2019).
* Kanter _et al._ [2002] G. Kanter, P. Kumar, R. Roussev, J. Kurz, K. Parameswaran, and M. Fejer, Optics Express 10, 177 (2002).
* Lu _et al._ [2021] J. Lu, A. Al Sayem, Z. Gong, J. B. Surya, C.-L. Zou, and H. X. Tang, Optica 8, 539 (2021).
* He _et al._ [2019b] L. He, M. Zhang, A. Shams-Ansari, R. Zhu, C. Wang, and L. Marko, Optics Letters 44, 2314 (2019b), arXiv:1902.08969 .
* Ying _et al._ [2021] P. Ying, H. Tan, J. Zhang, M. He, M. Xu, X. Liu, R. Ge, Y. Zhu, C. Liu, and X. Cai, Optics Letters 46, 1478 (2021).
* Weigel _et al._ [2016] P. O. Weigel, F. Valdez, J. Zhao, H. Li, and S. Mookherjea, Scientific Reports 6, 012001 (2016).
* Rao _et al._ [2016] A. Rao, A. Patil, P. Rabiei, A. Honardoost, R. DeSalvo, A. Paolella, and S. Fathpour, Opt. Lett. 41, 5700 (2016).
* Jin _et al._ [2016] S. Jin, L. Xu, H. Zhang, and Y. Li, IEEE Photonics Technology Letters 28, 736 (2016).
* Weigel _et al._ [2020] P. O. Weigel, F. Valdez, J. Zhao, H. Li, and S. Mookherjea, Journal of Physics: Photonics 3, 012001 (2020).
* Boynton _et al._ [2020] N. Boynton, H. Cai, M. Gehl, S. Arterburn, C. Dallo, A. Pomerene, A. Starbuck, D. Hood, D. C. Trotter, T. Friedmann, C. T. DeRose, and A. Lentine, Optics Express 28, 1868 (2020).
* Chang _et al._ [2017] L. Chang, M. H. P. Pfeiffer, N. Volet, M. Zervas, J. D. Peters, C. L. Manganelli, E. J. Stanton, Y. Li, T. J. Kippenberg, and J. E. Bowers, Opt. Lett. 42, 803 (2017).
* Ahmed _et al._ [2019] A. N. R. Ahmed, S. Shi, M. Zablocki, P. Yao, and D. W. Prather, Opt. Lett. 44, 618 (2019).
* Komljenovic _et al._ [2018] T. Komljenovic, D. Huang, P. Pintus, M. A. Tran, M. L. Davenport, and J. E. Bowers, Proceedings of the IEEE 106, 2246 (2018).
* Plößl and Kräuter [1999] A. Plößl and G. Kräuter, Materials Science and Engineering: R: Reports 25, 1 (1999).
* Pfeiffer _et al._ [2018] M. H. P. Pfeiffer, C. Herkommer, J. Liu, T. Morais, M. Zervas, M. Geiselmann, and T. J. Kippenberg, IEEE Journal of Selected Topics in Quantum Electronics 24, 1 (2018).
* Liu _et al._ [2018] J. Liu, A. S. Raja, M. H. P. Pfeiffer, C. Herkommer, H. Guo, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Opt. Lett. 43, 3200 (2018).
* Marin-Palomo _et al._ [2017] P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, Nature 546, 274 (2017).
* Obrzud _et al._ [2019] E. Obrzud, M. Rainer, A. Harutyunyan, M. H. Anderson, J. Liu, M. Geiselmann, B. Chazelas, S. Kundermann, S. Lecomte, M. Cecconi, A. Ghedina, E. Molinari, F. Pepe, F. Wildi, F. Bouchy, T. J. Kippenberg, and T. Herr, Nature Photonics 13, 31 (2019).
* Guo _et al._ [2018] H. Guo, C. Herkommer, A. Billat, D. Grassani, C. Zhang, M. H. P. Pfeiffer, W. Weng, C.-S. Brès, and T. J. Kippenberg, Nat. Photonics 12, 330 (2018).
* Shen _et al._ [2020] B. Shen, L. Chang, J. Liu, H. Wang, Q.-F. Yang, C. Xiang, R. N. Wang, J. He, T. Liu, W. Xie, J. Guo, D. Kinghorn, L. Wu, Q.-X. Ji, T. J. Kippenberg, K. Vahala, and J. E. Bowers, Nature 582, 365 (2020).
* Liu _et al._ [2016] J. Liu, V. Brasch, M. H. P. Pfeiffer, A. Kordts, A. N. Kamel, H. Guo, M. Geiselmann, and T. J. Kippenberg, Optics Letters 41, 3134 (2016), arXiv:1604.05149 .
* Schwesyg _et al._ [2010] J. R. Schwesyg, C. R. Phillips, K. Ioakeimidi, M. C. C. Kajiyama, M. Falk, D. H. Jundt, K. Buse, and M. M. Fejer, Opt. Lett. 35, 1070 (2010).
* Heinemeyer _et al._ [2006] U. Heinemeyer, M. C. Wengler, and K. Buse, Applied Physics Letters 89, 13 (2006).
* Mercante _et al._ [2016] A. J. Mercante, P. Yao, S. Shi, G. Schneider, J. Murakowski, and D. W. Prather, Optics Express 24, 15590 (2016).
* Hu _et al._ [2021] C. Hu, A. Pan, T. Li, X. Wang, Y. Liu, S. Tao, C. Zeng, and J. Xia, Opt. Express 29, 5397 (2021).
* Wang _et al._ [2016] Y. Wang, S. Gao, K. Wang, and E. Skafidas, Opt. Lett. 41, 2053 (2016).
* Wade _et al._ [2015] M. T. Wade, X. Zeng, and M. A. Popović, Optics Letters 40, 107 (2015).
* Zhang _et al._ [2019b] M. Zhang, C. Wang, Y. Hu, A. Shams-Ansari, T. Ren, S. Fan, and M. Lončar, Nature Photonics 13, 36 (2019b), arXiv:1809.08638 .
* Youssefi _et al._ [2021] A. Youssefi, I. Shomroni, Y. J. Joshi, N. R. Bernier, A. Lukashchuk, P. Uhrich, L. Qiu, and T. J. Kippenberg, Nature Electronics 4, 326 (2021), arXiv:2004.04705 .
* Tikan _et al._ [2022] A. Tikan, A. Tusnin, J. Riemensberger, M. Churaev, X. Ji, K. N. Komagata, R. N. Wang, J. Liu, and T. J. Kippenberg, Science Advances 8, eabm6982 (2022).
|
aainstitutetext: Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de
Madrid,
Calle de Nicolás Cabrera 13–15, Cantoblanco, E-28049 Madrid,
Spainbbinstitutetext: Center for Cosmology and AstroParticle Physics (CCAPP),
Ohio State University,
191 W. Woodruff Ave., Columbus, Ohio 43210, U.S.A.ccinstitutetext: Department
of Physics, Ohio State University,
191 W. Woodruff Ave., Columbus, Ohio 43210, U.S.A.ddinstitutetext: C. N. Yang
Institute for Theoretical Physics, Stony Brook University,
Stony Brook NY 11794-3840, U.S.A.eeinstitutetext: Institució Catalana de
Recerca i Estudis Avançats (ICREA),
Pg. Lluis Companys 23, E-08010 Barcelona, Spainffinstitutetext: Departament de
Fisica Quantica i Astrofisica and Institut de Ciencies del Cosmos,
Universitat de Barcelona, Diagonal 647, E-08028 Barcelona,
Spaingginstitutetext: Donostia International Physics Center (DIPC),
Paseo Manuel Lardizabal, 4, Donostia-San Sebastián, E-20018,
Spainhhinstitutetext: Ikerbasque, Basque Foundation for Science,
Plaza Euskadi 5, E-48013 Bilbao, Spainiiinstitutetext: Instituto de Física
Corpuscular (IFIC), CSIC & Universitat de València,
Parc Científic, C/ Catedrático José Beltrán 2, E-46980 Paterna, Spain
# Bounds on new physics with data of the Dresden-II reactor experiment and
COHERENT
Pilar Coloma<EMAIL_ADDRESS>b,c Ivan Esteban<EMAIL_ADDRESS>d,e,f M.C. Gonzalez-Garcia<EMAIL_ADDRESS>g
Leire Larizgoitia<EMAIL_ADDRESS>g,h Francesc Monrabal
<EMAIL_ADDRESS>i Sergio Palomares-Ruiz<EMAIL_ADDRESS>
###### Abstract
Coherent elastic neutrino-nucleus scattering was first experimentally
established five years ago by the COHERENT experiment using neutrinos from the
spallation neutron source at Oak Ridge National Laboratory. The first evidence
of observation of coherent elastic neutrino-nucleus scattering with reactor
antineutrinos has now been reported by the Dresden-II reactor experiment,
using a germanium detector. In this paper, we present constraints on a variety
of beyond the Standard Model scenarios using the new Dresden-II data. In
particular, we explore the constraints imposed on neutrino non-standard
interactions, neutrino magnetic moments, and several models with light scalar
or light vector mediators. We also quantify the impact of their combination
with COHERENT (CsI and Ar) data. In doing so, we highlight the synergies
between spallation neutron source and nuclear reactor experiments regarding
beyond the Standard Model searches, as well as the advantages of combining
data obtained with different nuclear targets. We also study the possible
signal from beyond the Standard Model scenarios due to elastic scattering off
electrons (which would pass selection cuts of the COHERENT CsI and the
Dresden-II experiments) and find more stringent constraints in certain parts
of the parameter space than those obtained considering coherent elastic
neutrino-nucleus scattering.
###### Keywords:
coherent neutrino-nucleus scattering, non-standard interactions, neutrino
magnetic moment, light mediators
††preprint:
IFT-UAM/CSIC-22-10 YITP-SB-2022-03 IFIC/22-06
## 1 Introduction
Low-energy neutrinos can elastically scatter off atomic nuclei via weak
neutral currents, with the initial and final states of the nucleus being
indistinguishable. For low enough momentum transfers, the interaction takes
place coherently with the whole nucleus, leading to an enhancement of the
cross section approximately proportional to the square of its number of
neutrons. Although the so-called coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) was first theoretically described in Freedman’s seminal paper
almost 50 years ago Freedman:1973yd , its detection has not been possible
until very recently. This is so because the single observable for this process
is a recoiling nucleus which generates a signal in the sub-keV to few keV
energy range, difficult to detect. An additional hindrance to CE$\nu$NS
detection is the limited number of suitable neutrino sources, which must be
sufficiently intense in yield and, at the same time, low enough in neutrino
energy for the coherence condition to be satisfied.
The detection of CE$\nu$NS was experimentally demonstrated by the COHERENT
experiment COHERENT:2017ipa using the currently most intense spallation
neutron source (SNS), sited at the Oak Ridge National Laboratory, U.S.A. The
original observation was obtained with a CsI[Na] scintillation detector
COHERENT:2017ipa ; COHERENT:2018imc , which was later followed by the
observation of CE$\nu$NS at a liquid Argon detector COHERENT:2020iec ;
COHERENT:2020ybo .
In addition to spallation sources, CE$\nu$NS is also searched for in a variety
of experiments using electron antineutrinos emitted by nuclear reactors, such
as TEXONO Wong:2004ru , $\nu$GeN Belov:2015ufh , CONNIE Aguilar-
Arevalo:2016khx , MINER Agnolet:2016zir , Ricochet Billard:2016giu ,
$\nu$-cleus Strauss:2017cuu , RED-100 Akimov:2019ogx , NEON Choi:2020gkm ,
CONUS CONUS:2020skt and NCC-1701 at Dresden-II Colaresi:2021kus . At present,
most of these reactor experiments have not yet been successful in the
CE$\nu$NS detection. The exception to this is the NCC-1701 experiment at the
Dresden-II nuclear reactor. In their first published data Colaresi:2021kus ,
the experimental collaboration presented an event spectrum with an excess of
events at low energies, compatible with expectations from a CE$\nu$NS signal
in the Standard Model (SM). Recently, the collaboration has released new
results with an increased exposure. The observation of CE$\nu$NS is reported
with strong to very strong preference (with respect to the only-background
hypothesis) in the Bayesian statistics sense, depending on the quenching
factor considered Colaresi2022suggestive .
A precision measurement of CE$\nu$NS provides a direct probe to both SM and
beyond the standard model (BSM) physics. Paradigmatic examples of the former
are the determination of the weak mixing angle at very low momentum transfer
Canas:2018rng ; Cadeddu:2018izq ; Huang:2019ene ; Cadeddu:2019eta ;
Cadeddu:2020lky ; Cadeddu:2021ijh and the study of nuclear structure
Cadeddu:2017etk ; Ciuffoli:2018qem ; Papoulias:2019lfi ; Cadeddu:2021ijh ;
Coloma:2020nhf . The program of BSM exploration with CE$\nu$NS is broad (see,
e.g., refs. Barranco:2005yy ; Formaggio:2011jt ; Anderson:2012pn ;
Dutta:2015nlo ; Cerdeno:2016sfi ; Dent:2016wcr ; Coloma:2017egw ;
Kosmas:2017zbh ; Ge:2017mcq ; Shoemaker:2017lzs ; Coloma:2017ncl ;
Liao:2017uzy ; Canas:2017umu ; Dent:2017mpr ; Papoulias:2017qdn ;
Farzan:2018gtr ; Billard:2018jnl ; Coloma:2019mbs ; Chaves:2021pey ;
AristizabalSierra:2018eqm ; Brdar:2018qqj ; Cadeddu:2018dux ; Blanco:2019vyp ;
Dutta:2019eml ; Miranda:2019wdy ; CONNIE:2019swq ; Dutta:2019nbn ;
Papoulias:2019txv ; Khan:2019cvi ; Cadeddu:2019eta ; Giunti:2019xpr ;
Baxter:2019mcx ; Canas:2019fjw ; Miranda:2020zji ; Flores:2020lji ;
Miranda:2020tif ; Hurtado:2020vlj ; Miranda:2020syh ; Cadeddu:2020nbr ;
Shoemaker:2021hvm ; delaVega:2021wpx ; Liao:2021yog ; CONUS:2021dwh ;
Flores:2021kzl ; Li:2022jfl ; AristizabalSierra:2019zmy ; Abdullah:2020iiv ;
Fernandez-Moroni:2021nap ; Bertuzzo:2021opb ; Bonet:2022imz for an incomplete
list) being most sensitive to a variety of scenarios leading to modified
neutrino interactions with nuclei – in particular at low momentum transfer –
but extending also to the production of new light neutral states, and sterile
neutrino searches, among others.
In this paper, we present the first analysis using the new data of the
Dresden-II reactor experiment in the context of BSM searches. In doing so, we
consider several new physics scenarios commonly studied in the literature:
effective four-fermion interactions, neutrino magnetic moments, and light
mediators. In order to place our results in a broader context, we also combine
the Dresden-II data with a germanium detector with the COHERENT data obtained
for neutrino scattering off CsI and Ar. We highlight and quantify the
synergies between SNS and reactor experiments using CE$\nu$NS, as well as the
advantages of the combination of data sets obtained with multiple nuclear
targets. Furthermore, we also study a potential signal from BSM scenarios due
to elastic electron scattering (ES). Although in the SM this contribution to
the event rates is negligible, it could be significantly enhanced in the
presence of new physics effects. The contribution from scattering off
electrons could pass selection cuts of both COHERENT CsI and Dresden-II
experiments and, as we discuss, its inclusion leads to stronger constraints in
certain parts of the parameter space than those obtained using CE$\nu$NS.
The paper is organized as follows. In section 2 we introduce the theoretical
frameworks we consider. Section 3 discusses the computation of the expected
event rates and the details regarding our fit to the experimental data, while
the results of our analysis are presented in section 4. Finally, we summarize
and conclude in section 5, and provide the binding energies for neutrino
scattering off electrons in appendix A.
## 2 Phenomenological frameworks
| $c_{V}$ | $c_{A}$
---|---|---
$\nu_{e}$ | $\frac{1}{2}+2\,\sin^{2}\theta_{W}$ | $+\frac{1}{2}$
$\nu_{\mu},\nu_{\tau}$ | $-\frac{1}{2}+2\,\sin^{2}\theta_{W}$ | $-\frac{1}{2}$
Table 1: Values of the SM effective couplings in neutrino-electron
scattering. For antineutrinos the vector couplings are the same, while the
axial ones change sign, $c_{A}\to-c_{A}$.
Let us start by introducing the differential cross sections for both CE$\nu$NS
and ES in the SM. The SM differential coherent elastic scattering cross
section, for a neutrino with energy $E_{\nu}$ scattering off a nucleus of mass
$m_{A}$, is given111This strictly applies to $J=1/2$ nuclei Lindner:2016wff .
Note that for $J=0$ nuclei the last term is not present Freedman:1973yd ,
although its contribution is negligible for the energies of interest. by
Freedman:1973yd ; Lindner:2016wff
$\frac{\mathrm{d}\sigma^{\rm CE\nu NS}_{\rm SM}}{\mathrm{d}E_{R}}={\cal
Q}_{W}^{2}\,\frac{G_{F}^{2}\,m_{A}}{2\,\pi}\,\left[2-\frac{m_{A}\,E_{R}}{E_{\nu}^{2}}-\frac{2\,E_{R}}{E_{\nu}}+\left(\frac{E_{R}}{E_{\nu}}\right)^{2}\right]\,\left|F(q)\right|^{2}~{},$
(1)
where $G_{F}$ is the Fermi constant, $E_{R}$ is the nuclear (kinetic) recoil
energy, and the weak hypercharge of the nucleus in the SM is ${\cal
Q}_{W}=\left(N\,g_{V,n}+Z\,g_{V,p}\right)$, where $N$ and $Z$ are its number
of neutrons and protons, respectively. The weak charges of the neutron and the
proton are $g_{V,n}=-1/2$ and $g_{V,p}=1/2-2\sin^{2}\theta_{W}$, with
$\sin^{2}\theta_{W}=0.23868$ at zero momentum transfer Erler:2004in ;
Erler:2017knj . For Ge we use $\\{N,Z\\}=\\{40.71,\,32\\}$ (weighted average
of natural isotopic abundances), while for CsI we use
$\\{N,Z\\}=\\{76,\,54\\}$ and for Ar, $\\{N,Z\\}=\\{22,\,18\\}$.
In eq. (1) above, the nuclear form factor $F(q)$ is assumed to be equal for
protons and neutrons. Some commonly used phenomenological parameterizations
are the Klein-Nystrand Klein:1999qj or the Helm Helm:1956zz form factors,
among others. Note that, at the energies of interest for the Dresden-II
experiment, $F(q)\simeq 1$, so the choice of form factor is completely
irrelevant. For COHERENT, we use two different form factors, depending on the
target nucleus. For CsI, we use the theoretical calculation from refs.
Klos:2013rwa (based on refs. Hoferichter:2018acd ; Hoferichter:2016nvd ),
while for Ar we assume a Helm form factor with $s=0.9~{}\mathrm{fm}$, $\langle
r_{n}^{2}\rangle=-0.1161~{}\mathrm{fm}^{2}$ and $\langle
r_{p}^{2}\rangle=0.70706~{}\mathrm{fm}^{2}$ ParticleDataGroup:2020ssz ,
$R_{\mathrm{ch}}=3.4274~{}\mathrm{fm}$ Angeli:2013epw and
$R_{n,\mathrm{pt}}=3.405~{}\mathrm{fm}$ Payne:2019wvy (following the same
notation as in ref. Coloma:2020nhf ).
The differential cross section for ES per atom within the SM can be expressed
as (see, e.g., refs. Vogel:1989iv ; Tomalak:2019ibg )
$\frac{\mathrm{d}\sigma^{\rm ES}_{\rm SM}}{\mathrm{d}T_{e}}=Z_{\rm eff}^{\rm
X}(T_{e})\,\frac{G_{F}^{2}\,m_{e}}{2\,\pi}\left[\left(c_{V}+c_{A}\right)^{2}+\left(c_{V}-c_{A}\right)^{2}\left(1-\frac{T_{e}}{E_{\nu}}\right)+\left(c_{A}^{2}-c_{V}^{2}\right)\frac{m_{e}\,T_{e}}{E_{\nu}^{2}}\right]~{},$
(2)
where $T_{e}$ stands for electron (kinetic) recoil energy. The effective
electron couplings $c_{V}$ and $c_{A}$ contain the contributions from the SM
neutral current (NC), which are equal for all flavors, plus the charged-
current (CC) contribution which only affects $\bar{\nu}_{e}e^{-}$ and
$\nu_{e}e^{-}$ scattering. For convenience, their values are given in table 1.
We assume that scattering off electrons is incoherent, so the total ES cross
section per atom is obtained by multiplying the ES cross section per electron
by an effective charge, $Z_{\rm eff}^{\rm X}(T_{e})$, where
$\textrm{X}\equiv^{A}_{Z}\\!\text{X}$ indicates the target nucleus. This takes
into account that, in this process, atomic binding effects have to be
considered for recoil energies comparable to atomic binding energies. In doing
so, we follow the procedure proposed in refs. Kopeikin:1997; Fayans:2000ns;
Mikaelyan:2002nv and assume $Z_{\rm eff}^{\rm X}(T_{e})$ to be a step
function of the binding energies of the different atomic levels, as provided
in appendix A.
Finally, both types of interactions (off nuclei and off electrons) are two-
body elastic scattering processes, so the kinematics is the same. Hence, the
minimum neutrino energy to produce a recoil with energy $T$ is $E_{\nu,{\rm
min}}=\left(T+\sqrt{T^{2}+2\,m\,T}\right)/2$ and the maximum recoil energy for
a given neutrino energy is $T_{\rm max}=2\,E_{\nu}^{2}/(2\,E_{\nu}+m)$, where
$T\equiv E_{R}$ and $m\equiv m_{A}$ for nuclei, while $T\equiv T_{e}$ and
$m\equiv m_{e}$ for electrons.
Throughout this work we will consider several phenomenological scenarios
leading to modifications to the interaction cross sections in eqs. (1) and
(2), as explained in more detail below.
### 2.1 Non-standard neutrino interactions
The so-called non-standard neutrino interaction (NSI) framework consists on
the addition of four-fermion effective operators to the SM Lagrangian at low
energies. For example, the effective Lagrangian
$\mathcal{L}_{\text{NSI,
NC}}=-\sum_{f,\alpha,\beta}2\sqrt{2}\,G_{F}\,\varepsilon_{\alpha\beta}^{f,P}\,\left(\bar{\nu}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\,\left(\bar{f}\gamma^{\mu}Pf\right)~{},$
(3)
would lead to new NC interactions with the rest of the SM fermions. Here,
$\alpha,\beta\equiv e,\mu,\tau$ while $f$ refers to SM fermions, and $P$ can
be either a left-handed or a right-handed projection operator ($P_{L}$ or
$P_{R}$, respectively). Such new interactions may induce lepton flavor-
changing processes (if $\alpha\neq\beta$), or may lead to a modified
interaction rate with respect to the SM result (if $\alpha=\beta$).
In presence of NC NSI, the effective charge of the nucleus in eq. (1) gets
modified,
$\mathcal{Q}_{W}^{2}\to\mathcal{Q}^{2}_{\alpha}(\boldsymbol{\varepsilon})$.
For real off-diagonal NSI parameters, it reads Barranco:2005yy
$\displaystyle\mathcal{Q}^{2}_{\alpha}(\boldsymbol{\varepsilon})$
$\displaystyle=$
$\displaystyle\left[Z\big{(}g_{p}^{V}+2\,\varepsilon_{\alpha\alpha}^{u}+\varepsilon_{\alpha\alpha}^{d}\big{)}+N\big{(}g_{n}^{V}+\varepsilon_{\alpha\alpha}^{u}+2\,\varepsilon_{\alpha\alpha}^{d}\big{)}\right]^{2}$
(4)
$\displaystyle+\sum_{\beta\neq\alpha}\left[Z\big{(}2\,\varepsilon_{\alpha\beta}^{u}+\varepsilon_{\alpha\beta}^{d})+N\big{(}\varepsilon_{\alpha\beta}^{u}+2\,\varepsilon_{\alpha\beta}^{d}\big{)}\right]^{2}~{}.$
where, in order to simplify notation, we have renamed
$\varepsilon_{\alpha\beta}\equiv\varepsilon_{\alpha\beta}^{q,V}=\varepsilon_{\alpha\beta}^{q,L}+\varepsilon_{\alpha\beta}^{q,R}$.
The weak charge may be rewritten in a more compact form as
$\displaystyle\mathcal{Q}^{2}_{\alpha}(\boldsymbol{\varepsilon})$
$\displaystyle=$
$\displaystyle\left[\mathcal{Q}_{W}+\varepsilon_{\alpha\alpha}^{\text{X}}\right]^{2}+\sum_{\beta\neq\alpha}\big{(}\varepsilon_{\alpha\beta}^{\text{X}}\big{)}^{2}~{},$
(5)
where we have defined
$\varepsilon_{\alpha\beta}^{\text{X}}\equiv
N\,\varepsilon_{\alpha\beta}^{n}+Z\,\varepsilon_{\alpha\beta}^{p}\,,\qquad\varepsilon_{\alpha\beta}^{n}\equiv\varepsilon_{\alpha\beta}^{u}+2\,\varepsilon_{\alpha\beta}^{d}~{},\qquad\varepsilon_{\alpha\beta}^{p}\equiv
2\,\varepsilon_{\alpha\beta}^{u}+\varepsilon_{\alpha\beta}^{d}~{}.$ (6)
The first consequence we observe of the inclusion of NSI effects is that the
weak charge may now depend on the incident neutrino flavor $\alpha$. As will
be discussed below, it is relevant to note that the COHERENT experiment
observes interactions of both electron and muon neutrinos. However, at first
approximation, it measures the linear combination
$\mathcal{Q}^{2}_{e}+2\,\mathcal{Q}^{2}_{\mu}$ and hence, both charges are
degenerate: a reduction in the value of $\mathcal{Q}^{2}_{e}$ can be
compensated by an increase in $\mathcal{Q}^{2}_{\mu}$ and vice versa. At
COHERENT, though, the addition of timing information partially breaks this
degeneracy for CsI, as discussed in detail in ref. Coloma:2019mbs . Including
reactor data, which is only sensitive to interactions with electron
antineutrinos, brings in additional complementary information in this respect
Dent:2017mpr . It provides an additional constraint on $\mathcal{Q}^{2}_{e}$,
which is independent of $\mathcal{Q}^{2}_{\mu}$.
The second relevant feature we observe from eqs. (4)-(6) is that the impact of
NSI on the weak charge depends on the values of $N$ and $Z$ in a non-trivial
manner. Because of this, the combination of data obtained for different nuclei
offers an additional handle to reduce the size of the allowed confidence
regions of this scenario (for earlier discussions see, e.g., refs.
Scholberg:2005qs ; Barranco:2005yy ; Coloma:2017egw ; Baxter:2019mcx ).
### 2.2 Neutrino magnetic moment
In the presence of a neutrino magnetic moment, $\mu_{\nu}$, the cross sections
for neutrino scattering off nuclei and electrons get additional contributions
which do not interfere with the SM ones. The scattering off protons can be
considered coherent and therefore its cross section is given, up to order
${\cal O}((E_{R}/E_{\nu})^{2},E_{R}/m_{A})$, by Vogel:1989iv
$\frac{\mathrm{d}\sigma^{\rm CE\nu
NS}_{\rm\mu_{\nu}}}{\mathrm{d}E_{R}}=Z^{2}\,\left(\frac{\mu_{\nu}}{\mu_{B}}\right)^{2}\,\frac{\alpha^{2}\,\pi}{m_{e}^{2}}\left[\frac{1}{E_{R}}-\frac{1}{E_{\nu}}\right]\;|F(q)|^{2}~{},$
(7)
where the form factor $F(q)$ is assumed to be the same as in the SM, which is
a reasonable approximation at the transfer momenta of interest
Hoferichter:2020osn .
Conversely, the scattering off electrons is incoherent and therefore, the
cross section in this case, up to order ${\cal O}((T_{e}/E_{\nu})^{2})$, is
Vogel:1989iv
$\frac{\mathrm{d}\sigma^{\rm ES}_{\rm\mu_{\nu}}}{\mathrm{d}T_{e}}=Z_{\rm
eff}^{\rm
X}(T_{e})\,\left(\frac{\mu_{\nu}}{\mu_{B}}\right)^{2}\,\frac{\alpha^{2}\,\pi}{m_{e}^{2}}\left[\frac{1}{T_{e}}-\frac{1}{E_{\nu}}\right]~{}.$
(8)
Notice that, in writing eqs. (7) and (8), we have denoted the neutrino
magnetic moment as $\mu_{\nu}$, without specifying the flavor of the neutrino.
However, neutrino magnetic moments arise in a variety of models of new physics
and, in particular, they do not need to be flavor-universal. Therefore, in
what follows, we will allow different magnetic moments for the different
neutrino flavors, reporting their bounds separately.
### 2.3 Light scalar mediators
The Lagrangian of the simplified model of interaction of a scalar $\phi$ with
the relevant fermions we consider is
${\cal
L}_{\phi}=g_{\phi}\,\phi\,\left(\sum_{q}q_{\phi}^{q}\,\bar{q}q+q_{\phi}^{e}\,\bar{e}e+q_{\phi}^{\nu}\,\bar{\nu}_{R}\nu_{L}+\text{h.c.}\right)-\frac{1}{2}\,M^{2}_{\phi}\,\phi^{2}~{},$
(9)
where $q_{\phi}^{i}$ are the charges of each fermion
($i=\\{\nu,\,e,\,u,\,d,\,s,\,c,\,b,\,t\\}$) under the new interaction. This
new interaction does not interfere with the SM one Rodejohann:2017vup ;
Farzan:2018gtr , and the corresponding neutrino-nucleus elastic scattering
cross section, up to order ${\cal O}((E_{R}/E_{\nu})^{2})$, reads
Cerdeno:2016sfi ; Farzan:2018gtr
$\frac{\mathrm{d}\sigma^{\rm CE\nu
NS}_{\phi}}{\mathrm{d}E_{R}}=\frac{g_{\phi}^{4}\,(q_{\phi}^{\nu})^{2}\,\mathcal{Q}_{\phi}^{2}}{4\,\pi}\,\frac{m_{A}^{2}\,E_{R}}{E_{\nu}^{2}\,(2\,m_{A}\,E_{R}+M_{\phi}^{2})^{2}}\,|F(q)|^{2}~{},$
(10)
where $F(q)$ is the form factor (which, with enough precision at the transfer
momenta of interest, can be assumed to be the same as in the SM
Hoferichter:2020osn ), and $\mathcal{Q}_{\phi}$ is the nuclear charge for the
scalar interaction Shifman:1978zn ; DelNobile:2013sia ; Ellis:2018dmb ,
$\mathcal{Q}_{\phi}=\sum_{N,q}q^{q}_{\phi}\,\frac{m_{N}}{m_{q}}\,f^{(N)}_{T,q}+\frac{2}{27}\,\left(1-\sum_{N,q}f^{(N)}_{T,q}\right)\sum_{N,\tilde{q}}q_{\phi}^{\tilde{q}}\,\frac{m_{N}}{m_{q}}~{}.$
(11)
Here $N=n,p$ stands for the nucleons, the superindex $q=u,d,s$ runs over the
light valence and sea quarks, $\tilde{q}=c,b,t$ runs over the heavy quarks,
and the coefficients $f_{T,q}^{(N)}$ incorporate the effective low-energy
couplings of the scalar to the nucleons Shifman:1978zn . For universal
coupling of the scalar to quarks,
$q_{\phi}^{u}=q_{\phi}^{d}=q_{\phi}^{s}=q_{\phi}^{c}=q_{\phi}^{b}=q_{\phi}^{t}\equiv
q_{\phi}^{q}$, the nuclear scalar charge takes the value
$\mathcal{Q}_{\phi}=q_{\phi}^{q}\,(14\,N+15.1\,Z)$ DelNobile:2013sia .222This
is an approximate expression, which actually does not coincide with any of the
sets of values in ref. DelNobile:2013sia (for updated values of the
coefficients, see ref. Ellis:2018dmb ). We use it for the sake of comparison
with the results from refs. Cerdeno:2016sfi ; CONUS:2021dwh . Current
uncertainties on the low-energy coefficients and masses lead to variations of
20%-30% in eq. (11); we have numerically checked that this leads to a
$\lesssim 10\%$ effect on our constraints.
Scalar exchange also gives a contribution to the cross section for neutrino-
electron elastic scattering as Cerdeno:2016sfi
$\frac{\mathrm{d}\sigma^{\rm ES}_{\phi}}{\mathrm{d}T_{e}}=Z_{\rm eff}^{\rm
X}(T_{e})\,\frac{g_{\phi}^{4}\,(q^{\nu}_{\phi})^{2}\,(q^{e}_{\phi})^{2}}{4\,\pi}\,\frac{m_{e}^{2}\,T_{e}}{E_{\nu}^{2}\,(2\,m_{e}\,T_{e}+M_{\phi}^{2})^{2}}~{}.$
(12)
In order to quantify the contribution of the scattering off electrons, in
section 4 we study the constraints on two specific models: a first one in
which the scalar couples universally to all relevant fermions (hereafter
referred to as _universal_), and another one in which it only couples to
leptons (dubbed _leptonic scalar_ model). For convenience, table 2 summarizes
the explicit values of charges considered for the different SM fermions.
Model | $q^{q}$ | $q^{e}$ | $q^{\nu_{e}}$ | $q^{\nu_{\mu}}$
---|---|---|---|---
Universal scalar or vector | 1 | 1 | 1 | 1
Leptonic ($\ell$) scalar | 0 | 1 | 1 | 1
$L_{e}$ vector | 0 | 1 | 1 | 0
$B-L$ vector | $\frac{1}{3}$ | -1 | -1 | -1
Table 2: Charges for the scalar and vector mediator models considered in this
work.
### 2.4 Light vector mediators
The Lagrangian of a simplified model of interaction of a neutral vector
$Z^{\prime}$ with the fermions of the first generation is given by
${\cal
L}_{Z^{\prime}}=g_{Z^{\prime}}\;Z^{\prime}_{\mu}\left(q_{Z^{\prime}}^{u}\,\bar{u}\gamma^{\mu}u+q_{Z^{\prime}}^{d}\,\bar{d}\gamma^{\mu}d+q_{Z^{\prime}}^{e}\,\bar{e}\gamma^{\mu}e+q_{Z^{\prime}}^{\nu}\,\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)+\frac{1}{2}M^{2}_{Z^{\prime}}{Z^{\prime}}^{\mu}Z^{\prime}_{\mu}~{},$
(13)
where $q_{Z^{\prime}}^{i}$ indicates the charges of each fermion
($i=\\{\nu,\,e,\,u,\,d\\}$) under the new interaction.
Unlike the magnetic-moment and scalar interactions, a neutral vector
interaction interferes with the SM. The additional contribution to the
neutrino-nucleus scattering cross section reads Cerdeno:2016sfi
$\Delta\frac{\mathrm{d}\sigma^{\rm CE\nu
NS}_{Z^{\prime}}}{\mathrm{d}E_{R}}=\frac{g^{2}_{Z^{\prime}}\,m_{A}}{2\,\pi}\left[\frac{g^{2}_{Z^{\prime}}\,(q_{Z^{\prime}}^{\nu})^{2}\,\mathcal{Q}^{2}_{Z^{\prime}}}{\left(2\,m_{A}\,E_{R}+M_{Z^{\prime}}^{2}\right)^{2}}-\frac{2\,\sqrt{2}\,G_{F}\,q_{Z^{\prime}}^{\nu}\,\mathcal{Q}_{Z^{\prime}}\,\mathcal{Q}_{W}}{\left(2\,m_{A}\,E_{R}+M_{Z^{\prime}}^{2}\right)}\right]\left(1-\frac{m_{A}\,E_{R}}{2\,E^{2}_{\nu}}\right)|F(q)|^{2}~{},$
(14)
where $\mathcal{Q}_{Z^{\prime}}$ is the weak charge of the nucleus for the
light vector interaction. Vector current conservation implies that only
valence quarks contribute by simply summing up their charges, so for universal
couplings ($q_{Z^{\prime}}^{u}=q_{Z^{\prime}}^{d}\equiv q_{Z^{\prime}}^{q}$),
$\mathcal{Q}_{Z^{\prime}}=3\,q_{Z^{\prime}}^{q}\,(Z+N)$ (see, e.g., ref.
DelNobile:2013sia ). As in the case of scalar mediators, we use the same
nuclear form factor as in the SM.
The corresponding contribution to the cross section for ES reads
Cerdeno:2016sfi
$\Delta\frac{\mathrm{d}\sigma^{\rm ES}_{Z^{\prime}}}{\mathrm{d}T_{e}}=Z_{\rm
eff}^{\rm
X}(T_{e})\,\frac{g^{2}_{Z^{\prime}}\,m_{e}}{2\,\pi}\left[\frac{{g^{2}_{Z^{\prime}}}\,(q_{Z^{\prime}}^{\nu})^{2}\,(q^{e}_{Z^{\prime}})^{2}}{\left(2\,m_{e}\,T_{e}+M_{Z^{\prime}}^{2}\right)^{2}}\,+\,\frac{2\,\sqrt{2}\,G_{F}\,q_{Z^{\prime}}^{\nu}\,q_{Z^{\prime}}^{e}\,c_{V}}{\left(2\,m_{e}\,T_{e}+M_{Z^{\prime}}^{2}\right)}\right]~{},$
(15)
where $c_{V}$ is the SM effective vector coupling introduced in eq.(2), which
depends on the flavor of the incident neutrino as given in table 1.
As in the case with light scalar mediators, in section 4 we study the
constraints on three specific models in which the vector couples universally
to all relevant fermions, another in which it only couples to electron flavor
($L_{e}$), and the anomaly-free flavor-universal model with coupling to $B-L$
(see table 2).
## 3 Data analysis
In this section we describe the procedure we follow to perform the data
analysis of the Dresden-II and COHERENT data.
### 3.1 The Dresden-II reactor experiment
We use a sample with 96.4-day exposure of a 2.924 kg ultra-low noise p-type
point contact (PPC) germanium detector (NCC-1701) to the high flux of electron
antineutrinos from the Dresden-II boiling water reactor (BWR). The new data
set spans the period between January 22 and May 8, 2021, when the reactor was
operated at its full nominal power of 2.96 GW. In the data release
Colaresi2022suggestive , which we follow closely, the data points and errors
are provided in the form of rate per day. The errors (per day) represent a
combination of statistical and signal acceptance uncertainties. With the new
information obtained from the reactor operator, the improved center-to-center
effective distance between the PPC crystal and the center point of the BWR
core is 10.39 m. The estimate of the $\bar{\nu}_{e}$ flux from the reactor is
$4.8\times 10^{13}~{}\bar{\nu}_{e}/\textrm{cm}^{2}\textrm{s}$ with a $\sim
2\%$ uncertainty Huber:2011wv ; Mueller:2011nm . We describe the reactor
$\bar{\nu}_{e}$ spectrum using the typical average values of fission fractions
of the four main isotopes (making up more than 99% of all reactor neutrinos),
for commercial power reactor using low-enriched uranium: ${}^{235}\rm{U}$
(58%), ${}^{239}\rm{Pu}$ (29%), ${}^{238}\rm{U}$ (8%) and ${}^{241}\rm{Pu}$
(5%) Qian:2018wid . We use the combination of the tabulated spectra for
$E_{\bar{\nu}_{e}}<2$ MeV from ref. Vogel:1989iv and for
$2~{}\textrm{MeV}\leq E_{\bar{\nu}_{e}}\leq 8$ MeV from ref. Mueller:2011nm .
We set the flux to zero beyond the maximum energy that is tabulated.
The background model consists of four components Colaresi2022suggestive ,
although only two of them really contribute to the SM signal region. The other
two allow constraining these components. The dominant source in the signal
region is the elastic scattering of epithermal neutrons, which is modeled by a
free exponential plus a free constant term, $R_{\rm epith}+A_{\rm
epith}\,e^{-E_{\rm rec}/E_{\rm epith}}$, where $E_{\rm rec}$ is the
reconstructed ionization energy. The other three components of the background
consist of electron capture (EC) peaks in 71Ge, which are all described as
Gaussian probability density functions (PDF) with three free parameters: the
amplitude $A_{\rm shell}$, the centroid $E_{\rm shell}$, and the standard
deviation $\sigma_{\rm shell}$. The $M$-shell EC peak is constrained by the
$L_{1}$-shell EC peak, with parameters
$\\{A_{L_{1}},\,E_{L_{1}},\,\sigma_{L_{1}}\\}$. Although the $L_{1}$-shell
peak is at a nominal energy of 1.297 keV, it is also allowed to vary freely.
The ratio of the amplitudes of the $L_{1}$-shell and $M$-shell has been
experimentally determined to be $0.16\pm 0.03$ SuperCDMS:2015eex , so
following the data release Colaresi2022suggestive , we model this ratio with a
Gaussian prior of width $\sigma_{M/L_{1}}=0.03$ and centered at
$A_{M}/A_{L_{1}}=0.16$, which we add to the likelihood. The centroid of the
$M$-shell EC contribution is fixed at $E_{M}=0.158$ keV SuperCDMS:2015eex ;
Firestone1996 and the standard deviation is set to be equal to the electronic
noise $\sigma_{M}=\sigma_{n}$, which is 68.5 eV during the 96.4 days of
reactor operation (Rx-ON) and 65.25 eV during the 25 days of reactor refueling
outage (Rx-OFF). Finally, the last contribution comes from the $L_{2}$-shell
EC, with the amplitude fixed by $A_{L_{2}}/A_{L_{1}}=0.008$, the centroid at
$E_{L_{2}}=1.142$ keV and the standard deviation
$\sigma_{L_{2}}=\sigma_{L_{1}}$. Explicitly, the background model, in terms of
the reconstructed ionization energy, $E_{\rm rec}$, is described by a
differential rate
$\frac{\mathrm{d}R_{\rm bkg}(\boldsymbol{\beta})}{\mathrm{d}E_{\rm
rec}}=R_{\rm epith}+A_{\rm epith}\,e^{-E_{\rm rec}/E_{\rm
epith}}+\sum_{i=L_{1},L_{2},M}\frac{A_{i}}{\sqrt{2\,\pi}\,\sigma_{i}}\,e^{-\frac{\left(E_{\rm
rec}-E_{i}\right)^{2}}{2\,\sigma_{i}^{2}}}~{},$ (16)
in the reconstructed energy region of interest (ROI) $E_{\rm
rec}=[0.2,1.5]~{}{\rm keV}$. The number of free background parameters to be
fitted in every analysis is seven, which are represented by the vector
$\boldsymbol{\beta}=\left\\{R_{\rm epith},A_{\rm epith},E_{\rm
epith},A_{L_{1}},E_{L_{1}},\sigma_{L_{1}},\beta_{M/L_{1}}\right\\}$. The
parameter $\beta_{M/L_{1}}$, defined as $A_{M}=\beta_{M/L_{1}}\,A_{L_{1}}$, is
added to the likelihood with a Gaussian prior, as described below.
The CE$\nu$NS signal rate from the reactor $\bar{\nu}_{e}$ flux is given by
$\frac{\mathrm{d}R_{\rm sig}^{\textrm{CE}\nu\textrm{NS}}}{\mathrm{d}E_{\rm
rec}}=N_{\rm T}\,\int_{E_{\nu,{\rm
min}}}^{\infty}\mathrm{d}E_{\bar{\nu}_{e}}\int_{E_{R,\rm min}}^{E_{R,\rm
max}}\mathrm{d}E_{R}\,\frac{\mathrm{d}\Phi_{\bar{\nu}_{e}}}{\mathrm{d}E_{\bar{\nu}_{e}}}\,\frac{\mathrm{d}\sigma^{\rm
CE\nu NS}}{\mathrm{d}E_{R}}\,{\cal R}(E_{\rm
rec},E_{I}=Q\,E_{R};\sigma_{I})~{},$ (17)
where $N_{T}=2.43\times 10^{25}$ is the number of germanium nuclei and
$\mathrm{d}\Phi_{\bar{\nu}_{e}}/\mathrm{d}E_{\bar{\nu}_{e}}$ is the reactor
electron antineutrino spectrum. Here $E_{R,\rm min}$ is the minimum nuclear
recoil energy which corresponds to a minimum average ionization energy,
$E_{I,\rm min}\simeq 2.98$ eV, required to produce a hole-pair in germanium
(at 77 K) Antman1966272 (see ref. Wei:2016xbw for a compilation of existing
data at other temperatures). The ionization energy is defined as
$E_{I}=Q(E_{R})\,E_{R}$, with $Q(E_{R})$ being the quenching factor, which
describes the reduction in ionization yield of a nuclear recoil when compared
to an electron recoil of same energy. We consider two models from the data
release, which are not in tension with CONUS data CONUS:2020skt , and are
denoted by ‘Fef’ (using iron-filtered monochromatic neutrons) and ‘YBe’ (based
on photoneutron source measurements) Collar:2021fcl . The spread between these
two models approximately represents the uncertainty on this parameter. Note
that the cross sections in eqs. (7) and (8) diverge as the recoil energy goes
to zero. Consequently, the contribution arising from the scattering due to the
neutrino magnetic moment (and similarly for models with very light mediators)
is larger in the lower energy bins, a contribution that can be divergently
large under the assumption that asymptotically low ionization energies could
trigger the detector. This unphysical behavior is cut-off by the physical
requirement of the average energy required to produce an electron-hole pair.
Thus, in what follows, we impose $E_{I,\rm min}=3$ eV when evaluating the
expected rate of events.
The energy resolution function, ${\cal R}(E_{\rm rec},E_{I};\sigma_{I})$, is
described as a truncated Gaussian ($E_{\rm rec}>0$),
${\cal R}(E_{\rm
rec},E_{I};\sigma_{I})=\left(\frac{2}{1+\textrm{Erf}\left(\frac{E_{I}}{\sqrt{2}\,\sigma_{I}}\right)}\right)\,\frac{1}{\sqrt{2\,\pi}\,\sigma_{I}}\,e^{-\frac{\left(E_{\rm
rec}-E_{I}\right)^{2}}{2\,\sigma_{I}^{2}}}~{},$ (18)
with $\sigma_{I}^{2}=\sigma_{n}^{2}+E_{I}\,\eta\,F$, where the electronic
noise is $\sigma_{n}=68.5~{}{\rm eV}$ (Rx-ON) and 65.25 eV (Rx-OFF), $\eta$ is
the average energy of electron-hole formation and $F$ is the Fano factor. For
Ge, $\eta=2.96$ eV and $F=0.11$ Colaresi2022suggestive . The prefactor with
the error function is included to guarantee the resolution function is
normalized to 1 for $E_{I}>0$. Note that, although the reconstructed energy
ROI is $E_{\rm rec}=[0.2,1.5]~{}{\rm keV}$, events with $E_{I}<0.2~{}{\rm
keV}$ are assumed to trigger the detector and be susceptible of filtering into
the ROI.
Moreover, it is important to point out that, as provided, all data points are
already corrected for signal acceptance, so this must not be included in the
calculation of the expected signal rate. Ideally, raw data and signal
acceptance as a function of the true ionization energy, $E_{I}$, must be used.
In this way, the signal acceptance must be included in eq. (17). Nevertheless,
the approach used here, following the data release, has a negligible impact on
signals that grow slowly at small ionization energies, as the SM or NSI cases.
Yet, it could have a non-negligible effect on the event rate in models with a
large neutrino magnetic moment or in the presence of light mediators (in the
case of interactions with nucleons). Furthermore, the impact of using signal
acceptance-corrected data also depends on the minimum ionization energy
capable of triggering the detector. Note that the minimum ionization energy at
which the signal acceptance is currently measured is 0.13 keV
Colaresi2022suggestive .
With all these ingredients, the expected event rate is given by
$\frac{\mathrm{d}R\left(\boldsymbol{\theta};\boldsymbol{\beta}\right)}{\mathrm{d}E_{\rm
rec}}=\frac{\mathrm{d}R_{\rm
bkg}\left(\boldsymbol{\beta}\right)}{\mathrm{d}E_{\rm
rec}}+\frac{\mathrm{d}R_{\rm
sig}^{\textrm{CE}\nu\textrm{NS}}\left(\boldsymbol{\theta}\right)}{\mathrm{d}E_{\rm
rec}}+\frac{\mathrm{d}R_{\rm sig}^{\rm
ES}\left(\boldsymbol{\theta}\right)}{\mathrm{d}E_{\rm rec}}~{},$ (19)
where our notation explicitly indicates that the event rates include both the
signal contribution from CE$\nu$NS and from ES.
In general, when considering different new physics scenarios, the signal rate
depends on a set of (one or two in this work) parameters
$\boldsymbol{\theta}$. We use the following $\chi^{2}$,
$\chi_{\rm
D-II}^{2}\left(\boldsymbol{\theta};\boldsymbol{\beta}\right)=\sum_{i=1}^{130}\frac{\left(P_{i}\left(\boldsymbol{\theta};\boldsymbol{\beta}\right)-D_{i}\right)^{2}}{\sigma_{i}^{2}}+\frac{\left(\beta_{M/L_{1}}-A_{M}/A_{L_{1}}\right)^{2}}{\sigma_{M/L_{1}}^{2}}~{},$
(20)
where $P_{i}$ refers to the prediction and $D_{i}$ to the measured rates (or
number of events in the exposure time) in energy bin $i$ (of width 10 eV and
with the center of the bin indicated in the data release), $\sigma_{i}$ is the
standard deviation of the rate (or of the measured number of events in the
exposure time) in bin $i$, which combines statistical and signal acceptance
uncertainties, and $\beta_{M/L_{1}}$ is the ratio between the amplitudes of
the $M$-shell and $L_{1}$-shell EC contributions to the background, with
central value $A_{M}/A_{L_{1}}=0.16$ and $\sigma_{M/L_{1}}=0.03$. We do not
include additional nuisance parameters. Nevertheless, we have checked that
even adding a 10% uncertainty on the normalization of the signal, the results
are only affected at the percent level. Finally, in order to study the
sensitivity to the new physics parameters, we profile the above likelihood,
$\left(\chi^{2}_{\rm D-II}\right)_{\rm
p}\left(\boldsymbol{\theta}\right)=\chi_{\rm
D-II}^{2}\left(\boldsymbol{\theta};\widehat{\widehat{\boldsymbol{\beta}\,}}(\boldsymbol{\theta})\right)\equiv\textrm{min}_{\boldsymbol{\beta}}\left\\{\chi_{\rm
D-II}^{2}\left(\boldsymbol{\theta};\boldsymbol{\beta}\right)\right\\}~{},$
(21)
where the profiled values of $\boldsymbol{\beta}$ that minimize $\chi^{2}$ for
each $\boldsymbol{\theta}$ are indicated by a double hat.
Figure 1: Left panel: Spectral rate of signal events from CE$\nu$NS and
$\bar{\nu}_{e}e^{-}$ scattering in the SM and from the electromagnetic
scattering induced by a neutrino magnetic moment
$\mu_{\nu_{e}}=10^{-10}\mu_{B}$. Right panel: Spectral rate of signal events
from CE$\nu$NS and $\bar{\nu}_{e}e^{-}$ scattering in the SM and for the
contribution induced by an interaction mediated by a massless scalar boson
(dashed lines) and by a massless vector boson (dotted lines), assuming
universal couplings (see table 2). For a vector mediator, we show the addition
of both contributions, the purely vector boson contribution and the
interference with the SM (see eq. (14)). In both panels we compare the spectra
to data, after subtracting the best-fit background model assuming only the SM
signal. All results in both panels are depicted for the Fef model of the
quenching factor (for interactions with nuclei) and assuming $E_{I,\rm min}=3$
eV.
In figure 1 we show the predicted event distributions for the SM and for the
models of new physics we consider. Along with these spectra, we also include
the Dresden-II data points, after subtracting the best-fit background model
assuming only the SM signal. In the left panel of figure 1 we show the
predicted event distributions for the SM and the extra contribution from a
non-zero magnetic moment. We see that the $\mu_{\nu_{e}}$-induced
electromagnetic scattering off protons dominates over the corresponding
scattering off electrons for $E_{\rm rec}\lesssim 0.35$ keV. This is so
because the coherent cross section off nuclei is enhanced by the factor
$Z^{2}/Z_{\rm eff}^{\rm Ge}$, which is large enough to compensate the
suppression due to the quenching factor (which makes the characteristic
$1/E_{R}$ of the nucleus smaller than $1/T_{e}$ of the electron, for the same
value of $E_{\rm rec}$). Also, from eqs. (10), (12), (14) and (15) we expect
different spectra of events for interactions mediated by light scalars or by
light vectors, as well as a different scaling with energy for scattering off
electrons and off nucleus. This can be seen from the comparison among the
different curves in the right panel of figure 1, where we have assumed
universal couplings (see table 2). The event spectra for interactions mediated
by a massless scalar are very similar to the $\mu_{\nu_{e}}$-induced ones, as
they are governed by the $1/E_{R}$ or $1/T_{e}$ dependence for CE$\nu$NS or
ES, respectively. In this case, the coherent cross section off nuclei is
enhanced by the factor $\mathcal{Q}_{\phi}^{2}/Z_{\rm eff}^{\rm Ge}$ with
respect to that of scattering off electrons. On the other hand, the event
spectra for ES mediated by a massless vector is much larger than the one for
scattering off nuclei. Unlike the cases of the magnetic moment or the scalar
mediator, the dominant contribution at small energies for CE$\nu$NS is a
factor $\sim m_{e}/m_{A}$ smaller than for ES.
### 3.2 The COHERENT experiment
For the COHERENT CsI analysis, we follow the same procedure as in ref.
Coloma:2019mbs . We use the nuclear form factor from ref. Klos:2013rwa , we
estimate the time dependence of the background directly from anti-coincidence
data, and for concreteness, we use the quenching factor from ref.
Collar:2019ihs . In the present work we also include scattering off electrons.
In this case, there is no quenching factor, the target mass is set to the
electron mass, and we use an effective number of electrons per nucleus,
$Z_{\mathrm{eff}}(T_{e})$, as provided in appendix A (see also section 2). The
rest of the details of the analysis are the same as in ref. Coloma:2019mbs .
Here, the minimum recoil energy considered can be safely set to zero, since
the efficiency vanishes at low energies.
For the analysis of Ar data, we follow the official data release
COHERENT:2020ybo . In this case we do _not_ include scattering off electrons,
as the experiment can discriminate nuclear from electron recoils
COHERENT:2020iec . To obtain the CE$\nu$NS event spectrum, we start by
computing the signal event spectrum in nuclear recoil energy, $E_{R}$, which
is given by
$\frac{\mathrm{d}N}{\mathrm{d}E_{R}}=\mathcal{N}\sum_{\alpha}\int_{E_{\nu,\mathrm{min}}}^{m_{\mu}/2}\frac{\mathrm{d}\sigma(E_{\nu},E_{R})}{\mathrm{d}E_{R}}\,\frac{\mathrm{d}\phi_{\alpha}}{\mathrm{d}E_{\nu}}\,\mathrm{d}E_{\nu}~{},$
(22)
where $E_{\nu}$ is the neutrino energy and the sum extends over the three
components of the neutrino flux $\\{\nu_{e},\nu_{\mu},\bar{\nu}_{\mu}\\}$. The
neutrino spectra $\mathrm{d}\phi_{\alpha}/\mathrm{d}E_{\nu}$ are normalized to
1 (see, e.g., eq. (2.1) in ref. Coloma:2019mbs for the expressions), and all
normalizations are absorbed into an overall constant $\mathcal{N}$ given by
$\mathcal{N}=\frac{1}{4\pi\ell^{2}}\,N_{\mathrm{PoT}}\,f_{\pi/p}\,N_{\mathrm{Ar}}~{},$
(23)
where $\ell=27.5~{}\mathrm{m}$ is the distance to the detector;
$N_{\mathrm{PoT}}=13.77\times 10^{22}$ is the number of protons on target
(PoT), corresponding to an integrated power of 6.12 GW$\cdot$hr;
$f_{\pi/p}=0.09$ is the number of pions produced per PoT; and
$N_{\mathrm{Ar}}=m_{\mathrm{det}}/m_{\rm Ar}$ is the number of nuclei in the
detector, with $m_{\mathrm{det}}=24.4~{}\mathrm{kg}$ the detector mass and
$m_{\rm Ar}$ the 40Ar mass.
However, the experimental collaboration does not bin their data in nuclear
recoil energy (keVnr), but in electron-equivalent recoil energy instead
(keVee). In addition, we have to account for the detection efficiency,
$\epsilon$, and the energy resolution. Introducing these effects, the expected
event rate in each bin $i$ (of width $\Delta E_{\rm rec}$) is computed as:
$N_{i}=\int_{E_{\rm rec,i}-\Delta E_{\rm rec}/2}^{E_{\rm rec,i}+\Delta E_{\rm
rec}/2}\mathrm{d}E_{\rm rec}\,\,\epsilon(E_{\rm
rec})\int_{E_{R,\mathrm{min}}}^{\infty}\mathrm{d}E_{R}\,\frac{\mathrm{d}N}{\mathrm{d}E_{R}}\,\mathcal{R}(E_{\rm
rec},E_{I};\sigma_{I})~{},$ (24)
where $E_{I}$ stands for the true electron-equivalent recoil energy and
$E_{\rm rec}$ is the reconstructed electron-equivalent recoil energy (that is,
after energy smearing effects). The function $\mathcal{R}$ accounts for the
energy resolution of the detector:
$\mathcal{R}(E_{\rm
rec},E_{I};\sigma_{I})=\frac{1}{\sqrt{2\,\pi}\,\sigma_{I}}\,e^{\frac{-\left(E_{\rm
rec}-E_{I}\right)^{2}}{2\,\sigma_{I}^{2}}}~{},$ (25)
with a width
$\sigma_{I}\equiv\sigma(E_{I})=0.58~{}\mathrm{keV}\,\sqrt{E_{I}/\mathrm{keV}}$,
as prescribed in the data release. As the energies in the ROI are much larger
than the standard deviation, this definition, unlike eq. (18), does not
require the extra factor to guarantee it is correctly normalized to 1. Also,
note that in eq. (24) the detection efficiency $\epsilon$ is obtained post-
triggering and it is a function of the reconstructed energy (and not of the
true energy).333Furthermore, contrary to the indications provided in the data
release, in our analysis we include this efficiency as a function of
reconstructed energy _before binning the event distribution_ , as otherwise we
do not find good agreement with the results of the collaboration.
We set the minimum nuclear recoil energy to $E_{I,\mathrm{min}}=19.5$ eV, the
average energy to produce a scintillation photon in Ar Creus:2013sau . As
indicated above, the relation between $E_{I}$ and $E_{R}$ is given by the
quenching factor, $E_{I}=Q(E_{R})\,E_{R}$, which is described as
$Q(E_{R})=a+b\,E_{R}$, with $a=0.246$ and $b=0.00078$ keV-1, as given in the
data release.
Following the procedure above, we obtain a nominal prediction of $135.3$
signal events. This is slightly higher than the rates predicted by the
collaboration, but well within their reported error bars (their nominal
prediction is $128\pm 17$ events). Our spectrum also shows good agreement with
the official one.
To further reduce backgrounds, the analysis includes information not only on
recoil energy, but also on timing and on the fraction of integrated amplitude
within the first 90 ns (this last variable is called F90 by the
collaboration). Once we compute the distribution in recoil energy as described
above, we can obtain the full 3D PDF. We do so by rescaling the original PDF
provided by the collaboration by the ratio of their projected PDF in $E_{\rm
rec}$ to our $E_{\rm rec}$ event distribution, i.e.,
$N_{i,j,k}=\mathrm{PDF}_{i,j,k}\times\frac{N_{i}}{\sum_{j,k}\mathrm{PDF}_{i,j,k}}~{},$
(26)
where $\mathrm{PDF}_{i,j,k}$ stands for the predicted PDF provided by the
collaboration as a function of recoil energy $i$, F90,j and time $k$. To do
this, we use their nominal prediction in absence of systematics.
Finally, we include systematic uncertainties by adding nuisance parameters as
prescribed in the data release.444We have checked that the impact of the
quenching factor uncertainties is negligible at the level of the uncertainties
quoted in the data release. Therefore, they are not included here. For each
nuisance parameter, we add a pull term to either the signal or background
prediction as
$n_{i,j,k}=\bar{n}_{i,j,k}\,(1+\xi\,\sigma_{i,j,k})~{},$ (27)
where $\bar{n}$ is the predicted signal or background event rates with no
systematic errors, $\xi$ is the nuisance parameter, and $\sigma_{i,j,k}$ is
obtained from the data release. We also add a signal normalization uncertainty
as
$s_{i,j,k}=\bar{s}_{i,j,k}\,(1+\xi_{\mathrm{norm}}\,\sigma_{\mathrm{norm}})~{},$
(28)
where we set $\sigma_{\mathrm{norm}}=0.1$ according to the data release (this
corresponds to the “neutrino flux” uncertainty listed in table 1 in ref.
COHERENT:2020iec ).
We use a Poissonian $\chi^{2}$ for statistical uncertainties,
$\left(\chi^{2}_{\rm
COH}\right)_{\mathrm{stat}}=\sum_{i,j,k}2\left(P_{i,j,k}-D_{i,j,k}+D_{i,j,k}\ln\frac{D_{i,j,k}}{P_{i,j,k}}\right)~{},$
(29)
where $D$ stands for the data and $P$ for the prediction (including the effect
of the nuisance parameters). A pull term is then added for each nuisance
parameter $\xi_{r}$, as well as for the normalization of the background
components,
$\chi_{\rm COH}^{2}\left({\boldsymbol{\xi}}\right)=\left(\chi^{2}_{\rm
COH}\right)_{\mathrm{stat}}+\sum_{r}\xi_{r}^{2}+\left(\frac{n_{\rm
pr}-\bar{n}_{\rm pr}}{\sigma_{\rm pr}}\right)^{2}+\left(\frac{n_{\rm
del}-\bar{n}_{\rm del}}{\sigma_{\rm del}}\right)^{2}+\left(\frac{n_{\rm
ss}-\bar{n}_{\rm ss}}{\sigma_{\rm ss}}\right)^{2}~{},$ (30)
where $\bar{n}$ and $\sigma$ are the central values and the uncertainties at
$1\sigma$ provided in the data release for the different background
components. The final $\left(\chi^{2}_{\rm COH}\right)_{\mathrm{p}}$ is
obtained after minimization over all nuisance parameters and the normalization
of the three background components.
## 4 Results
In this section we present the results of the analysis of the Dresden-II
reactor experiment data and its combination with COHERENT CsI and Ar data, for
the BSM frameworks presented in section 2.
### 4.1 Bounds on non-standard neutrino interactions
As described in section 2.1, in presence of NSI the Dresden-II reactor
experiment is sensitive to a unique combination of the $\varepsilon$
coefficients, $\mathcal{Q}^{\rm Ge}_{e}(\boldsymbol{\varepsilon})$, defined in
eq. (5), where we have introduced an explicit superindex indicating the
nucleus it refers to. We show in the left panel of figure 2 the dependence of
$\Delta\chi^{2}_{\rm D-II}$ on this effective NSI combination. In constructing
this $\Delta\chi^{2}_{\rm D-II}$, we have profiled over the background model
parameters for each value of $\mathcal{Q}^{\rm
Ge}_{e}(\boldsymbol{\varepsilon})^{2}$ and we have neglected small effects of
any additional systematic nuisance parameter. As mentioned above, the signal
acceptance uncertainties are included in the data provided by the experiment.
Figure 2: Left panel: Dependence of the profiled $\Delta\chi^{2}_{\rm D-II}$
on the combination of NSI coefficients relevant to CE$\nu$NS of
$\bar{\nu}_{e}$ scattering off the Ge detector of the Dresden-II experiment.
Right panel: Allowed regions at 90% CL (1 dof, two-sided,
$\Delta\chi^{2}=2.71$) in the $(\varepsilon_{ee}^{u},\varepsilon_{ee}^{d})$
plane. In both panels, the results are shown for two quenching factors,
denoted by Fef (in blue), using iron-filtered monochromatic neutrons, and YBe
(in red), based on photoneutron source measurements Collar:2021fcl . Although
not obvious, note that the allowed regions for the two quenching factors have
one common side, so the Fef constraints are more stringent.
From figure 2, we read the following ranges allowed at 90% confidence level
(CL) (1 dof)
$\left(g_{e}^{\rm Ge}\right)^{2}\equiv\left(\frac{\mathcal{Q}^{\rm
Ge}_{e}(\boldsymbol{\varepsilon})}{\mathcal{Q}^{\rm
Ge}_{e}(0)}\right)^{2}=0.91\pm 0.56\;\;(1.36\pm 0.97)~{},$ (31)
derived with the Fef (YBe) quenching factor for germanium. Notice that
$g_{e}^{\rm Ge}=1$ corresponds to the SM prediction and $g_{e}^{\rm Ge}=0$ to
no signal, so this implies that the absence of CE$\nu$NS in the Dresden-II
data is disfavored at 2.6$\sigma$ (2.3$\sigma$) for the analysis with the Fef
(YBe) quenching factor.
In the right panel of figure 2, we show the corresponding 90% CL allowed
regions for the flavor-diagonal NSI coefficients $\varepsilon_{ee}^{u}$ and
$\varepsilon_{ee}^{d}$ assuming vanishing non-diagonal NSI coefficients. The
shape of these bands can be understood directly from the expression of the
weak charge of the nucleus in eq. (4). They are defined as the two regions
around the points that satisfy
$\left[\mathcal{Q}_{W}+(2\,Z+N)\,\varepsilon_{ee}^{u}+(2\,N+Z)\,\varepsilon_{ee}^{d}\right]^{2}={\rm
constant}~{},$ (32)
which follow lines in the $(\varepsilon_{ee}^{u},\varepsilon_{ee}^{d})$ plane
with the slope given by $-(2Z+N)/(2N+Z)$. As seen in the figure, the analysis
employing the Fef quenching factor results in slightly stronger constraints.
In most scenarios we find that the analysis using the Fef quenching factor
reproduces slightly better the SM predictions and therefore leaves less room
for new physics. Exception, as we will see, is the case for some models with
light vector mediators for which a local non-standard minima appears in the
analysis with the YBe quenching factor, which results in slightly stronger
constraints (see section 4.4).
Figure 3: 90% CL allowed regions on flavor diagonal NSI with up-quarks (for
zero values of all other NSI coefficients) from the analysis of COHERENT CsI
and Ar data, the Dresden-II reactor data – with Fef (YBe) quenching factor in
left (right) panel –, and their combination. Note that, in the two-dimensional
panels, the results for the Dresden-II reactor experiment are obtained for 1
dof ($\Delta\chi^{2}=2.71$), while the rest of the regions are obtained for 2
dof ($\Delta\chi^{2}=4.61$).
CE$\nu$NS at the COHERENT experiment is sensitive to interactions of electron
and muon neutrinos and hence, it provides information on the corresponding
effective combinations $\mathcal{Q}^{\rm CsI}_{e}(\boldsymbol{\varepsilon})$,
$\mathcal{Q}^{\rm CsI}_{\mu}(\boldsymbol{\varepsilon})$, $\mathcal{Q}^{\rm
Ar}_{e}(\boldsymbol{\varepsilon})$, $\mathcal{Q}^{\rm
Ar}_{\mu}(\boldsymbol{\varepsilon})$. Generically, because both $\nu_{e}$ and
$\nu_{\mu}$ are present in the beam, degeneracies between NSI parameters
corresponding to $e$ and $\mu$ flavors appear. This is illustrated in figure
3, where we show the allowed regions obtained from our combined analysis of
CE$\nu$NS at COHERENT with both CsI and Ar targets for flavor-diagonal NSIs
with up-quarks only (the results for NSI with only down-quarks are similar).
The shape of these regions for a given nucleus leads to allowed regions
defined by a band around the points that approximately obey the equation of an
ellipse in the $(\varepsilon_{ee}^{u},\varepsilon_{\mu\mu}^{u})$ plane,
$\left[\mathcal{Q}_{W}+\left(2\,Z+N\right)\,\varepsilon_{ee}^{u}\right]^{2}+2\,\left[\mathcal{Q}_{W}+\left(2\,Z+N\right)\,\varepsilon_{\mu\mu}^{u}\right]^{2}={\rm
constant}~{}.$ (33)
Since $\mathcal{Q}_{W}$ depends on the target nucleus, the ellipse obtained is
different for different detector materials and, therefore, it may be broken by
adding information on different nuclei, provided they have a different ratio
of protons to neutrons (for earlier discussions on this, see, e.g., refs.
Scholberg:2005qs ; Barranco:2005yy ; Coloma:2017egw ; Baxter:2019mcx ). Most
importantly, the use of timing information at COHERENT translates into a
partial discrimination between the weak charges for the different flavors,
thanks to the distinct composition of the prompt ($\nu_{\mu}$) and delayed
($\bar{\nu}_{\mu}$ and $\nu_{e}$) neutrino flux. This leads to a partial
breaking of this degeneracy which, however, is not complete (see, e.g., ref.
Coloma:2019mbs for a more detailed explanation of this effect). This is what
explains the results shown in figure 3. The regions allowed by COHERENT data
present a double-wedge shape due to the partial breaking of the degeneracy
between those two parameters after the inclusion of the energy and, in
particular, the timing information.555For concreteness, the COHERENT analysis
shown in this plot was performed using the quenching factor from the Chicago
group Collar:2019ihs . See ref. Coloma:2019mbs for a discussion about the
small variation of the results obtained with other quenching factors. But yet,
a continuous wide range of values of $\varepsilon^{u}_{ee}$ remains allowed.
For the considered flavor-diagonal NSI with up-quarks only, CE$\nu$NS at the
Dresden-II reactor experiment provides information on $\varepsilon^{u}_{ee}$
only, so the allowed regions correspond to the vertical bands in the figure.
Consequently, the combined analysis of COHERENT + Dresden-II results in a
substantial reduction of the allowed values of $\varepsilon^{u}_{ee}$ (and
indirectly, also on $\varepsilon^{u}_{\mu\mu}$) as seen in the figure.
We finish by commenting that the results we show correspond to the case of
diagonal NSI with up-quarks only, but as mentioned above, similar results are
obtained for diagonal NSI with down-quarks only. We also notice that the
inclusion of flavor off-diagonal NSI couplings results in the enlargement of
the regions shown in figure 3. Nevertheless, within the current constraints
from other experiments, in particular from neutrino oscillations
Esteban:2018ppq ; Coloma:2019mbs , similar qualitative conclusions hold. For
NSI couplings to both up and down quarks, the complementarity between COHERENT
and Dresden-II data depends on the specific assumption about the ratio of
diagonal couplings to both quarks. If they are varied freely, meaningful
constraints cannot be obtained, as can be understood from figure 2. Therefore,
current data from these experiments is not enough by itself to impose
meaningful constraints on the complete parameter space of NSI with quarks, if
considered in full generality.
### 4.2 Bounds on neutrino magnetic moments
Figure 4: Left panel: Profiled $\Delta\chi^{2}_{\rm D-II}$ including events
induced by a magnetic moment for $\bar{\nu}_{e}$. We show results including
only $\mu_{\nu_{e}}$-induced scattering off electrons (dashed curves) and for
scattering off electrons and nucleons (solid curves). In both cases the SM
CE$\nu$NS contribution is included. Right panel: 90% CL excluded (one-sided)
regions from the combination of Dresden-II and COHERENT data (in color). We
also indicate (with arrows) the 90% CL (one-sided) allowed regions from
Dresden-II data with the Fef (blue dotted line) and YBe (red dotted line)
quenching factor, and from the combined analysis of COHERENT CsI and Ar data
(black dashed line). Notice that the vertical lines for the Dresden-II reactor
experiment are defined one-sided for 1 dof $(\Delta\chi^{2}=1.64$), while the
rest of the regions are defined for 2 dof ($\Delta\chi^{2}=3.22$).
The results of our analysis of data from the Dresden-II reactor experiment,
including the contribution induced by a neutrino magnetic moment, are shown in
figure 4 where, on the left panel, we plot the one-dimensional
$\Delta\chi^{2}_{\rm D-II}$ as a function $\mu_{\nu_{e}}$ after profiling over
all background parameters ${\boldsymbol{\beta}}$. As previously, we have
neglected small effects of any additional systematic nuisance parameter, while
the signal acceptance uncertainties are included in the data provided by the
experiment. For comparison, in the left panel we also show
$\Delta\chi^{2}_{\rm D-II}$ for the case of scattering off electrons only,
including the SM CE$\nu$NS contribution (dashed lines). As discussed in
section 3.1, the contribution from ES induced by a neutrino magnetic moment is
subdominant to that from CE$\nu$NS. One must notice, however, that a better
bound on $\mu_{\nu_{e}}$ (and similarly for models with light scalar
mediators, see below) could be attainable with a dedicated analysis aimed at
optimizing the sensitivity to the signal from scattering of electrons. The
signal acceptance at the lowest energies in the Dresden-II experiment is quite
small Colaresi2022suggestive , which significantly reduces statistics.
Furthermore, given the much flatter shape of the event spectrum for the
magnetic moment contribution than from the SM, as depicted in figure 1,
extending the ROI to higher energies could enhance the signal-to-noise ratio
(see, e.g., ref. Bonet:2022imz ).
The fit shows no evidence of a non-zero magnetic moment and therefore, the
analysis results in a bound which, at 90% CL (1 dof), reads
$\left|\mu_{\nu_{e}}\right|<\left\\{\begin{array}[]{c}1.9\;(2.9)\times
10^{-10}\;\mu_{B}\quad\textrm{(one-sided~{}limit)}\\\\[6.45831pt]
2.2\;(3.3)\times 10^{-10}\;\mu_{B}\quad\textrm{(two-
sided~{}limit)}\end{array}\right.~{},$ (34)
for the Fef (YBe) quenching factor for germanium. This is an order of
magnitude weaker than the current best limit from reactor antineutrinos
Beda:2012zz ; Beda:2013mta . Notice that here we report two different limits
according to different statistical criteria to derive constraints. The reason
is that the experiment is in fact sensitive to $|\mu_{\nu_{e}}|^{2}$, which
can only take non-negative values. Therefore, accounting for the physical
boundary, it is possible to report the limit on $|\mu_{\nu_{e}}|$ as a one-
sided limit, which at 90% CL for 1 dof (2 dof) corresponds to
$\Delta\chi^{2}=1.64\;(3.22)$. Conversely, if this restriction is not imposed,
the result obtained is what is denoted as a two-sided limit, which at 90%CL
for 1 dof (2 dof) corresponds to $\Delta\chi^{2}=2.71\;(4.61)$ and results
into less tight constraints. For the sake of comparison with different results
in the literature, we indicate in eq. (34) bounds obtained with both criteria.
As seen in eq. (34), the difference is at the level of 10%.
As mentioned in section 3.1 the $\mu_{\nu}$-induced cross sections in eqs. (7)
and (8) diverge as the recoil energy goes to zero and this unphysical behavior
is cut-off by the physical requirement of the average energy required to
produce an electron-hole pair, $E_{I,\rm min}=3$ eV. This raises the issue of
the possible dependence of the bounds on the exact value of this minimum
energy that could trigger the detector. The dependence on the cut energy,
however, is approximately logarithmic and we have verified that increasing it
by as much as one order of magnitude results in weakening the bounds in eq.
(34) by less than 25%.
CE$\nu$NS at the COHERENT experiment provides information on magnetic moments
for both $\nu_{e}$ and $\nu_{\mu}$. In the right panel of figure 4, we show
their 90% CL (2 dof, one-sided) excluded values, obtained with our combined
analysis of CE$\nu$NS at COHERENT with both CsI and Ar targets. Because there
is no interference between the contributions from $\mu_{\nu_{\mu}}$ and
$\mu_{\nu_{e}}$, the resulting allowed region is just a square with a rounded
upper-right corner. The 90% CL (1 dof, one-sided) upper bound from the
Dresden-II reactor experiment on $\mu_{\nu_{e}}$, eq. (34), is indicated by
vertical lines. As seen in the figure, the sensitivity of COHERENT to
$\mu_{\nu_{e}}$ is ${\cal O}(10)$ weaker. The combination of the two
experiments results in the 90% CL excluded regions (2 dof, one-sided) shown in
the figure in color. The corresponding combined bounds on each neutrino
magnetic moment (after profiling over the other) at 90% CL (1 dof) are
$\displaystyle\left|\mu_{\nu_{e}}\right|$ $\displaystyle<$
$\displaystyle\left\\{\begin{array}[]{c}1.8\;(2.8)\times
10^{-10}\;\mu_{B}\quad\textrm{(one-sided~{}limit)}\\\\[6.45831pt]
2.2\;(3.2)\times 10^{-10}\;\mu_{B}\quad\textrm{(two-
sided~{}limit)}\end{array}\right.~{},$ (37)
$\displaystyle\left|\mu_{\nu_{\mu}}\right|$ $\displaystyle<$
$\displaystyle\left\\{\begin{array}[]{c}2.4\;(2.4)\times
10^{-9}\;\mu_{B}\quad\;\,\textrm{(one-sided~{}limit)}\\\\[6.45831pt]
2.7\;(2.7)\times 10^{-9}\;\mu_{B}\quad\;\textrm{(two-
sided~{}limit)}\end{array}\right.~{},$ (40)
for the combination of data from the COHERENT and the Dresden-II reactor
experiments performed with the Fef (YBe) quenching factor for germanium.
### 4.3 Bounds on light scalar mediator models
Figure 5: Left panel: 90% CL (2 dof, two-sided, $\Delta\chi^{2}=4.61$)
excluded regions for models with light scalar mediators coupled universally to
all relevant fermions. Right panel: 90% CL (2 dof, two-sided,
$\Delta\chi^{2}=4.61$) excluded regions for models with light scalar mediators
coupled only to leptons. The thick blue and red dotted lines show the boundary
of the region excluded by the Dresden-II reactor experiment with the Fef and
YBe quenching factors, respectively. The black dashed line shows the boundary
of region excluded from the analysis of COHERENT CsI and Ar data. The filled
regions are those excluded by the combined COHERENT+Dresden-II analysis.
The results of the analysis of the data from the Dresden-II reactor experiment
and its combination with COHERENT, including the contribution of the events
induced by a new light scalar mediator, are shown in figure 5. We depict the
excluded region at 90% CL (2 dof, two-sided) for the parameter space of the
scalar models $\left(\left|g^{\rm mod}_{\phi}\right|,M_{\phi}\right)$, for the
two models with charges listed in table 2. On the left panel, we show the
results for a scalar coupling universally coupled to all relevant fermions,
while on the right panel we focus on a scalar which only interacts with
leptons.
The thick blue and red thick dotted lines in figure 5 show the boundaries of
the regions excluded by the Dresden-II reactor data with with the Fef and YBe
quenching factor, respectively. These results are obtained after profiling
over the background model parameters (i.e., for each value of
$\left(\left|g^{\rm mod}_{\phi}\right|,M_{\phi}\right)$). For light mediator
masses, the experiment has no sensitivity to the mediator mass and the limit
of the regions approaches a horizontal line of constant $\left|g^{\rm
mod}_{\phi}\right|$. For these effectively massless scalar mediators it is
possible to derive an upper bound on the coupling constant regardless the
mediator mass, which for the considered models, reads at 90% CL (1 dof),
$\displaystyle\left|g_{\phi}^{\rm univ}\right|$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{c}1.6\;(1.9)\times
10^{-6}\quad\textrm{(one-sided)}\\\\[6.45831pt] 1.7\;(2.0)\times
10^{-6}\quad\textrm{(two-sided)}\end{array}\right.~{},$ (43)
$\displaystyle\left|g_{\phi}^{\ell}\right|$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{c}4.9\;(5.5)\times
10^{-6}\quad\textrm{(one-sided)}\\\\[6.45831pt] 5.5\;(6.1)\times
10^{-6}\quad\textrm{(two-sided)}\end{array}\right.~{},$ (46)
for the Fef (YBe) quenching factor for germanium. As done above, we indicate
both, one-sided and two-sided (1 dof) bounds.
Conversely, for larger mediator masses, the boundary is a diagonal,
characteristic of the contact-interaction limit. In that case, the event rates
depend on $(g_{\phi}/M_{\phi})^{4}$, which we find to be well approximated by
$\displaystyle\frac{\left|g_{\phi}^{\rm univ}\right|}{M_{\phi}/{\rm MeV}}$
$\displaystyle\gtrsim$ $\displaystyle 3.4\;(3.6)\times 10^{-7}~{},\quad{\rm
for\;}M_{\phi}\gtrsim 10\;{\rm MeV}~{},$
$\displaystyle\frac{\left|g_{\phi}^{\ell}\right|}{M_{\phi}/{\rm MeV}}$
$\displaystyle\gtrsim$ $\displaystyle 1.1\;(1.1)\times 10^{-4}~{},\quad{\rm
for\;}M_{\phi}\gtrsim 0.1\;{\rm MeV}~{},$ (47)
for the Fef (YBe) quenching factor for germanium.
In summary, we find that for sufficiently light mediators ($M_{\phi}\lesssim
10^{-2}~{}\mathrm{MeV}$), the bound on the coupling constant is about a factor
${\cal O}(3)$ stronger if the interaction is coupled to quarks than if it
couples only to leptons. This is so because scalar-mediated CE$\nu$NS always
dominates over the corresponding scalar-mediated incoherent scattering off
electrons for the low-energy part of the spectrum, where statistics is best
(see figure 1). Equivalently, in the contact-interaction limit, the bound on
$\left|g^{\rm univ}_{\phi}\right|/M_{\phi}$ is more than two orders of
magnitude better than on $\left|g^{\ell}_{\phi}\right|/M_{\phi}$.
In figure 5, we also show the boundary of the regions excluded from the
analysis of COHERENT CsI and Ar data (black dashed lines). For most of the
parameter space we consider, the bounds imposed by the Dresden-II reactor
experiment dominate in either model. COHERENT bounds only become competitive
for the universal model for $\left|g_{\phi}^{\rm univ}\right|\gtrsim 5\times
10^{-5}$. Consequently, the results in eqs. (46) and (47) hold to good
approximation for the combination of COHERENT and Dresden-II reactor
experiments.
### 4.4 Bounds on light vector mediator models
The results of the analysis of the data from the Dresden-II reactor experiment
and its combination with COHERENT, including the contribution of the events
induced by a new light vector mediator, are shown in figure 6. The figure
displays the regions excluded at 90% CL (2 dof, two-sided) corresponding to
the parameter space of vector mediator models $\left(\left|g^{\rm
mod}_{Z^{\prime}}\right|,M_{Z^{\prime}}\right)$, for the three cases with
charges listed in table 2.
Figure 6: Upper left panel: 90% CL (2 dof, two-sided, $\Delta\chi^{2}=4.61$)
excluded regions for models with light vector mediators coupled universally to
all relevant fermions. Upper right panel: 90% CL (2 dof, two-sided,
$\Delta\chi^{2}=4.61$) excluded regions for models with light vector mediators
coupled only to $L_{e}$. Lower panel: 90% CL (2 dof, two-sided,
$\Delta\chi^{2}=4.61$) excluded regions for models with light vector mediators
coupled only to $B-L$. The thick blue and red dotted lines show the boundary
of the region excluded by the Dresden-II reactor experiment with the Fef and
YBe quenching factors, respectively. The black dashed line shows the boundary
of region excluded from the analysis of COHERENT CsI and Ar data. The filled
regions are those excluded by the combined COHERENT+Dresden-II analysis.
As above, the thick blue and red thick dotted lines in the figures indicate
the boundaries of the regions excluded at 90% CL by the Dresden-II reactor
data with the Fef and YBe quenching factor, respectively, after profiling over
the background model parameters. The black dashed lines indicate the boundary
of the regions excluded from the analysis of COHERENT CsI and Ar data and the
filled regions correspond to the results of the COHERENT + Dresden-II
combination. We see that COHERENT bounds only give some contribution for the
universal and $B-L$ models for $\left\\{\left|g_{Z^{\prime}}^{\rm
univ}\right|,\left|g_{Z^{\prime}}^{B-L}\right|\right\\}\gtrsim 5\times
10^{-4}$.
Again, we find that for sufficiently light mediator masses ($M_{\phi}\lesssim
10^{-2}~{}\mathrm{MeV}$) the region becomes independent of the mediator mass.
For these effectively massless vector mediators, the corresponding 90% CL (1
dof) upper bounds on the coupling constant read
$\displaystyle\left|g_{Z^{\prime}}^{\rm univ}\right|$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{c}0.87\;(1.0)\times
10^{-6}\quad\textrm{(one-sided)}\\\\[6.45831pt] 0.92\;(1.1)\times
10^{-6}\quad\textrm{(two-sided)}\end{array}\right.~{},$
$\displaystyle\left|g_{Z^{\prime}}^{L_{e}}\right|$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{c}0.90\;(1.0)\times
10^{-6}\quad\textrm{(one-sided)}\\\\[6.45831pt] 0.95\;(1.1)\times
10^{-6}\quad\textrm{(two-sided)}\end{array}\right.~{},$ (52)
$\displaystyle\left|g_{Z^{\prime}}^{B-L}\right|$ $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{c}0.90\;(1.1)\times
10^{-6}\quad\textrm{(one-sided)}\\\\[6.45831pt] 0.95\;(1.1)\times
10^{-6}\quad\textrm{(two-sided)}\end{array}\right.~{},$
for the Fef (YBe) quenching factor for germanium. As done above, we indicate
both, one-sided and two-sided (1 dof) bounds.
Unlike the case of scalar mediators, in the small-mass limit, the bounds for
vector mediators are very similar regardless of whether quarks are charged
under the new interaction. This is so because in this limit the $Z^{\prime}$
contribution to the event rates is dominated by the incoherent scattering off
electrons, as seen in figure 1. We also notice that for all models, the
regions obtained with the YBe quenching factor present a dip around
$M_{Z^{\prime}}\sim 0.03$ MeV and $\left|g_{Z^{\prime}}\right|\sim 10^{-6}$.
This arises because the inclusion of scattering off electrons mediated by such
a particle provides a fit which is better than the SM one at $\sim$
1.5$\sigma$. Comparing the two upper panels, we also see that constraints on
the universal model become more stringent than those on the $L_{e}$ model for
$M_{Z^{\prime}}\gtrsim 0.2$ MeV. For these masses, CE$\nu$NS starts dominating
over scattering off electrons.
For sufficiently large mediator masses, the contact-interaction approximation
is recovered and the event rates vary with a power of
$\left|g_{Z^{\prime}}\right|/M_{Z^{\prime}}$, which ranges from a power of
two, due the interference with the SM, to a power of four, when the new vector
contribution is the dominant one (the interplay between the interference and
quadratic contributions is responsible for the disconnected island observed in
the upper right excluded region of the $B-L$ model). From the figure, we read
that, in this regime, the boundaries of the 90% CL excluded regions for the
Dresden-II reactor experiment are approximately described by
$\displaystyle\frac{\left|g_{Z^{\prime}}^{\rm
univ}\right|}{M_{Z^{\prime}}/{\rm MeV}}$ $\displaystyle\gtrsim$ $\displaystyle
7.5\;(9.0)\times 10^{-7}~{},\quad{\rm for\;}M_{Z^{\prime}}\gtrsim 10\;{\rm
MeV}~{},$
$\displaystyle\frac{\left|g_{Z^{\prime}}^{L_{e}}\right|}{M_{Z^{\prime}}/{\rm
MeV}}$ $\displaystyle\gtrsim$ $\displaystyle 1.7\;(2.9)\times
10^{-5}~{},\quad{\rm for\;}M_{Z^{\prime}}\gtrsim 0.1\;{\rm MeV}~{},$ (55)
$\displaystyle\frac{\left|g_{Z^{\prime}}^{B-L}\right|}{M_{Z^{\prime}}/{\rm
MeV}}$ $\displaystyle\gtrsim$ $\displaystyle\quad\;\;(6.5)\times
10^{-7}~{},\quad{\rm for\;}M_{Z^{\prime}}\gtrsim 10\;{\rm MeV}~{},$
for the Fef (YBe) quenching factor for germanium. For the $B-L$ model, the
interplay between scattering off electrons and nucleus and between the
interference and quadratic pieces makes the region obtained with the Fef
quenching factor not well described by this approximation. Also, as mentioned
above, for the universal and $B-L$ models, the combination with COHERENT
slightly extends the large-mass part of the excluded region beyond these
values as seen in the figure.
## 5 Summary and conclusions
The first evidence of observation of CE$\nu$NS with reactor electron
antineutrinos has been recently reported by an experiment conducted at the
Dresden-II reactor site, using the NCC-1701 germanium detector
Colaresi2022suggestive . This adds to the previous measurements performed by
the COHERENT experiment, which uses neutrinos from a spallation neutron source
and has observed CE$\nu$NS using both CsI[Na] and Ar nuclei COHERENT:2017ipa ;
COHERENT:2018imc ; COHERENT:2020iec ; COHERENT:2020ybo .
The very low momentum transfer produced by CE$\nu$NS renders this process an
excellent probe of new interactions in the neutrino sector. In this paper, we
have performed a detailed analysis of the new data from the Dresden-II
experiment, with the aim of deriving powerful constraints on a variety of BSM
scenarios. In particular, we have derived new bounds on neutrino NSI with
quarks, magnetic moments for electron and muon neutrinos, and several models
with light neutral scalar or vector mediators. Our analysis includes the
contributions to the event rates from CE$\nu$NS and from elastic scattering
off electrons.
In our analysis of the Dresden-II data, we have taken special care in
profiling over the parameters required to accurately model the background,
closely following the data release of the experimental collaboration
Colaresi2022suggestive . We have also quantified the dependence of the results
on the quenching factor by considering two models: one based on the use iron-
filtered monochromatic neutrons (Fef) and another one based on photoneutron
source measurements (YBe). The impact on the results of the uncertainty on
this parameter is approximately indicated by the spread between the bounds
obtained for these two models. As for COHERENT, our analysis includes both
timing and energy information in the case of CsI, while in the case of the Ar
data, we perform a $\chi^{2}$ analysis using time, energy and F90 information.
A careful treatment of systematic uncertainties is performed, following the
prescriptions provided by the collaboration in refs. COHERENT:2018imc ;
COHERENT:2020ybo .
We have also quantified the impact of the combination of the new data from the
Dresden-II reactor experiment with COHERENT data. From the phenomenological
point of view, the main difference between these two experiments arises from
the different flavor composition of the neutrino flux: while a nuclear reactor
only emits $\bar{\nu}_{e}$, at a spallation neutron source neutrinos are
primarily produced from pion decay at rest and therefore, the flux contains an
equal admixture of $\nu_{\mu}$, $\bar{\nu}_{\mu}$, and $\nu_{e}$, with higher
energies than the reactor $\bar{\nu}_{e}$ flux. Therefore, generically, for
models with lepton-flavor dependent effects (such as NSI), COHERENT is a
priori sensitive to a larger number of parameters. Nevertheless, the flavor
discrimination with COHERENT comes from the timing information and is only
partial. Thus, within the current experimental precision, this results in
degeneracies among the $\mu$-flavor and $e$-flavor parameters. The new
Dresden-II data bring in complementary information in this respect, as they
provide independent constraints on the $e$-flavor parameters. As a result, the
combination of both experiments allows breaking (or at least alleviating)
degeneracies present in the case of NSI, as illustrated in figure 3. Moreover,
the combination of data obtained for scattering off different nuclei (Ar and
CsI for COHERENT, Ge for the Dresden-II experiment) also adds additional
synergies since, for a given set of NSI parameters, the impact on the weak
charge is different depending on the particular nucleus under consideration
(see eq. (4)).
In the case of neutrino magnetic moment and light mediators with masses below
${\cal O}(100\;{\rm MeV})$, we generically find that the Dresden-II experiment
outperforms the bounds obtained for COHERENT (an obvious exception is the
magnetic moment of muon neutrinos, which is only constrained by COHERENT,
since for the Dresden-II experiment only a $\bar{\nu}_{e}$ flux is available).
This is a priori expected, since the Dresden-II experiment is sensitive to
much lower values of the momentum transfer, a critical feature which drives
the experimental reach of these scenarios. On the other hand, for mediators
with masses above ${\cal O}(100\;{\rm MeV})$, we find that COHERENT and
Dresden-II lead to similar bounds for the universal and $B-L$ scenarios, while
if the new interaction is coupled only to leptons, the Dresden-II bound is
more restrictive.
We finish by commenting on the role of ES. While the event rates from this
process are insignificant in the SM, their contribution could be significantly
enhanced in the presence of new physics effects. Most notably, such
contribution could pass the signal selection cuts for both the Dresden-II
experiment and the COHERENT CsI detector. In fact, for the analysis of
COHERENT CsI, this is a novel point of our analysis with respect to previous
works in the literature. Our results show that the inclusion of ES is
particularly relevant in scenarios with light vector mediators. In the case of
COHERENT CsI, this improves the bound on the coupling obtained from CE$\nu$NS
alone for the $B-L$ and universal models by at least a factor $\sim 5$, for
mediator masses below $\sim 20$ MeV. And a similar improvement is obtained on
the bounds derived from the Dresden-II reactor data when taking ES into
account. Conversely, in the reconstructed energy range of the Dresden-II
experiment, the contribution of ES induced by a neutrino magnetic moment and
by light scalar mediators is smaller than that from CE$\nu$NS. At higher
energies, however, ES would also dominate over CE$\nu$NS at the Dresden-II
reactor experiment. Therefore, extending the reconstructed energy region to
higher energies could enhance the reach of the Dresden-II experiment within
these BSM scenarios.
###### Acknowledgements.
We warmly thank Juan Collar for all the help with the new Dresden-II data
release. We also thank Nicola Cargioli for noticing a plotting bug in figure
1. This project has received funding/support from the European Union’s Horizon
2020 research and innovation program under the Marie Skłodowska-Curie grant
agreement No 860881-HIDDeN, and from grants RTI2018-095979,
PID2019-108892RB-I00, PID2019-105614GB-C21, PID2020-113334GB-I00,
PID2020-113644GB-I00, CEX2019-000918-M and CEX2020-001007-S, funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. PC is
also supported by Grant RYC2018-024240-I funded by MCIN/AEI/
10.13039/501100011033 and by “ESF Investing in your Investing in your future”.
M.C.G-G is also supported by U.S.A.-NSF grant PHY-1915093, and by AGAUR
(Generalitat de Catalunya) grant 2017-SGR-929. LL is supported by the
predoctoral training program non-doctoral research personnel of the Department
of Education of the Basque Government. LL and FM are also supported by the
European Research Council (ERC) under Grant Agreement No. 101039048-GanESS and
the Severo Ochoa Program grant CEX2018-000867-S. SPR is also partially
supported by the Portuguese FCT (CERN/FIS-PAR/0004/2019).
## Appendix A Binding energies for neutrino scattering off electrons
For germanium we use x-ray
$Z_{\rm eff}^{\rm Ge}(T_{e})=\left\\{\begin{array}[]{lcl}32&&T_{e}>11.11\;{\rm
keV}\\\ 30&&11.11\;{\rm keV}>T_{e}>1.4146\;{\rm keV}\\\ 28&&1.4146\;{\rm
keV}>T_{e}>1.248\;{\rm keV}\\\ 26&&1.248\;{\rm keV}>T_{e}>1.217\;{\rm keV}\\\
22&&1.217\;{\rm keV}>T_{e}>0.1801\;{\rm keV}\\\ 20&&0.1801\;{\rm
keV}>T_{e}>0.1249\;{\rm keV}\\\ 18&&0.1249\;{\rm keV}>T_{e}>0.1208\;{\rm
keV}\\\ 14&&0.1208\;{\rm keV}>T_{e}>0.0298\;{\rm keV}\\\ 4&&0.0298\;{\rm
keV}>T_{e}\end{array}\right.$ (56)
while for caesium and for iodine we use x-ray
$\begin{array}[]{cc}Z^{\mathrm{Cs}}_{\rm
eff}(T_{e})=\left\\{\begin{array}[]{lcl}55&&T_{e}>35.99\;{\rm keV}\\\
53&&35.99\;{\rm keV}>T_{e}>5.71\;{\rm keV}\\\ 51&&5.71\;{\rm
keV}>T_{e}>5.36\;{\rm keV}\\\ 49&&5.36\;{\rm keV}>T_{e}>5.01\;{\rm keV}\\\
45&&5.01\;{\rm keV}>T_{e}>1.21\;{\rm keV}\\\ 43&&1.21\;{\rm
keV}>T_{e}>1.07\;{\rm keV}\\\ 41&&1.07\;{\rm keV}>T_{e}>1\;{\rm keV}\\\
37&&1\;{\rm keV}>T_{e}>0.74\;{\rm keV}\\\ 33&&0.74\;{\rm keV}>T_{e}>0.73\;{\rm
keV}\\\ 27&&0.73\;{\rm keV}>T_{e}>0.23\;{\rm keV}\\\ 25&&0.23\;{\rm
keV}>T_{e}>0.17\;{\rm keV}\\\ 23&&0.17\;{\rm keV}>T_{e}>0.16\;{\rm keV}\\\
19&&0.16\;{\rm keV}>T_{e}\end{array}\right.&Z^{\mathrm{I}}_{\rm
eff}(T_{e})=\left\\{\begin{array}[]{lcl}53&&T_{e}>33.17\;{\rm keV}\\\
51&&33.17\;{\rm keV}>T_{e}>5.19\;{\rm keV}\\\ 49&&5.19\;{\rm
keV}>T_{e}>4.86\;{\rm keV}\\\ 47&&4.86\;{\rm keV}>T_{e}>4.56\;{\rm keV}\\\
43&&4.56\;{\rm keV}>T_{e}>1.07\;{\rm keV}\\\ 41&&1.07\;{\rm
keV}>T_{e}>0.93\;{\rm keV}\\\ 39&&0.93\;{\rm keV}>T_{e}>0.88\;{\rm keV}\\\
35&&0.88\;{\rm keV}>T_{e}>0.63\;{\rm keV}\\\ 31&&0.63\;{\rm
keV}>T_{e}>0.62\;{\rm keV}\\\ 25&&0.62\;{\rm keV}>T_{e}>0.19\;{\rm keV}\\\
23&&0.19\;{\rm keV}>T_{e}>0.124\;{\rm keV}\\\ 21&&0.124\;{\rm
keV}>T_{e}>0.123\;{\rm keV}\\\ 17&&0.123\;{\rm
keV}>T_{e}\end{array}\right.\end{array}$ (57)
noting that $Z_{\rm eff}^{\mathrm{CsI}}(T_{e})=\frac{1}{2}\left[Z_{\rm
eff}^{\mathrm{Cs}}(T_{e})+Z_{\rm eff}^{\mathrm{I}}(T_{e})\right]$.
## References
* (1) D. Z. Freedman, _Coherent effects of a weak neutral current_ , _Physical Review D_ 9 (1974) 1389.
* (2) COHERENT collaboration, D. Akimov et al., _Observation of coherent elastic neutrino-nucleus scattering_ , _Science_ 357 (2017) 1123 [1708.01294].
* (3) COHERENT collaboration, D. Akimov et al., _COHERENT Collaboration data release from the first observation of coherent elastic neutrino-nucleus scattering_ , 1804.09459.
* (4) COHERENT collaboration, D. Akimov et al., _First measurement of coherent elastic neutrino-nucleus scattering on Argon_ , _Phys. Rev. Lett._ 126 (2021) 012002 [2003.10630].
* (5) COHERENT collaboration, D. Akimov et al., _COHERENT Collaboration data release from the first detection of coherent elastic neutrino-nucleus scattering on Argon_ , 2006.12659.
* (6) TEXONO collaboration, H. T. Wong, _The TEXONO research program on neutrino and astroparticle physics_ , _Mod. Phys. Lett. A_ 19 (2004) 1207.
* (7) $\nu$GeN collaboration, V. Belov et al., _The $\nu$GeN experiment at the Kalinin nuclear power plant_, _JINST_ 10 (2015) P12011.
* (8) CONNIE collaboration, A. Aguilar-Arevalo et al., _The CONNIE experiment_ , _J. Phys. Conf. Ser._ 761 (2016) 012057 [1608.01565].
* (9) MINER collaboration, G. Agnolet et al., _Background studies for the MINER coherent neutrino scattering reactor experiment_ , _Nucl. Instrum. Meth. A_ 853 (2017) 53 [1609.02066].
* (10) Ricochet collaboration, J. Billard et al., _Coherent neutrino scattering with low temperature bolometers at Chooz reactor complex_ , _J. Phys. G_ 44 (2017) 105101 [1612.09035].
* (11) $\nu$-cleus collaboration, R. Strauss et al., _The $\nu$-cleus experiment: a gram-scale fiducial-volume cryogenic detector for the first detection of coherent neutrino-nucleus scattering_, _Eur. Phys. J. C_ 77 (2017) 506 [1704.04320].
* (12) RED-100 collaboration, D. Y. Akimov et al., _First ground-level laboratory test of the two-phase Xenon emission detector RED-100_ , _JINST_ 15 (2020) P02020 [1910.06190].
* (13) J. J. Choi, _Neutrino elastic-scattering observation with NaI[Tl] (NEON)_ , _PoS_ NuFact2019 (2020) 047.
* (14) CONUS collaboration, H. Bonet et al., _Constraints on elastic neutrino nucleus scattering in the fully coherent regime from the CONUS experiment_ , _Phys. Rev. Lett._ 126 (2021) 041804 [2011.00210].
* (15) J. Colaresi et al., _First results from a search for coherent elastic neutrino-nucleus scattering at a reactor site_ , _Phys. Rev. D_ 104 (2021) 072003 [2108.02880].
* (16) J. Colaresi, J. I. Collar, T. W. Hossbach, C. M. Lewis and K. M. Yocum, _Suggestive evidence for coherent elastic neutrino-nucleus scattering from reactor antineutrinos_ , 2202.09672.
* (17) B. C. Cañas, E. A. Garcés, O. G. Miranda and A. Parada, _Future perspectives for a weak mixing angle measurement in coherent elastic neutrino nucleus scattering experiments_ , _Phys. Lett. B_ 784 (2018) 159 [1806.01310].
* (18) M. Cadeddu and F. Dordei, _Reinterpreting the weak mixing angle from atomic parity violation in view of the Cs neutron rms radius measurement from COHERENT_ , _Phys. Rev. D_ 99 (2019) 033010 [1808.10202].
* (19) X.-R. Huang and L.-W. Chen, _Neutron skin in CsI and low-energy effective weak mixing angle from COHERENT data_ , _Phys. Rev. D_ 100 (2019) 071301 [1902.07625].
* (20) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li and Y. Y. Zhang, _Neutrino, electroweak, and nuclear physics from COHERENT elastic neutrino-nucleus scattering with refined quenching factor_ , _Phys. Rev. D_ 101 (2020) 033004 [1908.06045].
* (21) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, E. Picciau and Y. Y. Zhang, _Physics results from the first COHERENT observation of coherent elastic neutrino-nucleus scattering in argon and their combination with cesium-iodide data_ , _Phys. Rev. D_ 102 (2020) 015030 [2005.01645].
* (22) M. Cadeddu et al., _New insights into nuclear physics and weak mixing angle using electroweak probes_ , _Phys. Rev. C_ 104 (2021) 065502 [2102.06153].
* (23) M. Cadeddu, C. Giunti, Y. F. Li and Y. Y. Zhang, _Average CsI neutron density distribution from COHERENT data_ , _Phys. Rev. Lett._ 120 (2018) 072501 [1710.02730].
* (24) E. Ciuffoli, J. Evslin, Q. Fu and J. Tang, _Extracting nuclear form factors with coherent neutrino scattering_ , _Phys. Rev. D_ 97 (2018) 113003 [1801.02166].
* (25) D. K. Papoulias, T. S. Kosmas, R. Sahu, V. K. B. Kota and M. Hota, _Constraining nuclear physics parameters with current and future COHERENT data_ , _Phys. Lett. B_ 800 (2020) 135133 [1903.03722].
* (26) P. Coloma, I. Esteban, M. C. Gonzalez-Garcia and J. Menéndez, _Determining the nuclear neutron distribution from coherent elastic neutrino-nucleus scattering: current results and future prospects_ , _JHEP_ 08 (2020) 030 [2006.08624].
* (27) J. Barranco, O. G. Miranda and T. I. Rashba, _Probing new physics with coherent neutrino scattering off nuclei_ , _JHEP_ 12 (2005) 021 [hep-ph/0508299].
* (28) J. A. Formaggio, E. Figueroa-Feliciano and A. J. Anderson, _Sterile neutrinos, coherent scattering and oscillometry measurements with low-temperature bolometers_ , _Phys. Rev. D_ 85 (2012) 013009 [1107.3512].
* (29) A. J. Anderson et al., _Measuring active-to-sterile neutrino oscillations with neutral current coherent neutrino-nucleus scattering_ , _Phys. Rev. D_ 86 (2012) 013004 [1201.3805].
* (30) B. Dutta, Y. Gao, R. Mahapatra, N. Mirabolfathi, L. E. Strigari and J. W. Walker, _Sensitivity to oscillation with a sterile fourth generation neutrino from ultra-low threshold neutrino-nucleus coherent scattering_ , _Phys. Rev. D_ 94 (2016) 093002 [1511.02834].
* (31) D. G. Cerdeño, M. Fairbairn, T. Jubb, P. A. N. Machado, A. C. Vincent and C. Bœhm, _Physics from solar neutrinos in dark matter direct detection experiments_ , _JHEP_ 05 (2016) 118 [1604.01025].
* (32) J. B. Dent, B. Dutta, S. Liao, J. L. Newstead, L. E. Strigari and J. W. Walker, _Probing light mediators at ultralow threshold energies with coherent elastic neutrino-nucleus scattering_ , _Phys. Rev. D_ 96 (2017) 095007 [1612.06350].
* (33) P. Coloma, P. B. Denton, M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, _Curtailing the dark side in non-standard neutrino interactions_ , _JHEP_ 04 (2017) 116 [1701.04828].
* (34) T. S. Kosmas, D. K. Papoulias, M. Tórtola and J. W. F. Valle, _Probing light sterile neutrino signatures at reactor and spallation neutron source neutrino experiments_ , _Phys. Rev. D_ 96 (2017) 063013 [1703.00054].
* (35) S.-F. Ge and I. M. Shoemaker, _Constraining photon portal dark matter with TEXONO and COHERENT data_ , _JHEP_ 11 (2018) 066 [1710.10889].
* (36) I. M. Shoemaker, _COHERENT search strategy for beyond standard model neutrino interactions_ , _Phys. Rev. D_ 95 (2017) 115028 [1703.05774].
* (37) P. Coloma, M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, _COHERENT enlightenment of the neutrino dark side_ , _Phys. Rev. D_ 96 (2017) 115007 [1708.02899].
* (38) J. Liao and D. Marfatia, _COHERENT constraints on nonstandard neutrino interactions_ , _Phys. Lett. B_ 775 (2017) 54 [1708.04255].
* (39) B. C. Cañas, E. A. Garcés, O. G. Miranda and A. Parada, _The reactor antineutrino anomaly and low energy threshold neutrino experiments_ , _Phys. Lett. B_ 776 (2018) 451 [1708.09518].
* (40) J. B. Dent, B. Dutta, S. Liao, J. L. Newstead, L. E. Strigari and J. W. Walker, _Accelerator and reactor complementarity in coherent neutrino-nucleus scattering_ , _Phys. Rev. D_ 97 (2018) 035009 [1711.03521].
* (41) D. K. Papoulias and T. S. Kosmas, _COHERENT constraints to conventional and exotic neutrino physics_ , _Phys. Rev. D_ 97 (2018) 033003 [1711.09773].
* (42) Y. Farzan, M. Lindner, W. Rodejohann and X.-J. Xu, _Probing neutrino coupling to a light scalar with coherent neutrino scattering_ , _JHEP_ 05 (2018) 066 [1802.05171].
* (43) J. Billard, J. Johnston and B. J. Kavanagh, _Prospects for exploring new physics in coherent elastic neutrino-nucleus scattering_ , _JCAP_ 11 (2018) 016 [1805.01798].
* (44) P. Coloma, I. Esteban, M. C. Gonzalez-Garcia and M. Maltoni, _Improved global fit to non-standard neutrino interactions using COHERENT energy and timing data_ , _JHEP_ 02 (2020) 023 [1911.09109].
* (45) M. Chaves and T. Schwetz, _Resolving the LMA-dark NSI degeneracy with coherent neutrino-nucleus scattering_ , _JHEP_ 05 (2021) 042 [2102.11981].
* (46) D. Aristizabal Sierra, V. De Romeri and N. Rojas, _COHERENT analysis of neutrino generalized interactions_ , _Phys. Rev. D_ 98 (2018) 075018 [1806.07424].
* (47) V. Brdar, W. Rodejohann and X.-J. Xu, _Producing a new fermion in coherent elastic neutrino-nucleus scattering: from neutrino mass to dark matter_ , _JHEP_ 12 (2018) 024 [1810.03626].
* (48) M. Cadeddu, C. Giunti, K. A. Kouzakov, Y. F. Li, A. I. Studenikin and Y. Y. Zhang, _Neutrino charge radii from COHERENT elastic neutrino-nucleus scattering_ , _Phys. Rev. D_ 98 (2018) 113010 [1810.05606].
* (49) C. Blanco, D. Hooper and P. Machado, _Constraining sterile neutrino interpretations of the LSND and MiniBooNE anomalies with coherent neutrino scattering experiments_ , _Phys. Rev. D_ 101 (2020) 075051 [1901.08094].
* (50) B. Dutta, S. Liao, S. Sinha and L. E. Strigari, _Searching for beyond the Standard Model physics with COHERENT energy and timing data_ , _Phys. Rev. Lett._ 123 (2019) 061801 [1903.10666].
* (51) O. G. Miranda, D. K. Papoulias, M. Tórtola and J. W. F. Valle, _Probing neutrino transition magnetic moments with coherent elastic neutrino-nucleus scattering_ , _JHEP_ 07 (2019) 103 [1905.03750].
* (52) CONNIE collaboration, A. Aguilar-Arevalo et al., _Exploring low-energy neutrino physics with the Coherent Neutrino Nucleus Interaction Experiment_ , _Phys. Rev. D_ 100 (2019) 092005 [1906.02200].
* (53) B. Dutta, D. Kim, S. Liao, J.-C. Park, S. Shin and L. E. Strigari, _Dark matter signals from timing spectra at neutrino experiments_ , _Phys. Rev. Lett._ 124 (2020) 121802 [1906.10745].
* (54) D. K. Papoulias, _COHERENT constraints after the COHERENT-2020 quenching factor measurement_ , _Phys. Rev. D_ 102 (2020) 113004 [1907.11644].
* (55) A. N. Khan and W. Rodejohann, _New physics from COHERENT data with an improved quenching factor_ , _Phys. Rev. D_ 100 (2019) 113003 [1907.12444].
* (56) C. Giunti, _General COHERENT constraints on neutrino nonstandard interactions_ , _Phys. Rev. D_ 101 (2020) 035039 [1909.00466].
* (57) D. Baxter et al., _Coherent elastic neutrino-nucleus scattering at the European Spallation Source_ , _JHEP_ 02 (2020) 123 [1911.00762].
* (58) B. C. Cañas, E. A. Garcés, O. G. Miranda, A. Parada and G. Sánchez García, _Interplay between nonstandard and nuclear constraints in coherent elastic neutrino-nucleus scattering experiments_ , _Phys. Rev. D_ 101 (2020) 035012 [1911.09831].
* (59) O. G. Miranda, D. K. Papoulias, M. Tórtola and J. W. F. Valle, _Probing new neutral gauge bosons with CE $\nu$NS and neutrino-electron scattering_, _Phys. Rev. D_ 101 (2020) 073005 [2002.01482].
* (60) L. J. Flores, N. Nath and E. Peinado, _Non-standard neutrino interactions in U(1)’ model after COHERENT data_ , _JHEP_ 06 (2020) 045 [2002.12342].
* (61) O. G. Miranda, D. K. Papoulias, G. Sánchez García, O. Sanders, M. Tórtola and J. W. F. Valle, _Implications of the first detection of coherent elastic neutrino-nucleus scattering (CE $\nu$NS) with Liquid Argon_, _JHEP_ 05 (2020) 130 [2003.12050].
* (62) N. Hurtado, H. Mir, I. M. Shoemaker, E. Welch and J. Wyenberg, _Dark matter-neutrino interconversion at COHERENT, direct detection, and the early Universe_ , _Phys. Rev. D_ 102 (2020) 015006 [2005.13384].
* (63) O. G. Miranda, D. K. Papoulias, O. Sanders, M. Tórtola and J. W. F. Valle, _Future CE $\nu$NS experiments as probes of lepton unitarity and light-sterile neutrinos_, _Phys. Rev. D_ 102 (2020) 113014 [2008.02759].
* (64) M. Cadeddu et al., _Constraints on light vector mediators through coherent elastic neutrino nucleus scattering data from COHERENT_ , _JHEP_ 01 (2021) 116 [2008.05022].
* (65) I. M. Shoemaker and E. Welch, _Sailing the CE $\nu$NS seas of non-standard neutrino interactions with the coherent CAPTAIN Mills experiment_, 2103.08401.
* (66) L. M. G. de la Vega, L. J. Flores, N. Nath and E. Peinado, _Complementarity between dark matter direct searches and CE $\nu$NS experiments in U(1)’ models_, _JHEP_ 09 (2021) 146 [2107.04037].
* (67) J. Liao, H. Liu and D. Marfatia, _Coherent neutrino scattering and the Migdal effect on the quenching factor_ , _Phys. Rev. D_ 104 (2021) 015005 [2104.01811].
* (68) CONUS collaboration, H. Bonet et al., _Novel constraints on neutrino physics beyond the standard model from the CONUS experiment_ , _JHEP_ 05 (2022) 085 [2110.02174].
* (69) L. J. Flores, N. Nath and E. Peinado, _CE $\nu$NS as a probe of flavored generalized neutrino interactions_, _Phys. Rev. D_ 105 (2022) 055010 [2112.05103].
* (70) Y.-F. Li and S.-y. Xia, _Constraining light mediators via detection of coherent elastic solar neutrino nucleus scattering_ , _Nucl. Phys. B_ 977 (2022) 115737 [2201.05015].
* (71) D. Aristizabal Sierra, J. Liao and D. Marfatia, _Impact of form factor uncertainties on interpretations of coherent elastic neutrino-nucleus scattering data_ , _JHEP_ 06 (2019) 141 [1902.07398].
* (72) M. Abdullah, D. Aristizabal Sierra, B. Dutta and L. E. Strigari, _Coherent elastic neutrino-nucleus scattering with directional detectors_ , _Phys. Rev. D_ 102 (2020) 015009 [2003.11510].
* (73) G. Fernandez-Moroni et al., _The physics potential of a reactor neutrino experiment with Skipper-CCDs: searching for new physics with light mediators_ , _JHEP_ 02 (2022) 127 [2108.07310].
* (74) E. Bertuzzo, G. Grilli di Cortona and L. M. D. Ramos, _Probing light vector mediators with coherent scattering at future facilities_ , _JHEP_ 06 (2022) 075 [2112.04020].
* (75) H. Bonet et al., _First limits on neutrino electromagnetic properties from the CONUS experiment_ , 2201.12257.
* (76) M. Lindner, W. Rodejohann and X.-J. Xu, _Coherent neutrino-nucleus scattering and new neutrino interactions_ , _JHEP_ 03 (2017) 097 [1612.04150].
* (77) J. Erler and M. J. Ramsey-Musolf, _The weak mixing angle at low energies_ , _Phys. Rev. D_ 72 (2005) 073003 [hep-ph/0409169].
* (78) J. Erler and R. Ferro-Hernández, _Weak mixing angle in the Thomson limit_ , _JHEP_ 03 (2018) 196 [1712.09146].
* (79) S. Klein and J. Nystrand, _Exclusive vector meson production in relativistic heavy ion collisions_ , _Phys. Rev. C_ 60 (1999) 014903 [hep-ph/9902259].
* (80) R. H. Helm, _Inelastic and elastic scattering of 187-MeV electrons from selected even-even nuclei_ , _Phys. Rev._ 104 (1956) 1466.
* (81) P. Klos, J. Menéndez, D. Gazit and A. Schwenk, _Large-scale nuclear structure calculations for spin-dependent WIMP scattering with chiral effective field theory currents_ , _Phys. Rev. D_ 88 (2013) 083516 [1304.7684].
* (82) M. Hoferichter, P. Klos, J. Menéndez and A. Schwenk, _Nuclear structure factors for general spin-independent WIMP-nucleus scattering_ , _Phys. Rev. D_ 99 (2019) 055031 [1812.05617].
* (83) M. Hoferichter, P. Klos, J. Menéndez and A. Schwenk, _Analysis strategies for general spin-independent WIMP-nucleus scattering_ , _Phys. Rev. D_ 94 (2016) 063505 [1605.08043].
* (84) Particle Data Group collaboration, P. A. Zyla et al., _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
* (85) I. Angeli and K. P. Marinova, _Table of experimental nuclear ground state charge radii: An update_ , _Atom. Data Nucl. Data Tabl._ 99 (2013) 69.
* (86) C. G. Payne, S. Bacca, G. Hagen, W. Jiang and T. Papenbrock, _Coherent elastic neutrino-nucleus scattering on 40Ar from first principles_, _Phys. Rev. C_ 100 (2019) 061304 [1908.09739].
* (87) P. Vogel and J. Engel, _Neutrino electromagnetic form-factors_ , _Phys. Rev. D_ 39 (1989) 3378.
* (88) O. Tomalak and R. J. Hill, _Theory of elastic neutrino-electron scattering_ , _Phys. Rev. D_ 101 (2020) 033006 [1907.03379].
* (89) L. A. Mikaelyan, _Investigation of neutrino properties in experiments at nuclear reactors: present status and prospects_ , _Phys.Atom.Nucl._ 65 (2002) 1173 [hep-ph/0210047].
* (90) K. Scholberg, _Prospects for measuring coherent neutrino-nucleus elastic scattering at a stopped-pion neutrino source_ , _Phys. Rev. D_ 73 (2006) 033005 [hep-ex/0511042].
* (91) M. Hoferichter, J. Menéndez and A. Schwenk, _Coherent elastic neutrino-nucleus scattering: EFT analysis and nuclear responses_ , _Phys. Rev. D_ 102 (2020) 074018 [2007.08529].
* (92) W. Rodejohann, X.-J. Xu and C. E. Yaguna, _Distinguishing between Dirac and Majorana neutrinos in the presence of general interactions_ , _JHEP_ 05 (2017) 024 [1702.05721].
* (93) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, _Remarks on Higgs boson interactions with nucleons_ , _Phys. Lett. B_ 78 (1978) 443.
* (94) M. Cirelli, E. Del Nobile and P. Panci, _Tools for model-independent bounds in direct dark matter searches_ , _JCAP_ 10 (2013) 019 [1307.5955].
* (95) J. Ellis, N. Nagata and K. A. Olive, _Uncertainties in WIMP dark matter scattering revisited_ , _Eur. Phys. J. C_ 78 (2018) 569 [1805.09795].
* (96) P. Huber, _On the determination of anti-neutrino spectra from nuclear reactors_ , _Phys. Rev. C_ 84 (2011) 024617 [1106.0687].
* (97) T. A. Mueller et al., _Improved predictions of reactor antineutrino spectra_ , _Phys. Rev. C_ 83 (2011) 054615 [1101.2663].
* (98) X. Qian and J.-C. Peng, _Physics with reactor neutrinos_ , _Rept. Prog. Phys._ 82 (2019) 036201 [1801.05386].
* (99) SuperCDMS collaboration, R. Agnese et al., _New results from the search for low-mass weakly interacting massive particles with the CDMS low ionization threshold experiment_ , _Phys. Rev. Lett._ 116 (2016) 071301 [1509.02448].
* (100) R. B. Firestone and V. S. Shirley. New York: Wiley, 1996.
* (101) S. O. W. Antman, D. A. Landis and R. H. Pehl, _Measurements of the Fano factor and the energy per hole-electron pair in Germanium_ , _Nucl. Instrum. Methods_ 40 (1966) 272.
* (102) W. Z. Wei, L. Wang and D. M. Mei, _Average energy expended per e-h pair for Germanium-based dark matter experiments_ , _JINST_ 12 (2017) P04022 [1602.08005].
* (103) J. I. Collar, A. R. L. Kavner and C. M. Lewis, _Germanium response to sub-keV nuclear recoils: a multipronged experimental characterization_ , _Phys. Rev. D_ 103 (2021) 122003 [2102.10089].
* (104) J. I. Collar, A. R. L. Kavner and C. M. Lewis, _Response of CsI[Na] to nuclear recoils: impact on coherent elastic neutrino-nucleus scattering (CE $\nu$NS)_, _Phys. Rev. D_ 100 (2019) 033003 [1907.04828].
* (105) W. Creus, _Light yield in liquid Argon for dark matter detection_ , Ph.D. thesis, Zurich U., 2013. 10.5167/uzh-86639.
* (106) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler and J. Salvado, _Updated constraints on non-standard interactions from global analysis of oscillation data_ , _JHEP_ 08 (2018) 180 [1805.04530].
* (107) GEMMA collaboration, A. G. Beda et al., _The results of search for the neutrino magnetic moment in GEMMA experiment_ , _Adv. High Energy Phys._ 2012 (2012) 350150.
* (108) GEMMA collaboration, A. G. Beda et al., _Gemma experiment: the results of neutrino magnetic moment search_ , _Phys. Part. Nucl. Lett._ 10 (2013) 139.
* (109) A. Thompson et al., _X-ray data booklet_ , _https://xdb.lbl.gov/_ (2009) .
|
Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
ISWC’22: The 21st International Semantic Web Conference, October 23–27, 2022,
Hangzhou, China
[orcid=0000-0001-8402-2853<EMAIL_ADDRESS>]
[orcid=0000-0003-4805-1467<EMAIL_ADDRESS>]
[orcid=0000-0003-1047-6339<EMAIL_ADDRESS>]
# SignalKG: Towards Reasoning about the Underlying Causes of Sensor
Observations
Anj Simmons Rajesh Vasa Antonio Giardina Applied Artificial Intelligence
Institute, Deakin University, Geelong, Australia
(2022)
###### Abstract
This paper demonstrates our vision for knowledge graphs that assist machines
to reason about the cause of signals observed by sensors. We show how the
approach allows for constructing smarter surveillance systems that reason
about the most likely cause (e.g., an attacker breaking a window) of a signal
rather than acting directly on the received signal without consideration for
how it was produced.
###### keywords:
knowledge graph ontology sensor surveillance
## 1 Introduction
Standards such as the Semantic Sensor Network (SSN/SOSA) ontology [1] allow
capturing the semantics of sensor observations, and emerging standards for
smart buildings [2, 3] and smart cities allow capturing the semantics of the
environments in which sensors operate. However, reasoning about the underlying
cause of sensor observations requires not only knowledge of the sensors and
their environment, but also an understanding of the signals they detect and
the possible causes of these signals. For example, inferring that the sound of
breaking glass may be due to a broken window requires knowledge of the fact
that glass windows produce a distinct sound when broken and that this sound
propagates as sound waves through the air to a sensor such as a human ear or
microphone. This paper proposes a signal knowledge graph (SignalKG) to support
machines to reason about the underlying cause of sensor observations. Sensors
and their environment are represented using existing standards, and then
linked to SignalKG. To reason about the cause of sensor observations, we
automatically generate a Bayesian network based on information in the
knowledge graph, and use this to infer the posterior probability of causes
given the sensor data.
Figure 1: High level overview of SignalKG ontology
Figure 1 presents an ontology describing the high level concepts. A category
of entities (e.g., humans) perform actions (e.g., walking) that act on a type
of object or place (e.g., in hallways), which in turn create a type of signal
(e.g., the sound of footsteps). A sensor observes a particular type of signal,
which usually reduces in strength with distance and can be distorted by
surrounding objects (e.g., a wall between a sound source and the receiver
attenuates the sound signal). The sensor may implement a classifier to detect
the presence of the signal (e.g., an acoustic detector may make use of a
binary machine learning classifier to detect if the sound of footsteps is
present or not). A formal RDF/OWL specification of the SignalKG ontology is
available online111SignalKG ontology: https://signalkg.visualmodel.org/skg as
well as an interactive demonstration222Interactive demonstration available at:
https://signalkg.visualmodel.org of how SignalKG can be applied and used to
reason about the underlying causes of sensor observations.
## 2 Related Work
Past research has utilised ontologies for the purpose of threat and situation
awareness using description logic [4] and rule based reasoning [5]. However,
these approaches assume that threats can be classified into classes according
to deterministic rules, whereas in reality threats may be probabilistic in
nature and there may be multiple possible explanations for a given set of
observations. To overcome this limitation, reasoning methods have been
proposed that combine deterministic rules with a Bayesian network for
reasoning probabilistically [6]. However, the structure of the Bayesian
network and associated probabilities need to be manually specified. In
contrast, we seek to express this knowledge in a reusable and extensible form.
There have been attempts to extend OWL to support representing/reasoning about
uncertainty [7]. However, general approaches do not directly specify concepts
for reasoning about sensor signals.
## 3 Demonstration
Figure 2: Knowledge graph of audio signals constructed using SignalKG ontology
showing link to building rooms/assets and sensors
Figure 2 demonstrates how concepts in the SignalKG ontology can be applied to
construct a knowledge graph of audio signals. Attackers and employees are
entities that are capable of producing an action, such as breaking a window or
dropping a tray of glasses in a dining room. Actions create signals, in this
case, the sound of breaking glass or the sound of dropped glass. The building
asset/room type on/in which an action can occur is represented using the
RealEstateCore ontology [2]. To group similar signals together, we use the
Simple Knowledge Organization System (SKOS) [8] ‘broader’ property to
represent a signal hierarchy, for example, the sound of breaking glass is
similar to the sound of dropped glass and thus are grouped under the same
category. The knowledge graph also includes information about how signals
propagate, for example, that sound intensity reduces with distance (according
to an inverse square law) and is attenuated by walls. To model the case in
which signals are classified on the sensing device, we allow for describing
different classification models, such as YAMNet, an audio classifier that
recognises 521 classes of sounds (e.g., glass).
Figure 3: Bayesian network before (left) and after (right) conditioning on
sensor observations. Green bars indicate the probability of each node value.
(Key: a=action, s=signal emitted due to action, r=received signal strength at
location of sensor, d=detected signal after classification). Visualisation
created with jsbayes-viz [9].
Sensor observations can be linked to the signal knowledge graph via the
property (signal) that the sensor observes. Our interactive demonstration
includes a simulator to generate sample sensor observations (represented using
the SSN/SOSA ontology) for hypothetical scenarios that could occur. The goal
is to infer what took place given only the sensor observations, knowledge of
the building and sensor placement, and our understanding of possible
underlying causes of observed signals (specified using SignalKG).
To support reasoning probabilistically about causes of sensor observations,
the knowledge graph also includes probabilities, such as the prior probability
that an entity will be present, and the probability that an entity (if
present) will perform an action. As our goal is to infer a cause given
observations, it lends itself to Bayesian reasoning. Rather than manually
specifying a Bayesian network for a particular scenario, we automatically
generate one based on the information in the knowledge graph. Encoding all the
information needed to reason about signals in the knowledge graph itself helps
facilite reuse and extension.
Nodes in the Bayesian network are generated for each entity, action at a
location, signal emitted/received, and sensor. While our simple example
results in a 1:1 mapping (shown in Figure 3), in more complex scenarios there
may be vastly more nodes due to all the possible permutations of entity,
action, location, signal and sensor. If multiple types of signals are present
(audio, vision, social, etc.) then these all appear as part of the same
generated Bayesian network, e.g., an attacker breaking a window will create
both audio (sound of breaking glass) and vision (suspicious behaviour in a
video feed) based signals. Prior probabilities for entities and actions need
to be specified in the knowledge graph. The probability that a signal emitted
by an action will be detected by a sensor is calculated based on the distance
from the location of the action to the location of the sensor, how the signal
intensity reduces with distance (e.g., the knowledge graph may specify an
inverse square law for sound signals), any barriers between the source and the
sensor that may attenuate/block the signal (e.g., the knowledge graph may
specify sound is attenuated by walls), and the sensitivity of the classifier
used by the sensor to detect presence of a signal.
Once we have generated a Bayesian network, we can condition it on sensor
observations to infer the posterior probability of the underlying cause. For
the demonstration, we estimate the posterior probability via likelihood weight
sampling, implemented by [9], drawing 20,000 samples (the number of samples to
draw is a trade-off between accuracy and computation time). In the example,
prior to conditioning on observations, there is a 50% chance of an attacker
being present. After conditioning on the observation that the microphone has
detected the sound of glass, the posterior probability of an attacker
increases to 97%.
## 4 Next Steps
Even for the simple example of detecting building intrusions, the space of
possible causes is large (e.g., an attacker could impersonate an employee then
ask someone to let them in, or suspicious sounds could be due to a movie
playing in the background). Furthermore, signals are not independent (as
assumed by our preliminary prototype), but rather occur in sequences (e.g.,
the sound of footsteps, followed by a weapon detected in video footage,
followed by a scream) that could help more reliably distinguish between
possible causes. Also, more realistic models of signal propagation are needed,
which may require continuous probability distributions rather than a
conditional probability table over a discrete set of values as in our example
Bayesian network. To support efficient reasoning about these more complex
scenarios, we plan to explore generation of probabilistic programs in place of
the discrete Bayesian networks used in this paper.
A practical barrier to uptake of our approach is the need to specify prior
probabilities for each action that an entity can perform. Ordinary behaviour
could potentially be learned from data. However, modelling prior probabilities
of actions intruders perform is more difficult, as attacks are rare events
(limited data to learn from) and an adversary will adjust their actions to
avoid detection. In future work, we plan to include the goals of the intruder
as part of the knowledge graph, then use a game theoretic approach to
determine probable actions they will take rather than manually specifying
prior probabilities for each action.
###### Acknowledgements.
This research was funded by National Intelligence Postdoctoral Grant
NIPG-2021-006.
## References
* Haller et al. [2018] A. Haller, K. Janowicz, S. J. Cox, M. Lefrançois, K. Taylor, D. Le Phuoc, J. Lieberman, R. García-Castro, R. Atkinson, C. Stadler, The modular SSN ontology: A joint W3C and OGC standard specifying the semantics of sensors, observations, sampling, and actuation, Semantic Web 10 (2018) 9–32. doi:10.3233/SW-180320.
* Hammar et al. [2019] K. Hammar, E. O. Wallin, P. Karlberg, D. Hälleberg, The RealEstateCore Ontology, in: The Semantic Web – ISWC 2019, 2019, pp. 130–145. doi:10.1007/978-3-030-30796-7_9.
* Rasmussen et al. [2021] M. H. Rasmussen, M. Lefrançois, G. F. Schneider, P. Pauwels, BOT: the building topology ontology of the W3C linked building data group, Semantic Web 12 (2021) 143–161. doi:10.3233/SW-200385.
* Roy and Guyard [2012] J. Roy, A. B. Guyard, Supporting threat analysis through description logic reasoning, in: 2012 IEEE International Multi-Disciplinary Conference on Cognitive Methods in Situation Awareness and Decision Support, IEEE, 2012, pp. 308–315. doi:10.1109/CogSIMA.2012.6188401.
* Pai et al. [2017] F. P. Pai, L. J. Yang, Y. C. Chung, Multi-layer ontology based information fusion for situation awareness, Applied Intelligence 46 (2017) 285–307. doi:10.1007/s10489-016-0834-7.
* Yao et al. [2022] H. Yao, C. Han, F. Xu, Reasoning Methods of Unmanned Underwater Vehicle Situation Awareness Based on Ontology and Bayesian Network, Complexity 2022 (2022) 7143974. doi:10.1155/2022/7143974.
* Carvalho et al. [2017] R. N. Carvalho, K. B. Laskey, P. C. Costa, PR-OWL – a language for defining probabilistic ontologies, International Journal of Approximate Reasoning 91 (2017) 56–79. doi:10.1016/j.ijar.2017.08.011.
* Miles and Bechhofer [2009] A. Miles, S. Bechhofer, SKOS simple knowledge organization system reference, W3C recommendation (2009). URL: https://www.w3.org/TR/skos-reference/.
* Vang [2016] J. Vang, jsbayes-viz, 2016. URL: https://github.com/vangj/jsbayes-viz/.
|
# Enhancing Intrinsic Features for Debiasing via Investigating
Class-Discerning Common Attributes in Bias-Contrastive Pair
Jeonghoon Park*1, Chaeyeon Chung*1, Juyoung Lee2, Jaegul Choo1
1Korea Advanced Institute of Science and Technology, South Korea, 2Kakao
Brain, South Korea.
1{jeonghoon_park, cy_chung<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
In the image classification task, deep neural networks frequently rely on bias
attributes that are spuriously correlated with a target class in the presence
of dataset bias, resulting in degraded performance when applied to data
without bias attributes. The task of debiasing aims to compel classifiers to
learn intrinsic attributes that inherently define a target class rather than
focusing on bias attributes. While recent approaches mainly focus on
emphasizing the learning of data samples without bias attributes (i.e., bias-
conflicting samples) compared to samples with bias attributes (i.e., bias-
aligned samples), they fall short of directly guiding models where to focus
for learning intrinsic features. To address this limitation, this paper
proposes a method that provides the model with explicit spatial guidance that
indicates the region of intrinsic features. We first identify the intrinsic
features by investigating the class-discerning common features between a bias-
aligned (BA) sample and a bias-conflicting (BC) sample (i.e., bias-contrastive
pair). Next, we enhance the intrinsic features in the BA sample that are
relatively under-exploited for prediction compared to the BC sample. To
construct the bias-contrastive pair without using bias information, we
introduce a bias-negative score that distinguishes BC samples from BA samples
employing a biased model. The experiments demonstrate that our method achieves
state-of-the-art performance on synthetic and real-world datasets with various
levels of bias severity.
††* indicates equal contribution.
## 1 Introduction
Deep neural networks in image classification [21, 5, 20, 26] are known to be
vulnerable to the dataset bias [23], which refers to a spurious correlation
between the target classes and the peripheral attributes. Basically, image
classification aims to learn intrinsic attributes — the visual features that
inherently define a target class — that generally appear across the samples in
the class. However, when the dataset bias exists in the training data, the
models tend to use the frequently appearing peripheral attribute (_i.e_., bias
attribute) to predict the class unintentionally. For instance, if airplanes in
the training images are mostly in the sky, a model can heavily rely on the sky
to predict an image as an airplane class due to its high correlation with the
airplane class. This indicates that the model is biased towards the bias
attribute (_e.g_., sky) rather than focusing on intrinsic features (_e.g_.,
the shape of wings or the body) when making decisions. As a result, even
though the biased model achieves high accuracy on the samples including bias
attributes (_e.g_., airplanes in the sky), termed as bias-aligned (BA)
samples, it may fail to accurately predict samples devoid of such bias
attributes (_e.g_., airplanes on the runway), referred to as bias-conflicting
(BC) samples.
In this regard, debiasing aims to encourage the model to focus on intrinsic
attributes rather than bias attributes when dataset bias exists. One
straightforward approach is utilizing prior knowledge regarding bias (_e.g_.,
labels for bias attribute) to inform the model which attributes to focus on or
not to focus on [9, 25, 2, 22]. However, acquiring such bias information is
often infeasible in real-world scenarios. Therefore, recent studies [15, 14,
13, 12, 8] have proposed debiasing methods that do not require bias
information. They identify and emphasize BC samples during the training using
an additional biased classifier that mainly learns the bias attributes.
However, such a training strategy fails to directly indicate where the model
should focus to learn the intrinsic features.
To address this issue, we present a debiasing approach that explicitly informs
the model of the region of the intrinsic features during the training while
not using bias labels. While the intrinsic features in the unbiased dataset
can simply be identified in generally appearing features in the training
samples, generally appearing features in the biased dataset inevitably include
bias features. Therefore, we identify the intrinsic features in the biased
dataset by investigating the common features between a BA and a BC sample
(i.e., a bias-contrastive pair). Here, the common features also need to be
class-discerning since the common features might include irrelevant
environmental features. For example, in the above scenario, the common feature
between an airplane in the sky (BA sample) and an airplane on the runway (BC
sample) might include the features of wings, the body, and trees. In this
case, the intrinsic features are the shape of the wings and the body that can
distinguish the airplane class from the others.
Specifically, we introduce an intrinsic feature enhancement (IE) weight that
identifies the spatial regions of intrinsic features commonly appearing in a
bias-contrastive pair. We leverage an auxiliary sample in addition to the
original input to construct the bias-contrastive pair. Since the majority of
the original input from training samples are BA samples, we mainly adopt the
BC samples as the auxiliary sample. To achieve this without bias information,
we present a bias-negative (BN) score that identifies BC samples by employing
a classification loss of a biased model. Our IE weight investigates common
features in the bias-contrastive pair and identifies the class-discerning
features among the common features. Within the identified intrinsic features,
we enhance the features that are relatively under-exploited in the BA samples
compared to the BC samples. In this way, we can explicitly provide our model
with spatial guidance for intrinsic attributes while not using bias labels.
We verify the effectiveness of our method on both synthetic and real-world
datasets with various levels of bias severity. Furthermore, the in-depth
analysis demonstrates that our method successfully guides the model to make
predictions based on the intrinsic features.
## 2 Related work
Debiasing with bias information. Previous approaches [9, 22, 18, 25, 4, 2]
utilize bias labels or predefined bias types to encourage the model to learn
intrinsic attributes for debiasing. Kim _et al_. [9], Tartaglione _et al_.
[22], and Sagawa _et al_. [18] employ bias labels to encourage the model not
to learn specific bias features. Wang _et al_. [25] and Bahng _et al_. [2]
predefine the bias type (e.g., color, texture, etc.) and utilize such prior
knowledge to supervise models to be robust against such predefined bias type.
However, obtaining bias information requires additional cost, which is often
infeasible in the real world.
Debiasing without bias information. Recent studies [3, 7, 15, 12, 13, 14, 8,
1, 28, 16, 11] propose debiasing strategies that do not require bias
information. Nam _et al_. [15] present an approach that encourages the model
to concentrate on BC samples during the training process considering that the
bias attributes are easier to learn than intrinsic attributes. Instead of
using bias information, they additionally train a biased model that mainly
learns bias attributes and regard the samples that are not easily trained by
the biased model as BC samples. Lee _et al_. [13] reveal that BC samples serve
as noisy samples when training the additional biased model and propose a
method to eliminate such BC samples using multiple biased models. Liu _et al_.
[14] regard the samples misclassified by the model trained with empirical risk
minimization as BC samples and emphasize them during training of a debiased
model. Also, MaskTune [1] expects the model to learn intrinsic features by
fine-tuning the model with the data whose already-explored area is masked out
using Grad-CAM [19].
Another stream of approaches [10, 12, 8] synthesize samples having similar
characteristics with BC samples and employ them to train a debiased model. Kim
_et al_. [10] synthesize images without bias attributes leveraging an image-
to-image translation model [17]. Lee _et al_. [12] and Hwang _et al_. [8]
augment BC samples in the feature space by employing the disentangled
representations and mixup [27], respectively. A recent pair-wise debiasing
method $\mathcal{X}^{2}$-model [28] encourages the model to retain intra-class
compactness using samples generated via feature-level interpolation between BC
and BA samples. However, such approaches lack explicit supervision about which
features to focus on to learn intrinsic features. To address this issue, we
present a debiasing method that provides spatial guidance to encourage a model
to learn intrinsic features during the training while not using bias labels.
We design our model architecture using bias-contrastive pairs referring to the
previous studies [8, 28].
Figure 1: Overview of our method. We provide explicit spatial guidance
$g(\mathbf{z})$ for a debiased model $f_{d}$, which is described with
$f_{d}^{\text{emb}}$ and $f_{d}^{\text{cls}}$, to learn intrinsic features. To
achieve this, we leverage a bias-contrastive pair, $\mathbf{x}$ and
$\mathbf{x}^{\text{BN}}$ from the same target class $y$. $g(\mathbf{z})$
highlights intrinsic features that are relatively under-exploited in
$\mathbf{z}$ compared to $\mathbf{z}^{\text{BN}}$, calculated by common
feature score $c$ and relative-exploitation score $r$. Here, we mainly adopt
BC samples from $\mathcal{D}^{\text{BN}}_{\text{cand}}$ to construct
$\mathcal{D}^{\text{BN}}$, where we sample $\mathbf{x}^{\text{BN}}$.
$\mathcal{D}^{\text{BN}}$ is updated every iteration using the BN score $S$,
which is also updated every iteration. At the inference, we only use $f_{d}$
in the gray-colored area.
## 3 Methodology
### 3.1 Overview
As shown in Fig. 1, our framework consists of a biased model $f_{b}$ that
focuses on bias attributes and a debiased model $f_{d}$ that learns debiased
representations. We use BiasEnsemble (BE) [13] as a backbone, where $f_{b}$ is
trained with bias-amplified dataset $\mathcal{D}^{\text{A}}$ which mainly
consists of BA samples, while $f_{d}$ concentrates on the samples that $f_{b}$
fails to learn. Our method provides $f_{d}$ with spatial guidance for
intrinsic features using a bias-contrastive pair: an input $\mathbf{x}$ and an
auxiliary input $\mathbf{x}^{\text{BN}}$. We denote the auxiliary input
$\mathbf{x}^{\text{BN}}$ as a bias-negative (BN) sample because we primarily
adopt samples devoid of bias attributes. We sample an image $\mathbf{x}$ from
the original training data $\mathcal{D}$, and $\mathbf{x}^{\text{BN}}$ from a
BN dataset $\mathcal{D}^{\text{BN}}$ which mainly consists of BC samples.
$\mathcal{D}^{\text{BN}}$ is updated every iteration to mainly include BC
samples using the BN score $S$ that employs $f_{b}$ to identify BC samples.
The BN score is also updated every iteration. Given the intermediate features
$\mathbf{z}$ and $\mathbf{z}^{\text{BN}}$, we first extract the common
features between the bias-contrastive pair ($c(\mathbf{z})$ in Fig. 1). Also,
we identify the class-discerning features that are relatively under-exploited
in $\mathbf{z}$ compared to $\mathbf{z}^{\text{BN}}$ ($r(\mathbf{z})$ in Fig.
1). Next, we calculate the IE weight that indicates relatively under-exploited
intrinsic features in $\mathbf{z}$ based on $c(\mathbf{z})$ and
$r(\mathbf{z})$ ($\text{IE}(\mathbf{z})$ in Fig. 1). Finally, we obtain the
guidance $g(\mathbf{z})$ that emphasizes the region of intrinsic feature in
$\mathbf{z}$ during the training. At the inference, we utilize $f_{d}$ without
$\mathbf{x}^{\text{BN}}$, as in a gray-colored area of Fig. 1.
### 3.2 Constructing bias-negative dataset
We construct a BN dataset $\mathcal{D}^{\text{BN}}$, where we sample
$\mathbf{x}^{\text{BN}}$ during the training. As the majority of the training
dataset is BA samples, we aim to mainly adopt BC samples as
$\mathbf{x}^{\text{BN}}$ to construct bias-contrastive pairs. To achieve this,
we first construct $\mathcal{D}^{\text{BN}}_{\text{cand}}$, a candidate
dataset for $\mathcal{D}^{\text{BN}}$, that contains roughly identified BC
samples. During the training, we dynamically update $\mathcal{D}^{\text{BN}}$
every iteration to mainly adopt BC samples from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ using our newly proposed BN score.
Constructing candidate dataset $\mathcal{D}^{\text{BN}}_{\text{cand}}$. To
roughly identify BC samples in $\mathcal{D}$, we filter out easily learned BA
samples from $\mathcal{D}$ using multiple biased models, following BE [13].
Since the bias features are easier to learn than the intrinsic features [15],
each biased model is trained only for a few iterations so that BC samples can
be distinguished from the easily learned BA samples. Inspired by JTT [14], we
regard the samples that are incorrectly predicted by the majority of the
biased models as BC samples. Finally, we construct
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ with the roughly identified BC
samples.
Adopting BC samples with BN score. We introduce a BN score to update
$\mathcal{D}^{\text{BN}}$ to primarily exploit BC samples as
$\mathbf{x}^{\text{BN}}$ from $\mathcal{D}^{\text{BN}}_{\text{cand}}$ during
training $f_{d}$. Considering the unavailability of bias labels, the BN score
employs $f_{b}$ to further exclude BA samples from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$. As training proceeds, $f_{b}$ is
overfitted to the bias attributes in $\mathcal{D}^{\text{A}}$ and outputs a
high probability on the ground-truth label for the samples that have similar
bias features with samples in $\mathcal{D}^{\text{A}}$. This indicates that
samples whose $f_{b}$ loss decreases as training proceeds are likely to have
bias attributes learned from $\mathcal{D}^{\text{A}}$. Such samples disturb
the extraction of intrinsic features when selected as
$\mathbf{x}^{\text{BN}}$. To validate this, we investigate the samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ whose $f_{b}$ loss at the later stage
of training (50K-th iteration) decreases compared to the early stage of
training (1K-th iteration). The result shows that 95.63% of them are BA
samples. We use the BFFHQ dataset [10] with a bias severity of 1% for the
analysis. Further details of the dataset are described in Sec. 4.1.
In this regard, we design a BN score to exclude the samples with decreasing
$f_{b}$ loss from the $\mathcal{D}^{\text{BN}}_{\text{cand}}$ to construct
$\mathcal{D}^{\text{BN}}$ by tracking the $f_{b}$ loss during training
$f_{d}$. First, the $f_{b}$ loss of $\mathbf{x}$ at the $t$-th iteration is
calculated as follows:
$l_{t}(\mathbf{x})=\alpha_{l}\cdot\mathcal{L}_{\text{CE}}(f_{b}(\mathbf{x}),y)+(1-\alpha_{l})\cdot
l_{t-1}(\mathbf{x}),$ (1)
where $\mathcal{L}_{\text{CE}}(f_{b}(\mathbf{x}),y)$ indicates the cross-
entropy (CE) loss of $\mathbf{x}$ on its ground-truth label $y$ and
$\alpha_{l}$ is a hyperparameter for the exponential moving average (EMA). We
employ EMA to enable a stable tracking of the classification losses. Note that
$l_{t}$ is updated only for the samples in a mini-batch at the $t$-th
iteration.
The BN score tracks $l_{t}(\mathbf{x})$ compared to the loss recorded at the
early stage of training. The BN score at the $t$-th iteration is formulated as
follows:
$s_{t}(\mathbf{x})=\alpha_{s}\cdot(l_{t}(\mathbf{x})-l_{\text{ref}}(\mathbf{x}))+(1-\alpha_{s})\cdot
s_{t-1}(\mathbf{x}),$ (2)
where $\alpha_{s}$ is a hyperparameter for the EMA and
$l_{\text{ref}}(\mathbf{x})$ denotes the reference loss of $\mathbf{x}$ that
is first recorded after a few iterations of training. We exploit EMA to
stabilize the tracking. Note that we update $s_{t}$ only for the samples in a
mini-batch at the $t$-th iteration. The negative value of $s_{t}(\mathbf{x})$
indicates that the loss of $\mathbf{x}$ decreased compared to the early stage
of training, which means that the sample is likely to contain bias attributes.
Updating $\mathcal{D}^{\text{BN}}$ with BN score. At every iteration, we
update $\mathcal{D}^{\text{BN}}$ to exclude the newly detected BA samples
whose BN score $s_{t}(\mathbf{x})$ is smaller than zero as follows:
$\mathcal{D}^{\text{BN}}_{t}=\\{\mathbf{x}\mid
s_{t}(\mathbf{x})>0,\mathbf{x}\sim\mathcal{D}^{\text{BN}}_{\text{cand}}\\},$
(3)
where $\mathcal{D}^{\text{BN}}_{t}$ indicates $\mathcal{D}^{\text{BN}}$ at the
$t$-th iteration. We employ
$\mathbf{x}^{\text{BN}}\sim\mathcal{D}^{\text{BN}}_{t}$ as an auxiliary input
at the $t$-th iteration. In this way, we can construct a bias-contrastive pair
that encourages the intrinsic attributes to be extracted as their common
features. We abbreviate $\mathcal{D}^{\text{BN}}_{t}$ as
$\mathcal{D}^{\text{BN}}$ for brevity in the rest of the paper.
### 3.3 Intrinsic feature enhancement
To emphasize the intrinsic features in $f_{d}$, we introduce the intrinsic
feature enhancement (IE) weight that imposes a high value on the intrinsic
features. The IE weight identifies the region of intrinsic features from bias-
contrastive pairs by investigating 1) their common features with common
feature score $c$ and 2) class-discerning features that are relatively under-
exploited in the input with relative-exploitation score $r$. For the
explanation, we split $f_{d}$ into two parts.
$f_{d}^{\text{emb}}:\mathbb{R}^{H\times W\times 3}\to\mathbb{R}^{h\times
w\times c}$ maps an input to the intermediate feature, and
$f_{d}^{\text{cls}}:\mathbb{R}^{h\times w\times c}\to\mathbb{R}^{C}$ is
composed of the average pooling and the linear classifier and outputs the
classification logits, where
$f_{d}(\mathbf{x})=f_{d}^{\text{cls}}\left(f_{d}^{\text{emb}}(\mathbf{x})\right)$.
First, given the input $\mathbf{x}$, common feature score $c$ identifies the
features that are similar to the features in $\mathbf{x}^{\text{BN}}$ that has
the same class label as $\mathbf{x}$ while not having bias attributes.
Specifically, we extract the intermediate features
$\mathbf{z}=f_{d}^{\text{emb}}(\mathbf{x})$ and
$\mathbf{z}^{\text{BN}}=f_{d}^{\text{emb}}(\mathbf{x}^{\text{BN}})$,
respectively. Next, we obtain the common feature score of $\mathbf{z}$
(_i.e_., $c(\mathbf{z})\in\mathbb{R}^{h\times w}$). Given the $n$-th feature
of $\mathbf{z}$ (_i.e_., $\mathbf{z}_{n}\in\mathbb{R}^{c}$), let $i^{*}$-th
feature of $\mathbf{z}^{\text{BN}}$ (_i.e_.,
$\mathbf{z}^{\text{BN}}_{i^{*}}\in\mathbb{R}^{c}$) be the most similar feature
to $\mathbf{z}_{n}$, where
$i^{*}=\underset{i}{\arg\max}\left(\mathbf{z}^{\text{BN}}_{i}\cdot\mathbf{z}_{n}\right)$.
Then, the $n$-th element of $c(\mathbf{z})$ denotes the similarity score
between $\mathbf{z}_{n}$ and $\mathbf{z}^{\text{BN}}_{i^{*}}$, which is
formulated as follows:
$c(\mathbf{z})_{n}=\frac{\mathbf{z}_{i^{*}}^{\text{BN}}\cdot\mathbf{z}_{n}}{\max_{i,j}(\mathbf{z}_{i}^{\text{BN}}\cdot\mathbf{z}_{j})},$
(4)
where $\cdot$ indicates a dot product operation. We adopt the dot product for
the similarity metric to consider both the scale and the direction of the
features. The max normalization is employed to limit the score to less than
one. We consider the features with a high common feature score $c$ in
$\mathbf{z}$ as features that have a high likelihood of being intrinsic
features.
Next, the relative-exploitation score $r$ identifies class-discerning features
that are relatively under-exploited in $\mathbf{x}$ compared to
$\mathbf{x}^{\text{BN}}$. Since most of the $\mathbf{x}^{\text{BN}}$ does not
contain bias attributes, we identify class-discerning intrinsic features by
investigating the features that are mainly used to predict
$\mathbf{x}^{\text{BN}}$ as its target label. At the same time, we identify
the features that are under-exploited in the $\mathbf{x}$ compared to the
$\mathbf{x}^{\text{BN}}$. To achieve this, we use a visual explanation map of
Grad-CAM [19] that imposes a higher value on the features that have more
contribution to predicting a specific label. We calculate the explanation map
$\text{E}(\mathbf{z})$ and $\text{E}(\mathbf{z}^{\text{BN}})$ with respect to
their ground-truth labels. We apply max normalization to the explanation maps
to compare the relative importance of the features in prediction. We compare
the $n$-th value of $\text{E}(\mathbf{z})$ (_i.e_.,
$\text{E}(\mathbf{z})_{n}$) with the $i^{*}$-th value of
$\text{E}(\mathbf{z}^{\text{BN}})$, where $i^{*}$ is the index of the feature
in $\mathbf{z}^{\text{BN}}$ that is the most similar with $\mathbf{z}_{n}$.
Accordingly, the $n$-th element of $r(\mathbf{z})\in\mathbb{R}^{h\times w}$ is
calculated as:
$r(\mathbf{z})_{n}=\left(\frac{2\text{E}(\mathbf{z}^{\text{BN}})_{i^{*}}}{\text{E}(\mathbf{z}^{\text{BN}})_{i^{*}}+\text{E}(\mathbf{z})_{n}}\right)^{\tau},$
(5)
where $\tau$ is the amplification factor. The score becomes larger than one
when the $\mathbf{z}_{n}$ is relatively under-exploited than
$\mathbf{z}^{\text{BN}}_{i^{*}}$ for prediction. When
$\mathbf{z}^{\text{BN}}_{i^{*}}$ is not used for discerning the class,
$\text{E}(\mathbf{z}^{\text{BN}})_{i^{*}}$ becomes close to zero and the score
converges to zero.
Finally, $n$-th element of the $\text{IE}(\mathbf{z})$ is defined as:
$\text{IE}(\mathbf{z})_{n}=\max(c(\mathbf{z})_{n}\odot r(\mathbf{z})_{n},1),$
(6)
where $\odot$ indicates the element-wise multiplication. The IE weight has a
large value on the features of $\mathbf{x}$ that commonly appear in
$\mathbf{x}^{\text{BN}}$ but has not exploited enough for the prediction of
$\mathbf{x}$. We clip the values to be larger than one to enhance only the
relatively under-exploited features in $\mathbf{z}$ while preserving the other
features.
Using the IE weight, we obtain the guidance $g(\mathbf{z})$ that emphasizes
the intrinsic features in $\mathbf{z}$ as follows:
$g(\mathbf{z})=\mathbf{z}\odot\text{IE}(\mathbf{z}).$ (7)
We broadcast $\text{IE}(\mathbf{z})$ to match its shape with $\mathbf{z}$
before multiplication. During the training, this spatial guidance informs the
model of where to focus to learn intrinsic features from training samples.
Algorithm 1 Debiasing with the intrinsic feature guidance
Input: pretrained biased models,
training dataset $\mathcal{D}$, biased model $f_{b}$, debiased model
$f_{d}$, reference iteration $T_{1}$ for BN score, starting
iteration $T_{2}(\geq T_{1})$ to apply intrinsic feature guidance
Output: trained debiased model $f_{d}$
1:Construct $\mathcal{D}^{\text{A}},\mathcal{D}^{\text{BN}}_{\text{cand}}$
from $\mathcal{D}$ using pretrained biased models
2:for every iteration $t$ do
3: Sample $\mathbf{x}\sim\mathcal{D}$
4: if $t\geq T_{1}$ and $l_{\text{ref}}(\mathbf{x})$ is not initialized then
5: $l_{\text{ref}}(\mathbf{x})\leftarrow l(\mathbf{x})$
6: end if
7: if $\mathbf{x}\in\mathcal{D}^{\text{A}}$ then
8: Train $f_{b}(\mathbf{x})$ with $\mathcal{L}_{\text{CE}}$
9: end if
10: if $t<$ $T_{2}$ then $\triangleright$ Train w/o guidance
11: Train $f_{d}(\mathbf{x})$ with $\mathcal{L}_{\text{main}}$
12: else if $t\geq$ $T_{2}$ then $\triangleright$ Train w/ guidance
13: Update $\mathcal{D}^{\text{BN}}$
14: Sample $\mathbf{x}^{\text{BN}}\sim\mathcal{D}^{\text{BN}}$
15: Train $f_{d}(\mathbf{x},\mathbf{x}^{\text{BN}})$ with
$\mathcal{L}_{\text{total}}$
16: end if
17:end for
### 3.4 Training with intrinsic feature guidance
We basically train $f_{d}$ with the CE loss as follows:
$\mathcal{L}_{\text{main}}=w(\mathbf{x})\mathcal{L}_{\text{CE}}(f_{d}(\mathbf{x}),y)$
(8)
where $w(\mathbf{x})$ is the sample reweighting value of $\mathbf{x}$ [15].
$w(\mathbf{x})$ emphasizes the samples that $f_{b}$ fails to learn, which are
mostly BC samples. The detailed description of $w(\mathbf{x})$ is included in
the Supplementary.
In addition, we guide the model to focus on the region of intrinsic features
through a guidance loss and a BN loss. We observe that the BN score has a
higher value on the BC samples compared to the BA samples as training $f_{b}$
proceeds (See Sec. 4.3). In this respect, we employ the BN score of
$\mathbf{x}^{\text{BN}}$ (_i.e_., $s(\mathbf{x}^{\text{BN}})$) to upweight the
loss when BC samples are adopted as $\mathbf{x}^{\text{BN}}$. Here, we clip
the value of loss weight $s(\mathbf{x}^{\text{BN}})$ to be larger than zero.
Guidance loss. To guide the model to exploit the intrinsic features from
$\mathbf{x}$, we minimize the L1 distance between $g(\mathbf{z})$ and
$\mathbf{z}$ as follows:
$\mathcal{L}_{\text{guide\\_sim}}=s(\mathbf{x}^{\text{BN}})\lVert\text{GAP}(\mathbf{z})-\text{GAP}\left({g(\mathbf{z})}\right)\rVert_{1},$
(9)
where GAP represents the global average pooling. $s(\mathbf{x}^{\text{BN}})$
is multiplied as a loss weight to impose a high weight on the loss when BC
samples are selected as $\mathbf{x}^{\text{BN}}$.
Also, we apply the CE loss to the guidance $g(\mathbf{z})$ to encourage it to
include the intrinsic features that contribute to the correct prediction as
follows:
$\mathcal{L}_{\text{guide\\_cls}}=w(\mathbf{x})\mathcal{L}_{\text{CE}}\left(f_{d}^{\text{cls}}(g(\mathbf{z})),y\right),$
(10)
where $w(\mathbf{x})$ is the reweighting value as in
$\mathcal{L}_{\text{main}}$.
Finally, our guidance loss $\mathcal{L}_{\text{guide}}$ is calculated as
follows:
$\mathcal{L}_{\text{guide}}=\lambda_{\text{sim}}\mathcal{L}_{\text{guide\\_sim}}+\mathcal{L}_{\text{guide\\_cls}},$
(11)
where $\lambda_{\text{sim}}$ is a hyperparameter to control the relative
significance between the losses. We set $\lambda_{\text{sim}}$ set as 0.1.
BN loss. We also employ the CE loss on $\mathbf{x}^{\text{BN}}$ to encourage
the model to learn class-discerning features. This enables the IE weight to
find intrinsic features among the common features. The BN loss is defined as:
$\mathcal{L}_{\text{BN}}=s(\mathbf{x}^{\text{BN}})\mathcal{L}_{\text{CE}}(f_{d}(\mathbf{x}^{\text{BN}}),y).$
(12)
Here, we also exploit $s(\mathbf{x}^{\text{BN}})$ to impose high weight on the
loss when $\mathbf{x}^{\text{BN}}$ is a BC sample.
Overall objective function. In summary, the overall objective function is
defined as follows:
$\mathcal{L}_{\text{total}}=\lambda_{\text{main}}\mathcal{L}_{\text{main}}+\mathcal{L}_{\text{guide}}+\mathcal{L}_{\text{BN}},$
(13)
$\lambda_{\text{main}}$ is the constant value that linearly increases from
zero to one during training $f_{d}$ with the guidance. This prevents the model
from focusing on bias features in $\mathbf{x}$ in the early phase. The overall
process of our method is provided in Algorithm 1. Here, we set $T_{1}$ and
$T_{2}$ as 1K and 10K, respectively. Note that all the hyperparameters are
identically applied across different datasets and bias severities. We provide
further details of the training and the implementation in the Supplementary.
Method | Waterbirds | BFFHQ | BAR
---|---|---|---
0.5 | 1.0 | 2.0 | 5.0 | 0.5 | 1.0 | 2.0 | 5.0 | 1.0 | 5.0
Vanilla [5] | 57.41 | 58.07 | 61.04 | 64.13 | 55.64 | 60.96 | 69.00 | 82.88 | 70.55 | 82.53
HEX [25] | 57.88 | 58.28 | 61.02 | 64.32 | 56.96 | 62.32 | 70.72 | 83.40 | 70.48 | 81.20
LNL [9] | 58.49 | 59.68 | 62.27 | 66.07 | 56.88 | 62.64 | 69.80 | 83.08 | - | -
EnD [22] | 58.47 | 57.81 | 61.26 | 64.11 | 55.96 | 60.88 | 69.72 | 82.88 | - | -
ReBias [2] | 55.44 | 55.93 | 58.53 | 62.14 | 55.76 | 60.68 | 69.60 | 82.64 | 73.04 | 83.90
LfF [15] | 60.66 | 61.78 | 58.92 | 61.43 | 65.19 | 69.24 | 73.08 | 79.80 | 70.16 | 82.95
DisEnt [12] | 59.59 | 60.05 | 59.76 | 64.01 | 62.08 | 66.00 | 69.92 | 80.68 | 70.33 | 83.13
LfF+BE [13] | 61.22 | 62.58 | 63.00 | 63.48 | 67.36 | 75.08 | 80.32 | 85.48 | 73.36 | 83.87
DisEnt+BE [13] | 51.65 | 54.10 | 53.43 | 54.21 | 67.56 | 73.48 | 79.48 | 84.84 | 73.29 | 84.96
Ours | 63.64 | 65.22 | 65.23 | 66.33 | 71.68 | 77.56 | 83.08 | 87.60 | 75.14 | 85.03
Table 1: Comparison to the baselines. We measure the classification accuracy
on test sets with different bias severities. The best accuracy values are in
bold. The hyphen mark ‘-’ means it is not applicable. Results with standard
deviations are provided in the Supplementary.
| $\mathcal{D}^{\text{BN}}_{\text{cand}}$ \- $\mathcal{D}^{\text{BN}}$ | $\mathcal{D}^{\text{BN}}$/$\mathcal{D}$ (%)
---|---|---
Dataset | BA | BC | BA | BC
Waterbirds | 26.50 $\pm$5.32 | 0.75 $\pm$0.83 | 2.75 $\pm$0.31 | 79.69 $\pm$3.72
BFFHQ | 199.80 $\pm$40.14 | 8.00 $\pm$2.76 | 0.46 $\pm$0.09 | 50.00 $\pm$1.04
BAR | 30.60 $\pm$3.83 | 3.20 $\pm$1.60 | 3.58 $\pm$0.14 | 47.14 $\pm$5.71
Table 2: Effectiveness of BN score on excluding BA samples.
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ \- $\mathcal{D}^{\text{BN}}$ presents
the number of excluded samples when constructing $\mathcal{D}^{\text{BN}}$
from $\mathcal{D}^{\text{BN}}_{\text{cand}}$.
$\mathcal{D}^{\text{BN}}$/$\mathcal{D}$ indicates that the ratio of samples in
$\mathcal{D}^{\text{BN}}$ to the samples in $\mathcal{D}$.
## 4 Experiments
### 4.1 Experimental settings
Dataset. We utilize Waterbirds [18], biased FFHQ (BFFHQ) [10], and BAR [15]
for the experiments. Each dataset contains different types of target class and
bias attributes: Waterbirds - {bird type, background}, BFFHQ - {age, gender},
and BAR - {action, background}. The former and the latter in the bracket
indicate the target class and the bias attribute, respectively. To be
specific, the Waterbirds dataset has two bird classes: waterbirds and
landbirds. Most of the waterbirds are in the water background, and most of the
landbirds are in the land background. In the training dataset of BFFHQ, most
young people are female while most old people are male. The word ‘young’
indicates an age ranging between 10 and 29, and ‘old’ indicates an age ranging
between 40 and 59. Lastly, the BAR dataset consists of six classes of action
(e.g., fishing), where the background (e.g., water surface) is highly
correlated with each class. Following the previous studies [15, 12, 13], we
validate our model’s effectiveness under different levels of bias severity,
_i.e_., a ratio of BC samples to the total training samples: 0.5%, 1%, 2%, and
5%. In the test sets, the spurious correlations found in the training set do
not exist. More details are provided in Supplementary.
Evaluation. We report the best accuracy of the test set averaged over five
independent trials with different random seeds. The Waterbirds dataset has an
extremely skewed test dataset composed of 4,600 landbirds and 1,194
waterbirds. This can mislead the debiasing performance as the model may
achieve high classification accuracy by simply predicting most images as
landbirds. We measure the classification accuracy for each class and report
their average value to obtain an accurate understanding of the effectiveness
of methods, regardless of class frequencies. Also, for the BFFHQ, we report
the best accuracy of BC samples in the test set, following the previous works
[12, 13]. For the analyses, we utilize the datasets with 1% bias severity.
### 4.2 Comparison to previous works
We compare the classification accuracy on the test sets between the baselines
and ours in Table 1. For baselines, we employ a vanilla model trained with the
CE loss, the methods using explicit bias label (_i.e_., LNL [9], EnD [22]),
presuming the type of the bias (_i.e_., HEX [25], ReBias [2]), and assuming
the bias information is unknown (_i.e_., LfF [15], DisEnt [12], LfF+BE [13],
DisEnt+BE [13]). Our approach achieves state-of-the-art performance in
comparison to the previous methods including those utilizing explicit bias
labels. The results exhibit that our method improves performance robustly
across various levels of bias severity, even under the constraints of extreme
bias severity (_e.g_., 0.5 %). This shows that providing explicit spatial
guidance for intrinsic features effectively encourages the model to learn
debiased representations, leading to performance improvement.
Figure 2: Visualization of BN scores of the samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ during the training. The red lines and
the blue lines indicate the BN scores of BA and BC samples, respectively.
Figure 3: Visualization of the spatial guidance using (a) Waterbirds and (b)
BAR dataset. Given bias-contrastive pairs, $\mathbf{x}$ and
$\mathbf{x}^{\text{BN}}$, $\text{E}(\mathbf{z})$ indicates the regions
originally focused on by $f_{d}$ and $\text{IE}(\mathbf{z})$ shows the regions
highlighted by our IE weight. Figure 4: Comparison of the region focused by a
debiased model trained with and without our method. We compare Grad-CAM
results on the test set of (a) Waterbirds and (b) BAR.
### 4.3 Analysis of BN score
We analyze our BN score that identifies and emphasizes BC samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ during training $f_{d}$. In this
section, we assess the effectiveness of the BN score on excluding BA samples
from $\mathcal{D}^{\text{BN}}_{\text{cand}}$. Also, we evaluate the efficacy
of the BN score as a loss weight by investigating the BN scores of the samples
in $\mathcal{D}^{\text{BN}}_{\text{cand}}$.
Effectiveness of BN score on excluding BA samples. Table 2 presents how the
BN score effectively filters out BA samples from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ while preserving BC samples when
constructing $\mathcal{D}^{\text{BN}}$. The first two columns present the
number of the BA and BC samples excluded from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ to construct
$\mathcal{D}^{\text{BN}}$, respectively. Also, the last two columns represent
the ratio of the number of the BA and BC samples in $\mathcal{D}^{\text{BN}}$
to that in $\mathcal{D}$, respectively. For the analysis, we use
$\mathcal{D}^{\text{BN}}$ at the 50K-th iteration and report the mean value of
the five independent trials. Here, we expect $\mathcal{D}^{\text{BN}}$ to
contain a maximal number of BC samples while including a minimal number of BA
samples. As shown in the first two columns of Table 2, our BN score excludes a
large number of BA samples while minimizing the loss of BC samples. As a
result, $\mathcal{D}^{\text{BN}}$ preserves around 50-80% of BC samples, while
containing a minimal number of BA samples compared to $\mathcal{D}$, as
presented in the last two columns.
Efficacy of BN score as loss weight. We utilize the BN score to upweight the
training loss when BC samples are chosen as $\mathbf{x}^{\text{BN}}$. To
verify its effectiveness as a loss weight, we compare the BN scores of BC
samples and BA samples in $\mathcal{D}^{\text{BN}}_{\text{cand}}$ during the
training for the Waterbirds, BFFHQ, and BAR dataset. In Fig. 2, we present the
average BN scores of BA (red line) and BC samples (blue line) in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ at every 500 iterations. Since BN
scores are recorded after the 1K-th iteration, the BN scores until the 1K-th
iteration are reported as zero. The BN scores of BC samples in the BFFHQ
dataset mostly range from 0.4 to 0.5 while the scores of BA samples are close
to 0. This indicates that the BN score as a loss weight in the BFFHQ imposes a
much larger value on the BC samples, while approximately zero values on the BA
samples. Also, the BN scores of BC samples in the Waterbirds and the BAR
dataset become twice larger than the scores of BA samples. The result shows
that the BN score effectively emphasizes BC samples compared to the BA samples
in $\mathcal{D}^{\text{BN}}_{\text{cand}}$ during the training. Further
analysis of the BN score is included in the Supplementary.
### 4.4 Analysis of intrinsic feature guidance
We conduct a qualitative analysis of the regions emphasized by our intrinsic
feature guidance during the training and the features learned by $f_{d}$ after
training. We use the Waterbirds and BAR datasets with 1% of bias severity for
the analysis.
Visualization of the guidance during training. In Fig. 3, we visualize the
features emphasized by our IE weight $\text{IE}(\mathbf{z})$. For comparison,
we also visualize $\text{E}(\mathbf{z})$, the features focused by the model
before applying $\text{IE}(\mathbf{z})$ for guidance. We select a BA sample as
$\mathbf{x}$ and a BC sample as $\mathbf{x}^{\text{BN}}$ from the training
data for the analysis. For the Waterbirds dataset in Fig. 3(a),
$\text{IE}(\mathbf{z})$ highlights the wings or the beak of the bird compared
to $\text{E}(\mathbf{z})$, where the forest or the water (_i.e_., bias
attributes) is highlighted. Also, in Fig. 3(b), $\text{E}(\mathbf{z})$ focuses
more on the bias attributes such as rocks or the water than the intrinsic
attributes. In contrast, $\text{IE}(\mathbf{z})$ emphasizes the action of the
human that is less exploited compared to the bias features in
$\text{E}(\mathbf{z})$. The results demonstrate that our guidance successfully
identifies and enhances under-exploited intrinsic features during the
training.
Effect of intrinsic feature guidance on debiasing. We qualitatively evaluate
the effectiveness of the intrinsic feature guidance by investigating the
visual explanation maps of the test samples. We compare the Grad-CAM [19]
results of the model trained with and without our method in Fig 4. The Grad-
CAM results highlight the features that the model employs to predict the input
as its ground-truth label. In Fig. 4 (a), while the model trained without
guidance focuses on the forest or the sea, ours focuses on the tail or a
curved shape of the bird’s body. Additionally, Fig. 4 (b) shows that ours
focuses on the motion of the human rather than the backgrounds that are
concentrated on by the model trained without our guidance. The results verify
that our method successfully encourages the model to learn intrinsic features
from the training dataset, improving the robustness of the model against
dataset bias.
### 4.5 Ablation study
As shown in Table 3, we perform ablation studies to verify the effectiveness
of the individual components in our method. The results of ours are reported
in the last row.
Importance of $\mathbf{x}^{\text{BN}}$ selection and BN score as loss weight.
We demonstrate the efficacy of adopting BC samples as
$\mathbf{x}^{\text{BN}}$. We train the model by sampling
$\mathbf{x}^{\text{BN}}$ from three different datasets: $\mathcal{D}$,
$\mathcal{D}^{\text{BN}}_{\text{cand}}$, and $\mathcal{D}^{\text{BN}}$. In the
first row of Table 3, we randomly sample $\mathbf{x}^{\text{BN}}$ from the
training dataset $\mathcal{D}$ without using the BN score. In the second row,
we train the model with $\mathbf{x}^{\text{BN}}$ sampled from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$, where BA samples are roughly filtered
out using the early-stopped biased models. The model in the third row is
trained with $\mathbf{x}^{\text{BN}}$ from $\mathcal{D}^{\text{BN}}$ that
mainly includes BC samples using our BN score. The results show a gradual
improvement in debiasing performance as more BC samples are selected as
$\mathbf{x}^{\text{BN}}$. This is because auxiliary samples without bias
attributes prevent the common features from including the bias features,
composing a bias-contrastive pair with the input. Therefore, our guidance
effectively enhances the intrinsic features during the training. Finally, the
last row in Table 3 presents that employing the BN score
$s(\mathbf{x}^{\text{BN}})$ to reweight the losses further enhances
performance by emphasizing the usage of BC samples as
$\mathbf{x}^{\text{BN}}$.
Training objectives. We examine the impact of each training objective,
$\mathcal{L}_{\text{guide}}$ and $\mathcal{L}_{\text{BN}}$, in our method. We
report the performance of the model trained without
$\mathcal{L}_{\text{guide}}$ (the fourth row) and without
$\mathcal{L}_{\text{BN}}$ (the fifth row) in Table 3. The model trained
without $\mathcal{L}_{\text{guide}}$ exhibits degraded performance, facing
difficulties in identifying where to focus to learn intrinsic features.
Similarly, training the model without $\mathcal{L}_{\text{BN}}$ also results
in a performance decrease. The results verify that $\mathcal{L}_{\text{BN}}$
successfully supports the IE weight to identify intrinsic features among the
common features by learning class-discerning features from
$\mathbf{x}^{\text{BN}}$. The model that incorporates both
$\mathcal{L}_{\text{guide}}$ and $\mathcal{L}_{\text{BN}}$ demonstrates the
best performance (the last row of Table 3).
$\mathcal{L}_{\text{guide}}$ | $\mathcal{L}_{\text{BN}}$ | $\mathbf{x}^{\text{BN}}$ | | $s(\mathbf{x}^{\text{BN}})$ as
---
loss weight
Waterbirds | BFFHQ | BAR
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | $\mathcal{D}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✗ | 62.79 $\pm$1.21 | 71.04 $\pm$2.55 | 73.36 $\pm$1.40
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | $\mathcal{D}^{\text{BN}}_{\text{cand}}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✗ | 64.65 $\pm$1.23 | 75.64 $\pm$1.87 | 74.27 $\pm$0.66
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | $\mathcal{D}^{\text{BN}}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✗ | 65.10 $\pm$0.87 | 77.08 $\pm$2.05 | 74.62 $\pm$1.07
TextRenderingMode=FillStroke, LineWidth=.5pt, ✗ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | $\mathcal{D}^{\text{BN}}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | 63.81 $\pm$1.24 | 76.92 $\pm$1.03 | 74.03 $\pm$1.13
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✗ | $\mathcal{D}^{\text{BN}}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | 62.10 $\pm$3.35 | 74.84 $\pm$2.00 | 74.87 $\pm$1.51
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | $\mathcal{D}^{\text{BN}}$ | TextRenderingMode=FillStroke, LineWidth=.5pt, ✓ | 65.22 $\pm$0.95 | 77.56 $\pm$1.24 | 75.14 $\pm$0.82
Table 3: Ablation study on the proposed training objectives, the dataset that
$\mathbf{x}^{\text{BN}}$ is sampled from, and the BN score of
$\mathbf{x}^{\text{BN}}$ as a loss weight. The check mark (
TextRenderingMode=FillStroke, LineWidth=.5pt, ✓) denotes the inclusion of the
corresponding method, while the cross mark ( TextRenderingMode=FillStroke,
LineWidth=.5pt, ✗) indicates the exclusion of the component in the experiment.
## 5 Conclusion
In this paper, we propose a debiasing method that explicitly provides the
model with spatial guidance for intrinsic features. Leveraging an auxiliary
sample, we first identify intrinsic features by investigating the class-
discerning features commonly appearing in a bias-contrastive pair. Our IE
weight enhances the intrinsic features that have not been focused on yet in
the input by a debiased model. To construct the bias-contrastive pair without
bias labels, we introduce a bias-negative (BN) score that tracks the
classification loss of a biased model to distinguish BC samples from BA
samples during the training. The effectiveness of our method is demonstrated
through experiments on synthetic and real-world datasets with varying levels
of bias severity. We believe this work sheds light on the significance of
providing explicit guidance on the intrinsic attributes for debiasing.
## Acknowledgments
This work was supported by the Institute for Information & communications
Technology Promotion(IITP) grant funded by the Korea government(MSIT)
(No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST), the
National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No. NRF-2022R1A2B5B02001913 & No. 2022R1A5A7083908) and
partly by Kakao Brain Corporation.
## References
* Asgari et al. [2022] Saeid Asgari, Aliasghar Khani, Fereshte Khani, Ali Gholami, Linh Tran, Ali Mahdavi Amiri, and Ghassan Hamarneh. Masktune: Mitigating spurious correlations by forcing to explore. In _Proc. the Advances in Neural Information Processing Systems (NeurIPS)_ , pages 23284–23296, 2022.
* Bahng et al. [2020] Hyojin Bahng, Sanghyuk Chun, Sangdoo Yun, Jaegul Choo, and Seong Joon Oh. Learning de-biased representations with biased representations. In _Proc. the International Conference on Machine Learning (ICML)_ , 2020.
* Darlow et al. [2020] Luke Darlow, Stanislaw Jastrzebski, and Amos Storkey. Latent adversarial debiasing: Mitigating collider bias in deep neural networks. _arXiv preprint arXiv:2011.11486_ , 2020.
* Geirhos et al. [2019] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A. Wichmann, and Wieland Brendel. Imagenet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness. In _Proc. the International Conference on Learning Representations (ICLR)_ , 2019.
* He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , 2016.
* Hu et al. [2018] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation networks. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , 2018.
* Huang et al. [2020] Zeyi Huang, Haohan Wang, Eric P. Xing, and Dong Huang. Self-challenging improves cross-domain generalization. In _Proc. of the European Conference on Computer Vision (ECCV)_ , 2020.
* Hwang et al. [2022] Inwoo Hwang, Sangjun Lee, Yunhyeok Kwak, Seong Joon Oh, Damien Teney, Jin-Hwa Kim, and Byoung-Tak Zhang. Selecmix: Debiased learning by contradicting-pair sampling. In _Proc. the Advances in Neural Information Processing Systems (NeurIPS)_ , 2022.
* Kim et al. [2019] Byungju Kim, Hyunwoo Kim, Kyungsu Kim, Sungjin Kim, and Junmo Kim. Learning not to learn: Training deep neural networks with biased data. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , 2019.
* Kim et al. [2021] Eungyeup Kim, Jihyeon Lee, and Jaegul Choo. Biaswap: Removing dataset bias with bias-tailored swapping augmentation. In _Proc. of the IEEE international conference on computer vision (ICCV)_ , pages 14992–15001, 2021.
* Kwon et al. [2022] Bum Chul Kwon, Jungsoo Lee, Chaeyeon Chung, Nyoungwoo Lee, Ho-Jin Choi, and Jaegul Choo. DASH: Visual Analytics for Debiasing Image Classification via User-Driven Synthetic Data Augmentation. _Eurographics Conference on Visualization (EuroVis)-Short Papers. The Eurographics Association_ , 2022.
* Lee et al. [2021] Jungsoo Lee, Eungyeup Kim, Juyoung Lee, Jihyeon Lee, and Jaegul Choo. Learning debiased representation via disentangled feature augmentation. In _Proc. the Advances in Neural Information Processing Systems (NeurIPS)_ , 2021.
* Lee et al. [2023] Jungsoo Lee, Jeonghoon Park, Daeyoung Kim, Juyoung Lee, Edward Choi, and Jaegul Choo. Revisiting the importance of amplifying bias for debiasing. In _Proc. the AAAI Conference on Artificial Intelligence (AAAI)_ , pages 14974–14981, 2023.
* Liu et al. [2021] Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training group information. In _Proc. the International Conference on Machine Learning (ICML)_ , pages 6781–6792, 2021.
* Nam et al. [2020] Junhyun Nam, Hyuntak Cha, Sungsoo Ahn, Jaeho Lee, and Jinwoo Shin. Learning from failure: Training debiased classifier from biased classifier. In _Proc. the Advances in Neural Information Processing Systems (NeurIPS)_ , 2020.
* Park et al. [2023] Geon Yeong Park, Sangmin Lee, Sang Wan Lee, and Jong Chul Ye. Training debiased subnetworks with contrastive weight pruning. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , pages 7929–7938, 2023.
* Park et al. [2020] Taesung Park, Jun-Yan Zhu, Oliver Wang, Jingwan Lu, Eli Shechtman, Alexei A. Efros, and Richard Zhang. Swapping autoencoder for deep image manipulation. In _Proc. the Advances in Neural Information Processing Systems (NeurIPS)_ , 2020.
* Sagawa et al. [2020] Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. _Proc. the International Conference on Learning Representations (ICLR)_ , 2020.
* Selvaraju et al. [2017] Ramprasaath R. Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In _Proc. of the IEEE international conference on computer vision (ICCV)_ , 2017.
* Simonyan and Zisserman [2015] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. _Proc. the International Conference on Learning Representations (ICLR)_ , 2015.
* Szegedy et al. [2015] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , pages 1–9, 2015.
* Tartaglione et al. [2021] Enzo Tartaglione, Carlo Alberto Barbano, and Marco Grangetto. End: Entangling and disentangling deep representations for bias correction. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , pages 13508–13517, 2021.
* Torralba and Efros [2011] Antonio Torralba and Alexei A. Efros. Unbiased look at dataset bias. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , pages 1521–1528, 2011.
* Wah et al. [2011] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011\.
* Wang et al. [2019] Haohan Wang, Zexue He, Zachary L. Lipton, and Eric P. Xing. Learning robust representations by projecting superficial statistics out. In _Proc. the International Conference on Learning Representations (ICLR)_ , 2019.
* Zagoruyko and Komodakis [2016] Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. _Proc. of the British Machine Vision Conference (BMVC)_ , 2016.
* Zhang et al. [2018] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In _Proc. the International Conference on Learning Representations (ICLR)_ , 2018.
* Zhang et al. [2023] Yi-Kai Zhang, Qi-Wei Wang, De-Chuan Zhan, and Han-Jia Ye. Learning debiased representations via conditional attribute interpolation. In _Proc. of the IEEE conference on computer vision and pattern recognition (CVPR)_ , pages 7599–7608, 2023.
* Zhang and Sabuncu [2018] Zhilu Zhang and Mert R Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. _arXiv preprint arXiv:1805.07836_ , 2018.
* Zhou et al. [2017] Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image database for scene recognition. _The IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_ , 40(6):1452–1464, 2017.
Supplementary Material
This supplementary material offers further analysis of our approach,
additional experimental results, the details of the datasets and
implementation, limitations, and future work. Appendix A and Appendix B
provide the analysis of the bias-negative (BN) score as a loss weight and
samples with negative BN score, respectively. Appendix C analyzes the effect
of BC samples in $\mathcal{D}^{\text{BN}}$ on debiasing performance. Also,
Appendix D compares the recent sample selection methods with ours. Moreover,
Appendix E and Appendix F present additional qualitative results regarding the
guidance and additional quantitative results, respectively. Appendix G and
Appendix H provide the details about the dataset and implementation. Lastly,
Appendix I discusses the limitations and future work.
## Appendix A Additional analysis of the BN score as a loss weight
Figure 5: The distributions of $f_{b}$’s classification loss of samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$. The red and blue lines denote the
losses of BA and BC samples, respectively. The dotted and solid lines indicate
the losses at the early and later stages of the training, respectively. Best
viewed in color.
As described in Sec. 3.4 in the main paper, we utilize the BN score of
$\mathbf{x}^{\text{BN}}$ (_i.e_., $s(\mathbf{x}^{\text{BN}})$) to reweight the
guidance loss $\mathcal{L}_{\text{guide\\_sim}}$ and the BN loss
$\mathcal{L}_{\text{BN}}$. The BN score as a loss weight is designed to
upweight the losses when bias-conflicting (BC) samples are selected as
$\mathbf{x}^{\text{BN}}$, which further encourages our IE weight to enhance
the intrinsic features. For verification, we present that the BN score has a
much larger value on the BC samples compared to bias-aligned (BA) samples
during the training in Fig. 2 in the main paper.
Since $s(\mathbf{x}^{\text{BN}})$ has a larger value when the current $f_{b}$
loss of $\mathbf{x}^{\text{BN}}$ is larger than that of the early stage of
training, the results imply that the $f_{b}$ loss of BC samples largely
increases as training proceeds compared to BA samples.
To further verify this, we present $f_{b}$’s classification loss of samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$ during the training in Fig. 5. The
BFFHQ dataset [10] with a bias severity of 1% is used for the analysis. In
Fig. 5, the dotted lines denote the distribution of $f_{b}$’s classification
loss at the early stage of training (1K-th iteration), and the solid lines
indicate that of the later stage of training (50K-th iteration). The results
show that the $f_{b}$ loss of BC samples (blue lines) largely increases at the
later stage of training compared to the early stage, unlike BA samples (red
lines). This demonstrates that the BN score as a loss weight can effectively
upweight the training losses when BC samples are chosen as
$\mathbf{x}^{\text{BN}}$.
## Appendix B Samples having negative BN score
Figure 6: The examples of samples that have negative BN scores at the later
stage of training. Figure 7: Additional visualization results of the spatial
guidance using (a) Waterbirds and (b) BAR dataset. Given bias-contrastive
pairs, $\mathbf{x}$ and $\mathbf{x}^{\text{BN}}$, $\text{E}(\mathbf{z})$
indicates the regions originally focused on by $f_{d}$ and
$\text{IE}(\mathbf{z})$ shows the regions highlighted by our IE weight.
As mentioned in Sec. 3.2 in the main paper, we further filter out the samples
with negative BN scores from $\mathcal{D}_{\text{cand}}^{\text{BN}}$ to mainly
exploit the BC samples as $\mathbf{x}^{\text{BN}}$. Here, we expect that the
samples with negative BN scores are mostly BA samples. To investigate the
samples with negative BN scores, we chose the samples that were erroneously
incorporated into $\mathcal{D}_{\text{cand}}^{\text{BN}}$ initially but
excluded at the later stage of training (_i.e_., 50K-th iteration), exhibiting
negative BN scores. This process is repeated five times, and we visualize the
samples chosen more than three times in Fig. 6. We use the BFFHQ dataset with
a 1% bias severity for the experiment.
We observe that the samples with negative BN scores in
$\mathcal{D}_{\text{cand}}^{\text{BN}}$ are mostly BA samples. As shown in the
figure, while the samples obviously contain bias attributes (_i.e_., features
representing female or male), the samples mostly have extreme shade, blur,
saturation, or unusual makeup, exhibiting non-typical appearance. Although the
bias attributes are known to be easy to learn, the non-typical appearance
prevents $f_{b}$ from detecting such bias attributes in the early stage of
training. In Sec. 4.5 of the main paper, we verify that employing such BA
samples as $\mathbf{x}^{\text{BN}}$ largely degrades the debiasing performance
by allowing the bias attributes to be included in the common features between
$\mathbf{x}$ and $\mathbf{x}^{\text{BN}}$. Our BN score effectively alleviates
this issue by filtering out such BA samples from
$\mathcal{D}_{\text{cand}}^{\text{BN}}$.
Figure 8: Additional comparison of the region focused by a debiased model
trained with and without our method. We compare Grad-CAM results on the test
set of (a) Waterbirds and (b) BAR.
## Appendix C Importance of BN sample selection
We analyze the effect of the BC sample ratio in $\mathcal{D}^{\text{BN}}$ on
debiasing performance. We measure the accuracy using the BFFHQ dataset with a
1% bias severity by varying the number of BA and BC samples in
$\mathcal{D}^{\text{BN}}$. Table 4 shows that higher accuracy is achieved for
more BC samples and a lower ratio of BA to BC samples in
$\mathcal{D}^{\text{BN}}$. Overall, our method constantly shows performance
gain, except for the last column ({${\text{\\#BC in
}\mathcal{D}^{\text{BN}}}/{\text{\\#BC in }\mathcal{D}}$, ${\text{\\#BA in
}\mathcal{D}^{\text{BN}}}/{\text{\\#BC in }\mathcal{D}^{\text{BN}}}$}-{1.0,
10.0}). It is crucial not to select too many BA samples as
$\mathbf{x}^{\text{BN}}$.
${\text{\\#BC in }\mathcal{D}^{\text{BN}}}/{\text{\\#BC in }\mathcal{D}}$ | 0.1 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
---|---|---|---|---|---|---|---
${\text{\\#BA in }\mathcal{D}^{\text{BN}}}/{\text{\\#BC in }\mathcal{D}^{\text{BN}}}$ | 0.0 | 0.0 | 0.0 | 0.1 | 1.0 | 2.0 | 10.0
Accuracy | 75.84 | 78.12 | 81.40 | 80.24 | 77.48 | 75.48 | 70.90
Table 4: Importance of $\mathbf{x}^{\text{BN}}$ selection.
## Appendix D Comparison to recent sample selection methods
Our BN score is designed to further filter out BA samples in
$\mathcal{D}^{\text{BN}}_{\text{cand}}$, improving debiasing performance (Sec.
4.5 in the main paper). We compare BC sample selection in recent methods with
ours using BFFHQ with a 1% bias severity. Let $\mathcal{S}$ be a set of
samples identified as BC samples from training data $\mathcal{D}$.
$\\{\text{\\#BC in }\mathcal{S}/\text{\\#BC in }\mathcal{D},\text{\\#BA in
}\mathcal{S}/\text{\\#BC in }\mathcal{S}\\}$ is {75.63, 10.18}-BE [13],
{27.29, 4.32}-DCWP [16], and {50.0, 0.89}-Ours, respectively. Our method has
the least number of BA compared to BC samples in $\mathcal{S}$ while
preserving half of the total BC samples.
## Appendix E Additional qualitative results
### E.1 Visualization of the guidance during training
In addition to Fig. 3 of the main paper, we provide supplementary qualitative
results that present the features that the current model $f_{d}$ focuses on
(_i.e_., $\text{E}(\mathbf{z})$) and the features emphasized by the guidance
(_i.e_., $\text{IE}(\mathbf{z})$) during the training in Fig. 7. We use the
Waterbirds and BAR datasets with a bias severity of 1% for the analysis. We
train $f_{d}$ during 10K iterations and obtain the visual explanation map
$\text{E}(\mathbf{z})$ for the ground-truth label using Grad-CAM [19]. The
min-max normalization is applied to the values of $\text{E}(\mathbf{z})$ and
$\text{IE}(\mathbf{z})$ for visualization. We intentionally select the BA
sample and BC sample as $\mathbf{x}$ and $\mathbf{x}^{\text{BN}}$,
respectively, to compose a bias-negative pair.
As shown in Fig. 7, our IE weight (_i.e_., $\text{IE}(\mathbf{z})$)
appropriately emphasizes the regions of the intrinsic features while
$\text{E}(\mathbf{z})$ shows that the current model $f_{d}$ relies on the bias
attributes for prediction. For example, in the Waterbirds dataset,
$\text{IE}(\mathbf{z})$ properly enhances the intrinsic features of a bird
such as wings, a body, or a neck, while $\text{E}(\mathbf{z})$ highlights the
background features such as the land, the forest or the water. Also, in the
BAR dataset, $\text{IE}(\mathbf{z})$ emphasizes the arm throwing the javelin,
the motion of a person vaulting, climbing, or diving, while the current
$f_{d}$ mainly focuses on the biased features such as the playing field, the
sky, the mountain, or the water. These results verify the validity of our IE
weight $\text{IE}(\mathbf{z})$ as guidance for emphasizing the intrinsic
features in $\mathbf{x}$ that are under-exploited yet.
### E.2 Effect of intrinsic feature guidance on debiasing
We present an additional qualitative analysis regarding the effectiveness of
the intrinsic feature guidance to supplement Fig. 4 in the main paper. Fig. 8
illustrates the Grad-CAM [19] results of the model trained with and without
our method. Here, the model trained without our method is the same as LfF+BE
[13]. We train the models with the Waterbids and the BAR datasets with a bias
severity of 1% and apply the Grad-CAM to the test samples for visualization.
The highlighted regions indicate the features that the model mainly employs
for prediction.
Fig. 8 (a) shows that our approach properly focuses on the intrinsic features
of the bird (_e.g_., wings, a beak, or feet), while the model trained without
our guidance mostly concentrates on the bias features (_e.g_., the water or
trees). For the BAR dataset in Fig. 8 (b), our model principally exploits the
action of a person (_e.g_., fishing, vaulting, or throwing) or the racing car
for prediction, while the model without our guidance focuses on the
backgrounds (_e.g_., the playing field or the water). The results demonstrate
the effectiveness of our method in guiding the model to learn intrinsic
features.
Method | Waterbirds
---|---
0.5 | 1.0 | 2.0 | 5.0
Vanilla [5] | 57.41 $\pm$0.74 | 58.07 $\pm$1.00 | 61.04 $\pm$0.55 | 64.13 $\pm$0.14
HEX [25] | 57.88 $\pm$0.83 | 58.28 $\pm$0.67 | 61.02 $\pm$0.48 | 64.32 $\pm$0.62
LNL [9] | 58.49 $\pm$0.81 | 59.68 $\pm$0.78 | 62.27 $\pm$0.91 | 66.07 $\pm$1.15
EnD [22] | 58.47 $\pm$0.97 | 57.81 $\pm$1.04 | 61.26 $\pm$0.54 | 64.11 $\pm$0.52
ReBias [2] | 55.44 $\pm$0.24 | 55.93 $\pm$0.66 | 58.53 $\pm$0.52 | 62.14 $\pm$1.03
LfF [15] | 60.66 $\pm$0.77 | 61.78 $\pm$1.53 | 58.92 $\pm$2.93 | 61.43 $\pm$1.92
DisEnt [12] | 59.59 $\pm$1.67 | 60.05 $\pm$0.82 | 59.76 $\pm$1.26 | 64.01 $\pm$1.36
LfF+BE [13] | 61.22 $\pm$2.54 | 62.58 $\pm$1.12 | 63.00 $\pm$1.18 | 63.48 $\pm$0.56
DisEnt+BE [13] | 51.65 $\pm$1.45 | 54.10 $\pm$1.04 | 53.43 $\pm$1.42 | 54.21 $\pm$1.36
Ours | 63.64 $\pm$1.63 | 65.22 $\pm$0.95 | 65.23 $\pm$1.06 | 66.33 $\pm$1.42
Table 5: Comparison to the baselines. We measure the classification accuracy
on test sets of the Waterbirds dataset with different bias severities. The
best accuracy values are in bold. Results with standard deviations are
provided in the Supplementary.
Method | BFFHQ | BAR
---|---|---
0.5 | 1.0 | 2.0 | 5.0 | 1.0 | 5.0
Vanilla [5] | 55.64 $\pm$0.44 | 60.96 $\pm$1.00 | 69.00 $\pm$0.50 | 82.88 $\pm$0.49 | 70.55 $\pm$0.87 | 82.53 $\pm$1.08
HEX [25] | 56.96 $\pm$0.62 | 62.32 $\pm$1.21 | 70.72 $\pm$0.89 | 83.40 $\pm$0.34 | 70.48 $\pm$1.74 | 81.20 $\pm$0.68
LNL [9] | 56.88 $\pm$1.13 | 62.64 $\pm$0.99 | 69.80 $\pm$1.03 | 83.08 $\pm$0.93 | - | -
EnD [22] | 55.96 $\pm$0.91 | 60.88 $\pm$1.17 | 69.72 $\pm$1.14 | 82.88 $\pm$0.74 | - | -
ReBias [2] | 55.76 $\pm$1.50 | 60.68 $\pm$1.24 | 69.60 $\pm$1.33 | 82.64 $\pm$0.64 | 73.04 $\pm$1.04 | 83.90 $\pm$0.82
LfF [15] | 65.19 $\pm$3.23 | 69.24 $\pm$2.07 | 73.08 $\pm$2.70 | 79.80 $\pm$1.09 | 70.16 $\pm$0.77 | 82.95 $\pm$0.27
DisEnt [12] | 62.08 $\pm$3.89 | 66.00 $\pm$1.33 | 69.92 $\pm$2.72 | 80.68 $\pm$0.25 | 70.33 $\pm$0.19 | 83.13 $\pm$0.46
LfF+BE [13] | 67.36 $\pm$3.10 | 75.08 $\pm$2.29 | 80.32 $\pm$2.07 | 85.48 $\pm$2.88 | 73.36 $\pm$0.97 | 83.87 $\pm$0.82
DisEnt+BE [13] | 67.56 $\pm$2.11 | 73.48 $\pm$2.12 | 79.48 $\pm$1.80 | 84.84 $\pm$2.11 | 73.29 $\pm$0.41 | 84.96 $\pm$0.69
DCWP [16] | 64.08 $\pm$1.08 | 67.44 $\pm$2.87 | 75.24 $\pm$1.73 | 85.00 $\pm$0.94 | 69.63 $\pm$0.85 | 81.89 $\pm$0.68
Ours | 71.68 $\pm$1.74 | 77.56 $\pm$1.24 | 83.08 $\pm$1.69 | 87.60 $\pm$1.68 | 75.14 $\pm$0.82 | 85.03 $\pm$0.64
Table 6: Comparison to the baselines. We measure the classification accuracy
on test sets of the BFFHQ and BAR datasets with different bias severities. The
best accuracy values are in bold. The hyphen mark (-) means it is not
applicable. Results with standard deviations are provided in the
Supplementary.
## Appendix F Additional quantitative results
### F.1 Quantitative results with standard deviations
In Table 1 of our main paper, we report the quantitative comparison results
with classification accuracies on the test set which are averaged across five
independent experiments with different random seeds. We additionally provide
the standard deviations of the classification accuracies in Table 5 and Table
6. Each table shows the results of the synthetic dataset (_i.e_., Waterbirds)
and the real-world dataset (_i.e_., BFFHQ and BAR), respectively. Since the
BAR dataset lacks explicit bias labels, approaches such as LNL and EnD that
necessitate explicit bias labels are not applicable to the BAR dataset. The
baseline results for the BFFHQ and the BAR dataset are from the results
reported in BE [13] except for DCWP [16].
### F.2 Comparison to recent baseline
Our primary contribution lies in providing the model with explicit spatial
guidance for intrinsic features by examining features that commonly appear in
bias-contrastive pairs. The intrinsic feature exists in generally appearing
features within a class, however, this property has not been tackled to
provide intrinsic feature guidance in prior studies to the best of our
knowledge. While recent debiasing approaches aim to encourage the model to
learn intrinsic features, they fail to directly indicate where the model
should focus to learn the features.
For instance, MaskTune [1] expects the model to learn intrinsic features by
fine-tuning the model with the data whose already-explored area is masked out
using GradCAM. However, simply exploring the unmasked area cannot inform the
model where exactly the intrinsic features are located. In this case, the
model may rather focus on non-intrinsic features during the fine-tuning. We
experiment on real-world datasets with a 1% bias severity: {58.00,
69.42}-MaskTune and {77.56, 75.14}-Ours for {BFFHQ, BAR}. Ours achieves better
debiasing performance by providing explicit spatial guidance for intrinsic
features based on common features in bias-contrastive pairs.
A recent pair-wise debiasing method $\mathcal{X}^{2}$-model [28] encourages
the model to retain intra-class compactness using samples generated via
feature-level interpolation between BC and BA samples. However,
$\mathcal{X}^{2}$-model does not inform the model where the intrinsic features
are located in the interpolated features. Simply making samples closer to the
interpolated samples does not assure the model to focus on the intrinsic
features. In contrast, our method directly encourages the model to focus on
the area of the intrinsic features.
Also, we conduct a quantitative comparison to the recently proposed debiasing
approach, DCWP [16], in Table 6. We use real-world datasets, BFFHQ and BAR,
with various bias severity. For a fair comparison, we utilize ResNet18, which
is the same architecture as ours. The ImageNet pretrained weight is employed
only for the BAR dataset. The results demonstrate the superiority of our
method over the DCWP, where ours provides the model with explicit guidance for
intrinsic features for debiasing, unlike DCWP.
### F.3 Worst accuracy between the accuracy of BA and BC samples in
Waterbirds
BS | Vanilla [5] | HEX [25] | LNL [9] | EnD [22] | ReBias [2] | LfF [15] | DisEnt [12] | LfF+BE [13] | DisEnt+BE [13] | Ours
---|---|---|---|---|---|---|---|---|---|---
0.5 | 24.08 $\pm$1.56 | 28.20 $\pm$3.07 | 26.08 $\pm$1.64 | 28.29 $\pm$3.53 | 27.00 $\pm$1.10 | 56.22 $\pm$6.07 | 38.07 $\pm$11.01 | 55.15 $\pm$2.78 | 36.60 $\pm$10.88 | 59.12 $\pm$3.67
1.0 | 24.78 $\pm$2.45 | 26.32 $\pm$2.90 | 29.72 $\pm$3.45 | 25.69 $\pm$2.41 | 27.95 $\pm$1.56 | 59.07 $\pm$3.40 | 47.02 $\pm$7.26 | 55.53 $\pm$1.60 | 28.35 $\pm$4.17 | 63.05 $\pm$1.97
2.0 | 34.39 $\pm$2.24 | 32.12 $\pm$2.89 | 33.92 $\pm$1.94 | 32.94 $\pm$1.48 | 32.16 $\pm$0.76 | 53.07 $\pm$6.74 | 44.93 $\pm$8.54 | 52.91 $\pm$2.62 | 31.08 $\pm$6.01 | 61.71 $\pm$4.94
5.0 | 38.34 $\pm$1.05 | 39.08 $\pm$0.92 | 43.22 $\pm$1.94 | 40.91 $\pm$1.11 | 39.72 $\pm$1.11 | 58.05 $\pm$2.37 | 52.96 $\pm$6.33 | 48.48 $\pm$3.72 | 37.92 $\pm$6.47 | 58.60 $\pm$3.32
Table 7: The worst accuracy between the accuracy of BA and BC samples in the
Waterbirds dataset. BS is bias severity.
To further analyze our model’s performance on the Waterbirds dataset, we
measure the accuracy of BA and BC samples separately, where the class accuracy
values are averaged. Then, we report the worst accuracy between them in Table
7. The results show that ours achieves the highest worst accuracy compared to
other baselines.
## Appendix G Detailed description of datasets
Figure 9: Visualization of datasets used in the experiments. A group of three
columns represents each class for (a) Waterbirds-{Landbird, Waterbird} and (b)
BFFHQ-{Young, Old}, and each column of (c) BAR-{Climbing, Diving, Fishing,
Vaulting, Racing, Throwing} represents a distinct class. The samples above the
dashed line are bias-aligned samples and the below ones are bias-conflicting
samples.
We utilize Waterbirds [18], BFFHQ [10], and BAR [15] dataset. First, the
Waterbirds dataset is composed of two classes of bird images and has
background bias. In the training set, the waterbirds are mostly with the water
background and the landbirds are with the land background. The number of BA
samples and that of BC samples are balanced in the test set. By following
Sagawa et al. [18], we utilize two datsets, the Caltech-UCSD Birds-200-2011
(CUB) dataset [24] and the Places111CC BY dataset [30], to construct the
Waterbirds dataset. The bird images are segmented from the CUB dataset, and
the segmented birds are combined with the background images from the Place
dataset. We employ the code released by Sagawa et al.
[18]222https://github.com/kohpangwei/group_DRO for constructing the dataset.
As mentioned in the repository, a few landbirds (Eastern Towhees, Western
Meadowlarks, and Western Wood Pewees) in the original dataset are incorrectly
labeled as waterbirds. Therefore, we correct their labels to landbirds for the
experiments.
The BFFHQ is initially presented by Kim et al. [10] and constructed by
modifying the FFHQ dataset 333BY-NC-SA 4.0. In the BFFHQ, the bias attribute
is the gender and the intrinsic attribute is the age. Specifically, most of
the young people are female, and most of the old people are male in the
training dataset.
Lastly, the BAR dataset is introduced by Nam et al. [15]. The dataset contains
six action classes (_i.e_., Climbing, Diving, Fishing, Vaulting, Racing,
Throwing) and each class is biased to a certain background (_i.e_., RockWall,
Underwater, WaterSurface, Sky, APavedTrack, PlayingField). In the test set,
such correlations do not exist. For the experiments, we use the BFFHQ dataset
and BAR dataset released by Lee et al.
[13]444https://github.com/kakaoenterprise/BiasEnsemble. The examples of the BA
samples and BC samples in each dataset are shown in Fig. 9.
## Appendix H Implementation details
Following the previous studies [15, 12, 13], we utilize ResNet18 [5] for the
biased model $f_{b}$ and the debiased model $f_{d}$. Also,
$f_{d}^{\text{emb}}$ indicates the subnetwork before the average pooling
layer, and $f_{d}^{\text{cls}}$ consists of an average pooling layer and a
linear classifier that outputs logits, where
$f_{d}(\mathbf{x})=f_{d}^{\text{cls}}\left(f_{d}^{\text{emb}}(\mathbf{x})\right)$.
Before training, $f_{b}$ and $f_{d}$ are initialized with the ImageNet
pretrained weight for the BAR dataset, while we randomly initialize the models
for the other datasets. This is because the size of the BAR dataset is
extremely small compared to the others [13].
During training $f_{d}$, we employ the sample reweighting value
$w(\mathbf{x})$ termed as relative difficulty score [15], as mentioned in Sec.
3.4 in the main paper. $w(\mathbf{x})$ is calculated as follows:
$w(\mathbf{x})=\frac{\mathcal{L}_{\text{CE}}(f_{b}(\mathbf{x}),y)}{\mathcal{L}_{\text{CE}}(f_{b}(\mathbf{x}),y)+\mathcal{L}_{\text{CE}}(f_{d}(\mathbf{x}),y)}.$
(14)
This score assigns a high weight to the BC samples and a low weight to the BA
samples. This encourages $f_{d}$ to mainly learn intrinsic features by
emphasizing BC samples with $w(\mathbf{x})$.
The models are trained for 50K iterations with a batch size of 64. The
horizontal flip and a random crop with a size of 224 are used for data
augmentation during the training. All the models are trained with the Adam
optimizer. The learning rate is set as 1e-4 for the Waterbirds and the BFFHQ
dataset, and 1e-5 for the BAR dataset. The hyperparameters of $\alpha_{l}$,
$\alpha_{s},$ and $\tau$ are set as $0.1$, $0.9$, and $2$, respectively, for
all the datasets. We apply class-wise max normalization to our BN score to
consider the different ranges of the scores across the classes for stability
of training.
During the training, we aim to select an auxiliary sample that has no bias
attribute but has the same class label with $\mathbf{x}$ as
$\mathbf{x}^{\text{BN}}$ from $\mathcal{D}^{\text{BN}}$. If no sample in
$\mathcal{D}^{\text{BN}}$ has the same label as $\mathbf{x}$, we select the
sample that has the same label with $\mathbf{x}$ from
$\mathcal{D}^{\text{BN}}_{\text{cand}}$. In a case where there’s no sample
with the same label as $\mathbf{x}$ in both $\mathcal{D}^{\text{BN}}$ and
$\mathcal{D}^{\text{BN}}_{\text{cand}}$, we sample $\mathbf{x}^{\text{BN}}$
that has the same label with $\mathbf{x}$ from $\mathcal{D}$.
As described in Sec. 3.1 in the main paper, we utilize the pretrained biased
models to construct $\mathcal{D}^{\text{A}}$, following the previous work
[13]. We utilize ResNet18 [5] for the pretrained biased models, and all the
pretrained biased models are randomly initialized before training. The models
are trained for 1K iterations with the generalized cross entropy (GCE) loss
[29]. Within each model, the samples with a high ground-truth probability
(i.e., $\geq$ 0.99) are considered as BA samples. Based on majority voting, we
collect the samples that are considered as the BA sample by the majority of
the models and construct the bias-amplified dataset $\mathcal{D}^{\text{A}}$.
We use five pretrained biased models following the study of Lee _et al_. [13].
Lee _et al_. demonstrate that adopting the additional biased models requires a
negligible amount of additional computational costs and memory space. Note
that the same biased models are utilized when constructing
$\mathcal{D}_{\text{cand}}^{\text{BN}}$.
## Appendix I Limitations and future work
Although our BN score effectively encourages BC samples to be mainly adopted
as auxiliary inputs, the auxiliary inputs still can include a few BA samples,
as shown in Table 2 of Sec. 4.3 in the main paper. Accordingly, such BA
samples may interfere with the model to capture the intrinsic features.
Identifying intrinsic attributes without relying on auxiliary inputs can be
one promising future work.
In addition, since our IE weight is designed to enhance intrinsic features by
imposing spatially different values on the features, our method might be more
effective especially when bias attributes are located in different regions
from the intrinsic attributes. In this regard, applying channel-wise re-
weighting [6] to our approach will further improve the general applicability
of our method.
Despite the limitations above, we believe that our work poses the importance
of enhancing intrinsic attributes for debiasing.
|
11institutetext: Physics Department, Columbia University, New York, NY 10027,
USA 22institutetext: Center for Astrophysics $|$ Harvard & Smithsonian,
Cambridge, MA 02138, USA 33institutetext: Department of Physics, Washington
University, St. Louis, MO 63130, USA 44institutetext: Physics Department,
California Polytechnic State University, San Luis Obispo, CA 94307, USA
55institutetext: Department of Astronomy and Astrophysics, 525 Davey Lab,
Pennsylvania State University, University Park, PA 16802, USA 66institutetext:
Department of Physics and Astronomy, Barnard College, Columbia University, NY
10027, USA 77institutetext: Department of Physics and Astronomy, Purdue
University, West Lafayette, IN 47907, USA 88institutetext: Department of
Physics and Astronomy and the Bartol Research Institute, University of
Delaware, Newark, DE 19716, USA 99institutetext: School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
1010institutetext: Department of Physics, California State University - East
Bay, Hayward, CA 94542, USA 1111institutetext: DESY, Platanenallee 6, 15738
Zeuthen, Germany 1212institutetext: Physics Department, McGill University,
Montreal, QC H3A 2T8, Canada 1313institutetext: Santa Cruz Institute for
Particle Physics and Department of Physics, University of California, Santa
Cruz, CA 95064, USA 1414institutetext: Department of Physics and Astronomy,
University of Utah, Salt Lake City, UT 84112, USA 1515institutetext:
Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL
35487, USA 1616institutetext: Department of Physics and Astronomy, University
of Iowa, Van Allen Hall, Iowa City, IA 52242, USA 1717institutetext: School of
Physics, National University of Ireland Galway, University Road, Galway,
Ireland 1818institutetext: Department of Physics, Engineering Physics, and
Astronomy, Queen’s University, Kingston, ON K7L 3N6, Canada 1919institutetext:
Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm,
Germany 2020institutetext: School of Physics, University College Dublin,
Belfield, Dublin 4, Ireland 2121institutetext: Department of Physical
Sciences, Munster Technological University, Bishopstown, Cork, T12 P928,
Ireland 2222institutetext: Department of Physics and Astronomy, University of
California, Los Angeles, CA 90095, USA 2323institutetext: Department of
Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
2424institutetext: Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Av. Complutense, 40, E-28040 Madrid, Spain
2525institutetext: Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain 2626institutetext: Universidad de La Laguna, Dept.
Astrofísica, E-38206 La Laguna, Tenerife, Spain
# The throughput calibration of the VERITAS telescopes
C. B. Adams 11 W. Benbow 22 A. Brill 11 J. H. Buckley 33 J. L. Christiansen 44
A. Falcone 55 Q. Feng 66 J. P. Finley 77 G. M Foote 88 L. Fortson 99 A.
Furniss 1010 C. Giuri 1111 D. Hanna 1212 T. Hassan 11112424 O. Hervet 1313 J.
Holder 88 B. Hona 1414 T. B. Humensky 11 W. Jin 1515 P. Kaaret 1616 T. K
Kleiner 1111 S. Kumar 1212 M. J. Lang 1717 M. Lundy 1212 G. Maier 11 and 22211
and 222 P. Moriarty 1717 R. Mukherjee 66 M. Nievas Rosillo 11 and 25 and 26
and 11111 and 25 and 26 and 111 S. O’Brien 1212 N. Park 1818 S. Patel 1616 K.
Pfrang 1111 M. Pohl 19 and 1119 and 11 R. R. Prado 1111 E. Pueschel 1111 J.
Quinn 2020 K. Ragan 1212 P. T. Reynolds 2121 D. Ribeiro 11 E. Roache 22 J. L.
Ryan 2222 M. Santander 1515 A. Weinstein 2323 D. A. Williams 1313 T. J
Williamson 88<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received ¡date¿ / Accepted ¡date¿)
###### Abstract
Context. The response of imaging atmospheric Cherenkov telescopes to incident
$\gamma$-ray-initiated showers in the atmosphere changes as the telescopes age
due to exposure to light and weather. These aging processes affect the
reconstructed energies of the events and $\gamma$-ray fluxes.
Aims. This work discusses the implementation of signal calibration methods for
the Very Energetic Radiation Imaging Telescope Array System (VERITAS) to
account for changes in the optical throughput and detector performance over
time.
Methods. The total throughput of a Cherenkov telescope is the product of
camera-dependent factors, such as the photomultiplier tube gains and their
quantum efficiencies, and the mirror reflectivity and Winston cone response to
incoming radiation. This document summarizes different methods to determine
how the camera gains and mirror reflectivity have evolved over time and how we
can calibrate this changing throughput in reconstruction pipelines for imaging
atmospheric Cherenkov telescopes. The implementation is validated against
seven years of observations with the VERITAS telescopes of the Crab Nebula,
which is a reference object in very-high-energy astronomy.
Results. Regular optical throughput monitoring and the corresponding signal
calibrations are found to be critical for the reconstruction of extensive air
shower images. The proposed implementation is applied as a correction to the
signals of the photomultiplier tubes in the telescope simulation to produce
fine-tuned instrument response functions. This method is shown to be effective
for calibrating the acquired $\gamma$-ray data and for recovering the correct
energy of the events and photon fluxes. At the same time, it keeps the
computational effort of generating Monte Carlo simulations for instrument
response functions affordably low.
###### Key Words.:
Throughput calibration, Instrument Response Functions, Cherenkov telescopes
## 1 Introduction
When energetic $\gamma$-rays or charged particles (typically protons, atomic
nuclei, or electrons) enter the atmosphere, they generate a cascade of
secondary particles (also known as an extensive air shower or EAS) through
pair production, bremsstrahlung emission, and, for the case of hadronic
showers, fragmentation and decay of unstable mesons ($\pi^{0}$, $\pi^{\pm}$,
and $K^{\pm}$). The resulting ultra-relativistic particles, traveling faster
than the local speed of light in the atmosphere, produce coherent polarization
of the dielectric medium. This in turn produces beamed Cherenkov radiation in
the forward direction, forming a light pool at ground level of approximately
$150\,\mathrm{m}$ radius.
Imaging atmospheric Cherenkov telescopes (IACTs) sample this Cherenkov light
pool where the optical assembly of each telescope collects and focuses the
light onto a corresponding camera, comprising an array of photo-sensitive
detectors. As the shower progresses from the top of the atmosphere to the
ground, Cherenkov light is produced with a duration of up to 100 nanoseconds.
From the ground, camera pixels register the flash of light to form an image of
the shower. The shape and time-gradient of that image depends on the
properties of the primary particle, on its arrival direction, and on the
distance of the air shower to the telescopes. Images due to pure
electromagnetic showers tend to be compact, well-defined, and of an
approximately elliptical shape. Hadronic showers, on the other hand, tend to
generate more penetrating secondary particles (e.g., muons) at higher
transverse momenta, resulting in images with more clumpy shapes. Particle
shower images, particularly of $\gamma$-ray origin, can be described in terms
of a first and second moment analysis whose parameters are used to derive the
properties of the primary particle (Hillas 1985).
The observed spectrum of Cherenkov light that is generated in extensive air
showers extends from about $200\,\mathrm{nm}$ to more than $700\,\mathrm{nm}$.
The generated Cherenkov light follows a $1/\lambda^{2}$ distribution;
absorption and scattering of the light in the atmosphere affects mostly the
ultraviolet (UV) part of the spectrum. As a result, the observed flux at
ground level from $\gamma$-ray showers is significantly reduced below a
wavelength of about $280\,\mathrm{nm}$. Even so, the bulk of the observed
emission happens at short wavelengths, decreasing in intensity as the
wavelength increases. This, in addition to the presence of strong airglow
lines and increasing night sky background (NSB) starting from
$550\,\mathrm{nm}$, explains why IACTs are designed to be mostly sensitive to
blue and near-UV radiation. The number of Cherenkov photons emitted in the
shower is approximately proportional to the energy of the primary particle
(Hillas 1985; de Naurois & Mazin 2015). This is particularly true for air
showers that are generated by primary electrons and $\gamma$-rays.
IACTs operate in harsh environments with variable weather conditions, wide
temperature ranges, and occasional snow, rain, or dust storms. Nevertheless,
these telescopes lack the protective buildings of other optical instruments,
such as astronomical domes. The optical components of IACTs, which are
designed to collect and focus the Cherenkov light, suffer from degradation
processes due to the aforementioned weather conditions. In addition, the
cameras of IACTs are usually made of photomultiplier tube (PMT) pixels
operating at high voltages and they degrade as the total accumulated charge
increases. During standard operations, PMTs are exposed to NSB light,
typically inducing pixel currents of $5-10\,\mathrm{\mu A}$, and up to
$15-20\,\mathrm{\mu A}$ during exceptionally bright moonlight conditions.
These factors combined can induce aging that impacts both the optical and
electronic response to incoming light, including Cherenkov light from EAS.
Monitoring and correcting for the varying instrument performance therefore
becomes a key element in the calibration and analysis of Cherenkov data.
Hillas & Patterson (1990) originally proposed to monitor and calibrate IACTs
using images generated from local high energy muons generated by hadronic
showers. First implemented for the calibration of the Whipple telescopes
(Fleury 1991; Rovero et al. 1996), various IACTs have used muon images to
study the changing responses of the telescopes (Vacanti et al. 1994; Aharonian
et al. 2004; Meyer 2005; Gaug 2006; Chalme-Calvet et al. 2014). The Cherenkov
emission generated by these particles is emitted in a cone with a roughly
constant opening angle, appearing as a ring when observed from the ground at
small incident angles. The radius of the ring depends on the properties of the
incident particle (e.g., energy) and is not affected by instrument throughput.
It can be compared with the number of photo-electrons seen by the telescopes
to provide a measure of the efficiency of the telescope detecting optical
light. In practice, the analysis of muon images is challenging for many
reasons. It involves a detailed geometrical reconstruction, particularly when
part of the muon ring image falls outside the camera’s field of view (FoV),
when the shower trajectory is tilted or displaced with respect to the center
of the FoV or if groups of pixels are malfunctioning. In addition, as
Cherenkov photons from muons are produced close to the telescopes, atmospheric
absorption is less severe and the Cherenkov photon spectrum is shifted to
shorter wavelengths compared with photons generated by $\gamma$-ray showers
(Chalme-Calvet et al. 2014). Therefore, the calibration method using the muon
images requires additional corrections or specific Monte Carlo (MC)
simulations in order to provide an accurate estimation of the throughput for
$\gamma$-ray showers (Gaug et al. 2019). Due to this added complexity, the
Very Energetic Radiation Imaging Telescope Array System (VERITAS) currently
uses muons mostly as a supporting technique to monitor the calibration
workflow (see section 2.3.1).
This work discusses a different method to measure the total throughput. It is
based on the characterization of mirror reflectivity and the continuous
monitoring of camera gains. The relative variations of these parameters are
used to correct the simulated $\gamma$-ray event pulse charges to provide
throughput-corrected response functions. We discuss how this method was
successfully implemented in VERITAS, providing a reliable way to determine the
total optical and camera throughput, and to reconstruct shower energies and
source fluxes.
This document is structured as follows. Section 2 describes the VERITAS
telescopes, giving a comprehensive overview of the main components that are
used for both the standard operation of VERITAS and the calibration of the
instrument. It also details the analysis techniques and workflows that are
used to produce the standard science products of the telescope. The
measurement of the total optical throughput and its evolution over time is
presented in section 3. In section 4 we show the implementation of throughput
corrections, which are applied to the instrument MC simulations to account for
instrument aging. The impact of the varying throughput on the performance of
the instrument is evaluated in section 4.3, using metrics such as the energy
threshold of the analysis, the differential sensitivity and the uncertainties
in the reconstruction of fluxes. The validation of the method using real data
collected over seven years is shown in section 5. Finally, we present a brief
discussion of systematic uncertainties, limitations and possible improvements
of the method in section 6.
## 2 The VERITAS telescopes
VERITAS is a ground-based very high energy (VHE) instrument operating at the
basecamp of the Fred Lawrence Whipple Observatory in southern Arizona, USA
($31^{\circ}40^{\prime}N$, $110^{\circ}57^{\prime}W$, $1268\,\mathrm{m}$
elevation). It consists of four $12\,\mathrm{m}$ IACTs which use a shower
imaging and moment analysis technique (Weekes 1996) to detect $\gamma$-ray
photons with energies above $85\,\mathrm{GeV}$.
VERITAS operates in stereoscopic mode, with the array only triggering when a
Cherenkov shower is detected by at least two telescopes. This allows a more
accurate reconstruction of the shower properties (in particular, the incident
direction and energy of the primary particle) while reducing the number of
accidental triggers of a single telescope by NSB fluctuations or local muons.
It also makes the discrimination of hadronic showers more efficient, which in
turn boosts the sensitivity. An example of a stereoscopic reconstruction of a
real EAS by VERITAS is shown in Figure 1.
The first VERITAS telescope (T1) started operations in February 2005 (Holder
et al. 2006). Three more telescopes (T2, T3 and T4) were added in the
following years. The full four-telescope array was commissioned by 2007
(Krennrich et al. 2007). The observatory was subsequently upgraded to an
optimized diamond-shaped array layout by moving T1 to its current position in
2009 (Perkins et al. 2009), and the camera and electronics were replaced with
improved hardware and higher quantum efficiency (QE) PMTs in 2011-2012 (Kieda
2013). With its current, post-upgrade, array configuration, VERITAS has been
able to characterize $\gamma$-rays with energies from $\sim 85\,\mathrm{GeV}$
to $>30\,\mathrm{TeV}$, and can detect a point-like source with $\sim 1\%$ of
the Crab Nebula flux in $25\,\mathrm{h}$.
Figure 1: Extensive air shower reconstructed by the VERITAS telescopes. The
shower images for each camera have been integrated over time and cleaned using
a two-level filter. Dead and disabled channels are shown in dark gray and
brown respectively. The signal registered by each PMT is color-coded, with red
colors representing higher signal than blue tones. An approximate geometrical
reconstruction of the shower core location is illustrated with red dashed
lines and a red star. The upper right plot shows the trace of one of PMT
(#255) of T1 for reference. There, the signal registered for each sample of
$2\,\mathrm{ns}$ is plotted in red and the integration window (six samples) is
shown in shadowed blue.
### 2.1 Telescope design
The VERITAS telescopes follow a Davies-Cotton design (Davies & Cotton 1957).
The primary mirror of each telescope, hereafter also referred to as the
“dish,” consists of 345 identical hexagonal facets. The resulting optical
focal ratio of the system is $f/1$, for a focal length of $12\,\mathrm{m}$.
Each facet is designed as a one-piece glass element with $\approx
61\,\mathrm{cm}$ sides (Roache et al. 2008). They are commercially ground and
slumped for a radius of curvature of $23.97\pm 0.01\,\mathrm{m}$ on average,
and a spot size of $<10\,\mathrm{mm}$. The coating is made of aluminum,
deposited at a rate of $3-8\,\mathrm{nm}/s$ under vacuum conditions with a
purity better than $99.999\,\%$. Its total thickness is $180\,\mathrm{nm}$,
the top $80\,\mathrm{nm}$ of which are oxidized during an anodizing process to
improve durability and ensure a peak reflectivity of $92\,\%$ at about
$320\,\mathrm{nm}$ (Roache et al. 2008).
Cherenkov light collected by the dish is focused onto the camera. In order to
detect the brief Cherenkov flashes, the cameras of the current configuration
are composed of 499 high-quantum-efficiency (Otte et al. 2011; Kieda 2011)
Hamamatsu R10560 PMTs located at the focal plane. Each PMT has a FoV of
$0.15^{\circ}$ and is equipped with a Winston cone, a nonimaging device
designed to minimize light collection gaps between the PMT photocathodes and
limit the acceptance angle (Winston 1974, 1976), therefore reducing the
background light intensity. Each Winston cone is made of plastic with an
evaporated aluminum coating and a protective overlayer. This provides a
reflectivity greater than $85\%$ above $260\,\mathrm{nm}$ (Nagai et al. 2007).
After a careful optical alignment of the system (McCann et al. 2010), the on-
axis point spread function (PSF) of the telescopes is about
$0.10-0.12^{\circ}$ in diameter, leading to the ability to concentrate $80\%$
of the light within one pixel in the camera. The optical PSF of the telescopes
is frequently monitored, with observed variations of less than $0.02^{\circ}$
over long periods of time and for most elevations. Up to $0.6^{\circ}$ off-
axis the degradation of the optical PSF with respect to on-axis observations
is small, $\lesssim 0.02^{\circ}$, but it quickly increases at larger angles
reaching $\sim 0.2^{\circ}$ at $1.2^{\circ}$ offset. The total FoV of the
telescopes is $3.5^{\circ}$.
### 2.2 Readout and trigger system
PMT signals are digitized using $500\,\mathrm{Msample/s}$ flash analog-to-
digital converters (FADCs) with 8-bit dynamic range, capable of storing the
waveforms in $64\,\mathrm{\mu s}$ memory buffers. VERITAS employs a three-
level trigger. A first-level trigger (L1) or constant fraction discriminator
(CFD) requires the PMT pulse height to be above a given threshold (typically
5-6 photo-electrons). A pattern trigger or telescope trigger (L2), requires L1
trigger signals to occur in three adjacent PMTs within the timing coincidence
window. This pattern trigger is based on $400\,\mathrm{MHz}$ Xilinx Virtex-5
FPGAs for pixel neighbor coincidence, allowing triggers to occur before
samples are read out. The time-aligning accuracy of this system is $\sim
0.2\,\mathrm{ns}$ and allows for a pixel-to-pixel coincidence window of $\sim
5\,\mathrm{ns}$. Finally, an array trigger (L3) requires a L2 trigger signal
from at least two telescopes to happen, once corrected for the propagation
time of the shower front across the array and the varying distance from each
telescope to the central control building, within a $50\,\mathrm{ns}$
coincidence window. A more detailed description of the VERITAS trigger system
can be found in Zitzer (2013).
Coincidence window widths and CFD thresholds are optimized to trigger on low-
energy gamma-ray showers while avoiding random coincident triggers from NSB.
The optimization process consists of scans of the trigger thresholds when the
instrument is exposed to NSB. The aim is to keep the L3 rates at a few hundred
Hz, and to balance a low trigger energy threshold with avoiding data losses
from dead-time. The typical dead-time for VERITAS, in its current
configuration, is roughly $15\%$ for a data acquisition rate of about
$300\,\mathrm{Hz}$.
### 2.3 Data analysis
VERITAS maintains two data analysis packages: Eventdisplay (Maier & Holder
2017) and VEGAS (Cogan 2008). They allow independent reconstruction of the
data, limiting the impact of systematic uncertainties due to the analysis
software implementation on the scientific results. Each package performs a
calibration of the signal collected for each shower by the four telescopes, a
second-order moment analysis, and parametrization of the shower images. The
resulting parameters are used to classify the showers as $\gamma$-ray-like
showers or hadronic events. Using the stereo shower reconstruction, the
arrival direction and the energy of the primary particle are estimated.
Finally, from a comparison with reconstructed MC shower simulations, the
effective collection area is evaluated and the events in excess of the
estimated background are converted into high-level analysis products such as
fluxes, light curves, and energy spectra. The results of the throughput
analysis shown in this work were obtained using the Eventdisplay package,
however they were also validated using VEGAS.
#### 2.3.1 Calibration
The calibration process comprises the determination of the electronics
baseline (pedestal) and its variation plus the measurement of the response of
each individual PMT to incident light, that is absolute and relative gain
differences between PMTs. FADC inputs are AC coupled and a DC voltage offset
is added to the signal inputs (pedestal) so that positive and negative
fluctuations around the mean value (pedestal variation) due to NSB variations
can be measured. Artificially triggered pedestal events are injected during
observation runs with a frequency of $1\,\mathrm{Hz}$ and used for the
estimation of pedestal level and variation. The sky brightness and therefore
the background light level might change during observations; pedestal levels
and variations are consequently updated every three minutes.
PMT gains are monitored and calibrated by using a flasher light source. The
VERITAS FADCs have two gain channels for a wide dynamic range of the readout
chain (Holder 2005). The calibration software reconstructs the relative gain
of the pixels, timing differences (e.g., due to differences in the cable
length for each channel), and relative calibration of the high voltage
settings and the high- and low-gain readout channels by using uniformly
illuminated camera events generated with flasher light sources (Hanna et al.
2010). Each flasher unit consist of blue ($370\,\mathrm{nm}$) light-emitting
diodes (LEDs), driver electronics, and a front face made of an opal diffuser
which spreads the light from the LEDs and distributes it almost homogeneously
across the entire PMT camera. The flasher pulses span eight brightness levels
(upgraded to fifteen in 2016), covering the dynamic range of VERITAS for both
high- and low-gain readout. Absolute gain calibration is determined on a
periodic basis following the procedure described in 3.1. Relative gain
differences between PMTs are monitored daily, and corrected during the data
analysis. The inter-calibration between the two gain channels is performed on
a monthly basis by recording calibration runs with a particularly long readout
window. This is needed to avoid truncation and provide samples at the end of
the trace that allow us to estimate the low-gain pedestal. For each run, half
the camera is operated at a reduced gain to force it to stay in high-gain
mode, while the other half operates in low-gain.
Finally, the results of the analysis of Cherenkov light from muons measured by
the VERITAS telescopes (Tyler 2013) are used to monitor the calibration, in
conjunction with the procedures described below (single photo-electron runs,
optical PSF, and mirror reflectivity measurements). This approach mitigates
several limitations of a calibration based on muons only, especially the
differences in the wavelength spectrum between Cherenkov light from muons and
from air shower events.
#### 2.3.2 Signal extraction and image analysis
Signals are extracted from the FADC traces using a two-pass integration
method. The first pass uses a long integration window to determine the pulse
arrival time and to derive the optimal position of the time integration
window, using a linear fit of the pulse arrival times along the image axis
(Holder et al. 2006). A second pass performs the trace integration, using a
short window of typically six samples ($12\,\mathrm{ns}$). An example of a PMT
trace for a real event and the optimized integration window is shown in Figure
1.
A two-level filter is then used to extract the signal pixels and clean the
image. First, the brightest parts of the shower image (core pixels) are
localized at five times the pedestal root mean square (RMS). If a Gaussian
distribution of pixel signals is assumed for images containing only noise,
this would correspond to a surviving rate of less than one in $3\times
10^{6}$. This is enough to remove most clusters of pixels containing only
noise that might exist in the full image. Then, to avoid cutting out the edges
of the shower images, adjacent (boundary) pixels with at least 2.5 times the
pedestal RMS are added to the core pixels. The resulting shower image, which
resembles an ellipse in the case of $\gamma$-rays, is parameterized using a
second moment analysis, based on the method originally developed by Hillas
(1985).
Images from showers with large inclinations, large offsets, or originating
from high energy showers may not be fully contained in the camera. These
“leaking events” cause a loss of signal, degrade energy resolution and take a
toll on the performance. Using a log-likelihood fitting algorithm, some of
these showers are recovered (Maier & Holder 2017), particularly improving the
performance of the array at high energies.
#### 2.3.3 Direction, shower core, and energy reconstruction
The direction of the primary particle and the impact parameter of the shower
core on the ground are derived using stereoscopic techniques (Hofmann et al.
1999; Maier & Holder 2017). The energy of each $\gamma$-ray event is estimated
from a comparison using lookup tables built with simulations. The lookup
tables encode the dependency of the distance of the air shower core to each
telescope, integrated signal of the shower image (size), noise level, azimuth,
zenith angle and array configuration.
#### 2.3.4 Gamma-hadron separation
Cosmic rays represent the largest fraction of the data for all $\gamma$-ray
observations111A notable exception are flaring events such as GRBs, for which
the rate of $\gamma$-rays may exceed the rate of background events for the
first few seconds after analysis cuts have been applied (Acciari et al.
2019).. In order to separate signal from background cosmic-ray events, it is
common practice to compare the shape of observed shower images with simulated
$\gamma$-ray showers. To break the dependency of length and width of the image
ellipse on the energy of each event, Eventdisplay uses the median reduced
scaled parameters (MSCP) of width (MSCW) and length (MSCL) (Krawczynski et al.
2006):
$\mathrm{MSCP}=\frac{1}{N_{tel}}\sum_{i=1}^{N_{tel}}{\frac{p_{i}-{\bar{p}}_{MC}(\theta,\phi,\alpha,\mathrm{NSB},\mathrm{ATM},\mathrm{size},r)}{\sigma_{90}}},$
(1)
where $p_{i}$ is the measured parameter (in our case width or length) for
telescope $i$, $\bar{p}_{MC}$ and $\sigma_{90}$ the expected value and width
of the distribution for $90\%$ of the events, which are read from lookup
tables that are filled using simulated gamma rays. Finally, $\theta$ is the
zenith angle, $\phi$ the azimuth angle, $\alpha$ the offset angle of the
telescope pointing direction with respect to the nominal source position, NSB
the night sky background, ATM the atmospheric profile, size the integrated
charge of the shower image, and $r$ the distance of the shower core to
telescope $i$. VEGAS, in contrast, uses nonreduced versions of these
parameters, centered at 1 instead of 0.
The separation of $\gamma$-rays from background cosmic-ray events is done
using boosted decision trees (BDTs) (Krause et al. 2017). BDTs are implemented
as part of the Toolkit for Multivariate Data Analysis (TMVA) package
(Speckmayer et al. 2010) and trained on a set of simulated $\gamma$-ray and
observed hadronic showers using reconstructed shower parameters: MSCW, MSCL,
shower core position, height of the maximum of the Cherenkov emission, and
quality of the energy reconstruction.
With different science goals in mind, three sets of cuts are usually defined
in VERITAS, each requiring different BDT scores and different minimum image
brightness. “Soft cuts” provide a low energy threshold but larger acceptance
of background events. They are ideal for sources with very steep energy
spectra which emit large fluxes from tens of GeV to a few hundreds of GeV.
“Moderate cuts” are the most widely used and are adequate for most analyses,
as they provide a good balance between energy threshold and sensitivity. “Hard
cuts”, particularly combined with higher multiplicity (e.g., 3-telescopes)
provide the best background rejection. Though the energy threshold is high
($\gtrsim 300\,\mathrm{GeV}$), this set of cuts is used when the source of
interest is weak but with emission extending well into the TeV energy range.
Unless otherwise stated, “moderate cuts” have been used throughout this work.
#### 2.3.5 Spatial and spectral reconstruction
Event counts are converted into energy spectra taking into account the
effective areas of the telescopes and dead-time corrected exposure time
(Acciari et al. 2008). The background at each point in the sky is estimated
using either the reflected region method or the ring-background model approach
(Berge et al. 2007).
### 2.4 Monte Carlo simulation
Instrument response functions (IRFs), required for both the separation of
$\gamma$-rays from background events and for the reconstruction of energy
spectra and light curves, are generated using large sets of simulations of
$\gamma$-ray showers.
The propagation of extensive air showers in the atmosphere is carried out
using the CORSIKA package (Heck et al. 1998) taking into account all relevant
particle interactions. The generation, propagation, scattering, and absorption
of Cherenkov light is simulated using measurements of molecular and aerosol
content of the atmosphere from a radiosonde (station number #72274) which
operates from Tucson, Arizona, approximately $60\,\mathrm{km}$ away from the
VERITAS site. From the radiosonde data, two sets of simulations are generated,
the first with an average winter atmospheric profile and the second for summer
atmosphere.
The response of the instrument is simulated using the
GrOptics222https://github.com/groptics/GrOptics and
CARE333https://github.com/nepomukotte/CARE packages for the optical and camera
simulations. The optical simulations take into account the properties of the
primary mirrors including facet alignment, reflectivity, diffuse scattering
and all relevant shadowing elements of the telescopes. The photo-detector and
electronic simulation code CARE is used to model the digitization, trigger,
and readout processes including noise components. The MC model is validated
through extensive comparisons with calibration measurements.
The generation of MC simulations for instrument response functions is an
important computational effort, given the large parameter space that needs to
be covered. The MC instrument model for the array configuration after the
upgrade of the cameras was generated to accurately reproduce the array
performance for the period October 2012 - June 2013.
## 3 Throughput measurements
The collection efficiency of Cherenkov light at ground level by the telescopes
and its conversion to measurable signals, that is telescope throughput,
strongly depends on the properties of the cameras and dishes, which change
over time. At the camera, PMT gains and the QE affect the conversion of
photons into charge pulses in each pixel. Similarly, the combined effect of
the reflectivity of the mirror facets and the collection efficiency of the
Winston cones affects the total optical light collected at each PMT
photocatode. Furthermore, some of these parameters vary on different time
scales and some are wavelength dependent, turning the characterization of the
different parts of the instrument into an important and very challenging step
in the analysis of data produced by IACTs. This section describes how these
components are studied in VERITAS.
### 3.1 Camera gains and gain factors
The current detector model of VERITAS assumes an average value of the absolute
PMT gain $G_{i,MC}$ for each telescope $i$, estimated after the upgrade of the
PMT cameras in 2011-2012:
* •
$G_{1,MC}=5.38\,\mathrm{dc/pe}$
* •
$G_{2,MC}=5.29\,\mathrm{dc/pe}$
* •
$G_{3,MC}=5.29\,\mathrm{dc/pe}$
* •
$G_{4,MC}=5.73\,\mathrm{dc/pe}$
where dc stands for digital counts and pe for photo-electrons.
In practice, the individual absolute gains are slightly different for every
PMT and they vary over time due to temperature fluctuations. Supply voltage
fluctuations are are kept stable at the subvolt scale, therefore are unlikely
to have a large impact in the absolute gains.
At the beginning of each observing campaign, usually in October, absolute and
relative gains are measured and adjusted by changing the HV settings of the
telescopes. The process is usually repeated once or twice during the season.
This brings the gains closer to the nominal values shown before and results in
absolute gain swings of, at most, $10\,\%$ over months. The resulting PMT
charge distribution is also optimized to have $\lesssim 5\%$ width. Part of
the aim of this work is to reduce the impact on the IRFs of the remaining gain
variations left after the flat fielding process.
Gains are measured in VERITAS in two independent ways, each with its own set
of assumptions. The first approach directly measures the absolute gain by
detecting the signal from single photo-electrons. This is achieved by placing
a cover to reduce the optical transmission in front of the camera. The second
method evaluates the absolute gains of the PMTs statistically, using “LED
flashers” to uniformly illuminate the camera. Flasher runs are collected daily
to perform a relative gain correction directly in the analysis chain.
#### 3.1.1 Single photo-electron (SPE) runs
Weak, pulsating sources of light can be used to study the response of PMTs to
a single photo-electron. The measured charge from a single photo-electron is,
on average, proportional to the number of electrons produced by the PMT. The
proportionality constant is the gain of the system, $G$ (see e.g., Kieda 2011;
Hanna et al. 2010; MacLeod 2007).
In VERITAS, single photo-electron light levels are achieved by attenuating the
light of the flasher with a custom-made camera cover with a small hole aligned
with the center of each PMT (Hanna 2008). Each of these holes allows only
about $1.6\%$ of the light to pass. With such a device, not only are the
required low-illumination conditions achieved, but NSB is suppressed by the
same amount.
A histogram of the accumulated charge shows a series of peaks. The first peak
describes the pedestal, the second the mean value of the SPE charge in digital
counts. The separation depends on the absolute gain set by the HV settings
used during the data acquisition, making it possible to derive the gain-
voltage relationship and measure the absolute gain. Since the absolute gain
can locally be approximated as having a linear dependence on voltage, VERITAS
takes SPE runs at both nominal and a slightly increased voltage ($110\%$) so
that the two peaks are better separated.
The shape of the single-electron peak follows a Polya distribution (López
Paredes et al. 2018), which is a special case of a negative-binomial
distribution. The comparison to photo-statistic gains, described in the next
section, requires a correction factor for each pixel, which quantifies the
deviation from Poisson statistics. SPE runs are taken in VERITAS approximately
once a month, as they require the manual installation of the custom cover on
each of the telescope cameras, temporarily interrupting the data taking and
leading to increased observer fatigue and safety risks.
#### 3.1.2 Photo-statistic gains from nightly flasher runs
The second method to measure absolute gains in VERITAS uses nightly flasher
runs. We refer to these gains as photo-statistic gains. During a flasher run,
the mean charge of a pulse on a PMT, $\mu$, is statistically proportional to
the mean number of photo-electrons at the first dynode, $N_{pe}$:
$\mu=G\times N_{pe}.$ (2)
The proportionality constant is the absolute gain $G$. It can be determined by
assuming that $N_{pe}$ follows Poisson statistics, which implies that the
variance in photo-electrons is approximately the mean charge $\mu$, hence
$\sigma_{pe}\simeq\sqrt{N_{pe}}$. After the dynodes and preamplifier have
amplified the signal, the variance becomes
$\sigma^{2}\simeq G^{2}\times N_{pe}=G\times\mu.$ (3)
Fitting a linear function to the charge variance $\sigma^{2}$ over $\mu$
provides the gain as the slope and the variance of the pedestal noise as the
intercept.
Figure 2: Time-dependent changes of photo-statistic gains normalized to the
nominal absolute gains used in the baseline MC model for the current detector
model. This detector model was constructed in 2012, and reproduces the
characteristics of VERITAS after the PMT camera upgrade. Black points show the
average gain factors for each instrument epoch. Gray points show the
individual unfiltered gain factor values per telescope and daily flasher run.
Blue, orange, green and red points show the result of a median filter of the
individual gain factors. Curves of the same colors represent a spline
interpolation of the filtered values. Even though spline interpolation can
locally diverge when there is no data available, we note that this occurs only
during Summer breaks, where no data is taken in VERITAS, therefore it is not a
concern for this study. Error bars represent statistical uncertainties.
#### 3.1.3 Comparison of absolute gain measurement methods
The two methods of measuring the gain agree within $5-6\%$, which is assumed
to be the systematic uncertainty for the absolute gain estimation. Photo-
statistic gains can be determined using runs that are taken once a night,
therefore allowing better monitoring. We consequently opted to use them for
our throughput calibration purposes, as opposed to SPE gains.
So far we have only discussed the evolution of the gains, but the QE of the
PMTs may also change as the detector ages. The possible aging of VERITAS PMTs
was covered in Gazda et al. (2016). In that work, a set of 20 PMTs were
removed from the telescopes and compared to 20 spare PMTs that were never
operated and therefore are expected to exhibit their original properties. The
absolute quantum efficiency was measured to be consistent for both samples at
the wavelength range in which VERITAS detects Cherenkov radiation from
$\gamma$-ray showers. More recently, during the Summer break of 2020, another
aging study was done using the same sets of PMTs. This time, the test
consisted of the determination of the HV-dependence of PMT gains, which is
related to their QE. Again the properties of both sets of PMTs were found to
be compatible within uncertainties.
#### 3.1.4 Gain factors
Once the average absolute camera gains ($G_{i}$) are measured for each
telescope $i$, they can be compared with the gains assumed in the reference MC
simulations ($G_{i,MC}$) to obtain the gain factors ($g_{i}=G_{i}/G_{i,MC}$)
needed to correct the simulated signals to account for changes in the camera
gains. Figure 2 shows the gain factors as derived from photostatic gains.
### 3.2 Optical throughput and mirror reflectivity
VERITAS is located in southern Arizona, where at least three months during
summer are under direct influence of the North American monsoon, in addition
to the all year round varying weather conditions. The mirror facets of the
telescopes are consequently affected by mechanical and chemical degradation of
their reflecting surfaces (e.g., due to fine dust grains which scratch the
reflective coating, pollutants chemically reacting with the mirror materials,
or temperature oscillations inducing deformations on the underlying mirror
substrate). Furthermore, periodic mirror cleaning, re-coating, and Winston
cone cleaning contribute to partially recover the reflectivity of VERITAS
telescopes, but it also adds an additional variability component to the
optical performance of the telescopes.
The degradation of the mirrors changes their reflective properties. The total
reflectivity is the combination of diffuse and specular reflectivity. The
first is the tendency of a surface to reflect the incident radiation in random
directions. Degraded mirror surfaces can cause high diffuse reflectivity,
scattering photons and making the mirrors unable to focus and properly form an
image. For imaging instruments, diffuse reflectivity is undesirable as it
increases background photon noise at the focal plane and degrades the optical
PSF. Specular reflectivity refers to the ability to form an image of a given
object at the focal plane. It is very sensitive to mirror aging since it
depends on the material of the substrate, how accurately the mirror is
figured, how smooth its surface is and the type of coating.
Mirror coating degradation causes a wavelength-dependent effect on the optical
throughput of the instrument (Garoli et al. 2020), with reflectivity at short-
wavelengths more severely affected than that for longer wavelengths. This is
relevant for IACTs because the bulk of the Cherenkov emission concentrates in
the near-UV and blue part of the visible spectrum.
The optical properties of the Winston cones are not expected to vary
dramatically over time as they are protected from the elements and the camera
is fully closed while the telescopes are stowed (i.e., during daylight and
during bad weather conditions). Nonetheless, these components were examined
over time and no evidence of degradation in their light collection efficiency
was detected. Therefore, we concentrate on the reflectivity of the telescope
dishes and treat Winston cone efficiency as a source of systematic
uncertainty.
#### 3.2.1 Laboratory measurements of specular reflectivity
Beginning with the VERITAS inauguration in 2007 and continuing until 2015,
fifteen facets from different parts of the dish (top, middle, bottom) were
regularly removed to monitor the changes of the wavelength-dependent
reflectivity over time. This was done in the laboratory using a broad-spectrum
light source, an adjustable filter wheel, and a photometer (Oriel 71610).
Measurements were compared to a calibration mirror made of pure aluminum,
periodically re-coated to ensure consistency in the measurements (Roache et
al. 2007).
Even though it was intented to measure specular reflectivity, this method had
the disadvantage of not being able to fully discriminate between diffuse and
specular reflectivity as it relied on a photometer and not on a real imaging
detector. It also did not evaluate the impact of PSF changes on the
performance of the instrument collecting Cherenkov light. Finally, the
evolution of the reflectivity of a few mirrors was measured, rather than that
for the entire dish.
Laboratory measurements of specular reflectivity were dropped in 2016 as the
achieved accuracy was limited by the complexity of determining the ratio of
specular to diffuse reflectivity.
#### 3.2.2 Whole dish reflectivity (WDR)
The WDR method was initially developed by MAGIC (Mirzoyan et al. 2007) and
after a few preliminary tests in 2010 and 2011, the method was adopted by the
VERITAS observatory starting from 2014 (Archambault et al. 2013, 2021). The
WDR technique uses a wide-field CCD camera attached to the dish, near its
center, which is pointed directly toward a target at the focal plane in front
of the PMT camera. In order to reproduce the response of the mirrors to
Cherenkov light combined with the QE of the PMT, which peaks at $\sim
340\,\mathrm{nm}$, the CCD (SBIG ST402ME) is equipped with a blue filter from
the same manufacturer (Diffraction Limited©). This effectively shifts the
response of the CCD camera to shorter wavelengths and provides a better match
to the spectral response of the PMT camera to extended air showers. The target
is made of Spectralon, a fluoropolymer which exhibits Lambertian behavior
(i.e., high diffuse reflectivity). This ensures that the apparent brightness
is the same regardless of the observing angle. The reflectivity of the
Spectralon plates used in VERITAS has been checked over the years and no
significant change could be detected, setting a conservative limit of
$\lesssim 5\%$ to their evolution, well within the total systematic
uncertainty of the reflectivity determination. With this set-up, the CCD
camera can be used to measure simultaneously the brightness of a reference
star in the FoV and its reflection generated by the dish on the Spectralon.
Flat and dark frames are used to correct for effects such as vignetting or
pixel-to-pixel response, and to subtract the electronic dark signal in the
image. The comparison of the brightness of the reference star and its
reflected spot yields an estimation of the total reflectivity of the dish. The
advantage of this method resides on its simplicity and robustness, using just
a standard CCD with a color filter and target plate over the focal plane.
Being based on differential photometry methods, it is insensitive to changes
in the QE or the gain of the CCD itself. The main disadvantage of the WDR
method is that the employed CCD is blind to UV radiation and therefore the
spectral averaged reflectivity does not fully match the reflectivity for
Cherenkov light.
#### 3.2.3 Optical throughput and optical throughput factors
In a manner similar to the case of the gains, this study focuses on the
relative changes of the reflectivity with respect to what is simulated for
each telescope $i$ in the detector model. From the two possible methods to
estimate the reflectivity of VERITAS mirrors, we adopted the WDR method
described in section 3.2.2.
VERITAS simulations take into account the complete spectral response of each
telescope as a function of wavelength (see Appendix A). On the other hand, the
WDR method only provides an averaged reflectivity for each telescope $i$,
measured in the blue part of the visible spectrum. The calibration of the
optical throughput requires the estimation of the same wavelength-averaged
reflectivity for the reflector model of the current instrument configuration
$R_{i,MC}$ and the assumption that the differences due to the spectral
mismatch that exists between the Cherenkov spectrum and the assumed blue
filter are small.
The spectral average is determined assuming a standard Bessel-B filter Bessell
(1990), similar to the one used with our CCD. The transmission of this filter
is multiplied by the reference reflectivity curves from the reflector model.
The resulting curve is integrated to obtain the average reflectivity
$R_{i,MC}$ for each telescope:
* •
$R_{1,MC}=0.828$
* •
$R_{2,MC}=0.841$
* •
$R_{3,MC}=0.852$
* •
$R_{4,MC}=0.846$
In order to estimate the collection efficiency error due to the use of a
B-filter and a CCD to characterize the response of the telescope to Cherenkov
light, we computed the wavelength-averaged reflectivities, this time
convolving each mirror reflectivity curve with a typical Cherenkov spectrum
from a $\gamma$-ray induced shower. The resulting average reflectivities are
between $0.79$ and $0.82$ for all telescopes. The differences are therefore
well within the assumed systematic uncertainty on the mirror reflectivity
($\lesssim 10\,\%$).
Figure 3: Time-dependent changes of the reflectivity factors obtained from WDR
measurements. Black points show the average reflectivity factors for each
instrument epoch. Blue, orange, green, and red points show the individual
reflectivity factor measurements for each telescope. Curves of the same colors
represent a spline interpolation, which removes outliers. The first point
(early 2012) is extracted from the simulations and it serves as the reference
for the calculation of the reflectivity factors, therefore adopting a value of
1 as reflectivity factor. We note that a value of the reflectivity factor
slightly greater than 1 just means that the reflectivity at that time was
slightly higher than at the time where simulations were produced.
Calculated based on the reflectivity measurements, the optical throughput
factors or reflectivity factors can be defined as $r_{i}=R_{i}/R_{i,MC}$.
Their evolution over time is shown in Figure 3. These reflectivity factors are
needed to correct our MC simulations so that they reflect the changes in the
mirror reflectivity. The first point is fixed at unity as it represents the
reflectivity values actually used in the simulations. Because the whole-dish
reflectivity only started to be measured routinely in 2014, the second and
third points of each panel of Figure 3 had to be interpolated. We artificially
assigned them a comparatively larger uncertainty of $\pm 5\%$ in absolute
reflectivity ($\pm 0.05$ in the plot). As can be seen from the figure,
reflectivity quickly degraded by $20-30\%$ from 2013 to 2015, coinciding with
a period of reduced cadence of re-coating degraded mirror facets. Since 2015,
the telescope reflectivity continues to decrease, but at a much slower pace.
Improvements in the reflectivity are also present in our data for specific
times, coinciding with cleaning and re-coating of the most damaged mirror
facets. This process usually involves the exchange of $\sim 100$ mirror
facets, and its effects are smoothed in the spline fitting case.
### 3.3 Total throughput
The total throughput of each telescope, normalized to the reference values in
the simulations, can be computed as the product of the gain factor $g_{i}$
(Figure 2) and the corresponding reflectivity factor $r_{i}$ (Figure 3). The
resulting values and their evolution for each telescope are shown in Figure 4
and table 1. We refer to this total throughput factor as $t_{i}$ or simply
throughput factor.
To ensure consistency, we computed the average throughput factors for each
time bin with three different strategies: i) using the product of spline
interpolations for gain factors and reflectivity factors and then computing
the mean for each season, ii) using the closest reflectivity and gain
measurement pairs to compute the total throughput and then estimating seasonal
averages, and iii) using the average values of gain factors and reflectivity
factors per instrument epoch and calculating their product. They all agree at
the $\sim 5\%$ level.
The application of these throughput factors to the Cherenkov signals of the
reference simulations allows us to produce IRFs that account for the
throughput degradation of the VERITAS telescopes. The actual implementation
and assessment of this calibration is discussed in the next section.
Figure 4: Time-dependent change of the throughput factors calculated from
photo-statistic gain factors and WDR reflectivity factors. Error bars show
only statistical uncertainties.
### 3.4 IRF period definition
VERITAS has undergone two major instrument upgrades. The first consisted on
the relocation of T1 to its current position, improving significantly the
trigger efficiency. The second upgrade targeted the camera, with improvements
to the electronics and the PMT QE. This naturally generated three major
instrument configurations, each with its own set of MC simulations, which
provide an accurate model of the telescope performance.
The current instrument configuration spans almost eight years of operations,
during which the instrument has evolved significantly due to aging and our
maintenance efforts. Consequently, we tune our IRFs and simulations in finer
time bins so that each time bin or IRF period has throughput drops of at most
$10\%$, comparable to the claimed systematic uncertainties on dish
reflectivity and gains. Similarly, we require the throughput values at the
limits of each bin to be covered by the statistical errors for that bin.
Finally, the duration of each epoch is at most a single year, and
approximately aligned with the observing cycle that begins with the end of the
monsoon in September or October.
With these criteria it is clear that for the last years, during which the
evolution of the throughput was less dramatic, one IRF period per year is
enough to provide the desired granularity and precision. For the first couple
of years, however, the total throughput was rapidly evolving and it required
finer time bins to fulfill our criteria. The resulting IRF periods are
summarized in Table 1.
Table 1: IRF periods for the current VERITAS instrument configuration.
season | MJD range | data runs | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$
---|---|---|---|---|---|---
2012-2013a | 56124-56367 | 63373-67410 | $0.986\pm 0.056$ | $1.039\pm 0.059$ | $0.972\pm 0.057$ | $0.914\pm 0.056$
2012-2013b | 56367-56588 | 67411-70170 | $0.941\pm 0.055$ | $1.023\pm 0.056$ | $0.885\pm 0.054$ | $0.823\pm 0.052$
2013-2014a | 56588-56778 | 70171-73235 | $0.887\pm 0.021$ | $0.958\pm 0.030$ | $0.818\pm 0.022$ | $0.803\pm 0.018$
2013-2014b | 56778-56968 | 73236-75021 | $0.809\pm 0.042$ | $0.878\pm 0.050$ | $0.782\pm 0.029$ | $0.775\pm 0.036$
2014-2015 | 56968-57242 | 75022-78239 | $0.713\pm 0.027$ | $0.821\pm 0.026$ | $0.730\pm 0.019$ | $0.715\pm 0.019$
2015-2016 | 57242-57600 | 78240-82587 | $0.708\pm 0.022$ | $0.772\pm 0.018$ | $0.690\pm 0.018$ | $0.731\pm 0.018$
2016-2017 | 57600-57966 | 82588-86848 | $0.653\pm 0.016$ | $0.774\pm 0.023$ | $0.647\pm 0.015$ | $0.688\pm 0.024$
2017-2018 | 57966-58331 | 86849-90608 | $0.677\pm 0.023$ | $0.748\pm 0.016$ | $0.683\pm 0.025$ | $0.695\pm 0.027$
2018-2019 | 58331-58696 | 90609-93829 | $0.608\pm 0.021$ | $0.678\pm 0.019$ | $0.619\pm 0.021$ | $0.635\pm 0.024$
2019-2020 | 58696-59061 | 93830-96048 | $0.608\pm 0.017$ | $0.698\pm 0.034$ | $0.732\pm 0.042$ | $0.640\pm 0.015$
* •
Note: Definition of IRF periods for the current VERITAS instrument
configuration with the covered dates, the corresponding data runs and the
values of the total throughput corrections that need to be applied to the
simulated event charges to obtain the calibrated MC and IRFs. Only statistical
uncertainties were considered in this study.
## 4 Implementation of throughput calibration
This section discusses various strategies to implement the throughput
calibration in IACTs, a description of the method adopted in the VERITAS
software packages and the tests we performed to ensure the consistency of the
results.
### 4.1 Implementation methods
MC simulations should accurately reproduce the response of VERITAS at any
given time. We discussed in the previous section the impact of the time-
dependent changes on the PMT gains and optical elements of the telescope.
Reproducing all the details with the required accuracy while using a single MC
production and a static set of IRFs is not an option if throughput is changing
over time. Here, we concentrate on how we can implement the throughput
calibration in our analysis framework to ensure consistent detector response
over the entire operating life of the telescopes.
There are a number of strategies one could adopt to implement this calibration
in the analysis, with varying degrees of accuracy that correlate with the
computational cost and the parameter space. In the case of VERITAS, we need to
cover the response of the instrument as a function of zenith angle of
observation, azimuth angle, atmospheric profiles, telescope multiplicity,
camera HV settings, wobble pointing offset, and NSB intensity.
Full MC productions that reproduce the changes in the throughput of the
instrument over time and corrections to an existing MC production are both
valid possibilities. Provided that the systematic errors on the calibration
measurements are small, the former is potentially the most accurate and
correctly reproduces the changes at the trigger level caused by throughput
variations. However, it is computationally very expensive. It also requires
producing a new complete MC model of the entire array and simulating
$\gamma$-ray showers with the required statistics. Consequently we chose to
explore corrections to the already existing MC simulations that were produced
for the baseline instrument configuration.
Calibration corrections could be applied on a run-wise basis (with typical
duration of an observing run shorter than one hour) or on longer periods of
time: from just one day to an entire year or even longer, during which the
instrument response is measured to be approximately stable. Because of the
monsoon shutdown in summer, which defines a natural observing season from
about mid-September to mid-June, the fact that camera voltages (and hence
gains) are adjusted at the beginning of each of these seasons and the slow
cadence of WDR measurements, such seasons seemed a natural choice for VERITAS
to define the time bins for the corrections. Exceptionally for the first two
years, for which the throughput evolved rapidly, each season was divided into
two shorter time bins. We refer to these time bins in which we apply the
throughput corrections as IRF periods.
Throughput corrections can be introduced at different steps in the analysis of
simulated showers and the production of the IRFs. A simple option, yet not
fully accurate, is to calibrate simulated event images after they have been
cleaned and the PMT traces have been integrated. The size of the event is
modified using the changes in the total throughput of the instrument. Noise
level changes can be either ignored or approximately corrected. Testing this
option in VERITAS resulted in a reasonable reconstruction of event energies
and $\gamma$-ray fluxes. However, the corrected simulated image shapes did not
match the shape of the observed shower images.
A natural improvement over this, and the solution adopted in this study,
consists of correcting the signals of the simulated events before the PMT
traces are integrated and the Cherenkov images are cleaned. Even though the
trigger changes are still not properly modeled, noise levels change as fewer
pixels containing only NSB are able to pass the image cleaning thresholds. Our
tests resulted in reconstructed simulated showers which are directly
comparable to observed data, making it easier to assess the accuracy of the
corrections by comparing the distributions for the different image parameters.
The main disadvantage is that the cleaning and Hillas parameterization of the
simulated showers had to be repeated once per IRF period with each throughput
factor $t_{i}$ on the signal integration, significantly increasing the
computing time, storage requirements and analysis complexity needed to cover
the entire parameter space. Producing IRFs for a single wobble offset with 10
time bins as described in this work, including two atmospheric profiles, and
all zenith angles (8) and noise levels required (11), took several weeks of
testing and computation time and required about $480\,\mathrm{GB}$ of storage
for the final products.
### 4.2 Robustness of shower parameter reconstruction
This work is based on the assumption that we can correct the simulated PMT
traces with just two global factors, one due to the camera gain $g_{i}$ and
the other due to the dish reflectivity $r_{i}$. In principle, this should not
distort the shape of the shower images; instead we expect it to alter the size
of the cleaned events. To check if this is the case, we extracted the same
simulated events corresponding to two different IRF periods (2012-2013a and
2019-2020) and calculated the ratio of values for some of their key
geometrical parameters (size, width, length, and time gradient along the major
axis) between the two periods. Figure 5 shows histograms of the obtained
ratios together with the mean and the standard deviation of the obtained
distribution. It can be seen that only the size of the events is significantly
shifted ($\mu\simeq 0.63$, $\sigma\simeq 0.34$) when we scale the simulated
pixel traces. As this is a statistical comparison between individual simulated
showers obtained with independently generated IRFs, the large width of the
distributions for some of these ratios is not a concern for this study.
Figure 5: Ratio between image parameters derived from MC simulations for the
season 2019-2020 vs season 2012-2013a (Telescope T1 only, winter simulations,
noise level of $100\,\mathrm{MHz}$, and no size cuts). The vertical axis of
each panel shows the number of events within a given ratio bin.
### 4.3 Impact of throughput changes on VERITAS performance
The evolution of the total throughput can affect the reconstructed fluxes in
two different ways. The relation between the size of an event and its energy
changes as discussed in section 4.3.1. Additionally, some events are no longer
reconstructed, as their integrated charges are not large enough to pass the
hardware trigger, analysis, and reconstruction cuts. This has a significant
impact for the lowest energies, as is shown in the effective areas of the
telescopes, discussed in section 4.3.2.
#### 4.3.1 Effect on reconstructed energies
Figure 6: Relative changes in energy reconstruction induced by the evolution
of the throughput of the instrument. The ratio of throughput-calibrated
energies with respect to the uncorrected energies is presented in two formats:
As 2D lookup tables, with the ratio as function of impact distance and size of
the simulated events (bottom panels); curves with the distance-averaged
lookup-tables, showing the ratio as a function only of size (top panels).
One of the main effects of the application of correction factors to the
measured digital traces is the improvement of the energy reconstruction.
Lowered gain and reflectivity naturally result in smaller images and
consequently lower estimated energy if the IRFs do not account for throughput
variations. The energy reconstruction method applied uses lookup tables,
similar to the ones described in section 2.3.4. They represent the energy as a
function of size and distance to the telescope in a coordinate system
perpendicular to the arrival direction of the air shower event, the NSB, the
wobble offset of the observation and the atmospheric profile. These tables are
filled using simulated $\gamma$-ray events by calculating the median and the
standard deviation of the distribution of true energies ($\mathrm{E_{MC}}$)
for each bin in the grid.
Figure 6 shows the ratio between the reconstructed energies of the events
whose traces have been corrected to account for the drop in telescope
throughput and the uncorrected case. That ratio is represented, following the
format of the lookup tables, as a function of impact distance and size of the
simulated event for a fixed set of values for the other parameters.
For a given noise level and zenith angle, the size of the event scales almost
linearly with its energy. The exception would be the highest energies. In that
regime, the accumulated charge may hit a maximum value due to FADC saturation,
affecting the linear relation between size and energy previously described.
Since the events migrate to lower sizes for a fixed energy as the telescope
throughput drops, one can easily see the effect of its evolution with time
once we calibrate the throughput in the analysis: for a given image size and
distance of the event, the reconstructed energy is higher in recent years than
it was right after the upgrade of VERITAS (years 2012-2013).
#### 4.3.2 Effect on flux reconstruction and effective areas
Figure 7: Evolution of the effective area for different instrument epochs, for
a zenith angle of observation of $20^{\circ}$, noise level of
$100\,\mathrm{MHz}$, moderate cuts and a minimum of two images per event.
The impact of the total throughput on the effective area is shown in Figure 7.
In that figure, the effective area is computed as a function of true energy in
each period for a fixed position in the parameter space (zenith angle of
$20^{\circ}$, noise level of $100\,\mathrm{MHz}$, moderate cuts, a minimum of
two images per event, and $0.5^{\circ}$ wobble offset).
There is insufficient background data to optimize the cuts individually for
every period. The reason is that we require data that were taken under very
good weather conditions, fields without strong $\gamma$-ray sources, and
dividing the remaining data sample into zenith angle bins. Consequently, we
used the same set of cuts optimized using data from the entire post-upgrade
array configuration. This artificially enhances the differences seen at low
energies in Figure 7, since fewer events have the size required to pass the
analysis cuts in the latest periods. Reducing the size cut over time to
palliate this effect is not an option as it would result in the inclusion of
worse quality events in the analysis. Above $\sim 300\,\mathrm{GeV}$, away
from the energy threshold of the analysis, almost no events are lost despite
the decreased optical throughput of the telescope.
#### 4.3.3 Effect on the energy threshold of the analysis
Due to the statistical nature of the reconstruction methods used for IACTs,
the energy threshold of the analysis is defined as the energy at which the
effective collection area is $10\%$ of the maximum effective area reached with
the telescope at any energy. This definition ensures that a significant
fraction of the events have passed the cuts and that the individual shower
images and traces are comparatively less affected by noise artifacts. It does
not imply that lower energy particles are undetectable, but the percentage of
particle showers below this energy quickly drops as the energy decreases. The
detected emission at energies lower than the energy threshold may still be
significant for sources with steep spectra.
Figure 8 shows the evolution of the energy threshold of the analysis with
VERITAS. Uncertainties of the energy thresholds shown in the figure are
calculated accounting for two independent contributions: i) the energy bin
width of the effective area histogram, ii) the uncertainty on the effective
area at the center of the bin.
Figure 8: Energy threshold of the analysis estimated from simulated events and
using moderate cuts, zenith angle of $20^{\circ}$, and noise level of
$100\,\mathrm{MHz}$. It is defined as the energy for which the effective area
becomes $10\%$ of the maximum effective area.
#### 4.3.4 Effect on the differential sensitivity
Figure 9: Differential sensitivity with $50\,\mathrm{h}$ of VERITAS
observations calculated using different low zenith angle Crab Nebula data,
with winter atmospheric profile conditions. The same cuts as in Figure 7 are
used. The sensitivity is defined in terms of the flux of the Crab Nebula (in
Crab units, CU) assuming it is a reference astrophysical object
at VHE energies.
We can define the sensitivity of a $\gamma$-ray instrument as the minimum flux
that such an instrument can detect for a given amount of exposure. Simulating
a realistic sample of background data is challenging, so it is common practice
to use the Crab Nebula data as a reference object and to use its flux as the
unit system for sensitivity studies. In the following, the differential
sensitivity is computed as the minimum photon flux required to achieve a
5$\,\sigma$ detection in $50\,\mathrm{h}$ of observation time (using the
statistics described in Li & Ma (1983)) in each energy bin, assuming five
equal-width bins per decade on the logarithmic energy axis. Additionally, at
least ten signal events are required per energy bin.
The differential sensitivity is presented in Figure 9, based on good quality
Crab Nebula data, which were collected under dark sky conditions, and at low
zenith angle to minimize the atmosphere’s influence. We show the results for
BDT-based moderate cuts with a minimum telescope multiplicity of two, which
provides a balance between optimizing sensitivity and maintaining a low energy
threshold. Results employing softer cuts and harder cuts were obtained and
compared between seasons, yielding similar conclusions.
It can be concluded that throughput degradation affects mostly the lowest
energies, where some dim shower events are lost or mistakenly classified as
hadronic showers. At high energies, the performance is similar between the
different periods, unaffected by the changes in the throughput.
## 5 Validation of the throughput calibration on real data
### 5.1 Comparison between real data and MC simulations
The validity of the estimated total throughput correction can be tested by
comparing image and stereo parameter distributions obtained with data and
throughput-calibrated simulated events for every IRF period. If MC showers are
not calibrated with the correct throughput, the shower shapes will be
distorted and the estimated shower parameter distributions will not match
those derived from data.
In this work, we compared MSCW and MSCL distributions of simulated and
observed data for six reconstructed energy bins: $\log_{10}E_{rec}\in$
$[-1.0,-0.7]$, $[-0.7,-0.3]$, $[-0.3,0.0]$, $[0.0,0.3]$, $[0.3,0.7]$,
$[0.7,1.0]$ (energy measured in TeV), and for all the considered IRF periods.
The results, with the last two energy bins combined to increase statistics,
can be seen in Figure 10. More detailed comparisons are shown in appendix B.
In all cases, data and simulated events have similar distributions for both
parameters, almost symmetric, with similar widths and centered at zero (as
expected for $\gamma$-ray events). No significant evolution was observed in
these parameters with time. The only exception is the highest energy bin,
which has a wider distribution of MSCW values for real shower events. This is
caused by the high energy events, $\sim 10\,\mathrm{TeV}$, being subject to
difficulties in calibrating of the low-gain readout channel, saturation
effects, and leakage of many shower images outside of the camera. Moreover,
its scientific impact is limited because statistical uncertainties almost
always dominate at $\sim 10\,\mathrm{TeV}$ due to the small photon fluxes
emitted at these energies from astrophysical sources.
Figure 10: Average distributions of MSCW values, in different energy bins, for
the data (blue points) and simulation (red curve) in the entire post-upgrade
period. They were both generated after throughput calibration. Data events
were obtained from a sample of Crab Nebula observations. Due to the small
total number of events, the last two energy bins are combined into a single
bin with energies $\mathrm{\log_{10}E_{rec}}\mathrm{(TeV)}\in[0.3,1.0]$.
### 5.2 Inter-telescope calibration
Each telescope that detects a $\gamma$-ray-induced shower provides an
independent estimation of the primary particle energy. This enables us to test
how well the throughput of each telescope is calibrated against that of the
other telescopes.
There are different ways of performing such a test (Hofmann 2003; Lebohec &
Holder 2003; Mitchell et al. 2016). One possibility involves selecting, for
each pair of telescopes, events with a similar distance to each telescope $R$
(in our case, within $\pm 20\%$). Such events should have approximately the
same reconstructed energy if the optical and camera calibration of the
telescopes is appropriate. Alternatively, one can compare the energy
reconstructed by a given telescope, $\mathrm{E}$, against the average energy
reconstructed by the array, $\mathrm{E_{mean}}$. Figure 11 illustrates the
first approach (telescope pairs compared against each other) for 2019-2020.
Figure 11: Inter-telescope calibration: reconstructed energies for
$\gamma$-ray-like events from Crab Nebula observations, with a similar
(difference less than $20\%$) distance from each telescope to the
reconstructed shower core. The density of events in the parameter space is
color-coded, with yellow values representing more events and blue values
representing fewer events. A 1:1 relationship for the event energies is
plotted in black, corresponding to perfect inter-telescope calibration. For
compactness, we only show the results for 2019-2020.
We repeated these measurements for all IRF periods. For each period, we
calculated the mean $\log_{10}\mathrm{(E_{i})}-\log_{10}\mathrm{(E_{mean})}$
and its standard deviation, shown in Figure 12. The reconstructed energy of
real $\gamma$-ray like events deviate, on average for each telescope and
season, no more than $\sim 10\%$ with respect to the mean energy reconstructed
by the entire array. If the throughput calibration had not been implemented
successfully, we would see larger differences in the energy measured by the
individual telescopes. Telescopes with faster throughput degradation would
provide smaller reconstructed energy estimates.
Figure 12: Mean of the $\log_{10}\mathrm{(E_{rec}[T_{i}]/E_{mean}})$ values
for selected $\gamma$-ray-like events and its standard deviation, for each
telescope and season, where $\mathrm{E_{rec}[T_{i}]}$ is the reconstructed
energy of telescope $i$ and $\mathrm{E_{mean}}$ the average reconstructed
energy from the entire array, using the same weights for all telescopes. For a
given epoch, the energies reconstructed by the different telescopes do not
deviate on average more than $\sim 10\%$ with respect to the mean energy
reconstructed with the entire array.
### 5.3 Reconstruction of the Crab Nebula flux
The reconstructed flux and spectral shape for any measurement depends on the
flux calibration of the instrument. The Crab Nebula is the closest object to a
standard reference object for VHE observations as it is one of the brightest
sources known that has a stable emission. Therefore, we used observations of
this source for the last tests to validate both the throughput measurements
and the actual implementation of the throughput calibration. This study was
performed using the same Crab Nebula dataset and the same cuts as used in
Section 4.3.4. Two figures summarizing the results of this study are obtained:
Figure 13: Reconstructed flux of the Crab Nebula above $200\,\mathrm{GeV}$,
using 1-day bins and IRFs that correctly match the instrument throughput for
each period. Results are obtained applying moderate cuts to runs with mean
elevation $>50^{\circ}$. Left panel: Each color represents an IRF period. The
blue dot-dashed curve represents the reference Crab Nebula spectrum of Meagher
(2015) integrated above $200\,\mathrm{GeV}$, horizontal solid black lines
represent the season-average fluxes, while dashed horizontal curves show the
standard deviation of the fluxes for each season. Right panel: Shown in gray
is the distribution of fluxes for all seasons combined, with a fit to a
Gaussian shape as a solid black line. The dashed black curve shows the
equivalent result when throughput changes are not taken into account in the
IRFs. The vertical blue dashed line shows the reference Crab Nebula flux from
Meagher (2015). Figure 14: Flux dispersion ($\sigma/\mu$) as a function of
threshold energy for light curves similar to the one of Figure 13, using
moderate cuts. Taking into account the throughput evolution brings down the
statistical uncertainties from $\sim 15\%$ baseline to $\sim 10\%$.
* •
Light curves with daily binned fluxes above $200\,\mathrm{GeV}$ for BDT-based
moderate cuts (see Figure 13). Datasets corresponding to the different IRF
sets are plotted with different colors. Flux points versus time are shown in
the left panel and a histogram of the flux points is presented in the right
panel. As can be seen from the figure, every season shows an average flux
compatible within $\pm 1\,\mathrm{\sigma}$ with the flux of the Crab Nebula
reported in Meagher (2015). In addition, the spread of the flux points
(standard deviation $\sigma$ over the mean $\mu$ of the distribution) shown in
the right panels remains on the level of $\lesssim 10\%$ for the selected
cuts. This value is significantly lower than the one obtained if no throughput
calibration is applied. In that case, the spread would be $\gtrsim 15\%$. The
relationship between the energy threshold of the light curve and flux
dispersion is shown in Figure 14.
* •
Energy spectra for the different periods, using the same cuts (Figure 15),
showing simultaneously for each period the energy spectrum obtained using the
uncorrected IRFs (blue open circles), the energy spectral points for the
corrected IRFs (filled red circles) and, for comparison purposes, the
published spectrum from the Fermi-LAT 3FHL catalog Ajello et al. (2017) and
the Crab Nebula spectrum published by the VERITAS collaboration in Meagher
(2015) (orange curved line), which is based mostly on data from seasons that
were minimally affected by the throughput degradation.
Figure 15: Collection of spectral energy distributions obtained with moderate
cuts for each IRF period. Purple open diamonds represent measurements of the
Crab Nebula spectrum from the 3FHL (Ajello et al. 2017), the orange curve
shows the Crab Nebula spectrum published by VERITAS in Meagher (2015), open
blue points show the fluxes that would be obtained if throughput is not
calibrated during the reconstruction of $\gamma$-ray showers and instead the
MC model from 2012 is employed. Finally, red solid points show the spectral
points that are obtained once the pulse signals are calibrated to generate
correct IRFs for each period.
## 6 Systematic uncertainties
The estimation of systematic uncertainties affecting the reconstructed
$\gamma$-ray energies and fluxes covered in this section is based on
calibration measurements, MC simulations, and reasonable assumptions. Overall,
we could identify the following components of the systematic uncertainty
budget impacting the throughput calibration and the determination of precise
IRFs:
* •
Accuracy and stability (time and temperature dependence) of the photo-electron
response, including the precision of the pixel-to-pixel relative gain
correction. Absolute pixel gains are corrected through high-voltage
adjustments (“flat-fielding”) every year at the beginning of the VERITAS
observing season, in September or October. This contributes to reduce the
photo-electron response drift of the PMTs over long periods of time. Relative
gains likely have a small contribution to the total systematic uncertainty if
the appropriate Flasher runs are used in the analysis.
* •
QE of the PMTs and collection efficiency at the first dynode. After several
years of operations, a comparison of used and unused PMTs shows no significant
evolution of these parameters over time.
* •
Mirror reflectivity and its wavelength dependence after correction for
degradation.
* •
The optical PSF, determined by mirror alignment accuracy and checked once a
month. Mirror facets are re-aligned after detection of significant deviation
from the expected position.
Independently of the throughput calibration, the overall systematic
uncertainty budget of the VERITAS instrument is affected by:
* •
The effect of broken pixels and electronic channels, which are not modeled in
the MC simulations. The fraction of unavailable pixels per camera is typically
less than 5%.
* •
The Winston cone efficiency, not covered in our study but included in the
simulations, with an estimated contribution to the optical throughput
uncertainty of $\sim 4\%$.
* •
The treatment of shadowing elements on the telescopes in the MC simulations,
including the camera housing and support structure. Smaller elements like
cross bars are not included in the simulation model. The impact of these
simplifications introduces a systematic uncertainty of less than 1%.
* •
The systematic limit of the VERITAS pointing monitor, which is less than 20
arc seconds, much smaller than the optical PSF of the telescopes. Its
contribution to the systematic uncertainty on the photon flux is therefore
¡3%.
* •
The ability to calibrate the linearity of the readout and the transition from
high to low gain using flasher pulses at eight (fifteen starting from 2016)
different levels of brightness. The pulse shape of the low-gain channel is
roughly twice as broad than that for high gain and exhibits a complex
broadening for extremely bright signals from nearby showers with energies of
tens of TeV. This complex behavior is not accurately reproduced in the current
MC simulations, leading to an increase of the systematic uncertainties for the
highest energies. The throughput changes discussed in this paper lead
additionally to a change of pulse brightness required to activate the low-gain
chain. Our correction method, which cannot account for this hardware effect,
leads to a mismatch in the activation point between the simulations and the
real data.
* •
The analysis and reconstruction pipelines, Eventdisplay and VEGAS. While the
code development is independent, the analysis and reconstruction techniques
employed in both pipelines are similar and they use the same MC simulations to
produce the IRFs. Based on a long-term comparison of scientific results
produced with both tools, we estimate a $10\%$ systematic uncertainty due to
the analysis pipeline used.
Table 2: Summary of systematic uncertainty estimations. Type | Flux
---|---
Photo-electron response and gain | 5%
Signal pulse shape | 5-10%
Low-gain modeling | 5-10%
Efficiency of photo detection | $<5$%
Unavailable pixels | 3-5%
Shadowing | 1%
Mirror reflectivity | 10%
Winston cone efficiency | 4%
Point spread function | 5-10%
Atmospheric profiles, |
Absorption, scattering | 10-15%
Analysis | 10%
Total | $\sim 25\%$
In general, the total systematic uncertainties are dominated by uncertainties
on telescope throughput, covered by this work, and the approximations in the
modeling of the atmosphere above the observatory.
As for the latter, the development of extensive air showers is determined by
the vertical density, temperature, and humidity profiles. The intensity of
Cherenkov light is influenced by the amount of absorption and scattering on
atmospheric molecules or aerosols. While VERITAS does not operate during the
warmest summer months, and two seasonal atmospheric profiles are used to
correct for the large atmospheric changes, the systematic error due to
inaccurate atmospheric models is approximately $10-15\,\%$. This includes day-
to-day variability of the atmosphere, which is monitored by a combination of a
weather station, a commercial ceilometer, and three infrared pyrometers. To
limit its impact on the total systematic errors, observing periods of inferior
quality (e.g., due to clouds) are flagged and removed from the analysis for
most publications.
The total systematic uncertainty is estimated by summing quadratically the
individual components listed in Table 2, ignoring possible correlations. As a
caveat, some of the mentioned sources of systematic uncertainty are energy
dependent and introduce an error on the reconstructed spectral slope,
resulting in larger uncertainties for the steepest spectra. The total
systematic uncertainty on the absolute flux level is estimated to be $\sim
25\%$ for a $\gamma$-ray source with a spectral index of $\approx 2.5$.
Starting from $\sim 10\mathrm{\,TeV}$, where the calibration of the low- and
high-gain channels and saturation effects begin to be important, an additional
$5-10\%$ would have to be added in quadrature. Similarly, we estimate the
systematic uncertainty on the spectral index to be $\pm 0.2$ for sources with
Crab Nebula-like spectra. For sources with steeper spectra, the corresponding
photon flux is dominated by the emission at the lowest energies, where many of
the systematic uncertainty components that we mentioned become most relevant.
In addition, the impact of the energy scale errors on the absolute flux become
significantly larger and their estimation requires a case-by-case study,
beyond the scope of this document.
Additional sources of systematic uncertainties not directly linked to the
telescope performance have not been included in this work. Nonetheless, some
might have relevant implications in long-lived ground-based astronomical
installations, including VERITAS. A good example of this is the long-term
evolution of NSB (Massey & Foltz 2000) due to an increased human activity over
time and changes in street lamp technology: from sodium lamps to LED-based
illumination, likely brighter at short wavelengths Sánchez de Miguel et al.
(2017), where IACTs are most sensitive. Such long-term variations are likely
more evident in the direction of largely populated areas, for example, Tucson
(to the north) and Nogales (to the southwest) areas, and might have a
significant impact during observations at low elevations. Since VERITAS
simulations are produced for different NSB levels and IRFs are interpolated to
match the NSB level of each data run, this effect is corrected to first order
in the standard analysis.
## 7 Conclusions
After almost 15 years of operations, VERITAS performance has changed due to a
combination of hardware upgrades (relocation of T1, camera and camera
electronics upgrades), aging of the different components, and maintenance
duties, such as recoating of the most degraded mirror facets. With this work,
we aim to document the calibration efforts that have been carried out by the
VERITAS collaboration during this time.
As a first step, we described in Section 3 the different approaches that are
used to monitor the behavior of the instrument, measuring the gains of each
pixel, the relative evolution of the average camera gain over time, and the
changes of reflectivity and optical throughput due to telescope aging. The
described methodology, now well defined, will continue to be used for the
upcoming years of VERITAS operations and we are confident it will serve for
other experiments as well.
In Section 4, we detailed the implementation of the proposed throughput
calibration method in our software pipelines. It is based on correction
factors for the optical throughput or reflectivity $r_{i}$ and the average
gain of the camera $g_{i}$, which we combined into a total throughput factor
$t_{i}$. We showed how the application of the throughput factors to the
measured signals of the simulated events could be used to produce throughput-
calibrated IRFs for VERITAS. These response functions can be used to analyze
real showers and derive the corrected particle shower energy and source
fluxes. Finally, using the time-dependent response functions obtained, we
evaluated the impact of the throughput changes on the performance of VERITAS
using different metrics such as the sensitivity and the energy threshold.
In order to validate this calibration method, we performed extensive tests,
detailed in Section 5, to ensure that all VERITAS telescopes are properly
calibrated against each other and that the geometrical parameters of simulated
and real shower events are comparable. In addition, we checked the stability
of the flux reconstruction using a multiyear sample of Crab Nebula
observations, often assumed to be a reference object in the very-high-energy
range.
Finally, in Section 6 we discussed the main systematic uncertainties on the
energy scale, fluxes, and spectral indices that affect the analysis and
reconstruction of data from the VERITAS telescopes.
As a final note, this work is based on direct measurements of the gains and
reflectivity of the dish as an alternative to the study of local muons or the
measurement of the cosmic electron spectrum. Both have been proposed and are
currently being actively discussed (Parsons et al. 2016; Gaug et al. 2019) as
possible calibrators for the upcoming Cherenkov Telescope Array (CTA). CTA
will be comprised of arrays several times bigger than that of VERITAS, and
have much tighter requirements on systematic uncertainties. It is clear that
having independent methods to calibrate the response of the instrument to
incident light can only help to reduce their individual limitations and
systematic uncertainties.
###### Acknowledgements.
This research is supported by grants from the U.S. Department of Energy Office
of Science, the U.S. National Science Foundation and the Smithsonian
Institution, by NSERC in Canada, and by the Helmholtz Association in Germany.
This research used resources provided by the Open Science Grid, which is
supported by the National Science Foundation and the U.S. Department of
Energy’s Office of Science, and resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under Contract No. DE-AC02-05CH11231. We
acknowledge the excellent work of the technical support staff at the Fred
Lawrence Whipple Observatory and at the collaborating institutions in the
construction and operation of the instrument. M.N. acknowledges the Young
Investigators Program of the Helmholtz Association for support during the
period of the project. Reproduced with permission from Astronomy &
Astrophysics, ©ESO.
## References
* Acciari et al. (2019) Acciari, V. A., Ansoldi, S., Antonelli, L. A., et al. 2019, Nature, 575, 455
* Acciari et al. (2008) Acciari, V. A., Beilicke, M., Blaylock, G., et al. 2008, ApJ, 679, 1427
* Aharonian et al. (2004) Aharonian, F., Akhperjanian, A. G., Aye, K. M., et al. 2004, Astroparticle Physics, 22, 109
* Ajello et al. (2017) Ajello, M., Atwood, W. B., Baldini, L., et al. 2017, ApJS, 232, 18
* Archambault et al. (2021) Archambault, S., Chernitsky, G., Griffin, S., & Hanna, D. 2021, Astroparticle Physics, 128, 102556
* Archambault et al. (2013) Archambault, S., Hanna, D., & Griffin, S. 2013, in 33rd International Cosmic Ray Conference, 0379
* Berge et al. (2007) Berge, D., Funk, S., & Hinton, J. 2007, A&A, 466, 1219
* Bessell (1990) Bessell, M. S. 1990, PASP, 102, 1181
* Chalme-Calvet et al. (2014) Chalme-Calvet, R., de Naurois, M., & Tavernet, J. P. 2014, in AtmoHEAD workshop, 2013
* Cogan (2008) Cogan, P. 2008, in International Cosmic Ray Conference, Vol. 3, International Cosmic Ray Conference, 1385–1388
* Davies & Cotton (1957) Davies, J. M. & Cotton, E. S. 1957, Solar Energy, 1, 16
* de Naurois & Mazin (2015) de Naurois, M. & Mazin, D. 2015, Comptes Rendus Physique, 16, 610
* Fleury (1991) Fleury, P. 1991, in International Cosmic Ray Conference, Vol. 2, International Cosmic Ray Conference, 595–598
* Garoli et al. (2020) Garoli, D., Rodriguez De Marcos, L. V., Larruquert, J. I., et al. 2020, Applied Sciences, 10
* Gaug (2006) Gaug, M. 2006, PhD thesis, Autonomous University of Barcelona
* Gaug et al. (2019) Gaug, M., Fegan, S., Mitchell, A. M. W., et al. 2019, ApJS, 243, 11
* Gazda et al. (2016) Gazda, E., Nguyen, T., Otte, N., & Richards, G. 2016, Journal of Instrumentation, 11, P11015
* Hanna (2008) Hanna, D. 2008, in International Cosmic Ray Conference, Vol. 3, International Cosmic Ray Conference, 1417–1420
* Hanna et al. (2010) Hanna, D., McCann, A., McCutcheon, M., & Nikkinen, L. 2010, Nuclear Instruments and Methods in Physics Research A, 612, 278
* Heck et al. (1998) Heck, D., Knapp, J., Capdevielle, J. N., Schatz, G., & Thouw, T. 1998, CORSIKA: a Monte Carlo code to simulate extensive air showers. (Forschungszentrum Karlsruhe GmbH))
* Hillas (1985) Hillas, A. M. 1985, in International Cosmic Ray Conference, Vol. 3, 19th International Cosmic Ray Conference (ICRC19), Volume 3, 445
* Hillas & Patterson (1990) Hillas, A. M. & Patterson, J. R. 1990, Journal of Physics G: Nuclear and Particle Physics, 16, 1271
* Hofmann (2003) Hofmann, W. 2003, Astroparticle Physics, 20, 1
* Hofmann et al. (1999) Hofmann, W., Jung, I., Konopelko, A., et al. 1999, Astroparticle Physics, 12, 135
* Holder (2005) Holder, J. 2005, in International Cosmic Ray Conference, Vol. 5, 29th International Cosmic Ray Conference (ICRC29), Volume 5, 379
* Holder et al. (2006) Holder, J., Atkins, R. W., Badran, H. M., et al. 2006, Astroparticle Physics, 25, 391
* Kieda (2011) Kieda, D. 2011, in International Cosmic Ray Conference, Vol. 9, International Cosmic Ray Conference, 14
* Kieda (2013) Kieda, D. B. 2013, in International Cosmic Ray Conference, Vol. 33, International Cosmic Ray Conference, 1124
* Krause et al. (2017) Krause, M., Pueschel, E., & Maier, G. 2017, Astroparticle Physics, 89, 1
* Krawczynski et al. (2006) Krawczynski, H., Carter-Lewis, D. A., Duke, C., et al. 2006, Astroparticle Physics, 25, 380
* Krennrich et al. (2007) Krennrich, F., Blaylock, G., Bradbury, S. M., et al. 2007, in Journal of Physics Conference Series, Vol. 60, Journal of Physics Conference Series, 34–39
* Lebohec & Holder (2003) Lebohec, S. & Holder, J. 2003, Astroparticle Physics, 19, 221
* Li & Ma (1983) Li, T. P. & Ma, Y. Q. 1983, ApJ, 272, 317
* López Paredes et al. (2018) López Paredes, B., Araújo, H. M., Froborg, F., et al. 2018, Astroparticle Physics, 102, 56
* MacLeod (2007) MacLeod, A. 2007, Master’s thesis, McGill University Libraries
* Maier & Holder (2017) Maier, G. & Holder, J. 2017, in International Cosmic Ray Conference, Vol. 301, 35th International Cosmic Ray Conference (ICRC2017), 747
* Massey & Foltz (2000) Massey, P. & Foltz, C. B. 2000, PASP, 112, 566
* McCann et al. (2010) McCann, A., Hanna, D., Kildea, J., & McCutcheon, M. 2010, Astroparticle Physics, 32, 325
* Meagher (2015) Meagher, K. 2015, in International Cosmic Ray Conference, Vol. 34, 34th International Cosmic Ray Conference (ICRC2015), 792
* Meyer (2005) Meyer, M. 2005, in American Institute of Physics Conference Series, Vol. 745, High Energy Gamma-Ray Astronomy, ed. F. A. Aharonian, H. J. Völk, & D. Horns, 774–778
* Mirzoyan et al. (2007) Mirzoyan, R., Garczarczyk, M., Hose, J., & Paneque, D. 2007, Astroparticle Physics, 27, 509
* Mitchell et al. (2016) Mitchell, A. M. W., Parsons, R. D., Hofmann, W., & Bernlöhr, K. 2016, Astroparticle Physics, 75, 1
* Nagai et al. (2007) Nagai, T., McKay, R., Sleege, G., & Petry, D. 2007, 30th International Cosmic Ray Conference, Merida, Mexico, July 2007, arXiv:0709.4517
* Otte et al. (2011) Otte, A. N., Gebremedhin, L., Kaplan, K., & Long, D. 2011, in 32nd International Cosmic Ray Conference
* Parsons et al. (2016) Parsons, R. D., Hinton, J. A., & Schoorlemmer, H. 2016, Astroparticle Physics, 84, 23
* Perkins et al. (2009) Perkins, J. S., Maier, G., & The VERITAS Collaboration. 2009, 2009 Fermi Symposium, eConf Proceedings C091122, arXiv:0912.3841
* Roache et al. (2007) Roache, E., Irvin, R., Perkins, J., et al. 2007, in 30th International Cosmic Ray Conference (ICRC2017), 1397–1400, 30th International Cosmic Ray Conference, ICRC 2007 ; Conference date: 03-07-2007 Through 11-07-2007
* Roache et al. (2008) Roache, E., Irvin, R., Perkins, J. S., et al. 2008, in International Cosmic Ray Conference, Vol. 3, International Cosmic Ray Conference, 1397–1400
* Rovero et al. (1996) Rovero, A. C., Buckley, J. H., Fleury, P., et al. 1996, Astroparticle Physics, 5, 27
* Sánchez de Miguel et al. (2017) Sánchez de Miguel, A., Aubé, M., Zamorano, J., et al. 2017, MNRAS, 467, 2966
* Speckmayer et al. (2010) Speckmayer, P., Höcker, A., Stelzer, J., & Voss, H. 2010, Journal of Physics: Conference Series, 219, 032057
* Tyler (2013) Tyler, J. 2013, in International Cosmic Ray Conference, Vol. 33, International Cosmic Ray Conference, 3096
* Vacanti et al. (1994) Vacanti, G., Fleury, P., Jiang, Y., et al. 1994, Astroparticle Physics, 2, 1
* Weekes (1996) Weekes, T. C. 1996, Space Sci. Rev., 75, 1
* Winston (1974) Winston, R. 1974, Solar Energy, 16, 89
* Winston (1976) Winston, R. 1976, Light collectors in cylindrical geometry, United States, https://www.osti.gov/biblio/7129508
* Zitzer (2013) Zitzer, B. 2013, in International Cosmic Ray Conference, Vol. 33, International Cosmic Ray Conference, 3076
## Appendix A Reference mirror reflectivities used in the simulations
The reflectivities of the mirror facets used in the simulation of the VERITAS
telescopes are based on measurements made in 2011, after T1 was relocated to
its current position in the post-upgrade array layout. Together with the
average camera gains of section 3.1, they are part of the current detector
model of VERITAS, defined in April 2012 and used as a reference throughout
this document. The reflectivities were measured from $260\,\mathrm{nm}$ to
$700\,\mathrm{nm}$ for several mirrors facets that were located on the top,
middle, and bottom of the dish of each telescope. The measured values were
then averaged to get a mean reflectivity representative of the entire dish.
Figure 16: Mean telescope reflectivity used to produce the baseline MC
simulations in the current instrument configuration of VERITAS, and the
reference IRFs discussed throughout this work.
## Appendix B Detailed MC-Data comparison
During the validation of the proposed method to calibrate the throughput, we
performed extensive tests to check the agreement between data and the modified
MC simulations. One of the most sensitive tests in VERITAS to detect possible
discrepancies between the measured Cherenkov signals and the corresponding
simulated showers is to check the agreement in the MSCW and MSCL parameters.
Both are shown, for each IRF period, in Figures 17 and 18.
Figure 17: Comparison between the MSCW values of simulated $\gamma$-ray
showers (red curves) and the ones for real $\gamma$-like events (bliue dots).
Each row corresponds to one IRF period and each column to a energy bin (in
log-scale, with the energy measured in TeV). Figure 18: Comparison between the
MSCL values of simulated $\gamma$-ray showers (red curves) and the ones for
real $\gamma$-like events (blue dots). Each row corresponds to one IRF period
and each column to a energy bin (in log-scale, with the energy measured in
TeV).
## Appendix C Stability of spectral parameters
As discussed in section 2.3, the Crab Nebula is often used as a source to
benchmark the performance of $\gamma$-ray instruments. It is not only one of
the brightest sources in the sky, but also visible from both hemispheres. The
Crab Nebula has a hard spectrum which reaches energies of a few tens of TeV
with photon fluxes that are detectable by VERITAS. Finally, it is thought to
be stable in year-timescales both in spectral shape and emitted luminosity, at
least in the very-high-energy band. Using eight one-year observation campaigns
of the Crab Nebula for the entire post-upgrade period of VERITAS, we could
evaluate the performance of the proposed throughput calibration method.
Figure 19 shows as a time series the run-wise values of the normalization flux
and the spectral index of the source, evaluated at $1\,\mathrm{TeV}$. It was
estimated by analysing each Crab Nebula run that survived quality cuts and had
more than $4\,\sigma$ excess. Then, the resulting run-wise spectra were fitted
locally with a power-law over the small energy range of $0.4-6\,\mathrm{TeV}$.
A histogram representation of the values of Figure 19 can be seen in Figure
20.
Figure 19: Run-wise measurement of the normalization flux at $1\,\mathrm{TeV}$
and the spectral index of the Crab Nebula between 0.6 and 4 TeV, including all
IRF periods starting from the upgrade of VERITAS. Only runs with a detection
significance of at least $4\,\sigma$ are included. Figure 20: Histogram of
run-wise measurements of normalization and spectral index of Crab Nebula
between 0.6 and 4 TeV including all IRF periods starting from the upgrade of
VERITAS. Only runs with a detection significance of at least $4\,\sigma$ are
included.
|
# Explore More Guidance: A Task-aware Instruction Network
for Sign Language Translation Enhanced with Data Augmentation
Yong Cao1* * Equal Contribution. Wei Li2* Xianzhi Li1* Min Chen1# #
Corresponding author: Min Chen. Guangyong Chen3
Long Hu1 Zhengdao Li4 and Hwang Kai4
1Huazhong University of Science and Technology 2Nanchang University
3Zhejiang University, Zhejiang Lab 4The Chinese University of Hong Kong,
Shenzhen
{yongcao_epic, weili_epic, xzli<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Sign language recognition and translation first uses a recognition module to
generate glosses from sign language videos and then employs a translation
module to translate glosses into spoken sentences. Most existing works focus
on the recognition step, while paying less attention to sign language
translation. In this work, we propose a task-aware instruction network, namely
TIN-SLT, for sign language translation, by introducing the isntruction module
and the learning-based feature fuse strategy into a Transformer network. In
this way, the pre-trained model’s language ability can be well explored and
utilized to further boost the translation performance. Moreover, by exploring
the representation space of sign language glosses and target spoken language,
we propose a multi-level data augmentation scheme to adjust the data
distribution of the training set. We conduct extensive experiments on two
challenging benchmark datasets, PHOENIX-2014-T and ASLG-PC12, on which our
method outperforms former best solutions by 1.65 and 1.42 in terms of BLEU-4.
Our code is published at https://github.com/yongcaoplus/TIN-SLT.
Explore More Guidance: A Task-aware Instruction Network
for Sign Language Translation Enhanced with Data Augmentation
## 1 Introduction
Figure 1: Comparing the sign language translation performance on two
challenging datasets, i.e., PHOENIX-2014-T (blue) and ASLG-PC12 (gray), in
terms of BLEU-1 and BLEU-4 metrics. Clearly, our approach achieves the highest
scores on both datasets compared with others. The experiments section contains
more results and analysis.
Sign language recognition and translation aims to transform sign language
videos into spoken languages, which builds a bridge for communication between
deaf and normal people. Considering the unique grammar of sign languages,
current effective recognition and translation systems involve two steps: a
tokenization module to generate glosses from sign language videos, and a
translation module to translate the recognized glosses into spoken natural
languages. Previous works (Li et al., 2020; Sincan and Keles, 2020; Sharma and
Kumar, 2021; Kumar et al., 2020; Camgoz et al., 2020) have proposed various
solutions to address the first step, but paid less attention to the
translation system. Hence, this paper aims to solve the problem of sign
language translation (SLT) with the goal of translating multiple recognized
independent glosses into a complete sentence.
To do so, most existing works (Ko et al., 2019; Stoll et al., 2018) directly
apply advanced techniques, e.g., Seq2Seq model (Sutskever et al., 2014) or
Transformer (Vaswani et al., 2017), from neural machine translation to SLT.
However, different from the lingual translation task in neural machine
translation, SLT poses several unique challenges. First, it is hard to collect
and annotate a large amount of sign language corpus. It is still an open
question that how to explore more guidance and external information for SLT
task by incorporating the pre-trained language models based on masses of
unlabeled corpus. Second, since sign languages are developed independently
from spoken languages with quite different linguistic features, the
discrepancy of representation space between glosses and spoken sentences is
significant, thus increasing the translation difficulty.
To address the above issues, we propose a novel task-aware instruction
network, called TIN-SLT for sign language translation, further enhanced with a
multi-level data augmentation scheme. Our TIN-SLT is capable of encoding pre-
trained language model’s ability into the translation model and also
decreasing the discrepancy between the representation space of glosses and
texts.
To begin with, we leverage the extracted hidden features from the pre-trained
model as extra information to guide the sign language translation. Besides, we
apply an instruction module to transform general token features into task-
aware features. In this way, we can fully utilize the language skills
originating from the external world, thus reducing the demand for sign
language training data.
Next, to better inject the information from pre-trained model into the SLT
model, we design a learning-based feature fusion strategy, which has been
analyzed and validated to be effective compared with existing commonly-used
fusion ways.
Finally, considering the large difference between the sign language glosses
and texts in terms of the representation space, we propose a multi-level data
augmentation scheme to enrich the coverage and variety of existing datasets.
In summary, our contributions are threefold: (i) a novel TIN-SLT network to
explore more guidance of pre-trained models, (ii) a learning-based feature
fusion strategy, and (iii) a multi-level data augmentation scheme. Extensive
experiments on challenging benchmark datasets validate the superiority of our
TIN-SLT over state-of-the-art approaches; see Figure 1 for example results.
## 2 Related Works
Methods for sign language recognition. SLR task mainly focuses on the
extraction of extended spatial and temporal multi-cue features (Zhou et al.,
2020; Koller et al., 2017). Most existing works (Yin et al., 2016; Qiu et al.,
2017; Wei et al., 2019; Cui et al., 2019) study the strong representation of
sign language videos such as multi-semantic (Cui et al., 2019) and multi-
modality (Koller et al., 2019) analysis. Although extracting representative
features from sign language videos is fully explored, how to effectively
conduct the subsequent translation by considering the unique linguistic
features of sign language is often ignored in these SLR works.
Methods for sign language translation. Early approaches for SLT rely on
seq2seq model and attention mechanism (Arvanitis et al., 2019), while facing
the limitation of long-term dependencies. Later, motivated by the ability of
the Transformer (Vaswani et al., 2017), many researchers utilize it to
effectively improve SLT performance. For example, the work in Camgoz et al.
(2020) tried to use Transformer for both recognition and translation, and
promote the joint optimization of sign language recognition and translation.
The subsequent work (Yin and Read, 2020) proposed the STMC-Transformer network
which first uses STMC networks (Zhou et al., 2020) to achieve better results
for SLR, and then exploits Transformer for translation to obtain better SLT
performance.
Figure 2: Comparing the sample distribution between the input sign glosses
(yellow dots) and the output translated texts (red dots) on two datasets.
General neural machine translation. Broadly speaking, sign language
translation belongs to the field of neural machine translation, with the goal
of carrying out automated text translation. Earlier approaches deployed
recurrent network (Bahdanau et al., 2014), convolutional network (Gehring et
al., 2017), or Transformer (Vaswani et al., 2017) as encoder-decoder module.
Among them, Transformer has achieved state-of-the-art results, but the
translation performance still needs to be improved due to the limited training
corpus. In addition, there are some explorations in bringing the pre-trained
models into neural machine translation (Imamura and Sumita, 2019; Shavarani
and Sarkar, 2021; Zhu et al., 2020).
## 3 Challenges
The goal of this work is to translate the recognized multiple independent
glosses (network input) into a complete spoken sentence (expected output).
Compared with general neural machine translation tasks, SLT faces two main
challenges:
Figure 3: Network architecture of TIN-SLT. As shown in the bottom row, we
first employ STMC model (Zhou et al., 2020) to recognize sign language videos
to independent glosses. Next, we design a multi-level data augmentation scheme
to enrich existing data pool for better feature embedding from glosses. Then,
we design a task-aware instruction network with a novel instruction module to
translate glosses into a complete spoken sentence.
Limited annotated corpus: Compared with natural languages, the data resources
of sign languages are scarce (Bragg et al., 2019). As a result, the SLT models
trained on limited data often suffer from the overfitting problem with poor
generalization (Moryossef et al., 2021; Yin et al., 2021).
Discrepancy between glosses (input) and texts (output): Figure 2 shows the
representation space of sign glosses (yellow dots) and translated texts (red
dots) using Word2Vec (Mikolov et al., 2013) on two different datasets. We can
observe that the representation space of sign glosses is clearly smaller than
that of the target spoken language, thus increasing the difficulty of network
learning.
## 4 Our Approach
To address the above challenges, we propose TIN-SLT by effectively introducing
the pre-trained model into SLT task and further designing a multi-level data
augmentation scheme. Figure 3 depicts the detailed network architecture. In
the following subsections, we will firstly introduce the network architecture
of TIN-SLT, followed by our solutions to address the above two challenges.
### 4.1 Network Architecture of TIN-SLT
Given a sign language video $\mathcal{V}=\\{V_{1},\dots,V_{T}\\}$ with $T$
frames, like existing approaches, we also adopt a two-step pipeline by first
(i) recognizing $\mathcal{V}$ into a sequence
$\mathcal{G}=\\{g_{1},\dots,g_{L}\\}$ with $L$ independent glosses and then
(ii) translating $\mathcal{G}$ into a complete spoken sentence
$\mathcal{S}=\\{w_{1},\dots,w_{M}\\}$ with $M$ words, but we pay more
attention to solve step (ii). Hence, for step (i), as shown in the bottom-left
part of Figure 3, we empirically use the spatial-temporal multi-cue (STMC)
network (Zhou et al., 2020), which consists of a spatial multi-cue module and
a temporal multi-cue module. For more technical details of STMC, please refer
to (Zhou et al., 2020). Below, we shall mainly elaborate on the details of
addressing step (ii).
After obtaining the sequence $\mathcal{G}$ of sign glosses, considering that
the representation space of glosses is much smaller than that of texts (see
Figure 2), we thus design a multi-level data augmentation scheme to expand the
gloss representation space; see the top-left part of Figure 3 as an
illustration and we shall present its details in Section 4.3.
Next, as shown in the bottom-middle part of Figure 3, the key of our design is
a task-aware instruction network, where we adopt Transformer as the network
backbone consisting of several encoder and decoder layers, whose objective is
to learn the conditional probabilities $p(\mathcal{S}|\mathcal{G})$. Since SLT
is an extremely low-data-resource task as we have discussed in Section 3, we
thus focus on exploring more task-aware guidance by learning external world
knowledge, which is dynamically incorporated into the Transformer backbone via
our designed task-aware instruction module. We shall present its details in
Section 4.2.
Lastly, the outputs of last decoder are passed through a non-linear point-wise
feed forward layer and we can obtain the predicted sentence $\mathcal{S}$ by a
linear transform and softmax layer.
### 4.2 Task-aware Instruction Module
As is shown in Figure 3, our task-aware instruction network is composed of a
series of encoder and decoder layers. To handle the limited training data, we
propose to leverage the learned external knowledge from natural language
datasets to guide the learning of sign languages. More specifically, we design
a task-aware instruction module to dynamically inject external knowledge from
pre-trained models into our encoder and decoder. Below, we shall present the
details.
Encoder. Given the recognized glosses,let $H_{I}$ denotes the instruction
features encoded by the pre-trained model (PTM), $H_{E}$ and $H_{E}^{\prime}$
denotes the input and output of encoder which is randomly initialized. As
shown in Figure 4, $H_{I}$ and $H_{E}$ are fed into the task-aware instruction
module for feature fusing. Then, the output of the instruction module is fed
into residual connection (Add&Norm) and feed forward network (FFN).
The light yellow box of Figure 4 shows the detailed design of task-aware
instruction module. Specifically, we feed $H_{E}$ into a self-attention module
to learn the contextual relationship between the features of glosses, while
$H_{I}$ is fed into a PTM-attention, which is the same architecture as self-
attention. Different from existing work which employ PTM in general neural
network (Zhu et al., 2020), we insert an adaptive layer to fine-tune PTM-
attention output for SLT task, to transform general gloss features into task-
aware features.
$h_{i}=\sigma(Attn_{I}(h_{t},H_{I},H_{I}))$ (1)
where $\sigma()$ denotes the adaptive layer (we set it as fully connection
layers here), and $h_{t}$ denotes the gloss features at time step $t$. Then,
the output of two modules are combined via $\alpha$ strategy. The whole
process is formulated as follows:
$\hat{h}_{t}=(1-\alpha)Attn_{E}(h_{t},H_{E},H_{E})+\alpha h_{i}$ (2)
where $Attn_{E}$ and $Attn_{I}$ are two attention layers with different
parameters, which follow (Vaswani et al., 2017). The way of setting an optimal
$\alpha$ will be introduced later.
Decoder. Let $S_{D}$ and $S^{\prime}_{D}$ denotes the input and output of
decoder, $s_{t}$ denote the hidden state at time step $t$, and $s_{0}$ denotes
the beginning token of a sentence, i.e., $<bos>$. The hidden states are passed
to a masked self-attention ensuring that each token may only use its
predecessors as follows:
$\tilde{s}_{t}=Attn_{D}(s_{t},s_{1:t},s_{1:t})$ (3)
Representations $H_{E}^{\prime}$ and $H_{I}$ extracted from encoder and PTM
are fed into the decoder-attention and PTM-attention module, respectively, as
shown in the right part of Figure 4. Similar to Encoder, we formulate this
decoding output as:
$\hat{s}_{t}=(1-\alpha)Attn_{D}(\tilde{s}_{t},H_{E}^{\prime},H_{E}^{\prime})+\alpha
h_{i}$ (4)
where $Attn_{D}$ represent decoder-attention, and $\hat{s}_{t}$ is the output
of decoder instruction module.
Learning-based feature fusion. As shown in Eq. (2), representations extracted
from both PTM- and self- attention are fused via a parameter $\alpha$. How to
set a reasonable and optimal $\alpha$ will directly affects the learning
performance, which is a problem worthy of exploration. Instead of manually
setting a constant $\alpha$, we propose a learning-based strategy to encourage
the network to learn the optimal $\alpha$ by itself for better feature fusion.
Specifically, learning-based strategy means that we adopt the back-propagation
learning algorithm to update $\alpha$ during the network training process:
$\alpha_{t+1}=\Gamma(\alpha_{t},g_{t})$ (5)
where $g_{t}$ indicates the gradient and $\Gamma(\cdot)$ represents the
optimization algorithm. Though the idea of self-learning is straightforward,
we shall show in the experiment section that it is quite effective compared
with many other strategies.
Figure 4: Details of Encoder layer, Decoder layer, and and Instruction Module.
### 4.3 Multi-level Data Augmentation
To decrease the discrepancy between glosses (input) and texts (output), we
propose a multi-level data augmentation scheme. Our key idea is that, besides
existing gloss-text pairs, we use upsampling as our data augmentation
algorithm and generate text-text pairs as extended samples to introduce texts
information into glosses, thus enlarging the feature distribution space of
glosses.
Actually, there is a trade-off between augmentation and overfitting, which
means the upsampling ratio $\Phi_{upsamp}$ should be determined by the degree
of gloss-text difference. We here propose four factors
$\phi=[\phi_{v},\phi_{r},\phi_{s},\phi_{d}]$ to calculate the difference in
terms of token, sentence and dataset level, and set weighted $\phi$ as
$\Phi_{upsamp}$.
Token level. Vocabulary Different Ratio (VDR, $\phi_{v}$) is used to measure
the difference of gloss vocabulary space and text’s, as calculated by Eq. (6).
$\phi_{v}=1-\frac{|W_{\mathcal{G}}|}{|W_{\mathcal{G}}\cup W_{\mathcal{S}}|}$
(6)
where $W_{\mathcal{G}}$ and $W_{\mathcal{S}}$ represent gloss and text
vocabularies, and $|\cdot|$ denotes the size of set.
We present Rare Vocabulary Ratio (RVR, $\phi_{r}$) to calculate the ratio of
the rare words:
$\phi_{r}=1-\frac{\sum_{\mathcal{G}\in
W_{\mathcal{G}}}\\#(Counter(\mathcal{G})<\tau_{r})}{|W_{\mathcal{G}}\cup
W_{\mathcal{S}}|}$ (7)
where $\\#(\cdot)$ is 1 if the value is true, else 0, $Counter(\mathcal{G})$
is to calculate the gloss vocabulary frequency, and $\tau_{r}$ means the
empirical thresh frequency determined by the vocabulary frequency, which is
empirically set to be 2.
Sentence level. We propose Sentence Cover Ratio (SCR, $\phi_{s}$) to compute
the gloss-text pair similarity and covered ratio, calculated as:
$r_{i}=\frac{|\mathcal{G}_{i}\cap\mathcal{S}_{i}|}{|\mathcal{S}_{i}|},\quad\phi_{s}=1-\frac{1}{N}\sum_{i,r_{i}>\tau_{c}}r_{i}$
(8)
where $r_{i}$ denotes the covered ratio of gloss-text pair $\mathcal{G}_{i}$
and $\mathcal{S}_{i}$, while $\tau_{c}$ means the empirical thresh (set
$\tau_{c}=0.5$). We labeled gloss-text pairs which satisfy $r_{i}>\tau_{c}$ as
candidates $\mathcal{C}$.
Dataset level. We use Dataset Length-difference Ratio (DLR, $\phi_{d}$) to
calculate the length of sentence distance, calculated as:
$\phi_{d}=1-\frac{\sum_{i}|\mathcal{G}_{i}|}{\sum_{i}|\mathcal{S}_{i}|}$ (9)
Then we can get the upsampling ratio by:
$\Phi_{upsamp}=\theta*\phi$ (10)
where the weight matrix $\theta$ is empirically set as $[0.1,0.1,0.6,0.2]$,
corresponding to the weight of $[\phi_{v},\phi_{r},\phi_{s},\phi_{d}]$, as we
suppose the sentence level matters the most and the weight of token level is
the same as dataset level. Lastly, we obtain the upsampling ratio and use
upsampling strategy among all candidates $\mathcal{C}$ to enrich the dataset.
## 5 Experiments
| Dev Set | Test Set
---|---|---
Model | BLEU-1 | BLEU-2 | BLEU-3 | BLEU-4 | ROUGE-L | METEOR | BLEU-1 | BLEU-2 | BLEU-3 | BLEU-4 | ROUGE-L | METEOR
| PHOENIX-2014-T Dataset Evaluation
Raw Data (Yin and Read 2020) | 13.01 | 6.23 | 3.03 | 1.71 | 24.23 | 13.69 | 11.88 | 5.05 | 2.41 | 1.36 | 22.81 | 12.12
Seq2seq (Camgoz et al. 2018) | 44.40 | 31.93 | 24.61 | 20.16 | 46.02 | - | 44.13 | 31.47 | 23.89 | 19.26 | 45.45 | -
Transformer (Camgoz et al. 2020) | 50.69 | 38.16 | 30.53 | 25.35 | - | - | 48.90 | 36.88 | 29.45 | 24.54 | - | -
Transformer (Yin and Read 2020) | 49.05 | 36.20 | 28.53 | 23.52 | 47.36 | 46.09 | 47.69 | 35.52 | 28.17 | 23.32 | 46.58 | 44.85
Transformer Ens. (Yin and Read 2020) | 48.85 | 36.62 | 29.23 | 24.38 | 49.01 | 46.96 | 48.40 | 36.90 | 29.70 | 24.90 | 48.51 | 46.24
DataAug (Moryossef et al. 2021b) | - | - | - | - | - | - | - | - | - | 23.35 | - | -
TIN-SLT(Ours) | 52.35 | 39.03 | 30.83 | 25.38 | 48.82 | 48.40 | 52.77 | 40.08 | 32.09 | 26.55 | 49.43 | 49.36
| ASLG-PC12 Dataset Evaluation
Raw data (Yin and Read 2020) | 54.60 | 39.67 | 28.92 | 21.16 | 76.11 | 61.25 | 54.19 | 39.26 | 28.44 | 20.63 | 75.59 | 61.65
Preprocessed data (Yin and Read 2020) | 69.25 | 56.83 | 46.94 | 38.74 | 83.80 | 78.75 | 68.82 | 56.36 | 46.53 | 38.37 | 83.28 | 79.06
Seq2seq (Arvanitis et al. 2019) | - | - | - | - | - | - | 86.70 | 79.50 | 73.20 | 65.90 | - | -
Transformer (Yin and Read 2020) | 92.98 | 89.09 | 83.55 | 85.63 | 82.41 | 95.93 | 92.98 | 89.09 | 85.63 | 82.41 | 95.87 | 96.46
Transformer Ens.(Yin and Read 2020) | 92.67 | 88.72 | 85.22 | 81.93 | 96.18 | 95.95 | 92.88 | 89.22 | 85.95 | 82.87 | 96.22 | 96.60
TIN-SLT (Ours) | 92.75 | 88.91 | 85.51 | 82.33 | 95.17 | 95.21 | 93.35 | 90.03 | 87.07 | 84.29 | 95.39 | 95.92
| | | | | | | | | | | |
Table 1: Comparing the translation performance of TIN-SLT against state-of-
the-art techniques on PHOENIX-2014-T and ASLG-PC12 datasets. Clearly, our TIN-
SLT achieves the best performance on most metrics.
### 5.1 Implementation Details
Datasets. We conduct our experiments on two popular benchmark datasets of
different languages and scales, including PHOENIX-2014-T (Camgoz et al., 2018)
dataset and ASLG-PC12 (Othman and Jemni, 2012) dataset.
Specifically, PHOENIX-2014-T, i.e., PH14, is an open-source German sign
language dataset, recorded from broadcast news about the weather. This dataset
contains parallel sign language videos from 9 different signers, gloss
annotations with a vocabulary of 1066 different signs, and their translations
with a vocabulary of 2887 different words.
ASLG-PC12, i.e., ASLG, is a parallel corpus of English written texts and
American Sign Language (ASL) glosses, which is constructed based on rule-based
approach. It contains more than one hundred million pairs of sentences between
English sentences and ASL glosses.
Evaluation metrics. To fairly evaluate the effectiveness of our TIN-SLT, we
follow (Yin and Read, 2020) to use the commonly-used BLEU-$N$ ($N$-grams
ranges from 1 to 4) (Papineni et al., 2002), ROUGE-L (Lin, 2004) and METEOR
(Banerjee and Lavie, 2005) as the evaluation metrics.
Experimental setup. The experiments are conducted on Ubuntu 18.04 system with
two NVIDIA V100 GPUs. Our Transformers are built using 2048 hidden units and 8
heads in each layer. Besides, we adopt Adam (Kingma and Ba, 2014) as
optimization algorithm with $\beta_{1}=0.9$, $\beta_{2}=0.998$ and use inverse
sqrt learning rate scheduler with a weight decay of $10^{-3}$. Please refer to
Appendix for more hyper-parameter settings.
### 5.2 Comparison with Others
To compare our TIN-SLT against state-of-the-art approaches on sign language
translation task, we conducted two groups of experiments, Gloss2Text (G2T) and
Sign2Gloss2Text (S2G2T).
Evaluation on G2T. G2T is a text-to-text translation task, whose objective is
to translate ground-truth sign glosses to spoken language sentences. In
specific, for PH14 dataset, we should output German spoken language sentences;
while for ASLG dataset, we should output English sentences. Table 1 summarizes
the comparison results. Clearly, our TIN-SLT achieves the highest values on
most evaluation metrics with a significant margin. Particularly, the
superiority of our method on PH14 dataset is more obvious, where almost all
the evaluation values are the highest. Thanks to our multi-level data
augmentation scheme, the integrity of translated sentences has been improved,
which is reflected in the significant improvement of BLEU-$N$ metric. In
addition, the strong guidance from external knowledge also encourages our
network to generate translated sentences in correct grammar, consistent tense
and appropriate word order. For the lower ROUGE-L metric, we think that
although the instruction module obviously help improve the accuracy and
fluency of translation results, it leads to a slight decrease of continuous
texts’ recall rate in this task.
(a) Comparing various $\alpha$ strategies on PH14 dataset
(b) Comparing various $\alpha$ strategies on ASLG dataset
(c) The learned value of $\alpha$ on PH14 dataset
(d) The learned value of $\alpha$ on ASLG dataset
(e) Effect of beam size
(f) Effect of layer number
(g) Effect of learning rate
(h) Effect of dropout rate
Figure 5: Various analysis results. (a) & (b) present the results by using
different feature fusion strategies on two datasets, respectively. (c) & (d)
show our learned value of $\alpha$ during the training process on the two
datasets, respectively. (e)-(h) explore how beam size, layer number, learning
rate, and dropout rate affect the model performance.
Evaluation on S2G2T. S2G2T is an extended task beyond G2T, which aims to
recognize sign language videos to sign glosses, and then translate the
recognized glosses to spoken sentences. Hence, unlike the task of G2T, in this
comparison, we focus on the evaluation of the whole two-step pipeline, that
is, obtaining spoken language sentences from sign language videos. Considering
that only PH14 contains sign language videos, we thus conduct experiments on
this dataset for S2G2T task, and the results are reported in Table 2. Note
that, for the recognition step, we employ STMC model to realize vision-based
sequence learning (Zhou et al., 2020). From the comparison we can see that,
our TIN-SLT still outperforms existing approaches on most evaluation metrics.
| Test Set
---|---
Model | BLEU-1 | BLEU-2 | BLEU-3 | BLEU-4 | ROUGL-L | METEOR
G2T | 44.13 | 31.47 | 23.89 | 19.26 | 45.45 | -
S2G-G2T | 41.54 | 29.52 | 22.24 | 17.79 | 43.45 | -
S2G2T | 43.29 | 30.39 | 22.82 | 18.13 | 43.80 | -
Sign2 | 46.61 | 33.73 | 26.19 | 21.32 | - | -
Bahdanau | 47.53 | 33.82 | 26.07 | 21.54 | 45.50 | 44.87
Luong | 47.08 | 33.93 | 26.31 | 21.75 | 45.66 | 44.84
Transformer Ens. | 50.63 | 38.36 | 30.58 | 25.40 | 48.78 | 47.60
TIN-SLT (Ours) | 51.06 | 38.85 | 31.23 | 26.13 | 48.56 | 47.83
| | | | | |
Table 2: Comparing the S2G2T performance by using our TIN-SLT and state-of-
the-art techniques on PHOENIX-2014-T dataset. The results of G2T, S2G-G2T and
S2G2T are from (Camgoz et al., 2018). The results of Sign2 are from (Camgoz et
al., 2020). The results of Bahdanau, Luong, and Transformer Ens. are from (Yin
and Read, 2020). Clearly, our TIN-SLT achieves the highest values on most
metrics.
### 5.3 Analysis and Discussions
Here, we conducted a series of detailed experiments to analyze our method and
give some insights behind our network design.
Effect of learning-based feature fusion. In this work, we propose a learning-
based strategy to set $\alpha$ dynamically. Here, we conducted experiments by
comparing this strategy with other four different strategies, including (1)
cosine annealing (Loshchilov and Hutter, 2016), (2) cosine increment, (3)
cosine decrement, and (4) constant value. The update of $\alpha$ by the three
cosine strategies are calculated as Eq. (11) with different settings of the
epoch cycle coefficient $T_{c}$:
$\alpha_{t+1}=\alpha_{min}+\frac{1}{2}(\alpha_{max}-\alpha_{min})(1-cos(\frac{T_{t}}{T_{c}}\pi+\gamma))$
(11)
where $\alpha$ is the fusion ratio, $T_{t}$ is current epoch step, and
$\gamma$ is the time-shift constant. We set $T_{c}$ as (25, 100, 100) and
$\gamma$ as (0, 0, $\pi$) for cosine annealing, cosine decrement, and cosine
increment, respectively. The minimum value $\alpha_{min}$ and maximum value
$\alpha_{max}$ of $\alpha$ are set to be 0 and 1.
Figures 5(a)-5(b) are the experimental results on the two datasets. We can
observe that the learning-based strategy (red line) gets the best result on
ASLG and comparable result with the constant setting ($\alpha$$=$$0.8$) on
PH14, but still better than other three cosine strategies. Moreover, we also
visualize the learned value of $\alpha$ during the training process as shown
in Figures 5(c)-5(d) to find out the contribution ratio of the BERT model to
the final performance. We can see that, the value of $\alpha$ is gradually
decreasing on PH14, meaning that the model depends more on the BERT pre-
trained knowledge at the beginning of the training process and gradually
inclines to our employed training corpus. The observation is just opposite on
ASLG, since it is a much larger dataset than PH14 and our model relies more on
BERT to further boost the performance near the end of training.
| Test Set
---|---
Model | BLEU-1 | BLEU-2 | BLEU-3 | BLEU-4 | ROUGE-L | METEOR
| PHOENIX-2014-T Dataset Evaluation
Baseline | 47.69 | 35.52 | 28.17 | 23.32 | 46.58 | 44.85
w/ DataAug | 50.77 | 37.85 | 29.88 | 24.57 | 47.39 | 46.95
w/ Encoder | 51.05 | 37.94 | 29.91 | 24.63 | 47.59 | 47.13
w/ Decoder | 50.99 | 38.47 | 30.48 | 25.08 | 48.78 | 48.20
Full pipeline | 52.77 | 40.08 | 32.09 | 26.55 | 49.43 | 49.36
| ASLG-PC12 Dataset Evaluation
Baseline | 92.98 | 89.09 | 85.63 | 82.41 | 95.87 | 96.46
w/ DataAug | 92.60 | 89.15 | 85.80 | 83.05 | 95.08 | 95.33
w/ Encoder | 92.77 | 89.22 | 86.23 | 83.40 | 95.22 | 96.87
w/ Decoder | 93.15 | 89.80 | 86.49 | 83.89 | 95.34 | 95.67
Full pipeline | 93.35 | 90.03 | 87.07 | 84.29 | 95.39 | 95.92
| | | | | |
Table 3: Ablation analysis of our major network components on the G2T task.
Analysis on major network components. In our TIN-SLT, there are two major
components: the multi-level data augmentation scheme and the instruction
module. To validate the effectiveness of each component, we conduct an
ablation analysis on the G2T task with the following cases.
* •
Baseline: We use two layers Transformer (Yin and Read, 2020) without data
augmentation and instruction module as baseline.
* •
w/ DataAug: Based on the baseline, we add our data augmentation scheme back.
* •
w/ Encoder: Based on w/ DataAug, we fuse instruction module only into the
encoder.
* •
w/ Decoder: Based on w/ DataAug, we fuse instruction module only into the
decoder.
As a contrast, in our full pipeline, the instruction module is inserted into
both encoder and decoder. Table 3 shows the evaluation results on both PH14
and ASLG. By comparing the results from Baseline and w/ DataAug, we can see
that our data augmentation improves the translation performance, especially
for the PH14 dataset. A reasonable interpretation is that the translation task
on PH14 dataset is more difficult than on ASLG, thus our data augmentation
contributes more. On the other hand, w/ Encoder, w/ Decoder and Full pipeline
explore the best location to introduce PTM information into the model. Results
in Table 3 show that our full model achieves the best performance.
Particularly, by comparing the results from w/ Encoder and w/ Decoder against
the results from SOTA methods (Tables 1 & 3), we can observe that as long as
we employ the pre-trained model, no matter where it is inserted into the
network, the performance is always better than existing methods.
Model111The pre-trained models links are listed in Appendix. | Size(MB) | Dataset | Gloss(%) | Text(%) | BLEU4
---|---|---|---|---|---
| PHOENIX-2014-T Dataset Evaluation
Multilingual | 641.10 | PH14 | 59.96 | 74.62 | 25.48
Distilbert | 257.30 | PH14 | 44.50 | 71.15 | 24.73
Gbert | 421.80 | PH14 | 44.50 | 71.15 | 25.13
Dbmdz | 421.80 | PH14 | 73.72 | 88.13 | 26.55
| ASLG-PC12 Dataset Evaluation
Base-Tiny | 16.90 | ASLG | 76.77 | 96.35 | 82.44
Electra | 51.70 | ASLG | 76.77 | 96.35 | 82.60
Distilbert | 255.60 | ASLG | 76.77 | 96.35 | 83.06
Base-uncased | 420.10 | ASLG | 76.77 | 96.35 | 84.29
| | | | |
Table 4: Comparing different pre-trained models in terms of BLEU-4.
Effect of different pre-trained models. We here explored the translation
performance by using different pre-trained models; see Table 4. We analyzed
the model size and vocabulary coverage of the pre-trained model with gloss and
text of our dataset. We can see that introducing a pre-trained model with
larger vocabulary coverage of the target dataset will gain better performance,
since a pre-trained model with larger vocabulary coverage can inject more
knowledge learned from another unlabeled corpus into the translation task. For
ASLG, although the vocabulary coverage is the same, we can see that the bigger
model has better performance since it can learn contextual representation
better.
Analysis on hyper-parameters. To search the best settings of our hyper-
parameters, we employed Neural Network Intelligence (NNI) (Microsoft, 2018), a
lightweight but powerful toolkit. As shown in Figures 5(e)-5(h), we explored
how beam size, layer number, learning rate and dropout rate affect the model
performance on PH14 dataset. First, beam search enables to explore more
possible candidates, but large beam widths do not always result in better
performance as shown in Figure 5(e). We obtain optimal beam size as 10 on
PH14. Second, the layer number decides the model size and capacity, where the
larger model would overfit on a small dataset. In Figure 5(f), we find the
optimal layer number to be 3 on PH14. Lastly, as shown in Figures 5(g) & 5(h),
we adopt an early-stopping strategy to avoid overfitting and find the best
learning rate and dropout rate are 0.0003 and 0.45, respectively.
Case study. Table 5 presents some intuitive translation results on ASLG by
reporting the translated spoken sentences. Overall, the translation quality is
good, even the translated sentences with low BLEU-4 still convey the same
information. Also, we can observe that our translated sentences are basically
the same with ground truth, although using different expressions, i.e.,
“decision making” vs. “decision made”. The translation results on PH14 are
reported in Appendix.
Type | Content | BLEU-4
---|---|---
GT Gloss | X-IT BE DESC-UP TO X-YOU TO CONSIDER | 100.00
| AND CHOOSE OUTCOME X-YOU WANT TO SEE .
GT Text | it is up to you to consider and choose
| the outcome you want to see .
Pred Text | it is up to you to consider and choose
| the outcome you want to see .
GT Gloss | X-I WANT IRELAND TO REMAIN AT | 57.58
| HEART DECISION MAKE IN EUROPE .
GT Text | i want ireland to remain at the
| heart of decision making in europe .
Pred Text | i want ireland to remain at the
| heart of the decision made in europe .
GT Gloss | X-I WILL DESC-NEVER FORGET WHAT X-I | 13.44
| EXPERIENCE . SHOULD BE ABOUT .
GT Text | that is what this european day of memorial should be
| about . i will never forget what i experienced .
Pred Text | i will never forget what i experienced .
| |
Table 5: Qualitative evaluation of translation performance in different
BLEU-4 scores on ASLG dataset.
## 6 Conclusion
In this paper, we proposed a task-aware instruction network for sign language
translation. To address the problem of limited data for SLT, we introduced a
pre-trained model into Transformer and designed an instruction module to adapt
SLT task. Besides, due to the discrepancy between the representation space of
sign glosses and spoken sentences, we proposed a multi-level data augmentation
scheme. Extensive experiments validate our superior performance compared with
state-of-the-art approaches. While there is obvious improvement among most
evaluation metrics, the complexity of our models is also increased, causing a
longer training period. In the future, we would like to explore the
possibility of designing a lightweight model to achieve real-time efficiency.
## Acknowledgements
We thank anonymous reviewers for the valuable comments. This work is supported
by the China National Natural Science Foundation (No. 62176101 & No. 62106094)
and Zhejiang Lab’s International Talent Fund for Young Professionals.
## References
* Arvanitis et al. (2019) Nikolaos Arvanitis, Constantinos Constantinopoulos, and Dimitrios Kosmopoulos. 2019\. Translation of sign language glosses to text using sequence-to-sequence attention models. In _2019 15th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS)_, pages 296–302. IEEE.
* Bahdanau et al. (2014) Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2014. Neural machine translation by jointly learning to align and translate. _arXiv preprint arXiv:1409.0473_.
* Banerjee and Lavie (2005) Satanjeev Banerjee and Alon Lavie. 2005. Meteor: An automatic metric for mt evaluation with improved correlation with human judgments. In _Proceedings of the acl workshop on intrinsic and extrinsic evaluation measures for machine translation and/or summarization_ , pages 65–72.
* Bragg et al. (2019) Danielle Bragg, Oscar Koller, Mary Bellard, Larwan Berke, Patrick Boudreault, Annelies Braffort, Naomi Caselli, Matt Huenerfauth, Hernisa Kacorri, Tessa Verhoef, et al. 2019. Sign language recognition, generation, and translation: An interdisciplinary perspective. In _The 21st international ACM SIGACCESS conference on computers and accessibility_ , pages 16–31.
* Camgoz et al. (2018) Necati Cihan Camgoz, Simon Hadfield, Oscar Koller, Hermann Ney, and Richard Bowden. 2018. Neural sign language translation. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 7784–7793.
* Camgoz et al. (2020) Necati Cihan Camgoz, Oscar Koller, Simon Hadfield, and Richard Bowden. 2020. Sign language transformers: Joint end-to-end sign language recognition and translation. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 10023–10033.
* Cui et al. (2019) Runpeng Cui, Hu Liu, and Changshui Zhang. 2019. A deep neural framework for continuous sign language recognition by iterative training. _IEEE Transactions on Multimedia_ , 21(7):1880–1891.
* Gehring et al. (2017) Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. 2017\. Convolutional sequence to sequence learning. In _International Conference on Machine Learning_ , pages 1243–1252. PMLR.
* Huggingface-community (2018) Huggingface-community. 2018. https://huggingface.co/models. Accessed: 2018-11-17.
* Imamura and Sumita (2019) Kenji Imamura and Eiichiro Sumita. 2019. Recycling a pre-trained bert encoder for neural machine translation. In _Proceedings of the 3rd Workshop on Neural Generation and Translation_ , pages 23–31.
* Kingma and Ba (2014) Diederik P Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_.
* Ko et al. (2019) Sang-Ki Ko, Chang Jo Kim, Hyedong Jung, and Choongsang Cho. 2019. Neural sign language translation based on human keypoint estimation. _Applied Sciences_ , 9(13):2683.
* Koller et al. (2019) Oscar Koller, Necati Cihan Camgoz, Hermann Ney, and Richard Bowden. 2019. Weakly supervised learning with multi-stream cnn-lstm-hmms to discover sequential parallelism in sign language videos. _IEEE transactions on pattern analysis and machine intelligence_ , 42(9):2306–2320.
* Koller et al. (2017) Oscar Koller, Sepehr Zargaran, and Hermann Ney. 2017. Re-sign: Re-aligned end-to-end sequence modelling with deep recurrent cnn-hmms. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_.
* Kumar et al. (2020) E Kiran Kumar, PVV Kishore, M Teja Kiran Kumar, and D Anil Kumar. 2020. 3d sign language recognition with joint distance and angular coded color topographical descriptor on a 2–stream cnn. _Neurocomputing_ , 372:40–54.
* Li et al. (2020) Dongxu Li, Cristian Rodriguez, Xin Yu, and Hongdong Li. 2020. Word-level deep sign language recognition from video: A new large-scale dataset and methods comparison. In _Proceedings of the IEEE/CVF winter conference on applications of computer vision_ , pages 1459–1469.
* Lin (2004) Chin-Yew Lin. 2004. Rouge: A package for automatic evaluation of summaries. In _Text summarization branches out_ , pages 74–81.
* Loshchilov and Hutter (2016) Ilya Loshchilov and Frank Hutter. 2016. Sgdr: Stochastic gradient descent with warm restarts. _arXiv preprint arXiv:1608.03983_.
* Microsoft (2018) Microsoft. 2018. https://github.com/microsoft/nni. Accessed: 2018-09-20.
* Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. 2013. Distributed representations of words and phrases and their compositionality. In _Advances in neural information processing systems_ , pages 3111–3119.
* Moryossef et al. (2021) A. Moryossef, K. Yin, G. Neubig, and Y. Goldberg. 2021. Data augmentation for sign language gloss translation.
* Othman and Jemni (2012) Achraf Othman and Mohamed Jemni. 2012. English-asl gloss parallel corpus 2012: Aslg-pc12. In _5th Workshop on the Representation and Processing of Sign Languages: Interactions between Corpus and Lexicon LREC_.
* Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002. Bleu: a method for automatic evaluation of machine translation. In _Proceedings of the 40th annual meeting of the Association for Computational Linguistics_ , pages 311–318.
* Qiu et al. (2017) Zhaofan Qiu, Ting Yao, and Tao Mei. 2017. Learning spatio-temporal representation with pseudo-3d residual networks. In _proceedings of the IEEE International Conference on Computer Vision_ , pages 5533–5541.
* Sharma and Kumar (2021) Shikhar Sharma and Krishan Kumar. 2021. Asl-3dcnn: American sign language recognition technique using 3-d convolutional neural networks. _Multimedia Tools and Applications_ , pages 1–13.
* Shavarani and Sarkar (2021) Hassan S Shavarani and Anoop Sarkar. 2021. Better neural machine translation by extracting linguistic information from bert. _arXiv preprint arXiv:2104.02831_.
* Sincan and Keles (2020) Ozge Mercanoglu Sincan and Hacer Yalim Keles. 2020. Autsl: A large scale multi-modal turkish sign language dataset and baseline methods. _IEEE Access_ , 8:181340–181355.
* Stoll et al. (2018) Stephanie Stoll, Necati Cihan Camgöz, Simon Hadfield, and Richard Bowden. 2018\. Sign language production using neural machine translation and generative adversarial networks. In _Proceedings of the 29th British Machine Vision Conference (BMVC 2018)_. University of Surrey.
* Sutskever et al. (2014) Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014. Sequence to sequence learning with neural networks. In _Advances in neural information processing systems_ , pages 3104–3112.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In _Advances in neural information processing systems_ , pages 5998–6008.
* Wei et al. (2019) Chengcheng Wei, Wengang Zhou, Junfu Pu, and Houqiang Li. 2019. Deep grammatical multi-classifier for continuous sign language recognition. In _2019 IEEE Fifth International Conference on Multimedia Big Data (BigMM)_ , pages 435–442. IEEE.
* Yin et al. (2016) Fang Yin, Xiujuan Chai, and Xilin Chen. 2016. Iterative reference driven metric learning for signer independent isolated sign language recognition. In _European Conference on Computer Vision_ , pages 434–450. Springer.
* Yin et al. (2021) K. Yin, A. Moryossef, J. Hochgesang, Y. Goldberg, and M. Alikhani. 2021. Including signed languages in natural language processing. In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_.
* Yin and Read (2020) Kayo Yin and Jesse Read. 2020. Better sign language translation with stmc-transformer. _arXiv preprint arXiv:2004.00588_.
* Zhou et al. (2020) Hao Zhou, Wengang Zhou, Yun Zhou, and Houqiang Li. 2020. Spatial-temporal multi-cue network for continuous sign language recognition. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 13009–13016.
* Zhu et al. (2020) Jinhua Zhu, Yingce Xia, Lijun Wu, Di He, Tao Qin, Wengang Zhou, Houqiang Li, and Tie-Yan Liu. 2020. Incorporating bert into neural machine translation. _arXiv preprint arXiv:2002.06823_.
## Appendix A Appendix
### A.1 Dataset Description
In this section, we will introduce two public benchmark datasets used in sign
language translation tasks, namely PHOENIX-2014-T and ASLG-PC12. We conducted
statistical analysis on the datasets and the results are shown in Table 6. It
is obvious that PHOENIX-2014-T is a small-scale dataset, while ASLG-PC12 is a
large-scale dataset.
Dataset | Gloss | Translation
---|---|---
Train | Dev | Test | Train | Dev | Test
PH14 | Samples | 7096 | 519 | 642 | 7096 | 519 | 642
Vocabs | 1066 | 393 | 411 | 2887 | 951 | 1001
Words | 67781 | 3745 | 4257 | 99081 | 6820 | 7816
ASLG | Samples | 82709 | 4000 | 1000 | 82709 | 4000 | 1000
Vocabs | 15782 | 4323 | 2150 | 21600 | 5634 | 2609
Words | 862046 | 41030 | 10503 | 975942 | 46637 | 11953
| | | | | | |
Table 6: The descriptive statistics of PHOENIX-2014-T and ASLG-PC12 datasets.
Samples row means the sample size of the dataset, Vocabs row represents the
total vocabularies contained in the dataset, and Words row means the total
words of the dataset.
### A.2 PHOENIX-2014-T Qulitative Result
BE-SLT performance of G2T task on PHOENIX-2014-T is shown in Table 7, from
which we can observe that sign language translation results are of good
quality with different BLEU-4 scores and the predicted sentences can convey
effective information even for low BLEU-4 scores.
Type | Content | BLEU-4
---|---|---
Gloss | BERG ORKAN MOEGLICH | 100.00
GT Text | auf den bergen sind orkanartige
| böen möglich .
Pred Text | auf den bergen sind orkanartige
| böen möglich .
Gloss | HEUTE NACHT ZWISCHEN NEUNZEHN ZWISCHEN | 57.58
| FUENFZEHN SUEDOST MAXIMAL ZWOELF
GT Text | heute nacht werte zwischen neunzehn und fünfzehn
| grad im südosten bis zwölf grad .
Pred Text | heute nacht neunzehn bis fünfzehn grad im
| südosten bis zwölf grad .
Gloss | RUSSLAND IX TROCKEN HEISS SCHEINEN FUENF | 13.44
| DREISSIG BIS VIERZIG GRAD
GT Text | ganz anders die trockene hitze über russland
| mit fünfunddreißig bis vierzig grad .
Pred Text | aber bei uns wird es auch noch ein bisschen
| heißer da sind es fünf bis vierzig grad .
| |
Table 7: PHOENIX-2014-T: Qualitatively evaluation of translation performance
in different BLEU-4 scores.
### A.3 Experiment Parameter
In order to help reproduce BE-SLT and its translation performance, as shown in
Table 8 and 9, we list the hyper-parameters of the best results on two
benchmark datasets. For G2T task on PHOENIX-2014-T, we list the best hyper-
parameter settings for the experiments which apply data augmentation scheme,
or fuse BERT-attention module into encoder, decoder, and both respectively
(namely,w/DataAug, w/Encoder, w/Decoder, w/All). W/All obtains the highest
BLEU-4 using the initial learning rate of 0.00025, dropout rate of 0.45, beam
search with width 5, and the max epoch size of 120. For G2T task on ASLG-PC12,
we also list the hyper-parameter settings for the four experiments that
achieve significant results, listed in Table 9. For more experiment details,
please refer to our code which will be published upon the publication of this
work.
| PHOENIX-2014-T
---|---
Parameter | w/DataAug | w/Encoder | w/Decoder | w/All
Embedding size | 512 | 512 | 512 | 512
Hidden size | 2048 | 2048 | 2048 | 2048
Head number | 8 | 8 | 8 | 8
Encoder BERT gate | 1 | 1 | 0 | 1
Decoder BERT gate | 1 | 0 | 1 | 1
Optimizer | Adam | Adam | Adam | Adam
Learning rate | 0.00025 | 0.00025 | 0.00025 | 0.0003
LR schedule | inverse sqrt | inverse sqrt | inverse sqrt | inverse sqrt
Weight decay | $10^{-3}$ | $10^{-3}$ | $10^{-3}$ | $10^{-3}$
Drop out | 0.45 | 0.45 | 0.45 | 0.45
Label smoothing | 0.3 | 0.3 | 0.3 | 0.3
BERT ratio | - | 0.6 | 0.6 | 0.65
Max epoch | 120 | 120 | 120 | 120
BERT model | bert-base-german-dbmdz-uncased
| | | |
Table 8: The hyper-parameters of the best results on PHOENIX-2014-T for the
G2T task.
| ASLG-PC12
---|---
Parameter | w/DataAug | w/Encoder | w/Decoder | w/All
Embedding size | 512 | 512 | 512 | 512
Hidden size | 2048 | 2048 | 2048 | 2048
Head number | 8 | 8 | 8 | 8
Encoder BERT gate | 1 | 1 | 0 | 1
Decoder BERT gate | 1 | 0 | 1 | 1
Optimizer | Adam | Adam | Adam | Adam
Learning rate | 0.00025 | 0.00025 | 0.00025 | 0.00045
LR schedule | inverse sqrt | inverse sqrt | inverse sqrt | inverse sqrt
Weight decay | $10^{-3}$ | $10^{-3}$ | $10^{-3}$ | $10^{-3}$
Drop out | 0.45 | 0.45 | 0.45 | 0.4
Label smoothing | 0.3 | 0.3 | 0.3 | 0.1
BERT ratio | - | 0.6 | 0.6 | 0.6
Max epoch | 70 | 70 | 70 | 70
BERT model | bert-base-uncased
| | | |
Table 9: The hyper-parameters of the best results on ASLG-PC12 for the G2T
task.
### A.4 Alpha Strategy Settings
Here we introduce the $\alpha$ value setting details corresponding to cosine
strategy and constant strategy adopted in this work as shown in Formula 2 and
Formula 4. The cosine annealing and cosine decrement strategies are calculated
according to Formula 11. To simplify the calculation, the cosine increment
strategy is calculated according to Formula 12. In order to be more intuitive,
we plotted the curve of $\alpha$ value during the training process, as shown
in Figure 6.
$\alpha_{t+1}=1-\alpha_{min}-\frac{1}{2}(\alpha_{max}-\alpha_{min})(1-cos(\frac{T_{t}}{T_{c}}\pi))$
(12)
(a) Cosine annealing strategy
(b) Constant strategy
(c) Cosine increment strategy
(d) Cosine decrement strategy
Figure 6: The $\alpha$ value during the training process in four setting
strategies, namely cosine annealing, cosine increment, cosine decrement and
constant.
### A.5 Pre-trained Models Download
All BERT pre-trainied models adopted in Table 4 are published by (Huggingface-
community, 2018). In order to help reproduce our work and use our code easily,
we summarize the download links of the pre-trained models as follows.
PHOENIX-2014-T Dataset
* •
Multilingual: _bert-base-multilingual-uncased_
https://huggingface.co/bert-base-multilingual-uncased
* •
Distilbert: _distilbert-base-german-cased_
https://huggingface.co/distilbert-base-german-cased
* •
Gbert: _gbert-base_
https://huggingface.co/deepset/gbert-base
* •
Dbmdz: _bert-base-german-dbmdz-uncased_
https://huggingface.co/bert-base-german-dbmdz-uncased
ASLG-PC12 Dataset
* •
Base-Tiny: _bert-tiny_
https://huggingface.co/prajjwal1/bert-tiny
* •
Electra: _electra-small-discriminator_
https://huggingface.co/google/electra-small-discriminator
* •
Distilbert: _distilbert-base-uncased_
https://huggingface.co/distilbert-base-uncased
* •
Base-uncased: _bert-base-uncased_
https://huggingface.co/bert-base-uncased
|
ampmtime
# Detailed Gender Wage Gap Decompositions:
Controlling for Worker Unobserved Heterogeneity Using Network Theory111Fogel
Opportunity Insights<EMAIL_ADDRESS>Modenesi: University of
Michigan<EMAIL_ADDRESS>This material is based upon work supported by the
National Science Foundation Graduate Research Fellowship Program under Grant
No. 1256260. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation. This research is also
supported by the Alfred P. Sloan Foundation through the CenHRS project at the
University of Michigan. This work is done in partnership with the Brazilian
Institute of Applied Economic Research (IPEA). We thank John Bound, Abigail
Jacobs, Matthew Shapiro, Mel Stephens, and Sebastian Sotelo for advice and
guidance throughout this project. We also thank Charlie Brown, Zach Brown, Raj
Chetty, Ying Fan, John Friedman, Florian Gunsilius, Nathan Hendren, Dhiren
Patki, Rafael Pereira, Matthew Staiger, Dyanne Vaught, and Jean-Gabriel Young
for helpful comments and discussions. We also received helpful feedback from
seminar participants at the University of Michigan, Labo(u)r Day, the Urban
Economics Association, Networks 2021, Yale University, Duke University, the
Federal Reserve Bank of Boston, Opportunity Insights, and JAM.
Jamie Fogel and Bernardo Modenesi
###### Abstract
Recent advances in the literature of decomposition methods in economics have
allowed for the identification and estimation of detailed wage gap
decompositions. In this context, building reliable counterfactuals requires
using tighter controls to ensure that similar workers are correctly identified
by making sure that important unobserved variables such as skills are
controlled for, as well as comparing only workers with similar observable
characteristics. This paper contributes to the wage decomposition literature
in two main ways: (i) developing an economic principled network based approach
to control for unobserved worker skills heterogeneity in the presence of
potential discrimination; and (ii) extending existing generic decomposition
tools to accommodate for potential lack of overlapping supports in covariates
between groups being compared, which is likely to be the norm in more detailed
decompositions. We illustrate the methodology by decomposing the gender wage
gap in Brazil.
## 1 Introduction
Significant attention has been paid to the gap in wages between men and women.
Researchers are interested in understanding how much of the gap is due to men
and women performing different work using different skills, and how much is
due to men and women being paid differently for similar work. A number of
methods exist for trying to answer this question. These methods decompose
gender wage gaps into a portion explained by differences in characteristics
between men and women, and a portion explained by differences in the return to
characteristics, or “discrimination”. However, all of these methods rely on
three assumptions. First, they assume that unobserved determinants of earnings
are independent of gender. To the extent that there exist unobserved worker
characteristics that are important for determining wages and are correlated
with gender, then researchers will obtain biased estimates of the return to
observable characteristics. As a result, decompositions of gender wage gaps
into a component explained by covariates and a component explained by the
return to covariates will be incorrect. Second, they assume a functional form
in order to estimate the function that maps observable characteristics into
wages and thus serves as the foundation for counterfactuals that ask what men
would earn if they had the same characteristics except their gender were
switched to female, and vice versa. Third, they assume that the covariates for
male workers and female workers share a common support. While this is likely
to hold when the number of covariates is small, as more covariates are added
(possibly to satisfy the independence assumption) the common support
assumption becomes more likely to be violated.222As more covariates are added
it becomes harder to find another worker who shares the same values of all
covariates.
In this paper, we (i) propose a new method for identifying unobserved
determinants of workers’ earnings from the information revealed by detailed
data on worker–job matching patterns, (ii) non-parametrically estimate
counterfactual wage functions for male and female workers, (iii) allow for a
relaxation of the common support assumption, and (iv) apply our methods by
decomposing the gender wage gap in Brazil using improved counterfactuals based
on (i), (ii) and (iii). We find that the Brazilian gender wage gap is almost
entirely explained by male and female workers who possess similar skills and
perform similar tasks being paid different wages, not women possessing skills
or tasks that pay relatively lower wages.
To understand the problem created by unobserved determinants of productivity,
suppose that there are three types of worker characteristics that are relevant
for determining wages: gender, other characteristics observable to
researchers, and characteristics that are observable to labor market
participants, but not to researchers. A naive wage decomposition would simply
compare male wages to female wages and attribute all differences to the effect
of gender. A more common approach would condition on observable
characteristics like age, experience, occupation, education, and union
membership and would attribute all differences in wages, conditional on these
characteristics, solely to being a woman as opposed to being a man. However,
this would miss the fact that even workers with identical observable
covariates may perform distinct labor. As Goldin (2014) shows, male lawyers
significantly outearn female lawyers largely because males are more likely to
work long, inflexible hours, which leads to high wages. Therefore, if we
simply compared the wages of male lawyers to the wages of female lawyers, we
might mistakenly conclude that male and female lawyers receive differential
pay for the same work, when in fact male and female lawyers perform different
types of legal work. In other words, male and female lawyers differ in terms
of covariates that are observed by labor market participants but not by
researchers.
The key to our approach is identifying information about worker
characteristics observable to labor market participants, but not to
researchers, directly from the behavior of labor market participants. If we
can identify groups of workers and groups of jobs who are similar _from the
perspective of labor market participants_ , then we can be confident that any
gender wage differentials within these groups are due to differential returns
to labor market activities by gender, rather than differences in the work done
by male and female workers.
We employ a revealed preference approach that relies on workers’ and jobs’
choices, rather than observable variables or expert judgments, to classify
workers and jobs into groups. Our key insight is that linked employer-employee
data contain a previously underutilized source of information: millions of
worker–job matches, each of which reflects workers’ and jobs’ perceptions of
the workers’ skills and the jobs’ tasks. Intuitively, if two workers are
employed by the same job, they probably have similar skills, and if two jobs
employ the same worker those jobs probably require workers to perform similar
tasks. However, since discrimination may lead men and women with similar
skills to sort into different jobs, our method includes a correction for
gender-based sorting into jobs that normalizes workers’ job match
probabilities by the match probabilities for their gender.
We formalize this intuition and apply it to large-scale data using a Roy
(1951) model in which workers supply labor to jobs according to comparative
advantage. Workers belong to a discrete set of latent _worker types_ defined
by having the same “skills” and jobs belong to a discrete set of latent
_markets_ defined by requiring employees to perform the same
“tasks.”333“Skills” and “tasks” should be interpreted broadly as any worker
and job characteristics that determine which workers match with which jobs.
Workers match with jobs according to comparative advantage, which is
determined by complementarities between skills and tasks at the worker
type–market level. Workers who have similar vectors of match probabilities
over markets are therefore revealed to have similar skills and belong to the
same worker type, and jobs that have similar vectors of match probabilities
over worker types are revealed to have similar tasks and belong to the same
market. Our model extends the model in Fogel and Modenesi (2023) to allow
firms to have labor market power, thereby rationalizing pay heterogeneity
among workers with the same skills in jobs requiring the same tasks and
microfounding the correction for gender-based sorting.
Once we have clustered workers with similar skills into worker types and jobs
requiring similar tasks into markets, we turn to estimating counterfactual
wage functions. Traditional decomposition methods estimate counterfactual
female earnings by fitting wage regressions using observations for male
workers only, but generating predicted values by multiplying average female
covariate values by the male regression coefficients. This approach suffers
from three main issues: (i) it requires the researcher to impose a restrictive
regression functional form; (ii) it does not necessarily allow for
heterogeneous returns to covariates in predictions; and (iii) it does not have
embedded tools to handle when workers do not share similar covariate support.
Taken together, these issues can potentially bias the counterfactual
estimation exercise, which is the foundation of gender wage gap
decompositions. In order to circumvent these issues, we make use of a flexible
matching estimator for counterfactual earnings.
We implement a matching estimator in which we match male and female workers
who belong to the same worker type and are employed by jobs in the same
market. In doing so, we implicitly assume that worker types and markets fully
account for all factors, other than gender, that affect workers’ wages,
although we also estimate specifications in which we include other observable
characteristics in addition to worker types and markets. Within these matched
groups, we use the male workers’ mean wages as counterfactuals for what the
female workers would have earned if they were male, and vice versa. We compare
our matching estimator to a standard estimator and find similar results,
although in some specifications the matching estimator is clearly preferable.
However, there may be some worker type–market cells with no male workers or no
female workers so we introduce a correction to account for this lack of common
support.
We address the issue of a lack of common covariate support between male and
female workers by decomposing the gender wage gap into four components: (i)
differences due to different covariate distributions between groups, i.e. the
composition factor, for observations that share the same support; (ii)
differences related to differential returns to covariates between groups over
a common support of the covariates, i.e. the structural factor, often
associated with labor market discrimination; (iii) a part due to observations
from male workers being out of the female workers’ support of the covariates;
and (iv) the last portion related to observations of female workers being out
of the male workers’ support of the covariates. This decomposition allows us
to perform counterfactuals similar to existing methods for the part of the
distribution of the covariates for which male and female workers have common
support, yet it still allows us to quantify how much of the gender wage gap
occurs outside the region of common support and would therefore be ignored by
standard decomposition methods.
We estimate our model and conduct empirical analyses using Brazilian
administrative records from the Annual Social Information Survey (RAIS) that
is managed by the Brazilian labor ministry. The RAIS data contain detailed
information about every formal sector employment contract, including worker
demographic information, occupation, sector, and earnings. Critically, these
data represent a network of worker–job matches in which workers are connected
to every job they have ever held, allowing us to identify job histories of
workers, their coworkers, their coworkers’ coworkers, and so on. We restrict
our analysis to the Rio de Janeiro metropolitan area both for computational
reasons and because restricting to a single metropolitan area enables us to
focus on skills and tasks dimensions of worker and job heterogeneity rather
than geographic heterogeneity.
In our data, the average male worker earns a wage 16.7% higher than the
average female worker. Our primary result is that almost the entire gender
wage gap is attributable to male and female workers who possess similar skills
and perform similar tasks being paid differently, or what is often referred to
as “discrimination.” This is true at the aggregate level, and remains true
when we perform wage decompositions within each worker type–market cell,
indicating that this is a widespread phenomenon, not one driven by large wage
differentials in small subsets of the labor market. We find that wage
decompositions based on standard observable variables suffer from omitted
variable bias, emphasizing the need for detailed worker and job
characteristics in the form of worker types and markets. We find that wage
decompositions based on linear regressions yield similar findings to those
based on matching when a lack of common support is not an issue, however when
male and female workers’ characteristics do not share a common support the
matching estimator with corrections for a lack of common support outperforms
alternatives.
Literature: The literature of decomposition methods in economics can be
classified into two main branches. The first decomposes average differences in
a variable of interest $Y$ — often wages — between two groups of workers. The
most widespread method in this class was developed by Oaxaca (1973) and
Blinder (1973). The second branch decomposes functionals of the variable of
interest $Y$ – e.g. its distribution or quantile function. Given that
functionals of a variable often provide more information than its average, the
second group of decompositions is referred to as “detailed decompositions”
(Fortin et al. 2011). A seminal paper in this group is DiNardo et al.
(1996)444Barsky et al. (2002) develop a methodology similar to DiNardo et al.
(1996), focusing on issues that arise from lack of common covariate support
between the groups in the decomposition. Modenesi (2022) discusses their
approach in light of alternatives to handle the lack of common support. and
their methodology and inference was further generalized and improved later by
Chernozhukov et al. (2013)555Firpo et al. (2018) later in this literature uses
influence functions to propose a detailed decomposition that is invariant to
the order of the decomposition.. We follow the first branch of the literature
in focusing on average differences, largely because our rich set of controls
introduces a curse of dimensionality that renders detailed decompositions
infeasible.
Our method for handling a lack of common covariate support follows Ñopo (2008)
and Garcia et al. (2009)666Garcia et al. (2009) and Morello and Anjolim (2021)
both study the evolution of the Brazilian gender gap. Garcia et al. (2009)
uses the same approach we use to handle the problem of lack of overlapping
supports, and Morello and Anjolim (2021) have a similar matching methodology
to decompose the gender gap. In addition to using similar methods for the
decomposition, we add the skills and tasks controls derived from the labor
market network, and we derive a distribution of gender gaps for different
clusters of similar workers performing similar tasks.. In concurrent work we
extend Ñopo (2008) to generic “detailed decompositions” (Modenesi 2022).
Our model of labor market power builds on Card et al. (2015), Card et al.
(2018) and Gerard et al. (2018) but allows for significantly more granular
worker and job heterogeneity. The way we model multidimensional worker–job
heterogeneity relates to papers that use a skills-tasks framework in the
worker-job matching literature (Autor et al. 2003; Acemoglu and Autor 2011;
Autor 2013; Lindenlaub 2017; Tan 2018; Kantenga 2018). Our method for
clustering workers and jobs fits into the relatively recent literature in
labor economics that extracts latent information from the network structural
of the labor market (Sorkin 2018; Nimczik 2018; Jarosch et al. 2019) and
directly extends Fogel and Modenesi (2023) by allowing for labor market power.
Methodologically, we draw from the community detection branch of network
theory (Larremore et al. 2014; Peixoto 2018; 2019)777More precisely, we employ
a variant of the SBM which makes use of network edge weights (Peixoto 2018),
which are key for us to model the presence of potential discrimination in the
labor market.. Our paper connects to this literature by formalizing a
theoretical link between monopsonistic labor market models and the stochastic
block model, providing microfoundations and economic interpretability of
network theory unsupervised learning tools in order to solve economic
problems.
By controlling for skills and tasks, our papers share common ground with
Goldin (2014) and Hurst et al. (2021). Goldin (2014) indicates that the
potential residual discrimination in the gender wage gap is due to the nature
of the tasks in some occupations, by using a linear regression approach
dummies for occupation interacted with the gender dummy. We add to her
approach by proposing an economic model for discrimination, which provides us
with both worker and job heterogeneity controls, in addition to performing the
gender gap decomposition while taking into account potential violations of
conventional decomposition assumptions. Hurst et al. (2021) on the other hand
are assessing the black-white wage gap over time as function of changes in the
taste vs statistical discrimination factors, as well as the result of workers
sorting after these changes.
Roadmap: The paper proceeds as follows. Section 2 introduces a simple
framework for decomposition methods. Section 3 presents our model of
worker–job matching and derives from it our algorithm for clustering workers
into worker types and jobs into markets. Section 4 provides greater detail on
the wage gap decomposition methods we employ. Section 5 describes our data.
Section 6 presents results. Finally, Section 7 concludes.
## 2 A framework for decomposition methods
We introduce a simple framework for decomposition methods to guide the
analysis in this paper. Define the actual wage of worker $i$ employed by job
$j$ as $Y_{ij}$, and let $G_{i}$ be a dummy denoting whether worker $i$ is
male. The difference between the average wage for male workers and the average
wage for female workers, which we call the “overall wage gap,” can be
expressed as:
$\Delta:=E[Y_{ij}|G_{i}=1]-E[Y_{ij}|G_{i}=0]$ (1)
The overall wage gap above can be decomposed into two factors: differences in
productivity between male and female workers, usually referred to as the
composition factor; and differences in pay between equally productive male and
female workers, known as the structural factor. We use the potential outcomes
framework in order to formally decompose the overall wage gap into these two
factors. Denote by $Y_{0ij}$ the potential wage of worker $i$ employed by job
$j$ when the worker is female, and $Y_{1ij}$ the potential wage of worker $i$
employed by job $j$ when the worker is male. Let $x$ be the vector of all
variables that determine workers’ productivity. We assume that the worker’s
gender may affect their pay, but does not directly affect their productivity.
We represent the potential outcomes as functions of $x$ as follows:
$Y_{gij}:=Y_{g}(x_{ij}),g\in\\{0,1\\}$. Notice that $x$ has both $i$ and $j$
subscripts, as the marginal product of worker $i$ at their current job $j$
depends on both the worker’s skills and the job’s tasks. The fact that there
is a different earnings function for men and women reflects the possibility
that male and female workers with identical productivities may be paid
differently. Furthermore, it is possible to use the dummy for gender to
represent observed wages as a function of potential outcomes using a switching
regression model $Y_{ij}:=G_{i}Y_{g}(x_{ij})-(1-G_{i})Y_{g}(x_{ij})$.
At this point we are able to decompose the overall wage gap, $\Delta$, into
the composition and structural components mentioned above by adding and
subtracting the quantity888Analogously, the overall decomposition can be
performed by adding and subtracting the male counterfactual quantity
$E[Y_{0}(x_{ij})|G_{i}=1]$ to $\Delta$. The main results in this paper use the
female counterfactual approach.
$E[Y_{1}(x_{ij})|G_{i}=0]:=\int Y_{1}(x_{ij})dF_{G=0}(x)$
from the overall wage gap $\Delta$, where $F_{G=0}(x)$ is the productivity
distribution for female workers. Intuitively, $E[Y_{1}(x_{ij})|G_{i}=0]$ is
the mean earnings for a counterfactual set of workers possessing the female
productivity distribution, but who are paid like men999Alternatively, this
counterfactual term can be interpreted as the mean earnings of male workers
whose productivity distribution was adjusted to match the female productivity
distribution.
$\Delta:=\underset{\Delta_{X}:=\text{Composition}}{\underbrace{E[Y_{ij}|G_{i}=1]-E[Y_{1}(x_{ij})|G_{i}=0]}}+\underset{\Delta_{0}:=\text{Structural}}{\underbrace{E[Y_{1}(x_{ij})|G_{i}=0]-E[Y_{ij}|G_{i}=0]}}$
(2)
The composition portion can be rewritten as
$E[Y_{1}(x_{ij})|G_{i}=1]-E[Y_{1}(x_{ij})|G_{i}=0]$101010We use the
representation of the observed $Y$ in terms of potential outcomes to write
$E[Y_{ij}|G_{i}=1]=E[G_{i}Y_{g}(x_{ij})-(1-G_{i})Y_{g}(x_{ij})|G_{i}=1]=E[Y_{1}(x_{ij})|G_{i}=1]$
and substitute it in $\Delta_{X}$.. It represents the difference between what
male workers actually earn and what male workers would have earned in a
counterfactual scenario in which their productivity distribution was
equivalent to the female productivity distribution. This quantity captures the
portion of the overall wage gap attributable to differences in the
composition, or distribution of productivity, between male and female workers.
The structural portion is equivalent to
$E[Y_{1}(x_{ij})-Y_{0}(x_{ij})|G_{i}=0]$111111Analogously to the previous
term, using the map from the potential outcomes to the observed $Y$, we can
write
$E[Y_{ij}|G_{i}=0]=E[G_{i}Y_{g}(x_{ij})-(1-G_{i})Y_{g}(x_{ij})|G_{i}=1]=E[Y_{0}(x_{ij})|G_{i}=0]$
and substitute it in $\Delta_{0}$.. This is the difference between female
earnings in a counterfactual state in which females were paid equivalently to
what equally productive male workers are paid and actual average female
earnings. This portion of the overall wage gap is due to structural
differences in how the two genders are paid, holding productivity constant,
which is why this term is often associated with a form of discrimination.
What we define as the structural component might reasonably be thought as
discrimination, where labor market discrimination is defined as workers with
similar productivity, performing similar tasks, and being paid differently
based on observables that do not influence productivity. Other forms of
discrimination may exist — including mistreatment or harassment, differential
pre-job human capital accumulation opportunities, or discriminatory hiring
practices — but we do not consider those in this paper. In our set up,
individual discrimination occurs when the wage for worker $i$ at job $j$ is
different if the individual’s gender changes, ceteris paribus, i.e.
$Y_{1}(x_{ij})-Y_{0}(x_{ij})\neq 0$. The problem is that, in order to measure
this quantity, we run into the fundamental problem of causal inference: it is
impossible to observe the potential wages in both states for the same
individual. Therefore we must make assumptions in order to construct
counterfactual values, i.e. the value of $Y_{1}$ for a female worker, or the
value of $Y_{0}$ for a male worker. In this paper, we break the assumptions
needed for the counterfactual estimation into two parts and we show how our
approach contributes to deal with limitations in each of them.
The first assumption is that workers with the same values of $x$ are equally
productive and would be paid equal wages if gender played no role in wage
determination, conditional on productivity. This is equivalent to assuming
that $x$ contains all factors that affect productivity and are correlated with
gender. This “conditional independence/ignorability” assumption, is the basis
of all decomposition methods in economics (Fortin et al. 2011), as it is a
requirement for consistency of its estimates for the gap decomposition
portions. However, not all factors that theoretically should be included in
$x$ are observable.
A problem would arise if certain factors that contribute to worker $i$’s
productivity in job $j$ are both unobserved by the econometrician and
correlated with gender. If such factors exist, our counterfactuals would be
invalid. Specifically, wage differentials due to unobserved differences in
skills and tasks between male and female workers would be attributed to the
effect of gender itself. For example, if women tend to have better social
skills but we do not observe social skills, then we would interpret women
outearning men in social skill-intensive jobs as discrimination against men,
when in fact it is simply the result of differences in unobserved skills.
Therefore, it is critical to come as close as possible to identifying groups
of male and female workers who have exactly the same skills and perform
exactly the same tasks. If we do so, then any gender wage differentials within
this group are attributable to the effect of gender per se. In Section 3 we
address this issue by identifying latent worker and job characteristics
relevant to productivity and wage determination using the network of
worker–job matches.
The second set of assumptions required to build the counterfactual $Y_{1}(x)$
for females in $\Delta$ are related to the choice of an estimation strategy
for the function $Y_{1}(\cdot)$121212Another approach decomposes the wage
distributions, as opposed to actual wages, which would be equivalent to
switching $Y$ for its distribution $F_{Y}$, but still needing the estimation
of the counterfactual $F_{Y_{1}}(y|x)$ for females (e.g. DiNardo et al. 1996
and Chernozhukov et al. 2013). We choose not to employ these decompositions in
this paper as our setup does not satisfy basic conditions for decomposing
distributions, such as having a low-dimensional vector of observable
characteristics $x$ – given curse of dimensionality – and having the
overlapping supports assumptions satisfied.. A common estimation strategy
requires fitting a linear wage regression for males and using its estimated
coefficients to predict wages, but inputting female workers’ covariates
(Oaxaca 1973 and Blinder 1973). This approach is highly tractable, however the
assumption of a linear functional form is to some extent arbitrary, and using
the same regression coefficients to predict counterfactual earnings for
distinct female workers (i.e. allowing no heterogeneous returns to observable
characteristics) could lead to biased estimates of counterfactual earnings. An
alternative approach relies on matching males to each female worker based on
similar observable characteristics, and uses the wages of matched male workers
in order to inform each female’s counterfactual wage. This less-parametric
approach has the advantage of not imposing any functional form assumption for
$Y_{1}(\cdot)$, however it requires us to observe a sufficiently rich set of
observable variables that male and female workers with the same observables
may be assumed to have similar productivity. Moreover, matching methods are
unreliable when we are unable to find a female worker with the same
observables as a male worker, or vice versa. In Section 3 we describe a new
method to enhance the set of observable characteristics available to the
researcher, reducing the scope for unobserved determinants of productivity to
cause biased estimates. In Section 4 we compare and contrast different methods
to decompose the gender wage gap given a set of observable characteristics,
circumventing issues present in counterfactual earnings estimation.
## 3 Revealing latent worker and job heterogeneity using network theory
In this section we present an economic model of monopsonistic wage setting,
which rationalizes a wage gap between two groups of workers who have different
demographic characteristics, but have the same skills and perform the same
tasks. Intuitively, otherwise identical male and female workers may supply
labor to individual jobs with different elasticities, and jobs respond by
offering them wages with different markdowns. If one group of workers supplies
labor to jobs more inelastically, then they will be paid less, holding
productivity constant. Moreover, the model microfounds our network-based
clustering algorithm, which identifies groups of male and female workers with
similar skills who perform similar tasks, and therefore can serve as good
counterfactuals for each other. The model builds on the model of the labor
market developed in Fogel and Modenesi (2023), with two important differences:
(i) in this paper workers have idiosyncratic preferences over individual jobs,
not just markets, causing jobs to face upward-sloping labor supply curves, and
(ii) firms may offer different wages to men and women, even if they have
identical skills and perform identical tasks. The model defines a probability
distribution that governs how workers match with jobs, forming the network of
worker-job matches observed in linked employer-employee data. We use this
probability distribution to assign similar workers to worker types and similar
jobs to markets, using a Bayesian method based on generative network theory
models, which we present after the economic model.
### 3.1 Economic model
We propose a model with two primary components: heterogeneous workers who
supply labor and firms that produce goods by employing labor to perform tasks.
Workers supply their skills to jobs, which are bundles of tasks embedded
within firms. Jobs’ tasks are combined by the firms’ production functions to
produce output. We assume that firms face an exogenously-determined demand for
their goods131313For an alternative version of the model with endogenous
product demand, see Fogel and Modenesi (2023).. Our model of the labor market
has the following components:
* •
Each worker is endowed with a “worker type,” and all workers of the same type
have the same skills.
* •
A job is a bundle of tasks within a firm. As we discuss in Section 5, we
define a job in our data as an occupation–establishment pair.
* •
Each job belongs to a “market,” and all jobs in the same market are composed
of the same bundle of tasks.
* •
There are $I$ worker types, indexed by $\iota$, and $\Gamma$ markets, indexed
by $\gamma$.
* •
The key parameter governing worker-job match propensity is an $I\times\Gamma$
productivity matrix, $\Psi$, where the ($\iota,\gamma$) cell,
$\psi_{\iota\gamma}$ denotes the number of efficiency units of labor a type
$\iota$ worker can supply to a job in market $\gamma$.141414We can think of
$\psi_{\iota\gamma}$ as $\psi_{\iota\gamma}=f(X_{\iota},Y_{\gamma})$, where
$X_{\iota}$ is an arbitrarily high dimensional vector of skills for type
$\iota$ workers, $Y_{\gamma}$ is an arbitrarily high dimensional vector of
tasks for jobs in market $\gamma$, and $f()$ is a function mapping skills and
tasks into productivity. This framework is consistent with Acemoglu and Autor
(2011)’s skill and task-based model, and is equivalent to Lindenlaub (2017)
and Tan (2018). A key difference is that Lindenlaub and Tan observe $X$ and
$Y$ directly and assume a functional form for $f()$, whereas we assume that
$X$, $Y$, and $f()$ exist but are latent. We do not identify $X$, $Y$, and
$f()$ directly because in our framework $\psi_{\iota\gamma}$ is a sufficient
statistic for all of them.
Time is discrete, with time periods indexed by $t\in\\{1,\dots,T\\}$ and
workers make idiosyncratic moves between jobs over time. Neither workers,
households, nor firms make dynamic decisions, meaning that the model may be
considered one period at a time. We do not consider capital as an input to
production.
#### 3.1.1 Firm’s problem
Each firm, indexed by $f$, has a production function $Y_{f}(\cdot)$ which
aggregates tasks from different labor markets, indexed by $\gamma$. Firm $f$
faces exogenously-determined demand for its output, $\bar{Y}_{f}$. The firm’s
only cost is labor. As we discuss in the next subsection, firms face upward-
sloping labor supply curves and therefore have wage-setting power. Firms
demand labor in each market, $\gamma\in\\{1,\dots,\Gamma\\}$ and offer a
different wage per efficiency unit of labor for each market. Firms also may
offer different wages to workers in different demographic groups
$g\in\\{A,B\\}$ (e.g. male and female workers), although type $A$ and type $B$
workers belonging to the same worker type $\iota$ are equally productive in
all jobs. We define a job $j$ as a firm $f$ – market $\gamma$ pair. We define
the wage per efficiency unit of labor for demographic group $g$ workers
employed in job $j$ $w_{j}^{g}$. Define $L_{j}^{g}$ as the quantity of
efficiency units of labor supplied by demographic group $g$ workers to job
$j$.
The firm’s problem is to choose the quantity of labor inputs in each job for
each demographic group in order to minimize costs subject to the constraint
that production is greater than or equal to the firm’s exogenous product
demand, $\bar{Y}_{f}$:
$\displaystyle\min_{\\{{w}_{j}^{A},{w}_{j}^{B}\\}_{j=1}^{\Gamma}}\sum_{j=1}^{\Gamma}w^{A}_{j}L^{A}_{j}+w^{B}_{j}L^{B}_{j}\quad\text{s.t.}\quad
Y_{f}\left(L_{1},\ldots,L_{\Gamma}\right)\geq\bar{Y}_{f}$
where $L_{j}=L^{A}_{j}+L^{B}_{j}$ is the total amount of efficiency units of
labor employed by job $j$ and $Y_{f}$ is a concave and differentiable
production function.
Taking the first order condition with respect to $w_{j}^{g}$ allows us to
solve for the wage paid by job $j$ to workers in demographic group $g$ as a
markdown relative to the marginal revenue product of labor:
$\displaystyle w_{j}^{g}=$ $\displaystyle\underset{\text{ Markdown
}}{\underbrace{\frac{e_{j}^{g}}{1+e_{j}^{g}}}}\qquad\times\underset{\text{Marg.
revenue product of labor}}{\underbrace{\mu_{f}\frac{\partial Y_{f}}{\partial
L_{j}}}}$ (3)
where $\mu_{f}$ is the shadow revenue associated with one more unit of output
and $e_{j}^{g}:=\frac{\partial L_{j}^{g}}{\partial
w_{j}^{g}}\frac{w_{j}^{g}}{L_{j}^{g}}$ is the labor supply elasticity of
workers from group $g$ to job $j$.
Equation (3) shows that the wage paid to demographic group $g$ workers
employed in job $j$ (equivalently, employed in market $\gamma$ by firm $f$) is
the product of a markdown and the marginal revenue product of labor in job
$j$. The markdown depends on the demographic group $g$’s elasticity of labor
supply to job $j$. As labor supply becomes more elastic, the markdown
converges to 1 and the wage converges to the marginal product of labor.
Conversely, as labor supply becomes less elastic, the wage declines further
below the marginal product of labor. This equation rationalizes different
demographic groups being paid different wages for the same labor: if one
demographic group supplies labor more inelastically, they will be paid
less.151515We are referring to the elasticity of labor supply _to a specific
job $j$_, which may differ from a group’s labor supply elasticity to the
overall labor market. For example, it could be the case that men supply labor
more inelastically at the extensive margin, but women have stronger
idiosyncratic preferences for specific jobs, making them less likely to change
jobs in response to a wage differential. In this case, women would supply
labor less elastically to a specific job $j$ and thus receive lower wages. The
firm employs workers in both demographic groups despite paying them different
wages because in order to attract the marginal worker from the lower-paid
demographic group, it must raise wages for all inframarginal workers in that
group. At some point the marginal cost (inclusive of the required raises for
inframarginal workers) of hiring workers from the lower-paid demographic group
exceeds the marginal cost of hiring workers from the higher-paid demographic
group, and the firm will switch to hiring the higher-paid workers.
#### 3.1.2 Worker’s problem
A worker belonging to worker type $\iota$ and demographic group
$g\in\\{A,B\\}$, has a two step decision. First, she chooses a market $\gamma$
in which to look for a job, and second she chooses a firm $f$ (and by
extension a job $j$). The worker’s type defines their skills. Type $\iota$
workers can supply $\psi_{\iota\gamma}$ efficiency units of labor to any job
in market $\gamma$. $\psi_{\iota\gamma}$ is a reduced form representation of
the skill level of a type $\iota$ worker in the various tasks required by a
job in market $\gamma$. Units of human capital are perfectly substitutable,
meaning that if type 1 workers are twice as productive as type 2 workers in a
particular market $\gamma$ (i.e. $\psi_{1\gamma}=2\psi_{2\gamma}$), firms
would be indifferent between hiring one type 1 worker and two type 2 workers
at a given wage per efficiency unit of labor, $w_{j}$. Therefore, the law of
one price holds within each demographic group for each job, and a type $\iota$
worker belonging to demographic group $g$ employed in a job in market $\gamma$
is paid $\psi_{\iota\gamma}w_{j}^{g}$. Because workers’ time is indivisible,
each worker may supply labor to only one job in each period and we do not
consider the hours margin.
Workers choose job $j$, equivalent to $\gamma f$, in order to maximize
utility, which is the sum of log earnings $\log(\psi_{\iota\gamma}w_{j}^{g})$
and an idiosyncratic preference for job $j$, $\varepsilon_{ij}^{g}$:
$\displaystyle j^{*}=$
$\displaystyle\arg\max_{j}\quad\log(\psi_{\iota\gamma}w_{j}^{g})+\varepsilon_{ij}^{g}.$
We assume that $\varepsilon_{ij}^{g}$ follows a nested logit distribution with
parameter $\nu_{\gamma}^{g}$, with the $\gamma$ subscript indicating that
nests are defined by $\gamma$:
$\displaystyle\varepsilon_{ij}^{g}\sim NestedLogit(\nu_{\gamma}^{g})$
It follows from this assumption about the distribution of
$\varepsilon_{ij}^{g}$ that the probability that worker $i$ belonging to
worker type $\iota$ and demographic group $g$ matches with job $j$ in market
$\gamma$ is161616Details for the derivation of the choice probability in the
Appendix A.:
$\displaystyle P(j=j^{*}|j\in\gamma,i\in\iota,g)$
$\displaystyle=\underset{\underset{\text{{1st step}: market
choice}}{\underbrace{\scriptstyle
P(\gamma=\gamma^{*}|i\in\iota,j\in\gamma,g)}}}{\underbrace{\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}}}\underset{\underset{\text{{2nd
step}: job choice}}{\underbrace{\scriptstyle
P(j=j^{*}|i\in\iota,j\in\gamma,\gamma=\gamma^{*},g)}}}{\underbrace{\frac{(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}{\sum_{j\in\gamma}(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}}}$
(4)
where
$I^{g}_{\iota\gamma}:=\sum_{j\in\gamma}(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}$,
also referred to as the inclusive value, is the expected utility a type
$\iota$ worker faces when choosing market $\gamma$. Intuitively, the nested
logit assumption decomposes the job choice probability into a first stage in
which the worker chooses a market and then a second stage in which the worker
chooses a job conditional on their choice of a market.
### 3.2 Identifying worker types and markets
#### 3.2.1 Deriving the likelihood
Now that we have derived the probability of worker $i$ matching with job $j$
from the primitives of our model, the next step is using this probability as
the basis for a maximum likelihood procedure that assigns workers to worker
types and jobs to markets based on the observed set of worker–job matches.
This procedure builds on Fogel and Modenesi (2023), by allowing workers in the
same worker type but different demographic groups to have different vectors of
match probabilities over jobs.
We decompose the choice probability in equation (4) into a component that
depends only on variation at the $\iota,\gamma,g$ level and a component that
depends on wages at individual jobs:
$\displaystyle P(j=j^{*}|j\in\gamma,i\in\iota,g)$
$\displaystyle=\underset{\underset{\iota-\gamma-g\text{
component}\quad}{\underbrace{=:\Omega_{\iota\gamma}^{g}}}}{\underbrace{\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}-1}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\psi_{\iota\gamma}^{\frac{1}{\nu_{\gamma}^{g}}}}}\underset{\underset{\quad
j-g\text{
component}}{\underbrace{=:d_{j}^{g}}}}{\underbrace{\vphantom{\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}-1}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\psi_{\iota\gamma}^{\frac{1}{\nu_{\gamma}^{g}}}}(w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}}.$
(5)
The first term reflects workers choosing markets according to comparative
advantage, while the second captures the fact that some jobs in market
$\gamma$ require more workers than others (due to exogenous product demand
differences), and since jobs face upward-sloping labor supply curves, they
must pay higher wages to attract greater numbers of workers. Isolating the
group-level ($\iota,\gamma,g$) variation from the idiosyncratic job-level
variation allows us to cluster workers into worker types and jobs into markets
on the basis of having the same group-level match probabilities, as we discuss
below.
The choice probabilities we have discussed thus far refer to a single job
search for worker $i$. In reality, we may observe workers searching for jobs
multiple times, and each of these searches is informative about the latent
worker skills and job tasks that define worker types $\iota$ and markets
$\gamma$. We incorporate repeated searches by assuming that workers
periodically receive exogenous separation shocks which arrive following a
Poisson process. Upon receiving a separation shock, the worker draws a new
$\varepsilon_{ij}^{g}$ shock and repeats the job choice process described
above. Assuming that $Poisson$-distributed exogenous separations happen at a
rate $d_{i}^{g}$ for the individual worker $i$, then the expected number of
times she will match with job $j$ throughout our sample period is given by
$\displaystyle d_{i}^{g}\cdot
P(j=j^{*}|j\in\gamma,i\in\iota,g)=\Omega_{\iota\gamma}^{g}d_{i}^{g}d_{j}^{g}.$
(6)
Equation 6 forms the basis of our algorithm for clustering workers into worker
types and jobs into markets, but before proceeding we must define some
notation. Let $N_{W}$ and $N_{J}$ denote the number of workers and jobs,
respectively, in our data. Define $A_{ij}$ as the number of times that worker
$i$ is observed to match with job $j$. Further, define $\bm{A}$ as the matrix
with typical element $A_{ij}$. $\bm{A}$ is a $N_{W}\times N_{J}$ matrix and
represents the full set of worker–job matches observed in our data. As
discussed previously, each individual worker belongs to a latent worker type
denoted by $\iota$ and each job belongs to a latent market denoted by
$\gamma$. The list of all latent worker type and market assignments is stored
in the $(N_{W}+N_{J})\times 1$ vector denoted by $\bm{b}$, known as the node
membership vector. We define $\bm{g}$ as the $N_{W}\times 1$ vector containing
each worker’s demographic group affiliation. The matrix of worker–job matches
$\bm{A}$ and workers’ demographic groups $\bm{g}$ are the data we use to
cluster workers and jobs, while the node membership vector $\bm{b}$ is the
latent object identified by the maximum likelihood procedure we discuss below.
Following equation (6), the expected number of matches between a worker–job
pair, $A_{ij}$, can be written as171717It is worth mentioning that: (i) the
information $i\in\iota,j\in\gamma$ is contained in $\bm{b}$; and (ii) $A_{ij}$
is the number of matches between worker $i$ and job $j$, which makes the event
that $j=j^{*}|i$ equivalent to the event that $A_{ij}=1$. These two facts
allow us to use more succinct notation that directly links theoretical objects
in our model to data:
$P(j=j^{*}|j\in\gamma,i\in\iota,g)=P(A_{ij}=1|\bm{b},g)$, which we know the
distributional form for. This connects notations from the economic model to
the network model, but it still lacks the precise definition of the likelihood
of interest, $P(\bm{A},\bm{g}|\bm{b})$, where $A_{ij}$ can assume values other
than just $1$.
$\displaystyle E[A_{ij}|\bm{b},g]=\Omega_{\iota\gamma}^{g}d_{i}^{g}d_{j}^{g}.$
(7)
We prove in Appendix C that our assumption of Poisson-distributed exogenous
separation shocks implies that $A_{ij}$ follows a Poisson distribution:
$\displaystyle A_{ij}|\bm{b},g\sim
Poisson(\Omega_{\iota\gamma}^{g}d_{i}^{g}d_{j}^{g})$ (8)
Finally, we incorporate equation (8) above to fully characterize the
likelihood of our data as a function of the unknown parameters, by applying
Bayes rule:
$\displaystyle
P(A_{ij},g|\bm{b})=\underset{Poisson(\Omega_{\iota\gamma}^{g}d_{i}^{g}d_{j}^{g})}{\underbrace{P(A_{ij}|\bm{b},g)}}\underset{\alpha_{\iota\gamma}^{g}}{\underbrace{P(g|\bm{b})}},$
(9)
where $\alpha_{\iota\gamma}^{g}\equiv P(g|\bm{b})$ is the fraction of type
$\iota$ workers employed in market $\gamma$ jobs who belong to the demographic
group $g$. Equation 9 corresponds to a commonly-used method from network
theory known as the bipartite degree-corrected stochastic block model with
edge weights (SBM). The SBM clusters nodes in a network (workers and jobs)
into groups (worker types and markets) based on patterns of connections
between nodes.181818Larremore et al. (2014) lays out the advantages of using
bipartite models over using one-sided network projections to fit SBMs; Karrer
and Newman (2011) presents the methodology for degree-correction as it
enhances significantly the ability of the SBM to fit large scale real world
networks; and Peixoto (2018) deal with weighted SBM inference, which is how we
accommodate discrimination influencing matches within the SBM.. The main
parameter of interest is the set of assignments of workers to worker types and
jobs to markets contained in $\bm{b}$, while all of the other parameters are
nuisance parameters that can be straightforwardly determined after $\bm{b}$ is
defined (Karrer and Newman 2011). The next step is to maximize the likelihood
defined in equation 9, which we address in the next subsection.
#### 3.2.2 A Bayesian approach to recovering worker types and markets
In order to make the estimation of worker types and markets feasible, together
with using a principled method for choosing the number of clusters, we employ
Bayesian methods from the network literature (Peixoto 2017). We can rewrite
equation (9) as
$\displaystyle P(\bm{b}|A_{ij},g)\quad\propto$ $\displaystyle\qquad
P(A_{ij},g|\bm{b})P(\bm{b})$ $\displaystyle=$
$\displaystyle\quad\underset{Poisson(\Omega_{\iota\gamma}^{g}d_{i}^{g}d_{j}^{g})}{\underbrace{P(A_{ij}|\bm{b},g)}}\underset{\alpha_{\iota\gamma}^{g}}{\underbrace{P(g|\bm{b})}}\underset{\text{Prior}}{\underbrace{P(\bm{b})}}$
(10)
Maximizing the posterior distribution means assigning individual workers to
worker types $\iota$ and jobs to markets $\gamma$. The basic intuition follows
from and is described in greater detail in Fogel and Modenesi (2023): workers
belong to the same worker type if they have approximately the same vector of
match probabilities over jobs, while jobs belong to the same market if they
have approximately the same vector of match probabilities over workers. The
key difference in this paper is that workers in the same worker type $\iota$
may belong to different demographic groups $g$ and each worker
type–demographic group pair may face its own wage and therefore have its own
match probability. Equation (3.2.2) allows for this by allowing the match
probabilities $P(A_{ij},g|\bm{b})$ to depend on the workers’ demographic group
$g$ in addition to the worker types and markets stored in $\bm{b}$.
If worker types are defined by having common vectors of match probabilities
over jobs, but match probabilities are allowed to vary by demographic group
within a worker type, how do we know that type $\iota$ workers in group $A$
belong to the same worker type as type $\iota$ workers in group $B$? The
answer is embedded in equation (3.2.2). The $\alpha_{\iota\gamma}^{g}$ term in
equation (3.2.2) adjusts workers’ match probabilities so that they are
relative to their own gender. Suppose women are significantly underrepresented
in construction jobs and overrepresented in nursing jobs, and vice versa for
men. Once we incorporate this adjustment, we would assign workers to a
construction-intensive worker type if they are disproportionately likely to
match with construction jobs, relative to other workers of their gender. Once
we adjust the raw match probabilities to account for this selection, we obtain
identical adjusted match probability vectors for this group of men and this
group of women, causing us to assign them to the same worker type, $\iota$.
Equation (3.2.2) assumes that we know the number of worker types and markets
_a priori_ , however this is rarely the case in real world applications.
Therefore we must choose the number of worker types and markets, $I$ and
$\Gamma$ respectively. We do so using the principle of minimum description
length (MDL), an information theoretic approach that is commonly used in the
network theory literature. MDL chooses the number of worker types and markets
to minimize the total amount of information necessary to describe the data,
where the total includes both the complexity of the model conditional on the
parameters _and_ the complexity of the parameter space itself. MDL will
penalize a model that fits the data very well but overfits by using a large
number of parameters (corresponding to a large number of worker types and
markets), and therefore requires a large amount of information to encode it.
MDL effectively adds a penalty term in our objective function, such that our
algorithm finds a parsimonious model. See Fogel and Modenesi (2023) for
greater detail.
Equation (3.2.2) defines a combinatorial optimization problem. If we had
infinite computing resources, we would test all possible assignments of
workers to worker types and jobs to markets and choose the one that maximizes
the likelihood in equation (3.2.2), however this is not computationally
feasible for large networks like ours. Therefore, we use a Markov chain Monte
Carlo (MCMC) approach in which we modify the assignment of each worker to a
worker type and each job to a market in a random fashion and accept or reject
each modification with a probability given as a function of the change in the
likelihood. We repeat the procedure for multiple different starting values to
reduce the chances of finding local maxima. We implement the procedure using a
Python package called graph-tool. (https://graph-tool.skewed.de/. See Peixoto
(2014) for details.) Now that we have dealt with the issue of important worker
and job characteristics being unobserved, we turn our attention to estimating
counterfactuals for wage gap decompositions.
## 4 Wage gap decomposition
This section lays out the estimation strategies we use to decompose the
Brazilian gender wage gap, while circumventing some of the issues associated
with conventional decomposition methods. We decompose the gender wage gap into
the quantities listed in equation (2): the composition component
$E[Y_{1}(x_{ij})|G_{i}=1]-E[Y_{1}(x_{ij})|G_{i}=0]$ and the structural
component $E[Y_{1}(x_{ij})-Y_{0}(x_{ij})|G_{i}=0]$. The quantity
$E[Y_{g}(x_{ij})|G_{i}=g]=E[Y_{ij}|G_{i}=g]$, $g\in\\{0,1\\}$ can be
consistently and straightforwardly estimated since it is directly observable.
The challenge is estimating the counterfactual wage function
$E[Y_{1}(x_{ij})|G_{i}=0]$, given that the potential outcome $Y_{1}(x_{ij})$
is not observed for female workers. Estimating $E[Y_{1}(x_{ij})|G_{i}=0]$
requires us to use data on male workers to estimate a relationship between
observable characteristics $x_{ij}$ and male earnings $Y_{1}$ and then
extrapolate this relationship to female workers.
In this paper, we consider two approaches to estimating counterfactual wage
functions. The first is the commonly-used Oaxaca-Blinder decomposition, which
we henceforth refer to as OB (Oaxaca 1973; Blinder 1973). For the OB
decomposition, we estimate two linear regressions — one for the set of male
workers and another for the set of female workers — to estimate the
functionals $Y_{1}(\cdot)$ and $Y_{0}(\cdot)$, respectively, as denoted in
equation (11). Values for $E[Y_{g}(x_{ij})|G_{i}=g]$ are obtained by averaging
out the fitted values of the respective linear regressions. Estimates for the
counterfactual $E[Y_{1}(x_{ij})|G_{i}=0]$ are obtained by using the
coefficients from the linear regression fitted for males, $\hat{\beta}_{G=1}$,
and multiplying them by the average female covariates, $\bar{x}_{G=0}$, as
defined in equation (11). This is equivalent to producing fitted values for
the males’ regression, while inputting females’ covariates.
OB regressions: $\displaystyle
Y_{g}(x_{ij})=x_{ij}^{T}\beta_{G=g}+\epsilon_{gij},\qquad g\in\\{0,1\\}$ (11)
OB counterfactual estimate:
$\displaystyle\widehat{E[Y_{1}(x_{ij})|G_{i}=0]}:=\bar{x}_{G=0}^{T}\hat{\beta}_{G=1},\quad\bar{x}_{G=0}:=\sum_{i|G_{i}=0}\frac{x_{ij}}{n}$
As discussed in section 2, the OB decomposition has several important
limitations. Although highly tractable, OB imposes potentially restrictive
assumptions on $Y_{1}(x_{ij})$. First, it assumes that its expectation is
linear in $x_{ij}$. Although linear regressions allow for flexible
transformations of its covariates, the functional form is still a somewhat
arbitrary researcher choice. Second, by using a linear regression to estimate
the potential outcome function, $Y_{1}(\cdot)$, as in equation (11), it uses
the same functional form to compute counterfactuals for all male workers. In
other words, it imposes the same average returns to covariates for all
workers, which would create biases in the counterfactual estimation if returns
to worker characteristics are heterogeneous. The third limitation of the OB is
related to the overlapping supports assumption, also referred to as the common
supports assumption. This assumption imposes that the support of $x$ for one
of the genders has to fully overlap with the support of $x$ for the other
gender, and is imposed by almost all decomposition methods in economics
(Fortin et al. 2011). The overlapping supports assumption is imposed to ensure
that the counterfactual function $Y_{1}(x)$ estimated using male data,
$x_{G_{i}=1}$, is only used to predict counterfactual earnings for females
whose values of $x$ lie within the male support of $x$. When this condition is
not satisfied in the data, observations that are outside of the common support
are typically trimmed or given virtually zero weight in the estimation
process, potentially eliminating significant numbers of workers from the
analysis and making the analysis representative of only a subset of the
population (Modenesi 2022). This is particularly salient when $x$ lies in a
high-dimensional space, as is the case in our application with high-
dimensional worker types and markets.
Our preferred decomposition strategy relies on matching male and female
workers with similar observable characteristics and using matched workers of
different genders as counterfactuals for each other. This approach was
initially proposed by Ñopo (2008) and was further extended by Modenesi (2022).
Not only does this approach avoid the strong functional form assumptions made
by OB, it includes a framework for handling a lack of common support. In this
paper, we choose to use the original estimation strategy laid out by Ñopo
(2008), given its tractability especially for a high-dimensional set of
covariates like ours, and we refer to it as the matching decomposition
henceforth.
The matching decomposition has two main components: (i) matching observations
and (ii) relaxing the overlapping supports assumption. First, counterfactual
female earnings $Y_{1}(x_{ij})|G_{i}=0$ — what female workers would have
earned if their gender were changed to male but nothing else about them
changed — are obtained by exact matching each female to one or more male
workers with similar observable characteristics and then taking a sample
average of the matched males191919In this paper we coarsened a few variables
such as years of education and age, and we use the coarsened version of these
variables instead to perform the exact matching. This serves the purpose of
matching more individuals, giving more statistical power to the method, since
workers with just e.g. 1 year difference in age, ceteris paribus, are roughly
the same in terms of productivity.. This method for building counterfactuals
is non-parametric, assuming no functional form for $Y_{1}(\cdot)$, it exerts
no extrapolations out of the support of $x$ and it avoids using data from all
workers to build counterfactuals for a specific worker. The matching
decomposition handles the lack of common support issue by allowing unmatched
workers, i.e. outside of the common support of $x$, to contribute to the
overall observed gap. In the matching decomposition, we add two terms,
$\Delta_{M}$ and $\Delta_{F}$, to the expression for the overall wage gap
$\Delta$ in equation (1) which captures the contributions of unmatched male
and female workers, respectively. The resulting expression is
$\displaystyle\Delta=$ $\displaystyle
E[Y_{ij}|G_{i}=1]-E[Y_{ij}|G_{i}=0]=:\Delta_{X}+\Delta_{0}+\Delta_{M}+\Delta_{F},$
(12)
where
$\displaystyle\Delta_{X}:=E\left[Y_{ij}|Matched,G_{i}=1\right]-E\left[Y_{1}(x_{ij})|Matched,G_{i}=0\right]$
$\displaystyle\Delta_{0}:=E\left[Y_{ij}|Matched,G_{i}=1\right]-E\left[Y_{1}(x_{ij})|Matched,G_{i}=0\right]$
$\displaystyle\Delta_{M}:=\left\\{E\left[Y_{ij}|Unmatched,G_{i}=1\right]-E\left[Y_{ij}|Matched,G_{i}=1\right]\right\\}P\left(Unmatched|G_{i}=1\right)$
$\displaystyle\Delta_{F}:=\left\\{E\left[Y_{ij}|Matched,G_{i}=0\right]-E\left[Y_{ij}|Unmatched,G_{i}=0\right]\right\\}P\left(Unmatched|G_{i}=0\right)$
Notice that if all observations are matched the $\Delta_{M}$ and $\Delta_{F}$
terms vanish and this method collapses back to the original decomposition we
have in equation (2). The terms $\Delta_{X}$ and $\Delta_{0}$ still have the
same interpretation as discussed in Section 2 — composition and structural,
respectively — but now only similar workers of one gender are used to build
counterfactuals for the other gender, using an agnostic functional form for
the counterfactual function. The extra terms $\Delta_{M}$ and $\Delta_{F}$
measure the contribution of unmatched male and female workers to the overall
observed gender gap. Each of them measures the difference between matched and
unmatched workers of a given gender, weighted by the proportion of unmatched
workers within that gender202020Precise definitions of each of the terms in
the NP decomposition can be found in the appendix section B. For example, if
unmatched male workers have an average log wage that is 0.2 higher than the
average log wage for matched male workers and 10% of male workers are
unmatched, then $\Delta_{M}=0.2\times 0.1=0.02$.
To understand how the matching decomposition handles a lack of common support,
consider male workers employed as professional football players. These workers
will not be matched to female workers and therefore would be omitted from the
analysis if we simply restrict it to the region of common support. However,
the male workers do contribute meaningfully to the overall gender wage gap
because they earn significantly more than the average female worker. The
matching decomposition would handle this by including these workers in the
$\Delta_{M}$ term. Intuitively, it would say that some of the gender wage gap
can be decomposed within the region of common support, while some of it is
explained by male workers outside the region of common support earning more
than male workers within the region of common support, and similarly for
female workers.
Our preferred specifications in this paper use the matching decomposition in
conjunction with the latent skills and tasks clusters revealed by our network
methodology developed in Section 3. Since we define labor market gender
discrimination as workers with similar skills performing similar tasks with
similar productivity but being paid differently based on gender, our worker
type–market clusters serve as natural cells within which workers are
considered as equivalent in terms of productivity. With the matching
decomposition we are able to ensure that only similar workers are used when
estimating counterfactual earnings, mitigating counterfactual biases, and also
avoid dropping unmatched workers from the estimation procedure as mentioned
above. Although the original matching decomposition is not considered to be a
“detailed decomposition” by the literature of decompositions in economics, in
combination with our network clusters, it is possible to compute an
economically principled distribution of the gender gap (and its components)
for a vast amount of cells of workers in the labor market, mapping how
discrimination is spread in different parts of the market.
## 5 Data
### 5.1 Administrative Brazilian data
We use the Brazilian linked employer-employee data set RAIS. The data contain
detailed information on all employment contracts in the Brazilian formal
sector, going back to the 1980s. The sample we work with includes all workers
between the ages of 25 and 55 employed in the formal sector in the Rio de
Janeiro metro area at least once between 2009 and 2018. These workers are
defined as matching with the unemployment (or informal sector) in years we do
not observe them. We also exclude the public sector because institutional
barriers make flows between the Brazilian public and private sectors rare, as
well as the military. Finally, we exclude the small number of jobs that do not
pay workers on a monthly basis.
Our wage variable is the real hourly log wage in December, defined as total
December earnings divided by hours worked. We deflate wages using the national
inflation index. We exclude workers who were not employed for the entire month
of December because we do not have accurate hours worked information for such
workers. We define a job as an occupation-establishment pair. This implicitly
assumes that all workers employed in the same occupation at the same
establishment are performing approximately the same tasks.
Our data contain 4,578,210 unique workers, 289,836 unique jobs, and 7,940,483
unique worker–job matches. The average worker matches with 1.73 jobs and the
average job matches with 27.4 workers. 42% of workers match with more than one
job during our sample. Figure 1(b) presents histograms of the number of
matches for workers and jobs, respectively. In network theory parlance, these
are known as degree distributions.
Figure 1: Distributions of Number of Matches Per Worker and Job
(a) Workers
(b) Jobs
Our network-based classification algorithm identifies 187 worker types
($\iota$) and 341 markets ($\gamma$). Figure 2(b) presents histograms of the
number of workers per worker type and jobs per market. The average worker
belongs to a worker type with 20,896 workers and the median worker belongs to
a worker type with 14,211 workers. The average job belongs to a market with
1,156 jobs and the median job belongs to a market with 1,127 jobs.
Figure 2: Worker Type ($\iota$) and Market ($\gamma$) Size Distributions
(a) Number of Workers Per Worker Type ($\iota$)
(b) Number of Jobs Per Market ($\gamma$)
## 6 Results
### 6.1 Aggregate wage gap decomposition
Table 1 presents the results of performing gender wage decompositions using
each of our two methods: OB and matching. For each method, we have three
specifications. The first, presented in columns (1) and (4), estimates
counterfactual earnings distributions using a standard set of observable
characteristics: experience, education, race, industry and union status. The
second, presented in columns (2) and (5), estimates counterfactual earnings
distributions using the worker types and markets identified by the SBM. The
third specification, presented in columns (3) and (6) uses both standard
observable characteristics and worker types and markets. The first row of each
column presents the overall wage gap: the average male worker earns 16.7
percent more than the average female worker in our sample. The second row
presents the wage gap that would exist if male and female workers with the
same productivity were paid equivalently but the observed differences between
the distributions of male and female productivity — as proxied by observable
characteristics and/or worker types and markets — remained, the composition
component. The third row presents the wage gap that would exist if male and
female workers had identical productivity distributions, but the observed
earnings differences conditional on productivity remained, the structural
component. The fourth and fifth rows present the wage gap explained by male
and female workers outside the region of common support, respectively. For the
OB method the composition and structural components add up to the overall wage
gap; for the matching method the overall wage gap equals the sum of the
composition and structural components and the components due to a lack of
common support.
The qualitative stories told by both the OB method and the matching method are
similar. When we define counterfactual earnings using observable
characteristics (columns 1 and 4), we find that if male and female workers
with the same productivity were paid similarly, then female workers would
significantly outearn male workers (structural effect): by 12.7% using the OB
method and 8.8% using the matching method. By contrast, female workers would
be paid significantly less if they possessed the male’s productivity
distribution (composition effect): 29.4% less using the OB method and 25.6%
less using the matching method. When we define counterfactuals using worker
types and markets instead of observable characteristics (columns 2 and 5) we
find that the wage gap would nearly disappear if male and female workers with
the same productivity were paid similarly. By contrast, the wage gap that
would exist if male and female workers had the same productivity distribution
— 17.9% according to OB and 17.8% according to matching — is almost equal to
the overall wage gap of 16.7%. In other words, when we compute counterfactuals
using worker types and markets we find that differential pay for similar
productivity explains roughly the entire gender wage gap. This tells us that
the results of gender wage gap decompositions are highly sensitive to the way
in which we define counterfactuals. If, as we argue, worker types and markets
do a better job of capturing the latent productivity of worker–job matches
than do standard observable characteristics, then these results imply that
gender wage gaps are almost entirely due to similarly productive male and
female workers being paid differently, not male and female workers having
different productivity distributions.
Columns (3) and (6) of Table 1 use both observable characteristics and worker
types and markets to form counterfactuals for the gender wage gap
decompositions. The OB method finds that female workers have covariates that
would imply that they would outearn male workers if equally productive workers
were paid equivalently, similar to the findings when we included only
observable characteristics, not worker and job types, in column (1). By
contrast, the matching method finds that male workers’ covariates imply 3.4%
higher earnings than female workers’ covariates and that male workers are paid
18.5% more than similarly productive female workers. Why do we observe a
discrepancy between the OB and matching methods once we include observable
characteristics and worker types and markets? The answer lies in the final two
rows of Table 1, which present the fraction of male and female workers,
respectively, for whom we are unable to find a counterfactual. Once we try to
match workers on such a large set of variables, many workers are unable to be
matched, and a significant part of the gender wage gap occurs among such
workers. The matching method allows us to take this into account, while the OB
method simply makes a linear extrapolation. However, a linear extrapolation
outside the region of common support is likely to lead to incorrect
inferences. Furthermore, the fact that the matching estimator yields similar
results when we use worker types and markets as it does when we use worker
types, markets, and other observable characteristics, but not when we use
other observables alone, implies that worker types and markets capture
significant determinants of productivity, and omitting them leads to incorrect
inferences. This highlights the importance of using a sufficiently set of
worker characteristics when estimating counterfactuals, and our method for
identifying previously unobserved heterogeneity enhances our ability to do so.
All of the results presented in this section correspond to the aggregate
gender wage gap. In the next section, we consider heterogeneity in wage gaps
within different subsets of the labor market.
Table 1: Gap decomposition using Oaxaca-Blinder vs Matching | Oaxaca-Blinder | Matching
---|---|---
| Observables | $\iota\times\gamma$ | Full model | Observables | $\iota\times\gamma$ | Full model
| (1) | (2) | (3) | (4) | (5) | (6)
Gap | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167
Composition | -0.127 | -0.011 | -0.084 | -0.088 | -0.006 | 0.034
structural | 0.294 | 0.179 | 0.250 | 0.256 | 0.178 | 0.185
Males unmatched | - | - | - | 0.000 | -0.005 | -0.076
Females unmatched | - | - | - | 0.000 | 0.000 | 0.024
% of males matched | - | - | - | 1.00 | 0.98 | 0.57
% of females matched | - | - | - | 1.00 | 0.99 | 0.74
### 6.2 Wage gaps within worker type–market cells
An appealing feature of our worker types and markets is that they allow us to
further decompose gender wage gaps and identify heterogeneity in gender wage
gaps across the labor market. We do so by computing overall wage gaps,
$\Delta$, and then decomposing them following the matching decomposition,
within each worker type–market cell.
For each worker type–market cell we decompose the overall wage gap (Row 1 of
Table 1) into its four components: composition, structural, males unmatched,
and females unmatched (Rows 2–5 of Table 1). Figure 3 presents kernel density
plots of the resulting distributions of overall wage gaps and their four
components. Several clear patterns emerge. First, the overall wage gaps
$\Delta$ are almost universally positive, meaning that male workers outearning
their female counterparts is a widespread phenomenon. Specifically, 91% of
workers are in clusters where males outearn females. Second, the distribution
of the structural component, $\Delta_{0}$, is similar to the distribution of
the overall wage gap. This suggests that the result from the aggregate
decomposition in Section 6.1 that almost the entire overall gender wage gap is
explained by the structural component holds within worker type–market cells as
well. The fact that the structural component roughly coincides with the
overall wage gap implies that the other three components — composition, males
outside the common support, and females outside the common support — must
contribute relatively little to the overall gender wage gap, which is
confirmed by the fact that the distributions for these three components are
centered close to zero and have low variances. We present the same results
quantitatively in Table 2. Together, these results tell us that while there is
significant variability in gender wage gaps across different worker
type–market pairs, the overall qualitative pattern of male workers outearning
their female counterparts, and almost all of this gap being explained by
differential returns to the same skills rather than different skills, is true
in the disaggregated results as well as the aggregated results.
Figure 3: Distribution of Components of Overall Wage Gap, Disaggregated Table 2: Summary Statistics of Components of Overall Wage Gap, Disaggregated | mean | sd | min | max | count
---|---|---|---|---|---
$\Delta$ | 0.215 | 0.240 | -1.183 | 6.228 | 4791014
$\Delta_{0}$ | 0.196 | 0.172 | -2.506 | 9.384 | 4791014
$\Delta_{M}$ | 0.016 | 0.134 | -3.577 | 3.448 | 4783255
$\Delta_{F}$ | -0.011 | 0.116 | -2.632 | 4.684 | 4724863
$\Delta_{X}$ | 0.013 | 0.153 | -1.150 | 2.418 | 4791014
Frac. Male Workers Matched | 0.766 | 0.238 | 0.004 | 1.000 | 4791014
Frac. Female Workers Matched | 0.875 | 0.199 | 0.008 | 1.000 | 4791014
Frac. Workers that Are Male | 0.617 | 0.162 | 0.037 | 0.999 | 4791014
## 7 Conclusion
In this paper we reconsider the wage gap decomposition literature and make
three key contributions. First, we propose a new method for identifying
unobserved determinants of workers’ earnings from the information revealed by
detailed data on worker–job matching patterns. The method builds on Fogel and
Modenesi (2023) and provides a blueprint for incorporating observable
variables into the clustering algorithm, while also relaxing the assumption of
perfect competition in labor markets. Second, we non-parametrically estimate
counterfactual wage functions for male and female workers and use them to
decompose gender wage gaps into a composition component in which male and
female workers earn different wages because they possess different skills and
perform different tasks, and a structural component in which male and female
workers who possess similar skills and perform similar tasks nonetheless earn
different wages. Third, we address the issue of male workers’ observables
characteristics falling outside the support of female workers’ observable
characteristics, and vice versa, by augmenting the wage decomposition with
components attributable to male and female workers, respectively, outside the
region of common support.
We apply these methods to Brazilian administrative data and find that almost
the entire gender wage gap is attributable to male and female workers who
possess similar skills and perform similar tasks being paid differently. This
is true at the aggregate level, and remains true when we perform wage
decompositions within each worker type–market cell, indicating that this is a
widespread phenomenon, not one driven by large wage differentials in small
subsets of the labor market. We find that wage decompositions based on
standard observable variables suffer from omitted variable bias, emphasizing
the need for detailed worker and job characteristics in the form of worker
types and markets. We find that wage decompositions based on linear
regressions yield similar findings to those based on matching when a lack of
common support is not an issue, however when male and female workers’
characteristics do not share a common support the matching estimator with
corrections for a lack of common support outperforms alternatives.
While this paper focuses on gender wage gaps, the methods are applicable to
other wage gaps, for instance race. Moreover, our strategy for using
worker–job matching patterns to control for previously-unobserved, but
potentially confounding, covariates may be applied in a wide variety of
contexts.
## References
* (1)
* Acemoglu and Autor (2011) Acemoglu, Daron and David Autor, “Skills, tasks and technologies: Implications for employment and earnings,” 2011, 4, 1043–1171.
* Autor (2013) Autor, David H, “The ‘task approach’ to labor markets: an overview,” 2013\.
* Autor et al. (2003) Autor, David H., Frank Levy, and Richard J. Murnane, “The Skill Content of Recent Technological Change: An Empirical Exploration,” The Quarterly Journal of Economics, 2003, 118 (4), 1279–1333.
* Barsky et al. (2002) Barsky, Robert, John Bound, Kerwin Kofi Charles, and Joseph P. Lupton, “Accounting for the Black-White Wealth Gap: A Nonparametric Approach,” Journal of the American Statistical Association, 2002, 97 (459), 663–673.
* Blinder (1973) Blinder, Alan S., “Wage Discrimination: Reduced Form and Structural Estimates,” The Journal of Human Resources, 1973, 8 (4), 436–455.
* Card et al. (2015) Card, David, Ana Rute Cardoso, and Patrick Kline, “ Bargaining, Sorting, and the Gender Wage Gap: Quantifying the Impact of Firms on the Relative Pay of Women *,” The Quarterly Journal of Economics, 10 2015, 131 (2), 633–686.
* Card et al. (2018) , , Joerg Heining, and Patrick Kline, “Firms and Labor Market Inequality: Evidence and Some Theory,” Journal of Labor Economics, 2018, 36 (S1), S13–S70.
* Chernozhukov et al. (2013) Chernozhukov, Victor, Iván Fernández-Val, and Blaise Melly, “Inference on Counterfactual Distributions,” Econometrica, 2013, 81 (6), 2205–2268.
* DiNardo et al. (1996) DiNardo, John, Nicole M. Fortin, and Thomas Lemieux, “Labor Market Institutions and the Distribution of Wages, 1973-1992: A Semiparametric Approach,” Econometrica, 1996, 64 (5), 1001–1044.
* Firpo et al. (2018) Firpo, Sergio P., Nicole M. Fortin, and Thomas Lemieux, “Decomposing Wage Distributions Using Recentered Influence Function Regressions,” Econometrics, May 2018, 6 (2), 1–40.
* Fogel and Modenesi (2023) Fogel, Jamie and Bernardo Modenesi, “What is a Labor Market? Classifying Workers and Jobs Using Network Theory,” 2023.
* Fortin et al. (2011) Fortin, Nicole, Thomas Lemieux, and Sergio Firpo, “Chapter 1 - Decomposition Methods in Economics,” in Orley Ashenfelter and David Card, eds., Orley Ashenfelter and David Card, eds., Vol. 4 of Handbook of Labor Economics, Elsevier, 2011, pp. 1–102.
* Garcia et al. (2009) Garcia, Luana Marquez, Hugo Nopo, and Paola Salardi, “Gender and Racial Wage Gaps in Brazil 1996-2006: Evidence Using a Matching Comparisons Approach,” Research Department Publications 4626, Inter-American Development Bank, Research Department May 2009.
* Gerard et al. (2018) Gerard, François, Lorenzo Lagos, Edson Severnini, and David Card, “Assortative Matching or Exclusionary Hiring? The Impact of Firm Policies on Racial Wage Differences in Brazil,” Working Paper 25176, National Bureau of Economic Research October 2018.
* Goldin (2014) Goldin, Claudia, “A Grand Gender Convergence: Its Last Chapter,” American Economic Review, April 2014, 104 (4), 1091–1119.
* Hurst et al. (2021) Hurst, Erik, Yona Rubinstein, and Kazuatsu Shimizu, “Task-Based Discrimination,” Working Paper 29022, National Bureau of Economic Research July 2021.
* Jarosch et al. (2019) Jarosch, Gregor, Jan Sebastian Nimczik, and Isaac Sorkin, “Granular search, market structure, and wages,” Technical Report, National Bureau of Economic Research 2019.
* Kantenga (2018) Kantenga, Kory, “The effect of job-polarizing skill demands on the US wage structure,” 2018.
* Karrer and Newman (2011) Karrer, Brian and Mark EJ Newman, “Stochastic blockmodels and community structure in networks,” Physical review E, 2011, 83 (1), 016107.
* Larremore et al. (2014) Larremore, Daniel B, Aaron Clauset, and Abigail Z Jacobs, “Efficiently inferring community structure in bipartite networks,” Physical Review E, 2014, 90 (1), 012805.
* Lindenlaub (2017) Lindenlaub, Ilse, “Sorting multidimensional types: Theory and application,” The Review of Economic Studies, 2017, 84 (2), 718–789.
* McFadden (1978) McFadden, Daniel, “Modeling the choice of residential location,” Transportation Research Record, 1978, (673).
* Modenesi (2022) Modenesi, Bernardo, “Advancing Distribution Decomposition Methods Beyond Common Supports: Applications to Racial Wealth Disparities,” 2022.
* Morello and Anjolim (2021) Morello, Thiago and Jacqueline Anjolim, “Gender wage discrimination in Brazil from 1996 to 2015: A matching analysis,” EconomiA, 2021.
* Nimczik (2018) Nimczik, Jan Sebastian, “Job Mobility Networks and Endogenous Labor Markets,” 2018.
* Ñopo (2008) Ñopo, Hugo, “Matching as a Tool to Decompose Wage Gaps,” The Review of Economics and Statistics, 2008, 90 (2), 290–299.
* Oaxaca (1973) Oaxaca, Ronald, “Male-Female Wage Differentials in Urban Labor Markets,” International Economic Review, 1973, 14 (3), 693–709.
* Peixoto (2014) Peixoto, Tiago P, “Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models,” Physical Review E, 2014, 89 (1), 012804.
* Peixoto (2017) , “Nonparametric Bayesian inference of the microcanonical stochastic block model,” Physical Review E, 2017, 95 (1), 012317\.
* Peixoto (2018) Peixoto, Tiago P., “Nonparametric weighted stochastic block models,” Phys. Rev. E, Jan 2018, 97, 012306.
* Peixoto (2019) Peixoto, Tiago P, “Bayesian stochastic blockmodeling,” Advances in network clustering and blockmodeling, 2019, pp. 289–332.
* Roy (1951) Roy, Andrew Donald, “Some thoughts on the distribution of earnings,” Oxford economic papers, 1951, 3 (2), 135–146.
* Sorkin (2018) Sorkin, Isaac, “Ranking firms using revealed preference,” The quarterly journal of economics, 2018, 133 (3), 1331–1393.
* Tan (2018) Tan, Joanne, “Multidimensional heterogeneity and matching in a frictional labor market - An application to polarization,” 2018.
* Train (2010) Train, Kenneth E, Discrete choice methods with simulation, Cambridge university press, 2010.
## Appendix A Nested Logit Choice Probability
According to Train (2010), and originally developed by McFadden (1978),
maximizing the utility choosing $j$, which is nested within a group $\gamma$
$j^{*}:=\arg\max_{j}\quad W_{\gamma}+Y_{j}+\varepsilon_{j}$ (13)
with $\varepsilon_{j}\sim NestedLogit(\nu_{\gamma})$ results in the following
choice probability:
$\displaystyle P(j=j^{*})$ $\displaystyle=P(\text{Choose
}\gamma)P(j=j^{*}|\gamma)$
$\displaystyle=\frac{\exp(W_{\gamma}+\nu_{\gamma}I_{\gamma})}{\sum_{\gamma}\exp(W_{\gamma}+\nu_{\gamma}I_{\gamma})}\frac{\exp(Y_{j})^{\frac{1}{\nu_{\gamma}}}}{\sum_{j\in\gamma}\exp(Y_{j})^{\frac{1}{\nu_{\gamma}}}}$
where
$I_{\gamma}=\log\left(\sum_{j\in\gamma}\exp(Y_{j})^{\frac{1}{\nu_{\gamma}}}\right)$.
Our problem is similar, with workers choosing job $j$ within a market $\gamma$
in order to maximize the sum of log earnings
$\log(\psi_{\iota\gamma}w_{j}^{g})$ and an idiosyncratic preference for job
$j$, $\varepsilon_{ij}^{g}$:
$\displaystyle j^{*}=$
$\displaystyle\arg\max_{j}\quad\log(\psi_{\iota\gamma}w_{j}^{g})+\varepsilon_{ij}^{g}.$
(14)
We also assume that $\varepsilon_{ij}^{g}\sim NestedLogit(\nu_{\gamma}^{g})$.
One of the differences from our setup to what is covered by Train (2010) is
that we add extra worker indexes $\iota$ for her/his skills and $g$ for her
gender and we condition our probabilities on knowing $\iota$ and $g$. Notice
that when comparing equations 13 and 14, $W_{\gamma}=0$ and
$Y_{j}=\log(\psi_{\iota\gamma}w_{j}^{g})$, which results in the following
choice probabilities:
$\displaystyle P(j=j^{*}|j\in\gamma,i\in\iota,g)$
$\displaystyle=P(\gamma=\gamma^{*}|i\in\iota,j\in\gamma,g)P(j=j^{*}|i\in\iota,j\in\gamma,\gamma=\gamma^{*},g)$
$\displaystyle=\frac{\exp(\nu_{\gamma}^{g}I_{\iota\gamma}^{g})}{\sum_{\gamma}\exp(\nu_{\gamma}^{g}I_{\iota\gamma}^{g})}\frac{\exp(\log(\psi_{\iota\gamma}w_{j}^{g}))^{\frac{1}{\nu_{\gamma}^{g}}}}{\sum_{j\in\gamma}\exp(\log(\psi_{\iota\gamma}w_{j}^{g}))^{\frac{1}{\nu_{\gamma}^{g}}}}\quad\text{\tiny(plugging
objects in)}$
$\displaystyle=\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\frac{(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}{\sum_{j\in\gamma}(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}\quad\text{\tiny(similar
to equation \ref{eq_worker_choice})}$
$\displaystyle=\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\frac{(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}{\exp(I_{\iota\gamma}^{g})}\quad\text{\tiny(by
definition of $I_{\iota\gamma}^{g}$)}$
$\displaystyle=\underset{\underset{\iota-\gamma-g\text{
component}\quad}{\underbrace{=:\Omega_{\iota\gamma}^{g}}}}{\underbrace{\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}-1}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\psi_{\iota\gamma}^{\frac{1}{\nu_{\gamma}^{g}}}}}\underset{\underset{\quad
j-g\text{
component}}{\underbrace{=:d_{j}^{g}}}}{\underbrace{\vphantom{\frac{\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}-1}}{\sum_{\gamma}\exp(I_{\iota\gamma}^{g})^{\nu_{\gamma}^{g}}}\psi_{\iota\gamma}^{\frac{1}{\nu_{\gamma}^{g}}}}(w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}}}\quad\text{\tiny(similar
to equation \ref{eq_worker_choice_separation})}$
where
$I_{\iota\gamma}^{g}=\log\left[\sum_{j\in\gamma}\exp(\log(\psi_{\iota\gamma}w_{j}^{g}))^{\frac{1}{\nu_{\gamma}^{g}}}\right]=\log\left[\sum_{j\in\gamma}(\psi_{\iota\gamma}w_{j}^{g})^{\frac{1}{\nu_{\gamma}^{g}}}\right]$.
## Appendix B Terms in the NP decomposition
The terms in the NP decomposition from equation 12 can be more formally
defined as follows:
$\displaystyle\Delta_{M}:=\left[\int_{\bar{S}_{F}}Y_{1}(x)\frac{dF_{M}(x)}{\mu_{M}(\bar{S}_{F})}-\int_{S_{F}}Y_{1}(x)\frac{dF_{M}(x)}{\mu_{M}(S_{F})}\right]\mu_{M}(\bar{S}_{F})$
(15) $\displaystyle\Delta_{X}:=\int_{S_{M}\cap
S_{F}}Y_{1}(x)\left[\frac{dF_{M}(x)}{\mu_{M}(S_{F})}-\frac{dF_{F}(x)}{\mu_{F}(S_{M})}\right]$
$\displaystyle\Delta_{0}:=\int_{S_{M}\cap
S_{F}}\left[Y_{1}(x)-Y_{0}(x)\right]\frac{dF_{F}(x)}{\mu_{F}(S_{M})}$
$\displaystyle\Delta_{F}:=\left[\int_{S_{M}}Y_{0}(x)\frac{dF_{F}(x)}{\mu_{F}(S_{M})}-\int_{\bar{S}_{M}}Y_{0}(x)\frac{dF_{F}(x)}{\mu_{F}(\bar{S}_{M})}\right]\mu_{F}(\bar{S}_{M})$
where: $F_{M}(x)$ and $F_{F}(x)$ denote the distributions of $x$ for both
males and females, respectively; $\mu_{M}$ and $\mu_{F}$ measure the
proportions of males and females over regions of the supports of $x$; and the
support of $x$ for a gender $g$, $supp_{(}X_{g})$, is partitioned as
$supp_{(}X_{g}):=S_{g}\cup\bar{S}_{g}$, with $S_{g}\cap\bar{S}_{g}=\emptyset$,
for $g\in\\{F,M\\}$.
## Appendix C Proof that $A_{ij}$ follows a Poisson distribution
If an individual worker $i$ only searched for a job once, then the probability
of worker $i$ matching with job $j$ would be equal to
$\mathbb{P}_{ij}=\mathcal{P}_{\iota\gamma}d_{j}$ and $A_{ij}$ would follow a
Bernoulli distribution:
$A_{ij}\sim Bernoulli(\mathcal{P}_{\iota\gamma}d_{j}).$
However, since worker $i$ searches for jobs $c_{i}\equiv\sum_{t=1}^{T}c_{it}$
times, $A_{ij}$ is actually the sum of $c_{i}$ Bernoulli random variables, and
is therefore a Binomial random variable. Conditional on knowing $c_{i}$,
$A_{ij}|c_{i}\sim Binomial(c_{i},\mathcal{P}_{\iota\gamma}d_{j}).$
However, we still need to take into account the fact that $c_{i}$ is a
Poisson-distributed random variable with arrival rate $d_{i}$. Consequently,
the unconditional distribution of $A_{ij}$ is Poisson as well:
$A_{ij}\sim Poisson(d_{i}d_{j}\mathcal{P}_{\iota\gamma}).$
We prove this fact by multiplying the conditional density of $A_{ij}|c_{i}$ by
the marginal density of $c_{i}$ to get the joint density of $A_{ij}$ and
$c_{i}$, and then integrating out $c_{i}$.
$\displaystyle
P(A_{ij},c_{i})=\underset{Bin(c_{i},d_{j}P_{\iota\gamma})}{\underbrace{P(A_{ij}|c_{i})}}\quad\times\quad\underset{Poisson(d_{i})}{\underbrace{P(c_{i})}}$
Deriving the joint distribution:
$\displaystyle P(A_{ij},c_{i})=$
$\displaystyle\binom{c_{i}}{A_{ij}}(d_{j}P_{\iota\gamma})^{A{ij}}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}\times\frac{d_{i}^{c_{i}}\exp{(-d_{i}})}{c_{i}!}$
We want to find out the marginal distribution of $A_{ij}$:
$\displaystyle P(A_{ij})$
$\displaystyle=\sum_{c_{i}=0}^{\infty}P(A_{ij},c_{i})$
$\displaystyle=\sum_{c_{i}=0}^{\infty}\binom{c_{i}}{A_{ij}}(d_{j}P_{\iota\gamma})^{A{ij}}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}\times\frac{d_{i}^{c_{i}}\exp{(-d_{i}})}{c_{i}!}$
$\displaystyle=\sum_{c_{i}=0}^{\infty}\frac{c_{i}!}{A_{ij}!(di-
A_{ij})!}(d_{j}P_{\iota\gamma})^{A{ij}}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}\times\frac{d_{i}^{c_{i}}\exp{(-d_{i}})}{c_{i}!}$
$\displaystyle=\frac{(d_{j}P_{\iota\gamma})^{A{ij}}\exp{(-d_{i}})}{A_{ij}!}\sum_{c_{i}=0}^{\infty}\frac{1}{(di-
A_{ij})!}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}d_{i}^{c_{i}}$
If the summation term is equal to
$\sum_{c_{i}=0}^{\infty}\frac{1}{(di-
A_{ij})!}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}d_{i}^{c_{i}}=d_{i}^{A_{ij}}\exp{(d_{i}(1-d_{j}P_{\iota\gamma}))}$
(16)
then
$P(A_{ij})=\frac{(d_{i}d_{j}P_{\iota\gamma})^{A{ij}}\exp{(-d_{i}d_{j}P_{\iota\gamma}})}{A_{ij}!}$,
i.e. $A_{ij}$ would be Poisson distributed:
$A_{ij}\sim Poisson(d_{i}d_{j}P_{\iota\gamma})$
Proving (16) is equivalent to proving the following equality:
$\displaystyle 1=$
$\displaystyle\frac{1}{d_{i}^{A_{ij}}\exp{(d_{i}(1-d_{j}P_{\iota\gamma}))}}\sum_{c_{i}=0}^{\infty}\frac{1}{(di-
A_{ij})!}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}d_{i}^{c_{i}}$
Proof:
$\displaystyle
d_{i}^{-A_{ij}}\exp{(-d_{i}(1-d_{j}P_{\iota\gamma}))}\sum_{c_{i}=0}^{\infty}\frac{1}{(di-
A_{ij})!}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}d_{i}^{c_{i}}=$
$\displaystyle=\sum_{c_{i}=0}^{\infty}\frac{\exp{(-d_{i}(1-d_{j}P_{\iota\gamma}))}}{(di-
A_{ij})!}(1-d_{j}P_{\iota\gamma})^{c_{i}-A{ij}}d_{i}^{c_{i}-A_{ij}}$
$\displaystyle=\sum_{c_{i}=0}^{\infty}\frac{\exp{(-d_{i}(1-d_{j}P_{\iota\gamma}))}}{(di-
A_{ij})!}(d_{i}(1-d_{j}P_{\iota\gamma}))^{c_{i}-A{ij}}$ We assume
$\lambda=d_{i}(1-d_{j}P_{\iota\gamma})$ for simplicity and we apply a change
of variables $z=c_{i}-A_{ij}$
$\displaystyle=\sum_{z=0}^{\infty}\frac{\exp{(-\lambda)}}{z!}\lambda^{z}\text{,
knowing that in our problem $c_{i}\geq A_{ij}$, i.e. $z\geq 0$}.$
$\displaystyle=1$ $\displaystyle\text{Since we have the p.d.f. of a Poisson
r.v. inside the summation, i.e. $z\sim Poisson(\lambda)$ }\square$
Therefore, we have
$A_{ij}\sim Poisson(d_{i}d_{j}P_{\iota\gamma})\qed$
## Appendix D Soft assignment workers and jobs to worker types and markets
In section 3, at the maximum of our posterior in equation 3.2.2, each worker
is assigned to only one skill cluster, a process of hard assignments. However,
it is possible that, given the pattern of worker matches, a particular worker
could be revealed to possess certain skills $\iota_{1}$ in most of her
matches, and skills $\iota_{2}$ in a few other of her matches. Creating a
single worker skill group to accommodate her hybrid skills might not improve
model fit if there are only a few workers who exhibit similar matches.
Instead, allowing her to have mixed skills $\iota_{1}$ and $\iota_{2}$, i.e.
soft assignment, with weights according to her matching history, provides
further nuanced information to the researcher. In fact, we propose using the
Bayesian setup in order to recover these weights.
It turns out that the posterior $P(\bm{b}|\bm{A},\bm{g})$ ultimately carries
the desired measure of workers’ skill profile needed to control for workers’
unobserved skills in the wage gap estimation. Given a total of $I$ clusters of
workers competing for the same jobs in the labor market network, i.e. with
similar skills, the posterior distribution provides the chance of each worker
to belong to a certain skill cluster, given the worker demographic group $g$
and the entire network $\bm{A}$. More formally, for worker $i$, her skills
profile is defined as:
$\displaystyle\vec{P}_{i}:=\left[P(i\in\iota_{1}|\bm{A},\bm{g})\qquad
P(i\in\iota_{2}|\bm{A},\bm{g})\qquad\cdots\qquad
P(i\in\iota_{I}|\bm{A},\bm{g})\right]^{T}$ (17)
|
# Regret Analysis in Threshold Policy Design††thanks: I am grateful to Ivan
Canay, Joel Horowitz and Charles Manski for their guidance in this project. I
am also thankful to Anastasiia Evdokimova, Amilcar Velez and all the
participants of the Econometric Reading Group at Northwestern University for
their comments and suggestions.
Federico Crippa
Department of Economics
Northwestern University
<EMAIL_ADDRESS>
(April 17, 2024)
Threshold policies are targeting mechanisms that assign treatments based on
whether an observable characteristic exceeds a certain threshold. They are
widespread across multiple domains, such as welfare programs, taxation, and
clinical medicine. This paper addresses the problem of designing threshold
policies using experimental data, when the goal is to maximize the population
welfare. First, I characterize the regret—a measure of policy optimality—of
the Empirical Welfare Maximizer (EWM) policy, popular in the literature. Next,
I introduce the Smoothed Welfare Maximizer (SWM) policy, which improves the
EWM’s regret convergence rate under an additional smoothness condition. The
two policies are compared studying how differently their regrets depend on the
population distribution, and investigating their finite sample performances
through Monte Carlo simulations. In many contexts, the welfare guaranteed by
the novel SWM policy is larger than with the EWM. An empirical illustration
demonstrates how the treatment recommendation of the two policies may in
practice notably differ.
Keywords: Threshold policies, heterogeneous treatment effects, statistical
decision theory, randomized experiments.
JEL classification codes: C14, C44
## 1 Introduction
Treatments are rarely universally assigned. When their effects are
heterogeneous across individuals, policymakers aim to target those who would
benefit the most from specific interventions. Scholarships, for example, are
awarded to students with high academic performance or financial need; tax
credits are provided to companies engaged in research and development
activities; medical treatments are prescribed to sick patients. Despite the
potential complexity and multidimensionality of heterogeneous treatment
effects, treatment eligibility criteria are often kept quite simple. This
paper studies one of the most common of these simple assignment mechanisms:
threshold policies, where the decision to assign the treatment is based on
whether a scalar observable characteristic — referred to as the index —
exceeds a specified threshold.
Threshold policies are ubiquitous, ranging across multiple domains. In welfare
policies, they regulate the qualification for public health insurance programs
through age (Card et al., 2008; Shigeoka, 2014) and anti-poverty programs
through income (Crost et al., 2014). In taxation, they determine marginal
rates through income brackets (Taylor, 2003). In clinical medicine, the
referral for liver transplantation depends on whether a composite of
laboratory values obtained from blood tests is beyond a certain threshold
(Kamath and Kim, 2007). Even criminal offenses are defined through threshold
policies: sanctions for Driving Under the Influence are based on whether the
Blood Alcohol Content exceeds specific values.
The regression discontinuity design has been developed to study the treatment
effect at the point of discontinuity of threshold policies: its popularity in
econometrics and applied economics is a further indication of how widespread
threshold policies are. Regression discontinuity design focuses on an ex-post
evaluation. In this paper, my perspective is different: I consider the ex-ante
problem faced by a policymaker wanting to implement a threshold policy and
interested in maximizing the average social welfare, targeting individuals who
would benefit from the treatment. Experimental data are available: how should
they be used to implement the threshold policy in the population?
Answering this question requires defining a criterion by which policies are
evaluated. Since the performance of a policy depends on the unknown data
distribution, the policy maker searches for a policy that behaves uniformly
well across a specified family of data distribution (the state space). The
regret of a policy is the (possibly) random difference between the maximum
achievable welfare and the welfare it generates in the population. Policies
can be evaluated considering their maximum expected regret (Manski, 2004;
Hirano and Porter, 2009; Kitagawa and Tetenov, 2018; Athey and Wager, 2021;
Mbakop and Tabord-Meehan, 2021; Manski, 2023), or other worst-case statistics
of the regret distribution (Manski and Tetenov, 2023; Kitagawa et al., 2022).
Once the criterion has been established, optimal policy learning aims to
pinpoint the policy that minimizes it. Rather than directly tackling the
functional minimization problem, following the literature, I consider
candidate threshold policy functions and characterize some properties of their
regret.
The first contribution of this paper is to show how to derive the asymptotic
distribution of the regret for a given threshold policy. The underlying
intuition is simple: threshold policies use sample data to choose the
threshold, which is hence a random variable with a certain asymptotic
behavior. A Taylor expansion establishes a map between the regret of a policy
and its threshold, allowing one to characterize the asymptotic distribution of
the regret through the asymptotic behavior of the threshold. This shifts the
problem to characterizing the asymptotic of the threshold estimator,
simplifying the analysis as threshold estimators can be studied with common
econometric tools.
I start considering the Empirical Welfare Maximizer (EWM) policy studied by
kitagawa2018should. They derive uniform bounds for the expected regret of the
policy for various policy classes, where the policy class impacts the findings
only in terms of its VC dimensionality. My approach is more specific,
considering only threshold policies, but also more informative: leveraging the
knowledge of the policy class, I characterize the entire asymptotic
distribution of the regret. As mentioned above, this requires the derivation
of the asymptotic distribution of the threshold for the EWM policy, which is
non-standard: it exhibits the “cube root asymptotics” behavior studied in
kim1990cube. The convergence rate is $n^{1/3}$, and the asymptotic
distribution is of chernoff1964estimation type. The non-standard behavior and
the unusual convergence rate are due to the discontinuity in the objective
function and are reflected in the asymptotic distribution of the regret, and
in its $n^{2/3}$ convergence rate.
My second contribution is hence to propose a novel threshold policy, the
Smoothed Welfare Maximizer (SWM) policy. This policy replaces the indicator
function in the EWM policy’s objective function with a smooth kernel. Under
certain regularity assumptions, the threshold estimator for the SWM policy is
asymptotically normal and its regret achieves a $n^{4/5}$ convergence rate.
This implies that the regret’s convergence rate with the SWM is faster than
with the EWM policy.
Building on these asymptotic results, I extend the comparison of the regrets
with the EWM and the SWM policies beyond their convergence rates. My findings
allow to compare the asymptotic distributions and investigate how differently
they depend on the data distribution; theoretical results are helpful to
inform and guide the Monte Carlo simulations, which confirm that the
asymptotic results approximate the actual finite sample behaviors. Notably,
the simulations confirm that the SWM policy may guarantee lower expected
regret in finite samples.
To demonstrate the practical differences between the two policies, I present
an empirical illustration considering a job-training treatment. In that
context, the SWM threshold policy would recommend treating 79% of unemployed
workers, as opposed to 83% with the EWM policy. This difference of 4
percentage points is economically non-negligible.
### 1.1 Related Literature
This paper relates to the statistical decision theory literature studying the
problem of policy assignment with covariates (manski2004statistical;
stoye2012minimax; kitagawa2018should; athey2021policy; mbakop2021model;
sun2021treatment; sun2021empirical; viviano2023fair). My setting is mainly
related to kitagawa2018should and athey2021policy, with some notable
differences. kitagawa2018should study the EWM policy for policy classes with
finite VC dimension. They derive finite sample bounds for regret without
relying on smoothness assumptions. athey2021policy consider a double robust
version of the EWM and allow for observational data. Under smoothness
assumptions analogous to mine, they derive asymptotic bounds for the expected
regret for policy classes with finite VC dimensions. Conversely, results in
this paper apply exclusively to threshold policies, relying on a combination
of the assumptions in kitagawa2018should and athey2021policy. The narrower
focus allows for more comprehensive results: the entire asymptotic
distribution of the regret is derived rather than providing some bounds for
the expected regret.
Another critical distinction lies in the different nature of the uniform
convergence rates: my state space is a subset of theirs, which is why the
rates I derive for the EWM and the SWM policies are faster than the $\sqrt{n}$
rate reported as the optimal by kitagawa2018should and athey2021policy. Their
$\sqrt{n}$ rate is, in fact, uniformly valid for a family of data
distributions that, at least for threshold policies, include extreme cases
(e.g. discontinuous conditional ATE), which determine the slower rate of
convergence. Their uniform results may be viewed as a benchmark: when more
structure is added to the problem and some distribution in the family
excluded, they can be improved.
Optimal policy learning finds its empirical counterpart in the literature
dealing with targeting, especially common in development economics. Recent
studies rely on experimental evidence to decide who to treat in a data-driven
way (hussam2022targeting; aiken2022machine), even if haushofer2022targeting
pointed out the need for a more formalized approach to the targeting decision
problem. The availability for the policy maker of appropriate tools to use the
data in the decision process is probably necessary to guarantee a broader
adoption of data-driven targeting strategies. Focusing on threshold policies,
this paper explicitly formulates the decision problem, introduces
implementable policies (the EWM and the SWM policy), and compares their
asymptotic properties.
Turning to the threshold estimators, I already mentioned that the EWM policy
exhibits the cube root of $n$ asymptotics studied by kim1990cube, distinctive
of several estimators in different contexts. Noteworthy examples are the
maximum score estimator in choice models (manski1975maximum), the split point
estimator in decision trees (banerjee2007confidence), and the risk minimizer
in classification problems (mohammadi2005asymptotics), among others. Specific
to my analysis is the emergence of the cube root asymptotic within a causal
inference problem relying on the potential outcomes model, which is then
mirrored in the regret’s asymptotic distribution.
Addressing the cube root problem by smoothing the indicator in the objective
function aligns closely with the strategy proposed by horowitz1992smoothed for
studying the asymptotic behavior of the maximum score estimator. Objective
functions are nonetheless different, and in my context, I derive the
asymptotic distribution for both the unsmoothed (EWM) and the smoothed (SWM)
policies. This is convenient, as it allows me to compare not only the
convergence rates but also the entire asymptotic distributions of the
estimators and their regrets and study the asymptotic approximations in Monte
Carlo simulations.
The rest of the paper is structured as follows. Section 2 introduces the
problem and outlines my analytical approach. Section 3 derives formal results
for the asymptotic of the EWM and SWM policies and their regrets. In Section
4, I investigate finite sample performance of the EWM and SWM policies through
Monte Carlo simulations, while in Section 5 I consider the analysis of
experimental data from the National Job Training Partnership Act (JTPA) Study
to compare the practical implications of the policies. Section 6 concludes.
## 2 Overview of the Problem
I consider the problem of a policymaker who wants to implement a binary
treatment in a population of interest. Individuals are characterized by a
vector of observable characteristics $\textbf{X}\in\mathbb{R}^{d}$, on which
the policy maker bases the treatment assignment choice. A policy is hence a
map $\pi(\textbf{x}):\mathbb{R}^{d}\rightarrow\\{0,1\\}$, from observable
characteristics to the binary treatment status. The policy maker is
utilitarian: its goal is to maximize the average welfare in the population.
Indicating by $Y_{1}$ and $Y_{0}$ the potential outcomes with and without the
treatment, population average welfare generated by a policy $\pi$ can be
written as
$\displaystyle W(\pi)=\mathbb{E}[Y_{1}\pi(X)+Y_{0}(1-\pi(X))].$ (1)
When treatment effects are heterogeneous, the same treatment can have opposite
average effects across individuals with different X. For this reason, the
policy assignment may vary with X: the policymaker wants to target only those
who benefit from being treated, to maximize the average welfare.
The policy learning literature has considered several classes $\Pi$ of policy
functions, such for example linear eligibility indexes, decision trees, or
monotone rules, discussed in kitagawa2018should; athey2021policy;
mbakop2021model. This paper focuses on threshold policies, a specific class of
policy functions that can be represented as
$\displaystyle\pi(\textbf{X})=\pi(X,t)=\mathbf{1}\\{X>t\\}.$ (2)
Treatment is assigned whenever the scalar observable index $X\in\mathbb{R}$
exceeds threshold $t$, a parameter that must be chosen. These threshold
policies are widespread: they regulate, beyond others, organ transplants
kamath2007model, taxation (taylor2003corporation), and access to social
welfare programs (card2008impact; crost2014aid). This paper aims not to
justify or advocate for the use of threshold policies, and the restriction to
this policy class is taken as exogenous.
I will focus on the case when the index $X$ is chosen before the experiment.
Population welfare depends only on threshold $t$, and can be written as
$\displaystyle
W(\pi)=W(t)=\mathbb{E}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}].$ (3)
Choosing the policy is equivalent to choosing the threshold. If the joint
distribution of $Y_{1}$, $Y_{0}$, and $X$ were known, the policy maker would
implement the policy with threshold $t^{*}$ defined as:
$\displaystyle t^{*}\in\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}]$ (4)
which would guarantee the maximum achievable welfare $W(t^{*})$.
The problem described in equation (4) is unfeasible since the joint
distribution of $Y_{1}$, $Y_{0}$, and $X$ is unknown. The policy maker
observes an experimental sample
$Z=\\{Z_{i}\\}_{i=1}^{n}=\\{Y_{i},D_{i},X_{i}\\}$, where $Y$ is the outcome of
interest, $D$ the randomly assigned treatment status, and $X$ the policy
index. Experimental data, which allows to identify the conditional average
treatment effect, are used to learn the threshold policy
$\hat{t}_{n}=\hat{t}_{n}(Z)$, function of the observed sample.
Statistical decision theory deals with the problem of choosing the map
$\hat{t}_{n}$. First, it is necessary to specify the decision problem the
policymaker faces. For any threshold policy $\hat{t}_{n}$, define the regret
$\mathcal{R}(\hat{t}_{n})$:
$\displaystyle\mathcal{R}(\hat{t}_{n})=W(t^{*})-W(\hat{t}_{n}),$ (5)
a measure of welfare loss indicating the suboptimality of policy
$\hat{t}_{n}$. The regret depends on the unknown data distribution: the
policymaker specifies a state space, and searches for a policy that does well
uniformly for all the data distributions in the state space. Following
manski2004statistical, statistical decision theory has mainly focused on the
problem of minimizing the maximum expected regret, looking for a policy
$\hat{t}_{n}$ that does uniformly well on average across repeated samples.
Directly solving the constrained minimization problem of the functional
$\sup\mathbb{E}[\mathcal{R}(\hat{t}_{n})]$ is impractical: the literature
instead focuses on considering a specific policy map and studying its
properties, for example showing its rate optimality, through finite sample
valid (kitagawa2018should) or asymptotic (athey2021policy) arguments.
Following this approach, I characterize and compare some properties for the
regret of two different threshold policies, the Empirical Welfare Maximizer
(EWM) policy, commonly studied in the literature, and the novel Smoothed
Welfare Maximizer (SWM) policy.
kitagawa2018should derive bounds for the expected regret of the EWM policy for
a wide range of policy function classes. In their results, the policy class
$\Pi$ affects the bounds only through its VC dimension, and the knowledge of
$\Pi$ is not further exploited. Conversely, I leverage the additional
structure from the knowledge of the policy class and characterize the entire
asymptotic distribution of the regret for the EWM and the SWM threshold
policies, comparing how their regrets depend on the data distribution. My
results could hence be of interest also when decision problems not involving
the expected regret are considered, as in manski2023statistical and
kitagawa2022treatment: I characterize the asymptotic behavior of regret
quantiles, and the asymptotic distributions can be used to simulate
expectations of their non-linear functions.
To derive my results, I take advantage of the link between a threshold policy
function $\hat{t}_{n}$ and its regret $\mathcal{R}(\hat{t}_{n})$. Let
$\\{r_{n}\\}$ be a sequence such that $r_{n}\rightarrow\infty$ for
$n\rightarrow\infty$, and suppose that $r_{n}(\hat{t}_{n}-t^{*})$ converges to
a non degenerate limiting distribution, i.e
$(\hat{t}_{n}-t^{*})=O_{p}(r_{n}^{-1})$.
Assume function $W(t)$ to be twice continuously differentiable, and consider
its second-order Taylor expansion around $t^{*}$:
$\displaystyle
W(\hat{t}_{n})=W(t^{*})+\underbrace{W^{\prime}(t^{*})}_{=0}\left(\hat{t}_{n}-t^{*}\right)+\frac{1}{2}W^{\prime\prime}(\tilde{t})\left(\hat{t}_{n}-t^{*}\right)^{2}$
where $|\tilde{t}-t^{*}|\leq|\hat{t}_{n}-t^{*}|$, and $W^{\prime}(t^{*})=0$ by
optimality of $t^{*}$. The previous equation can be written as
$\displaystyle
r_{n}^{2}\mathcal{R}(\hat{t}_{n})=\frac{1}{2}W^{\prime\prime}(\tilde{t})\left[r_{n}\left(\hat{t}_{n}-t^{*}\right)\right]^{2},$
(6)
establishing a relationship between the convergence rates of $\hat{t}_{n}$ and
$\mathcal{R}(\hat{t}_{n})$, and between their asymptotic distributions.
Equation (6) therefore shows how the rate of convergence and the asymptotic
distribution of regret $\mathcal{R}(\hat{t}_{n})$ can be studied through the
rate of convergence and the asymptotic distribution of policy $\hat{t}_{n}$.
In the next section, I consider the EWM policy $\hat{t}^{e}_{n}$ and the SWM
policy $\hat{t}^{s}_{n}$: through their asymptotic behaviors, I characterize
the asymptotic distributions of their regrets $\mathcal{R}(\hat{t}^{e}_{n})$
and $\mathcal{R}(\hat{t}^{s}_{n})$.
## 3 Formal Results
Let $Y_{0}$ and $Y_{1}$ be scalar potential outcomes, $D$ the binary treatment
assignment in the experiment, $\textbf{X}\in\mathbb{R}^{d}$ a vector of $d$
observable characteristics, and $X$ the observable index.
$\\{Y_{0},Y_{1},D,\textbf{X}\\}$ are random variables distributed according to
the distribution $P$. They satisfy the following assumptions, which guarantee
the identification of the optimal threshold:
###### Assumption 1.
(Identification)
1. 1.1
(No interference) Observed outcome $Y$ is related with potential outcomes by
the expression $Y=DY_{1}+(1-D)Y_{0}$.
2. 1.2
(Unconfoundedness) Distribution $P$ satisfies
$D\perp\\!\\!\\!\perp(Y_{0},Y_{1})|\textbf{X}$.
3. 1.3
(Overlap) Propensity score $p(\textbf{x})=\mathbb{E}[D|\textbf{X}=\textbf{x}]$
is assumed to be known and such that $p(\textbf{x})\in(\eta,1-\eta)$, for some
$\eta\in(0,0.5)$.
4. 1.4
(Joint distribution) Potential outcomes $(Y_{0},Y_{1})$ and index $X$ are
continuous random variables with joint probability density function
$\varphi(y_{0},y_{1},x)$, and marginal densities $\varphi_{0}$, $\varphi_{1}$,
and $f_{x}$ respectively. Expectations $\mathbb{E}[Y_{0}|x]$ and
$\mathbb{E}[Y_{1}|x]$, for $x$ in the support of $X$, exist.
Assumptions 1.1, 1.2, and 1.3 are standard assumptions in many causal models.
Assumption 1.1 requires the outcome of each unit to depend only on their
treatment status, excluding spillover effects. Assumption 1.2 requires random
assignment of the treatment, conditionally on X. Assumption 1.3 requires that,
for any value of X, there is a positive probability of observing both treated
and untreated units. Probabilities of being assigned to the treatment may vary
with X, allowing for stratified experiments.
Assumption 1.4 specifies the focus on continuous outcome and index. While it
would be possible to accommodate discrete $Y_{0}$ and $Y_{1}$, maintaining the
continuity of $X$ remains essential. The arguments developed in this paper, in
fact, are not valid for a discrete index: my focus is on studying optimal
threshold policies in contexts where the probability of observing any value on
the support of the index $X$ is zero, and the threshold must be chosen from a
continuum of possibilities.
Under Assumption 1, optimal policy $t^{*}$ defined in (4) can be written as
$\displaystyle t^{*}\in$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}_{P}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}]$ (7) $\displaystyle=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}_{P}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\mathbf{1}\\{X>t\\}\right]$
(8)
and is hence identified. This standard result specifies under which conditions
an experiment allows to identify $t^{*}$.
### 3.1 Empirical Welfare Maximizer Policy
Policymaker observes an i.i.d. random sample $Z=\\{Y_{i},D_{i},X_{i}\\}$ of
size $n$ from $P$, and considers the Empirical Welfare Maximizer policy
$\hat{t}^{e}_{n}$, the sample analog of $t^{*}$ in equation
(4)111Computationally, the problem requires to evaluate the function inside
the argmax $n+1$ times; the solution is the convex set of points in
$\mathbb{R}$ that give the maximum of these values.:
$\displaystyle\hat{t}^{e}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}\mathbf{1}\\{X_{i}>t\\}+\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\mathbf{1}\\{X_{i}\leq
t\\}\right).$ (9)
Policy $\hat{t}^{e}_{n}$ can be seen as an extremum estimator, maximizer of a
function not continuous in $t$.
#### 3.1.1 Consistency of $\hat{t}^{e}_{n}$
First, I will prove that $\hat{t}^{e}_{n}$ consistently estimates the optimal
threshold $t^{*}$, implying that
$\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{p}0$. To prove this result, I need
the following assumptions on the data distribution.
###### Assumption 2.
(Consistency)
1. 2.1
(Maximizer $t^{*}$) Maximizer $t^{*}\in\mathcal{T}$ of
$\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ exists and is unique. It is an
interior point of the compact parameter space
$\mathcal{T}\subseteq\mathbb{R}$.
2. 2.2
(Square integrability) Conditional expectations $\mathbb{E}[Y_{0}^{2}|X]$ and
$\mathbb{E}[Y_{1}^{2}|X]$ exist.
3. 2.3
(Smoothness) In a neighbourhood of $t^{*}$, density $f_{x}(x)$ is positive,
and function $\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ is at least
$s$-times continuously differentiable in $t$.
By requiring the existence of the optimal threshold in the interior of the
parameter space, Assumption 2.1 is assuming heterogeneity in the sign of the
conditional average treatment effect $\mathbb{E}[Y_{1}-Y_{0}|X]$. It is
because of this heterogeneity that the policymaker implements the threshold
policy, targeting groups that would benefit from being treated. The assumption
neither excludes the multiplicity of local maxima, as long as the global one
is unique, nor excludes unbounded support for $X$, but requires the parameter
space to be compact. A sufficient condition for Assumption 2.1, easy to
interpret and plausible in many applications, is that the conditional average
treatment effect has negative and positive values, and crosses zero exactly
once.
Assumption 2.2 requires that the conditional potential outcomes have finite
second moments and is satisfied when $Y$ is assumed to be bounded (as in
kitagawa2018should).
Assumption 2.3 will be used with increasing values of $s$ to prove different
results. To prove consistency, it needs to hold for $s=0$, requiring the
continuity of the objective function $W(t)$ in a neighborhood of $t^{*}$. The
derivative of $\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ with respect to
$t$ is equal to $f_{x}(t)\tau(t)$, where $\tau(x)=\mathbb{E}[Y_{1}-Y_{0}|X=x]$
is the conditional average treatment effect. Assumption 2.3 with $s\geq 1$
hence requires smoothness of $f_{x}(x)$ and $\tau(x)$, in a neighborhood of
$t^{*}$.
The following theorem proves the consistency of $\hat{t}^{e}_{n}$ for $t^{*}$.
###### Theorem 1.
Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1 and 2 with
$s=0$,
$\displaystyle\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$
i.e. $\hat{t}^{e}_{n}$ is a consistent estimator for $t^{*}$.
Note that, since $\hat{t}^{e}_{n}$ is not continuous in $t$, the proof of
Theorem 1 does not resort to the standard arguments for consistency of
extremum estimators. Conversely, it relies on the approach used to prove
consistency for estimators exhibiting the “cube root asymptotics” behavior,
discussed in the next section.
#### 3.1.2 Asymptotic Distribution for $\hat{t}^{e}_{n}$
The fact that $\hat{t}^{e}_{n}$ is not continuous in $t$ directly affects the
convergence rate and the asymptotic distribution. The EWM policy
$\hat{t}^{e}_{n}$ exhibits the “cube root asymptotics” behavior studied in
kim1990cube, the same as, beyond others, the maximum score estimator
(manski1975maximum), and the split point estimator in decision trees
(banerjee2007confidence).
The limiting distribution is not Gaussian, and its derivation requires an
additional regularity condition on the tails of the probability density
distribution of $Y_{1}$ and $Y_{0}$:
###### Assumption 3.
(Tail condition) Let $\varphi_{1}$ and $\varphi_{0}$ be the probability
density functions of $Y_{1}$ and $Y_{0}$. Assume that, as
$|y|\rightarrow\infty$, $\varphi_{1}(y)=o(|y|^{-(4+\delta)})$ and
$\varphi_{0}(y)=o(|y|^{-(4+\delta)})$, for $\delta>0$.
The following theorem gives the asymptotic distribution of $\hat{t}^{e}_{n}$.
###### Theorem 2.
Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=2$ and 3, as $n\rightarrow\infty$,
$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (10)
where $B(r)$ is the two-sided standard Brownian motion process, and $K$ and
$H$ are
$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$
The limiting distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$ is a
of Chernoff type (chernoff1964estimation). The Chernoff’s distribution is the
probability distribution of the random variable $\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}$, where $B(r)$ is the two-sided standard
Brownian motion process. The process $B(r)-r^{2}$ can be simulated, and the
distribution of $\mathop{\rm arg~{}max}\limits_{r}B(r)-r^{2}$ numerically
studied. groeneboom2001computing report values for selected quantiles.
It’s worth noticing how the variance of $\hat{t}^{e}_{n}$ depends on the data
distribution. $K$ and $H$ are functions of the density of $X$, the variance of
the Conditional Average Treatment Effect, and the derivative of the CATE at
$t^{*}$. The optimal threshold is estimated with more precision when more data
around the optimal threshold are available (larger density), when the
treatment effect changes more rapidly (larger derivative of CATE), and when
the outcomes have less variability (smaller variance of CATE). Exponents on
$K$ and $H$, determined by the cube root behavior, will be crucial for
comparing $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$.
Results in Theorem 2 can be used to derive asymptotic valid confidence
intervals for $\hat{t}^{e}_{n}$, as discussed in appendix A. More
interestingly, they can be combined with equation 6 to characterize the
asymptotic distribution of the regret $\mathcal{R}(\hat{t}^{e}_{n})$, as
derived in the following corollary.
###### Corollary 2.1.
The asymptotic distribution of regret $\mathcal{R}(\hat{t}^{e}_{n})$ is:
$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$
The expected value of the asymptotic distribution is
$K^{\frac{2}{3}}H^{-\frac{1}{3}}C^{e}$, where
$C^{e}=\sqrt[3]{2}\mathbb{E}\left[\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}\right]$
is a constant not dependent on $P$.
For the regret of the EWM policy, Corollary 2.1 establishes a
$n^{\frac{2}{3}}$ rate, faster than the $\sqrt{n}$ rate found to be the
optimal for the EWM expected regret (kitagawa2018should). It is essential to
highlight how the two results are different: the rate derived by
kitagawa2018should for the expected regret is valid uniformly over a family of
distributions that may violate the assumptions in Theorem 4, for example
including distributions where the CATE is discontinuous at the threshold. On
the other hand, Corollary 2.1 is derived for distributions satisfying
Assumptions 1, 2 and 3. They imply that $K$ is bounded and $H$ is bounded away
from zero: the state space can be further characterized setting an upper bound
$\overline{K}<\infty$ for $K$ and a lower bound $\underline{H}>0$ for $H$,
implying a maximum for the expectation of the asymptotic regret equal to
$\overline{K}^{\frac{2}{3}}\underline{H}^{-\frac{1}{3}}C^{e}$.
### 3.2 Smoothed Welfare Maximizer Policy
Corollary 2.1 shows how the cube root of $n$ convergence rate of the EWM
policy directly impacts the convergence rate of its regret. In this section, I
propose an alternative threshold policy, the Smoothed Welfare Maximizer
policy, that achieves a faster rate of convergence and hence guarantees a
faster rate of convergence for its regret. My approach exploits some
additional smoothness assumptions on the distribution $P$: Corollary 2.1 holds
when $f_{x}(x)$ and $\tau(x)$ are assumed to be at least once differentiable;
if they are at least twice differentiable, the SWM policy guarantees a
$n^{\frac{4}{5}}$ convergence rate for the regret. Note that asking the
density of the index and the conditional average treatment effect to be twice
differentiable seems plausible for many applications. In the context of policy
learning, it is, for example, assumed by athey2021policy to derive their
results222This assumption is implied by the high-level assumptions made in the
paper, as the authors discuss in footnote 15..
My approach involves smoothing the objective function in (9), in the same
spirit as the smoothed maximum score estimator proposed by
horowitz1992smoothed to deal with inference for the maximum score estimator
(manski1975maximum). The Smoothed Welfare Maximizer (SWM) policy
$\hat{t}^{s}_{n}$ is defined as:
$\displaystyle\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$
(11)
where $\sigma_{n}$ is a sequence of positive real numbers such that
$\lim_{n\rightarrow\infty}\sigma_{n}=0$, and the function $k(\cdot)$
satisfies:
###### Assumption 4.
(Kernel function) Kernel function $k(\cdot):\mathbb{R}\rightarrow\mathbb{R}$
is continuous, bounded, and with limits $\lim_{x\rightarrow-\infty}k(x)=0$ and
$\lim_{x\rightarrow\infty}k(x)=1$.
In practice, the indicator function found in $\hat{t}^{e}_{n}$ is here
substituted by a smooth function $k(\cdot)$ with the same limiting behavior,
which guarantees the differentiability of the expression in the argmax. The
bandwidth $\sigma_{n}$, decreasing with the sample size, ensures that when
$n\to\infty$ the policy converges to the optimal one, as proved in the next
section.
#### 3.2.1 Consistency of $\hat{t}^{s}_{n}$
I start showing consistency of $\hat{t}^{s}_{n}$ for $t^{*}$, which implies
$\mathcal{R}(\hat{t}^{s}_{n})\rightarrow^{p}0$.
###### Theorem 3.
Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=0$ and 4, as $n\rightarrow\infty$,
$\displaystyle\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$
i.e. $\hat{t}^{s}_{n}$ is a consistent estimator for $t^{*}$.
Theorems 3 and 1 are analogous: they rely on the same assumptions on the data
(Assumptions 1, 2) to prove the consistency of $\hat{t}^{e}_{n}$ and
$\hat{t}^{s}_{n}$. Where the two policies differ is in the asymptotic
distributions: smoothness in the objective function for $\hat{t}^{s}_{n}$
guarantees asymptotic normality, but also introduces a bias, since the
bandwidth $\sigma_{n}$ equals zero only in the limit, which pops up in the
limiting distribution.
#### 3.2.2 Asymptotic Distribution for $\hat{t}^{s}_{n}$
Deriving this asymptotic behavior of $\hat{t}^{s}_{n}$ requires an additional
assumption on the rate of bandwidth $\sigma_{n}$ and the kernel function $k$.
Since both are chosen by the policy maker, the assumption is not a restriction
on unobservables but a condition on properly picking $\sigma_{n}$ and $k$.
###### Assumption 5.
(Bandwidth and kernel)
1. 5.1
(Rate of $\sigma_{n}$) $\frac{\log n}{n\sigma_{n}^{4}}\rightarrow 0$ as
$n\rightarrow\infty$.
2. 5.2
(Kernel function) Kernel function $k(\cdot):\mathbb{R}\rightarrow\mathbb{R}$
satisfies Assumption 4 and the following:
* •
$k(\cdot)$ is continuous, bounded, and with limits
$\lim_{x\rightarrow-\infty}k(x)=0$ and $\lim_{x\rightarrow\infty}k(x)=1$.
* •
$k(\cdot)$ is twice differentiable, with uniformly bounded derivatives
$k^{\prime}$ and $k^{\prime\prime}$.
* •
$\int k^{\prime}(x)^{4}dx$, $\int k^{\prime\prime}(x)^{2}dx$, and
$\int|x^{2}k^{\prime\prime}(x)|dx$ are finite.
* •
For some integer $h\geq 2$ and each integer $i\in[1,h]$,
$\int|x^{i}k^{\prime}(x)|dx=0$ for
$i<h$ and $\int|x^{h}k^{\prime}(x)|dx=d\neq 0$, with $d$ finite.
* •
For any integer $i\in[0,h]$, any $\eta>0$, and any sequence
$\sigma_{n}\rightarrow 0$,
$\lim_{n\rightarrow\infty}\sigma_{n}^{i-h}\int_{|\sigma_{n}x|>\eta}|x^{i}k^{\prime}(x)|dx=0$,
and
$\lim_{n\rightarrow\infty}\sigma_{n}^{-1}\int_{|\sigma_{n}x|>\eta}|k^{\prime\prime}(x)|dx=0$.
* •
$\int xk^{\prime\prime}(x)dx=1$,
$\lim_{n\rightarrow\infty}\int_{|\sigma_{n}x|>\eta}|xk^{\prime\prime}(x)|dx=0$.
An example of a function $k$ satisfying Assumption 5.2 with $h=2$ is the
cumulative distribution function of the standard normal distribution.
I can now derive the asymptotic distribution of $\hat{t}^{s}_{n}$.
###### Theorem 4.
Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=h+1$ for some $h\geq 2$, and 5, as $n\rightarrow\infty$:
1. 1.
if $n\sigma_{n}^{2h+1}\rightarrow\infty$,
$\sigma_{n}^{-h}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{p}H^{-1}A;$
2. 2.
if $n\sigma_{n}^{2h+1}\rightarrow\lambda<\infty$,
$(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{d}\mathcal{N}(\lambda^{1/2}H^{-1}A,H^{-2}\alpha_{2}K);$
where $A$, $\alpha_{1}$, and $\alpha_{2}$ are:
$\displaystyle A=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{x}(y,t^{*})dy$
(12) $\displaystyle\alpha_{1}=$
$\displaystyle\int_{\zeta}\zeta^{h}k^{\prime}\left(\zeta\right)d\zeta$ (13)
$\displaystyle\alpha_{2}=$
$\displaystyle\int_{\zeta}k^{\prime}\left(\zeta\right)^{2}d\zeta.$ (14)
The asymptotic distribution of $(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})$ is
normal, centered at the asymptotic bias $\lambda^{1/2}H^{-1}A$ introduced by
the smoothing function, which exploits local information giving non-zero
weights to treated units in the untreated region and vice versa. The bias and
the variance of the distribution depend on the population distribution through
$K$ and $H$, as for the EWM policy, and also through $A$, a new term that
determines the bias. In the definition of $A$, $\varphi^{h}_{x}$ is the $h$
derivative with respect to $x$ of $\varphi(y_{0},y_{1},x)$, the joint density
distribution of $Y_{0}$, $Y_{1}$, and $X$: the integral in the expression for
$A$ is the expectation of the $h$-derivative of $f_{x}(X)\tau(X)$ computed in
$X=t^{*}$, whose existence is guaranteed by Assumption 2.3 with $s=h+1$.
$\alpha_{1}$ and $\alpha_{2}$ depends only on kernel function $k$, and are
hence known.
The difference between Theorems 4 and 2 is the degree of smoothness required
by Assumption 2.3. In Theorem 2, the objective function $W(t)$ is required to
be twice differentiable, while Theorem 4 requires $h+1$ continuous
derivatives, where $h$ directly impacts rate of convergence of the policy: to
achieve a rate faster than the EWM policy, $\hat{t}^{s}_{n}$ requires $W(t)$
to be three times differentiable.
As for the EWM policy, results in Theorem 4 can be used to derive asymptotic
valid confidence intervals for $\hat{t}^{s}_{n}$ (see appendix A), and,
combined with equation 6, to characterize the asymptotic distribution of the
regret $\mathcal{R}(\hat{t}^{s}_{n})$, as derived in the next corollary.
###### Corollary 4.1.
Asymptotic distribution of regret $\mathcal{R}(\hat{t}^{s}_{n})$ is:
$\displaystyle
n\sigma_{n}\mathcal{R}(\hat{t}^{s}_{n})\rightarrow^{d}\frac{1}{2}\frac{\alpha_{2}K}{H}\chi^{2}\left(1,\frac{\lambda
A^{2}}{\alpha_{2}K}\right)$
where $\chi^{2}\left(1,\frac{\lambda A^{2}}{\alpha_{2}K}\right)$ is a non-
centered chi-squared distribution with 1 degree of freedom and non-central
parameter $\frac{\lambda A^{2}}{\alpha_{2}K}$. The expected value of the
asymptotic distribution is:
$\displaystyle\frac{1}{2}\frac{\alpha_{2}K}{H}\left(1+\frac{\lambda
A^{2}}{\alpha_{2}K}\right)=\frac{\alpha_{2}}{2}\frac{K}{H}+\frac{1}{2}\frac{\lambda
A^{2}}{H}.$ (15)
Let $\sigma_{n}=(\lambda/n)^{1/(2h+1)}$ with $\lambda\in(0,\infty)$. The
expectation of the asymptotic regret is minimized by setting
$\lambda=\lambda^{*}=\frac{\alpha_{2}K}{2hA^{2}}$: in this case, the
expectation of the asymptotic distribution scaled by $n^{\frac{2h}{2h+1}}$ is
$A^{\frac{2}{2h+1}}K^{\frac{2h}{2h+1}}H^{-1}C^{s}$, where
$C^{s}=\frac{2h+1}{2}\left(\frac{\alpha_{2}}{2h}\right)^{\frac{2h}{2h+1}}$ is
a constant not dependent on $P$.
With the optimal bandwidth $\sigma_{n}=O_{p}(n^{-\frac{1}{2h+1}})$, the regret
converges at $n^{\frac{2h}{2h+1}}$ rate. For $h\geq 2$, this implies that the
regret converges faster with the SWM than with the EWM policy: the extra
smoothness assumption has been exploited to achieve a better rate for the
asymptotic regret. The corollary is valid for distributions satisfying
Assumptions 1, 2 and 3, which imply a bounded $A$: the state space can be
characterized setting $\overline{A}<\infty$ as the upper bound for $|A|$,
implying a maximum for the expectation of the asymptotic regret equal to
$\overline{A}^{\frac{2}{2h+1}}\overline{K}^{\frac{2h}{2h+1}}\underline{H}^{-1}C^{s}$.
### 3.3 Comparison of Regrets $\mathcal{R}(\hat{t}^{e}_{n})$ and
$\mathcal{R}(\hat{t}^{s}_{n})$
Corollaries 2.1 and 4.1 showed that the regret with the SWM policy has a
faster convergence rate than with the EWM policy. Assuming the accuracy of the
asymptotic approximations, the comparison can extend beyond the rates to
investigate how differently the asymptotic distributions depend on $P$,
through $H$, $K$, and $A$. Since it is more intuitive to compare expectations
than entire distributions, I will focus on this comparison, with the
understanding that similar reasoning also applies to alternative statistics of
the asymptotic distributions, such as medians or different quantiles.
The findings in the corollaries suggest that, for any sample size, one can
select distributions $P$ in a manner that makes each expectation of both the
asymptotic regrets smaller. Hence, both the EWM and the SWM can be the better
policy. If the asymptotic behavior is reflected in finite sample, this implies
that in a given application where $P$ is unknown it is impossible to determine
which policy would guarantee a smaller expected regret.
In statistical decision theory, this ambiguity is recognized: focusing on the
worst case scenario, the policy maker should choose the policy that guarantees
the smaller regret when the expectation of the asymptotic regret is maximized.
In this case, for any sample size, it means to compare
$n^{-\frac{2}{3}}\overline{K}^{\frac{2}{3}}\underline{H}^{-\frac{1}{3}}C^{e}$
and
$n^{-\frac{2h}{2h+1}}\overline{A}^{\frac{2}{2h+1}}\overline{K}^{\frac{2h}{2h+1}}\underline{H}^{-1}C^{s}$,
and choose the policy accordingly. Which policy is uniformly better does hence
depends on the application, and specifically on values of parameters
$\overline{K}$, $\underline{H}$, and $\overline{A}$, which the policy maker
must set in advance.
It is interesting to investigate for which distributions $P$ one policy is
expected to do relatively better than the other, in a pointwise sense. It’s
clear that the SWM policy does relatively better with increasing sample size
due to its faster convergence rate, for any $P$. Conversely, for any sample
size, when $h=2$, the ratio of the expectations of asymptotic regrets is
$\frac{H^{\frac{2}{3}}}{A^{\frac{2}{5}}K^{\frac{2}{15}}}\frac{C^{e}}{C^{s}}$:
the SWM policy does relatively better for larger $H$ and smaller $K$ and $A$.
The negative impact of $A$ is straightforward, since it only affects the bias
for the SWM policy, without influencing the distribution of the EWM. The role
of $H$ and $K$ is less intuitive, since they appear in both distributions but
with different exponents. It’s worth noting that the SWM policy holds a
relative advantage when the derivative of the CATE is higher, and its variance
is lower- scenarios where the benefits of selecting a threshold closer to the
optimal are larger. Further comparisons of the regrets could explore the
interplay between sample size and the distribution $P$, focusing on sequences
$P_{n}$ that change with sample size. Appendix B delves into this
investigation.
It is important to acknowledge that the relevance of these discussions on
asymptotic results relies on their ability to approximate behaviors in finite
samples, as, after all, the policymaker has only access to a finite sample of
experimental data. Guided by the theoretical results just derived, Monte Carlo
simulations in the next section allow to analyze the finite sample regrets
associated with the EWM and the SWM policies.
## 4 Monte Carlo Simulations
I examine the finite sample properties of the EWM and SWM policies using Monte
Carlo simulations. The scope of this section is twofold: first, I will provide
examples of data generating processes which lead different rankings for the
two policies in terms of asymptotic regret. Then, I will verify how the
asymptotic results approximate the finite sample distributions, and compare
the finite sample regrets of the two policies.
As data generating process, consider the following distribution $P$ of
$(Y_{0},Y_{1},D,X)$:
$\displaystyle X$ $\displaystyle\sim\mathcal{N}(0,1)$
$\displaystyle\epsilon_{1}$ $\displaystyle\sim\mathcal{N}(0,\gamma)$
$\displaystyle Y_{1}$
$\displaystyle=X^{3}+\beta_{2}X^{2}+\beta_{1}X+\epsilon_{1}$ $\displaystyle
Y_{0}$ $\displaystyle\sim\mathcal{N}(0,\gamma)$ $\displaystyle D$
$\displaystyle\sim\text{Bern}(p).$
Under $P$, the potential outcome $Y_{0}$ does not depend on the index $X$, and
the treatment is randomly assigned with constant probability $p$. Parameter
values are chosen such that $\mathbb{E}[Y_{1}|X=x]$ and hence
$\mathbb{E}[Y_{1}-Y_{0}|X=x]$ are increasing function of $x$, and the optimal
threshold $t^{*}$ is 0. It can be verified that such $P$ implies the
following:
$\displaystyle K=$
$\displaystyle\phi(0)\left(\frac{\gamma^{2}}{p}+\frac{\gamma^{2}}{1-p}\right)$
$\displaystyle H=$ $\displaystyle\phi(0)\beta_{1}$ $\displaystyle A=$
$\displaystyle-\phi(0)\beta_{2}$ $\displaystyle W(t)=$
$\displaystyle\beta_{2}\left(1-\Phi(t)+t\phi(t)\right)+\beta_{1}\phi(t)+\left(t^{2}\phi(t)+2\phi(t)\right)$
where $\phi(t)$ and $\Phi(t)$ are the pdf and the cdf of the standard normal
distribution, respectively.
I consider two models characterized by different parameter values:
Model | $\gamma$ | $\beta_{1}$ | $\beta_{2}$ | $p$
---|---|---|---|---
1 | 1 | 1 | -0.5 | 0.5
2 | 3 | 0.5 | -1 | 0.5
Table 1 reports values for $K$, $H$, and $A$, and asymptotic expected regrets
for both policies for sample of size $n\in\\{500,1000,2000,3000\\}$. For the
SWM policy, asymptotic regrets are computed using the infeasible optimal
bandwidth $\sigma_{n}^{*}$. Compared to Model 1, Model 2 entails larger $K$
and $A$, and smaller $H$. Results from the previous section hence imply the
regret with the EWM policy being relatively lower in Model 2. Consider the
case with $n=500$. In model 1, the asymptotic expected regret is higher with
the EWM policy, whereas in model 2, it’s higher with the SWM policy: this
confirms that the ranking of asymptotic expected regrets depends on the
unknown data distribution $P$. Because of the fastest rate, though, as $n$
increases the SWM policy exhibits relatively better performance. Regardless of
the specific distribution $P$, there exists a certain sample size beyond which
the asymptotic expected regret with the SWM policy becomes smaller. In model
2, when $n=1,000$, the inversion of ranking already occurs.
Table 1: Values of $K$, $H$, and $A$, and asymptotic expected regret using both EWM and SWM policies across different models, are presented. The regret for the SWM policy is computed using the optimal bandwidth $\sigma_{n}^{*}$. To facilitate the reading, asymptotic expected regrets have been scaled by a factor of 10,000. Model | n | EWM | SWM | K | H | A
---|---|---|---|---|---|---
1 | $500$ | $96.190$ | $39.714$ | $1.596$ | $0.399$ | $0.199$
$1,000$ | $60.596$ | $22.809$ | $1.596$ | $0.399$ | $0.199$
$2,000$ | $38.173$ | $13.101$ | $1.596$ | $0.399$ | $0.199$
$3,000$ | $29.131$ | $9.471$ | $1.596$ | $0.399$ | $0.199$
2 | $500$ | $400.166$ | $439.442$ | $9.575$ | $0.199$ | $0.399$
$1,000$ | $252.089$ | $252.393$ | $9.575$ | $0.199$ | $0.399$
$2,000$ | $158.806$ | $144.962$ | $9.575$ | $0.199$ | $0.399$
$3,000$ | $121.192$ | $104.805$ | $9.575$ | $0.199$ | $0.399$
To investigate the finite sample distributions of regret, I draw samples of
size $n$ from $P$ 5,000 times for each model. Each sample is used to estimate
the thresholds $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$. Estimating
$\hat{t}^{s}_{n}$ requires specifying a bandwidth $\sigma_{n}$, for which I
adopt the following method: I use the estimated policy $\hat{t}^{e}_{n}$ to
compute $\hat{A}_{n}$ and $\hat{K}_{n}$, which are then used to compute the
optimal $\hat{\lambda}_{n}^{*}$, and the optimal bandwidth
$\hat{\sigma}_{n}^{*}$. The SWM policy $\hat{t}^{s}_{n}$ is hence estimated
with this bandwidth $\hat{\sigma}_{n}^{*}$, which consistently estimates the
optimal one if $\hat{A}_{n}$ and $\hat{K}_{n}$ consistently estimate $A$ and
$K$. I also consider $\hat{t}^{s}_{n}$ with the infeasible optimal
$\sigma_{n}^{*}$ computed from the data generating process.
Estimates for $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$ are used to compute
regrets $\mathcal{R}(\hat{t}^{e}_{n})$ and $\mathcal{R}(\hat{t}^{s}_{n})$. I
thus obtain the finite sample distributions of the regret, which can be
compared with the asymptotic distributions derived in Corollaries 2.1 and 4.1.
Table 2 presents the mean of these finite sample and asymptotic distributions,
also depicted in Figures 1 and 2. Corresponding tables and figures for the
median regret are provided in Appendix C.
The last column of each table reports the ratio between the finite sample mean
regrets for the EWM and SWM policy, facilitating the comparison: a ratio
larger than one indicates that the SWM policy outperforms the EWM policy.
These ratios increase with the sample size, reflecting the faster asymptotic
convergence rate of the SWM policy. Similar to the asymptotic results, in
finite sample the SWM policy does relatively better as the sample size
increases.
Table 2: Finite sample and asymptotic expected regrets with the EWM and SWM policies across different models are presented. Finite sample (empirical) values are computed through 5,000 Monte Carlo simulations. The last column displays the ratio between finite sample expected regrets with the EWM and SWM policies. Model | n | EWM | SWM | Ratio
---|---|---|---|---
| | empirical | asymptotic | empirical $\sigma_{n}^{*}$ | empirical $\hat{\sigma}_{n}^{*}$ | asymptotic |
1 | $500$ | $89.635$ | $96.190$ | $31.492$ | $77.747$ | $39.714$ | $1.153$
$1,000$ | $58.799$ | $60.596$ | $18.248$ | $44.240$ | $22.809$ | $1.329$
$2,000$ | $37.297$ | $38.173$ | $10.887$ | $29.176$ | $13.101$ | $1.278$
$3,000$ | $28.615$ | $29.131$ | $7.872$ | $16.189$ | $9.471$ | $1.768$
2 | $500$ | $276.867$ | $400.166$ | $215.876$ | $291.654$ | $439.442$ | $0.949$
$1,000$ | $201.182$ | $252.089$ | $151.181$ | $209.069$ | $252.393$ | $0.962$
$2,000$ | $149.572$ | $158.806$ | $107.733$ | $151.069$ | $144.962$ | $0.990$
$3,000$ | $124.168$ | $121.192$ | $90.251$ | $125.565$ | $104.805$ | $0.989$
Figure 1: The figure illustrates asymptotic and finite sample expected regrets
for the EWM and SWM policies in Model 1, corresponding to the values reported
in Table 2.
Figure 2: The figure illustrates asymptotic and finite sample expected regrets
for the EWM and SWM policies in Model 2, corresponding to the values reported
in Table 2.
Simulations enable comparison between finite sample regrets and their
asymptotic counterparts. Across all models and sample sizes, the asymptotic
approximation for the feasible SWM policy (with the estimated bandwidth
$\hat{\sigma}_{n}^{*}$) is relatively less accurate. Simulations suggest that
this is partly attributable to the need for estimating an additional tuning
parameter, the bandwidth $\sigma_{n}$. When the SWM policy is estimated using
the infeasible optimal bandwidth $\sigma^{*}_{n}$, in fact, the asymptotic
approximation is more accurate and the regret is smaller.
In Model 1, as illustrated in Figure 1, both the finite sample and the
asymptotic expected regrets are lower for the SWM policy. Conversely, in Model
2 (illustrated in Figure 2), the finite sample expected regret is lower with
the EWM policy. This confirms the impossibility of ranking the policies in a
pointwise sense: different distributions $P$ result in different rankings for
the finite sample expected regrets.
It is important to note that the ranking of the EWM and the SWM policies
indicated by the asymptotic results may differ from the actual finite sample
comparison. Consider, for example, Model 2 with $n=2,000$: despite the
asymptotic analysis suggesting a smaller expected regret with the SWM policy,
the EWM actually guarantees a smaller regret. In this scenario, even if $P$
were known in advance, choosing according to the asymptotic approximation
would not have been optimal. As $n$ increases, the approximation improves, and
the rankings based on asymptotic analysis and finite sample comparisons
coincide.
Monte Carlo simulations have confirmed that the asymptotic results can
approximate some finite sample behavior of the regrets, highlighting some
caveats to consider when applying conclusions from asymptotic analysis to
finite sample regrets with the EWM and SWM policies. However, they have not
yet provided insight into the practical significance of the differences
between the two policies, whether these differences are relevant or negligible
in real-world scenarios. An empirical illustration is useful to answer these
questions, illustrating the different implications that the policies may have.
## 5 Empirical Illustration
I consider the same empirical setting as kitagawa2018should: experimental data
from the National Job Training Partnership Act (JTPA) Study. bloom1997benefits
describes the experiment in detail. The study randomized whether applicants
would be eligible to receive a mix of training, job-search assistance, and
other services provided by the JTPA for a period of 18 months. Background
information on the applicants was collected before treatment assignment,
alongside administrative and survey data on the applicants’ earnings over the
subsequent 30 months.
I consider the same sample of 9,223 observations as in kitagawa2018should. The
outcome variable $Y$ represents the total individual earnings during the 30
months following program assignment, and the treatment variable $D$ is a
binary indicator denoting whether the individual was assigned to the program
(intention-to-treat). The threshold policy is implemented by considering the
individual’s earnings in the year preceding the assignment as the index $X$.
Treatment is exclusively assigned to workers with prior earnings below the
threshold, based on the expectation that program services yield a more
substantial positive effect for individuals who previously experienced lower
earnings. Experimental data are employed to determine the threshold beyond
which the treatment, on average, harms the recipients.
The bandwidth to estimate the SWM policy is chosen as in the Monte Carlo
simulation, using the EWM policy $\hat{t}^{e}_{n}$ to compute $\hat{A}_{n}$,
$\hat{K}_{n}$, the optimal $\hat{\lambda}_{n}^{*}$, and then the optimal
bandwidth $\hat{\sigma}_{n}^{*}$. Table 3 reports the threshold estimates,
including the confidence intervals constructed as discussed in Appendix A. The
threshold with the SWM policy is 700 dollars lower than with the EWM (5,924 vs
6,614 dollars), a drop of $10.5\%$. The lower threshold implies that the
treatment would target fewer workers: if the EWM policy were implemented,
$82.9\%$ of the workers in the sample would receive the program services,
compared to the $78.9\%$ with the SWM policy, resulting in a $4$ percentage
point decrease.
Table 3: Summary of the Empirical Welfare Maximizer (EWM) and Smoothed Welfare Maximizer (SWM) policies. | EWM | SWM
---|---|---
Optimal threshold | 6614 | 5924
Confidence interval | (4880.6, 8347.4) | (5832.7, 6411.3)
Bootstrapped confidence interval | (4748.5, 8060) |
Expected asymptotic $\mathcal{R}$ | 41.299 | 5.296
Median asymptotic $\mathcal{R}$ | 19.47 | 2.597
% of workers treated | 82.912 | 78.911
Results in corollaries 2.1 and 4.1 allow to estimate the expected and the
median regret with the two policies: the asymptotic approximation suggests
that the average regret drops from $42$ to $5.5$ dollars per worker, when
comparing the EWM and SWM. In this context, the SWM would guarantee an average
gain of $37.6$ dollars per worker over the 30-month period under study,
equating to $909$ dollars on average for the workers who would change their
treatment assignment under the two policies.
The numbers in the table should be considered with care, and clearly, the
intention of this empirical illustration was not to advocate for a specific
new job-training policy. Rather, the application aimed to assess if the EWM
and SWM policies may have implications with relevant economic differences.
Results suggest this is the case: together with theoretical and simulation
findings, this implies that the choice between the EWM and the SWM policy
should be thoughtfully considered, as it may determine relevant improvement in
population welfare.
## 6 Conclusion
In this paper, I addressed the problem of using experimental data to estimate
optimal threshold policies when the policymaker seeks to minimize the regret
associated with implementing the policy in the population. I first examined
the Empirical Welfare Maximizer threshold policy, deriving its asymptotic
distribution, and showing how it links to the asymptotic distribution of its
regret. I then introduced the Smoothed Welfare Maximizer policy, replacing the
indicator function in the EWM policy with a smooth kernel function. Under the
assumptions commonly made in the policy learning literature, the convergence
rate for the worst-case regret of the SWM is faster than with the EWM policy.
A comparative analysis of the asymptotic distributions of the two policies was
conducted, to investigate how differently their regrets depend on the data
distribution $P$. Monte Carlo simulations corroborated the asymptotic finding
that the SWM policy may perform better than the commonly studied EWM policy
also in finite sample. An empirical illustration displayed that the
implications of the two policies can remarkably differ in real-world
application.
Three sets of problems remain open for future research, to extend the results
of this paper in diverse directions. While this study compared the EWM policy
with its smoothed counterpart SWM, the literature in statistical decision
theory has also examined alternative policy functions. Consider, for example,
the Augmented Inverse Propensity Weighted policy proposed by athey2021policy:
what is the asymptotic distribution of the AIPW policy’s regret in the context
of threshold policies? How differently does it depend on $P$? Is it possible,
analogously to what I did in this paper for the EWM policy, to modify the AIPW
policy by smoothing the indicator function in its definition?
Second, it would be interesting to extend the smoothing approach from the EWM
policy to other policy classes. Threshold policies are convenient as they
depend on only one parameter. Still, besides more convoluted derivations, the
same intuition of smoothing the indicator function also seems valid for the
linear index or the multiple indices policy. The questions to explore include
adapting the theory developed in this paper to these policy classes and
whether this approach could be generalized even to all cases where the EWM
policy is applied.
Lastly, the framework developed in this paper for using experimental data to
estimate optimal policies could inform experimental design. While the existing
literature mainly focuses on optimal design for estimating the average
treatment effect, it could be valuable to consider scenarios when estimating
the threshold policy is the goal: how should the experimental design be
adapted? How the allocation of units to treatment and control groups would
change? The results presented in this paper, elucidating the connection
between the distribution $P$ and the regret of the policy, provide a natural
foundation for exploring experimental designs optimal for threshold policy
estimation.
## References
* Abrevaya and Huang (2005) Abrevaya, J. and J. Huang (2005). On the bootstrap of the maximum score estimator. Econometrica 73(4), 1175–1204.
* Aiken et al. (2022) Aiken, E., S. Bellue, D. Karlan, C. Udry, and J. E. Blumenstock (2022). Machine learning and phone data can improve targeting of humanitarian aid. Nature 603(7903), 864–870.
* Amemiya (1985) Amemiya, T. (1985). Advanced econometrics. Harvard university press.
* Athey and Wager (2021) Athey, S. and S. Wager (2021). Policy learning with observational data. Econometrica 89(1), 133–161.
* Banerjee and McKeague (2007) Banerjee, M. and I. W. McKeague (2007). Confidence sets for split points in decision trees. The Annals of Statistics 35(2), 543–574.
* Banerjee and Wellner (2001) Banerjee, M. and J. A. Wellner (2001). Likelihood ratio tests for monotone functions. Annals of Statistics, 1699–1731.
* Bloom et al. (1997) Bloom, H. S., L. L. Orr, S. H. Bell, G. Cave, F. Doolittle, W. Lin, and J. M. Bos (1997). The benefits and costs of jtpa title ii-a programs: Key findings from the national job training partnership act study. Journal of human resources, 549–576.
* Card et al. (2008) Card, D., C. Dobkin, and N. Maestas (2008). The impact of nearly universal insurance coverage on health care utilization: evidence from medicare. American Economic Review 98(5), 2242–2258.
* Cattaneo et al. (2020) Cattaneo, M. D., M. Jansson, and K. Nagasawa (2020). Bootstrap-based inference for cube root asymptotics. Econometrica 88(5), 2203–2219.
* Chernoff (1964) Chernoff, H. (1964). Estimation of the mode. Annals of the Institute of Statistical Mathematics 16(1), 31–41.
* Crost et al. (2014) Crost, B., J. Felter, and P. Johnston (2014). Aid under fire: Development projects and civil conflict. American Economic Review 104(6), 1833–1856.
* Groeneboom and Wellner (2001) Groeneboom, P. and J. A. Wellner (2001). Computing chernoff’s distribution. Journal of Computational and Graphical Statistics 10(2), 388–400.
* Haushofer et al. (2022) Haushofer, J., P. Niehaus, C. Paramo, E. Miguel, and M. W. Walker (2022). Targeting impact versus deprivation. Technical report, National Bureau of Economic Research.
* Hirano and Porter (2009) Hirano, K. and J. R. Porter (2009). Asymptotics for statistical treatment rules. Econometrica 77(5), 1683–1701.
* Horowitz (1992) Horowitz, J. L. (1992). A smoothed maximum score estimator for the binary response model. Econometrica: journal of the Econometric Society, 505–531.
* Hussam et al. (2022) Hussam, R., N. Rigol, and B. N. Roth (2022). Targeting high ability entrepreneurs using community information: Mechanism design in the field. American Economic Review 112(3), 861–98.
* Kamath and Kim (2007) Kamath, P. S. and W. R. Kim (2007). The model for end-stage liver disease (meld). Hepatology 45(3), 797–805.
* Kim and Pollard (1990) Kim, J. and D. Pollard (1990). Cube root asymptotics. The Annals of Statistics, 191–219.
* Kitagawa et al. (2022) Kitagawa, T., S. Lee, and C. Qiu (2022). Treatment choice with nonlinear regret. arXiv preprint arXiv:2205.08586.
* Kitagawa and Tetenov (2018) Kitagawa, T. and A. Tetenov (2018). Who should be treated? empirical welfare maximization methods for treatment choice. Econometrica 86(2), 591–616.
* Leboeuf et al. (2020) Leboeuf, J.-S., F. LeBlanc, and M. Marchand (2020). Decision trees as partitioning machines to characterize their generalization properties. Advances in Neural Information Processing Systems 33, 18135–18145.
* Léger and MacGibbon (2006) Léger, C. and B. MacGibbon (2006). On the bootstrap in cube root asymptotics. Canadian Journal of Statistics 34(1), 29–44.
* Manski (1975) Manski, C. F. (1975). Maximum score estimation of the stochastic utility model of choice. Journal of econometrics 3(3), 205–228.
* Manski (2004) Manski, C. F. (2004). Statistical treatment rules for heterogeneous populations. Econometrica 72(4), 1221–1246.
* Manski (2021) Manski, C. F. (2021). Econometrics for decision making: Building foundations sketched by haavelmo and wald. Econometrica 89(6), 2827–2853.
* Manski (2023) Manski, C. F. (2023). Probabilistic prediction for binary treatment choice: with focus on personalized medicine. Journal of Econometrics 234(2), 647–663.
* Manski and Tetenov (2023) Manski, C. F. and A. Tetenov (2023). Statistical decision theory respecting stochastic dominance. The Japanese Economic Review 74(4), 447–469.
* Mbakop and Tabord-Meehan (2021) Mbakop, E. and M. Tabord-Meehan (2021). Model selection for treatment choice: Penalized welfare maximization. Econometrica 89(2), 825–848.
* Mohammadi et al. (2005) Mohammadi, L., S. Van De Geer, and J. Shawe-Taylor (2005). Asymptotics in empirical risk minimization. Journal of Machine Learning Research 6(12).
* Rai (2018) Rai, Y. (2018). Statistical inference for treatment assignment policies. Unpublished Manuscript.
* Shigeoka (2014) Shigeoka, H. (2014). The effect of patient cost sharing on utilization, health, and risk protection. American Economic Review 104(7), 2152–2184.
* Stoye (2012) Stoye, J. (2012). Minimax regret treatment choice with covariates or with limited validity of experiments. Journal of Econometrics 166(1), 138–156.
* Sun et al. (2021) Sun, H., E. Munro, G. Kalashnov, S. Du, and S. Wager (2021). Treatment allocation under uncertain costs. arXiv preprint arXiv:2103.11066.
* Sun (2021) Sun, L. (2021). Empirical welfare maximization with constraints. arXiv preprint arXiv:2103.15298.
* Taylor (2003) Taylor, J. (2003). Corporation income tax brackets and rates, 1909-2002. Statistics of Income. SOI Bulletin 23(2), 284–291.
* Viviano and Bradic (2023) Viviano, D. and J. Bradic (2023). Fair policy targeting. Journal of the American Statistical Association, 1–14.
## Appendix A Confidence Intervals for Threshold Policies
Results derived in Section 3 can be used to construct confidence intervals
that asymptotically cover the optimal threshold policy with a given
probability, and to conduct hypotheses tests. It is important to remark that,
in a decision problem setting, hypotheses testing does not have a clearly
motivated justification, and indeed, statistical decision theory is the
alternative approach to deal with decisions under uncertainty, as pointed out
in manski2021econometrics. Rather than advocating for confidence intervals and
hypothesis tests for threshold policies, this appendix aims to provide a
procedure agnostic on why one may be interested in it.
For the EWM policy, rai2018statistical proposes some confidence intervals
uniformly valid for several policy classes. They rely on test inversion of a
certain bootstrap procedure, which compares the welfare generated by all the
policies in the class. My procedure is much simpler for the EWM threshold
policies, and I directly construct confidence intervals from the asymptotic
distributions derived in Theorem 2. An analogous approach, built over results
in Theorem 4, is then used to construct confidence intervals for the SWM
policy.
### A.1 Empirical Welfare Maximizer Policy
Consider the asymptotic distribution for the EWM threshold policy derived in
Theorem 2:
$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right).$ (16)
If $H$ and $K$ were known, confidence intervals for the optimal policy $t^{*}$
with asymptotic coverage $1-\alpha$ could be constructed as
$(\hat{t}^{e}_{n}-w^{e}_{n},\hat{t}^{e}_{n}+w^{e}_{n})$, where
$\displaystyle w^{e}_{n}=n^{-1/3}(2\sqrt{K}/H)^{2/3}c_{\alpha/2}$ (17)
and $c_{\alpha/2}$ is the critical value, the upper $\alpha/2$ quantile of the
distribution of $\max_{r}B(r)-r^{2}$.
In practice, $H$ and $K$ are unknown and should be estimated. They are defined
as:
$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$
and can be estimated by a plug-in method: consider kernel density estimator
$\hat{f}_{x}(x)$ for $f_{x}(x)$, and local linear estimators
$\hat{\kappa}_{j}(x)$ and $\hat{\nu}_{j}^{\prime}(x)$ for
$\kappa_{j}(x)=\mathbb{E}\left[Y_{j}^{2}|X=x,D=j\right]$ and
$\nu_{j}^{\prime}(x)=\frac{\partial\nu_{j}(x)}{\partial
x}=\frac{\partial\mathbb{E}\left[Y_{j}|X=x,D=j\right]}{\partial x}$. Define
estimators $\hat{K}_{n}$ and $\hat{H}_{n}$ by:
$\displaystyle\hat{K}_{n}=\hat{f}_{x}(\hat{t}^{e}_{n})\left(\frac{1}{p(\textbf{X})}\hat{\kappa}_{1}(\hat{t}^{e}_{n})+\frac{1}{1-p(\textbf{X})}\hat{\kappa}_{0}(\hat{t}^{e}_{n})\right)$
(18)
and
$\displaystyle\hat{H}_{n}=\hat{f}_{x}(\hat{t}^{e}_{n})(\hat{\nu}_{1}^{\prime}(\hat{t}^{e}_{n})-\hat{\nu}_{0}^{\prime}(\hat{t}^{e}_{n})).$
(19)
Under the additional assumption that the second derivatives of $f_{x}$,
$\nu_{1}$ and $\nu_{0}$ are continuous and bounded in a neighborhood of
$t^{*}$, and with the proper choice of bandwidth sequences, $\hat{K}_{n}$ and
$\hat{H}_{n}$ are consistent estimators for $K$ and $H$.
Feasible confidence intervals with asymptotic coverage $1-\alpha$ can hence be
constructed as
$(\hat{t}^{e}_{n}-\hat{w}^{e}_{n},\hat{t}^{e}_{n}+\hat{w}^{e}_{n})$, where
$\displaystyle\hat{w}^{e}_{n}=n^{-1/3}(2\sqrt{\hat{K}_{n}}/\hat{H}_{n})^{2/3}c_{\alpha/2}.$
(20)
#### A.1.1 Bootstrap
To avoid relying on tabulated values for $c_{\alpha/2}$ and on estimation of
$K$, an alternative approach to inference for the EWM policy is the bootstrap.
Nonparametric bootstrap is not valid for $\hat{t}^{e}_{n}$ and, more
generally, for “cube root asymptotics” estimators (abrevaya2005bootstrap;
leger2006bootstrap). Nonetheless, cattaneo2020bootstrap provide a consistent
bootstrap procedure for estimators of this type. Consistency is achieved by
altering the shape of the criterion function defining the estimator whose
distribution must be approximated. The standard nonparametric bootstrap is
inconsistent for $Q_{0}(r)=-\frac{1}{2}Hr$ as defined in the proof of Theorem
2, and hence the procedure in cattaneo2020bootstrap directly estimates this
non-random part.
Let $\\{Z_{i}^{b}\\}$ be a random sample from the empirical distribution
$P_{n}$, and define the estimator $\hat{t}^{b}_{n}$ as:
$\displaystyle\hat{t}^{b}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}^{b}Y_{i}^{b}}{p(\textbf{X}_{i}^{b})}-\frac{(1-D_{i}^{b})Y_{i}^{b}}{(1-p(\textbf{X}_{i}^{b}))}\right)\mathbf{1}\\{X_{i}^{b}>t\\}\right]-$
(21)
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\mathbf{1}\\{X_{i}>t\\}\right]-\frac{1}{2}(t-\hat{t}_{n})^{2}\hat{H}_{n}.$
(22)
The bootstrap procedure proposed by cattaneo2020bootstrap is the following:
1. 1.
Compute $\hat{t}^{e}_{n}$ as described in equation (9).
2. 2.
Using $\hat{t}^{e}_{n}$, compute $\hat{H}_{n}$ as described in equation (19).
3. 3.
Using $\hat{t}^{e}_{n}$, $\hat{H}_{n}$, and the bootstrap sample
$\\{Z_{i}^{b}\\}$, compute $\hat{t}^{b}_{n}$ as described in equation (21).
4. 4.
Iterate step 3 to obtain the distribution of
$n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$, and use it as an
estimate for the distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$.
To be valid, the procedure needs an additional assumption.
###### Assumption 6.
(Bounded 4th moment) Potential outcomes distribution are such that
$\frac{1}{n^{2/3}}\mathbb{E}[Y_{1}^{4}|X=t^{*}]=o(1)$ and
$\frac{1}{n^{2/3}}\mathbb{E}[Y_{0}^{4}|X=t^{*}]=o(1)$.
Assumption 6 guarantees that the envelope $G_{R}$ is such that
$PG_{R}^{4}=o(R^{-1})$. Theorem 5 proves that the distribution of
$n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$ consistently estimates
the distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$, and validate
the bootstrap procedure.
###### Theorem 5.
Consider estimators $\hat{t}^{e}_{n}$ defined in equation (9) and
$\hat{t}^{b}_{n}$ defined in equation (21) and the estimand $t^{*}$ defined in
equation (4). Under Assumptions 1, 2 with $s=2$, 3, and 6, as
$\hat{H}_{n}\rightarrow^{p}H$ and $n\rightarrow\infty$,
$\displaystyle
n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}_{n}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (23)
where the limiting distribution is the same as in Theorem 2.
Distribution of $n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$ can
hence be used to construct asymptotic valid confidence intervals and run
hypothesis tests for $\hat{t}^{e}_{n}$.
### A.2 Smoothed Welfare Maximizer Policy
Consider the asymptotic distribution for the SWM threshold policy derived in
Theorem 4, for $n\sigma_{n}^{2h+1}\rightarrow\lambda<\infty$:
$\displaystyle(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{d}\mathcal{N}(\lambda^{1/2}H^{-1}A,H^{-2}\alpha_{2}K).$
(24)
$\lambda$, $\sigma_{n}$, and $\alpha_{2}$ are known. If also $K$, $H$, and $A$
were known, confidence intervals for the optimal policy $t^{*}$ with
asymptotic coverage $1-\alpha$ could be constructed as
$(\hat{t}^{s}_{n}-b_{n}-w^{s}_{n},\hat{t}^{s}_{n}-b_{n}+w^{s}_{n})$, where
$\displaystyle b_{n}=(n\sigma_{n})^{-1/2}\lambda^{1/2}\frac{A}{H}$ (25)
$\displaystyle
w^{s}_{n}=(n\sigma_{n})^{-1/2}(\sqrt{\alpha_{2}K}/H)c_{\alpha/2}$ (26)
and $c_{\alpha/2}$ the upper $\alpha/2$ quantile of the standard normal
distribution.
In practice, $K$, $H$, and $A$ are unknown and should be estimated. As usual
with inference involving bandwidths and kernels, two approaches are available:
estimate and remove the asymptotic bias, or undersmooth.
For the first approach, consider estimators in equation (18) for $\hat{K}_{n}$
and in equation (19) for $\hat{H}_{n}$, substituting $\hat{t}^{e}_{n}$ with
$\hat{t}^{s}_{n}$. For $A$, recall that
$\displaystyle A=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{x}(y,t^{*})dy$
(27) $\displaystyle=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\left[2f^{\prime}_{x}(t^{*})[\lambda^{\prime}_{1}(t^{*})-\lambda^{\prime}_{0}(t^{*})]+f_{x}(t^{*})[\lambda^{\prime\prime}_{1}(t^{*})-\lambda^{\prime\prime}_{0}(t^{*})]\right]$
(28)
where $f_{x}(x)$ is the probability density function of $X$ and
$\nu_{j}(x)=\mathbb{E}[Y_{j}|X=x,D=j]$. Consider kernel density estimator
$\hat{f}_{x}(x)$ and $\hat{f}^{\prime}_{x}(x)$ for $f_{x}(x)$ and
$f^{\prime}_{x}(x)$, and local linear estimators $\hat{\nu}_{j}^{\prime}(x)$
and $\hat{\nu}_{j}^{\prime\prime}(x)$ for
$\nu_{j}^{\prime}(x)=\frac{\partial\nu_{j}(x)}{\partial x}$ and
$\nu_{j}^{\prime\prime}(x)=\frac{\partial^{2}\nu_{j}(x)}{\partial^{2}x}$.
Define estimator $\hat{A}_{n}$ by:
$\displaystyle\hat{A}_{n}=-\frac{1}{h!}\alpha_{1}\left[2\hat{f}^{\prime}_{x}(\hat{t}^{s}_{n})[\hat{\lambda}^{\prime}_{1}(\hat{t}^{s}_{n})-\hat{\lambda}^{\prime}_{0}(\hat{t}^{s}_{n})]+\hat{f}_{x}(\hat{t}^{s}_{n})[\hat{\lambda}^{\prime\prime}_{1}(\hat{t}^{s}_{n})-\hat{\lambda}^{\prime\prime}_{0}(\hat{t}^{s}_{n})]\right]$
(29)
which consistently estimate $A$ the additional assumption that the third
derivatives of $f_{x}$, $\nu_{1}$ and $\nu_{0}$ are continuous and bounded in
a neighborhood of $t^{*}$, and with the proper choice of bandwidth sequences.
Confidence intervals with asymptotic coverage $1-\alpha$ can hence be
constructed as
$(\hat{t}^{s}_{n}-\hat{b}_{n}-\hat{w}^{s}_{n},\hat{t}^{s}_{n}-\hat{b}_{n}+\hat{w}^{s}_{n})$,
where
$\displaystyle\hat{b}_{n}=(n\sigma_{n})^{-1/2}\lambda^{1/2}\frac{\hat{A}_{n}}{\hat{H}_{n}}$
(30)
$\displaystyle\hat{w}^{s}_{n}=(n\sigma_{n})^{-1/2}(\sqrt{\alpha_{2}\hat{K}_{n}}/\hat{H}_{n})c_{\alpha/2}$
(31)
The second approach relies on undersmoothing, and chooses a suboptimally small
$\sigma_{n}$ to eliminate the asymptotic bias, with no need to estimate $A$.
Instead of a bandwidth sequence $\sigma_{n}=O_{p}(n^{-\frac{1}{2h+1}})$, it
considers a sequence $\sigma_{n}=o_{p}(n^{-\frac{1}{2h+1}})$ such that
$n\sigma_{n}^{2h+1}\rightarrow\lambda=0$, and ensures $b_{n}\to 0$. Confidence
intervals with asymptotic coverage $1-\alpha$ can hence be constructed as
$(\hat{t}^{s}_{n}-\hat{w}^{s}_{n},\hat{t}^{s}_{n}+\hat{w}^{s}_{n})$, with
$\hat{w}^{s}_{n}$ defined as above.
## Appendix B Local Asymptotics
The interplay between the population distribution $P$ and the sample size can
be studied considering a local asymptotic framework, considering a sequence of
population distributions $\\{P_{n}\\}$ that varies with $n$. I focus on the
following two sequences.
###### Definition 1.
(Sequence $\\{P^{1}_{n}\\}$) The sequence of distributions $\\{P^{1}_{n}\\}$
is such that $\frac{\sqrt{K_{n}}}{H_{n}}=n^{\gamma}$, with
$\gamma\in[0,\frac{h}{2h+1})$, and $\frac{A_{n}}{H_{n}}=1$.
###### Definition 2.
(Sequence $\\{P^{2}_{n}\\}$) The sequence of distributions $\\{P^{2}_{n}\\}$
is such that $\frac{\sqrt{K_{n}}}{H_{n}}=1$, and
$\frac{A_{n}}{H_{n}}=n^{\gamma}$, with $\gamma\in[0,\frac{h}{2h+1})$.
Sequence $\\{P^{1}_{n}\\}$ mimics a scenario where $\sqrt{K}$ is large
compared to $H$. This occurs when the variance of the conditional treatment
effect, $\operatorname{\mathrm{Var}}(Y_{1}-Y_{0}|X=t^{*})$, is large compared
to the derivative of the conditional ATE,
$\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial X}$,
or compared to the density of the index, $f_{x}(t^{*})$. In these situations,
the population welfare remains relatively stable for thresholds in the
neighborhood of the optimal one. The limit case with $\gamma=\frac{1}{2}$
coincides with the local asymptotic framework proposed by
hirano2009asymptotics and studied by athey2021policy.
Sequence $\\{P^{2}_{n}\\}$, instead, mimics a situation where $A$ is large
compared to $H$. When $h=2$, this happens when the second derivative of the
conditional ATE,
$\frac{\partial^{2}\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial^{2}X}$,
is large compared to the first derivative,
$\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial X}$.
Since the asymptotic bias of $\hat{t}^{s}_{n}$ is
$\lambda\frac{A_{n}}{H_{n}}$, sequence $\\{P^{2}_{n}\\}$ represents situation
where this bias is large relatively to the sample size.
Let $r_{n}^{e}$ and $r_{n}^{s}$ denote the rates of convergence of
$\mathcal{R}(\hat{t}^{e}_{n})$ and $\mathcal{R}(\hat{t}^{s}_{n})$, i.e. let
$r_{n}^{e}$ and $r_{n}^{s}$ be sequences such that
$\mathcal{R}(\hat{t}^{e}_{n})=O_{p}(r_{n}^{e})$ and
$\mathcal{R}(\hat{t}^{s}_{n})=O_{p}(r_{n}^{s})$. The following theorem
establishes a relationship between $r_{n}^{e}$ and $r_{n}^{s}$ under
$\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$.
###### Theorem 6.
Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, 3, and 5 hold. Consider
sequences $\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$ of data generating
processes. The limit of the ratio $\frac{r_{n}^{e}}{r_{n}^{s}}$ depends on
$\gamma$ as follows:
* •
$\frac{r_{n}^{e}}{r_{n}^{s}}\to\infty$ if
$\gamma\in(\bar{\gamma},\frac{h}{2h+1})$
* •
$\frac{r_{n}^{e}}{r_{n}^{s}}=O_{p}(1)$ if $\gamma=\bar{\gamma}$
* •
$\frac{r_{n}^{e}}{r_{n}^{s}}\to 0$ if $\gamma\in[0,\bar{\gamma})$.
where $\bar{\gamma}=\frac{h-1}{2h+1}$ under $\\{P^{1}_{n}\\}$, and
$\bar{\gamma}=\frac{1}{3}\frac{h-1}{2h+1}$ under $\\{P^{2}_{n}\\}$.
Theorem 6 shows how, under sequences $\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$,
the EWM policy guarantees a regret convergence rate faster than the SWM policy
whenever the parameter $\gamma$ exceeds a certain value $\bar{\gamma}$. For
sequence $\\{P^{1}_{n}\\}$, the result depends on the fact that the term
$\frac{\sqrt{K_{n}}}{H_{n}}$ enters the asymptotic distributions of
$\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$ with different powers ($\frac{2}{3}$
and 1, respectively): if $\frac{\sqrt{K_{n}}}{H_{n}}$ is large enough
relatively to the sample size, the exponent lower than one makes $r^{e}_{n}$
faster (and hence in the limit $\hat{t}^{e}_{n}$ preferable). For sequence
$\\{P^{2}_{n}\\}$, the result is due to the asymptotic bias of the SWM policy,
proportional to $\frac{A_{n}}{H_{n}}$. Since the asymptotic distribution of
the regret of the EWM policy remains constant in $\\{P^{2}_{n}\\}$, when the
bias of $\hat{t}^{s}_{n}$ is large enough compared to sample size,
$\hat{t}^{e}_{n}$ becomes preferable. The value of $\bar{\gamma}$ is
increasing in $h$, the order of the kernel $k$: a smoother objective function
$\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ amplifies the benefit of using
the SWM policy, expanding the region of values of $\gamma$ where the SWM
policy has a faster rate compared to the EWM.
Considering the sequence $\\{P^{1}_{n}\\}$ with $\gamma=\frac{1}{2}$,
athey2021policy show that their AIPW policy achieves the uniform fastest
asymptotic convergence rate. Results in Theorem 6 are different: without
focusing on a single specific sequence, their goal is to shed light on how the
asymptotic behavior of the EWM and the SWM policies depends on $P$ and sample
size.
## Appendix C Monte Carlo Simulation
I report the analogous of Table 2 and Figures 1 and 2 for the median regret.
Comments made for the mean also extend to the median regret.
Table 4: Finite sample and asymptotic median regret with the EWM and SWM policies across different models. Finite sample (empirical) values are computed through 5,000 Monte Carlo simulations. The last column reports the ratio between finite sample median regrets with the EWM and SWM policies. Model | n | EWM | SWM | Ratio
---|---|---|---|---
| | empirical | asymptotic | empirical $\sigma_{n}^{*}$ | empirical $\hat{\sigma}_{n}^{*}$ | asymptotic |
1 | $500$ | $39.809$ | $45.347$ | $15.113$ | $24.169$ | $18.459$ | $1.647$
$1,000$ | $27.547$ | $28.567$ | $8.692$ | $14.145$ | $10.602$ | $1.948$
$2,000$ | $18.816$ | $17.996$ | $5.073$ | $8.507$ | $6.089$ | $2.212$
$3,000$ | $14.043$ | $13.733$ | $3.471$ | $5.491$ | $4.402$ | $2.558$
2 | $500$ | $147.029$ | $188.651$ | $116.784$ | $159.876$ | $204.255$ | $0.920$
$1,000$ | $110.372$ | $118.843$ | $82.473$ | $118.695$ | $117.314$ | $0.930$
$2,000$ | $87.325$ | $74.866$ | $55.731$ | $79.436$ | $67.379$ | $1.099$
$3,000$ | $72.122$ | $57.134$ | $46.007$ | $68.656$ | $48.714$ | $1.050$
Figure 3: Figure reports asymptotic and finite sample median regrets for the
EWM and SWM policies, illustrating the values reported in Table 4.
Figure 4: Figure reports asymptotic and finite sample median regrets for the
EWM and SWM policies, illustrating the values reported in Table 4.
## Appendix D Proofs
### Theorem 1
###### Theorem 1.
Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1 and 2 with
$s=0$,
$\displaystyle\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$
i.e. $\hat{t}^{e}_{n}$ is a consistent estimator for $t^{*}$.
###### Proof.
Estimator $\hat{t}^{e}_{n}$ in (9) can be written as
$\displaystyle\hat{t}^{e}_{n}=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)(\mathbf{1}\\{X_{i}>t\\}-\mathbf{1}\\{X_{i}>t^{*}\\})\right]$
(32) $\displaystyle=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}m\left(Z_{i},t\right)$ (33)
where
$\displaystyle
m\left(Z,t\right)=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\}\right)$
(34)
and $\\{Z_{i}\\}$ is a sample of $n$ observation from distribution $P$.
Define $P_{n}m\left(\cdot,t\right)=\sum_{i=1}^{n}m\left(Z_{i},t\right)$ and
$Pm\left(Z,t\right)=\mathbb{E}_{P}[m\left(Z,t\right)]$. With this notation,
equation (4) and (9) can be written as
$\displaystyle t^{*}=\mathop{\rm arg~{}max}\limits_{t}Pm\left(Z,t\right)$ (35)
$\displaystyle\hat{t}^{e}_{n}=\mathop{\rm
arg~{}max}\limits_{t}P_{n}m\left(\cdot,t\right).$ (36)
Threshold policies I am considering can be seen as a tree partition of depth
1. Tree partitions of finite depth are a VC class (leboeuf2020decision), and
hence $m\left(\cdot,t\right)$ is a manageable class of functions. Consider the
envelope function
$F=2\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|$.
Note that
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|^{2}\right]=\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}]$:
Assumption 2.2 guarantees the existence of
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|^{2}\right]$.
It follows from corollary 3.2 in kim1990cube that
$\displaystyle\sup_{t}\left|P_{n}m(\cdot,t)-Pm(Z,t)\right|\rightarrow^{a.s.}0.$
(37)
Under Assumptions 2.1 and 2.3, $Pm(Z,t)$ is continuous in $t$, and $t^{*}$ is
the unique maximizer. Hence,
$\displaystyle\sup_{t}\left|P_{n}m(\cdot,t)-Pm(Z,t)\right|+Pm(Z,t^{*})\geq$
(38)
$\displaystyle\left|P_{n}m(\cdot,\hat{t}^{e}_{n})-Pm(Z,\hat{t}^{e}_{n})\right|+Pm(Z,t^{*})\geq$
(39)
$\displaystyle\left|P_{n}m(\cdot,\hat{t}^{e}_{n})-Pm(Z,\hat{t}^{e}_{n})\right|+Pm(Z,\hat{t}^{e}_{n})\geq$
(40) $\displaystyle P_{n}m(\cdot,\hat{t}^{e}_{n})\geq
P_{n}m(\cdot,t^{*})\rightarrow Pm(Z,t^{*})$ (41)
where the second inequality is due to the fact that $t^{*}$ is the maximizer
of $Pm(Z,t)$, the third comes from the triangular inequality, the fourth from
$\hat{t}^{e}_{n}$ being the maximizer of $P_{n}m(\cdot,t)$, and the last limit
comes from LLN. This prove that
$P_{n}m(\cdot,\hat{t}^{e}_{n})\rightarrow^{a.s.}Pm(Z,t^{*})$, and hence
$Pm(Z,\hat{t}^{e}_{n})\rightarrow^{a.s.}Pm(Z,t^{*})$. Since $t^{*}$ is the
unique maximizer of $Pm(Z,t)$ and $Pm(Z,t)$ is continuous,
$\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$. It means that $\hat{t}^{e}_{n}$ is a
consistent estimator for $t^{*}$. ∎
### Theorem 2
###### Theorem 2.
Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=2$ and 3, as $n\rightarrow\infty$,
$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (42)
where $B(r)$ is the two-sided Brownian motion process, and $K$ and $H$ are
$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$
###### Proof.
The proof shows that conditions for the main theorem in kim1990cube hold;
hence, their result is valid for $\hat{t}^{e}_{n}$. For completeness, I report
the theorem.
###### Theorem Kim and Pollard.
Consider estimators defined by maximization of processes
$\displaystyle
P_{n}g(\cdot,\theta)=\frac{1}{n}\sum_{i=1}^{n}g\left(\xi_{i},\theta\right)$
(43)
where $\left\\{\xi_{i}\right\\}_{i}$ is a sequence of i.i.d. observations from
a distribution $P$ and $\\{g(\cdot,\theta):\theta\in\Theta\\}$ is a class of
functions indexed by a subset $\Theta$ in $\mathbb{R}^{k}$. The envelope
$G_{R}(\cdot)$ is defined as the supremum of $g(\cdot,\theta)$ over the class
$\displaystyle\mathcal{G}_{R}=\left\\{|g(\cdot,\theta)|:\left|\theta-\theta_{0}\right|\leq
R\right\\},\quad R>0.$ (44)
Let $\left\\{\theta_{n}\right\\}$ be a sequence of estimators for which
1. 1.
$P_{n}g\left(\cdot,\theta_{n}\right)\geq\sup_{\theta\in\Theta}P_{n}g(\cdot,\theta)-o_{P}\left(n^{-2/3}\right)$.
2. 2.
$\theta_{n}$ converges in probability to the unique $\theta_{0}$ that
maximizes $Pg(\cdot,\theta)$.
3. 3.
$\theta_{0}$ is an interior point of $\Theta$.
Let the functions be standardized so that
$g\left(\cdot,\theta_{0}\right)\equiv 0$. If the classes $\mathcal{G}_{R}$ for
$R$ near 0 are uniformly manageable for the envelopes $G_{R}$ and satisfy:
1. 4.
$Pg(\cdot,\theta)$ is twice differentiable with second derivative matrix $-H$
at $\theta_{0}$.
2. 5.
$K(s,r)=\lim_{\alpha\rightarrow\infty}\alpha
Pg\left(\cdot,\theta_{0}+r/\alpha\right)g\left(\cdot,\theta_{0}+s/\alpha\right)$
exists for each $s,r$ in $\mathbb{R}^{k}$ and
$\lim_{\alpha\rightarrow\infty}\alpha
Pg\left(\cdot,\theta_{0}+r/\alpha\right)^{2}\left\\{\left|g\left(\cdot,\theta_{0}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=0$
for each $\varepsilon>0$ and $r$ in $\mathbb{R}^{k}$.
3. 6.
$PG_{R}^{2}=O(R)$ as $R\rightarrow 0$ and for each $\varepsilon>0$ there is a
constant C such that $PG_{R}^{2}\mathbf{1}\\{G_{R}>C\\}\leq\varepsilon R$ for
$R$ near 0.
4. 7.
$P\left|g\left(\cdot,\theta_{1}\right)-g\left(\cdot,\theta_{2}\right)\right|=O\left(\left|\theta_{1}-\theta_{2}\right|\right)$
near $\theta_{0}$.
Then, the process $n^{2/3}P_{n}g\left(\cdot,\theta_{0}+rn^{-1/3}\right)$
converges in distribution to a Gaussian process $Q(r)$ with continuous sample
paths, expected value $-\frac{1}{2}r^{\prime}Hr$ and covariance kernel $K$. If
$H$ is positive definite and if $Q$ has nondegenerate increments, then
$n^{1/3}\left(\theta_{n}-\theta_{0}\right)$ converges in distribution to the
(almost surely unique) random vector that maximizes $Q$.
I apply Theorem Theorem Kim and Pollard by taking
$\xi_{i}=Z_{i},\theta=t,\theta_{n}=\hat{t}^{e}_{n},\theta_{0}=t^{*}$,
$g(\cdot,\theta)=m(\cdot,t)$, where $m(\cdot,t)$ is standardized:
$\displaystyle
m\left(Z_{i},t\right)=\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\left(\mathbf{1}\\{X_{i}>t\\}-\mathbf{1}\\{X_{i}>t^{*}\\}\right).$
(45)
First, I will verify that conditions 1-7 apply to my setting:
1. 1.
$P_{n}m(\cdot,t)$ takes only finite ($n+1$) values; hence, condition 1 is
satisfied with the equality.
2. 2.
In Theorem 1, I proved that $\hat{t}^{e}_{n}$ is a consistent estimator for
$t^{*}$.
3. 3.
$t^{*}$ is an interior point of $\mathcal{T}$ by Assumption 2.1.
I need to prove that the classes $\mathcal{G}_{R}$ for $R$ near 0 are
uniformly manageable for the envelopes $G_{R}$. The envelope function
$G_{R}(\cdot)$ is defined as
$\displaystyle G_{R}(z)$
$\displaystyle=\sup\left\\{m(z,t):\left|t-t^{*}\right|\leq R\right\\}$ (46)
$\displaystyle=\sup_{\left|t-t^{*}\right|\leq
R}\left[\left(\frac{dy}{p(\textbf{x})}-\frac{(1-d)y}{(1-p(\textbf{x}))}\right)\left(\mathbf{1}\\{x>t\\}-\mathbf{1}\\{x>t^{*}\\}\right)\right]$
(47)
$\displaystyle=\left|\frac{dy}{p(\textbf{x})}-\frac{(1-d)y}{(1-p(\textbf{x}))}\right|\mathbf{1}\\{\left|x-t^{*}\right|<R\\}$
(48)
and I have:
$\displaystyle PG_{R}^{2}$
$\displaystyle=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\right]$
(49)
$\displaystyle=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Bigg{|}X\in(t^{*}-R,t^{*}+R)\right]$
(50)
$\displaystyle=2R\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Bigg{|}X=t^{*}\right]+o(1)=O(R)$
(51)
where the second to last equality comes from Assumption 2.2. The envelope
function is uniformly square-integrable for $R$ near 0, and therefore, the
classes $\mathcal{G}_{R}$ are uniformly manageable.
1. 4.
Define $h(t)=Pm(Z,t)$ and consider derivatives:
$\displaystyle h(t)=$
$\displaystyle\mathbb{E}_{P}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\})\right]=$
(52)
$\displaystyle\mathbb{E}_{P}\left[\left(Y_{1}-Y_{0}\right)(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\})\right]$
(53) $\displaystyle h^{\prime}(t)=$ $\displaystyle-
f_{x}(t)\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t\right]$ (54) $\displaystyle
h^{\prime\prime}(t)=$
$\displaystyle-f^{\prime}_{x}(t)\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t\right]-f_{x}(t)\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t\right]}{\partial
X}\right).$ (55)
Assumption 2.3 with $s=2$ guarantees the existence of $h^{\prime}$ and
$h^{\prime\prime}$. Since
$\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t^{*}\right]=0$, $H$ is given by
$\displaystyle
H=-h^{\prime\prime}(t^{*})=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$ (56)
2. 5.
This condition is divided into two parts. First, I prove the existence of
$K(s,r)=\lim_{\alpha\rightarrow\infty}\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)$ for
each $s,r$ in $\mathbb{R}$. Covariance $K$ is:
$\displaystyle Pm$
$\displaystyle\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(57)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})(\mathbf{1}\\{X>t^{*}+s/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})\Big{]}.$
(58)
If $rs<0$, covariance and $K(s,r)$ are 0. If $rs>0$, and suppose $r>0$:
$\displaystyle
Pm\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)=$ (59)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X\in(t^{*},t^{*}+\min\\{r,s\\}/\alpha)\right]$
(60)
and hence
$\displaystyle K(s,r)$
$\displaystyle=\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X\in(t^{*},t^{*}+\min\\{r,s\\}/\alpha)\right]$
(61)
$\displaystyle=\min\\{r,s\\}f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X=t^{*}\right].$
(62)
where the equality is due to continuity of $f_{x}$ (Assumption 2.3 with
$s=2$). Boundedness of the quantity follows from Assumptions 2.2 and 2.3.
Now, I will prove that $\lim_{\alpha\rightarrow\infty}\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=0$
for each $\varepsilon>0$ and $r$ in $\mathbb{R}$. I have:
$\displaystyle\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=$
(63)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(64)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}\Big{]}=$
(65)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(66)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\varepsilon\alpha\right\\}\Big{]}\leq$
(67)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\varepsilon\alpha\right\\}\right].$
(68)
Some algebra gives
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(69)
$\displaystyle\mathbb{E}\left[\left(\frac{DY_{1}^{2}}{p(\textbf{X})^{2}}+\frac{(1-D)Y_{0}^{2}}{(1-p(\textbf{X}))^{2}}\right)\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(70)
$\displaystyle\mathbb{E}\left[\frac{Y_{1}^{2}}{p(\textbf{X})}\mathbf{1}\left\\{\left|\frac{Y_{1}}{p(\textbf{X})}\right|>\epsilon\right\\}\right]+\mathbb{E}\left[\frac{Y_{0}^{2}}{(1-p(\textbf{X}))}\mathbf{1}\left\\{\left|\frac{Y_{0}}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(71)
$\displaystyle\frac{1}{p(\textbf{X})}\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon_{1}\right\\}\right]+\frac{1}{(1-p(\textbf{X}))}\mathbb{E}\left[Y_{0}^{2}\mathbf{1}\left\\{\left|Y_{0}\right|>\epsilon_{0}\right\\}\right]$
(72)
and hence the condition is satisfied if
$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon\alpha\right\\}\right]=0$
(73)
$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{0}^{2}\mathbf{1}\left\\{\left|Y_{0}\right|>\epsilon\alpha\right\\}\right]=0.$
(74)
Consider the limit for $Y_{1}$:
$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon\alpha\right\\}\right]=\lim_{\alpha\rightarrow\infty}\frac{\int_{\epsilon\alpha}Y_{1}^{2}\varphi_{1}(y_{1})dy_{1}}{\alpha^{-1}}=\lim_{\alpha\rightarrow\infty}\frac{\epsilon^{3}\alpha^{2}\varphi_{1}(\epsilon\alpha)}{\alpha^{-2}}=$
(75)
$\displaystyle\lim_{\alpha\rightarrow\infty}\epsilon^{3}\alpha^{4}\varphi_{1}(\epsilon\alpha)=\lim_{\alpha\rightarrow\infty}\epsilon^{3}\alpha^{4}|\epsilon\alpha|^{-(4+\delta)}o(1)=0$
(76)
where the second to last equality follows from Assumption 3.
3. 6.
I showed that $PG_{R}^{2}=O(R)$ as $R\rightarrow 0$ in equation (49). I need
to prove that for each $\varepsilon>0$ there is a constant C such that
$PG_{R}^{2}\mathbf{1}\\{G_{R}>C\\}\leq\varepsilon R$ for $R$ near 0. For any
$\varepsilon>0$ and $C>0$:
$\displaystyle PG_{R}^{2}$
$\displaystyle\mathbf{1}\\{G_{R}>C\\}=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\mathbf{1}\\{G_{R}>C\\}\right]\leq$
(77)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>C\right\\}\right]\leq$
(78)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>C\right\\}\right]\rightarrow
0$ (79)
where the last limit is taken for $C\rightarrow\infty$, and follows from
Assumption 3.
4. 7.
I need to show that
$P\left|m\left(\cdot,t_{1}\right)-m\left(\cdot,t_{2}\right)\right|=O\left(\left|t_{1}-t_{2}\right|\right)$
near $t^{*}$. Consider $t_{2}>t_{1}$:
$\displaystyle
P\left|m\left(\cdot,t_{1}\right)-m\left(\cdot,t_{2}\right)\right|=\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)(\mathbf{1}\\{X>t_{1}\\}-\mathbf{1}\\{X>t_{2}\\})\right|\right]\leq$
(80) $\displaystyle
M_{x}\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\right|\Big{|}X\in(t_{1},t_{2})\right]\leq
M_{x}M_{y}|t_{2}-t_{1}|$ (81)
where $M_{x}=\max_{x\in(t_{1},t_{2})}f_{x}(x)$ and
$M_{y}=\max_{x\in(t_{1},t_{2})}\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\right|\Big{|}X\right]$.
$M_{x}<\infty$ and $M_{y}<\infty$ because of Assumption 2.3.
Assumptions 1-7 of Theorem Theorem Kim and Pollard by kim1990cube are hence
satisfied. It follows that, for $n\rightarrow\infty$,
$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}\mathop{\rm
arg~{}max}\limits_{r}Q(r)$ (82)
where $Q(r)=Q_{1}(r)+Q_{0}(r)$, and $Q_{1}$ is a non degenerate zero-mean
Gaussian process with covariance $K$, while $Q_{0}(r)$ is non-random and
$Q_{0}(r)=-\frac{1}{2}r^{2}H$.
Limiting distribution $\mathop{\rm arg~{}max}\limits_{r}Q(r)$ is of
chernoff1964estimation type. It can be shown (banerjee2001likelihood) that
$\displaystyle\mathop{\rm
arg~{}max}\limits_{r}Q(r)=^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}$ (83)
where $B(r)$ is the two-sided Brownian motion process, $K$ is:
$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X=t^{*}\right]$
(84) $\displaystyle=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
(85)
and $H$ is:
$\displaystyle
H=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$ (86)
This completes the proof of the theorem. ∎
### Corollary 2.1
###### Corollary 2.1.
The asymptotic distribution of regret $\mathcal{R}(\hat{t}^{e}_{n})$ is:
$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$
The expected value of the asymptotic distribution is
$K^{\frac{2}{3}}H^{-\frac{1}{3}}C^{e}$, where
$C^{e}=\sqrt[3]{2}\mathbb{E}\left[\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}\right]$
is a constant not dependent on $P$.
###### Proof.
Result in equation (6) for $\hat{t}^{e}_{n}$ implies
$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})=\frac{1}{2}W^{\prime\prime}(\tilde{t})\left(n^{\frac{1}{3}}\left(\hat{t}^{e}_{n}-t^{*}\right)\right)^{2},$
where $|\tilde{t}-t^{*}|\leq|\hat{t}_{n}-t^{*}|$. By continuous mapping
theorem
$\displaystyle
W^{\prime\prime}(\tilde{t})\rightarrow^{p}W^{\prime\prime}(t^{*})=H$
and hence by Slutsky’s theorem
$\displaystyle
n^{\frac{2}{3}}\left(W(\hat{t}_{n})-W(t^{*})\right)\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$
∎
### Theorem 3
###### Theorem 3.
Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=0$ and 4,
$\displaystyle\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$
i.e. $\hat{t}^{s}_{n}$ is a consistent estimator for $t^{*}$.
###### Proof.
To prove the result, I show that conditions for Theorem 4.1.1 in
amemiya1985advanced hold, and hence $\hat{t}^{s}_{n}$ is consistent for
$t^{*}$.
First, define function $m^{s}(Z,t)$:
$\displaystyle
m^{s}\left(Z,t\right)=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(k\left(\frac{X-t}{\sigma_{n}}\right)-k\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right)$
and recall definitions of $m(Z,t)$, $P_{n}$, and $P$ introduced in the proof
of Theorem 2:
$\displaystyle m\left(Z,t\right)$
$\displaystyle=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\}\right)$
(87) $\displaystyle P_{n}m\left(\cdot,t\right)$
$\displaystyle=\sum_{i=1}^{n}m\left(Z_{i},t\right)$ (88) $\displaystyle
Pm\left(Z,t\right)$ $\displaystyle=\mathbb{E}_{P}[m\left(Z,t\right)].$ (89)
In this notation, $\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}P_{n}m^{s}(.,t)$ and $t^{*}=\mathop{\rm
arg~{}max}\limits_{t}Pm(Z,t)$. I can now show that conditions $A$, $B$, and
$C$ for Theorem 4.1.1 in amemiya1985advanced hold:
* A)
Parameter space $\mathcal{T}$ is compact by Assumption 2.1.
* B)
Function $P_{n}m^{s}\left(Z_{i},t\right)$ is continuous in $t$ for all $Z$ and
is a measurable function of $Z$ for all $t\in\mathcal{T}$, as $k(\cdot)$ is
continuous by Assumption 4.
* C1)
I need to prove that $P_{n}m^{s}\left(Z_{i},t\right)$ converges a.s. to
$Pm(Z,t)$ uniformly in $t\in\mathcal{T}$ as $n\rightarrow\infty$, i.e.
$\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm(Z,t)\right|\rightarrow^{a.s.}0$. Note
that:
$\displaystyle\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm(Z,t)\right|\leq$ (90)
$\displaystyle\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm^{s}(Z,t)\right|+\sup_{t}\left|Pm^{s}(\cdot,t)-Pm(Z,t)\right|.$
(91)
I need to show that the two addends on the right-hand side converge to zero.
To show uniform convergence of $P_{n}m^{s}(\cdot,t)$ to $Pm^{s}(Z,t)$, I
consider sufficient conditions provided by Theorem 4.2.1 in
amemiya1985advanced. $m^{s}(Z,t)$ is continuous in $t\in\mathcal{T}$ with
$\mathcal{T}$ compact, and measurable in $Z$. I only need to show that
$\mathbb{E}[\sup_{t\in\mathcal{T}}|m^{s}(Z,t)|]<\infty$. By Assumption 4,
$k(\cdot)$ is a bounded function, i.e. it exists an $M$ such that $|k(x)|<M$
for all $x$. Hence $\mathbb{E}[\sup_{t\in\mathcal{T}}|m^{s}(Z,t)|]\leq
M\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|\right]$,
and
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|\right]<\infty$
by Assumption 2.2.
To show uniform convergence of $Pm^{s}(\cdot,t)$ to $Pm(Z,t)$, note that
$t\in\mathcal{T}$, where $\mathcal{T}$ is bounded, and hence the result holds
for $\sigma_{n}\rightarrow 0$.
* C2)
By Assumption 2.1, $t^{*}$ is the unique global maximum of $Pm(Z,t)$.
Assumptions $A$, $B$, $C$ of Theorem 4.1.1 in amemiya1985advanced are
satisfied, and hence $\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$. ∎
### Lemmas
Proof of Theorem 4 requires some intermediate lemmas, stated and proved below.
Arguments follows the ideas in horowitz1992smoothed, but are adapted to my
context. I report the entire proof for completeness, even when it overlaps
with the original in horowitz1992smoothed.
To make the notation simpler, define:
$\displaystyle\hat{S}_{n}(t,\sigma_{n})=\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$
and note that $\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\hat{S}_{n}(t,\sigma_{n})$. Then define:
$\displaystyle\hat{S}_{n}^{1}(t,\sigma_{n})=\frac{\partial\hat{S}_{n}(t,\sigma_{n})}{\partial
t}=-\frac{1}{\sigma_{n}}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$
$\displaystyle\hat{S}_{n}^{2}(t,\sigma_{n})=\frac{\partial\hat{S}_{n}(t,\sigma_{n})}{\partial^{2}t}=\frac{1}{\sigma_{n}^{2}}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right].$
Indicate with $\varphi_{y,x}(y,x)$ the joint distribution of $Y_{1}$, $Y_{0}$,
and $X$, and with $\varphi_{y|x}(y|x)$ the conditional distribution, where
$\varphi_{y,x}(y,x)=\varphi_{y|x}(y|x)f_{x}(x)$.
#### Lemma 1
###### Lemma 1.
Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. Then
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=A$
(92)
$\displaystyle\lim_{n\rightarrow\infty}\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\alpha_{2}K.$
(93)
###### Proof.
First, I will prove that
$\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=A$:
$\displaystyle\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]$
$\displaystyle=-\frac{\sigma_{n}^{-h}}{\sigma_{n}}\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]=$
(94)
$\displaystyle=-\frac{\sigma_{n}^{-h}}{\sigma_{n}}\mathbb{E}\left[\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]=$
(95)
$\displaystyle=-\sigma_{n}^{-h}\int_{x}\int_{y}\left(Y_{1}-Y_{0}\right)\frac{1}{\sigma_{n}}k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\varphi_{y,x}(y,x)dydx=$
(96)
$\displaystyle=-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\varphi_{y,x}(y,t^{*}+\zeta\sigma_{n})dyd\zeta$
(97)
where in the last line I made the substitution
$\zeta=\frac{X_{i}-t^{*}}{\sigma_{n}}$. Consider the Taylor expansion of
$\varphi$ around $\varphi(y,t^{*})$:
$\displaystyle\varphi(y,t^{*}+\zeta\sigma_{n})$
$\displaystyle=\varphi(y,t^{*})+\zeta\sigma_{n}\varphi^{1}_{2}(y,t^{*})+\frac{1}{2}(\zeta\sigma_{n})^{2}\varphi^{2}_{2}(y,t^{*})+\dots$
(98)
$\displaystyle=\varphi(y,t^{*})+\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\zeta^{i}\sigma_{n}^{i}\right)+\frac{1}{h!}\varphi^{h}_{2}(y,\tilde{t})\zeta^{h}\sigma_{n}^{h}$
(99)
with $|\tilde{t}-t^{*}|\leq|t^{*}+\zeta\sigma_{n}-t^{*}|$. Existence of
$\varphi^{m}_{2}(y,x)$, the $m$-derivatives of $\varphi(y,x)$ with respect to
its second argument, is guaranteed by Assumption 2.3 with $s=h+1$.
Write
$\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]$ as
$I_{n1}+I_{n2}+I_{n3}$, where:
$\displaystyle I_{n1}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\varphi(y,t^{*})dyd\zeta$
(100) $\displaystyle=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(Y_{1}-Y_{0}\right)dy}_{=\mathbb{E}[Y_{1}-Y_{0}|X=t^{*}]=0}=0$
(101) $\displaystyle I_{n2}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\zeta^{i}\sigma_{n}^{i}\right)dyd\zeta$
(102) $\displaystyle=$
$\displaystyle\sigma_{n}^{-h}\int_{y}\left(Y_{1}-Y_{0}\right)\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\sigma_{n}^{i}\underbrace{\int_{\zeta}k^{\prime}\left(\zeta\right)\zeta^{i}d\zeta}_{=0}\right)dy=0.$
(103)
Result on $I_{n1}$ follows from definition of $t^{*}$, while Assumption 5.2
guarantees result on $I_{n2}$. Finally, consider $I_{n3}$:
$\displaystyle I_{n3}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\frac{1}{h!}\varphi^{h}_{2}(y,\tilde{t})\zeta^{h}\sigma_{n}^{h}dyd\zeta$
(104) $\displaystyle=$
$\displaystyle-\frac{1}{h!}\int_{\zeta}k^{\prime}\left(\zeta\right)\zeta^{h}d\zeta\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{2}(y,\tilde{t})dy$
(105)
and conclude that:
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=-\frac{1}{h!}\int_{\zeta}\zeta^{h}k^{\prime}\left(\zeta\right)d\zeta\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{2}(y,t^{*})dy=A.$
(106)
Now I will prove that
$\lim_{n\rightarrow\infty}\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\alpha_{2}K$.
Note that:
$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=$
$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{-1/2}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]\right]=$
(107) $\displaystyle=$
$\displaystyle\sigma_{n}\operatorname{\mathrm{Var}}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right]=$
(108) $\displaystyle=$
$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}^{2}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\right]-$
(109)
$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right]^{2}.$
(110)
Second term in the last expression goes to 0 as $n\rightarrow\infty$. For the
first term, observe that:
$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\right]=$
(111)
$\displaystyle\int_{x}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\frac{1}{\sigma_{n}}\varphi_{y,x}(y,x)dydx=$
(112)
$\displaystyle\int_{\zeta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\zeta\right)^{2}\varphi_{y,x}(y,t^{*}+\zeta\sigma_{n})dyd\zeta$
(113)
where in the last line I made the substitution
$\zeta=\frac{X_{i}-t^{*}}{\sigma_{n}}$. Conclude that
$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=$
$\displaystyle\int_{\zeta}k^{\prime}\left(\zeta\right)^{2}d\zeta
f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}|X=t^{*}\right]$
(114) $\displaystyle=$ $\displaystyle\alpha_{2}K.$ (115)
Note that $\alpha_{2}K$ is bounded by Assumptions 2.2, 2.3, and 5.2. ∎
#### Lemma 2
###### Lemma 2.
Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. If
$n\sigma_{n}^{2h+1}\rightarrow\infty$,
$\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})$ converges in probability to
A. If $n\sigma_{n}^{2h+1}$ has a finite limit $\lambda$,
$(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})$ converges in
distribution to $\mathcal{N}(\lambda^{1/2}A,\alpha_{2}K)$.
###### Proof.
Note that
$\operatorname{\mathrm{Var}}\left[(\sigma_{n})^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\underbrace{(n\sigma_{n}^{2h+1})^{-1}}_{\rightarrow
0}\underbrace{\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]}_{\rightarrow\alpha_{2}K}.$
So the first result follows from lemma 1 and Chebyshev’s inequality.
For the second result, first note that under the stated assumptions and from
lemma 1,
$\mathbb{E}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\underbrace{(n\sigma_{n}^{2h+1})^{1/2}}_{\rightarrow\lambda^{1/2}}\underbrace{\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]}_{A}$
and so the result follows if I show that
$U_{n}=(n\sigma_{n})^{1/2}\left(\hat{S}_{n}^{1}(t^{*},\sigma_{n})-\mathbb{E}\left[\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]\right)\rightarrow^{d}\mathcal{N}(0,\alpha_{2}K).$
Note that
$\displaystyle U_{n}=$
$\displaystyle(n\sigma_{n})^{1/2}\frac{1}{n}\sum_{i=1}^{n}\left[\underbrace{\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\frac{1}{\sigma_{n}}}_{=B}-\mathbb{E}[B]\right]=$
(116) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])$
(117)
and hence $U_{n}$ has characteristic function $\psi(\tau)^{n}$, where
$\psi(\tau)=\mathbb{E}\left[\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right]$
and
$\displaystyle\psi^{{}^{\prime}}(\tau)=\mathbb{E}\left[i\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right]$
(118)
$\displaystyle\psi^{{}^{\prime\prime}}(\tau)=\mathbb{E}\left[-\frac{\sigma_{n}}{n}(B-\mathbb{E}[B])\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right].$
(119)
Note that $\psi^{{}^{\prime}}(0)=0$ and
$\psi^{{}^{\prime\prime}}(0)=-\frac{\sigma_{n}}{n}\operatorname{\mathrm{Var}}(B)=-\frac{1}{n}(\alpha_{2}K+o(1))$,
since lemma 1 proved that
$\lim_{n\rightarrow\infty}\sigma_{n}\operatorname{\mathrm{Var}}(B)=\alpha_{2}K$.
A Taylor series expansion of $\psi(\tau)$ about $\tau=0$ yields:
$\psi(\tau)=\underbrace{\psi(0)}_{=1}+\underbrace{\psi^{{}^{\prime}}(0)}_{=0}\tau+\frac{1}{2}\underbrace{\psi^{{}^{\prime\prime}}(0)}_{=-\frac{\alpha_{2}K}{n}}\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)=1-\frac{1}{2n}\alpha_{2}K\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)$
and hence the characteristic function of $U_{n}$ has limit:
$\displaystyle\lim_{n\rightarrow\infty}\left[1-\frac{1}{2n}\alpha_{2}K\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)\right]^{n}=\exp\left(-\alpha_{2}K\frac{\tau^{2}}{2}\right).$
(120)
Since $\exp\left(-\alpha_{2}K\frac{\tau^{2}}{2}\right)$ is the characteristic
function of $\mathcal{N}(0,\alpha_{2}K)$, the second result of the lemma
holds. ∎
#### Lemma 3
###### Lemma 3.
Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. Let $\eta>0$
be such that $\varphi_{y,x}(y,x)$ has second derivative uniformly bounded for
almost every $X$ if $|X-t^{*}|<\eta$. For $\theta\in\mathbb{R}$, define
$\hat{S}^{\theta}_{n}(\theta)$ by
$\displaystyle\hat{S}^{\theta}_{n}(\theta)=-(n\sigma_{n}^{2})^{-1}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)\right].$
(121)
Define the sets $\Theta_{n}(n=1,2,\dots)$ by
$\Theta_{n}=\\{\theta:\theta\in\mathbb{R},\sigma_{n}|\theta|\leq\frac{\eta}{2}\\}$.
Then
$\displaystyle\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|\hat{S}^{\theta}_{n}(\theta)-\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]|=0.$
(122)
In addition, there are finite numbers $\alpha_{1}$ and $\alpha_{2}$ such that
for all $\theta\in\Theta_{n}$
$|\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]-H\theta|\leq
o(1)+\alpha_{1}\sigma_{n}|\theta|+\alpha_{2}\sigma_{n}\theta^{2}$
uniformly over $\theta\in\Theta_{n}$.
###### Proof.
To prove the first result, first define
$\displaystyle-g_{i}(\theta)=$
$\displaystyle\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)-$
(123)
$\displaystyle\mathbb{E}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)\right].$
(124)
It is necessary to prove that for any $\varepsilon>0$
$\lim_{n\rightarrow\infty}\operatorname{Pr}\left[\sup_{\theta\in\Theta_{n}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{n=1}^{N}g_{i}(\theta)\right|>\varepsilon\right]=0.$
Given any $\delta>0$, divide each set $\Theta_{n}$ into nonoverlapping subsets
$\Theta_{nj}(j=1,2,\ldots)$ such that the distance between any two points in
the same subset does not exceed $\delta\sigma_{n}^{2}$ and the number
$\Gamma_{n}$ of subsets does not exceed $C\sigma_{n}^{-3(q-1)}$ for some
$C>0$. Let $\left\\{\theta_{Ni}\right\\}$ be a set of vectors such that
$\theta_{nj}\in\Theta_{nj}$. Then
$\displaystyle\operatorname{Pr}\left[\sup_{\theta\in\Theta_{n}}\left(n\sigma_{n}^{2}\right)^{-1}\right.$
$\displaystyle\left.\left|\sum_{n=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]=$
(125) $\displaystyle=$
$\displaystyle\operatorname{Pr}\left[\bigcup_{j=1}^{\Gamma_{n}}\sup_{\theta\in\Theta_{nj}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]$
(126) $\displaystyle\leqslant$
$\displaystyle\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\sup_{\theta\in\Theta_{nj}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]$
(127) $\displaystyle\leqslant$
$\displaystyle\underbrace{\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]}_{=B_{1}}$
(128)
$\displaystyle+\underbrace{\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\sum_{i=1}^{n}\sup_{\theta\in\Theta_{nj}}\left|g_{i}(\theta)-g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]}_{=B_{2}},$
(129)
where the last two lines follow from the triangle inequality. By Hoeffding’s
inequality, there are finite numbers $c_{1}>0$ and $c_{2}>0$ such that
$\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]\leqslant
c_{1}\exp\left(-c_{2}n\sigma_{n}^{4}\right).$
Therefore, $B_{1}$ is bounded by
$Cc_{1}\sigma_{n}^{-3(q-1)}\exp\left(-c_{2}n\sigma_{n}^{4}\right)$, which
converges to 0 as $n\rightarrow\infty$ by Assumption 5.1. In addition, by
Assumption 4 there is a finite $c_{3}>0$ such that if $\theta\in\Theta_{nj}$,
$\displaystyle\left|-\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\left[k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)-k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta_{nj}\right)\right]\right|$
(130) $\displaystyle\leqslant c_{3}\left|\theta-\theta_{nj}\right|\leqslant
c_{3}\delta\sigma_{n}^{2}.$ (131)
So
$\left(n\sigma_{n}^{2}\right)^{-1}\sum_{i=1}^{n}\sup_{\theta\in\Theta_{nj}}\left|g_{i}(\theta)-g_{i}\left(\theta_{nj}\right)\right|\leqslant
2c_{3}\delta.$
Choose $\delta<\varepsilon/4c_{3}$. Then $B_{2}$ is 0. This establishes
$\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|\hat{S}^{\theta}_{n}(\theta)-\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]|=0$.
To prove the second result, start noting that
$\displaystyle\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]=-\sigma_{n}^{-2}\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\right]=I_{n1}+I_{n2}$
(132)
where
$\displaystyle
I_{n1}=-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx$
(133) $\displaystyle
I_{n2}=-\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx.$
(134)
First, consider $I_{n2}$ and observe that
$\displaystyle
I_{n2}=-\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi(y|x)dy}_{=\mathbb{E}[Y_{1}-Y_{0}|X]}f_{x}(x)dx$
(135)
and since $\mathbb{E}[Y_{1}-Y_{0}|X]$ is bounded by Assumption 2.2,
$|I_{n2}|\leq\left|C\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(x)dx\right|.$
Define $\zeta=\frac{X-t^{*}}{\sigma_{n}}+\theta$. Since
$\sigma_{n}|\theta|\leq\frac{\eta}{2}$, when $|X-t^{*}|>\eta$
$\displaystyle|\zeta|=$
$\displaystyle\left|\frac{X-t^{*}}{\sigma_{n}}+\theta\right|=\pm\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\geq\pm\left(\frac{X-t^{*}}{\sigma_{n}}\right)-|\theta|=\left|\frac{X-t^{*}}{\sigma_{n}}\right|-|\theta|$
(136) $\displaystyle\geq$
$\displaystyle\left|\frac{X-t^{*}}{\sigma_{n}}\right|-\frac{\eta}{2\sigma_{n}}>\frac{\eta}{\sigma_{n}}-\frac{\eta}{2\sigma_{n}}=\frac{\eta}{2\sigma_{n}}.$
(137)
and so the event $|X-t^{*}|>\eta$ implies $|\zeta|>\frac{\eta}{2\sigma_{n}}$.
Then
$\displaystyle|I_{n2}|\leq$
$\displaystyle\left|C\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(x)dx\right|$
(138) $\displaystyle=$
$\displaystyle\left|C\sigma_{n}^{-1}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\zeta\right)f_{x}(t^{*}-\theta\sigma_{n})d\zeta\right|$
(139) $\displaystyle\leq$
$\displaystyle\left|C\underbrace{f_{x}(t^{*}-\theta\sigma_{n})}_{\rightarrow
f_{x}(t^{*})}\underbrace{\sigma_{n}^{-1}\int_{|\zeta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta}_{\rightarrow
0}\right|.$ (140)
The fact that $f_{x}(t^{*}-\theta\sigma_{n})\rightarrow f_{x}(t^{*})$ bounded
by Assumption 2.3 with $s=h+1$ and
$\sigma_{n}^{-1}\int_{|\zeta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\rightarrow
0$ by Assumption 5.2 implies
$\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|I_{n2}|=0.$
Recall that $I_{n1}$ is defined as
$I_{n1}=-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx.$
Consider a Taylor expansion of $\varphi(y,x)$ about $x=t^{*}$:
$\varphi(y,x)=\varphi(y,t^{*})+\varphi^{\prime}(y,t^{*})(x-t^{*})+\frac{1}{2}\varphi^{\prime\prime}(y,\tilde{t})(x-t^{*})^{2}$
with $|\tilde{t}-t^{*}|\leq|x-t^{*}|$. Write $I_{n1}$ as
$J_{n1}+J_{n2}+J_{n3}$ where
$\displaystyle J_{n1}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,t^{*})dydx$
(141) $\displaystyle=$
$\displaystyle-\sigma_{n}^{-2}\underbrace{\mathbb{E}[Y_{1}-Y_{0}|X=t^{*}]}_{=0}\int_{|X-t^{*}|\leq\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(t^{*})dx=0$
(142) $\displaystyle J_{n2}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi^{\prime}(y,t^{*})(x-t^{*})dydx$
(143) $\displaystyle J_{n3}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\frac{1}{2}\varphi^{\prime\prime}(y,\tilde{t})(x-t^{*})^{2}dydx.$
(144)
Consider $J_{n2}$ and the substitution
$\zeta=\frac{X-t^{*}}{\sigma_{n}}+\theta$:
$\displaystyle J_{n2}=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)(\zeta-\theta)\varphi^{\prime}(y,t^{*})dyd\zeta$
(145) $\displaystyle=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)\zeta\varphi^{\prime}(y,t^{*})dyd\zeta+$
(146)
$\displaystyle\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)\theta\varphi^{\prime}(y,t^{*})dyd\zeta$
(147) $\displaystyle=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi^{\prime}(y,t^{*})dy}_{=H}+$
(148)
$\displaystyle\theta\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi^{\prime}(y,t^{*})dy}_{=H}$
(149)
where
$H=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right)$ is bounded by Assumption 2.3. Since $\int\zeta
k^{\prime}\left(\zeta\right)d\zeta=0$ and
$\sigma_{n}|\theta|\leq\frac{\eta}{2}$:
$\displaystyle\left|\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|=\left|\int_{|\zeta-\theta|>\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|\leq\left|\int_{|\zeta|>\eta/2\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|.$ (150)
By Assumption 5.2, $\left|\int_{|\zeta|>\eta/2\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|$ converges to 0 uniformly over
$\theta\in\Theta_{n}$. It means that
$\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta$ converges uniformly to 0.
Consider $\theta
H\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta$,
and note that, since $\int k^{\prime}\left(\zeta\right)d\zeta=1$,
$\displaystyle\left|\theta H-\theta
H\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\right|=\left|\theta
H\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\right|\leq$
(151) $\displaystyle|\sigma_{n}\theta
H|\sigma_{n}^{-1}\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\leq\frac{\eta}{2}\sigma_{n}^{-1}\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta.$
(152)
The last term is bounded uniformly over $n$ and $\theta\in\Theta_{n}$ and
converges to 0 by Assumption 5.2. It means that
$\displaystyle\lim_{n\rightarrow\infty}\left|\sup_{\theta\in\Theta_{n}}J_{n2}-\theta
H\right|=0.$ (153)
Finally, consider $J_{n3}$:
$\displaystyle|J_{n3}|=$
|
# Weakly Supervised Co-training with Swapping Assignments
for Semantic Segmentation
Xinyu Yang1 Hossein Rahmani1 Sue Black2 Bryan M. Williams1
1Lancaster University 2St John’s College of the University of Oxford
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Class activation maps (CAMs) are commonly employed in weakly supervised
semantic segmentation (WSSS) to produce pseudo-labels. Due to incomplete or
excessive class activation, existing studies often resort to offline CAM
refinement, introducing additional stages or proposing offline modules. This
can cause optimization difficulties for single-stage methods and limit
generalizability. In this study, we aim to reduce the observed CAM
inconsistency and error to mitigate reliance on refinement processes. We
propose an end-to-end WSSS model incorporating guided CAMs, wherein our
segmentation model is trained while concurrently optimizing CAMs online. Our
method, Co-training with Swapping Assignments (CoSA), leverages a dual-stream
framework, where one sub-network learns from the swapped assignments generated
by the other. We introduce three techniques: i) soft perplexity-based
regularization to penalize uncertain regions; ii) a threshold-searching
approach to dynamically revise the confidence threshold; and iii) contrastive
separation to address the coexistence problem. CoSA demonstrates exceptional
performance, achieving mIoU of 76.2% and 51.0% on VOC and COCO validation
datasets, respectively, surpassing existing baselines by a substantial margin.
Notably, CoSA is the first single-stage approach to outperform all existing
multi-stage methods including those with additional supervision.
## 1 Introduction
The objective of weakly supervised semantic segmentation (WSSS) is to train a
segmentation model without relying on pixel-level labels but utilizing only
weak and cost-effective annotations, such as image-level classification labels
[3, 26, 50], object points [4, 45], and bounding boxes [16, 28, 57, 36]. In
particular, image-level classification labels have commonly been employed as
weak labels due to their minimal or negligible annotation effort involved [2,
60]. With the absence of precise localization information, image-level WSSS
often necessitates the use of the coarse localization offered by class
activation maps (CAMs) [72]. CAMs pertain to the intermediate outputs derived
from a classification network. They can visually illustrate the activation
regions corresponding to each individual class. Thus, they are often used to
generate pseudo masks for training. However, CAMs suffer from i) Inconsistent
Activation: CAMs demonstrate variability and lack robustness in accommodating
geometric transformations of input images [60], resulting in inconsistent
activation regions for the same input. ii) Inaccurate Activation: activation
region accuracy is often compromised, resulting in incomplete or excessive
class activation, only covering the discriminative object regions [1]. Despite
enhanced localization mechanisms in the variants GradCAM [55] and GradCAM++
[7], they still struggle to generate satisfactory pseudo-labels for WSSS [60].
Thus, many WSSS works are dedicated to studying CAM refinement or post-
processing [1, 15, 31].
In general, WSSS methods [2, 65, 18, 50] comprise three stages: CAM
generation, refinement, and segmentation training with pseudo-labels. This
multi-stage framework is known to be time-consuming and complex as several
models must be trained at different stages. In contrast, single-stage models
[3, 70, 52], which include a unified network of all stages, are more
efficient. They are trained to optimize the segmentation and classification
tasks at the same time; however, their CAMs are not explicitly trained. As a
result, they need refinement to produce high-quality pseudo-labels, often
leveraging hand-crafted modules, such as CRF in [70], PAMR in [3], PAR in [52,
53]. As the refinement modules are predefined and offline, they decouple the
CAMs from the primary optimization. When the refined CAMs are employed as
learning objectives for segmentation, the optimization of the segmentation
branch may deviate from that of the classification branch. Hence, it is
difficult for a single-stage model to optimize its segmentation task while
yielding satisfactory CAM pseudo-labels. This optimization difficulty
underlies the inferior performance in single-stage approaches compared to
multi-stage ones. Further, such hand-crafted refinement modules require
heuristic tuning and empirical changes, thereby limiting their adaptability to
novel datasets. Despite the potential benefits of post-refinement in
addressing the aforementioned two issues associated with CAMs, which have been
extensively discussed in WSSS studies, there has been limited exploration in
explicit online optimization for CAMs.
The absence of fully optimized CAMs is an important factor in the existing
indispensability of this refinement. In this paper, we take a different
approach, proposing a model that optimizes CAMs in an end-to-end fashion,
resulting in reliable, consistent and accurate CAMs for WSSS without the
necessity for subsequent refinements in two respects. First, we note that even
though CAM is differentiable, it is not robust to variation. As the
intermediate output of a classification model, CAMs are not fully optimized
for segmentation purpose since the primary objective is to minimize
classification error. This implies that within an optimized network, numerous
weight combinations exist that can yield accurate classification outcomes,
while generating CAMs of varying qualities. To investigate this, we conduct
oracle experiments, training a classification model while simultaneously
guiding the CAMs with segmentation ground truth. A noticeable enhancement in
quality is observed in guided compared to vanilla CAMs, without compromising
classification accuracy. Second, we demonstrate the feasibility of
substituting the oracle with segmentation pseudo-labels (SPL) in the context
of weak supervision. Consequently, we harness the potential of SPL for WSSS by
co-training both CAMs and segmentation through mutual learning.
We explore an effective way to substitute the CAM refinement process, i.e.
guiding CAMs in an end-to-end fashion. Our method optimizes the CAMs and
segmentation prediction simultaneously thanks to the differentiability of
CAMs. To achieve this, we adopt a dual-stream framework that includes an
online network (ON) and an assignment network (AN), inspired by self-
supervised frameworks [5, 22]. The AN is responsible for producing CAM pseudo-
labels (CPL) and segmentation pseudo-labels (SPL) to train the ON. Since CPL
and SPL are swapped for supervising segmentation and CAMs, respectively, our
method is named Co-training with Swapping Assignments (CoSA).
Our end-to-end framework enables us to leverage the quantified reliability of
pseudo-labels for training online, as opposed to relying on offline hard
pseudo-labels as the existing methodologies [2, 65, 15, 50]. Thus, our model
can incorporate soft regularization driven by uncertainty, where the CPL
perplexity is continually assessed throughout training. This regularization is
designed to adaptively weight the segmentation loss, considering the varying
reliability levels of different regions. By incorporating the soft CPL, CoSA
enables dynamic learning at different time-steps rather than the performance
being limited by the predefined CPL.
The threshold is a key parameter for generating the CPL [60, 50, 53]. It not
only requires tuning but also necessitates dynamic adjustment to align with
the model’s learning state at various time-steps. CoSA integrates threshold
searching to dynamically adapt its learning state, as opposed to the commonly
used hard threshold scheme [18, 13, 52]. With the proposed dynamic
thresholding, we eliminate the laborious task of manual parameter tuning and
enhance performance.
We further address a common issue in WSSS with CAMs, known as the coexistence
problem, whereby certain class activations display extensive false positive
regions that inaccurately merge the objects with their surroundings. In
response, we introduce a technique to leverage low-level CAMs enriched with
object-specific details to contrastively separate the foreground regions from
the background.
The proposed CoSA greatly surpasses existing WSSS methods. Our approach
achieves 76.2% and 75.1% mIoU on the VOC val and test splits, and 51.0% on the
COCO val split, which are +5.1%, +3.9%, and +8.7% ahead of the previous
single-stage SOTA [53].
The contributions of this paper are as follows: 1) We are the first to propose
SPL as a substitute for guiding CAMs in WSSS. We present compelling evidence
showcasing its potential to produce more reliable, consistent and accurate
CAMs. 2) We introduce a dual-stream framework with swapped assignments in
WSSS, which co-optimizes the CAMs and segmentation predictions in an end-to-
end fashion. 3) We develop a reliability-based adaptive weighting in
accommodating the learning dynamics. 4) We incorporate threshold searching to
automatically adjust the threshold, ensuring alignment with the learning state
at different training time-steps. 5) We address the CAM coexistence issue and
propose a contrastive separation approach to regularize CAMs. This greatly
alleviates the coexistence problem, significantly enhancing the results of
affected classes. 6) We demonstrate CoSA’s SOTA results on key challenging
WSSS benchmarks, significantly surpassing existing methods. Our source code
and model weights will be available.
## 2 Related Work
Multi-Stage WSSS. The majority of image-level WSSS work is multi-stage,
typically comprising three stages: classification training (CAM generation),
CAM refinement, and segmentation training. Some approaches employ heuristic
strategies to address incomplete activation regions. For example, adversarial
erasing [71, 58, 30, 68], feature map optimization [32, 14, 13, 12], self-
supervised learning [60, 11, 56], and contrastive learning [27, 64, 73, 15]
are employed. Some methods focus on post-refining the CAMs by propagating
object regions from the seeds to their semantically similar pixels.
AffinityNet [2], for instance, learns pixel-level affinity to enhance CPL.
This has motivated other work [1, 20, 9, 38] that utilize additional networks
to generate more accurate CPL. Other work is dedicated to studying
optimization given the coarse pseudo-labels: [40] explores uncertainty of
noisy labels, [43] adaptively corrects CPL during early learning, and [50]
enhances boundary prediction through co-training. Since image-level labels
alone do not yield satisfactory results, several methods incorporate
additional modalities, such as saliency maps [37, 38, 73, 18] and CLIP models
[67, 42, 63]. More recently, vision transformers [17] have emerged as
prominent models for various vision tasks. Several WSSS studies benefit from
vision transformers: [21] enhance CAMs by incorporating the attention map from
ViT; [65] introduces class-specific attention for discriminative object
localization; [42] and [67] leverage multi-modal transformers to enhance
performance.
Single-Stage WSSS. In contrast, single-stage methods are much more efficient.
They contain a shared backbone with heads for classification and segmentation
[3, 70, 52, 53]. The pipeline involves generating and refining the CAMs,
leveraging an offline module, such as PAMR [3], PAR [52], or CRF [70].
Subsequently, the refined CPL are used for segmentation. Single-stage methods
exhibit faster speed and a lower memory footprint but are challenging to
optimize due to the obfuscation in offline refinement. As a result, they often
yield inferior performance compared to multi-stage methods. More recently,
with the success of ViT, single-stage WSSS has been greatly advanced. AFA [52]
proposes learning reliable affinity from attention to refine the CAMs.
Similarly, ToCo [53] mitigates the problem of over-smoothing in vision
transformers by contrastively learning from patch tokens and class tokens.
The existing works depend heavily on offline refinement of CAMs. In this
study, we further explore the potential of single-stage approaches and
showcase the redundancy of offline refinement. We propose an effective
alternative for generating consistent, and accurate CAMs in WSSS.
## 3 Method
### 3.1 Guiding Class Activation Maps
Class activation maps are determined by the feature map $F$ and the weights
$W_{\text{fc}}$ for the last FC layer [72]. Let us consider a $C$ classes
classification problem:
$\small\mathcal{L}_{\text{cls}}(Z,Y)\\!=\\!\frac{-1}{C}\sum_{c=1}^{C}\\!\Big{[}Y^{c}\log\sigma_{Z}^{c}+(1-Y^{c})\log\left(1-\sigma_{Z}^{c}\right)\Big{]},$
(1)
where $\sigma_{Z}^{c}\triangleq\sigma({Z^{c}})$ represents Sigmoid activation,
$Y\triangleq Y_{\text{gt}}$ denotes the one-hot multi-class label, and
$Z\triangleq GW_{\text{fc}}^{\top}\\!\in\\!\mathbb{R}^{C}$ represents the
prediction logits, derived from the final FC layer, where
$G\\!=\\!\texttt{Pooling}(F)\\!\in\\!\mathbb{R}^{D}$ is a spatial pooled
feature from $F\\!\in\\!\mathbb{R}^{HW\times D}$. During training, Eq. 1 is
optimized with respect to the learnable parameters $\theta$ in the backbone.
When gradients flow backwards from $G$ to $F$, only a fraction of elements in
$F$ get optimized, implying that a perturbation in $F$ does not guarantee
corresponding response in $G$ due to the spatial pooling, resulting in non-
determinism in the feature map $F$. This indeterminate nature can lead to
stochasticity of the generated CAMs.
Figure 1: Oracle Experiments on VOC. CAMs are guided by the ground truth (GT),
proposed segmentation pseudo-labels (SPL), no guidance (NO) and random noise
(NS). (a): classification performance; (b): CAM quality; (c) CAMs
visualization. All experiments employ 2k-iters warm-up before guidance is
introduced.
To demonstrate this, we conduct oracle experiments in which we supervise the
output CAMs from a classifier directly with the ground truth segmentation
(GT). This enables all elements in $F$ to be optimized. For comparison, we
conduct experiments wherein the CAMs are (i) not guided (NO), (ii) guided with
masks of random noise (NS).
Results, shown in Fig. 1, demonstrate that different guidance for $M$ does not
affect classification even for the NS group, as all experiment groups achieved
over 97% classification precision. However, drastic differences can be
observed w.r.t. the quality of the CAMs. The GT group results in a notable
quality improvement over the NO group, as shown in Fig. 1(b)(c). In contrast,
the NS group sabotages the CAMs. This suggests the stochasticity of CAMs and
explains their variability and lack robustness, something also observed in [2,
60, 13].
Figure 2: Co-training with Swapping Assignments (CoSA). We propose an end-to-
end dual-stream weakly-supervised segmentation framework, capable of co-
optimizing the segmentation prediction and CAMs by leveraging the swapped
assignments, namely CAM pseudo-labels (CPL) and segmentation pseudo-labels
(SPL). Our framework comprises two networks: an assignment network (AN) and an
online network (ON), where the AN is responsible for generating pseudo-labels
for training the ON. While the AN has identical architecture to the ON, it is
updated through exponential moving average (EMA) of the ON. The diagram on the
right provides an illustration of the architecture. Given weak-augmented
images as input, the AN produces CPL to supervise segmentation in the ON
($\mathcal{L}_{\text{c2s}}$). During training, the CPL is softened by
reliability-based adaptive weighting (RAW), formed based on CAM perplexity
estimation and dynamic thresholding. The AN also generates SPL which is
utilized to supervise the CAMs ($\mathcal{L}_{\text{s2c}}$). Further, the CAMs
are regularized to contrastively separate the foreground from the background
regions ($\mathcal{L}_{\text{csc}}$). Note that the ON is also trained for
classification using the image-level class labels
($\mathcal{L}_{\text{cls}}$).
Since relying on GT segmentation is not feasible in WSSS, we propose an
alternative for guiding CAMs, employing mask predictions as segmentation
pseudo-labels (SPL). As shown in Fig. 1, an SPL-guided classifier yields CAMs
that significantly outperform vanilla CAMs (NO group), performing close to the
oracle trained with the GT. Motivated by this, we introduce a co-training
mechanism in which CAMs and mask predictions are optimized mutually without
any additional CAM refinement.
### 3.2 Co-training with Swapping Assignments
Overall Framework. As shown in Fig. 2, CoSA contains two networks: an online
network (ON) and an assignment network (AN). ON, parameterized by $\Theta$,
comprises three parts: a backbone encoder, FC layers, and a segmentation head.
AN has the same architecture as ON but uses different weights, denoted
$\Theta^{\prime}$. ON is trained with the pseudo assignments generated by AN,
while AN is updated by the exponential moving average of ON:
$\Theta^{\prime}\leftarrow m\Theta^{\prime}+(1-m)\Theta,$ where $m\in[0,1]$
denotes a momentum coefficient. Consequently, the weights of AN represent a
delayed and more stable version of the weights of ON, which helps to yield a
consistent and stabilized learning target [22].
An image and class label pair $(x,Y_{\text{gt}})$ is randomly sampled from a
WSSS dataset $\mathcal{D}$. CoSA utilizes two augmented views
$\mathcal{T}_{s}(x)$ and $\mathcal{T}_{w}(x)$ as input for ON and AN,
respectively, representing strong and weak image transformations. During
training, AN produces CAMs $\mathcal{M}^{\prime}$ and segmentation predictions
$\mathcal{S}^{\prime}$. The CAM pseudo-labels (CPL) and segmentation pseudo-
labels (SPL) are generated by $\mathcal{M}^{\prime}$ and
$\mathcal{S}^{\prime}$ after filtering with respect to $Y_{\text{gt}}$. CPL
and SPL are subsequently used as learning targets for supervising the
segmentation predictions $\mathcal{S}$ and CAMs $\mathcal{M}$ from ON,
respectively.
Swapping Assignments. Our objective is to co-optimize $\mathcal{S}$ and
$\mathcal{M}$. A naive approach could enforce the learning objectives
$\mathcal{S}\triangleq\mathcal{S}^{\prime}$ and
$\mathcal{M}\triangleq\mathcal{M}^{\prime}$ as a knowledge distillation
process [25], where AN and ON play the roles of teacher and student. However,
this assumes availability of a pretrained teacher which is not possible in
WSSS settings. Instead, we setup a swapped self-distillation with the
objective:
$\small\mathcal{L}_{\text{swap}}=\mathcal{L}_{\text{c2s}}(\mathcal{S},\mathcal{M}^{\prime})+\mathcal{L}_{\text{s2c}}(\mathcal{M},\mathcal{S}^{\prime})~{},\vspace{-5pt}$
(2)
where $\mathcal{L}_{\text{c2s}}$ optimizes the segmentation performance given
the CPL, and $\mathcal{L}_{\text{s2c}}$ considers the CAM quality with respect
to SPL. Building on self-distillation [6, 48], we present this swapped self-
distillation framework tailored to facilitate information exchange between the
CAMs and segmentation.
Figure 3: CPL Analysis on val splits of VOC (a, c, e) and COCO (b, d, f). (a)
and (b): heatmap of CPL accuracy vs. confident ranges (x-axis) for different
time-steps (y-axis). (c) and (d): correlation between perplexity and accuracy
of CPL for different time-steps. (e) and (f): distribution of CAMs’ confidence
categorized by the proposed dynamic threshold. (best viewed under zoom)
### 3.3 Segmentation Optimization.
CAM2Seg Learning. As the CAMs in CoSA are inherently guided, extra refinement
[29, 2, 52] is not required, and they can be directly employed as learning
targets. Nonetheless, CAMs primarily concentrate on the activated regions of
the foreground while disregarding the background. As per the established
convention [60, 15, 53], a threshold value $\xi$ is employed for splitting the
foreground and the background. Formally, the CAM pseudo-label (CPL) is given
by:
$\small\hat{\mathcal{Y}}^{\text{CPL}}_{x,y}=\left\\{\begin{aligned}
&\mathtt{argmax}(\mathcal{M}_{x,y}^{\prime})+1,&\text{if $\nu\geq\xi$,}\\\
&0,&\text{if $\nu<\xi$,}\\\ \end{aligned}\right.,\vspace{-5pt}$ (3)
where $\nu\triangleq\mathtt{max}(\mathcal{M}_{x,y}^{\prime})$ denotes the the
maximum activation, $0$ denotes the background index. Then, the CAM2seg
learning objective $\mathcal{L}_{\text{c2s}}$ is cross entropy between
${\mathcal{Y}}^{\text{CPL}}$ and $\mathcal{S}$, as with the general supervised
segmentation loss [10].
Reliability based Adaptive Weighting. Segmentation performance depends heavily
on the pseudo-labels. Despite the high-quality CPL generated by our guided
CAMs, their reliability must be assessed, particularly in the initial training
phases. Existing methods use post-refinement to increase reliability [70, 3].
As CoSA can generate online pseudo-labels, we propose to leverage confidence
information to compensate the CAM2Seg loss during training.
Specifically, we propose to assess the perplexity scores for each pixel in
$\hat{\mathcal{Y}}^{\text{CPL}}$ and leverage these scores to re-weight
$\mathcal{L}_{\text{c2s}}$ for penalizing unreliable regions. However,
estimating per-pixel perplexity is non-trivial. Through empirical analysis, we
observe a noteworthy association between the confidence values of CAMs and
their accuracy at each time-step. This correlation suggests that regions with
extremely low or high confidence exhibit higher accuracy throughout training,
as shown in Fig. 3(a)(b). To quantitatively model perplexity, we make two
assumptions: i) the reliability of pseudo-labels is positively correlated with
their accuracy, and ii) the perplexity score is negatively correlated with the
reliability. Then, per-pixel perplexity of
$\hat{\mathcal{Y}}^{\text{CPL}}_{x,y}$ is defined as:
$\small\mathcal{P}_{x,y}=\left\\{\begin{aligned}
&\left[-\log\left(\lambda_{\alpha}(\nu-\xi)/(1-\xi)\right)\right]^{\lambda_{\beta}}&\text{if
$\nu\geq\xi$,}\\\
&\left[-\log\left(\lambda_{\alpha}(\xi-\nu)/\xi\right)\right]^{\lambda_{\beta}}&\text{if
$\nu<\xi$,}\\\ \end{aligned}\right.$ (4)
where the term within the logarithm denotes the normalized distance to $\xi$
in $[0,1]$. The logarithm ensures $\mathcal{P}_{x,y}\\!\rightarrow\\!+\infty$
as distance $\\!\rightarrow\\!0$, and $\mathcal{P}_{x,y}\\!\rightarrow\\!0$ as
distance $\rightarrow\\!1$. $\lambda_{\alpha}\in\mathbb{R}^{+}$ controls the
perplexity score’s minimum value and $\lambda_{\beta}\in\mathbb{R}^{+}$
determines the sharpness or smoothness of the distribution. Higher
$\mathcal{P}_{x,y}$ indicates confidence of
$\hat{\mathcal{Y}}^{\text{CPL}}_{x,y}$ closer to threshold $\xi$. This
observation is substantiated by Fig. 3 (a)(b), where confidence values near
$\xi\\!=\\!0.5$ exhibit lower reliability. Furthermore, the correlation
between perplexity and accuracy remains significant across various training
time-steps and datasets, as depicted in Fig. 3(c)(d).
Since we hypothesize negative reliability-perplexity correlation, the
reliability score can be defined as the reciprocal of perplexity. To
accommodate reliability variation for different input, we use the normalized
reliability as the per-pixel weights for $\mathcal{L}_{\text{c2s}}$. Thus,
Reliability based Adaptive Weighting (RAW) is defined as:
$\mathcal{W}_{x,y}^{\text{raw}}=|\mathcal{R}|/\sum_{i,j\in\mathcal{R}}{(\mathcal{P}_{i,j}\mathcal{P}_{x,y})}^{-1}$,
where $|\mathcal{R}|$ represents total number of pixels in a batch. Then, the
re-weighted CAM2seg loss for each position $(x,y)$ can be defined as
$\small\mathcal{L}_{\text{c2s}}(x,y)\\!\\!=\\!\\!-\mathcal{W}_{x,y}^{\text{raw}}\\!\sum_{c=0}^{C}\\!\Bigg{[}\\!\mathbbm{1}\\!\big{[}\hat{\mathcal{Y}}^{\text{CPL}}_{x,y}\\!\\!=\\!\\!c\big{]}\\!\\!\log\\!\\!\Bigg{(}\\!\frac{\exp{\mathcal{S}^{c}_{x,y}}}{\sum_{k=0}^{C}\exp\mathcal{S}^{k}_{x,y}}\\!\\!\Bigg{)}\\!\\!\Bigg{]}.$
(5)
Dynamic Threshold. Existing WSSS work [52, 53] prescribes a fixed threshold to
separate foreground and background, which neglects inherent variability due to
prediction confidence fluctuation during training. Obviously, applying a fixed
threshold in Fig. 3(a)(b) is sub-optimal. To alleviate this, we introduce
dynamic thresholding. As shown in Fig. 3(e)(f), the confidence distribution
reveals discernible clusters. We assume the foreground and background pixels
satisfy a bimodal Gaussian Mixture distribution. Then, the optimal dynamic
threshold $\xi^{\star}$ is determined by maximizing the Gaussian Mixture
likelihood:
$\displaystyle\xi^{\star}=\underset{\xi}{\mathtt{argmax}}$
$\displaystyle\prod_{x\in\\{\mathcal{M}^{\prime}\geq\xi\\}}\tilde{\pi}_{fg}\mathcal{N}\left(x\mid\tilde{\mu}_{fg},\tilde{\Sigma}_{fg}\right)$
(6) $\displaystyle+$
$\displaystyle\prod_{x\in\\{\mathcal{M}^{\prime}<\xi\\}}\tilde{\pi}_{bg}\mathcal{N}\left(x\mid\tilde{\mu}_{bg},\tilde{\Sigma}_{bg}\right)~{},$
where $\mathcal{N}(x\mid\mu,\Sigma)$ denotes the Gaussian function and $\pi$,
$\mu$, $\Sigma$ represent the weight, mean and covariance. To avoid mini-batch
bias, we maintain a queue to fit GMM, with the current $\mathcal{M}^{\prime}$
batch enqueued and the oldest dequeued. This practice facilitates the
establishment of a gradually evolving threshold, contributing to learning
stabilization.
### 3.4 CAM Optimization.
Seg2CAM Learning. To generate SPL, segmentation predictions
$\mathcal{S}^{\prime}$ are filtered by the weak labels $Y_{\text{gt}}$ and
transformed into probabilities:
$\small\mathcal{S}^{\prime~{}c}_{x,y}=\left\\{\begin{aligned}
&-\infty,\\!\\!\\!&\text{if $Y_{\text{gt}}^{c}=0$,}\\\
&\mathcal{S}^{\prime~{}c}_{x,y},\\!\\!\\!&\text{if $Y_{\text{gt}}^{c}\neq
0$,}\\\
\end{aligned}\right.\;\;\;\;\hat{\mathcal{Y}}^{\text{SPL}}_{x,y}=\mathtt{Softmax}(\frac{\mathcal{S}^{\prime}_{x,y}}{\tau})~{},\vspace{-5pt}$
(7)
where $\tau$ represents the softmax temperature to sharpen the
$\hat{\mathcal{Y}}^{\text{SPL}}$. Then, we arrive Seg2CAM learning objective:
$\displaystyle\mathcal{L}_{\text{s2c}}=-\frac{1}{C|\mathcal{R}|}$
$\displaystyle\sum_{c=1}^{C}\sum_{x,y\in\mathcal{R}}\bigg{[}\hat{\mathcal{Y}}^{\text{SPL}}_{x,y}[c]\log(\sigma(\mathcal{M}^{c}_{x,y}))$
(8) $\displaystyle+$
$\displaystyle(1-\hat{\mathcal{Y}}^{\text{SPL}}_{x,y}[c])\log(1-\sigma(\mathcal{M}^{c}_{x,y}))\bigg{]}~{},$
where $\mathcal{R}$ represents all the positions in SPL.
Coexistence Problem in CAMs. Certain class activations often exhibit large
false positive regions, where objects are incorrectly merged with
surroundings, as shown in Fig. 8 (1st row). For instance, the classes ’bird’
and ’tree branches’, ’train’ and ’railways’, etc. frequently appear together
in VOC. We refer to this issue as the coexistence problem. We hypothesize that
the coexistence problem is attributed to three factors: i) Objects that
coexist in images, such as ’tree branches’, are not annotated w.r.t. weak
labels, which makes it challenging for a model to semantically distinguish
coexistence. ii) Training datasets lack sufficient samples for such classes.
iii) High-level feature maps, though rich in abstract representations and
semantic information, lack essential low-level features such as edges,
textures, and colors [24]. Thus, CAMs generated from the last layer are poor
in low-level information for segmenting objects. Conversely, segmenting
objects with high-level semantics is hindered due to factors i) and ii).
Figure 4: $\mathcal{M}$ and $\mathcal{M}^{\dagger}$ Comparisons. (a): mIoU vs.
time-steps for $\mathcal{M}$ and $\mathcal{M}^{\dagger}$ on VOC val. (b): same
as (a) but filtered by perplexity. (c): cases of coexistence issue in
$\mathcal{M}$ but not in $\mathcal{M}^{\dagger}$.
Contrastive Separation in CAMs. We posit that the effective usage of low-level
information can alleviate the coexistence problem. Since shallower-layer
feature maps contain more low-level information, we propose to extract CAMs
from earlier in the backbone, denoted $\mathcal{M}^{\dagger}$, and present a
comparison with $\mathcal{M}$ in Fig. 4, showing that substituting
$\mathcal{M}$ with $\mathcal{M}^{\dagger}$ is not feasible due to the lower
mIoU upperbound of $\mathcal{M}^{\dagger}$. If we consider the confident
regions in $\mathcal{M}$ and $\mathcal{M}^{\dagger}$, i.e. filter by a low-
pass perplexity $\mathcal{P}$,
$\\{\mathcal{M}^{\dagger}_{x,y}\\!\mid\\!\mathcal{P}_{x,y}\\!\leq\\!\epsilon\\}$
are better than
$\\{\mathcal{M}_{x,y}\\!\mid\\!\mathcal{P}_{x,y}\\!\leq\\!\epsilon\\}$, as
shown in Fig. 4(b), where $\epsilon$ represents a low-pass coefficient. Fig.
4(c) illustrates the presence of coexistence issues in $\mathcal{M}$ but
absence in $\mathcal{M}^{\dagger}$. Those findings suggest that
$\mathcal{M}^{\dagger}$ is worse than $\mathcal{M}$ in general, but better for
those regions with low perplexity.
In CoSA, we propose to regularize $\mathcal{M}$ by
$\mathcal{M}^{\dagger\prime}$ (from AN). We define positive
$\mathcal{R}^{+}_{i,j}$ and negative $\mathcal{R}^{-}_{i,j}$ regions as
$\displaystyle\mathcal{R}^{+}_{i,j}$
$\displaystyle=\Big{\\{}(x,y)\mid\mathcal{P}_{x,y}\leq\epsilon,~{}\hat{y}^{\text{CPL}}_{x,y}=\hat{y}^{\text{CPL}}_{i,j},(x,y)\neq(i,j)\Big{\\}}$
(9) $\displaystyle\mathcal{R}^{-}_{i,j}$
$\displaystyle=\Big{\\{}(x,y)\mid\mathcal{P}_{x,y}\leq\epsilon,~{}\hat{y}^{\text{CPL}}_{x,y}\neq\hat{y}^{\text{CPL}}_{i,j}\Big{\\}}~{},$
where $(i,j)\\!\in\\!\Omega$,
$\Omega\\!=\\!\\{(x,y)\\!\mid\\!\mathcal{P}_{x,y}\\!\leq\\!\epsilon\\}$ is
low-perplexity region in $\mathcal{M}^{\dagger\prime}$, and
$\hat{y}^{\text{CPL}}$ represents the CPL of $\mathcal{M}^{\dagger\prime}$.
Then, we arrive the contrastive separation loss for $\mathcal{M}$:
$\small\mathcal{L}_{\text{csc}}=-\frac{1}{\lvert\Omega\rvert}\sum_{i,j\in\Omega}\frac{1}{\lvert\mathcal{R}^{+}_{i,j}\rvert}\sum_{x,y\in\mathcal{R}^{+}_{i,j}}\log\frac{L_{x,y}^{i,j}}{L_{x,y}^{i,j}+K_{n,m}^{i,j}}~{},$
(10)
where
$L_{x,y}^{i,j}\\!=\\!\exp(l_{d}(\mathcal{M}_{i,j},\mathcal{M}_{x,y})/\tau)$,
$l_{d}(a,b)$ measures the (a,b) distance, $\tau$ denotes the InfoNCE loss [47]
temperature, and
$K_{n,m}^{i,j}\\!=\\!\sum_{n,m\in\mathcal{R}^{-}_{i,j}}L_{n,m}^{i,j}$.
Overall Objectives. The objectives encompass the aforementioned losses and a
further $\mathcal{L}_{\text{c2s}}^{\mathcal{M}^{\dagger}}$ to stabilize
training and accelerate convergence, resulting in the CoSA objective:
$\small\mathcal{L}_{\text{CoSA}}\\!\\!=\\!\mathcal{L}_{\text{cls}}\\!+\mathcal{L}_{\text{cls}}^{\mathcal{M}^{\dagger}}\\!\\!+\\!\lambda_{\text{c2s}}\big{(}\mathcal{L}_{\text{c2s}}\\!+\\!\mathcal{L}_{\text{c2s}}^{\mathcal{M}^{\dagger}}\big{)}\\!+\\!\lambda_{\text{s2c}}\mathcal{L}_{\text{s2c}}+\\!\lambda_{\text{csc}}\mathcal{L}_{\text{csc}}.$
(11)
## 4 Experiments
### 4.1 Experiment Details and Results
Datasets. We evaluate on two benchmarks: PASCAL VOC 2012 [19] and MS-COCO 2014
[41]. VOC encompasses 20 categories with train, val, and test splits of 1464,
1449, and 1456 images. Following WSSS practice [2, 3, 65], SBD [23] is used to
augment the train split to 10,582. COCO contains 80 categories with train and
val splits of approx. 82K and 40K images. Our model is trained and evaluated
using _only_ the image-level classification labels111Not available for VOC
test split and so not used in evaluation., and employing mIoU as evaluation
metrics.
Implementation Details. Following [53], we use ImageNet pretrained ViT-base
(ViT-B) [17] as the encoder. For classification, we use global max pooling
(GMP) [51] and the CAM approach [72]. For the segmentation decoder, we use
LargeFOV [10], as with [53]. ON is trained with AdamW [46]. The learning rate
is set to 6E-5 in tandem with polynomial decay. AN is updated with a momentum
of $0.9994$. For preprocessing, the images are cropped to $448^{2}$, then
weak/strong augmentations are applied (see Supp. Materials). The perplexity
constants $(\lambda_{\alpha},\lambda_{\beta})$ are set to $(0.8,1)$, GMM-
fitting queue length is $100$, and softmax temperature $\tau$ is $0.01$. The
low perplexity threshold $\epsilon$ is set to $1$ and the loss weight factors
$(\lambda_{\text{c2s}},\lambda_{\text{s2c}},\lambda_{\text{csc}})$ to
$(0.1,0.05,0.1)$.
Method | Backbone | train | val
---|---|---|---
RRM [70] AAAI’2020 | WR38 | – | 65.4
1Stage [3] CVPR’2020 | WR38 | 66.9 | 65.3
AFA [52] CVPR’2022 | MiT-B1 | 68.7 | 66.5
MCT [65] CVPR’2022 | MCT | 69.1 | –
ViT-PCM [51] ECCV’2022 | ViT-B | 71.4 | 69.3
Xu _et al_. [66] CVPR’2023 | ViT-B | 66.3 | –
ACR-ViT [31] CVPR’2023 | ViT-B | 70.9 | –
CLIP-ES [42] CVPR’2023 | ViT-B | 75.0 | –
ToCo [53] CVPR’2023 | ViT-B | 73.6 | 72.3
CoSA | ViT-B | 78.5 | 76.4
CoSA∙ | ViT-B | 78.9 | 77.2
Table 1: Comparisons of CPL. CAM pseudo-labels evaulation on VOC dataset. Backbone denotes the encoder used for generating the CAMs. $\bullet$ represents the ensemble of $\mathcal{M}^{\prime}$ and $\mathcal{M}^{\dagger\prime}$ in CoSA. Methods | Sup. | Net. | VOC | COCO
---|---|---|---|---
val | test | val
Supervised Upperbounds. | |
Deeplab [10] TPAMI’2017 | $\mathcal{F}$ | R101 | 77.6 | 79.7 | –
WideRes38 [61] PR’2019 | $\mathcal{F}$ | WR38 | 80.8 | 82.5 | –
ViT-Base [17] ICLR’2021 | $\mathcal{F}$ | ViT-B | 80.5 | 81.0 | –
UperNet-Swin [44] ICCV’2021 | $\mathcal{F}$ | SWIN | 83.4 | 83.7 | –
Multi-stage Methods. | |
L2G [26] CVPR’2022 | $\mathcal{I}+\mathcal{S}$ | R101 | 72.1 | 71.7 | 44.2
Du _et al_. [18] CVPR’2022 | $\mathcal{I}+\mathcal{S}$ | R101 | 72.6 | 73.6 | –
CLIP-ES [42] CVPR’2023 | $\mathcal{I}+\mathcal{L}$ | R101 | 73.8 | 73.9 | 45.4
ESOL [39] NeurIPS’2022 | $\mathcal{I}$ | R101 | 69.9 | 69.3 | 42.6
BECO [50] CVPR’2023 | $\mathcal{I}$ | R101 | 72.1 | 71.8 | 45.1
Mat-Label [59] ICCV’2023 | $\mathcal{I}$ | R101 | 73.0 | 72.7 | 45.6
CoSA-MS | $\mathcal{I}$ | R101 | 76.5 | 75.3[1] | 50.9
Xu _et al_. [66] CVPR’2023 | $\mathcal{I}+\mathcal{L}$ | WR38 | 72.2 | 72.2 | 45.9
W-OoD [35] CVPR’2022 | $\mathcal{I}$ | WR38 | 70.7 | 70.1 | –
MCT [65] CVPR’2022 | $\mathcal{I}$ | WR38 | 71.9 | 71.6 | 42.0
ex-ViT [69] PR’2023 | $\mathcal{I}$ | WR38 | 71.2 | 71.1 | 42.9
ACR-ViT [31] CVPR’2023 | $\mathcal{I}$ | WR38 | 72.4 | 72.4 | –
MCT+OCR [15] CVPR’2023 | $\mathcal{I}$ | WR38 | 72.7 | 72.0 | 42.0
CoSA-MS | $\mathcal{I}$ | WR38 | 76.6 | 74.9[2] | 50.1
ReCAM [14] CVPR’2022 | $\mathcal{I}$ | SWIN | 70.4 | 71.7 | 47.9
LPCAM [12] CVPR’2023 | $\mathcal{I}$ | SWIN | 73.1 | 73.4 | 48.3
CoSA-MS | $\mathcal{I}$ | SWIN | 81.4 | 78.4[3] | 53.7
Single-stage (End-to-end) Methods. | |
1Stage [3] CVPR’2020 | $\mathcal{I}$ | WR38 | 62.7 | 64.3 | –
RRM [70] AAAI’2020 | $\mathcal{I}$ | WR38 | 62.6 | 62.9 | –
AFA [52] CVPR’2022 | $\mathcal{I}$ | MiT-B1 | 66.0 | 66.3 | 38.9
RRM [70]† AAAI’2020 | $\mathcal{I}$ | ViT-B | 63.1 | 62.4 | –
ViT-PCM [51] ECCV’2022 | $\mathcal{I}$ | ViT-B | 69.3 | – | 45.0
ToCo [53] CVPR’2023 | $\mathcal{I}$ | ViT-B | 71.1 | 72.2 | 42.3
CoSA | $\mathcal{I}$ | ViT-B | 76.2 | 75.1[4] | 51.0
CoSA∗ | $\mathcal{I}$ | ViT-B | 76.4 | 75.2[5] | 51.1
Table 2: Weakly Supervised Semantic Segmentation Results. Sup.: supervision
type. Net.: segmentation backbone. $\mathcal{F}$: Fully supervised,
$\mathcal{I}$: Image-level labels, $\mathcal{S}$: Saliency maps,
$\mathcal{L}$: language models. $*$ represents CRF [10] postprocessing
results.
CAM Quality Comparison. Tab. 1 shows CoSA CPL results on VOC compared with
existing WSSS methods, using our $\hat{\mathcal{Y}}^{\text{CPL}}$
($\xi\\!\\!=\\!\\!0.5$). Our method yields 78.5% and 76.4% mIoU on train and
val. Notably, an ensemble of $\mathcal{M}^{\prime}$ and
$\mathcal{M}^{\dagger\prime}$ improves performance to 78.9% and 77.2%,
suggesting the activation of $\mathcal{M}^{\prime}$ is orthogonal to that of
$\mathcal{M}^{\dagger\prime}$.
Semantic Segmentation Comparison. We compare our method with existing SOTA
WSSS methods on VOC and COCO for semantic segmentation in Tab. 2. CoSA
achieves 76.2% and 75.1% on VOC12 val and test, respectively, surpassing the
highest-performing single-stage model (ToCo) by 5.1% and 2.9%, as well as all
multi-stage methods, including those with additional supervision. In the COCO
evaluation, CoSA consistently outperforms other approaches, demonstrating a
significant increase of 8.7% over the top-performing single-stage methods.
Further, there is a also 2.7% improvement over the leading multi-stage method
[12]. While our primary goal is to provide an end-to-end WSSS solution, we
also offer a multi-stage version of CoSA, denoted as CoSA-MS in Tab. 2, where
various standalone segmentation networks are trained using our CPL. Our CoSA-
MS models can also attains SOTA performance in multi-stage scenarios.
Qualitative Comparison. Fig. 5 presents CAMs and segmentation visualizations,
comparing with recent methods: MCT, BECO, and ToCo. As shown, our method can
generate improved CAMs and produce well-aligned segmentation, exhibiting
superior results in challenging segmentation problems with intra-class
variation and occlusions. In addition, CoSA performs well w.r.t. the
coexistence cases (Fig. 5 R1, R2), while existing methods struggle. Moreover,
CoSA reveals limitations in the GT segmentation (Fig. 5 R4).
Figure 5: Qualitative Comparison. The results are reported on the val splits
of VOC (in R1 \- R3) and COCO (in R4 \- R6). The official codebases and
provided weights for MCT [65], BECO [50], and ToCo [53] are used for this
comparison. (best viewed under zoom; see Supp. Materials for more high-res
Comparisons).
### 4.2 Ablation Studies
CoSA Module Analysis. We begin by employing CAMs directly as the supervision
signal for segmentation, akin to [70], albeit without refinement, and
gradually apply CoSA modules to this baseline. As shown in Tab. 3(a), the
mIoUs progressively improve with addition of our components. Further, we
examine the efficacy of each CoSA component. As shown in Tab. 3(b), the
elimination of each component results in deteriorated performance, most
notably for CSC.
(a)
mIoU (inc.) Base. GC SA RAW CSC DT VOC COCO ✓ 55.96 37.32 ✓ ✓ 63.09 (+7.13)
42.55 (+5.23) ✓ ✓ ✓ 64.41 (+8.45) 43.92 (+6.60) ✓ ✓ ✓ ✓ 68.22 (+12.26) 45.39
(+8.07) ✓ ✓ ✓ ✓ 71.66 (+15.70) 47.10 (+9.78) ✓ ✓ ✓ ✓ ✓ 75.54 (+19.58) 49.67
(+12.35) ✓ ✓ ✓ ✓ ✓ ✓ 76.19 (+20.23) 51.00 (+13.68)
(b)
mIoU (dec.) CoSA GC SA RAW CSC DT VOC COCO ✓ 76.19 51.00 ✓ ✗ 75.54 (-0.65)
49.67 (-1.33) ✓ ✗ 69.89 (-6.30) 45.95 (-5.05) ✓ ✗ 72.45 (-3.74) 47.83 (-3.17)
✓ ✗ 72.10 (-4.09) 49.04 (-1.96) ✓ ✗ 74.12 (-2.07) 49.67 (-1.33)
Table 3: Ablation Study on Contribution of Each Component. (a): gradually add
proposed components to baseline. (b): systematically exclude components from
CoSA. GC: Guided CAMs, SA: Swapping Assignments, RAW: Reliability based
Adaptive Weighting, CSC: Contrastive Separation in CAMs, and DT: Dynamic
Threshold. mIoU is reported on PASCAL VOC12 and COCO val splits.
(a) Source Detach train val GT None 83.99 80.16 NO – 72.28 71.38 SPL $F$ 74.19
73.36 SPL $W_{\text{fc}}$ 78.05 76.15 SPL None 78.54 76.37
(b) Method C-mIoU mIoU FPR [8] 53.09 53.34 MCT [65] 58.46 61.24 ToCo [53]
63.62 72.33 w/o CSC 62.61 67.82 w/ CSC 82.34 76.37
Table 4: Ablation Study of GC & CSC. (a): CPL performance comparison on VOC
for CAMs. Detach: stop gradient in GC for feature map $F$ or FC weights
$W_{\text{fc}}$. Source: guidance sources. (b): CPL performance comparison on
VOC. FPR, MCT and ToCo results are based on provided code and weights. C-mIoU:
mIoU for classes with coexistence issues.
Figure 6: Ablative Study of SA. The performance of SPL (left) and CPL (right)
w.r.t. iterations on VOC val set are shown for CoSA with or without SA.
Figure 7: Ablation Study of RAW. (left) boxplot of mIoU, perplexity and MAE to
(1,0) for individual CPLs on VOC val. (right) perplexity reduction over times.
Figure 8: Coexistence Problems & Effect of CSC. The class activation for bird,
train, airplane, and boat are presented from left to right. (best viewed under
zoom)
Impact of Guided CAMs. We evaluate the impact of including guided CAMs w.r.t.
CAM quality, comparing a baseline using vanilla CAMs [72] as CPL following
[70, 52] with the proposed guided CAMs. As shown in Tab. 4(a), our guided CAMs
notably enhance CPL quality by 6.26% and 4.99% for train and val splits.
Further, we conduct experiments to ascertain the extent to which the two CAM
components, feature map $F$ and classification weights $W_{\text{fc}}$, exert
greater impact on guiding CAMs. As shown, disconnection of $F$ from the
gradient chains results in 74.19% and 73.36%, while the detachment of
$W_{\text{fc}}$ decrease the results slightly to 78.05% and 76.37%. This
suggests that guiding CAMs primarily optimizes the feature maps, verifying our
initial hypothesis of the inherent non-deterministic feature map contributing
to the stochasticity of CAMs in Sec. 3.1.
Impact of Swapping Assignments (SA). Tab. 3(b) suggests that eliminating SA
results in significant mIoU decreases, highlighting the importance of this
training strategy. Further examination of the ON and AN w.r.t. SPL and CPL
indicates that, in later training stages, AN consistently outperforms ON for
both SPL and CPL, as shown in Fig. 8, due to AN performing a form of model
ensembling similar to Polyak-Ruppert averaging [49, 54]. We observe a
noticeable disparity of mIoUs between two ONs (solid orange line vs. solid
blue line in Fig. 8), which may be attributed to the superior quality of CPL
and SPL from the AN facilitating a more robust ON for CoSA. The momentum
framework, originally introduced to mitigate noise and fluctuations of the
online learning target [22, 6], is used for information exchange across CAMs
and segmentation in CoSA. To the best of our knowledge, we are the first to
apply and demonstrate the efficacy of this type of training scheme in WSSS.
Impact of RAW. Tab. 3(b) shows notable mIoU reduction without RAW. We conduct
further studies to investigate its effect on perplexity reduction. The boxplot
in Fig. 8 suggests that RAW leads to higher mIoU but lower perplexity. Fig.
8(right) illustrates a faster decrease in perplexity when RAW is used,
affirming its impact on perplexity reduction.
Impact of CSC. Our CSC is introduced to address the coexistence issue. We
establish C-mIoU to measure the CAM quality for those coexistence-affected
classes. As shown in Tab. 4(b), applying CSC sees a boost in C-mIoU and mIoU,
which surpass the existing methods. Some visual examples demonstrating these
enhancements are given in Fig. 8.
Impact of Dynamic Threshold. We evaluate CoSA using some predetermined
thresholds, comparing them with one employing dynamic threshold on VOC val
split (see Supp. Materials for results). The performance is sensitive to the
threshold, but dynamic thresholding achieves 0.65% increased performance over
the best manual finetuning while saving 80% of hyper-parameter searching time.
## 5 Conclusion
This paper presents an end-to-end WSSS method: Co-training with Swapping
Assignments (CoSA), which eliminates the need for CAM refinement and enables
concurrent CAM and segmentation optimization. Our empirical study reveals the
non-deterministic behaviors of CAMs and that proper guidance can mitigate such
stochasticity, leading to substantial quality enhancement. We propose explicit
CAM optimization leveraging segmentation pseudo-labels in our approach, where
a dual-stream model comprising an online network for predicting CAMs and
segmentation masks, and an ancillary assignment network providing swapped
assignments (SPL and CPL) for training, is introduced. We further propose
three techniques within this framework: RAW, designed to mitigate the issue of
unreliable pseudo-labels; contrastive separation, aimed at resolving
coexistence problems; and a dynamic threshold search algorithm. Incorporating
these techniques, CoSA outperforms all SOTA methods on both VOC and COCO WSSS
benchmarks while achieving exceptional speed-accuracy trade-off.
## References
* [1] Jiwoon Ahn, Sunghyun Cho, and Suha Kwak. Weakly supervised learning of instance segmentation with inter-pixel relations. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 2209–2218, 2019.
* [2] Jiwoon Ahn and Suha Kwak. Learning pixel-level semantic affinity with image-level supervision for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4981–4990, 2018.
* [3] Nikita Araslanov and Stefan Roth. Single-stage semantic segmentation from image labels. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4253–4262, 2020.
* [4] Amy Bearman, Olga Russakovsky, Vittorio Ferrari, and Li Fei-Fei. What’s the point: Semantic segmentation with point supervision. In European Conference on Computer Vision (ECCV), pages 549–565. Springer, 2016.
* [5] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. Neural Information Processing Systems (NeurIPS), 33:9912–9924, 2020.
* [6] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 9650–9660, 2021.
* [7] Aditya Chattopadhay, Anirban Sarkar, Prantik Howlader, and Vineeth N Balasubramanian. Grad-cam++: Generalized gradient-based visual explanations for deep convolutional networks. In IEEE Winter Conference on Applications of Computer Vision (WACV), pages 839–847. IEEE, 2018.
* [8] Liyi Chen, Chenyang Lei, Ruihuang Li, Shuai Li, Zhaoxiang Zhang, and Lei Zhang. Fpr: False positive rectification for weakly supervised semantic segmentation. In IEEE International Conference on Computer Vision (ICCV), pages 1108–1118, 2023.
* [9] Liyi Chen, Weiwei Wu, Chenchen Fu, Xiao Han, and Yuntao Zhang. Weakly supervised semantic segmentation with boundary exploration. In European Conference on Computer Vision (ECCV), pages 347–362. Springer, 2020.
* [10] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 40(4):834–848, 2017.
* [11] Qi Chen, Lingxiao Yang, Jian-Huang Lai, and Xiaohua Xie. Self-supervised image-specific prototype exploration for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4288–4298, 2022.
* [12] Zhaozheng Chen and Qianru Sun. Extracting class activation maps from non-discriminative features as well. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 3135–3144, 2023.
* [13] Zhang Chen, Zhiqiang Tian, Jihua Zhu, Ce Li, and Shaoyi Du. C-cam: Causal cam for weakly supervised semantic segmentation on medical image. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 11676–11685, 2022.
* [14] Zhaozheng Chen, Tan Wang, Xiongwei Wu, Xian-Sheng Hua, Hanwang Zhang, and Qianru Sun. Class re-activation maps for weakly-supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 969–978, 2022.
* [15] Zesen Cheng, Pengchong Qiao, Kehan Li, Siheng Li, Pengxu Wei, Xiangyang Ji, Li Yuan, Chang Liu, and Jie Chen. Out-of-candidate rectification for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 23673–23684, 2023.
* [16] Jifeng Dai, Kaiming He, and Jian Sun. Boxsup: Exploiting bounding boxes to supervise convolutional networks for semantic segmentation. In IEEE International Conference on Computer Vision (ICCV), pages 1635–1643, 2015.
* [17] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations (ICLR), 2021.
* [18] Ye Du, Zehua Fu, Qingjie Liu, and Yunhong Wang. Weakly supervised semantic segmentation by pixel-to-prototype contrast. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4320–4329, 2022.
* [19] Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision (IJCV), 88:303–338, 2010.
* [20] Junsong Fan, Zhaoxiang Zhang, Tieniu Tan, Chunfeng Song, and Jun Xiao. Cian: Cross-image affinity net for weakly supervised semantic segmentation. In AAAI Conference on Artificial Intelligence (AAAI), volume 34, pages 10762–10769, 2020.
* [21] Wei Gao, Fang Wan, Xingjia Pan, Zhiliang Peng, Qi Tian, Zhenjun Han, Bolei Zhou, and Qixiang Ye. Ts-cam: Token semantic coupled attention map for weakly supervised object localization. In IEEE International Conference on Computer Vision (ICCV), pages 2886–2895, 2021.
* [22] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent-a new approach to self-supervised learning. Neural Information Processing Systems (NeurIPS), 33:21271–21284, 2020.
* [23] Bharath Hariharan, Pablo Arbeláez, Lubomir Bourdev, Subhransu Maji, and Jitendra Malik. Semantic contours from inverse detectors. In IEEE International Conference on Computer Vision (ICCV), pages 991–998. IEEE, 2011.
* [24] Bharath Hariharan, Pablo Arbeláez, Ross Girshick, and Jitendra Malik. Hypercolumns for object segmentation and fine-grained localization. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 447–456, 2015.
* [25] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
* [26] Peng-Tao Jiang, Yuqi Yang, Qibin Hou, and Yunchao Wei. L2g: A simple local-to-global knowledge transfer framework for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 16886–16896, 2022.
* [27] Tsung-Wei Ke, Jyh-Jing Hwang, and Stella Yu. Universal weakly supervised segmentation by pixel-to-segment contrastive learning. In International Conference on Learning Representations (ICLR), 2020.
* [28] Anna Khoreva, Rodrigo Benenson, Jan Hosang, Matthias Hein, and Bernt Schiele. Simple does it: Weakly supervised instance and semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 876–885, 2017.
* [29] Philipp Krähenbühl and Vladlen Koltun. Efficient inference in fully connected crfs with gaussian edge potentials. Neural Information Processing Systems (NeurIPS), 24, 2011.
* [30] Hyeokjun Kweon, Sung-Hoon Yoon, Hyeonseong Kim, Daehee Park, and Kuk-Jin Yoon. Unlocking the potential of ordinary classifier: Class-specific adversarial erasing framework for weakly supervised semantic segmentation. In IEEE International Conference on Computer Vision (ICCV), pages 6994–7003, 2021.
* [31] Hyeokjun Kweon, Sung-Hoon Yoon, and Kuk-Jin Yoon. Weakly supervised semantic segmentation via adversarial learning of classifier and reconstructor. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 11329–11339, 2023.
* [32] Jungbeom Lee, Eunji Kim, Sungmin Lee, Jangho Lee, and Sungroh Yoon. Ficklenet: Weakly and semi-supervised semantic image segmentation using stochastic inference. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 5267–5276, 2019.
* [33] Jungbeom Lee, Eunji Kim, Jisoo Mok, and Sungroh Yoon. Anti-adversarially manipulated attributions for weakly supervised semantic segmentation and object localization. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 2022.
* [34] Jungbeom Lee, Eunji Kim, and Sungroh Yoon. Anti-adversarially manipulated attributions for weakly and semi-supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4071–4080, 2021.
* [35] Jungbeom Lee, Seong Joon Oh, Sangdoo Yun, Junsuk Choe, Eunji Kim, and Sungroh Yoon. Weakly supervised semantic segmentation using out-of-distribution data. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 16897–16906, 2022.
* [36] Jungbeom Lee, Jihun Yi, Chaehun Shin, and Sungroh Yoon. Bbam: Bounding box attribution map for weakly supervised semantic and instance segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 2643–2652, 2021.
* [37] Seungho Lee, Minhyun Lee, Jongwuk Lee, and Hyunjung Shim. Railroad is not a train: Saliency as pseudo-pixel supervision for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 5495–5505, 2021.
* [38] Jing Li, Junsong Fan, and Zhaoxiang Zhang. Towards noiseless object contours for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 16856–16865, 2022.
* [39] Jinlong Li, Zequn Jie, Xu Wang, Lin Ma, et al. Expansion and shrinkage of localization for weakly-supervised semantic segmentation. In Neural Information Processing Systems (NeurIPS), 2022.
* [40] Yi Li, Yiqun Duan, Zhanghui Kuang, Yimin Chen, Wayne Zhang, and Xiaomeng Li. Uncertainty estimation via response scaling for pseudo-mask noise mitigation in weakly-supervised semantic segmentation. In AAAI Conference on Artificial Intelligence (AAAI), volume 36, pages 1447–1455, 2022.
* [41] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In European Conference on Computer Vision (ECCV), pages 740–755. Springer, 2014.
* [42] Yuqi Lin, Minghao Chen, Wenxiao Wang, Boxi Wu, Ke Li, Binbin Lin, Haifeng Liu, and Xiaofei He. Clip is also an efficient segmenter: A text-driven approach for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 15305–15314, 2023.
* [43] Sheng Liu, Kangning Liu, Weicheng Zhu, Yiqiu Shen, and Carlos Fernandez-Granda. Adaptive early-learning correction for segmentation from noisy annotations. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 2606–2616, 2022.
* [44] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hierarchical vision transformer using shifted windows. In IEEE International Conference on Computer Vision (ICCV), pages 10012–10022, 2021.
* [45] Zhengzhe Liu, Xiaojuan Qi, and Chi-Wing Fu. One thing one click: A self-training approach for weakly supervised 3d semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 1726–1736, 2021.
* [46] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.
* [47] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
* [48] Maxime Oquab, Timothée Darcet, Théo Moutakanni, Huy Vo, Marc Szafraniec, Vasil Khalidov, Pierre Fernandez, Daniel Haziza, Francisco Massa, Alaaeldin El-Nouby, et al. Dinov2: Learning robust visual features without supervision. arXiv preprint arXiv:2304.07193, 2023.
* [49] Boris T Polyak and Anatoli B Juditsky. Acceleration of stochastic approximation by averaging. SIAM journal on control and optimization, 30(4):838–855, 1992.
* [50] Shenghai Rong, Bohai Tu, Zilei Wang, and Junjie Li. Boundary-enhanced co-training for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 19574–19584, 2023.
* [51] Simone Rossetti, Damiano Zappia, Marta Sanzari, Marco Schaerf, and Fiora Pirri. Max pooling with vision transformers reconciles class and shape in weakly supervised semantic segmentation. In European Conference on Computer Vision (ECCV), pages 446–463. Springer, 2022.
* [52] Lixiang Ru, Yibing Zhan, Baosheng Yu, and Bo Du. Learning affinity from attention: end-to-end weakly-supervised semantic segmentation with transformers. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 16846–16855, 2022.
* [53] Lixiang Ru, Heliang Zheng, Yibing Zhan, and Bo Du. Token contrast for weakly-supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 3093–3102, 2023.
* [54] David Ruppert. Efficient estimations from a slowly convergent robbins-monro process. Technical report, Cornell University Operations Research and Industrial Engineering, 1988.
* [55] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In IEEE International Conference on Computer Vision (ICCV), pages 618–626, 2017.
* [56] Wataru Shimoda and Keiji Yanai. Self-supervised difference detection for weakly-supervised semantic segmentation. In IEEE International Conference on Computer Vision (ICCV), pages 5208–5217, 2019.
* [57] Chunfeng Song, Yan Huang, Wanli Ouyang, and Liang Wang. Box-driven class-wise region masking and filling rate guided loss for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 3136–3145, 2019.
* [58] Kunyang Sun, Haoqing Shi, Zhengming Zhang, and Yongming Huang. Ecs-net: Improving weakly supervised semantic segmentation by using connections between class activation maps. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 7283–7292, 2021.
* [59] Changwei Wang, Rongtao Xu, Shibiao Xu, Weiliang Meng, and Xiaopeng Zhang. Treating pseudo-labels generation as image matting for weakly supervised semantic segmentation. In IEEE International Conference on Computer Vision (ICCV), pages 755–765, 2023.
* [60] Yude Wang, Jie Zhang, Meina Kan, Shiguang Shan, and Xilin Chen. Self-supervised equivariant attention mechanism for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 12275–12284, 2020.
* [61] Zifeng Wu, Chunhua Shen, and Anton Van Den Hengel. Wider or deeper: Revisiting the resnet model for visual recognition. Pattern Recognition, 90:119–133, 2019.
* [62] Tete Xiao, Yingcheng Liu, Bolei Zhou, Yuning Jiang, and Jian Sun. Unified perceptual parsing for scene understanding. In European Conference on Computer Vision (ECCV), pages 418–434, 2018.
* [63] Jinheng Xie, Xianxu Hou, Kai Ye, and Linlin Shen. Clims: Cross language image matching for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4483–4492, 2022.
* [64] Jinheng Xie, Jianfeng Xiang, Junliang Chen, Xianxu Hou, Xiaodong Zhao, and Linlin Shen. C2am: Contrastive learning of class-agnostic activation map for weakly supervised object localization and semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 989–998, 2022.
* [65] Lian Xu, Wanli Ouyang, Mohammed Bennamoun, Farid Boussaid, and Dan Xu. Multi-class token transformer for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4310–4319, 2022.
* [66] Lian Xu, Wanli Ouyang, Mohammed Bennamoun, Farid Boussaid, and Dan Xu. Learning multi-modal class-specific tokens for weakly supervised dense object localization. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 19596–19605, 2023.
* [67] Rongtao Xu, Changwei Wang, Jiaxi Sun, Shibiao Xu, Weiliang Meng, and Xiaopeng Zhang. Self correspondence distillation for end-to-end weakly-supervised semantic segmentation. In AAAI Conference on Artificial Intelligence (AAAI). AAAI Press, 2023.
* [68] Sung-Hoon Yoon, Hyeokjun Kweon, Jegyeong Cho, Shinjeong Kim, and Kuk-Jin Yoon. Adversarial erasing framework via triplet with gated pyramid pooling layer for weakly supervised semantic segmentation. In European Conference on Computer Vision (ECCV), pages 326–344. Springer, 2022.
* [69] Lu Yu, Wei Xiang, Juan Fang, Yi-Ping Phoebe Chen, and Lianhua Chi. ex-vit: A novel explainable vision transformer for weakly supervised semantic segmentation. Pattern Recognition, page 109666, 2023.
* [70] Bingfeng Zhang, Jimin Xiao, Yunchao Wei, Mingjie Sun, and Kaizhu Huang. Reliability does matter: An end-to-end weakly supervised semantic segmentation approach. In AAAI Conference on Artificial Intelligence (AAAI), volume 34, pages 12765–12772, 2020.
* [71] Xiaolin Zhang, Yunchao Wei, Jiashi Feng, Yi Yang, and Thomas S Huang. Adversarial complementary learning for weakly supervised object localization. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 1325–1334, 2018.
* [72] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 2921–2929, 2016.
* [73] Tianfei Zhou, Meijie Zhang, Fang Zhao, and Jianwu Li. Regional semantic contrast and aggregation for weakly supervised semantic segmentation. In IEEE Computer Vision and Pattern Recognition (CVPR), pages 4299–4309, 2022.
## Appendix A Additional Results
### A.1 Hyper-parameter Finetuning
Here, we examine the impact of hyper-parameter variation with CoSA resulting
from our finetuning. The fine-tuning of each hyper-parameter is demonstrate
with the remaining parameters fixed at their determined optimal values.
Loss Weights. We demonstrate the finetuning of the Seg2CAM and CAM2Seg loss
weights in Tab. 5(a)(b). A significant mIoU decrease is observed as
$\lambda_{c2s}$ reduces the influence of the segmentation branch, as expected.
The mIoU reaches its peak when $\lambda_{\text{s2c}}\\!=\\!0.05$ and
$\lambda_{\text{c2s}}\\!=\\!0.1$.
Low-perplexity Filter. We finetune the coefficient for the low-pass perplexity
filter $\epsilon$, described in eq. (9) of the main paper. The corresponding
findings are illustrated in Tab. 5(c). Optimum performance is obtained when
$\epsilon$ is set to $1$, either decreasing or increasing this value can
impair the performance of our model.
EMA Momentum. Here, the momentum used for updating the assignment network is
finetuned. Results presented in Tab. 5(d) indicate that the optimal
performance is achieved when $m=0.9994$. Additionally, we find that setting
$m=1$ freezes the assignment network, breaking the training of online network
and leading to framework collapse.
Fixed Threshold vs. Dynamic Threshold. In this study, we evaluate CoSA with
predetermined thresholds. The results are presented in Fig. 9. As shown, the
performance peaks when this threshold is set to $0.45$, with an mIoU of
$75.54\%$. However, our dynamic threshold can outperform the best manual
finetuning by $0.65\%$. Despite the incurred additional $10\%$ computation
overhead, our threshold searching algorithm obviates time-consuming finetuning
efforts, resulting in nearly 80% reduction in hyper-parameter searching time
in this case and $(1-1.1n^{-1})\%$ in general where $n$ thresholds are
considered. In addition, the adoption of dynamic thresholding can enhance the
generalizability to novel datasets.
(a) Seg2CAM weight $\lambda_{\text{s2c}}$
$\lambda_{\text{s2c}}$ | mIoU
---|---
$0.2$ | 73.79
$0.1$ | 74.67
$0.05$ | 76.19
$0.025$ | 75.25
$0.0125$ | 74.33
(b) CAM2Seg weight $\lambda_{\text{c2s}}$
$\lambda_{\text{c2s}}$ | mIoU
---|---
$0.4$ | 74.67
$0.2$ | 75.56
$0.1$ | 76.19
$0.05$ | 73.95
$0.025$ | 61.55
(c) Perplexity filter $\epsilon$
$\epsilon$ | mIoU
---|---
$\infty$ | 73.66
$2$ | 75.52
$1$ | 76.19
$0.5$ | 74.30
$0.1$ | 70.63
(d) Momentum $m$
$m$ | mIoU
---|---
$0.9990$ | 73.79
$0.9992$ | 75.40
$0.9994$ | 76.19
$0.9996$ | 75.58
$0.9999$ | 71.42
$1.0000$ | 15.99
Table 5: Hyper-parameters Finetuning Results. Parameter searching for (a) Loss
weight for CAM2Seg $\lambda_{\text{c2s}}$; (b) Loss weight for Seg2CAM
$\lambda_{\text{s2c}}$; (c) Low-pass perplexity filter coefficient $\epsilon$;
(e) EMA Momentum $m$ for updating assignment network. mIoU represents semantic
segmentation result on PASCAL VOC val split. Figure 9: Threshold finetuning.
(left) determined dynamic threshold during training. (right) mIoU comparison
of fixed threshold vs. the purposed dynamic threshold on VOC val.
### A.2 Per-class Segmentation Comparisons
We show the per-class semantic segmentation results on VOC val and test splits
as well as COCO val split.
Comparisons on VOC. Tab. 12 illustrates the CoSA per-class mIoU results
compared with recent works: AdvCAM [33], MCT [65], ToCo [53], Xu _et al_.
[66], BECO [50]. To be fair in comparison, we include CoSA with CRF [10]
postprocessing results, denoted as CoSA∗, same as other SOTA models. Notably,
CoSA dominates in 10 out of 21 classes. In particular, categories like boat
($5.9\%\\!\uparrow$), chair ($8.2\%\\!\uparrow$), and sofa
($17.2\%\\!\uparrow$), demonstrate substantial lead over the SOTA models. In
the VOC test split (depicted in Tab. 12), we still observe its superiority
over other SOTA methods, where CoSA dominates in 15 out of 21 classes.
Comparisons on COCO. We compare CoSA with recent WSSS works for individual
class performance on the COCO val set. As illustrated in Tab. 13, CoSA
outperforms its counterparts in 56 out of 81 classes. Particularly, classes
such as truck ($10.6\%\\!\uparrow$), tie ($14.3\%\\!\uparrow$), kite
($12.4\%\\!\uparrow$), baseball glove ($20.3\%\\!\uparrow$), knife
($14.5\%\\!\uparrow$), ($10.6\%\\!\uparrow$), carrot ($13.0\%\\!\uparrow$),
donuts ($10.0\%\\!\uparrow$), couch ($13.9\%\\!\uparrow$), oven
($13.0\%\\!\uparrow$), and toothbrush ($10.0\%\\!\uparrow$) exhibit remarkable
leading performance.
### A.3 Further Qualitative Comparisons
More visualizations of our CoSA results are given in Fig. 10 for VOC and Fig.
11, Fig. 12 for COCO. When compared to other SOTA models, CoSA exhibits i)
better foreground-background separation (evidenced in R2–R3 in Fig. 10 and
R1–R10 in Fig. 11); ii) more robust to inter-class variation and occlusion
(affirmed in R4–R7 in Fig. 10 and R1–R4 in Fig. 12). iii) less coexistence
problem (demonstrated in R9–R11 in Fig. 10 and R8– R10 in Fig. 12); Last but
not least, our CoSA can reveal certain limitations in manual GT segmentation,
as depicted in R8 in Fig. 10 and R5–R7 in Fig. 12. We also show our CoSA
results on VOC test set in Fig. 14 and some failure cases in Fig. 14.
## Appendix B Further Analysis
Impact of CRF. The conditional random field (CRF) proposes to optimize the
segmentation by utilizing the low-level information obtained from the local
interactions of pixels and edges [10]. Traditionally, a manually designed CRF
postprocessing step has been widely adopted for refining segmentation [15, 50]
or CAMs [51, 65, 70] in WSSS. As our aim is to develop a fully end-to-end WSSS
solution, incorporating CRF postprocessing contradicts this principal. Through
our experiments, we demonstrate that CoSA, unlike other single-stage methods,
does not heavily depend on CRF. Our results indicate that incorporating CRF
results in marginal improvement of $0.2\%$, $0.1\%$, and $0.1\%$ for VOC val,
VOC test, and COCO val, respectively, as presented in Tab. 2 of the main
paper. Tab. 6(a) suggests that in comparison to other SOTA models, our CoSA
exhibits a lesser dependency on the CRF postprocessing. On the contrary,
eliminating the CRF step leads to a noteworthy enhancement of 165% in terms of
inference speed, as demonstrated in Tab. 6(b).
(a)
Method w/o CRF w/ CRF AFA [52] 63.8 66.0 (+2.2) VIT-PCM [51] 67.7 71.4 (+3.7)
ToCo [53] 69.2 71.1 (+0.9) CoSA 76.2 76.4 (+0.2)
(b)
CoSA Speed w/o CRF 4.11 imgs/s w/ CRF 1.83 imgs/s
Table 6: CRF Impact. (a): Comparisons of CRF impact on SOTA single-stage WSSS methods on VOC val. (b): Inference speed with and without CRF. Speed tested using a single 3090 GPU. | CAMs | CAMs | Seg. | Total | mIoU
---|---|---|---|---|---
Generation | Refinement | Training
MCT [65] | 2.2hrs (21M) | 11.1hrs (106M) | 15.5hrs (124M) | 28.8hrs (251M) | 71.6
BECO [50] | 0.9hrs (23M) | 6.5hrs (24M) | 22.2hrs (91M) | 29.6hrs (138M) | 71.8
ToCo [53] | 9.9hrs (98M) | 9.9hrs (98M) | 72.2
| CAMs and Seg. Co-optimization | |
CoSA | 8.7hrs (92M) | 8.7hrs (92M) | 75.1
Table 7: Training Speed and Parameters Comparisons. We report the detailed
training time, parameters and final mIoU on VOC test split for MCT, BECO, ToCo
and our CoSA. All methods are tested using the same machine with a single 3090
GPU. The official MCT, BECO and ToCo code repositories are utilized in this
study.
Transformation | Description | Parameter Setting
---|---|---
RandomRescale | Rescale the image by $r$ times, $r$ randomly sampled from $r\sim U(r_{min},r_{max})$. | $r_{min}=0.5,r_{max}=2$
RandomFlip | Randomly horizontally flip a image with probability of $p$. | $p=0.5$
RandomCrop | Randomly crop a image by a hight $h$ and a width $w$. | $w=448,h=448$
GaussianBlur | Randomly blur a image with probability of $p$. | $p=0.5$
Table 8: Weak data augmentation $\mathcal{T}_{w}$ for the input of assignment
network.
Transformation | Description | Parameter Setting
---|---|---
RandomRescale | Rescale the image by $r$ times, $r$ randomly sampled from $r\sim U(r_{min},r_{max})$. | $r_{min}=0.5,r_{max}=2$
RandomFlip | Randomly horizontally flip a image with probability of $p$. | $p=0.5$
RandomCrop | Randomly crop a image by a hight $h$ and a width $w$. | $w=448,h=448$
GaussianBlur | Randomly blur a image with probability of $p$. | $p=0.5$
OneOf | Select one of the transformation in a transformation set $T$. | $T=$ TransAppearance
Table 9: Strong data augmentation $\mathcal{T}_{s}$ for the input of online
network image.
Transformation | Description | Parameter Setting
---|---|---
Identity | Returns the original image. |
Autocontrast | Maximizes the image contrast by setting the darkest (lightest) pixel to black (white). |
Equalize | Equalizes the image histogram. |
RandSolarize | Invert all pixels above a threshold value $T$. | $T\in U(0,1)$
RandColor | Adjust the color balance. $C=0$ returns a black&white image, $C=1$ returns the original image. | $C\in U(0.05,0.95)$
RandContrast | Adjust the contrast. $C=0$ returns a solid grey image, $C=1$ returns the original image. | $C\in U(0.05,0.95)$
RandBrightness | Adjust the brightness. $C=0$ returns a black image, $C=1$ returns the original image. | $C\in U(0.05,0.95)$
RandSharpness | Adjust the sharpness. $C=0$ returns a blurred image, $C=1$ returns the original image. | $C\in U(0.05,0.95)$
RandPolarize | Reduce each pixel to $C$ bits. | $C\in U(4,8)$
Table 10: Appearance transformations, called TransAppearance, used in strong
data augmentation.
Efficiency Study. Unlike multi-stage approaches, CoSA is extremely efficient
in training. It can be trained end-to-end efficiently. When training a
semantic segmentation model with weak labels on the VOC dataset, our method
requires a mere 8.7 hours of training time and a total of 92M parameters. In
contrast, MCT [65] would necessitate approximately 231% more time (20.1hrs
$\uparrow$) and 173% more parameters (159M $\uparrow$) for the same task, and
BECO [50] would require around 240% more time (20.9hrs $\uparrow$) and 50%
more parameters (46M $\uparrow$). When compared to the single-stage method,
CoSA also demonstrate its advantage in speed-accuracy trade-off. Further
details regarding the efficiency study can be found in Tab. 7.
Algorithm 1 CoSA Training Pseudo Code
1:Require: $\mathcal{D}$ $\triangleright$ image-level classification dataset
2:Require: $\mathcal{F}_{\Theta},~{}\mathcal{F}_{\Theta^{\prime}}$
$\triangleright$ online network parameterized by $\Theta$ and assignment
network by $\Theta^{\prime}$
3:$\mathcal{F}_{\Theta}$ $\leftarrow$ Init, $\mathcal{F}_{\Theta^{\prime}}$
$\leftarrow$ Init $\triangleright$ initialize networks with pretrained
backbone
4:do
5: $x$, $Y_{\text{gt}}$ $\leftarrow$ Sample($\mathcal{D}$) $\triangleright$
sample a mini-batch of image and weak-label pairs
6: $x_{s},x_{w}$ $\leftarrow$ $\mathcal{T}_{s}(x),\mathcal{T}_{w}(x)$
$\triangleright$ apply strong and weak augumentations
7: $\\{x_{w}^{s}\\}$ $\leftarrow$ $\texttt{multiscale}(x_{w})$
$\triangleright$ generate a set of $x_{w}$ with different scales
8:
$\\{\mathcal{M}^{\prime},~{}\mathcal{M}^{\dagger\prime},~{}\mathcal{S}^{\prime}\\}$
$\leftarrow$ $\mathcal{F}_{\Theta^{\prime}}(\\{x_{w}^{s}\\})$ $\triangleright$
forward a set of $x_{w}$ in assignment network
9:
$\mathcal{M}^{\prime},~{}\mathcal{M}^{\dagger\prime},~{}\mathcal{S}^{\prime}$
$\leftarrow$ Maxpool$(\\{\mathcal{M}^{\prime}\\})$,
Maxpool$(\\{\mathcal{M}^{\dagger\prime}\\})$,
Avgpool$(\\{\mathcal{S}^{\prime}\\})$ $\triangleright$ ensemble multiscale
outputs
10:
$\mathcal{M}^{\prime},~{}\mathcal{M}^{\dagger\prime},~{}\mathcal{S}^{\prime}$
$\leftarrow$
Filter$(\mathcal{M}^{\prime},~{}\mathcal{M}^{\dagger\prime},~{}\mathcal{S}^{\prime})$,
$\triangleright$ filter CAMs and segmentation prediction with $Y_{\text{gt}}$
11: $Z,~{}Z^{\dagger},~{}\mathcal{M},~{}\mathcal{M}^{\dagger},~{}\mathcal{S}$
$\leftarrow$ $\mathcal{F}_{\Theta}(x_{s})$ $\triangleright$ forward $x_{s}$ in
online network
12:
$\mathcal{L}_{\text{cls}}+\mathcal{L}_{\text{cls}}^{\mathcal{M}^{\dagger}}$
$\leftarrow$
$\mathcal{L}_{\text{cls}}(Z,Y_{\text{gt}})+\mathcal{L}_{\text{cls}}(Z^{\dagger},Y_{\text{gt}})$
$\triangleright$ get classification losses for $\mathcal{M}$ and
$\mathcal{M}^{\dagger}$ by eq. (1)
13: $\xi^{\star}$ $\leftarrow$ solve eq. (6) with $\mathcal{M}^{\prime}$
$\triangleright$ get dynamic threshold
14: $\hat{\mathcal{Y}}^{\text{CPL}}$ $\leftarrow$ eq. (2) with
$\mathcal{M}^{\prime},~{}\xi^{\star}$ $\triangleright$ obtain CPL
15: ${\mathcal{P}}$ $\leftarrow$ eq. (4) with
$\mathcal{M}^{\prime},~{}\xi^{\star}$ $\triangleright$ estimate perplexity
score
16: ${\mathcal{L}_{\text{c2s}}}$ $\leftarrow$ eq. (5) with
$\hat{\mathcal{Y}}^{\text{CPL}},\mathcal{S},~{}{\mathcal{P}}$ $\triangleright$
get CAM2seg loss
17: ${\mathcal{L}_{\text{c2s}}^{\mathcal{M}^{\dagger}}}$ $\leftarrow$ follow
14 – 17 but with $\mathcal{M}^{\dagger\prime}$ $\triangleright$ get another
CAM2seg loss
18: $\hat{\mathcal{Y}}^{\text{SPL}}$ $\leftarrow$ eq. (7) with
$\mathcal{S}^{\prime}$ $\triangleright$ obtain SPL
19: ${\mathcal{L}_{\text{s2c}}}$ $\leftarrow$ eq. (8) with
$\hat{\mathcal{Y}}^{\text{SPL}},~{}\mathcal{M}$ $\triangleright$ get Seg2CAM
loss
20: $\mathcal{R}^{+},~{}\mathcal{R}^{-}$ $\leftarrow$ eq. (9) with
$\mathcal{P},~{}\hat{\mathcal{Y}}^{\text{CPL}}$ $\triangleright$ define
positive and negative correlation matrix
21: ${\mathcal{L}_{\text{csc}}}$ $\leftarrow$ eq. (10) with
$\mathcal{M},~{}\mathcal{R}^{+},~{}\mathcal{R}^{-}$ $\triangleright$ get
contrastive seperation loss
22: $\mathcal{L}_{\text{CoSA}}$ $\leftarrow$
$\\!\mathcal{L}_{\text{cls}}\\!+\mathcal{L}_{\text{cls}}^{\mathcal{M}^{\dagger}}\\!\\!+\\!\lambda_{\text{c2s}}\big{(}\mathcal{L}_{\text{c2s}}\\!+\\!\mathcal{L}_{\text{c2s}}^{\mathcal{M}^{\dagger}}\big{)}\\!+\\!\lambda_{\text{s2c}}\mathcal{L}_{\text{s2c}}+\\!\lambda_{\text{csc}}\mathcal{L}_{\text{csc}}.$
$\triangleright$ weighted sum as the overall training objective
23: $\Delta\Theta$ $\leftarrow$ $-\nabla_{\mathcal{L}_{\text{CoSA}}}\Theta$
$\triangleright$ backpropagate the overall loss
24: $\Theta$ $\leftarrow$ $\Theta+\Delta\Theta$ $\triangleright$ undate online
network with gradient
25: $\Theta^{\prime}$ $\leftarrow$
$m\Theta^{\prime}+(1-m)\Theta$$\triangleright$ undate assignment network via
EMA
26:until $\mathcal{L}_{\text{CoSA}}$ converge
27:end
Method | bkg | plane | bike | bird | boat | bottle | bus | car | cat | chair | cow
---|---|---|---|---|---|---|---|---|---|---|---
AdvCAM [34] CVPR21 | 90.0 | 79.8 | 34.1 | 82.6 | 63.3 | 70.5 | 89.4 | 76.0 | 87.3 | 31.4 | 81.3
MCT [65] CVPR22 | 91.9 | 78.3 | 39.5 | 89.9 | 55.9 | 76.7 | 81.8 | 79.0 | 90.7 | 32.6 | 87.1
ToCo [53] CVPR23 | 91.1 | 80.6 | 48.7 | 68.6 | 45.4 | 79.6 | 87.4 | 83.3 | 89.9 | 35.8 | 84.7
Xu _et al_. [66] CVPR23 | 92.4 | 84.7 | 42.2 | 85.5 | 64.1 | 77.4 | 86.6 | 82.2 | 88.7 | 32.7 | 83.8
BECO [50] CVPR23 | 91.1 | 81.8 | 33.6 | 87.0 | 63.2 | 76.1 | 92.3 | 87.9 | 90.9 | 39.0 | 90.2
CoSA* (Ours) | 93.1 | 85.5 | 48.5 | 88.7 | 70.0 | 77.6 | 90.4 | 86.4 | 90.3 | 47.2 | 88.7
Method | table | dog | horse | mbike | person | plant | sheep | sofa | train | tv | mIoU
AdvCAM [34] CVPR21 | 33.1 | 82.5 | 80.8 | 74.0 | 72.9 | 50.3 | 82.3 | 42.2 | 74.1 | 52.9 | 68.1
MCT [65] CVPR22 | 57.2 | 87.0 | 84.6 | 77.4 | 79.2 | 55.1 | 89.2 | 47.2 | 70.4 | 58.8 | 71.9
ToCo [53] CVPR23 | 60.5 | 83.7 | 83.7 | 76.8 | 83.0 | 56.6 | 87.9 | 43.5 | 60.5 | 63.1 | 71.1
Xu _et al_. [66] CVPR23 | 59.0 | 82.4 | 80.9 | 76.1 | 81.4 | 48.0 | 88.2 | 46.4 | 70.2 | 62.5 | 72.2
BECO [50] CVPR23 | 41.6 | 85.9 | 86.3 | 81.8 | 76.7 | 56.7 | 89.5 | 54.7 | 64.3 | 60.6 | 72.9
CoSA* (Ours) | 54.1 | 87.3 | 87.1 | 79.6 | 85.6 | 53.2 | 89.9 | 71.9 | 65.1 | 63.4 | 76.4
Table 11: Per-class Segmentation on VOC val Split. Comparison of per-class
segmentation results on VOC val. CoSA is compared with AdvCAM, MCTformer,
ToCo, Xu et al. and BECO. Best results are in bold.
Method | bkg | plane | bike | bird | boat | bottle | bus | car | cat | chair | cow
---|---|---|---|---|---|---|---|---|---|---|---
AdvCAM [34] CVPR21 | 90.1 | 81.2 | 33.6 | 80.4 | 52.4 | 66.6 | 87.1 | 80.5 | 87.2 | 28.9 | 80.1
MCT [65] CVPR22 | 90.9 | 76.0 | 37.2 | 79.1 | 54.1 | 69.0 | 78.1 | 78.0 | 86.1 | 30.3 | 79.5
ToCo [53] CVPR23 | 91.5 | 88.4 | 49.5 | 69.0 | 41.6 | 72.5 | 87.0 | 80.7 | 88.6 | 32.2 | 85.0
CoSA* (Ours) | 93.3 | 88.1 | 47.0 | 84.2 | 60.2 | 75.0 | 87.7 | 81.7 | 92.0 | 34.5 | 87.8
Method | table | dog | horse | mbike | person | plant | sheep | sofa | train | tv | mIoU
AdvCAM [34] CVPR21 | 38.5 | 84.0 | 83.0 | 79.5 | 71.9 | 47.5 | 80.8 | 59.1 | 65.4 | 49.7 | 68.0
MCT [65] CVPR22 | 58.3 | 81.7 | 81.1 | 77.0 | 76.4 | 49.2 | 80.0 | 55.1 | 65.4 | 54.5 | 68.4
ToCo [53] CVPR23 | 68.4 | 81.4 | 85.6 | 83.2 | 83.4 | 68.2 | 88.9 | 55.0 | 49.3 | 65.0 | 72.2
CoSA* (Ours) | 59.6 | 86.2 | 86.3 | 84.9 | 82.8 | 68.2 | 87.4 | 63.9 | 67.7 | 61.6 | 75.2
Table 12: Per-class Segmnetation on VOC test Split. Comparison of per-class
segmentation results on VOC test. Results from AdvCAM, MCT, and ToCo are used
for this comparison. Best results are in bold.
Class | MCT[65] (CVPR22) | Xu _et al_. [66] (CVPR23) | ToCo[53] (CVPR23) | CoSA (Ours) | Class | MCT[65] (CVPR22) | Xu _et al_. [66] (CVPR23) | ToCo[53] (CVPR23) | CoSA (Ours)
---|---|---|---|---|---|---|---|---|---
background | 82.4 | 85.3 | 68.5 | 84.0 | wine glass | 27.0 | 33.8 | 20.6 | 42.1
person | 62.6 | 72.9 | 28.1 | 70.3 | cup | 29.0 | 35.8 | 26.0 | 33.1
bicycle | 47.4 | 49.8 | 39.7 | 52.4 | fork | 23.4 | 20.0 | 7.6 | 24.2
car | 47.2 | 43.8 | 38.9 | 54.3 | knife | 12.0 | 12.6 | 18.4 | 32.9
motorcycle | 63.7 | 66.2 | 55.1 | 71.9 | spoon | 6.6 | 6.7 | 3.0 | 9.0
airplane | 64.7 | 69.2 | 62.1 | 74.0 | bowl | 22.4 | 23.7 | 19.8 | 22.8
bus | 64.5 | 69.1 | 39.0 | 77.2 | banana | 63.2 | 64.4 | 71.5 | 69.3
train | 64.5 | 63.7 | 48.7 | 60.0 | apple | 44.4 | 50.8 | 55.5 | 61.3
truck | 44.8 | 43.4 | 37.3 | 55.4 | sandwich | 39.7 | 47.0 | 41.2 | 48.3
boat | 42.3 | 42.3 | 49.1 | 52.1 | orange | 63.0 | 64.6 | 70.6 | 69.2
traffic light | 49.9 | 49.3 | 47.3 | 55.1 | broccoli | 51.2 | 50.6 | 56.7 | 52.8
fire hydrant | 73.2 | 74.9 | 69.6 | 78.8 | carrot | 40.0 | 38.6 | 46.4 | 59.4
stop sign | 76.6 | 77.3 | 70.1 | 82.2 | hot dog | 53.0 | 54.0 | 60.1 | 59.9
parking meter | 64.4 | 67.0 | 67.9 | 71.5 | pizza | 62.2 | 64.1 | 54.9 | 56.5
bench | 32.8 | 34.1 | 43.9 | 50.2 | donut | 55.7 | 59.7 | 61.1 | 71.1
bird | 62.6 | 63.1 | 58.6 | 65.4 | cake | 47.9 | 50.6 | 42.5 | 57.0
cat | 78.2 | 76.2 | 74.0 | 79.8 | chair | 22.8 | 24.5 | 24.1 | 33.8
dog | 68.2 | 70.6 | 64.0 | 72.8 | couch | 35.0 | 40.0 | 44.2 | 58.1
horse | 65.8 | 67.1 | 66.1 | 71.4 | potted plant | 13.5 | 13.0 | 27.4 | 23.5
sheep | 70.1 | 70.8 | 67.9 | 74.3 | bed | 48.6 | 53.7 | 54.0 | 61.5
cow | 68.3 | 71.2 | 69.0 | 74.0 | dining table | 12.9 | 19.2 | 25.6 | 29.2
elephant | 81.6 | 82.2 | 79.7 | 81.9 | toilet | 63.1 | 66.6 | 62.0 | 69.7
bear | 80.1 | 79.6 | 76.8 | 85.3 | tv | 47.9 | 50.8 | 49.1 | 53.2
zebra | 83.0 | 82.8 | 77.5 | 76.3 | laptop | 49.5 | 55.4 | 55.7 | 63.9
giraffe | 76.9 | 76.7 | 66.1 | 68.5 | mouse | 13.4 | 14.4 | 8.6 | 16.4
backpack | 14.6 | 17.5 | 20.3 | 28.6 | remote | 41.9 | 47.1 | 56.6 | 49.1
umbrella | 61.7 | 66.9 | 70.9 | 73.4 | keyboard | 49.8 | 57.2 | 41.8 | 49.6
handbag | 4.5 | 5.8 | 8.1 | 11.9 | cellphone | 54.1 | 54.9 | 58.5 | 66.2
tie | 25.2 | 31.4 | 33.4 | 47.7 | microwave | 38.0 | 46.1 | 55.5 | 53.2
suitcase | 46.8 | 51.4 | 55.3 | 63.8 | oven | 29.9 | 35.3 | 36.2 | 49.2
frisbee | 43.8 | 54.1 | 39.6 | 63.1 | toaster | 0.0 | 2.0 | 0.0 | 0.0
skis | 12.8 | 13.0 | 4.0 | 22.5 | sink | 28.0 | 36.1 | 19.0 | 41.9
snowboard | 31.4 | 30.3 | 15.5 | 40.5 | refrigerator | 40.1 | 52.7 | 51.9 | 62.0
sports ball | 9.2 | 36.1 | 11.0 | 33.1 | book | 32.2 | 34.8 | 31.5 | 37.8
kite | 26.3 | 47.5 | 40.7 | 59.9 | clock | 43.2 | 51.5 | 32.9 | 55.2
baseball bat | 0.9 | 7.0 | 1.8 | 3.8 | vase | 22.6 | 25.8 | 33.3 | 33.8
baseball glove | 0.7 | 10.4 | 17.6 | 37.9 | scissors | 32.9 | 30.7 | 49.8 | 54.7
skateboard | 7.8 | 15.2 | 13.3 | 12.5 | teddy bear | 61.9 | 61.4 | 67.5 | 69.3
surfboard | 46.5 | 51.5 | 21.5 | 16.5 | hair drier | 0.0 | 1.3 | 10.0 | 0.3
tennis racket | 1.4 | 26.4 | 6.8 | 7.2 | toothbrush | 12.2 | 19.0 | 29.3 | 39.3
bottle | 31.1 | 37.1 | 25.7 | 35.1 | mIoU | 42.0 | 45.9 | 42.4 | 51.1
Table 13: Per-class Segmentation on COCO val Split. Comparison of per-class
segmentation results on the COCO 2014 val set. CoSA is compared with
MCTformer, Xu et al. and ToCo. Best results are in bold.
## Appendix C Further Implementation Details
CoSA Implementation Details. For image preprocessing, weak transformation
$\mathcal{T}_{w}$ and strong transformation $\mathcal{T}_{s}$ are employed in
CoSA for the input of assignment network and online network, respectively.
$\mathcal{T}_{w}$ and $\mathcal{T}_{s}$ details are given in Tab. 10 and Tab.
10. Following [53], we use the multi-scale inference in assignment network to
produce CPL and SPL. For VOC training, CoSA is warmed up with 6K iterations,
where $\lambda_{\text{c2s}}$, $\lambda_{\text{c2s}}$, and
$\lambda_{\text{csc}}$ are set to $0$. In practice, we train CoSA for 20K
iterations on 2 GPUs, with 2 images per GPU, or for 40K iterations on 1 GPU
for some ablation experiments. For COCO training, CoSA is warmed up with 10K
iterations and is trained on 2 GPUs, handling 4 images per GPU across 40K
iterations.
CoSA-MS Implementation Details. Tab. 2 in the main paper presents the
segmentation results of the multi-stage version of our approach, known as
CoSA-MS. In those experiments, we leverage the CAM pseudo-labels generated by
our CoSA to directly train standalone segmentation networks. It is important
to note that we do not use PSA [2], which is widely used in [65, 15], nor IRN
[1], extensively used in [50, 12, 59, 31], for CPL post-refinement. For our
R101 segmentation network, we use a ResNet101 version of DeepLabV3+ model,
same as BECO [50]. As for the CoSA-MS with WR38 network, we utilize a encoder-
decoder framework, where encoder is WideResNet38 [61] and decoder is LargeFoV
[10], following the final step described in MCT [65]. Regarding the SWIN
implementation, we use the SWIN-Base encoder [44] in conjunction with UperNet
decoder [62], following the description in [14, 12].
Training Pseudo Code. we present the pseudo code for training CoSA in
Algorithm 1.
Figure 10: Qualitative Comparisons on VOC Dataset. CoSA exhibits 1) better
foreground-background separation (R1–R3); 2) more robust to inter-class
variation and occlusion (R4–R7); 3) limitations in the ground troth
annotations (R8); 4) less coexistence problem (R9–R11). Different colors
represent different categories: black: background; white: ignore areas; :
chair; : plant; : cat; : person; : bottle; : sofa; : dog; : cow. : bird; :
boat; The activated classes in the demonstration from top to bottom are:
chair, cat, bottle, person, dog, person, cow, person, bird, boat, boat.
Figure 11: Qualitative Comparisons on COCO Dataset. CoSA demonstrates superior
quality in terms of foreground-background separation (R1–R10). Categories
involved – R1: person, tie; R2: person, umbrella; R3: person, skis; R4:
person, tie; R5: person, train, umbrella; R6: person, hot dog; R7: person, hot
dog; R8: dog, frisbee; R9: bottle, toilet; R10: person, teddy bear; Categories
in Bold denotes the activated classes in CAMs.
Figure 12: More Qualitative Comparisons on COCO Dataset. CoSA shows 1) more
robust to inter-class variation and occlusion (R1–R4); 2) limitations in the
ground troth annotations (R5–R7); 3) less coexistence problem (R8–R10).
Categories involved – R1: person, donuts; R2: person, surfboard. R3: person,
car, motorcycle, bus; R4: toilet; R5: person, kite; R6: person, kite; R7:
person, cell phone; R8: clock; R9: clock; R10: clock. Categories in Bold
denotes the activated classes in CAMs.
Figure 13: Visualization on VOC test. Different colors represent different
categories: black: background; : car; : person; : boat; : plant; : dog; : cow.
: dining-table. : bird; : sofa; : sheep; : house; : airplane; : cat.
Figure 14: Illustrations of CoSA failure Cases. Different colors represent
different categories: black: background; white: ignore areas; : plant; :
person; : sofa; : dog; : cat; : chair; : motorbike; : bicycle. The activated
classes in the demonstration from left to right and from top to bottom are:
plant, sofa, dog, plant, person, cat, person, bicycle.
|
# DynaConF: Dynamic Forecasting of Non-Stationary Time-Series
Siqi Liu
Borealis AI &Andreas Lehrmann
Borealis AI
###### Abstract
Deep learning models have shown impressive results in a variety of time series
forecasting tasks, where modeling the conditional distribution of the future
given the past is the essence. However, when this conditional distribution is
non-stationary, it poses challenges for these models to learn consistently and
to predict accurately. In this work, we propose a new method to model non-
stationary conditional distributions over time by clearly decoupling
stationary conditional distribution modeling from non-stationary dynamics
modeling. Our method is based on a Bayesian dynamic model that can adapt to
conditional distribution changes and a deep conditional distribution model
that can handle large multivariate time series using a factorized output
space. Our experimental results on synthetic and popular public datasets show
that our model can adapt to non-stationary time series better than state-of-
the-art deep learning solutions.
## 1 Introduction
Time series forecasting is a cornerstone of modern machine learning and has
applications in a broad range of domains, such as operational processes [1],
energy [2], and transportation [3]. In recent years, models based on deep
neural networks have shown particularly impressive results [4, 5, 3, 1] and
demonstrated the effectiveness of deep feature and latent state
representations.
Despite this exciting progress, current time series forecasting methods often
make the implicit assumption that training and test data follow the same
distribution. In real-world applications this assumption is often violated,
which is known as _non-stationarity_ and poses serious practical challenges to
a model’s robustness and predictive power. The statistics literature contains
several related concepts of (non-)stationarity for time series, with weak and
strong stationarity being the most widely used ones [6, 7]. Common to these
variants of (non-)stationarity is that they are defined in terms of a
stochastic process’ joint or marginal distribution. For example, given a
discrete time series $\\{y_{t}\in\mathbb{R}\\}_{t\in\mathbb{Z}}$ and any
subset of time points $\\{t_{1},t_{2},\ldots,t_{k}\\}$, we call the time
series _strongly stationary_ if
$\forall\tau\in\mathbb{Z}:\mathrm{p}(y_{t_{1}},y_{t_{2}},\ldots,y_{t_{k}})=\mathrm{p}(y_{t_{1}+\tau},y_{t_{2}+\tau},\ldots,y_{t_{k}+\tau})$.
While non-stationarity in a stochastic process’ joint or marginal distribution
is important and has been widely studied [7, 8, 9, 10], we argue that temporal
_forecasting_ relies more heavily on the properties of a time-series’
_conditional_ distribution $\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t})$,
where $y_{t-B:t-1}=(y_{t-B},y_{2},\ldots,y_{t-1})$, $B\in\mathbb{Z}_{>0}$ can
be arbitrarily large, and $\bm{x}_{t}\in R^{Q}$ contains auxiliary
information. Most forecasting methods, from traditional approaches (e.g.,
Autoregressive Integrated Moving Average (ARIMA; [11]), Generalized
Autoregressive Conditional Heteroskedasticity (GARCH; [12, 13]), state-space
models (SSMs; [14])) to more recent models (e.g., recurrent neural networks
(RNNs; [15, 1]), temporal convolutional networks (TCNs; [16]), Transformers
[17, 4], Neural Processes (NPs; [18, 19])), rely on this conditional
distribution for predictions, but many of them implicitly assume its
stationarity by using time-invariant model parameters. A model with a
stationary conditional distribution can still handle non-stationarity in the
joint or marginal distribution, such as seasonality and trend, by conditioning
on extra features in $\bm{x}_{t}$, such as day of the week, but the
conditional distribution itself may also change due to (1) unobserved causes
and (2) new causes. For example, the daily number of posts from each user on a
social media platform is unlikely to be robustly predictable from historical
data, even with input features like day of the week, because (1) user activity
is often affected by events that are not reflected in the observable data
(e.g., illness); and (2) events that have not been seen before, such as a new
functionality being added to the platform, may occur and change the functional
pattern of the input-output relation between the conditioning and target
variables in an unpredictable way. How to deal with these changes in the
model’s _conditional_ distribution, which are based on a dynamic cause-effect
structure, is the main focus of this work.
Autoregressive (AR) models, TCNs, and Transformers model
$\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t};\bm{\psi})$ with time-invariant
parameters $\bm{\psi}$ and therefore assume stationarity in
$\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t})$. In contrast, SSMs, RNNs, and
NPs model $\mathrm{p}(y_{t}|y_{1:t-1},\bm{x}_{1:t};\bm{\psi})$, which has a
growing number of conditioning variables (note the time range $1\\!:\\!t$) and
therefore can potentially model different conditional distributions
$\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t})$ at different time points $t$.
However, these models need to achieve two goals using the same time-invariant
structure: (1) modeling $\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t})$; and
(2) modeling its changes over time. Because they do not incorporate explicit
inductive biases for changes in
$\mathrm{p}(y_{t}|y_{t-B:t-1},\bm{x}_{t-B:t})$, they either cannot learn
different (seemingly inconsistent) relations between the conditioning and
target variables (if the model capacity is limited) or tend to memorize the
training data and are not able to generalize to new changes at test time.
In this work we take a different approach to dealing with non-stationary
conditional distributions in time series. The core of our model, called
DynaConF, is a clean decoupling of the time-variant (non-stationary) part and
the time-invariant (stationary) part of the distribution. The time-invariant
part models a stationary conditional distribution, given some control
variables, while the time-variant part focuses on modeling the changes in this
conditional distribution over time through those control variables. Using this
separation, we build a flexible time-invariant conditional model and make
efficient inferences about how the model changes over time. At test time, our
model takes both the uncertainty of the conditional distribution and non-
stationarity into account when making predictions and can adapt to new changes
in the conditional distribution over time in an online manner.
## 2 Related Work
Time-series forecasting models have a rich history [7, 11], with many recent
advances due to deep neural networks [20]. Here we discuss relations of
related works to non-stationarity and our work.
Non-Stationary Marginal Distribution. There are three common ways of dealing
with non-stationarity in the marginal distribution, such as seasonality and
trend: (1) Data transformation. In ARIMA, taking the difference of the time
series over time can remove trend and seasonality. More advanced approaches
based on exponential smoothing [21] or seasonal-trend decomposition [22] have
also been combined with deep neural networks and achieve good performance [23,
24]. More recently, Kim et al. [10] propose to use reversible
normalization/denormalization on the input/output of the time series model to
account for (marginal) distribution shifts over time. (2) Inductive bias. As
an alternative to data transformations, the underlying ideas of exponential
smoothing and decomposition can also be incorporated into deep learning models
as inductive biases to deal with seasonality/trend end-to-end [25, 26, 27, 2].
Similarly, for models based on Gaussian processes, the inductive biases can be
added as specific (e.g. periodic/linear) kernel choices [28]. (3) Conditioning
information. Adding features such as relative time (e.g, day of the week) and
absolute time to the model input as conditional information is commonly used
in deep probabilistic forecasting models [1, 3, 5, 29, 4]. In our work, we
focus on proper handling of changes in the _conditional_ distribution. To deal
with marginal distribution shifts, we simply add conditional information as in
approach (3), although we could potentially utilize approach (1) and (2) as
well.
Non-Stationary Conditional Distribution. State-space models [30, 31, 32, 5,
33] and recurrent neural networks [1, 3, 4, 29] are among the most popular
choices to model temporal dependencies in time series. When these models are
applied to, and therefore conditioned on, the entire history of the time
series, they can theoretically “memorize” different conditional distributions
at different time points. However, for these models to generalize and adapt to
new changes in the future, it is critical to have appropriate inductive biases
built into the model. A common approach is to allow the state space model to
switch between a discrete set of dynamics, which can be learned from training
data [34, 35]. However, the model cannot adapt to continuous changes or
generalize to new changes that have not been observed in the training data. In
contrast, our model has explicit inductive biases to account for both
continuous and discontinuous changes, and to adapt to new changes.
Observation Model. The expressivity and flexibility of the observation model
is a topic that is especially relevant in case of multivariate time series.
Different observational models have been employed in time series models,
including low-rank covariance structures [3], auto-encoders [30, 36],
normalizing flows [31, 4], determinantal point processes [37], denoising
diffusion models [29], and probabilistic circuits [38]. In this work, to deal
with multivariate time series, we take a simple approach by assuming
conditional independence in the output dimensions for scalability and consider
more expressive observation models as orthogonal to our work.
Online Approach. Continual learning [39, 40, 41, 42, 43, 44] also addresses
(conditional) distribution changes in an online manner, but usually in a
multi-task supervised learning setting and not in time series. In this work,
our focus is on conditional distribution changes in time series, and the
conditional distribution can change either continuously or discontinuously
over time.
## 3 Method
We study the problem of modeling and forecasting time series with changes in
the conditional distribution
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ over time $t$, where
$\bm{y}_{t}\in\mathbb{R}^{P}$ is the target time series, and
$\bm{x}_{t}\in\mathbb{R}^{Q}$ is the input containing contextual information,
under the following assumption:
###### Assumption 1
$\bm{y}_{t}$ only depends on a bounded history of $\bm{y}$ and $\bm{x}$ for
all $t$. That is, there exists $B\in\mathbb{Z}_{>0}$ such that for all $t$,
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{<t},\bm{x}_{\leq
t})=\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$.
Although we assume $\bm{y}_{t}$ can only depend on the history up to $B$ time
steps, its conditional distribution can change over time based on information
beyond $B$ steps. Notice that this is not particularly restrictive in
practice, since we usually have a finite amount of training data while $B$ can
be arbitrarily large, although usually not needed. Specifically, given the
historical data $\\{(\bm{x}_{t},\bm{y}_{t})\\}_{t=1}^{T}$, the task is to fit
a model to the data and use it to forecast
$\\{\bm{y}_{t}\\}_{t=T+1}^{T+H},\\{\bm{y}_{t}\\}_{t=T+H+1}^{T+H+H},\ldots,$
continually with a horizon and step size $H$. For notational simplicity, we
assume that $\bm{x}_{t}$ only contains information known in advance at time
$t$, such as day of the week.
### 3.1 Decoupled Model
We assume the distribution
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ to be parameterized
by $\bm{\theta}_{t}\in\mathbb{R}^{M}$:
$\bm{\theta}_{t}=f(\bm{y}_{t-B:t-1},\bm{x}_{t-B:t};\bm{\phi}_{t},\bm{\psi}).$
(1)
$f:\mathbb{R}^{B\times P}\times\mathbb{R}^{(B+1)\times Q}\to\mathbb{R}^{M}$
models the conditional distribution, with its own static parameters, denoted
collectively as $\bm{\psi}$, and is modulated by a dynamic control variable
$\bm{\phi}_{t}\in\mathbb{R}^{F}$, which can change over time according to a
dynamic process we define later. A property we try to guarantee in our model
is that if $\bm{\phi}_{t}$ stays the same at different time points, then the
conditional distribution
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ stays the same as
well. Figure 1 shows an overview of our model.
Our key idea is to separate the time-variant part of the model from the time-
invariant part, instead of allowing all components to vary across time. This
simplifies the probabilistic inference and improves the generalization
capabilities of the learned model by allowing the time-invariant part to learn
from time-invariant input-output relations with time-variant modulations.
### 3.2 Conditional Distribution at One Time Point
First we describe how we model the conditional distribution
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ at each time point
$t$ without accounting for non-stationary effects (Figure 1, bottom). We use a
neural network $g$ to summarize the historical and contextual information into
a vector $\bm{h}_{t}\in\mathbb{R}^{D}$ as
$\bm{h}_{t}=g(\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$. For example, $g$ could be a
multi-layer perceptron (MLP) or a recurrent neural network (RNN). The
parameters of $g$ are time-invariant and included in $\bm{\psi}$ (Eq. 1).
We note that a key distinction between our model’s use of an RNN and a typical
deep time series model using an RNN is that the latter keeps unrolling the RNN
over time to model the dynamics of the time series. In contrast, we unroll the
RNN for $B+1$ steps to summarize $(\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ in the
exact same way at each time point $t$, i.e., we apply it in a _time-invariant_
manner.
We construct the distribution of $\bm{y}_{t}$ such that each dimension $i$ of
$\bm{y}_{t}$, denoted as $y_{t,i}$, is conditionally independent from the
others given $\bm{h}_{t}$, as this helps the learning and inference algorithms
to scale better with the dimensionality of $\bm{y}_{t}$. First we transform
$\bm{h}_{t}$ into $P$ vectors of dimension $E$,
$\bm{z}_{t,i}\in\mathbb{R}^{E}$, $E<D$, as
$\bm{z}_{t,i}=\tanh(\bm{W}_{z,i}\bm{h}_{t}+\bm{b}_{z,i}),\quad\forall
i=1,\ldots,P$ (2)
where $\bm{W}_{z,i}\in\mathbb{R}^{E\times D}$ and
$\bm{b}_{z,i}\in\mathbb{R}^{E}$, so $\bm{z}_{t,i}$ corresponds to $y_{t,i}$.
Then, from $\bm{z}_{t,i}$, we obtain the distribution parameters
$\bm{\theta}_{t,i}$ of $y_{t,i}$. Specifically, we assume a normal
distribution with a diagonal covariance for $\bm{y}_{t}$, so
$y_{t,i}\sim\mathcal{N}(\mu_{t,i},\sigma^{2}_{t,i})$ and
$\bm{\theta}_{t,i}:=(\mu_{t,i},\sigma^{2}_{t,i})$, with
$\mu_{t,i}=\bm{w}_{\mu,i}^{T}\bm{z}_{t,i}+b_{\mu,i},\quad\sigma_{t,i}=s(\bm{w}_{\sigma,i}^{T}\bm{z}_{t,i}+b_{\sigma,i}),$
(3)
where $\bm{w}_{\mu,i}\in\mathbb{R}^{E}$, $\bm{w}_{\sigma,i}\in\mathbb{R}^{E}$,
and $s:\mathbb{R}\to\mathbb{R}_{>0}$ is the soft-plus function. We use
$\bm{W}_{\mu}\in\mathbb{R}^{P\times E}$ and $\bm{b}_{\mu}\in\mathbb{R}^{P}$ to
denote the result of stacking $\bm{w}_{\mu,i}^{T}$ and $b_{\mu,i}$ along $i$,
and similarly $\bm{W}_{\sigma},\bm{b}_{\sigma}$, $\bm{z}_{t}$, $\bm{\mu}_{t}$,
$\bm{\sigma}_{t}$. As we will see in Section 3.5, the use of a Normal
distribution enables more efficient inference.
### 3.3 Conditional Distributions Across Time Points
We have explained how we model the conditional distribution
$\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$ at each time $t$. To
model changes in the conditional distribution over time, we first specify
which parameters to include in the control variable
$\bm{\phi}_{t}\in\mathbb{R}^{F}$, which changes over time and modulates the
conditional distribution (Figure 1, top).
We choose $\bm{\phi}_{t}:=\operatorname{vec}(\bm{W}_{\mu})$, i.e.,
$\bm{\phi}_{t}$ is the vectorization of $\bm{W}_{\mu}$. Recall that
$\bm{W}_{\mu}$ transforms $\bm{z}_{t}$ into the mean $\bm{\mu}_{t}$ of
$\bm{y}_{t}$, where $\bm{z}_{t}$ is essentially a summary of the information
in the conditioning variables $(\bm{y}_{t-B:t-1},\bm{x}_{t-B:t})$. By allowing
$\bm{W}_{\mu}$ to be different at each time point $t$, the conditional mean of
$\bm{y}_{t}$, $\mathrm{E}[\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t}]$, can
change as well. We could allow $\bm{W}_{\sigma}$ to change over time to model
a time-variant conditional variance as well, but focusing on $\bm{W}_{\mu}$
reduces the dimensionality of $\bm{\phi}_{t}$ and enables more efficient
inference utilizing Rao-Blackwellization (see Section 3.5).
We propose to decompose $\bm{\phi}_{t}$ into a dynamic stochastic process
$\bm{\chi}_{t}\in\mathbb{R}^{F}$ and a static vector
$\bm{b}_{\phi}\in\mathbb{R}^{F}$ as
$\bm{\phi}_{t}=\bm{\chi}_{t}+\bm{b}_{\phi}.$ (4)
The intuition is that $\bm{b}_{\phi}$ captures the global information of
$\bm{\phi}_{t}$ and acts as a baseline, while $\bm{\chi}_{t}$ captures the
time-variant changes of $\bm{\phi}_{t}$ relative to $\bm{b}_{\phi}$.
We assume that $\bm{\chi}_{t}$ follows the following generative process
$\displaystyle\pi_{t}\sim\mathcal{B}(\lambda);$
$\displaystyle\bm{\chi}_{t}\sim\mathcal{N}(\bm{0},\bm{\Sigma}_{0}),\text{ if
}\pi_{t}=0;$ (5)
$\displaystyle\bm{\epsilon}_{t}\sim\mathcal{N}(\bm{0},\bm{\Sigma}_{d});$
$\displaystyle\bm{\chi}_{t}=\bm{\chi}_{t-1}+\bm{\epsilon}_{t},\text{ if
}\pi_{t}=1.$
$\mathcal{B}$ and $\mathcal{N}$ denote the Bernoulli and normal distributions,
respectively.
$\pi_{t}\in\\{0,1\\}$ is a binary random variable that either generates the
current $\bm{\chi}_{t}$ as a new sample drawn from a global distribution
$\mathcal{N}(\bm{0},\bm{\Sigma}_{0})$, or as a continuation from the previous
$\bm{\chi}_{t-1}$ following a simple stochastic process in the form of a
random walk. The intention is to allow $\bm{\chi}_{t}$ to change both
continuously (when $\pi_{t}=1$) through the random walk, and discontinuously
(when $\pi_{t}=0$) through the global distribution, which captures the variety
of possible changes of $\bm{\chi}_{t}$ in its parameter $\bm{\Sigma}_{0}$.
We assume $\bm{\chi}_{t}$ to start at $t=B$, since it controls the
_conditional_ distribution, whose first observation occurs at $t=B+1$. For the
initial $\bm{\chi}_{B}$, we assume generation from
$\mathcal{N}(\bm{0},\bm{\Sigma}_{0})$ as well. Our intention is that
$\mathcal{N}(\bm{0},\bm{\Sigma}_{0})$ should be the distribution to generate
new $\bm{\chi}_{t}$ whenever there is a drastic change in the conditional
distribution of $\bm{y}_{t}$, so at $t=B$ it is natural to use that
distribution.
Recall that $\bm{\chi}_{t}\in\mathbb{R}^{F}$, with $F=P\times E$. We propose
to separate $\bm{\chi}_{t}$ along the $P$ dimensions of $\bm{y}_{t}$ into $P$
groups. For each $i=1,\ldots,P$, we define $\bm{\chi}_{t,i}\in\mathbb{R}^{E}$
as in Eq. 5. The final $\bm{\chi}_{t}$ is the concatenation of
$\bm{\chi}_{t,i}$ for all $i$. The intuition is to allow the group of
components of $\bm{\chi}_{t}$ modulating each dimension of $\bm{y}_{t}$ to
change independently from the others, corresponding to the conditional
independence assumption we made in Section 3.2, so we can sample a subset of
dimensions of $\bm{y}$ in each iteration during training to scale to high-
dimensional time series.
Figure 1: Overview. DynaConF is built on the principle of a clean decoupling
of stationary conditional distribution modeling (red) and non-stationary
dynamics modeling (blue). We predict the parameters ($\bm{\theta}$) of the
conditional distribution (green) by aggregating time-invariant local context
($\bm{z}$) and modulating this context with time-variant global dynamics
($\bm{\phi}$) driven by a random walk ($\bm{\chi}$) with Bernoulli-restarts
($\bm{\pi}$).
### 3.4 Learning
All parameters of the conditional distribution model and the prior are learned
by fitting the model to the historical training data
$\\{(\bm{y}_{t},\bm{x}_{t})\\}_{t=1}^{T}$. We train our model by maximizing
the marginal log-likelihood, where the latent variables $\bm{\chi}_{t}$ are
marginalized out. Given a trajectory of $\bm{\chi}_{B:T}$, the conditional
log-likelihood is
$\log\mathrm{p}(\bm{y}_{B+1:T}|\bm{y}_{1:B},\bm{x}_{1:T},\bm{\chi}_{B:T})=\sum_{t=B+1}^{T}\log\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t},\bm{\chi}_{t}).$
(6)
Marginalizing out $\bm{\chi}_{t}$ gives us the log-likelihood objective
$\log\mathrm{p}(\bm{y}_{B+1:T}|\bm{y}_{1:B},\bm{x}_{1:T})=\log\int\mathrm{p}(\bm{y}_{B+1:T}|\bm{y}_{1:B},\bm{x}_{1:T},\bm{\chi}_{B:T})\mathrm{p}(\bm{\chi}_{B:T})\mathrm{d}\bm{\chi}_{B:T}.$
(7)
Since the integral is intractable, we instead introduce a variational
distribution $\mathrm{q}(\bm{\chi}_{B:T})$ and maximize the following
variational lower-bound $\mathcal{L}$ of the log-likelihood in Eq. 7:
$\mathcal{L}:=\mathrm{E}_{q}[\log\mathrm{p}(\bm{y}_{B+1:T}|\bm{y}_{1:B},\bm{x}_{1:T},\bm{\chi}_{B:T})]+\mathrm{E}_{q}\left[\log\frac{p(\bm{\chi}_{B:T})}{\mathrm{q}(\bm{\chi}_{B:T})}\right]\leq\log\mathrm{p}(\bm{y}_{B+1:T}|\bm{y}_{1:B},\bm{x}_{1:T}).$
(8)
Based on the conditional independence structure of the prior process, we
construct the variational distribution $\mathrm{q}(\bm{\chi}_{B:T})$ similarly
as an autoregressive process, but we assume a simple Normal distribution at
each time step for efficient sampling and back-propagation. At the beginning,
we assume $\mathrm{q}(\bm{\chi}_{B})=\mathrm{p}(\bm{\chi}_{B})$. Then,
conditioning on the previous $\bm{\chi}_{t-1}$, we recursively define
$\mathrm{q}(\bm{\chi}_{t}|\bm{\chi}_{t-1})=\mathcal{N}(\bm{a}_{t}\odot\bm{\chi}_{t-1}+(1-\bm{a}_{t})\odot\bm{m}_{t},\mathrm{diag}(\bm{s}_{t}^{2})),\quad\forall
t=1,\ldots,T,$ (9)
where $\bm{a}_{t}$, $\bm{m}_{t},\bm{s}_{t}\in\mathbb{R}^{F}$ are variational
parameters. Intuitively, $\bm{a}_{t}$ is a gate that chooses between
continuing from the previous $\bm{\chi}_{t-1}$, with noise
$\mathcal{N}(0,\mathrm{diag}(\bm{s}^{2}_{t})$, and using a new distribution
$\mathcal{N}(\bm{m}_{t},\mathrm{diag}(\bm{s}_{t}^{2}))$.
We note that both terms in Eq. 8 factorize over time $t=B+1,\ldots,T$ as
follows:
$\displaystyle\mathcal{L}$
$\displaystyle=\mathrm{E}_{q(\bm{\chi}_{B:T})}\left[\sum_{t=B+1}^{T}\log\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t},\bm{\chi}_{t})\right]+\mathrm{E}_{q(\bm{\chi}_{B:T})}\left[\sum_{t=B+1}^{T}\log\frac{p(\bm{\chi}_{t}|\bm{\chi}_{t-1})}{\mathrm{q}(\bm{\chi}_{t}|\bm{\chi}_{t-1})}\right]$
(10)
$\displaystyle=\sum_{t=B+1}^{T}\mathrm{E}_{q(\bm{\chi}_{t})}\left[\log\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t},\bm{\chi}_{t})\right]+\sum_{t=B+1}^{T}\mathrm{E}_{q(\bm{\chi}_{t-1:t})}\left[\log\frac{p(\bm{\chi}_{t}|\bm{\chi}_{t-1})}{\mathrm{q}(\bm{\chi}_{t}|\bm{\chi}_{t-1})}\right].$
In the above derivation, $\mathrm{p}(\bm{\chi}_{B})=\mathrm{q}(\bm{\chi}_{B})$
were canceled out due to the definition of $\mathrm{q}(\bm{\chi}_{B})$. The
expectations in this equation can be evaluated by Monte-Carlo sampling from
$\mathrm{q}(\bm{\chi})$ with the reparameterization trick [45] for back-
propagation.
In cases where training efficiency is critical, sequential sampling in the
autoregressive posterior might not be feasible, and we leverage Inverse
Autoregressive Flows (IAFs; [46]) for parallel sampling. While the latter does
not reflect the structure of the prior and true posterior as closely as the
autoregressive posterior, it is a viable option in cases where parallel
sampling is necessary. In the IAF-based posterior,
$\mathrm{q}(\bm{\chi}_{B:T})$ can be sampled over $t=B:T$ in parallel by
sampling from a standard Normal distribution of dimension $T-B+1$ and
transforming the sample through several MADE layers [47]. In our case, each
MADE layer not only takes the output from the previous layer as input but is
also conditioned on a learnable embedding representing each dimension of
$\bm{\chi}_{t}$. Details of this posterior model can be found in Appendix A.
We develop optimization procedures based on stochastic gradient descent (SGD)
that work in practice on large datasets utilizing our modeling assumptions
from Section 3.2 and Section 3.3. Specifically, we alternate between
optimizing the conditional distribution model and the prior and posterior of
the dynamic model with different sampling strategies to accommodate high
dimensionalities and long time spans. More details of our optimization
procedures are in Appendix B.
### 3.5 Forecasting
At test time we are given the past observations $\bm{y}_{1:T}$ as well as the
input features $\bm{x}_{1:T+H}$, including $H$ future steps, and infer the
conditional distribution
$\mathrm{p}(\bm{y}_{T+1:T+H}|\bm{y}_{1:T},\bm{x}_{1:T+H})$. Based on our
modeling assumptions, the latter can be computed as
$\displaystyle\mathrm{p}(\bm{y}_{T+1:T+H}|\bm{y}_{1:T},\bm{x}_{1:T+H})$ (11)
$\displaystyle=$
$\displaystyle\int\mathrm{p}(\bm{y}_{T+1:T+H}|\bm{y}_{T+1-B:T},\bm{x}_{T+1-B:T+H},\bm{\chi}_{T+1:T+H})\mathrm{p}(\bm{\chi}_{T+1:T+H}|\bm{y}_{1:T},\bm{x}_{1:T})\mathrm{d}\bm{\chi}_{T+1:T+H}.$
The first factor in the integrand can be computed recursively by step-by-step
predictions based on our conditional distribution model given
$\bm{\chi}_{T+1:T+H}$,
$\mathrm{p}(\bm{y}_{T+1:T+H}|\bm{y}_{T+1-B:T},\bm{x}_{T+1-B:T+H},\bm{\chi}_{T+1:T+H})=\prod_{t=T+1}^{T+H}\mathrm{p}(\bm{y}_{t}|\bm{y}_{t-B:t-1},\bm{x}_{t-B:t},\bm{\chi}_{t}).$
(12)
The second factor in the integrand can be further factorized into
$\mathrm{p}(\bm{\chi}_{T+1:T+H}|\bm{y}_{1:T},\bm{x}_{1:T})=\int\mathrm{p}(\bm{\chi}_{T+1:T+H}|\bm{\chi}_{T})\mathrm{p}(\bm{\chi}_{T}|\bm{y}_{1:T},\bm{x}_{1:T})\mathrm{d}\bm{\chi}_{T}.$
(13)
We use Rao-Blackwellized particle filters [48] to infer
$\mathrm{p}(\bm{\chi}_{T}|\bm{y}_{1:T},\bm{x}_{1:T})$, so our model can keep
adapting to new changes in an online manner. Specifically, we jointly infer
$\bm{\pi}_{t}$ and $\bm{\chi}_{t}$ with particles representing $\bm{\pi}_{t}$
and closed-form inference of $\bm{\chi}_{t}$. Once we have the posterior
samples of $\mathrm{p}(\bm{\chi}_{T}|\bm{y}_{1:T},\bm{x}_{1:T})$, we use the
prior model to sample trajectories of $\bm{\chi}_{T+1:T+H}$ conditioned on the
samples of $\bm{\chi}_{T}$. With the sample trajectories of
$\mathrm{p}(\bm{\chi}_{T+1:T+H}|\bm{y}_{1:T},\bm{x}_{1:T})$, we sample the
trajectories of
$\mathrm{p}(\bm{y}_{T+1:T+H}|\bm{y}_{T+1-B:T},\bm{x}_{T+1-B:T+H},\bm{\chi}_{T+1:T+H})$
using the aforementioned step-by-step predictions with our conditional
distribution model.
## 4 Experiments
Table 1: Baselines Models.
Method | S | R | MV | Implementation
---|---|---|---|---
DeepAR [1] | ✓ | | | GluonTS[49, 50]
DeepSSM [5] | ✓ | | | GluonTS
TransformerMAF [4] | ✓ | ✓ | ✓ | PyTorchTS[51]
DeepVAR (I)nd. [3] | ✓ | ✓ | ✓ | GluonTS
DeepVAR (C)opula [3] | | ✓ | ✓ | GluonTS
GP-Scaling [3] | | ✓ | ✓ | GluonTS
GP-Copula [3] | | ✓ | ✓ | GluonTS
LSTM-NVP [4] | | ✓ | ✓ | PyTorchTS
LSTM-MAF [4] | | ✓ | ✓ | PyTorchTS
TimeGrad [29] | | ✓ | ✓ | PyTorchTS
[S = Synthetic; R = Real-World; MV = Multivariate]
We compare our approach with $2$ univariate and $8$ multivariate time series
models, both on synthetic (Section 4.1) and real-world (Section 4.2) datasets;
see Table 1 for an overview, including references to the relevant literature
and implementations. We note that DeepVAR is also called Vec-LSTM in previous
works [3, 4]. For our own model we consider two variants: an ablated model
without the dynamic updates to the conditional distribution described in
Section 3.3 (StatiConF); and our full model including those contributions
(DynaConF). In both cases we experiment with different encoder architectures.
For synthetic data, we use either a two-layer MLP with $32$ hidden units (* –
MLP) or a point-wise linear + $\tanh$ mapping (* – PP) as the encoder. For
real-world data, we use an LSTM as the encoder.
### 4.1 Experiments on Synthetic Data
#### Datasets
For our experiments on synthetic data we simulate four conditionally non-
stationary stochastic processes for $T=2500$ time steps. We use the first
$1000$ steps as training data, the next $500$ steps as validation data, and
the remaining $1000$ steps as test data:
* •
(AR(1) – Flip) We simulate an AR(1) process,
$y_{t}=w_{t}y_{t-1}+\epsilon,\epsilon\sim\mathcal{N}(0,1)$, but resample its
coefficient $w_{t}$ from a uniform categorical distribution over
$\\{-0.5,+0.5\\}$ every 100 steps to introduce non-stationarity.
* •
(AR(1) – Dynamic) We simulate the same process as above but now resample
$w_{t}$ from a continuous uniform distribution $\mathcal{U}(-1,1)$ every 100
steps.
* •
(AR(1) – Sin) We simulate the same process as above but now resample $w_{t}$
according to $w_{t}=\sin(2\pi t/T)$. Different from the two processes above,
this process has a continuously changing non-stationary conditional
distribution with its own time-dependent dynamics.
* •
(VAR(1) – Dynamic) This process can be viewed as a multivariate generalization
of AR(1) – Dynamic and is used in our comparisons with multivariate baselines.
We use a four-dimensional VAR process with an identity noise matrix. Similar
to the univariate case, we resample the entries of the coefficient matrix from
a continuous uniform distribution $\mathcal{U}(-0.8,0.8)$ every 250 steps. In
addition, we ensure the stability of the resulting process by computing the
eigenvalues of the coefficient matrix and discard it if its largest absolute
eigenvalue is greater than $1$.
#### Experimental Setup
For univariate data (AR(1) – Flip/Sin/Dynamic), we compare our approach with
the two univariate baselines DeepAR and DeepSSM. For multivariate data (VAR(1)
– Dynamic), most baselines are redundant because their focus is on better
observation models, while the underlying temporal backbone is similar. Since
our synthetic observation distributions are simple, we compare with two model
families that differ in their temporal backbone: DeepVAR (RNN backbone) and
TransformerMAF (Transformer backbone). We use a context window size of $200$
to give the model access to the information needed to infer the current
parameter of the true conditional distribution. We tried increasing the window
size to $500$ on VAR(1) – Dynamic but did not see performance improvements. We
also removed the unnecessary default input features of these models to prevent
overfitting. For details of the setup, please see Appendix D. DynaConF is
trained with the autoregressive posterior.
For evaluation we use a rolling-window approach with a window size of $10$
steps. The final evaluation metrics are the aggregated results from all $100$
test windows. For univariate data we report the continuous ranked probability
score (CRPS) [52], a commonly used score to measure how close the predicted
distribution is to the true distribution. For multivariate data we also report
$\textrm{CRPS}_{\Sigma}$, a popular variant of CRPS for multivariate
distributions [3, 29, 4]. Additional results, including an evaluation based on
mean squared error (MSE), can be found in the Appendix E. In all cases lower
values indicate better performance.
#### Results
The results on synthetic data are shown in Table 2. For univariate data, our
full model (DynaConF) outperforms its ablated counterpart (StatiConF) by 2.0%
– 12.3%, validating the importance of our dynamic adaptation to non-starionary
effects. DynaConF – PP is also superior to all univariate baselines, with its
closest competitor DeepAR – 10 behind by an average of 6.1%. Furthermore, we
note that our model with the pointwise encoder tends to outperform the MLP
encoder, both for the ablated and full model. For multivariate data we observe
similar trends. Here, our full model (DynaConF – PP) performs 24.3% (CRPS) /
33.8% ($\textrm{CRPS}_{\Sigma}$) better than the ablated model with static
conditional distribution (StatiConF – PP) and 22.6% (CRPS) / 31.0%
($\textrm{CRPS}_{\Sigma}$) better than the best-performing baseline (DeepVAR –
160). As before, pointwise encoders tend to perform better than MLP encoders.
Since for synthetic data we also have access to the ground-truth models, we
include the corresponding scores as a reference and upper bound in terms of
performance.
Figure 2 shows qualitative results of our model on AR(1) – Flip/Sin/Dynamic.
Note that because the encoder in our model is non-linear, the inferred
$\bm{\phi}_{t}$ does not necessarily correspond to the original parameter.
However, we can clearly see how it differs for Sin (continuous changes) vs
Flip/Dynamic (discrete jumps) and how it tracks the ground-truth up to
scale/sign.
(a) Flip
(b) Sin
(c) Dynamic
Figure 2: Qualitative Results. We show one dimension of $\bm{\phi}_{t}$ of
DynaConF – PP inferred with particle filters at test time on (a) AR(1) – Flip,
(b) AR(1) – Sin, and (c) AR(1) – Dynamic. The red dashed lines show the
ground-truth parameters of the conditional distribution varying over time. The
blue curves and bands are the medians and 90% confidence intervals of the
posterior. Table 2: Quantitative Evaluation (Synthetic Data). We compare
StatiConF/DynaConF to (a) $4$ univariate and (b) $6$ multivariate baselines.
For univariate processes (AR(1)-Flip/Sin/Dynamic) we report CRPS. For
multivariate processes (VAR(1)-Dynamic) we report both CRPS and
$\textrm{CRPS}_{\Sigma}$. Values behind models indicate the number of hidden
units.
| Method | CRPS
---|---|---
| AR(1)-F | AR(1)-S | AR(1)-D
| GroundTruth | 0.731$\scriptscriptstyle\pm\text{0.001}$ | 0.710$\scriptscriptstyle\pm\text{0.001}$ | 0.624$\scriptscriptstyle\pm\text{0.001}$
| DeepAR – 10 | 0.741$\scriptscriptstyle\pm\text{0.005}$ | 0.764$\scriptscriptstyle\pm\text{0.004}$ | 0.768$\scriptscriptstyle\pm\text{0.011}$
| DeepAR – 40 | 0.740$\scriptscriptstyle\pm\text{0.003}$ | 0.776$\scriptscriptstyle\pm\text{0.002}$ | 0.820$\scriptscriptstyle\pm\text{0.053}$
| DeepAR – 160 | 0.740$\scriptscriptstyle\pm\text{0.001}$ | 0.774$\scriptscriptstyle\pm\text{0.004}$ | 0.801$\scriptscriptstyle\pm\text{0.047}$
| DeepSSM | 0.755$\scriptscriptstyle\pm\text{0.001}$ | 0.761$\scriptscriptstyle\pm\text{0.001}$ | 0.803$\scriptscriptstyle\pm\text{0.001}$
(ours) | StatiConF – MLP | 0.753$\scriptscriptstyle\pm\text{0.001}$ | 0.784$\scriptscriptstyle\pm\text{0.002}$ | 0.764$\scriptscriptstyle\pm\text{0.003}$
StatiConF – PP | 0.752$\scriptscriptstyle\pm\text{0.002}$ | 0.763$\scriptscriptstyle\pm\text{0.001}$ | 0.783$\scriptscriptstyle\pm\text{0.002}$
DynaConF – MLP | 0.750$\scriptscriptstyle\pm\text{0.001}$ | 0.727$\scriptscriptstyle\pm\text{0.019}$ | 0.691$\scriptscriptstyle\pm\text{0.006}$
DynaConF – PP | 0.737$\scriptscriptstyle\pm\text{0.001}$ | 0.721$\scriptscriptstyle\pm\text{0.005}$ | 0.687$\scriptscriptstyle\pm\text{0.010}$
[AR(1)-*: F = Flip; S = Sin; D = Dynamic]
(a) Univariate
| Method | CRPS | $\textrm{CRPS}_{\Sigma}$
---|---|---|---
| GroundTruth | 0.496$\scriptscriptstyle\pm\text{0.001}$ | $0.437\scriptscriptstyle\pm\text{0.001}$
| DeepVAR – 10 | 0.797$\scriptscriptstyle\pm\text{0.009}$ | $0.824\scriptscriptstyle\pm\text{0.015}$
| DeepVAR – 40 | 0.792$\scriptscriptstyle\pm\text{0.000}$ | $0.821\scriptscriptstyle\pm\text{0.002}$
| DeepVAR – 160 | 0.787$\scriptscriptstyle\pm\text{0.001}$ | $0.816\scriptscriptstyle\pm\text{0.002}$
| TransformerMAF – 8 | 0.800$\scriptscriptstyle\pm\text{0.001}$ | $0.831\scriptscriptstyle\pm\text{0.008}$
| TransformerMAF – 32 | 0.806$\scriptscriptstyle\pm\text{0.008}$ | $0.855\scriptscriptstyle\pm\text{0.011}$
| TransformerMAF – 128 | 0.866$\scriptscriptstyle\pm\text{0.077}$ | $0.873\scriptscriptstyle\pm\text{0.025}$
(ours) | StatiConF – MLP | 0.806$\scriptscriptstyle\pm\text{0.004}$ | $0.843\scriptscriptstyle\pm\text{0.005}$
StatiConF – PP | 0.805$\scriptscriptstyle\pm\text{0.002}$ | $0.850\scriptscriptstyle\pm\text{0.003}$
DynaConF – MLP | 0.762$\scriptscriptstyle\pm\text{0.036}$ | 0.777$\scriptscriptstyle\pm\text{0.047}$
DynaConF – PP | 0.609$\scriptscriptstyle\pm\text{0.012}$ | 0.563$\scriptscriptstyle\pm\text{0.019}$
(b) Multivariate: VAR(1) – Dynamic
### 4.2 Experiments on Real-World Data
#### Datasets
We evaluate the proposed method on six publicly available datasets [53, 2, 3]:
(Exchange) daily exchange rates of 8 different countries from 1990 to 2016;
(Solar) [54] solar power production in 10-minute intervals of 137 PV plants in
2006; (Electricity) [55] hourly electricity consumption of 370 customers from
2012 to 2014; (Traffic) [56] hourly occupancy data at 963 sensor locations in
the San Francisco Bay area; (Taxi) rides taken in 30-minute intervals at 1214
locations in New York City in January 2015/2016; (Wikipedia) daily page views
of 2000 Wikipedia articles.
#### Experimental Setup
We use the same train/test splits and the same input features, such as time of
the day, as previous works with published code and results [3, 4, 29]. For our
method, we first train StatiConF and then reuse its learned encoder in
DynaConF, so the learning of DynaConF is focused on the dynamic model.
DynaConF is trained with the IAF posterior for efficiency. Our models use a
two-layer LSTM with 128 hidden units as the encoder, except for the
8-dimensional Exchange data, where the hidden size is 8. We stress again that,
different from DeepVAR or LSTM-MAF, we use LSTM as an encoder of
$(\bm{y}_{t-B:t-1},\bm{x}_{t-B,t})$ only, so we actually “restart” it at every
time step. More details of our hyperparameters can be found in Appendix C.
#### Results
The results are shown in Table 3. As we can see, for different datasets and
different evaluation metrics, the relative performance of each method can be
different. This shows the diversity of these datasets and that different
models may benefit from dataset-specific structure in different ways. However,
DynaConF achieves the best performance more often than all the other
baselines. Where it does not outperform, its performance is competitive
consistently, unlike the baselines, whose relative performance (compared to
others) varies significantly across datasets. We also note that our full
model, DynaConF, which adapts to changes in the conditional distribution
consistently outperforms our ablated model, StatiConF, which only models a
static conditional distribution, except for electricity, where the performance
is similar. This shows the effectiveness of modeling the dynamic changes in
the conditional distribution. Further results, including CRPSΣ, and standard
deviations of MSE and CRPS are in Appendix E.
Table 3: Quantitative Evaluation (Real-World Data). We compare the CRPS and
MSE scores of StatiConF/DynaConF and 7 baselines on 6 publicly available
datasets. Lower values are better.
Method | Exchange | Solar | Electricity | Traffic | Taxi | Wikipedia
---|---|---|---|---|---|---
CRPS | MSE | CRPS | MSE | CRPS | MSE | CRPS | MSE | CRPS | MSE | CRPS | MSE
| [e-4] | | [e+2] | | [e+5] | | [e-4] | | [e+1] | | [e+7]
DeepVAR (I) | 0.013 | 1.6 | 0.434 | 9.3 | 1.059 | 2.1 | 0.168 | 6.3 | 0.586 | 7.3 | 0.379 | 7.2
DeepVAR (C) | 0.009 | 1.9 | 0.384 | 29 | 0.084 | 55 | 0.165 | 15 | 0.416 | 5.1 | 0.247 | 3.8
GP-scaling | 0.017 | 2.9 | 0.415 | 11 | 0.053 | 1.8 | 0.140 | 5.2 | 0.346 | 2.7 | 1.549 | 5.5
GP-Copula | 0.008 | 1.7 | 0.371 | 9.8 | 0.056 | 2.4 | 0.133 | 6.9 | 0.360 | 3.1 | 0.236 | 4.0
LSTM-NVP | 0.010 | 2.4 | 0.365 | 9.1 | 0.059 | 2.5 | 0.172 | 6.9 | 0.327 | 2.6 | 0.333 | 4.7
LSTM-MAF | 0.012 | 3.8 | 0.378 | 9.8 | 0.051 | 1.8 | 0.124 | 4.9 | 0.314 | 2.4 | 0.282 | 3.8
TransformerMAF | 0.012 | 3.4 | 0.368 | 9.3 | 0.052 | 2.0 | 0.134 | 5.0 | 0.377 | 4.5 | 0.274 | 3.1
StatiConF (ours) | 0.011 | 4.4 | 0.343 | 6.6 | 0.059 | 2.0 | 0.143 | 4.1 | 0.311 | 2.2 | 0.427 | 4.1
DynaConF (ours) | 0.010 | 2.6 | 0.338 | 6.4 | 0.058 | 2.1 | 0.140 | 3.9 | 0.308 | 2.2 | 0.296 | 3.8
## 5 Discussion
In this work, we addressed the problem of modeling and forecasting time series
with non-stationary conditional distributions. We proposed a new model,
DynaConF, that explicitly decouples the time-invariant conditional
distribution modeling and the time-variant non-stationarity modeling. We
designed specific architectures, developed two types of variational
posteriors, and employed Rao-Blackwellized particle filters to allow the model
to train efficiently on large multivariate time series and adapt online at
test time. Results on synthetic and real-world data show that our model can
learn and adapt to different types of changes in the conditional distribution
parameters and perform competitively or better than state-of-the-art time
series forecasting models.
Our model currently has the following limitations. (1) Since the variational
posterior model complexity scales in $O(T)$, it is challenging to train the
model on very long time series. (2) We used a simple observation distribution
family in this work for efficient inference, but there are cases where this
may impact performance. For future work, it would be interesting to develop
new variational posteriors and training algorithms that can scale better and
more flexible inference algorithms that can deal with more complex observation
distributions.
## References
* Salinas et al. [2020] David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. DeepAR: Probabilistic forecasting with autoregressive recurrent networks. _International Journal of Forecasting_ , 2020.
* Lai et al. [2018] Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_, 2018.
* Salinas et al. [2019] David Salinas, Michael Bohlke-Schneider, Laurent Callot, Roberto Medico, and Jan Gasthaus. High-dimensional multivariate forecasting with low-rank Gaussian Copula Processes. In _Advances in Neural Information Processing Systems_ , 2019\.
* Rasul et al. [2021a] Kashif Rasul, Abdul-Saboor Sheikh, Ingmar Schuster, Urs M. Bergmann, and Roland Vollgraf. Multivariate probabilistic time series forecasting via conditioned normalizing flows. In _International Conference on Learning Representations_ , 2021a.
* Rangapuram et al. [2018] Syama Sundar Rangapuram, Matthias W. Seeger, Jan Gasthaus, Lorenzo Stella, Yuyang Wang, and Tim Januschowski. Deep state space models for time series forecasting. In _Advances in Neural Information Processing Systems_ , 2018.
* Brockwell and Davis [2009] Peter J. Brockwell and Richard A. Davis. _Time Series: Theory and Methods_. 2009\.
* Hamilton [1994] James Douglas Hamilton. _Time Series Analysis_. 1994\.
* Kwiatkowski et al. [1992] Denis Kwiatkowski, Peter CB Phillips, Peter Schmidt, and Yongcheol Shin. Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? _Journal of Econometrics_ , 1992.
* Dickey and Fuller [1979] David A. Dickey and Wayne A. Fuller. Distribution of the estimators for autoregressive time series with a unit root. _Journal of the American Statistical Association_ , 1979.
* Kim et al. [2022] Taesung Kim, Jinhee Kim, Yunwon Tae, Cheonbok Park, Jang-Ho Choi, and Jaegul Choo. Reversible instance normalization for accurate time-series forecasting against distribution shift. In _International Conference on Learning Representations_ , 2022.
* Box et al. [2015] George EP Box, Gwilym M. Jenkins, Gregory C. Reinsel, and Greta M. Ljung. _Time Series Analysis: Forecasting and Control_. 2015\.
* Engle [1982] Robert F. Engle. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. _Econometrica: Journal of the Econometric Society_ , 1982.
* Bollerslev [1986] Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity. _Journal of Econometrics_ , 1986.
* Kalman [1960] Re E Kalman. A new approach to linear filtering and prediction problems. _Transactions of the ASME-Journal of Basic Engineering_ , 1960.
* Hochreiter and Schmidhuber [1997] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. _Neural Computation_ , 1997.
* Bai et al. [2018] Shaojie Bai, J. Zico Kolter, and Vladlen Koltun. An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. _arXiv preprint arXiv:1803.01271_ , 2018.
* Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. _Advances in Neural Information Processing Systems_ , 2017.
* Garnelo et al. [2018a] Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and S. M. Ali Eslami. Conditional neural processes. In _International Conference on Machine Learning_ , 2018a.
* Garnelo et al. [2018b] Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J. Rezende, S. M. Eslami, and Yee Whye Teh. Neural processes. _arXiv preprint arXiv:1807.01622_ , 2018b.
* Lim and Zohren [2020] Bryan Lim and Stefan Zohren. Time series forecasting with deep learning: A survey. _arXiv:2004.13408 [cs, stat]_ , 2020.
* Holt [2004] Charles C. Holt. Forecasting seasonals and trends by exponentially weighted moving averages. _International Journal of Forecasting_ , 2004.
* Cleveland et al. [1990] Robert B. Cleveland, William S. Cleveland, Jean E. McRae, and Irma Terpenning. STL: A seasonal-trend decomposition procedure based on loess. _Journal of Official Statistics_ , 1990.
* Smyl [2020] Slawek Smyl. A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting. _International Journal of Forecasting_ , 2020.
* Bandara et al. [2020] Kasun Bandara, Christoph Bergmeir, and Slawek Smyl. Forecasting across time series databases using recurrent neural networks on groups of similar series: A clustering approach. _Expert Systems with Applications_ , 2020.
* Oreshkin et al. [2020] Boris N. Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-BEATS: Neural basis expansion analysis for interpretable time series forecasting. _arXiv:1905.10437 [cs, stat]_ , 2020.
* Wu et al. [2021] Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting. In _Advances in Neural Information Processing Systems_ , 2021\.
* Woo et al. [2022] Gerald Woo, Chenghao Liu, Doyen Sahoo, Akshat Kumar, and Steven Hoi. ETSformer: Exponential Smoothing Transformers for Time-series Forecasting. _arXiv preprint arXiv:2202.01381_ , 2022.
* Corani et al. [2021] Giorgio Corani, Alessio Benavoli, and Marco Zaffalon. Time series forecasting with Gaussian Processes needs priors. In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , 2021.
* Rasul et al. [2021b] Kashif Rasul, Calvin Seward, Ingmar Schuster, and Roland Vollgraf. Autoregressive denoising diffusion models for multivariate probabilistic time series forecasting. In _Proceedings of the 38th International Conference on Machine Learning_ , 2021b.
* Fraccaro et al. [2017] Marco Fraccaro, Simon Kamronn, Ulrich Paquet, and Ole Winther. A disentangled recognition and nonlinear dynamics model for unsupervised learning. In _Advances in Neural Information Processing Systems_ , 2017\.
* de Bézenac et al. [2020] Emmanuel de Bézenac, Syama Sundar Rangapuram, Konstantinos Benidis, Michael Bohlke-Schneider, Richard Kurle, Lorenzo Stella, Hilaf Hasson, Patrick Gallinari, and Tim Januschowski. Normalizing Kalman filters for multivariate time series analysis. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, _Advances in Neural Information Processing Systems_ , 2020.
* Tang and Matteson [2021] Binh Tang and David S Matteson. Probabilistic transformer for time series analysis. In _Advances in Neural Information Processing Systems_ , 2021\.
* Klushyn et al. [2021] Alexej Klushyn, Richard Kurle, Maximilian Soelch, Botond Cseke, and Patrick van der Smagt. Latent matters: Learning deep state-space models. In _Advances in Neural Information Processing Systems_ , 2021\.
* Ansari et al. [2021] Abdul Fatir Ansari, Konstantinos Benidis, Richard Kurle, Ali Caner Turkmen, Harold Soh, Alexander J. Smola, Bernie Wang, and Tim Januschowski. Deep explicit duration switching models for time series. _Advances in Neural Information Processing Systems_ , 2021.
* Kurle et al. [2020] Richard Kurle, Syama Sundar Rangapuram, Emmanuel de Bézenac, Stephan Günnemann, and Jan Gasthaus. Deep Rao-Blackwellised particle filters for time series forecasting. _Advances in Neural Information Processing Systems_ , 2020.
* Nguyen and Quanz [2021] Nam Nguyen and Brian Quanz. Temporal latent auto-encoder: A method for probabilistic multivariate time series forecasting. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , 2021.
* Le Guen and Thome [2020] Vincent Le Guen and Nicolas Thome. Probabilistic time series forecasting with shape and temporal diversity. _Advances in Neural Information Processing Systems_ , 2020.
* Yu et al. [2021] Zhongjie Yu, Fabrizio G. Ventola, and Kristian Kersting. Whittle networks: A deep likelihood model for time series. In _International Conference on Machine Learning_ , 2021.
* De Lange et al. [2021] Matthias De Lange, Rahaf Aljundi, Marc Masana, Sarah Parisot, Xu Jia, Ales Leonardis, Gregory Slabaugh, and Tinne Tuytelaars. A continual learning survey: Defying forgetting in classification tasks. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2021.
* Parisi et al. [2019] German I. Parisi, Ronald Kemker, Jose L. Part, Christopher Kanan, and Stefan Wermter. Continual lifelong learning with neural networks: A review. _Neural Networks_ , 2019.
* Kirkpatrick et al. [2017] James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A. Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, Demis Hassabis, Claudia Clopath, Dharshan Kumaran, and Raia Hadsell. Overcoming catastrophic forgetting in neural networks. _arXiv:1612.00796 [cs, stat]_ , 2017.
* Gupta et al. [2022] Vibhor Gupta, Jyoti Narwariya, Pankaj Malhotra, Lovekesh Vig, and Gautam Shroff. Continual learning for multivariate time series tasks with variable input dimensions. _arXiv:2203.06852 [cs]_ , 2022.
* Nguyen et al. [2018] Cuong V. Nguyen, Yingzhen Li, Thang D. Bui, and Richard E. Turner. Variational continual learning. In _International Conference on Learning Representations_ , 2018.
* Kurle et al. [2019] Richard Kurle, Botond Cseke, Alexej Klushyn, Patrick van der Smagt, and Stephan Günnemann. Continual learning with bayesian neural networks for non-stationary data. In _International Conference on Learning Representations_ , 2019.
* Kingma and Welling [2013] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. _arXiv preprint arXiv:1312.6114_ , 2013.
* Kingma et al. [2017] Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving variational inference with inverse autoregressive flow. _arXiv:1606.04934 [cs, stat]_ , 2017.
* Germain et al. [2015] Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. MADE: Masked autoencoder for distribution estimation. In _Proceedings of the 32nd International Conference on Machine Learning_ , 2015.
* Doucet et al. [2000] Arnaud Doucet, Nando de Freitas, Kevin Murphy, and Stuart Russell. Rao-Blackwellised particle filtering for dynamic Bayesian networks. In _Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence_ , UAI’00, 2000.
* Alexandrov et al. [2020] Alexander Alexandrov, Konstantinos Benidis, Michael Bohlke-Schneider, Valentin Flunkert, Jan Gasthaus, Tim Januschowski, Danielle C. Maddix, Syama Rangapuram, David Salinas, Jasper Schulz, Lorenzo Stella, Ali Caner Türkmen, and Yuyang Wang. GluonTS: Probabilistic and Neural Time Series Modeling in Python. _Journal of Machine Learning Research_ , 2020.
* glu [2021] GluonTS, 2021. URL https://github.com/awslabs/gluon-ts.
* [51] Kashif Rasul. PytorchTS. URL https://github.com/zalandoresearch/pytorch-ts.
* Matheson and Winkler [1976] James E. Matheson and Robert L. Winkler. Scoring rules for continuous probability distributions. _Management Science_ , 1976.
* [53] Public time series datasets. URL https://github.com/mbohlkeschneider/gluon-ts/tree/mv_release/datasets.
* [54] Solar power dataset. URL http://www.nrel.gov/grid/solar-power-data.html.
* [55] Electricity dataset. URL https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014.
* [56] Traffic dataset. URL http://pems.dot.ca.gov.
* Smith and Topin [2017] Leslie N. Smith and Nicholay Topin. Super-convergence: Very fast training of neural networks using large learning rates. _arXiv preprint arXiv:1708.07120_ , 2017.
* Kingma and Ba [2014] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_ , 2014.
## Appendix A Normalizing-Flow-Based Variational Posterior
We develop an alternative variational posterior model based on Inverse
Autoregressive Flows (IAFs; [46]). In this model,
$\mathrm{q}(\bm{\chi}_{B:T})$ can be sampled jointly over $t=B:T$ in parallel
as follows. We again let $\mathrm{q}(\bm{\chi}_{B})=\mathrm{p}(\bm{\chi}_{B})$
and only use IAF to model $\mathrm{q}(\bm{\chi}_{B+1:T})$. Let
$T^{\prime}=T-B$. To sample the $i$-th dimension of $\bm{\chi}_{B+1:T}$, we
first sample a vector $\bm{z}_{0}$ of dimension $T^{\prime}$ from a standard
multivariate normal distribution
$\bm{z}_{0}\sim\mathcal{N}(\bm{0},\bm{I}).$ (14)
Then, this sample is transformed sequentially for $l=1,\ldots,L$ as
$\bm{z}_{l}=\bm{\mu}_{l}+\bm{\sigma}_{l}\odot\bm{z}_{l-1},$ (15)
where $\bm{\mu}_{l}$ and $\bm{\sigma}_{l}$ are the output from a neural
network, specifically MADE [47], with a soft-plus transformation on the latter
to make sure it is positive. The final output $\bm{z}_{L}$ is taken as the
sample of the $i$-th dimension of $\bm{\chi}_{B+1:T}$. The input of the MADE
at the $l$-th iteration consists of not only the previous output
$\bm{z}_{l-1}$ but also an additional learnable embedding for
$\bm{\chi}_{B+1:T,i}$, the $i$-th dimension of $\bm{\chi}_{B+1:T}$. We set
$L=3$ and use MADE of 2 hidden layers of size 1024 and learnable embeddings of
size 512 for all our experiments.
To compute $\mathrm{q}(\bm{\chi}_{B+1:T})$, we also need to compute the
determinants of the Jacobians
$|{\mathrm{d}\bm{z}_{l}}/{\mathrm{d}\bm{z}_{l-1}}|,l=1,\ldots,L$, in addition
to the density of the base distribution, which is the standard normal
$\mathcal{N}(\bm{0},\bm{I})$. Due to the structure of MADE and the definition
of IAF (Eq. 15), the Jacobian determinants can be easily computed from
$\bm{\sigma}_{l}$.
## Appendix B Optimization Procedure
In contrast to existing time series models, we utilize a flexible variational
posterior with a large number of parameters in the order of $O(T)$ and a
structured prior model to account for conditional distribution changes over
time. However, jointly optimizing over these variational parameters and the
parameters in the conditional distribution model itself using stochastic
gradient descent can be prohibitively demanding on the computational
resources, especially GPU memory. Instead, we propose an alternative
optimization procedure to learn these parameters.
Specifically, instead of optimizing by stochastic gradient descent (SGD) over
all parameters in both the conditional distribution model and the prior and
variational posterior model, we propose to learn the model by alternating the
optimization of the parameters in the conditional distribution model and the
prior and variational posterior models. For the former, we condition on the
samples from the current posterior model while optimizing the conditional
distribution model on randomly sampled sub-sequences from the time series
using SGD. For the latter, we fix the conditional distribution model, sample
from the variational posterior model over the entire time series either
sequentially or in parallel, depending on the variational posterior model we
use, and compute the loss using the samples through the conditional
distribution, prior, and posterior models. Then we perform an update on the
prior and posterior model parameters using the gradient from the loss.
When the time series is high-dimensional and GPU memory becomes a constraint
during training, we randomly sample a subset of _observation_ dimensions for
each batch, since the loss decomposes over the observation dimensions in our
model.
## Appendix C Hyperparameters
On synthetic data, we use the validation set to early stop and choose the best
model for both the baselines and StatiConF. For DynaConF, we keep training as
long as the loss is decreasing on the training set. We perform 50 updates per
epoch. We use 32 hidden units for our 2-layer MLP encoder. For the baselines,
we report their results with different hidden sizes, including the default
ones.
On real-world data, we use the “1cycle” learning rate scheduler [57] and
manually choose the number of epochs for StatiConF. After training StatiConF,
we reuse its encoders in DynaConF, so it only needs to learn the dynamic
model. For DynaConF, we train until the loss on the training set converges.
For our models, we use Adam [58] as the optimizer with the default learning
rate of $0.001$ unless the learning rate is controlled by the “1cycle”
learning rate scheduler. The dimension of the latent vector $\bm{z}_{t,i}$
(see Section 3.2) is set to $E=4$ across all the experiments.
Because of the diversity of the real-world datasets, we further apply
techniques to stablize training. Specifically, for all datasets, we use the
mean and standard deviation to shift and scale each dimension of the time
series. For Exchange, we use the mean and standard deviation of the recent
past data in a moving context window. For the other datasets, we simply use
the global mean and standard deviation of each dimension computed using the
whole training set. In all cases, for forecasting, the output from the model
is inversely scaled and shifted back for evaluation.
Different from the rest of the datasets, Taxi and Wikipedia consist of counts.
The time series in Taxi are relatively small counts, while the time series in
Wikipedia are very large and can have extreme values. To stabilize training,
we apply the dequantization technique used in [4] to Taxi, where we add a
small noise from the uniform distribution $\mathcal{U}(-0.5,0.5)$ to the
target time series $\bm{y}$. On Wikipedia, we use the quantiles (0.02 and
0.95) computed from the recent past data in a moving context window to
Winsorize extreme values.
## Appendix D Experiment Setup Details
On synthetic data, we keep most baseline hyperparameters to their default
values but make the following changes to account for properties of our
synthetic data: (1) To reduce overfitting we remove any unnecessary input
features from the models and use only past observations with time lag $1$ as
input. (2) To allow the models to adapt to changes in the conditional
distribution we increase the context window size to $200$. This allows the
models to see enough observations generated with the latest ground-truth
distribution parameters, so the models have the necessary information to adapt
to the current distribution. For VAR(1) – Dynamic, we also tried extending it
to $500$, but it did not improve the performance. (3) DeepSSM allows modeling
of trend and seasonality, but since our synthetic data do not have those, we
explicitly remove those components from the model specification to avoid
overfitting; (4) DeepVAR allows modeling of different covariance structures in
the noise, such as diagonal, low-rank, and full-rank. Since our synthetic data
follow a diagonal covariance structure in the noise, we explicitly specify
that in DeepVAR.
We run all experiments for three different random seeds independently and
calculate and report the mean and standard deviation of each evaluation metric
for each model. On synthetic datasets, we use 1000 sample paths to empirically
estimate the predicted distributions for all models. On real-world datasets,
we use 100 sample paths.
We use three evaluation metrics: mean squared error (MSE), continuous ranked
probability score (CRPS)[52] and CRPSΣ. Assume that we observe $y$ at time $t$
but a probabilistic forecasting model predicts the distribution of $y$ to be
$F$. MSE is widely used for time series forecasting, and for a probabilistic
forecasting model, where the mean of the distribution is used for point
prediction, it is defined as
$\text{MSE}(F,y)=(\mathrm{E}_{z\sim F}[z]-y)^{2}$ (16)
for a single time point $t$ and averaged over all the time points in the test
set.
CRPS has been used for evaluating how close the predicted probability
distribution is to the ground-truth distribution and is defined as
$\text{CRPS}(F,y)=\int_{\mathbb{R}}(F(z)-\mathbb{I}[y\leq
z])^{2}\mathrm{d}{z},$ (17)
for a single time point $t$ and averaged over all the time points in the test
set, where $\mathbb{I}$ denotes the indicator function. Generally, $F(z)$ can
be approximated by the empirical distribution of the samples from the
predicted distribution. Both MSE and CRPS can be applied to multivariate time
series by computing the metric on each dimension and then averaging over all
the dimensions.
CRPSΣ is another metric that has been used in recent works [3, 4, 29] to
evaluate multivariate probabilistc forecasting results. To compute it, the sum
of the time series across all the dimensions is computed and then compared
with the sum of the prediction using CRPS. That is
$\text{CRPS}_{\Sigma}(F_{\Sigma},\sum_{i}y_{i})=\int_{\mathbb{R}}(F_{\Sigma}(z)-\mathbb{I}[\sum_{i}y_{i}\leq
z])^{2}\mathrm{d}{z},$ (18)
where $F_{\Sigma}$ is the distribution of the sum of the predicted values
$\sum_{i}z_{i},z_{i}\sim F_{i}(z)$ and $i$ denotes the dimension $i$ of the
time series. $F_{\Sigma}$ is usually approximated by the empirical
distribution of the samples summed across the dimensions.
## Appendix E Additional Experiment Results
Table 4 and 5 show the MSE results of the baselines and our models on the
univariate and multivariate processes respectively. Table 6 shows the CRPSΣ
scores of our models on the real-world datasets, while Table 7 and 8 show the
full CRPS and MSE results of our models with means and standard deviations.
The results of the baselines on the real-world data are from [4, 29].
Table 4: Quantitative Evaluation (Synthetic Data). MSE results on univariate processes AR(1)-Flip/Sin/Dynamic. Values behind architectures indicate the number of hidden units. | Method | MSE
---|---|---
| AR(1)-Flip | AR(1)-Sin | AR(1)-Dynamic
| GroundTruth | 1.2$\scriptscriptstyle\pm\text{9.2e-4}$ | 1.6$\scriptscriptstyle\pm\text{3.0e-3}$ | 2.1$\scriptscriptstyle\pm\text{5.3e-3}$
| DeepAR – 10 | 1.3$\scriptscriptstyle\pm\text{1.7e-2}$ | 1.8$\scriptscriptstyle\pm\text{1.8e-2}$ | 3.2$\scriptscriptstyle\pm\text{7.2e-2}$
| DeepAR – 40 | 1.3$\scriptscriptstyle\pm\text{8.9e-3}$ | 1.8$\scriptscriptstyle\pm\text{1.3e-2}$ | 3.6$\scriptscriptstyle\pm\text{3.8e-1}$
| DeepAR – 160 | 1.3$\scriptscriptstyle\pm\text{2.8e-3}$ | 1.8$\scriptscriptstyle\pm\text{9.6e-3}$ | 3.5$\scriptscriptstyle\pm\text{3.3e-1}$
| DeepSSM | 1.3$\scriptscriptstyle\pm\text{2.2e-3}$ | 1.8$\scriptscriptstyle\pm\text{2.8e-3}$ | 3.3$\scriptscriptstyle\pm\text{3.0e-3}$
(ours) | StatiConF – MLP | 1.3$\scriptscriptstyle\pm\text{4.0e-3}$ | 1.9$\scriptscriptstyle\pm\text{7.7e-3}$ | 3.3$\scriptscriptstyle\pm\text{3.5e-2}$
StatiConF – PP | 1.3$\scriptscriptstyle\pm\text{4.7e-3}$ | 1.8$\scriptscriptstyle\pm\text{5.4e-3}$ | 3.3$\scriptscriptstyle\pm\text{6.2e-3}$
DynaConF – MLP | 1.3$\scriptscriptstyle\pm\text{4.0e-3}$ | 1.6$\scriptscriptstyle\pm\text{9.2e-2}$ | 2.6$\scriptscriptstyle\pm\text{5.2e-2}$
DynaConF – PP | 1.2$\scriptscriptstyle\pm\text{5.5e-3}$ | 1.6$\scriptscriptstyle\pm\text{2.7e-2}$ | 2.6$\scriptscriptstyle\pm\text{5.1e-2}$
Table 5: Quantitative Evaluation (Synthetic Data). MSE results on multivariate process VAR(1)-Dynamic. Values behind architectures indicate the number of hidden units. | Method | MSE
---|---|---
| GroundTruth | 2.8$\scriptscriptstyle\pm\text{7.6e-3}$
| DeepVAR – 10 | 8.4$\scriptscriptstyle\pm\text{5.6e-2}$
| DeepVAR – 40 | 8.4$\scriptscriptstyle\pm\text{2.7e-3}$
| DeepVAR – 160 | 8.3$\scriptscriptstyle\pm\text{2.2e-2}$
| TransformerMAF – 8 | 8.5$\scriptscriptstyle\pm\text{2.9e-2}$
| TransformerMAF – 32 | 8.5$\scriptscriptstyle\pm\text{3.7e-2}$
| TransformerMAF – 128 | 9.4$\scriptscriptstyle\pm\text{1.2e0}$
(ours) | StatiConF – MLP | 8.5$\scriptscriptstyle\pm\text{8.5e-2}$
StatiConF – PP | 8.5$\scriptscriptstyle\pm\text{2.3e-2}$
DynaConF – MLP | 7.9$\scriptscriptstyle\pm\text{5.7e-1}$
DynaConF – PP | 4.5$\scriptscriptstyle\pm\text{3.0e-1}$
Table 6: Quantitative Evaluation (Real-World Data). Full CRPSΣ results of StatiConF/DynaConF and 8 baselines with means and standard deviations on 6 publicly available datasets. Lower values are better. Method | Exchange | Solar | Electricity | Traffic | Taxi | Wikipedia
---|---|---|---|---|---|---
DeepVAR (I) | $0.008\scriptscriptstyle\pm\text{0.001}$ | $0.391\scriptscriptstyle\pm\text{0.017}$ | $0.025\scriptscriptstyle\pm\text{0.001}$ | $0.087\scriptscriptstyle\pm\text{0.041}$ | $0.506\scriptscriptstyle\pm\text{0.005}$ | $0.133\scriptscriptstyle\pm\text{0.002}$
DeepVAR (C) | $0.007\scriptscriptstyle\pm\text{0.000}$ | $0.319\scriptscriptstyle\pm\text{0.011}$ | $0.064\scriptscriptstyle\pm\text{0.008}$ | $0.103\scriptscriptstyle\pm\text{0.006}$ | $0.326\scriptscriptstyle\pm\text{0.007}$ | $0.241\scriptscriptstyle\pm\text{0.033}$
GP-Scaling | $0.009\scriptscriptstyle\pm\text{0.000}$ | $0.368\scriptscriptstyle\pm\text{0.012}$ | $0.022\scriptscriptstyle\pm\text{0.000}$ | $0.079\scriptscriptstyle\pm\text{0.000}$ | $0.183\scriptscriptstyle\pm\text{0.395}$ | $1.483\scriptscriptstyle\pm\text{1.034}$
GP-Copula | $0.007\scriptscriptstyle\pm\text{0.000}$ | $0.337\scriptscriptstyle\pm\text{0.024}$ | $0.024\scriptscriptstyle\pm\text{0.002}$ | $0.078\scriptscriptstyle\pm\text{0.002}$ | $0.208\scriptscriptstyle\pm\text{0.183}$ | $0.086\scriptscriptstyle\pm\text{0.004}$
LSTM-NVP | $0.006\scriptscriptstyle\pm\text{0.003}$ | $0.331\scriptscriptstyle\pm\text{0.020}$ | $0.024\scriptscriptstyle\pm\text{0.001}$ | $0.078\scriptscriptstyle\pm\text{0.001}$ | $0.175\scriptscriptstyle\pm\text{0.001}$ | $0.078\scriptscriptstyle\pm\text{0.001}$
LSTM-MAF | 0.005$\scriptscriptstyle\pm\text{0.003}$ | $0.315\scriptscriptstyle\pm\text{0.023}$ | $0.021\scriptscriptstyle\pm\text{0.000}$ | $0.069\scriptscriptstyle\pm\text{0.002}$ | $0.161\scriptscriptstyle\pm\text{0.002}$ | $0.067\scriptscriptstyle\pm\text{0.001}$
TransformerMAF | 0.005$\scriptscriptstyle\pm\text{0.003}$ | $0.301\scriptscriptstyle\pm\text{0.014}$ | $0.021\scriptscriptstyle\pm\text{0.000}$ | $0.056\scriptscriptstyle\pm\text{0.001}$ | $0.179\scriptscriptstyle\pm\text{0.002}$ | $0.063\scriptscriptstyle\pm\text{0.003}$
TimeGrad | $0.006\scriptscriptstyle\pm\text{0.001}$ | $0.287\scriptscriptstyle\pm\text{0.020}$ | $0.021\scriptscriptstyle\pm\text{0.001}$ | $0.044\scriptscriptstyle\pm\text{0.006}$ | 0.114$\scriptscriptstyle\pm\text{0.020}$ | 0.049$\scriptscriptstyle\pm\text{0.002}$
StatiConF | 0.006$\scriptscriptstyle\pm\text{0.001}$ | 0.255$\scriptscriptstyle\pm\text{0.045}$ | 0.020$\scriptscriptstyle\pm\text{0.004}$ | 0.033$\scriptscriptstyle\pm\text{0.002}$ | 0.149$\scriptscriptstyle\pm\text{0.013}$ | 0.069$\scriptscriptstyle\pm\text{0.009}$
DynaConF | 0.006$\scriptscriptstyle\pm\text{0.001}$ | 0.242$\scriptscriptstyle\pm\text{0.052}$ | 0.021$\scriptscriptstyle\pm\text{0.003}$ | 0.032$\scriptscriptstyle\pm\text{0.002}$ | 0.146$\scriptscriptstyle\pm\text{0.014}$ | 0.081$\scriptscriptstyle\pm\text{0.006}$
Table 7: Quantitative Evaluation (Real-World Data).. Full CRPS results with means and standard deviations on 6 publicly available datasets. Lower values are better. Method | Exchange | Solar | Electricity | Traffic | Taxi | Wikipedia
---|---|---|---|---|---|---
DeepVAR (I) | 0.013$\scriptscriptstyle\pm\text{0.000}$ | 0.434$\scriptscriptstyle\pm\text{0.012}$ | 1.059$\scriptscriptstyle\pm\text{0.001}$ | 0.168$\scriptscriptstyle\pm\text{0.037}$ | 0.586$\scriptscriptstyle\pm\text{0.004}$ | 0.379$\scriptscriptstyle\pm\text{0.004}$
DeepVAR (C) | 0.009$\scriptscriptstyle\pm\text{0.000}$ | 0.384$\scriptscriptstyle\pm\text{0.010}$ | 0.084$\scriptscriptstyle\pm\text{0.006}$ | 0.165$\scriptscriptstyle\pm\text{0.004}$ | 0.416$\scriptscriptstyle\pm\text{0.004}$ | 0.247$\scriptscriptstyle\pm\text{0.001}$
GP-Scaling | 0.017$\scriptscriptstyle\pm\text{0.000}$ | 0.415$\scriptscriptstyle\pm\text{0.009}$ | 0.053$\scriptscriptstyle\pm\text{0.000}$ | 0.140$\scriptscriptstyle\pm\text{0.002}$ | 0.346$\scriptscriptstyle\pm\text{0.348}$ | 1.549$\scriptscriptstyle\pm\text{1.017}$
GP-Copula | 0.008$\scriptscriptstyle\pm\text{0.000}$ | 0.371$\scriptscriptstyle\pm\text{0.022}$ | 0.056$\scriptscriptstyle\pm\text{0.002}$ | 0.133$\scriptscriptstyle\pm\text{0.001}$ | 0.360$\scriptscriptstyle\pm\text{0.201}$ | 0.236$\scriptscriptstyle\pm\text{0.000}$
LSTM-NVP | 0.010$\scriptscriptstyle\pm\text{0.001}$ | 0.365$\scriptscriptstyle\pm\text{0.020}$ | 0.059$\scriptscriptstyle\pm\text{0.001}$ | 0.172$\scriptscriptstyle\pm\text{0.001}$ | 0.327$\scriptscriptstyle\pm\text{0.001}$ | 0.333$\scriptscriptstyle\pm\text{0.001}$
LSTM-MAF | 0.012$\scriptscriptstyle\pm\text{0.003}$ | 0.378$\scriptscriptstyle\pm\text{0.032}$ | 0.051$\scriptscriptstyle\pm\text{0.000}$ | 0.124$\scriptscriptstyle\pm\text{0.002}$ | 0.314$\scriptscriptstyle\pm\text{0.003}$ | 0.282$\scriptscriptstyle\pm\text{0.002}$
TransformerMAF | 0.012$\scriptscriptstyle\pm\text{0.003}$ | 0.368$\scriptscriptstyle\pm\text{0.001}$ | 0.052$\scriptscriptstyle\pm\text{0.000}$ | 0.134$\scriptscriptstyle\pm\text{0.001}$ | 0.377$\scriptscriptstyle\pm\text{0.002}$ | 0.274$\scriptscriptstyle\pm\text{0.007}$
StatiConF | 0.011$\scriptscriptstyle\pm\text{0.001}$ | 0.343$\scriptscriptstyle\pm\text{0.031}$ | 0.059$\scriptscriptstyle\pm\text{0.003}$ | 0.143$\scriptscriptstyle\pm\text{0.001}$ | 0.311$\scriptscriptstyle\pm\text{0.006}$ | 0.427$\scriptscriptstyle\pm\text{0.001}$
DynaConF | 0.010$\scriptscriptstyle\pm\text{0.000}$ | 0.338$\scriptscriptstyle\pm\text{0.035}$ | 0.058$\scriptscriptstyle\pm\text{0.004}$ | 0.140$\scriptscriptstyle\pm\text{0.002}$ | 0.308$\scriptscriptstyle\pm\text{0.005}$ | 0.296$\scriptscriptstyle\pm\text{0.009}$
Table 8: Quantitative Evaluation (Real-World Data).. Full MSE results with means and standard deviations on 6 publicly available datasets. Lower values are better. Method | Exchange | Solar | Electricity | Traffic | Taxi | Wikipedia
---|---|---|---|---|---|---
[e-4] | [e+2] | [e+5] | [e-4] | [e+1] | [e+7]
DeepVAR (I) | 1.6 | 9.3 | 2.1 | 6.3 | 7.3 | 7.2
DeepVAR (C) | 1.9 | 29 | 55 | 15 | 5.1 | 3.8
GP-Scaling | 2.9 | 11 | 1.8 | 5.2 | 2.7 | 5.5
GP-Copula | 1.7 | 9.8 | 2.4 | 6.9 | 3.1 | 4.0
LSTM-NVP | 2.4 | 9.1 | 2.5 | 6.9 | 2.6 | 4.7
LSTM-MAF | 3.8 | 9.8 | 1.8 | 4.9 | 2.4 | 3.8
TransformerMAF | 3.4 | 9.3 | 2.0 | 5.0 | 4.5 | 3.1
StatiConF | 4.4$\scriptscriptstyle\pm\text{2.0e-4}$ | 6.6$\scriptscriptstyle\pm\text{1.2e2}$ | 2.0$\scriptscriptstyle\pm\text{2.8e4}$ | 4.1$\scriptscriptstyle\pm\text{3.1e-6}$ | 2.2$\scriptscriptstyle\pm\text{7.1e-1}$ | 4.1$\scriptscriptstyle\pm\text{4.6e5}$
DynaConF | 2.6$\scriptscriptstyle\pm\text{1.6e-5}$ | 6.4$\scriptscriptstyle\pm\text{1.3e2}$ | 2.1$\scriptscriptstyle\pm\text{5.1e4}$ | 3.9$\scriptscriptstyle\pm\text{4.0e-6}$ | 2.2$\scriptscriptstyle\pm\text{6.9e-1}$ | 3.8$\scriptscriptstyle\pm\text{1.6e5}$
|
# Composition of rough singular integral operators on rearrangement invariant
Banach type spaces
Jiawei Tan Jiawei Tan: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>and Qingying Xue∗ Qingying
Xue: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.
Let $\Omega$ be a homogeneous function of degree zero and enjoy the vanishing
condition on the unit sphere $\mathbb{S}^{n-1}(n\geq 2)$. Let $T_{\Omega}$ be
the convolution singular integral operator with kernel
${\Omega(x)}{|x|^{-n}}$. In this paper, when $\Omega\in
L^{\infty}(\mathbb{S}^{n-1})$, we consider the quantitative weighted bounds of
the composite operators of $T_{\Omega}$ on rearrangement invariant Banach
function spaces. These spaces contain the classical Lorentz spaces and Orlicz
spaces as special examples. Weighted boundedness of the composite operators on
rearrangement invariant quasi-Banach spaces were also given.
###### Key words and phrases:
rough singular integral operator, composite operator, rearrangement invariant
Banach function spaces , bilinear sparse operators.
2010 Mathematics Subject Classification. Primary 42B20, Secondary 42B35.
The authors were partly supported by the National Key R&D Program of China
(No. 2020YFA0712900) and NNSF of China (No. 12271041).
∗ Corresponding author, e-mail address<EMAIL_ADDRESS>
## 1\. Introduction and main results
This paper aims to establish the quantitative weighted boundedness for the
composition of rough singular integral operators in rearrangement invariant
Banach spaces and quasi-Banach spaces. It is worthy to pointing out that the
classical Lorentz spaces, Orlicz spaces and the Marcinkiewicz spaces are
special examples of these spaces. The study of these rearrangement invariant
function spaces has a long history. Indeed, in 1955, Lorentz [40] first showed
that the Hardy-Littlewood maximal operator $M$ is bounded on rearrangement
invariant Banach function space $\mathbb{X}$ if and only if
$p_{\mathbb{X}}>1$. Subsequently, Boyd [5] proved that the Hilbert transform
$H$ is also bounded on $\mathbb{X}$ if and only if $1<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<\infty.$ Here $p_{\mathbb{X}}$ and $q_{\mathbb{X}}$ denote the
Boyd indices of $\mathbb{X}$ (see Section 2.1 below). These results were
originally proved for Banach spaces, but they were later generalized to the
quasi-Banach case with the same restrictions on Boyd’s index in [42]. Since
then many other contributions came to enrich the literature on this subject,
we refer the readers to [4, 21, 18] and the references therein. In particular,
by using sparse domination, Anderson and Hu [2] obtained the quantitative
weighted bounds for the maximal truncated singular integral operator on
rearrangement invariant Banach spaces.
We now give a brief review on the study of rough singular integrals. Let
$\Omega$ be a homogeneous function of degree zero, $\Omega\in
L^{1}(\mathbb{S}^{n-1})$ and satisfy the vanishing condition on the unit
sphere $\mathbb{S}^{n-1}(n\geq 2)$ as follows
(1.1) $\int_{\mathbb{S}^{n-1}}\Omega(y)d\sigma(y)=0,$
where $d\sigma(y)$ denotes the Lebesgue measure with restrictions on
$\mathbb{S}^{n-1}$. In 1956, Calderón and Zygmund [8] introduced the following
rough homogeneous singular integral operator
(1.2)
$T_{\Omega}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega\left(y/|y|\right)}{|y|^{n}}f(x-y)dy.$
Using the method of rotation, Calderón and Zygmund [8] demonstrated the
$L^{p}$ $(1<p<\infty)$ boundedness of $T_{\Omega}$ whenever $\Omega$ is odd
and $\Omega\in L^{1}(\mathbb{S}^{n-1})$, or $\Omega$ is even, $\Omega\in L\log
L(\mathbb{S}^{n-1})$ and satisfies (1.1). In 1979, Connett [16], Ricci and
Weiss [47] independently showed that $\Omega\in H^{1}({\mathbb{S}^{n-1}})$ is
sufficient to warrant the $L^{p}$ boundedness of $T_{\Omega}$. Here
$H^{1}({\mathbb{S}^{n-1}})$ denotes the Hardy space on ${\mathbb{S}^{n-1}}$
which contains $L\log L({\mathbb{S}^{n-1}})$ as a proper subspace.
Conside the weak endpoint case and the weighted case. This area has flourished
and has been enriched by several important works. Among them are the
celebrated works of Christ, Christ and Rubio de Francia, Hofmann, Seeger, and
Tao for weak type $(1,1)$ bounds of $T_{\Omega}$. In particular, in 1996,
Seeger [48] proved that $T_{\Omega}$ is bounded from $L^{1}(\mathbb{R}^{n})$
to $L^{1,\infty}(\mathbb{R}^{n})$ with a sufficient condition $\Omega\in L\log
L\left({\mathbb{S}^{n-1}}\right).$ For the weighted cases, it was
Duoandikoetxea and Rubio de Francia [19] who obtained the
$L^{p}(\mathbb{R},w(x)dx)$-boundedness of $T_{\Omega}$ for $1<p<\infty$
provided that $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ and $w$ is a
Muckenhoupt $A_{p}$ weight. This result was later improved in [20] and [51].
It is worth mentioning that, in 2017, Hytönen et al. [33] obtained the
quantitative weighted boundedness of $T_{\Omega}.$ For other works related to
$T_{\Omega}$, we refer the readers to see [12, 47, 39, 25] and the references
therein.
In general, there are two distinct approaches in the study of singular
integral operators. Consider it as a principal value operator of convolution
type or as an operator of Fourier multipliers defined by
$\widehat{T_{m}f}(\xi)=m(\xi)\hat{f}(\xi),$
where $m\in L^{\infty}\left(\mathbb{R}^{n}\right)$ and $\hat{f}$ denotes the
Fourier transform of $f$. Let $\Omega\in L\log L(\mathbb{S}^{n-1})$ be
homogeneous of degree zero and satisfy the vanishing condition (1.1). Then the
following identity transformation relationship holds between $m$ and $\Omega$
$m(\xi)=\int_{\mathbb{S}^{n-1}}\Omega\left(y^{\prime}\right)\left(\log\frac{1}{\left|\xi\cdot
y^{\prime}\right|}-\frac{i\pi}{2}\operatorname{sgn}\left(\xi\cdot
y^{\prime}\right)\right)d\sigma(y).$
However, unfortunately, this identity does not provide an exact correspondence
between various auxiliary conditions assumed on $m$ and $\Omega$.
One of the fundamental questions in operator theory is what the composition of
two operators is. This question is of great importance and attracts lots of
attention. Indeed, it was known that the composition of singular integral
operators arises typically in the study of algebra of singular integral (see
[6, 9]) and the non-coercive boundary-value problems for elliptic equations
(see [43, 45]). The answer to this question for $T_{\Omega}$ is easily
obtained by using the Fourier multiplier representation, which states that the
composition of two singular integral operators is an operator of the same form
and the multiplier of the composition is the product of the two multipliers
(see [49]). This answer is so elegant and useful that it actually forms the
basis for the calculus of pseudo-differential operators. What if it is in the
form of a principal value integral? Coifman and Meyer [14] considered the
composition of classical Calderón-Zygmund operators and pointed out that if
$T_{1},T_{2}$ are two Calderón-Zygmund operators, $T_{2}^{*}$ be the adjoint
operator of $T_{2}$ and $T_{1}(1)=T_{2}^{*}(1)=0$, then the composite operator
$T_{1}T_{2}$ is also a Calderón-Zygmund operator. It then follows that the
composition of Calderón-Zygmund operators is still strong $(p,p)$ type and
weak $(1,1)$ type. In 2018, Benea and Bernicot [3] used the sparse domination
method to reduce the above conditions to $T_{1}(1)=0$ or $T^{*}_{2}(1)=0,$ and
the weighted boundedness of the composite operator for the Calderón-Zygmund
operators can also be obtained. In addition, previous result for Hardy-
Littlewood maximal operators $M$ were obtained by Carozza and Passarelli di
Napoli [10]. They showed that the following weak type endpoint estimates hold
for the composition of $M$,
$\left|\left\\{x\in\mathbb{R}^{n}:M^{k}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\Psi_{k-1}\left(\frac{|f(x)|}{\lambda}\right)dx,\quad
0\leq\beta<\infty,$
where $M^{k}$ is the $k$-th iterations of $M$ and
$\Psi_{\beta}(t)=t\log^{\beta}(\mathrm{e}+t).$
By sparse domination, the results in [3] imply the conclusion that if
$T_{1},T_{2}$ be two Calderón-Zygmund operators with $T_{1}(1)=0$, then for
any $1<p<\infty,1<q<p$, and $w\in A_{p/q}\left(\mathbb{R}^{n}\right)$,
$\left\|T_{1}T_{2}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim[w]_{A_{p/q}}^{\max\left\\{\frac{1}{p-q},1\right\\}}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)},$
where the precise definitions of $A_{p}\left(\mathbb{R}^{n}\right)$ weight and
$A_{p}$ constants are listed in Section 2. It was Hu [27] who proved the
weighted bound for the composite operator $T_{1}T_{2}$ without the assumption
$T_{1}(1)=0.$ Recently, Hu [26] established the weighted weak type endpoint
estimate for $T_{1}T_{2}.$ Still more recently, using the method of sparse
domination, more accurate weighted estimate for the composition of rough
homogeneous singular integral operators were obtained by Hu, Lai and Xue [29].
###### Theorem A ([29]).
Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Then for $p\in(1,\infty)$ and $w\in A_{p}\left(\mathbb{R}^{n}\right)$,
$\displaystyle\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim$
$\displaystyle{[w]_{A_{p}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{\frac{1}{p^{\prime}}}+[\sigma]_{A_{\infty}}^{\frac{1}{p}}\right)\left([\sigma]_{A_{\infty}}+[w]_{A_{\infty}}\right)}$
$\displaystyle\times\min\left\\{[\sigma]_{A_{\infty}},[w]_{A_{\infty}}\right\\}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)},$
where $p^{\prime}=p/(p-1),\sigma=w^{-1/(p-1)}.$
As we mentioned in the beginning of this paper, the main purpose of this paper
is to obtain the boundedness of the composition for rough singular integral
operators $T_{\Omega_{1}}T_{\Omega_{2}}$ on rearrangement invariant Banach
spaces (RIBFS in the sequel) and quasi-Banach spaces (RIQBFS in the sequel)
whenever both $\Omega_{1}$ and $\Omega_{2}$ belong to
$L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
We summary our first main result as follows.
###### Theorem 1.1.
Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have the vanishing
moment (1.1) and $\Omega_{1},\Omega_{2}\in
L^{\infty}\left(\mathbb{S}^{n-1}\right)$. Let $1<r<\infty$ and $\mathbb{X}$ be
a RIBFS with $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q,q_{0}>1$ such that
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}},w\in A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}<q_{0},w\in A_{p_{\mathbb{X}}}.\end{array}\right.$
###### Remark 1.2.
If we set $\mathbb{X}=L^{p}$ with $1<p<\infty$, then
$p_{\mathbb{X}}=q_{\mathbb{X}}=p$ and the result in Theorem 1.1 covers the
conclusion in Theorem A as a special case. Furthermore, to the best knowledge
of the author, even the quantitative weighted estimates for a single rough
singular integral operator $T_{\Omega}$ on $\mathbb{X}$ is new.
As a consequence of Theorem 1.1, it follows that:
###### Corollary 1.3.
Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $1<r<\infty$ and $\mathbb{X}$ be a RIQBFS, which is $p$-convex for some
$p>0$. If $p<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q,q_{0}>1$ such that
$\left\||T_{\Omega_{1}}T_{\Omega_{2}}f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{\frac{2}{p}}[w]_{A_{\frac{p_{\mathbb{X}}}{pr}}}^{\frac{2}{prq}}\left\||f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)},&pr<p_{\mathbb{X}},w\in
A_{\frac{p_{\mathbb{X}}}{pr}};\\\
{[w]_{A_{\infty}}^{\frac{2}{p}}}[w]_{A_{\frac{p_{\mathbb{X}}}{p}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]^{\frac{1}{p}}_{A_{\infty}}+[w]_{A_{\frac{p_{\mathbb{X}}}{p}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\||f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)},&p_{\mathbb{X}}<pq_{0},w\in
A_{\frac{p_{\mathbb{X}}}{p}}.\end{array}\right.$
It is well known that some important facts may be significantly different
between Banach spaces and quasi-Banach spaces. For example, the direction of
the Hölder’s inequality in $L^{p}(0<p<1)$ and $L^{q}(q\geq 1)$ is opposite.
When $\mathbb{X}$ is a space of rearrangement invariant quasi-Banach type, we
obtain the following quantitative weighted bounds for the composition of rough
singular integral operators.
###### Theorem 1.4.
Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $\mathbb{X}$ be a RIQBFS, which is p-convex with $0<p\leq 1$. If
$1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<2p-\frac{1}{1+p_{\mathbb{X}}}p$, then for
every $w\in A_{{p_{\mathbb{X}}}}$,
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left([w]_{A_{\infty}}^{1+\frac{1}{p}}+[w]_{A_{\infty}}^{2+\frac{1}{p}}\right)\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$
This paper is organized as follows. In Section 2, we present some lemmas and
related definitions, such as RIBFS and RIQBFS, dyadic cubes, some maximal
operators and bi-sublinear sparse operators. The proofs of Theorem 1.1 and
Corollary 1.3 will be given in Section 3. In Section 4, we demonstrate Theorem
1.4. An application of Theorem 1.1 will be given in Section 5.
In what follows, $C$ always denotes a positive constant that is independent of
the main parameters involved but whose value may differ from line to line. For
any $a,b\in\mathbb{R},a\lesssim b$ $(a\gtrsim b,$ respectively) denotes that
there exists a constant $C>0$ such that $a\leq Cb;$ and $a\simeq b$ denotes
$a\lesssim b$ and $b\lesssim a.$ $p^{\prime}$ will always denote the conjugate
of $p,$ namely, $1/p+1/p^{\prime}=1$.
## 2\. Preliminary
First, we recall some basic properties for RIBFS, RIQBFS, sparse family and
Orlicz maximal operators.
### 2.1. RIBFS and RIQBFS
Let’s start with some simple definitions.
$\circ$ Basic definitions of RIBFS. Let $\mathcal{M}$ be the set of measurable
functions on $\left(\mathbb{R}^{n},dx\right)$ and $\mathcal{M}^{+}$ be the
nonnegative ones. A rearrangement invariant Banach norm is a mapping
$\rho:\mathcal{M}^{+}\mapsto[0,\infty]$ such that the following properties
hold:
1. i).
$\rho(f)=0\Leftrightarrow f=0,$ a.e.;
$\rho(f+g)\leq\rho(f)+\rho(g);\rho(af)=a\rho(f)$ for $a\geq 0$;
2. ii).
If $0\leq f\leq g,$ a.e., then $\rho(f)\leq\rho(g)$;
3. iii).
If $f_{n}\uparrow f,$ a.e., then $\rho\left(f_{n}\right)\uparrow\rho(f)$;
4. iv).
If $E$ is a measurable set such that $|E|<\infty,$ then
$\rho\left(\chi_{E}\right)<\infty,$ and $\int_{E}fdx\leq$ $C_{E}\rho(f),$ for
some constant $0<C_{E}<\infty,$ depending on $E$ and $\rho,$ but independent
of $f$;
5. v).
$\rho(f)=\rho(g)$ if $f$ and $g$ are equimeasurable, that is,
$d_{f}(\lambda)=d_{g}(\lambda),\lambda\geq 0$ where $d_{f}\left(d_{g}\right.$
respectively) denotes the distribution function of $f$ ($g$ respectively).
By means of $\rho,$ a rearrangement invariant Banach function space (RIBFS)
$\mathbb{X}$ can be defined:
$\mathbb{X}=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}}:=\rho(|f|)<\infty\right\\}.$
Let $\mathbb{X}^{\prime}$ be the associated space of $\mathbb{X}$, which is
also a Banach function space given by
$\mathbb{X}^{\prime}=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}^{\prime}}:=\sup\left\\{\int_{\mathbb{R}^{n}}fgdx:g\in\mathcal{M}^{+},\rho(g)\leq
1\right\\}<\infty\right\\}.$
Note that in the present setting, $\mathbb{X}$ is a RIBFS if and only if
$\mathbb{X}^{\prime}$ is a $\mathrm{RIBFS}$ ([4, Chapter 2, Corollary 4.4]).
By definition, the following generalized Hölder’s inequality holds for any
$f\in\mathbb{X},g\in\mathbb{X}^{\prime}$ :
$\int_{\mathbb{R}^{n}}|fg|dx\leq\|f\|_{\mathbb{X}}\|g\|_{\mathbb{X}^{\prime}}.$
A key fact in a RIBFS $\mathbb{X}$ is that the Lorentz-Luxemburg theorem
holds:
$\|f\|_{\mathbb{X}}=\sup\left\\{\left|\int_{\mathbb{R}^{n}}fgdx\right|:g\in\mathbb{X}^{\prime},\|g\|_{\mathbb{X}^{\prime}}\leq
1\right\\}.$
Recall that the decreasing rearrangement function $f^{*}$ is defined by
$f^{*}(t)=\inf\left\\{\lambda\geq 0:d_{f}(\lambda)\leq t\right\\},t\geq 0.$
An important property of $f^{*}$ is that it is equimeasurable with $f$. This
allows one to obtain a representation of $\mathbb{X}$, i.e., Luxemburg’s
representation theorem ([4, Chapter 2, Theorem 4.10]), which asserts that
there exists a RIBFS $\overline{\mathbb{X}}$ over
$\left(\mathbb{R}^{+},dt\right)$ such that $f\in\mathbb{X}$ if and only if
$f^{*}\in\overline{\mathbb{X}}$, and in this case
$\|f\|_{\mathbb{X}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}}$. From this
it can be seen that the mapping $f\mapsto f^{*}$ is an isometry. In addition,
notice that $\overline{\mathbb{X}}^{\prime}=\overline{\mathbb{X}^{\prime}}$
and
$\|f\|_{\mathbb{X}^{\prime}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}^{\prime}}$
hold for the associated space.
$\circ$ Weighted versions of RIBFS $\mathbb{X}$. Before we consider weighted
versions of RIBFS $\mathbb{X}$, we need to recall the definitions of
Muckenhoupt weights. Let $w$ be a non-negative locally integrable function
defined on $\mathbb{R}^{n}.$ We say that a weight $w$ belongs to the
Muckenhoupt class $A_{p}$ with $1<p<\infty,$ if
$[w]_{A_{p}}:=\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}w(x)dx\right)\left(\frac{1}{|Q|}\int_{Q}w(x)^{1-p^{\prime}}dx\right)^{p-1}<\infty,$
and for $w\in A_{\infty},$
$[w]_{A_{\infty}}:=\sup_{Q}\frac{1}{w(Q)}\int_{Q}M\left(w\chi_{Q}\right)(x)dx,$
where the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}.$ The
$A_{1}$ constant is defined by
$[w]_{A_{1}}:=\sup_{x\in\mathbb{R}^{n}}\frac{Mw(x)}{w(x)},$
where $M$ is the Hardy-Littlewood maximal operator.
Some important properties of weights are listed in the following lemmas.
###### Lemma 2.1 ([44]).
Let $1<p<\infty$ and let $w\in A_{p}.$ Then $w\in A_{p-\varepsilon}$ with
$\varepsilon:=\varepsilon(p)=\frac{p-1}{1+2^{n+1}[\sigma]_{A_{\infty}}}$
where $\sigma=w^{1-p^{\prime}}$ is the dual weight. Furthermore
$[w]_{A_{p-\varepsilon}}\leq 2^{p-1}[w]_{A_{p}}.$
###### Lemma 2.2 ([32]).
Let $w\in A_{\infty}\left(\mathbb{R}^{n}\right)$. Then for any cube $Q$ and
$\delta\in\left(1,1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}\right]$,
$\left(\frac{1}{|Q|}\int_{Q}w^{\delta}(x)\mathrm{d}x\right)^{\frac{1}{\delta}}\leq\frac{2}{|Q|}\int_{Q}w(x)\mathrm{d}x.$
We now give the weighted version of the RIBFS $\mathbb{X}$. First, the
distribution function and the decreasing rearrangement with respect to $w$ are
defined by
$w_{f}(\lambda)=w\left(\\{x\in\mathbb{R}^{n}:|f(x)|>\lambda\\}\right);\quad
f_{w}^{*}(t)=\inf\left\\{\lambda\geq 0:w_{f}(\lambda)\leq t\right\\}.$
In this way, the weighted version of the space $\mathbb{X}$ is given by
$\mathbb{X}(w)=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}(w)}:=\left\|f_{w}^{*}\right\|_{\overline{\mathbb{X}}}<\infty\right\\}.$
Then, the same procedure applying on the associate spaces yields
$\mathbb{X}^{\prime}(w)=\mathbb{X}(w)^{\prime}$ (see [18, p. 168]).
$\circ$ Boyd indices and $r$ exponent. Next, we recall the Boyd indices of a
RIBFS, which are closely related to some interpolation properties, see [4,
Chapter 3] for more details. We start with the dilation operator $D_{t}$ of
$\mathbb{X}$,
$D_{t}f(s)=f\left(\frac{s}{t}\right),0<t<\infty,f\in\overline{\mathbb{X}},$
and its norm
$h_{\mathbb{X}}(t)=\left\|D_{t}\right\|_{\overline{\mathbb{X}}\mapsto\overline{\mathbb{X}}},0<t<\infty.$
The lower and upper Boyd indices are defined, respectively, by the following
form:
$p_{\mathbb{X}}=\lim_{t\rightarrow\infty}\frac{\log t}{\log
h_{\mathbb{X}}(t)}=\sup_{1<t<\infty}\frac{\log t}{\log
h_{\mathbb{X}}(t)},\quad q_{\mathbb{X}}=\lim_{t\rightarrow 0^{+}}\frac{\log
t}{\log h_{\mathbb{X}}(t)}=\inf_{0<t<1}\frac{\log t}{\log h_{\mathbb{X}}(t)}.$
A simple calculation shows that $1\leq p_{\mathbb{X}}\leq
q_{\mathbb{X}}\leq\infty,$ which follows from the fact that
$h_{\mathbb{X}}(t)$ is submultiplicative. In order to give the above
definition a general explanation, we consider a special case. If
$\mathbb{X}=L^{p}$ with $1<p<\infty$, then $h_{\mathbb{X}}(t)=t^{\frac{1}{p}}$
and thus $p_{\mathbb{X}}=q_{\mathbb{X}}=p.$
The relationship between the Boyd indices of $\mathbb{X}$ and
$\mathbb{X}^{\prime}$ are as follows :
$p_{\mathbb{X}^{\prime}}=\left(q_{\mathbb{X}}\right)^{\prime}$ and
$q_{\mathbb{X}^{\prime}}=\left(p_{\mathbb{X}}\right)^{\prime},$ where $p$ and
$p^{\prime}$ are conjugate exponents.(see [41, Chapter 11, Corollary 11.6]).
Now, we consider the following $r$ exponent of RIBFS $\mathbb{X}$ with
$0<r<\infty$:
$\mathbb{X}^{r}=\left\\{f\in\mathcal{M}:|f|^{r}\in\mathbb{X}\right\\},$
and the norm
$\|f\|_{\mathbb{X}^{r}}=\left\||f|^{r}\right\|_{\mathbb{X}}^{\frac{1}{r}}$. By
the definition of Boyd indices it is easy to verify that
$p_{\mathbb{X}^{r}}=p_{\mathbb{X}}\cdot r$ and
$q_{\mathbb{X}^{r}}=q_{\mathbb{X}}\cdot r$.
$\circ$ The case of RIQBFS. Analogy to $L^{p}$ space, for each $r\geq 1,$
$\mathbb{X}^{r}$ is still a RIBFS when $\mathbb{X}$ is a RIBFS. However, if
$0<r<1,$ the space $\mathbb{X}^{r}$ is not necessarily a Banach space (see
[18, p. 269]). Hence, it is natural to consider the quasi-Banach case.
To see this, we first give the definition of the quasi-Banach function norm.
We say a mapping $\rho^{\prime}:\mathcal{M}^{+}\mapsto[0,\infty)$ is a
rearrangement invariant quasi-Banach function norm if $\rho^{\prime}$
satisfies the basic condition i),ii),iii) and v) with the triangle inequality
replaced by the quasi-triangle inequality as follows:
$\rho^{\prime}(f+g)\leq C\left(\rho^{\prime}(f)+\rho^{\prime}(g)\right),$
where $C$ is an absolute positive constant. Similarly, a rearrangement
invariant quasi-Banach function space is a collection that consist of all
measurable functions which satisfies $\rho^{\prime}(|f|)<\infty$. In order to
get a better study in transformation between RIBFS and RIQBFS, we need to
consider the following $p$-convex property with $p>0$ on $\mathbb{X}$ which is
a RIQBFS (see [24, p. 3]):
$\left\|\left(\sum_{j=1}^{N}\left|f_{j}\right|^{p}\right)^{\frac{1}{p}}\right\|_{\mathbb{X}}\lesssim\left(\sum_{j=1}^{N}\left\|f_{j}\right\|_{\mathbb{X}}^{p}\right)^{\frac{1}{p}}.$
A very important conclusion states that the $p$-convex property is equivalent
to the fact that $\mathbb{X}^{\frac{1}{p}}$ is a RIBFS (see [18, p. 269]).
According to the above results and using Lorentz-Luxemburg’s theorem again, we
have
$\|f\|_{\mathbb{X}}\simeq\sup\left\\{\left(\int_{\mathbb{R}^{n}}|f(x)|^{p}g(x)dx\right)^{\frac{1}{p}}:g\in\mathcal{M}^{+},\|g\|_{\mathbb{Y}^{\prime}}\leq
1\right\\},$
where $\mathbb{Y}^{\prime}$ is the associated space of the RIBFS
$\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$. Simultaneously, for each $w\in
A_{\infty}$ and $0<r<\infty$, one may define $\mathbb{X}(w)$ for a RIQBFS
$\mathbb{X}$ and it enjoys that $\mathbb{X}(w)^{r}=\mathbb{X}^{r}(w)$(see [18,
p. 269]).
###### Remark 2.3.
It is necessary for us to make a remark about the newly added $p$-convex
property. As Grafakos and Kalton ([24]) said “all practical spaces are
$p$-convex for some $p>0$”. This is because there are only very few spaces
that do not satisfy the $p$-convex property for any $p>0$ (see [34]).
### 2.2. Young function and Orlicz maximal operators
In this subsection, we present some fundamental facts about Young functions
and Orlicz local averages which will play an important role in our analysis.
We refer the readers to [46] for more information.
Firstly, let $\Phi$ be the set of functions
$\phi:[0,\infty)\longrightarrow[0,\infty)$ which are non-negative, increasing
and such that $\lim_{t\rightarrow\infty}\phi(t)=\infty$ and
$\lim_{t\rightarrow 0}\phi(t)=0.$ If $\phi\in\Phi$ is convex we say that
$\phi$ is a Young function. Next, we can define the average of the Luxemburg
norm, namely $\phi$-norm, of $f$ over a cube $Q$ as
$\|f\|_{\phi(\mu),Q}:=\inf\left\\{\lambda>0:\frac{1}{\mu(Q)}\int_{Q}\phi\left(\frac{|f(x)|}{\lambda}\right)d\mu\leq
1\right\\}.$
For the sake of notation, if $\mu$ is the Lebesgue measure, we write
$\|f\|_{\phi,Q}$, and we denote $\|f\|_{\phi(w),Q},$ if $\mu=wdx$ is an
absolutely continuous measure with respect to the Lebesgue measure.
Each Young function $\phi$ enjoys the following generalized Hölder’s
inequality:
$\frac{1}{\mu(Q)}\int_{Q}|fg|d\mu\leq
2\|f\|_{\phi(\mu),Q}\|g\|_{\bar{\phi}(\mu),Q},$
where $\bar{\phi}(t)=\sup_{s>0}\\{st-\phi(s)\\}$ is the complementary function
of $\phi$.
Then we can naturally represent the Orlicz maximal operator $M_{\phi}f$
associated to the Young function $\phi$ with the following form:
$M_{\phi}f(x):=\sup_{x\in Q}\|f\|_{\phi,Q}.$
Finally, we present some particular examples of maximal operators related to
certain Young functions.
* •
If $\phi(t)=t^{r}$ with $r>1,$ then $M_{\phi}=M_{r}$.
* •
If we consider $\phi(t)=t\log^{\alpha}(e+t)$ with $\alpha>0,$ then
$\bar{\phi}(t)\simeq e^{t^{1/\alpha}}-1$ and we denote $M_{\phi}=M_{L(\log
L)^{\alpha}}$. We have that $M\leq M_{\phi}\lesssim M_{r}$ for all
$1<r<\infty,$ moreover, it can be proved that $M_{\phi}\simeq M^{l+1},$ where
$\alpha=l\in\mathbb{N}$ and $M^{l+1}$ is $M$ iterated $l+1$ times.
* •
$M_{\phi}=M_{L(\log L)^{\alpha}(\log\log L)^{\beta}}$ given by the function
$\phi(t)=t\log^{\alpha}(e+t)\log^{\beta}(e+\log(e+t))$ with $\alpha,\beta>0.$
### 2.3. Sparse family
We also need a system of dyadic calculus from [37, 38], so in this subsection,
we introduce a part of it.
###### Definition 2.4.
By a dyadic lattice $\mathcal{D},$ we mean a collection of cubes which
satisfies the following properties:
1. (i).
For any $Q\in\mathcal{D}$ its sidelength $\ell_{Q}$ is of the form
$2^{k},k\in\mathbb{Z}$;
2. (ii).
$Q\cap R\in\\{Q,R,\emptyset\\}$ for any $Q,R\in\mathcal{D}$;
3. (iii).
The cubes of a fixed sidelength $2^{k}$ form a partition of $\mathbb{R}^{n}$.
An interesting and crucial theorem in dyadic calculus is Three Lattice
Theorem, which asserts that given a dyadic lattice $\mathcal{D},$ there exist
$3^{n}$ dyadic lattices $\mathcal{D}_{1},\ldots,\mathcal{D}_{3^{n}}$ such that
for each cube $Q\in\mathcal{D}$, we can find a cube $R_{Q}$ in some
$\mathcal{D}_{j}$ such that $Q\subseteq R_{Q}$ and $3l_{Q}=l_{R_{Q}}$.
According to the method of taking dyadic lattice $\mathcal{D}$ in Definition
2.4, we can give the definition of sparse family $\mathcal{S}$ as follows.
###### Definition 2.5.
Let $\mathcal{D}$ be a dyadic lattice. $\mathcal{S}\subset\mathcal{D}$ is
called a $\eta$-sparse family with $\eta\in(0,1)$ if for every cube
$Q\in\mathcal{S},$
$\left|\bigcup_{P\in\mathcal{S},P\subsetneq
Q}P\right|\leq(1-\eta)\left|Q\right|.$
There is another equivalent definitions of sparsity for a collection of sets.
If we define
$E(Q)=Q\backslash\bigcup_{P\in\mathcal{S},P\subsetneq Q}P,$
then it is easy to deduce that the sets $E(Q)$ are pairwise disjoint and
$|E(Q)|\geq\eta|Q|$.
Let $\mathcal{D}$ be a dyadic lattice and $\mathcal{S}\subseteq\mathcal{D}$ be
a $\eta$-sparse family, the sparse operator is defined by
$\mathcal{A}_{r,\mathcal{S}}f(x)=\sum_{Q\in\mathcal{S}}\langle|f|\rangle_{r,Q}\chi_{Q}(x)=\sum_{Q\in\mathcal{S}}\left(\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy\right)^{\frac{1}{r}}\chi_{Q}(x),$
where $r>0$ and
$\langle|f|\rangle_{r,Q}^{r}=\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy$. Furthermore,
associated with the constants $\beta\in[0,\infty)$ and
$r_{1},r_{2},r\in[1,\infty)$, we can define the bilinear sparse operators
$\mathcal{A}_{\mathcal{S};L(\log L)^{\beta},L^{r}}$ and
$\mathcal{A}_{\mathcal{S};L^{r_{1}},L^{r_{2}}}$ by
$\mathcal{A}_{\mathcal{S};L(\log
L)^{\beta},L^{r}}(f,g)=\sum_{Q\in\mathcal{S}}|Q|\|f\|_{L(\log
L)^{\beta},Q}\langle|g|\rangle_{r,Q},$
$\mathcal{A}_{\mathcal{S};L^{r_{1}},L^{r_{2}}}(f,g)=\sum_{Q\in\mathcal{S}}|Q|\langle|f|\rangle_{r_{1},Q}\langle|g|\rangle_{r_{2},Q}.$
According to the above definitions, we can give the condition that the
operator $T$ satisfies the bilinear sparse domination. More specifically, for
$\beta,q\in(0,\infty)$, we say that a sublinear operator $T$ acting on
$\cup_{p\geq 1}L^{p}\left(\mathbb{R}^{n}\right)$ enjoys a $\left(L(\log
L)^{\beta},L^{q}\right)$-bilinear sparse domination with bound $A$, if for
each bounded function $f$ with compact support, there exists a sparse family
$\mathcal{S}$ of cubes, such that
$\left|\int_{\mathbb{R}^{n}}g(x)Tf(x)dx\right|\leq
A\mathcal{A}_{\mathcal{S},L(\log L)^{\beta},L^{q}}(f,g),$
holds for all bounded function $g$.
The following results of bilinear sparse domination are crucial in our
analysis ([29, Corollary 5.1]).
###### Lemma 2.6 ([29]).
Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $r\in(1,3/2]$. Then for each bounded function $f$ with compact support,
there exists a $\frac{1}{2}\frac{1}{9^{n}}$-sparse family of cubes
$\mathcal{S}=\\{Q\\}$, and functions $J_{1}$ and $J_{2}$, such that for each
function $g$,
(2.1)
$\displaystyle\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)\mathrm{d}x\right|\lesssim
r^{\prime}\mathcal{A}_{\mathcal{S};L(\log L),L^{r}}(f,g),$
$\displaystyle\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)\mathrm{d}x\right|\lesssim
r^{\prime 2}\mathcal{A}_{\mathcal{S};L^{1},L^{r}}(f,g),$
and for a. e. $x\in\mathbb{R}^{n}$,
(2.2) $T_{\Omega_{1}}T_{\Omega_{2}}f(x)=J_{1}(x)+J_{2}(x).$
Here, we give a brief introduction to sparse domination. The study of sparse
domination is a very active research area in Harmonic analysis in recent
years. It helps to simplify the proof of some well known results and even can
be used to obtain the dependence of the norm constant on the weight functions.
For example, Lerner [36] introduced a class of sparse operators and gave an
alternative and simple proof of the $A_{2}$ conjecture. Later on, Lerner [37]
obtained quantitative weighted bounds for Calderón-Zygmund operator $T$ which
satisfies a Hölder-Lipschitz condition. Since then, great attentions have been
paid to the study of the sparse bounds. We refer the readers to [11, 15, 17]
and the references therein for more informations.
## 3\. Proofs of Theorems 1.1
This section is devoted to prove Theorem 1.1 and Corollary 1.3. To demonstrate
Theorem 1.1, we need the following lemma in [2, Lemma 3.3].
###### Lemma 3.1 ([2]).
Let $\mathbb{X}$ be an RIQBFS which is $p$-convex for some $0<p\leq 1$. If
$1<p_{\mathbb{X}}<\infty$, then for all $w\in A_{p_{\mathbb{X}}},$ we have
$\|M\|_{\mathbb{X}(w)\mapsto\mathbb{X}(w)}\leq
C[w]_{A_{p_{\mathbb{X}}}}^{1/p_{\mathbb{X}}},$
where $C$ is an absolute constant only depending on $p_{\mathbb{X}}$ and $n$.
###### Proof of Theorem 1.1.
We define the weighted dyadic Hardy-Littlewood maximal operators
$M_{w}^{\mathcal{D}}$ and $M_{w,r}^{\mathcal{D}}$ by
$M_{w}^{\mathcal{D}}f(x):=\sup_{x\in
R,R\in\mathcal{D}}\frac{1}{w(R)}\int_{R}|f(y)|w(y)dy,\quad f\in
L_{\operatorname{loc}}^{1}\left(\mathbb{R}^{n}\right),$
$M_{w,r}^{\mathcal{D}}f(x):=\sup_{x\in
R,R\in\mathcal{D}}\left(\frac{1}{w(R)}\int_{R}|f(y)|^{r}w(y)dy\right)^{\frac{1}{r}},\quad
f\in L_{\operatorname{loc}}^{r}\left(\mathbb{R}^{n}\right),$
where $w\in A_{\infty}$ and $\mathcal{D}$ is the given dyadic grid.
First, we consider the case of $1<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<2-\frac{1}{1+p_{\mathbb{X}}}=:q_{0}.$ For a fixed $w\in
A_{p_{\mathbb{X}}}$, by (2.2), we obtain
$\displaystyle\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\Omega_{1}}T_{\Omega_{2}}f(x)g(x)w(x)dx\right|$
$\displaystyle\leq\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)w(x)dx\right|+\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)w(x)dx\right|$
$\displaystyle=:I_{1}+I_{2},$
where $I_{i}=\sup\limits_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{i}(x)g(x)w(x)dx\right|,i=1,2.$
Consider the contribuion of $I_{1}$. Take and fix any
$g\in\mathbb{X}^{\prime}(w)$ with $\|g\|_{\mathbb{X}^{\prime}(w)}\leq 1$, by
(2.1),
$\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)w(x)dx\right|\lesssim
r^{\prime}\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log{L}),Q}{\langle|gw|\rangle}_{r,Q}|Q|.$
A direct computation gives that
$\displaystyle{\langle|gw|\rangle}_{r,Q}$
$\displaystyle=\left(\frac{1}{|Q|}\int_{Q}|g(x)w(x)|^{r}dx\right)^{\frac{1}{r}}=\left(\frac{1}{|Q|}\int_{Q}|g(x)|^{r}|w(x)|^{\frac{1}{s}}|w(x)|^{r-\frac{1}{s}}dx\right)^{\frac{1}{r}}$
$\displaystyle\leq\left(\frac{1}{|Q|}\int_{Q}|g(x)|^{rs}w(x)dx\right)^{\frac{1}{rs}}\left(\frac{1}{|Q|}\int_{Q}|w(x)|^{(r-\frac{1}{s})s^{\prime}}dx\right)^{\frac{1}{rs^{\prime}}}.$
Take appropriate $r$ and $s$ such that
$(r-\frac{1}{s})s^{\prime}<1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}.$ For example,
let us choose
$r=1+\frac{1}{2^{13+n}p_{\mathbb{X}}[w]_{A_{\infty}}},s=1+\frac{1}{2p_{\mathbb{X}}},$
then
$(r-\frac{1}{s})s^{\prime}=1+\frac{1}{2^{13+n}p_{\mathbb{X}}[w]_{A_{\infty}}}+\frac{1}{2^{12+n}[w]_{A_{\infty}}}<1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}.$
Thus, combining Lemma 2.2, we deduce that
$\displaystyle\frac{1}{|Q|}\int_{Q}(w(x))^{(r-\frac{1}{s})s^{\prime}}dx$
$\displaystyle\leq\left(\frac{2}{|Q|}\int_{Q}w(x)dx\right)^{1-\frac{1}{rs}}.$
Now one can get
$\displaystyle{\langle|gw|\rangle}_{r,Q}\cdot|Q|$
$\displaystyle\leq\left(\frac{1}{|w(Q)|}\int_{Q}|g(x)|^{rs}w(x)dx\right)^{\frac{1}{rs}}w(Q)=:g_{Q,w}^{rs}\cdot
w(Q).$
Taking into account the generalized Hölder’s inequality and recalling that
$M^{2}$ is $M$ iterated 2 times, we have
$\displaystyle\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L),Q}{\langle|gw|\rangle}_{r,Q}|Q|$
$\displaystyle\leq\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L),Q}w(Q)g_{Q,w}^{rs}$
$\displaystyle\leq\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}g_{B,w}^{2s}\sum_{\begin{subarray}{c}R\in\mathcal{S}\\\
\pi(R)=B\end{subarray}}w(R)$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}w(B)g_{B,w}^{2s}$
$\displaystyle=C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}g_{B,w}^{2s}\chi_{B}(x)w(x)dx$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M_{L(\log
L)}f(x)M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M^{2}f(x)M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\leq
C(n)[w]_{A_{\infty}}\left\|M^{2}f\right\|_{\mathbb{X}(w)}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)},$
where $\mathcal{B}$ is the family of the principal cubes in the usual sense
and $\pi(R)$ is the minimal principal cube which contains $R$. That is
$\mathcal{B}=\cup_{k=0}^{\infty}\mathcal{B}_{k}$
with $\mathcal{B}_{0}:=\\{$ maximal cubes in $\mathcal{S}\\}$ and
$\mathcal{B}_{k+1}:=\underset{B\in\mathcal{B}_{k}}{\cup}\operatorname{ch}_{\mathcal{B}}(B),\quad\operatorname{ch}_{\mathcal{B}}(B)=\\{R\subsetneq
B\text{ maximal s.t. }\tau(R)>2\tau(B)\\},$
where $\tau(R)=\|f\|_{{L(\log L)},R}g_{R,w}^{2s}$.
Now we observe that, using Lemma 3.1,
$\displaystyle I$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\|\left(M_{w}^{\mathcal{D}}(|g|^{2s})\right)^{\frac{1}{2s}}\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle=C(n)[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\|\left(M_{w}^{\mathcal{D}}(|g|^{2s})\right)\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\||g|^{2s}\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\leq[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)},$
where in the second inequality of the countdown, we have used the fact that
the boundedness of $M_{w}^{\mathcal{D}}$( see [18, Theorem 3.2]). Indeed,
using the condition $q_{\mathbb{X}}<\frac{2s}{2s-1}$, it’s easy to deduce that
$p_{\mathbb{X}^{\prime\frac{1}{2s}}}=\frac{p_{{\mathbb{X}}^{\prime}}}{2s}=\frac{{(q_{\mathbb{X}})}^{{}^{\prime}}}{2s}>1.$
Now we turn our attention to the estimate of $I_{2}.$ For this case, arguing
as in the first case, we obtain
$\displaystyle\sum_{Q\in\mathcal{S}}\langle|f|\rangle_{Q}\langle|gw|\rangle_{r,Q}|Q|$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}}\langle|f|\rangle_{Q}g_{Q,w}^{rs}w(Q)$
$\displaystyle\leq\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}.$
That fact together with Lemma 2.6 yields
(3.1) $\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)w(x)dx\right|\lesssim r^{\prime
2}\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}.$
Now we note that $\mathcal{S}$ is a $\frac{1}{2\cdot 9^{n}}$ -sparse family
(i.e., for any $Q\in\mathcal{S},$ there exists $E(Q)$ such that
$|E(Q)|\geq\frac{1}{2\cdot 9^{n}}|Q|$ ). Hence, for each dyadic cube
$Q\in\mathcal{S}$, we have
(3.2) $\displaystyle\sum_{R\subseteq Q}w(R)=\sum_{R\subseteq
Q}\frac{w(R)}{|R|}|R|$ $\displaystyle\leq 2\cdot 9^{n}\sum_{R\subseteq
Q}\frac{w(R)}{|R|}\cdot|E(R)|$ $\displaystyle\leq 2\cdot 9^{n}\sum_{R\subseteq
Q}\int_{E(R)}M(w\chi_{R})(x)dx$ $\displaystyle\leq 2\cdot
9^{n}\int_{Q}M(w\chi_{Q})(x)dx$ $\displaystyle\leq 2\cdot
9^{n}[w]_{A_{\infty}}w(Q).$
Combining (3.2) with (3.1) and the Carleson embedding theorem, together with
the generalized Hölder’s inequality, one may obtain
$\displaystyle I_{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{2}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{3}\left\|Mf\right\|_{\mathbb{X}(w)}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\leq[w]_{A_{\infty}}^{3}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left\||g|^{2s}\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\leq[w]_{A_{\infty}}^{3}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$
This inequality, combined with the estimate of $I_{1}$ yields
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$
To end the proof we consider the case of $r<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<\infty,$ where $1<r<\infty$ is any fixed constant.
If $w\in A_{p_{\mathbb{X}}/r},$
$\displaystyle\left\|T_{\Omega_{1}}f\right\|_{\mathbb{X}(w)}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\Omega_{1}}f(x)g(x)w(x)dx\right|$
$\displaystyle\lesssim\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\langle|gw|\rangle_{1,Q}|Q|,$
where the last step follows from [15, Theorem A] or [35, Corollary 3.4].
By a direct computation and the generalized Hölder’s inequality, it follows
that
$\displaystyle\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\langle|gw|\rangle_{1,Q}|Q|$
$\displaystyle=\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\frac{1}{w(Q)}\int_{Q}|g(x)|w(x)dxw(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M_{r}f(x)M_{w}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|M_{r}f\right\|_{\mathbb{X}(w)}\left\|M_{w}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|M_{r}f\right\|_{\mathbb{X}(w)}\left\|g\right\|_{\mathbb{X}^{\prime}(w)},$
where in the first inequality we apply the Carleson embedding theorem.
Hence, by the fact that
$\left\|M_{r}f\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)}$
(see [50, Lemma 3.1]), where
$q=p_{\mathbb{X}}/{r}-\varepsilon(p_{\mathbb{X}}/{r})$ and $\varepsilon(p)$ is
defined in Lemma 2.1, we obtain
$\left\|T_{\Omega_{1}}f\right\|_{\mathbb{X}(w)}\leq
C[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)}.$
This inequality, combining with the definition of
$T_{\Omega_{1}}T_{\Omega_{2}}$ gives
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\leq
C[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},$
This finish the proof of the second case.
Therefore, we obtain
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}},w\in A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}<q_{0},w\in A_{p_{\mathbb{X}}},\end{array}\right.$
where $q_{0}=2-\frac{1}{1+p_{\mathbb{X}}}$. ∎
Using Theorem 1.1, we can easily get Corollary 1.3.
###### Proof of Corollary 1.3.
Recall that if $\mathbb{X}$ be a RIQBFS with $p$-convex then
$\mathbb{X}^{\frac{1}{p}}$ is a RIBFS. This together with Theorem 1.1 lead to
the desired result. As a matter of fact, it suffices for us to prove that for
any $r>0$,
$\left\||f|^{r}\right\|^{1/r}_{\mathbb{X}(w)}=\left\|f\right\|_{\mathbb{X}^{r}(w)}$
and $p_{\mathbb{X}^{r}}=r\cdot p_{\mathbb{X}},q_{\mathbb{X}^{r}}=r\cdot
q_{\mathbb{X}}$ hold for RIQBFS $\mathbb{X}$. To see this, using the fact that
$\mathbb{X}^{1/p}$ is a RIBFS, we have
$p_{\mathbb{X}^{r}}=p_{\mathbb{X}^{\frac{1}{p}pr}}=pr\cdot
p_{\mathbb{X}^{\frac{1}{p}}}.$
On the other hand,
$p_{\mathbb{X}}=p_{\mathbb{X}^{\frac{1}{p}p}}=p\cdot
p_{\mathbb{X}^{\frac{1}{p}}}.$
Therefore $p_{\mathbb{X}^{r}}=r\cdot p_{\mathbb{X}}$ and it also works for
$q_{\mathbb{X}}$.
From the definitions of the norm of $\mathbb{X}^{r},$ it is easy to check that
$\left\|f\right\|_{\mathbb{X}^{r}(w)}=\left\||f|^{pr}\right\|^{\frac{1}{pr}}_{\mathbb{X}^{\frac{1}{p}}(w)}$
and
$\left\||f|^{r}\right\|_{\mathbb{X}(w)}=\left\||f|^{pr}\right\|^{\frac{1}{p}}_{\mathbb{X}^{\frac{1}{p}}(w)}.$
Then we have
$\left\||f|^{r}\right\|^{1/r}_{\mathbb{X}(w)}=\left\|f\right\|_{\mathbb{X}^{r}(w)}.$
∎
## 4\. Proofs of Theorem 1.4
The section will be devoted to prove Theorem 1.4. Using sparse domination
method, we will show that the bilinear sparse domination of the composition of
rough singular integral operators holds, which implies Theorem 1.4.
We start with some definitions. For a linear operator $T$ and $1\leq
r<\infty$, we define the corresponding grand maximal truncated operator
$\mathscr{M}_{T,r}$ by
$\mathscr{M}_{T,r}f(x):=\sup_{Q\ni
x}|Q|^{-\frac{1}{r}}\left\|T\left(f\chi_{\mathbb{R}^{n}\backslash
3Q}\right)\chi_{Q}\right\|_{L^{r}\left(\mathbb{R}^{n}\right)},$
where the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}$ containing
$x$. It is well known that the operator $\mathscr{M}_{T,r}$ was introduced by
Lerner [35] and it plays a key role in establishing bilinear sparse domination
of rough operator. Let $T_{1},T_{2}$ be two linear operators. We define the
grand maximal operator $\mathscr{M}_{T_{1}T_{2},r}^{*}$ by
$\mathscr{M}_{T_{1}T_{2},r}^{*}f(x):=\sup_{Q\ni
x}\left(\frac{1}{|Q|}\int_{Q}\left|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q}T_{2}\left(f\chi_{\mathbb{R}^{n}\backslash
9Q}\right)\right)(\xi)\right|^{r}\mathrm{~{}d}\xi\right)^{\frac{1}{r}}.$
Now, we can state bilinear sparse domination of the composition operator
associated with RIQBFS $\mathbb{X}.$
###### Lemma 4.1.
Let $1<r<\frac{3}{2}$, $0\leq\beta_{1},\beta_{2}<\infty.$ Let $T_{1},T_{2}$ be
two linear operators satisfying additionally the following conditions
1. (i).
the operator $T_{1}$ is bounded on $L^{r^{\prime}}\left(\mathbb{R}^{n}\right)$
with bound $A$;
2. (ii).
there exists $A_{0}>0$ such that for each $\lambda>0$,
$\left|\left\\{x\in\mathbb{R}^{n}:\left|T_{1}T_{2}f(x)\right|>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{0}|f(x)|}{\lambda}\log^{\beta_{1}}\left(\mathrm{e}+\frac{A_{0}|f(x)|}{\lambda}\right)\mathrm{d}x;$
3. (iii).
there exists $A_{1},A_{2}>0$ such that for each $\lambda>0$,
$\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}_{T_{1},r^{\prime}}T_{2}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{1}|f(x)|}{\lambda}\log^{\beta_{1}}\left(\mathrm{e}+\frac{A_{1}|f(x)|}{\lambda}\right)\mathrm{d}x$
and
$\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}_{T_{2},r^{\prime}}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{2}|f(x)|}{\lambda}\log^{\beta_{2}}\left(\mathrm{e}+\frac{A_{2}|f(x)|}{\lambda}\right)\mathrm{d}x.$
Then for each $0<p\leq 1$ and a bounded function $f$ with compact support,
there exists a $\frac{1}{2\cdot 9^{n}}$-sparse family of cubes
$\mathcal{S}=\\{Q\\}$, and functions $H_{1}$ and $H_{2}$, such that for each
function $g$,
$\int_{\mathbb{R}^{n}}|H_{1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{1}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,g),$
$\int_{\mathbb{R}^{n}}|H_{2}(x)|^{p}|g(x)|dx\lesssim
A^{p}A_{2}^{p}\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{2}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,g),$
and for a.e. $x\in\mathbb{R}^{n}$,
$T_{1}T_{2}f(x)=H_{1}(x)+H_{2}(x),$
where
$\mathcal{A}^{(a,b)}_{\mathcal{S};L(\log
L)^{\beta},L^{t}}(f,g):=\sum_{Q\in\mathcal{S}}|Q|\|f\|^{a}_{L(\log
L)^{\beta},Q}\langle|g|\rangle^{b}_{t,Q}$
with $0\leq a,b,\beta<\infty,1\leq t<\infty.$
###### Proof of Lemma 4.1.
For any $0<p\leq 1$, in order to establish bilinear sparse domination over
operator $(T_{1}T_{2}f)^{p}$, we are going to follow the scheme of the proof
of [35, Theorem 3.1], together with some ideas in [26]. For a fixed cube
$Q_{0}$, we should also consider a local version of operators
$\mathscr{M}_{T_{2},r^{\prime}}$ and $\mathscr{M}_{T_{1}T_{2},r^{\prime}}^{*}$
by
$\mathscr{M}_{T_{2};r^{\prime};Q_{0}}f(x)=\sup_{Q\ni x,Q\subset
Q_{0}}|Q|^{-\frac{1}{r^{\prime}}}\left\|\chi_{Q}T_{2}\left(f\chi_{3Q_{0}\backslash
3Q}\right)\right\|_{L^{r^{\prime}}\left(\mathbb{R}^{n}\right)}$
and
$\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x)=\sup_{Q\ni x,Q\subset
Q_{0}}\left(\frac{1}{|Q|}\int_{Q}\left|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q}T_{2}\left(f\chi_{9Q_{0}\backslash
9Q}\right)\right)(\xi)\right|^{r^{\prime}}\mathrm{d}\xi\right)^{\frac{1}{r^{\prime}}},$
respectively. Then we define three sets $E_{i}$ with $i=1,2,3$ by
$E_{1}=\left\\{x\in
Q_{0}:\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|>DA_{0}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}\right\\};$ $E_{2}=\left\\{x\in
Q_{0}:\mathscr{M}_{T_{2},r^{\prime};Q_{0}}f(x)>DA_{2}\|f\|_{L(\log
L)^{\beta_{2},},9Q_{0}}\right\\};$ $E_{3}=\left\\{x\in
Q_{0}:\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x)>DA_{1}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}\right\\},$
with $D$ a positive constant. Let $E=\cup_{i=1}^{3}E_{i}.$ Hence, by our
hypothesis $(ii),$ $(iii)$ and [29, Lemma 4.3], taking $D$ large enough, we
deduce that
$|E|\leq\frac{1}{2^{n+2}}\left|Q_{0}\right|.$
Now, by using the Calderón-Zygmund decomposition to the function $\chi_{E}$ on
$Q_{0}$ at height $\lambda=$ $\frac{1}{2^{n+1}}$, we obtain pairwise disjoint
cubes $\left\\{P_{l}\right\\}_{l}\subset\mathcal{D}\left(Q_{0}\right)$ such
that
$\chi_{E}(x)\leq\frac{1}{2^{n+1}}$
for a.e. $x\notin\bigcup_{l}P_{l}$. Together with this we immediately obtain
$\left|E\backslash\bigcup_{l}P_{l}\right|=0.$ At the same time, for each
$l\geq 1$, we also have
$\frac{1}{2^{n+1}}\left|P_{l}\right|\leq\left|P_{l}\cap
E\right|\leq\frac{1}{2}\left|P_{l}\right|.$
Observe that $\sum_{l}\left|P_{l}\right|\leq
2^{n+1}|E|\leq\frac{1}{2}\left|Q_{0}\right|,$ $P_{l}\cap E^{c}\neq\emptyset$.
Let
$\displaystyle G_{1}(x):=$ $\displaystyle
T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}\backslash\cup_{l}P_{l}}(x)$
$\displaystyle+\sum_{l}T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\chi_{P_{l}}(x).$
Then we have the following claim:
$\int_{\mathbb{R}^{n}}|G_{1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$
Indeed, noting that $0<p\leq 1$ and the definition of $G_{1}$ we have
$\displaystyle|G_{1}(x)|^{p}\leq$
$\displaystyle\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|^{p}\chi_{Q_{0}\backslash\cup_{l}P_{l}}(x)$
$\displaystyle+\sum_{l}|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)|^{p}\chi_{P_{l}}(x)$
$\displaystyle=:L_{1}(x)+\sum_{l}L^{p}_{2,l}(x),$
where $L_{2,l}(x)=|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)|\chi_{P_{l}}(x)$ with $l=1,2,\ldots$.
Therefore,
$\displaystyle\int_{\mathbb{R}^{n}}|G_{1}(x)|^{p}|g(x)|dx$
$\displaystyle\leq\int_{Q_{0}\backslash\cup_{l}P_{l}}L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle\leq\left(\int_{Q_{0}\backslash
E}+\int_{E\backslash\cup_{l}P_{l}}\right)L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle=\int_{Q_{0}\backslash
E}L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx,$
where the last inequality is due to the fact that
$\left|E\backslash\bigcup_{l}P_{l}\right|=0.$
Note that for any $x\in Q_{0}\backslash E$ yields that $x\notin E_{1}$ which
implies that
$\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|\leq DA_{0}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}.$
This estimate combined with Hölder’s inequality for any $t\geq 1$ yields
(4.1) $\displaystyle\int_{Q_{0}\backslash E}L_{1}(x)|g(x)|dx$
$\displaystyle\leq D^{p}A_{0}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}{\langle|g|\rangle}_{1,Q_{0}}|Q_{0}|$ $\displaystyle\leq
D^{p}A_{0}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}{\langle|g|\rangle}_{t,Q_{0}}|Q_{0}|.$
For each $l$, using the fact that $P_{l}\cap E^{c}\neq\emptyset,$ we observe
that there exists some $x_{l}\in P_{l}\cap E^{c}$ such that
$\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\leq\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x_{l}).$
Combining this inequality and by the Hölder’s inequality with
$\frac{r^{\prime}}{p}\geq r^{\prime}>1$, it gives
(4.2) $\displaystyle\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle\lesssim\sum_{l}\left(\int_{P_{l}}L_{2,l}^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle=\sum_{l}\left(\frac{1}{|P_{l}|}\int_{P_{l}}L_{2,l}^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq\sum_{l}\left(\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\right)^{p}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq D^{p}A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\sum_{l}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq D^{p}A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\left(\sum_{l}|P_{l}|\right)^{\frac{p}{r^{\prime}}}\left(\sum_{l}\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\left|Q_{0}\right|^{1-\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}\left(\int_{Q_{0}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle=A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$
Combining this bounds with (4.1), we see that our claim holds with
$t=(\frac{r^{\prime}}{p})^{\prime}$.
Let $G_{2}(x)$ be the function defined by
$G_{2}(x):=\sum_{l}T_{1}\left(\chi_{3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\chi_{P_{l}}(x).$
For each function $g$, applying the Hölder’s inequality and the
$L^{r^{\prime}}$ boundedness of $T_{1}$ yield
$\displaystyle\int_{\mathbb{R}^{n}}|G_{2}(x)|^{p}|g(x)|dx$
$\displaystyle\leq\sum_{l}\left(\int_{P_{l}}\left|T_{1}\left(\chi_{3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\right|^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq
A^{p}\sum_{l}\left(\int_{3P_{l}}\left|T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)(x)\right|^{r^{\prime}}dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq
3^{n}A^{p}\sum_{l}\left(\frac{1}{|3P_{l}|}\int_{3P_{l}}\left|T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)(x)\right|^{r^{\prime}}dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|.$
The same reasoning as what we have done for $G_{1}$ then gives
(4.3) $\displaystyle\int_{\mathbb{R}^{n}}|G_{2}(x)|^{p}|g(x)|dx$
$\displaystyle\lesssim A^{p}\sum_{l}\left(\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\right)^{p}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\lesssim A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\sum_{l}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\left(\sum_{l}|P_{l}|\right)^{\frac{p}{r^{\prime}}}\left(\sum_{l}\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$
We note that at each point $x\in\mathbb{R}^{n},$
$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x)=G_{1}(x)+G_{2}(x)+\sum_{l}T_{1}T_{2}\left(f\chi_{9P_{l}}\right)(x)\chi_{P_{l}}(x).$
Observe that the last term on the right-hand side is consistent with the form
on the left-hand side, so we can iterate with
$T_{1}T_{2}\left(f\chi_{9P_{l}}\right)(x)\chi_{P_{l}}(x)$ instead of
$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x),$ and so on. For
fixed $j_{1},\ldots,j_{m-1}\in\mathbb{Z}^{+}$, let $\\{Q_{0}^{j_{1}\ldots
j_{m-1}j_{m}}\\}_{j_{m}}$ be the cubes obtained at the $m$-th stage of the
decomposition process to the cube $Q_{0}^{j_{1}\ldots j_{m-1}}$, where
$\\{Q_{0}^{j_{1}}\\}=\\{P_{j}\\}$ . For each fixed $j_{1}\ldots,j_{m}$, define
the functions $G_{Q_{0},1}^{j_{1}\ldots j_{m}}f$ and $G_{Q_{0},2}^{j_{1}\ldots
j_{m}}f$ by
$G_{Q_{0},1}^{j_{1}\ldots j_{m}}f(x)=T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q_{0}^{j_{1}\ldots j_{m}}}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots
j_{m-1}}\backslash 9Q_{0}^{j_{1}\ldots
j_{m}}}\right)\right)(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x)$
and
$G_{Q_{0},2}^{j_{1}\ldots j_{m}}f(x)=T_{1}\left(\chi_{3Q_{0}^{j_{1}\ldots
j_{m}}}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots j_{m-1}}\backslash
9Q_{0}^{j_{1}\ldots j_{m}}}\right)\right)(x)\chi_{Q_{0}^{j_{1}\ldots
j_{m}}}(x),$
respectively.
Let
$\mathcal{F}=\left\\{Q_{0}\right\\}\cup_{m=1}^{\infty}\cup_{j_{1},\ldots,j_{m}}\left\\{Q_{0}^{j_{1}\ldots
j_{m}}\right\\}$. It is easy to check that
$\mathcal{F}\subset\mathcal{D}\left(Q_{0}\right)$ is a $\frac{1}{2}$-sparse
family with $\sum_{l}\left|P_{l}\right|\leq\frac{1}{2}\left|Q_{0}\right|$.
Then
(4.4) $\displaystyle G_{Q_{0},1}(x)=T_{1}$ $\displaystyle
T_{2}\left(f\chi_{9Q_{0}}\right)\chi_{Q_{0}\backslash\cup_{j_{1}}Q_{0}^{j_{1}}}(x)$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{j_{1},\ldots,j_{m}}T_{1}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots
j_{m}}}\right)\chi_{Q_{0}^{j_{1}\ldots
j_{m}}\backslash\cup_{j_{m+1}}Q_{0}^{j_{1}\ldots j_{m+1}}}(x)$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{j_{1},\ldots,j_{m}}G_{Q_{0},1}^{j_{1}\ldots
j_{m}}f(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x).$
Similar as what we have done for $G_{2}$, we may define function $G_{Q_{0},2}$
by
$G_{Q_{0},2}(x)=\sum_{m=1}^{\infty}\sum_{j_{1}\ldots
j_{m}}G_{Q_{0},2}^{j_{1}\ldots j_{m}}f(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x).$
Then for a. e. $x\in\mathbb{R}^{n}$,
$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x)=G_{Q_{0},1}(x)+G_{Q_{0},2}(x).$
We are now ready to combine all our ingredients to finish the proof. In fact,
by applying (4.2), (4.3) and (4.1) with $t=(\frac{r^{\prime}}{p})^{\prime},$
we obtain
(4.5)
$\int_{\mathbb{R}^{n}}|G_{Q_{0},1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\sum_{Q\in\mathcal{F}}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\left|Q\right|,$
(4.6)
$\int_{\mathbb{R}^{n}}|G_{Q_{0},2}(x)|^{p}|g(x)|dx\lesssim\left(A^{p}A_{2}^{p}\right)\sum_{Q\in\mathcal{F}}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\left|Q\right|.$
Observe that $\bigcup_{l}Q_{l}=\mathbb{R}^{n}$, where the cubes $Q_{l}$’s have
disjoint interiors and $\operatorname{supp}f\subset 9Q_{j}$ for each $l$. To
see this, we begin by taking a cube $Q_{0}$ such that
$\operatorname{supp}f\subset Q_{0}$. And cover $9Q_{0}\backslash Q_{0}$ by
$9^{n}-1$ congruent cubes $Q_{l}$. For every $l,Q_{0}\subset 9Q_{l}$. We
continue to do the same way for $27Q_{0}\backslash 9Q_{0}$ and so on. It’s
easy to check that the union of the cubes $Q_{l}$ of this process, including
$Q_{0}$, satisfies our requirement. Applying to each $Q_{l}$, we have that the
above estimates (4.5) and (4.6) hold for $\mathcal{F}_{l}.$ Moreover, for a.e.
$x\in\mathbb{R}^{n}$,
(4.7)
$T_{1}T_{2}f(x)=\sum_{l}G_{Q_{l},1}f(x)+\sum_{l}G_{Q_{l},2}f(x)=:H_{1}f(x)+H_{2}f(x).$
Let $\mathcal{S}$ denote $\left\\{9Q:Q\in\bigcup_{l}\mathcal{F}_{l}\right\\}.$
From the definitions of $\mathcal{F}_{l}$, it is easy to check that
$\mathcal{S}$ is a $\frac{1}{2\cdot 9^{n}}$-sparse family. This fact, together
with (4.7) gives us the desired results. ∎
Applying Lemma 4.1 to the rough singular integral operators $T_{\Omega_{1}}$
and $T_{\Omega_{2}}$, we can complete the proof of Theorem 1.4.
###### Proof of Theorem 1.4.
Let $T_{1},T_{2}$ be two linear operators satisfying the conditions in Lemma
4.1. Note that $\mathbb{X}$ is a RIQBFS with $p$-convex, which implies that
$\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$ is a RIBFS. Then Lemma 4.1 tells us that
for a.e.$x\in\mathbb{R}^{n},$ $T_{1}T_{2}f(x)=H_{1}(x)+H_{2}(x).$ Therefore,
(4.8) $\displaystyle\left\|T_{1}T_{2}f\right\|_{\mathbb{X}(w)}$
$\displaystyle\lesssim\left\|H_{1}\right\|_{\mathbb{X}(w)}+\left\|H_{2}\right\|_{\mathbb{X}(w)}$
$\displaystyle\backsimeq\sum_{i=1}^{2}\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\int_{\mathbb{R}^{n}}|H_{i}(x)|^{p}g(x)w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle=:\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\mathcal{L}_{1}(g)\right)^{\frac{1}{p}}+\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\mathcal{L}_{2}(g)\right)^{\frac{1}{p}},$
where
$\mathcal{L}_{i}(g)=\int_{\mathbb{R}^{n}}|H_{i}(x)|^{p}g(x)w(x)dx,i=1,2.$
Consider first the estimate of $\mathcal{L}_{1}(g).$ By Lemma 4.1, we have
$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{1}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,gw)$
$\displaystyle=(A_{0}^{p}+A_{1}^{p})\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L)^{\beta_{1}},Q}^{p}\langle|gw|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}|Q|.$
Observe that $(\frac{r^{\prime}}{p})^{\prime}\leq r$ for $0<p\leq 1,$ and then
a direct calculation shows that
$|Q|\langle|gw|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\leq\langle|gw|\rangle_{r,Q}|Q|\leq
w(Q)\cdot g_{Q,w}^{rs},$
where the definitions of $s$ and $g_{Q,w}^{rs}$ are as the same as in the
proof of Theorem 1.1.
Hence, by the Carleson embedding theorem,
$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L)^{\beta_{1}},Q}^{p}w(Q)g_{Q,w}^{rs}$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\left(M_{L(\log
L)^{\beta_{1}}}f(x)\right)^{p}M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\left(M^{\beta_{1}+1}f(x)\right)^{p}M_{w,2s}^{\mathcal{D}}g(x)w(x)dx,$
which, together with the generalized Hölder’s inequality, implies that
$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\left\|(M^{\beta_{1}+1}f)^{p}\right\|_{\mathbb{Y}(w)}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{Y}^{\prime}(w)}$
$\displaystyle=(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\left\|M^{\beta_{1}+1}f\right\|_{\mathbb{X}(w)}^{p}\left\|M_{w}^{\mathcal{D}}(g^{2s})\right\|_{\mathbb{Y}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}p}\left\|f\right\|^{p}_{\mathbb{X}(w)},$
where the last inequality follows from Lemma 3.1 and the fact
$p_{\mathbb{Y}^{\prime\frac{1}{2s}}}=\frac{p_{\mathbb{Y}^{\prime}}}{2s}=\frac{1}{2s}\frac{q_{\mathbb{X}}}{q_{\mathbb{X}}-p}>1.$
Therefore
(4.9)
$\left\|H_{1}\right\|_{\mathbb{X}(w)}\lesssim(A_{0}^{p}+A_{1}^{p})^{\frac{1}{p}}[w]_{A_{\infty}}^{\frac{1}{p}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$
Now we turn to the proof of $\left\|H_{2}\right\|_{\mathbb{X}(w)}.$ The same
reasoning as what we have done for $\mathcal{L}_{1}(g)$ yields that
$\mathcal{L}_{2}(g)=\int_{\mathbb{R}^{n}}|H_{2}(x)|^{p}|g(x)|w(x)dx\lesssim
A^{p}A_{2}^{p}[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}p}\left\|f\right\|_{\mathbb{X}(w)}^{p}\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}.$
By taking the supermum over $\|g\|_{\mathbb{Y}^{\prime}(w)}\leq 1,$ we obtain
(4.10) $\left\|H_{2}\right\|_{\mathbb{X}(w)}\lesssim
AA_{2}[w]_{A_{\infty}}^{\frac{1}{p}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$
Combining the above estimates (4.8), (4.9) and (4.10), we conclude that
$\left\|T_{1}T_{2}f\right\|_{\mathbb{X}(w)}\lesssim\left[(A_{0}^{p}+A_{1}^{p})^{\frac{1}{p}}+AA_{2}\right][w]_{A_{\infty}}^{\frac{1}{p}}\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$
Finally, we set
$T_{1}=T_{\Omega_{1}},T_{2}=T_{\Omega_{2}},A_{0}=1,A_{1}=A=A_{2}=r^{\prime},\beta_{1}=1,\beta_{2}=0.$
This together with the proof of Corollary 5.1 in [29] yields
$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left([w]_{A_{\infty}}^{1+\frac{1}{p}}+[w]_{A_{\infty}}^{2+\frac{1}{p}}\right)\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$
This finishes the proof of Theorem 1.4. ∎
## 5\. Applications
This section will be devoted to give an application of Theorem 1.1. We
consider the boundedness of certain non-standard Calderón-Zygmund operators
with rough kernels in RIBFS $\mathbb{X}$. For fixed $n\geq 2$, let $\Omega$ be
a function with homogeneous of degree zero, integrable on the unit sphere
$\mathbb{S}^{n-1}$ and satisfy the vanishing moment condition that for all
$1\leq j\leq n$,
(5.1) $\int_{\mathbb{S}^{n-1}}\Omega\left(x\right)x_{j}d\sigma(x)=0,$
where $x_{j}(1\leq j\leq n)$ denote the $j$-th variable of
$x\in\mathbb{R}^{n}$. Note that the vanishing condition here is different from
(1.1).
Let $A$ be a function on $\mathbb{R}^{n}$ whose derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right),$ namely, $\nabla A\in\mathrm{BMO}$.
Then we can define the non-standard rough Calderón-Zygmund operator
$T_{\Omega,A}$ by
(5.2)
$T_{\Omega,A}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n+1}}\left(A(x)-A(y)-\nabla
A(y)\cdot(x-y)\right)f(y)dy.$
The dual operator of $T_{\Omega,A}$ has the following form
$\widetilde{T}_{\Omega,A}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n+1}}(A(x)-A(y)-\nabla
A(x)\cdot(x-y))f(y)dy.$
The operator ${T}_{\Omega,A}$ is closely related to the Calderón commutator,
of interest in PDE, and was first studied by Cohen [13]. An interesting aspect
of this operator is that it may not satisfy the classical standard kernel
condition, even if the kernel $\Omega$ is a smooth kernel. This is also the
main reason why people call it the non-standard singular integral operator. We
refer the reader to [13, 31, 28] and their references for more details on this
topic. It is worth mentioning that Hu et al. [30] recently got the endpoint
$L\log L$ type estimate and the $L^{p}$ boundedness of ${T}_{\Omega,A}$ with
$\Omega\in L(\log L)^{2}(\mathbb{S}^{n-1}).$ In addition, they also obtained
the following results:
###### Theorem C ([30]).
Let $\Omega\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$ be homogeneous of
degree zero, satisfy the vanishing condition (5.1), and $A$ be a function on
$\mathbb{R}^{n}$ with derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right)$. Then for $p\in(1,\infty)$ and $w\in
A_{p}\left(\mathbb{R}^{n}\right)$, the following weighted norm inequality
holds
$\left\|T_{\Omega,A}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim[w]_{A_{p}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{\frac{1}{p}}+[\sigma]_{A_{\infty}}^{\frac{1}{p}}\right)[\sigma]_{A_{\infty}}\min\left\\{[\sigma]_{A_{\infty}},[w]_{A_{\infty}}\right\\}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)}.$
As in [30, Theorem 4.11] and [30, Theorem 5.6], by Theorem 1.1, we obtain
###### Theorem 5.1.
Let $\Omega$ be homogeneous of degree zero, have the vanishing moment (5.1)
and $\Omega\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$, and $A$ be a function
on $\mathbb{R}^{n}$ with derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right)$. Let $1<r<\infty$ and $\mathbb{X}$
be a RIBFS with $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q>1$ such that
$\left\|T_{\Omega,A}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty,w\in
A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<2-\frac{1}{1+p_{\mathbb{X}}},w\in
A_{p_{\mathbb{X}}}.\end{array}\right.$
## References
* [1] M. Akcoglu, R. L. Jones and P. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math. 42 (1) (1998), 154-177.
* [2] T. C. Anderson and B. Hu, A unified method for maximal truncated Calderón-Zygmund operators in general function spaces by sparse domination, Proc. Edinb. Math. Soc. 63 (2) (2020), 229-247.
* [3] C. Benea and F. Bernicot, Conservation de certaines propriétés á travers un contrôle épars d’un opérateur et applications au projecteur de Leray–Hopf, Ann. Inst. Fourier(Grenoble), 68 (2018), 2329-2379.
* [4] C. Bennett and R. Sharply, Interpolation of Operators, Academic Press, New York, (1988).
* [5] D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19 (1967), 599-616.
* [6] A. P. Calderón, Algebras of singular integral operators, Proceedings of the International Congress Mathematicians, (1968), 393-395.
* [7] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139.
* [8] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309.
* [9] A. P. Calderón and A. Zygmund, Algebras of certain singular operators, Am. J. Math. 78 (1956), 310-320.
* [10] N. Carozza and A. Passarelli di Napoli, Composition of maximal operators, Publ. Mat. 40 (1996), 397-409.
* [11] M. E. Cejas, K. Li, C. Pérez and I.P. Rivera-Ríos, Vector-valued operators, optimal weighted estimates and the $C_{p}$ condition, Sci. China Math. 63 (7) (2020), 1339-1368.
* [12] M. Christ and J. -L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), 225-237.
* [13] J. Cohen, A sharp estimate for a multilinear singular integral in $\mathbb{R}^{n}$, Indiana Univ. Math. J. 30 (1981), 693-702.
* [14] R. R. Coifman and Y. Meyer, Wavelets: Calderón-Zygmund Operators and Multilinear Operators, Cambridge University Press, Cambridge, 1997.
* [15] J. M. Conde-Alonso, A. Culiuc, F. Di Plinio and Y. Ou, A sparse domination principle for rough singular integrals, Anal. PDE. 10 (5) (2017), 1255-1284.
* [16] W. C. Connett, Singular integral near $L^{1}$, Proc. Sympos. Pure Math. 35 (1979), 163-165.
* [17] A. Culiuc, F. Di Plinio and Y. Ou, Uniform sparse domination of singular integrals via dyadic shifts, Math. Res. Lett. 2 (1) (2018), 21-42.
* [18] G. Curbera, J.Cuerva, J. Martell and C. Pérez, Extrapolation with weights, rearrangement invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math. 203 (2006), 256-318.
* [19] J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integrals via Fourier transform estimates, Invent. Math. 84 (1986), 541-561.
* [20] J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Am. Math. Soc. 336 (1993), 869-880.
* [21] D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer, Berlin, 2004.
* [22] L. Grafakos, Classical Fourier Analysis, 3rd ed., Graduate Texts in Mathematics, 249, Springer, New York, 2014.
* [23] L. Grafakos, Modern Fourier Analysis, 3rd ed., Graduate Texts in Mathematics, 250, Springer, New York, 2014.
* [24] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), 151-180.
* [25] L. Grafakos, A. Stefanov, Convolution Calderón-Zygmund singular integral operators with rough kernels, Analysis of Divergence, Appl. Numer. Harmon. Anal. (1999), 119-143.
* [26] G. Hu, Weighted weak type endpoint estimates for the composition of Calderón-Zygmund operators, J. Aust. Math. Soc. 109 (2020), 320-339.
* [27] G. Hu, Quantitative weighted bounds for the composition of Calderón-Zygmund operators, Banach J. Math. Anal. 13 (2019), 133-150.
* [28] Y. Hu, An estimate for multilinear singular integrals on $\mathbb{R}^{d}$, Beijing Daxue Xuebao, (1985), no. 3, 19-26. 552 (Chinese. English summary)
* [29] G. Hu, X. Lai and Q. Xue, On the composition of rough singular integral operators, J. Geom. Anal., 31 (3) (2021), 2742-2765.
* [30] G. Hu, X. Tao, Z. Wang and Q. Xue, On the boundedness of non-standard rough singular integral operators, arXiv: 2203.05249 [math.CA] (2022).
* [31] G. Hu and D. Yang, Sharp function estimates and weighted norm inequalities for multilinear singular integral operators, Bull. London Math. Soc. 35 (2003), 759-769.
* [32] T. P. Hytönen and C. Pérez, Sharp weighted bounds involving $A_{\infty}$, Anal. PDE. 6 (2013), 777-818.
* [33] T. Hytönen, L. Roncal and O.Tapiola, Quantitative weighted estimates for rough homogeneous singular integrals, Israel J. Math. 218 (2017), 133-164.
* [34] W. B. Johnson and G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989), 789-808.
* [35] A. K. Lerner, A weak type estimates for rough singular integrals, Rev. Mat. Iberoam. 35 (5) (2019), 1583-1602.
* [36] A. K. Lerner, A simple proof of the $A_{2}$ conjecture, Int. Math. Res. Not. 24 (14) (2013), 3159-3170.
* [37] A. K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141-161.
* [38] A. K. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics, Expo. Math. 37(3) (2019), 225-265.
* [39] K. Li, C. Pérez, I. P. Rivera-Rios and L. Roncal, Weighted norm inequalities for rough singular integral operators, J. Geom. Anal. 29 (2019), 2526-2564.
* [40] G. G. Lorentz, Majorants in spaces of integrable functions, Amer. J. Math. 77 (1955), 484-492.
* [41] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Univ. Estadual de Campinas, Campinas, 1989.
* [42] S. J. Montgomery-Smith, The Hardy Operator and Boyd Indices, Interaction between Functional Analysis, Harmonic Analysis, and Probability, Lecture Notes in Pure and Appl. Math. 175 (1996), 359-364.
* [43] A. Nagel, F. Ricci, E. M. Stein and S. Wainger, Algebras of singular integral operators with kernels controlled by multiple norms, Mem. Am. Math. Soc. 256 (1230) (2018), vii+141.
* [44] C. Pérez, Singular integrals and weights. (English summary) Harmonic and geometric analysis, 91-143, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer Basel AG, Basel, 2015.
* [45] D. H. Phone and E. M. Stein, Some further classes of pseudo-differential and singular integral operators arising in boundary-value problems I, composition of operators, Amer. J. Math. 104 (1982), 141-172.
* [46] M. M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
* [47] F. Ricci and G. Weiss, A characterization of $H^{1}(\Sigma_{n-1})$, Proc. Symp. Pure Math. 35 (part I) (1979), 289-294.
* [48] A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95-105.
* [49] R. S. Strichartz, Compositions of singular integral operators, J. Funct. Anal. 49 (1982), 91-127.
* [50] J. Tan and Q. Xue, Weighted variation inequalities for singular integrals and commutators in rearrangement invariant Banach and quasi-Banach spaces, Acta Math. Sin. (Engl. Ser.) 39 (2023), no. 7, 1389-1413.
* [51] D. K. Watson, Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389-399.
* [52] Y. Wen, H. Wu and Q. Xue, Sparse dominations and weighted variation inequalities for singular integrals and commutators, J. Geom Anal. 32 (2022), Paper No. 297, 30 pp.
|
# Temporal-Spatial Processing of Event Camera Data via Delay-Loop Reservoir
Neural Network
Richard Lau ([email protected]), Anthony Tylan-Tyler (anthony.tylan-
[email protected]),
Lihan Yao ([email protected]), Rey de Castro Roberto
([email protected]),
Robert Taylor ([email protected]), Isaiah Jones
<EMAIL_ADDRESS>
(Peraton Labs
331 Newman Springs Road, Red Bank NJ, 07701, USA)
###### Abstract
This paper describes a temporal-spatial model for video processing with
special applications to processing event camera videos. We propose to study a
conjecture motivated by our previous study of video processing with delay loop
reservoir (DLR) neural network [1, 2], which we call _Temporal-Spatial
Conjecture_ (TSC). The TSC postulates that there is significant information
content carried in the temporal representation of a video signal and that
machine learning algorithms would benefit from separate optimization of the
spatial and temporal components for intelligent processing. To verify or
refute the TSC, we propose a _Visual Markov Model_ (VMM) which decompose the
video into spatial and temporal components and estimate the mutual information
(MI) of these components. Since computation of video mutual information is
complex and time consuming, we use a Mutual Information Neural Network[10] to
estimate the bounds of the mutual information. Our result shows that the
temporal component carries significant MI compared to that of the spatial
component. This finding has often been overlooked in neural network
literature. In this paper, we will exploit this new finding to guide our
design of a _delay-loop reservoir_ neural network for event camera
classification, which results in a _18% improvement_ on classification
accuracy.
_Keywords_ Temporal-Spatial Conjecture, Video Markov Model, Mutual
Information, Mutual Information Neural Estimate, Delay Loop Reservoir, Event
Camera, Edge Application.
## 1 Introduction
While there are many Artificial Intelligence/Machine Learning (AI/ML)
techniques in literature that are used in practice for classification and
prediction of video signals, most employ techniques that are handcrafted by
experts in the field, often without theoretical justifications [5, 6]. In this
paper, which is based upon work supported by the Defense Advanced Research
Projects Agency (DARPA) on “Reservoir-based Event Camera for Intelligent
Processing on the Edge (RECIPE)”, we decompose a video signal into temporal
and spatial component and postulate a Temporal-Spatial Conjecture (TSC) that
suggests benefits by separating temporal and spatial processing to reduce
training time and complexity. To validate or refute the TSC, we propose a
Video Markov Model (VMM) that captures inter-dependence of key components of
the visual signal. The VMM quantifies the dependence by measuring by their
mutual information (MI). Results from the VMM provide insight to help explain
why certain approaches work and possibly guidance for the construction of
improved AI/ML algorithms. The TSC study was motivated by a prior result from
our previous work in HyDDENN [1, 2], where we had found that seperate
temporal/spatial processing in a drone/bird classifier leads to an improvement
of Signal-to-Noise ratio compared to a classifier that processes the combined
temporal-spatial information. In this paper, we aim to quantify such
advantage.
To demonstrate the TSC, we apply the lessons learned from the VMM to process
event camera videos. Recent years have seen growing research efforts in “Event
Cameras”, which provides a new way of capturing visual sensing in the emerging
field of neuromorphic engineering. Event cameras are fundamentally different
from traditional visual cameras, and has created a paradigm shift in how
visual information is acquired. A traditional camera samples the scene
periodically (e.g. 30 frames/sec) and captures light intensity on all the
pixels. In contrast, an event camera records only differential pixel intensity
information upon scene changes. This represents a fundamentally different
method of capturing visual information and leads to the following benefits: 1)
the sensed data is sparse and significantly smaller and thus reduces storage
and transport requirements by a factor of 1000×; 2) ultra-high temporal
resolution and low latency (in microseconds); and 3) very high dynamic range
(140dB), and 4) low power consumption. These characteristics have great impact
on the next generation of sensors deployed in both military and commercial
systems, including self-driving automobiles, high-speed Unmanned Aerial
Vehicles (UAV), robotics, ultra-high-speed data transfer, storage reduction,
and smart surveillance.
To qualify the potential improvement, we will apply the TSC-inspired
architecture to process event video via a Delay-Loop Reservoir Neural Network
(DLR) for classifying and predicting spike signals from event-based cameras.
Advantages of DLR for edge networks have been studied and reported in [2]. In
this paper, we will demonstrate how TSC is used in improving DLR for
processing event videos.
This paper is organized as follows. In Section 2, we describe the TSC and its
motivation, followed by defining the VMM structure. Section 2 also includes a
description of the DVS event dataset that we use to perform the study. Section
3 goes into detailed description of the Mutual Information Neural Estimate
approach for estimating the Mutual Information for the VMM. It then describes
the results when the VMM is applied to the DVS data. Motivated by the TSC
result and insight, Section 4 applies the Delay-Loop Reservoir (DLR) for
classification of the DVS data. It compares performance results between a
direct DLR vs. a modified DLR as inspired by the TSC and explains how lessons
learned from the TSC helps to design a better architecture for the DLR.
Section 5 discusses future research in extension of the study and implications
of the TSC. Section 6 gives the final conclusion.
## 2 Temporal Spatial Conjecture
The study of the TSC is motivated by understanding the fundamental theory
behind processing video signals. Video processing algorithms often process the
video signal as a 3-dimensional tensor of space and time, which requires large
amount of memory and computational resource. Since a video signal is made up
of consecutive video frames, each of which are static images, the spatial
content of nearby frames is often highly correlated or even repetitive.
However, most current AI/ML techniques used for classification and prediction
of video signals do not exploit this correlation. While well-designed AI/ML
algorithms, often handcrafted by experts in the field, can extract this
correlation to achieve the goal, the algorithms are excessively
computationally demanding, which becomes a significant problem for efficient
edge applications.
The current study asks the following question: Are there efficient ways to
break up the video signal before processing which may lead to more efficient
resource usage? We start with postulating a conjecture, called Temporal-
Spatial Conjecture (TSC), which states that separating the temporal and
spatial processing is beneficial in visual processing to reduce training time
and complexity. The TSC was motivated by a prior result from our previous work
in HyDDENN [1, 2], where we had found that separating temporal/spatial
processing in a drone/bird classifier leads to an improvement of Signal-to-
Noise ratio of 4dB compared to a classifier that processes the combined
temporal-spatial information. In this paper, we will build an analytic tool to
quantify such advantage.
### 2.1 Video Model
To analyze the TSC, we build a Video Markov Model (VMM) for analysis of
information content of the components of the video signal. We first represent
an event video ($V$) as a set of contiguous event frames, $F_{t},$for
$t=1,...,D$ are the time samples. The frames are sampled in time, which can be
non-uniform. For notational simplicity we assume uniform sampling in the
following. Thus, we write,
$V={F_{t}}\hskip 150.0ptt=1,...,D$
The VMM is created via two procedures. First it decomposes the event-based
video signal ($V$) into a spatial component ($S$) and a temporal component
($T$). The decomposition should satisfy two criteria:
1. 1.
Each component $S$, $T$, is a subset of the events of the original signal $V$.
2. 2.
The union of $S$ and $T$, carries information that approaches that of the
entire signal. Such information is measured with respect to an AI/ML
application such as predicting or classifying the label of the video.
Figure 1: Video Markov Model
As shown in Figure 1, the spatial component $S$ is defined to be a set of
sampled event frames such that the frames have negligible correlation among
them. Thus, we write
$S=\\{F_{t_{u}}\\}\hskip 100.0ptt_{u}=\text{time index of uncorrelated
frames}$
There are different ways to obtain the spatial component. One way is to
randomly shuffle the event frames so that they are uncorrelated. Another
method is to sparsely sample the event frames such that there is little
correlation between the samples. Note that both methods satisfy criteria 1
given above. However, the later method is usually preferred since it reduces
processing and storage complexity.
While the spatial component modifies and reduces temporal content, the
temporal component retains temporal information but allows reduction of the
spatial information. There are numerous ways that $T$ can be defined to
satisfy the VMM criteria. We propose to define $T$ as clusters of events in
contiguous frames. Thus, we write
$T=\\{C_{t}\\}\hskip 150.0ptt=1,...,D$
where $C_{t}s$ are clusters of the events at frame of time $t$. This
definition does not place stringent requirements on the structure of the
clusters but requires that the temporal information be unchanged from that of
the original video signal. The second part of the VMM is to compute or learn
the mutual information for 3 configurations as shown in Figure 2. We describe
in more detailed in the following section on the computation and estimation of
these MIs _I_. The amount of MI carried in these models provide answer to the
TSC and hints on how to design efficient AI/ML architectures.
Figure 2: VMM sub structure diagrams for study of TSC
### 2.2 Event Data
For testing of the VMM and testing of subsequent even data classification, we
will use Dynamic Vision Sensor (DVS) dataset [7]. DVS is a low-power, real-
time, event-based hand gesture recognition system. It is the first implemented
entirely on event-based hardware. DVS captures changes in pixel brightness,
transmitting data only when a change is detected, which significantly reduces
power consumption compared to traditional frame-based cameras. The DVS dataset
comprises of 11 hand gesture categories from 29 subjects under 3 illumination
conditions (see Figure 3).
Figure 3: Examples from 8 of the 11 classes in the DVS dataset
Figure 4: VMM RGB captures of gestures, the corresponding event stream, and
the aggregation of events into frames.
To process the data, the event stream is first partitioned in time as shown in
Figure 4, where events in a partition, i.e. a time window, is aggregated into
one frame. In the DVS gesture dataset, each frame collects events over a time
window of 30 milliseconds. By sequencing 200 consecutive frames, an
observation represents $\sim$6 seconds in time.
## 3 Mutual Information Extraction
Solving the TSC requires computation of Mutual Information, $I$, which is a
difficult task due to the large size of the data. This section describes how
we use the Mutual Information Neural Estimate (MINE) [10] method to estimate
$I$.
### 3.1 Mutual Information
The mutual information between two variables is a measure of how dependent
they are on each other. For a pair of variables where one is known, the mutual
information quantifies how much is known about the unknown variable from the
known variable. If the mutual information is high, then the unknown variable
is well constrained by the known variable. If the mutual information is low,
then the unknown variable is relatively unconstrained by the known variable.
Applying this to the TSC, we can use the mutual information to see how well
the separate spatial and temporal information can be associated with the video
label.
The mutual information is defined as the difference between the entropy of a
single variable, $H)X=\mathbb{E}_{p}[logp(x)]$, where $\mathbb{E}_{P}[x]$ is
the expectation value of $x$ for the distribution $P$, and the joint entropy
of both, $H(X|Y):$
$I(X,Y)=H(X)-H(X|Y)$
The mutual information between the two variables is constrained from above by
$I(X,Y)\leq min(H(X),H(Y))$ This constraint arises from $I(X,Y)=I(Y,X)$ and
that $min(H(X|Y))=0$ when $X$ is fully determined by $Y$. Thus the maximum
mutual information occurs when one variable is fuly determined by the other
and is then constrained by which variable has the lowest entropy. This
constraint will allow us to determine when the mutual information is ’high’ as
well as compare $I(V_{S},V_{C})$ and $I(V_{T},V_{C})$ where $V_{S}$ is the
spacial information of the video, $V_{T}$ is the temporal information of the
video, and $V_{C}$ is the video class label.
### 3.2 Mutual Information Neural Estimate
Determining the mutual information $I(X,Y)$ can be done when the joint
distribution of the variables is known. In our case, these distributions are
not known and, further complicating matters, the video data is very high
dimensional. For the individual frames, for example, the probability
distribution is over a 32768 dimensional space. Due to its high
dimensionality, the sample frames are likely to lie very far from each other,
making the estimation of the whole distribution unfeasible. In order to
overcome this limitation, we use the Mutual Information Neural Estimation
(MINE) [10] to estimate the mutual information between pairs of variables.
Figure 5: A diagrammatic representation of the MINE algorithm
MINE is a neural network which estimates the mutual information using an
approximate calculation of the Kullback-Leiber (KL) divergence. To begin, the
mutual information can be rewritten in terms of the KL diverence between the
joint distribution $P(X|Y)$ and the product of the marginal distribution
$P_{X}\oplus P_{Y}$
$I(X,Y)=D_{KL}(P(X|Y)\hskip 2.0pt||\hskip 2.0ptP(X)\oplus P(Y))$
The KL divergence between two distributions is defined as
$I(X,Y)=D_{KL}(P||Q)=\mathbb{E}_{P}[log(\frac{P}{Q})]$
$D_{KL}$ can be estimated using the Donsker-Varadhan representation as
$D_{KL}(P||Q)\geq\sup_{T^{\prime}\in\mathcal{F}}\mathbb{E}_{P}[T^{\prime}]-log(\mathbb{E}_{Q}[e^{T^{\prime}-1}])$
where $T^{\prime}$ is a function and $\mathcal{F}$ is a subset of functions.
Using this inequality, the neural network shown in Figure 2 attempts to learn
the function $T^{\prime}$ which maximizes the Donsker-Varadhan representation
of the KL divergence.
Figure 6: An example of the trajectories making up $V_{T}$ from a video of the
‘clapping hands’ class of the DVS128 Gesture dataset
Applying MINE to video data requires us to rigorously define the video spatial
($V_{S}$) and temporal ($V_{T}$) information. We choose to represent the
$V_{S}$ as a segment of 5 full resolution images covering 0.5s of video (a
flattened 5x2x128x128 tensor). $V_{T}$, is represented as a flattened 8x200
matrix of the trajectories of 4 centroids defined by $k$-means clustering on
the event stream which has been sliced along time into segments of 0.027s (see
Figure 6).
Using these definitions, the MINE can be used to calculate the entropy of each
variable as $I(X,X)=H(X)$. This gives us $H(V_{S})>24$, and $H(V_{C})\geq
2.49$. As the MINE estimate is established to be tight, we use $H(V_{T})$ as
the maximum value for both $I(V_{S},V_{C})$ and $I(V_{T},V_{C})$. Thus we can
directly compare the values of $I(V_{S},V_{C})$ and $I(V_{T},V_{C})$ to
understand the relative information content of each. Next, we can use MINE to
estimate $I(V_{S},V_{C})\geq 2.13$ (85% of maximum) and $I(V_{T},V_{C})\geq
2.31$ (92% of maximum)
### 3.3 Implication of Mutual Information on TSC
These results suggest that for event camera data, the bulk of the information
for labelling the video data is in the temporal information. As we highlighted
before, it is difficult to separate the spatial and temporal information. For
example, $V_{T}$ is composed of the spatial location of cluster centroids over
time which still includes spatial information. Similarly, our definition of
$V_{S}$ covers 0.5s of events over 5 frames (0.1s of events compiled into each
frame), which includes changes over time. If, instead, a single frame compiled
from 0.1s of events is used for $V_{S}$, the mutual information drops to 1.26
(50% of maximum), suggesting that the spatial information alone contains
significantly less information necessary for labelling the videos.
An additional complication of the spatial information is its high
dimensionality. At a resolution of 2x128x128, the resulting vector has 32768
components. In order to simplify the spatial information while retaining the
temporal information which appears to contain the bulk of the video
information, we propose to down sample the spatial resolution to 2x8x8 over
200 frames, giving vectors of 25600 a 21% reduction in size to cover the whole
video compared to a single frame while still retaining the bulk of the spatial
information. Labelling this the full video information, $V$, we find that
$I(V,V_{C})\geq 2.44$ (98% of maximum). Thus, by simplifying the spatial
information through down sampling while retaining all of the temporal
information, we are able to improve our result by 6%. This result is
significant as it motivates us to design a corresponding DLR to take advange
of the insight learned. This will be verified in an DLR application in the
following section.
## 4 DLR Classification of Event Camera Data
This section will illustrate how the insight gained from the TSC study
motivate an improvement on the Delay Loop Reservoir (DLR) algorithm for event
camera processing.
### 4.1 DLR Algorithm
Figure 7: Delay-Loop Reservoir structure
As shown in Figure 7, the Delay Loop Reservoir (DLR) [2] is a nonlinear, high
dimensional model within the reservoir computing framework. The DLR operates
on serialized stream data by sequentially upsampling each sample into high
dimensions, via random weights. The resulting high dimensional representation
is then processed by the core delay loop, which serves two functions: 1)
inject nonlinearity using hyperbolic tangent; and 2) allows processing
temporal processing via a leaky factor which combine past temporal data with
current data. The leaky factor thus controls the degree of “memory” of the
DLR, and can be used to select frequency components for processing. After loop
processing, the data is classified by a linear classifier such as a ridge
regressor. Note that the entire DLR has only training at the ridge regressor
stage and the delay loop itself does not required training. This is a main
difference between reservoir based NN and other NN’s. The simplicity of the
DLR architecture leads to practical implementation in hardware, which has been
demonstrated in FPGA implementation as reported in [2].
### 4.2 Initial Result DLR Algorithm
We first apply the DLR on the 11-class classification task of the DVS dataset
with tuning parameters including the leaky factor, loop dimensionality, and
ridge regression regularization. Our initial result showed an accuracy of 76%.
We noted that this sub-par performance can be attributed to an overfitting
problem. Overfitting occurs when a model learns the training data too well,
including its noise and outliers, and therefore performs poorly on unseen
data. This is especially prevalent for event camera data, where noise among
events in an event stream may become a ‘crutch’ for the model. As an
illustration, Figure 8 shows the spurious noisy events that cause overfitting.
The red blob in the event stream has been memorized by the model during
training. Whenever the spurious event appears, the model predicts class $i$.
However, in the test set, an event stream of class $i$ does not contain this
specific event.
Figure 8: Example of spurious events in the event stream
### 4.3 TSC-inspired modification of the DLR Algorithm
As illustrated above, the problem of overfitting in event classification is
that the DLR tries to classify using the noisy data. It does a good job for
the training data and indeed this is what had found. During training, the DLR
performs close to 100% accuracy while its accuracy falls to 76
In traditional NN, methods to mitigate overfitting in neural networks include
reducing the number of weights in the NN or generation of new training data by
data augmentation. These would not work in DLR as the loop does not have
training parameter and regularization in the ridge regressor was already used.
Data augmentation is also not desirable as it adds complexity to the model.
Figure 9: TSC-inspired preprocessing for DLR improvement
Instead, we will apply a data reduction module to the DLR as inspired by the
TSC result. This modification is shown in Figure 9. The Sparse Spatial module
suggests lowering the sampling rate of the events while the Temporal module
captures frames from the event. To tackle the overfitting problem, we will
substantially reduce the spatial sampling rate, but retaining sufficient
temporal resolution, which is consistent with the TSC result that the temporal
component carries more information for label classification. Using this
modification, the new DLR architecture is shown in Figure 10.
Figure 10: Modified DLR Architecture
The main addition of Figure 10 is the Spatial Filter and Down-sample module.
We apply a low-pass filter to remove high-frequency noise that could lead to
aliasing artifacts before subsampling. The subsampling process reduces the
image size by keeping every $n$-th pixel in each row and column, and
discarding the rest. The procedure $g$ is applied to pixel coordinate $(i,j)$,
given by
$g(i,j)=\sum_{k,l}^{n}f(k,l)h(i-\frac{k}{r},j-\frac{l}{r})$
where $f(k,l)$ is the smoothing filter, $h(i-\frac{k}{r},j-\frac{l}{r})$
subsamples the blurred image to dimensions r by r. Figure 11 shows effect of
the sampling process for various subsampling rates.
Figure 11: Image of subsampling effect
Figure 12: DLR Accuracy with respect to different frame sub-sampling
Figure 12 shows the performance improvement of the modified DLR architecture.
It shows that the largest improvement is achieved by downsampling the event
frame to an 8 × 8 event pixel frames. This result cannot be adequately
explained by traditional methods to tackle overfitting in machine learning, as
traditional NN methods focus on spatial learning. However, this result can be
fully explained from the viewpoint of the TSC as temporal component carries
more information. Since noise is most prominent in the spatial component, we
can substantially reduce the frame size and thus reducing overfitting to a
minimum without affecting the temporal component.
Using the modified DLR architecture, with fine tuning of other DLR parameters
including leaky factor, loop dimensionality, we were able to achieve an
accuracy of 89.6% for the 11-class DVS data classification. This is in
comparison with an accuracy of 96% [11] when attention neural networks are
used. Thus, the DLR has achieved on-par performance with State of the Art
Attention Neural Network but with substantial reduction in training time and
computation complexity as measured by Multiply-and-Accumulate units [2].
## 5 Future Work
This paper is motivated by exploring processing of the temporal component of a
video signal as this area of research seems to have not been the focus in
machine learning algorithm development. The proposed VMM validates an
important result for applying DLR to event data by optimizing the spatial
component. We have not investigated optimization with respect to the temporal
domain. For example, what would be an optimum frame structure? And what would
be a joint spatial-temporal optimum structure? Also, we would also like to
study the impact of TSC on other applications such as object segmentation and
object recognition.
## 6 Conclusion
This paper describes a temporal-spatial model for video processing with
special applications to processing event camera videos. We first describe the
Temporal-Spatial Conjecture which postulates that there is significant
information content carried in the temporal representation of a video signal.
TSC also advocates that machine learning algorithms would benefit from
separate optimization of the spatial and temporal components for intelligent
processing. We described the Visual Markov Model which decomposed videos into
spatial and temporal components and estimate the mutual information of these
components with respect to machine learning functions. The VMM study found
that temporal component carries more information than its spatial counterpart.
The finding is important, which was exploited to help design a modification to
the delay-loop reservoir algorithm for classification of DVS event data. The
modified DLR algorithm tackles the overfitting problem successfully and
achieved 18% improvement on classification accuracy, thereby validating the
TSC.
## 7 Acknowledgement
This paper is based upon work supported by the DARPA MTO contract
N6600122C4025 on “Reservoir-based Event Camera for Intelligent Processing on
the Edge (RECIPE)". We thank Dr. Whitney Mason and Dr. Timothy Klausutis,
Program Managers, for their vision on the TSC and DLR and guidance of the
program, Greg Jones for his guidance and technical discussions on the program,
and Justin Mauger (CIV USN NIWC PACIFIC CA (USA)) for many technical
discussions. The views, opinions and/or findings expressed are those of the
authors and should not be interpreted as representing the official views or
policies of DARPA or the U.S. Government.
## References
* [1] DARPA AIE. on Hyper Dimensional Data Enabled Neural Networks (HyDDENN) A.I. Exploratory PA-19-03-03, December 2019.
* [2] Richard Lau, Lihan Yao, Todd Huster, William Johnson, Stephen Arleth, Justin Wong, Devin Ridge, Michael Fletcher, William C. Headley. Scaled-Time-Attention Robust Edge Network. CoRR abs/2107.04688 (2021)
* [3] DARPA AIE. on Photonic Edge AI Compact Hardware (PEACH) DARPA-PA-18-02-05
* [4] Guy Van der Sande, Daniel Brunner and Miguel C. Soriano Advances in photonic reservoir computing Nanophotonics, 6(3), 2017.
* [5] He Wang, Edmond S. L. Ho, Hubert P. H. Shum, Zhanxing Zhu Spatio-Temporal Manifold Learning for Human Motions via Long-Horizon Modeling IEEE Trans. Vis. Comput. Graph. 27(1): 216-227 (2021).
* [6] Li Yao, Atousa Torabi, Kyunghyun Cho “Describing Videos by Exploiting Temporal Structure,” arXiv:1502.08029v5 [stat.ML] 1 Oct 2015.
* [7] A. Amir, B. Taba, D. Berg, T. Melano, J. McKinstry, C. Di Nolfo, T. Nayak, A. Andreopoulos, G. Garreau, M. Mendoza, J. Kusnitz, M. Debole, S. Esser, T. Delbruck, M. Flickner, and D. Modha. "A Low Power, Fully Event-Based Gesture Recognition System," 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, 2017.
* [8] Pierot et al Prophesee 1 Megapixel Automotive Detection Dataset. "Learning to Detect Objects with a 1 Megapixel Event Camera," NeuroIPS 2020.
* [9] Lukoševičius, M., & Jaeger, H Reservoir computing approaches to recurrent neural network training. Computer Science Review, 3(3), 127-149. 2009.
* [10] Mohamed Ishmael Belghazi, Aristide Baratin, Sai Rajeshwar, Sherjil Ozair, Yoshua Bengio, Aaron Courville, Devon Hjelm. "Mutual Information Neural Estimation," Proceedings of the 35th International Conference on Machine Learning, PMLR 80:531-540, 2018.
* [11] Sabater, L. Montesano, A. Murillo "Event Transformer. A sparse-aware solution for efficient event data processing" arXiv preprint arXiv:1804.09028, 2018.
* [12] T. J. O’Shea, T. Roy and T. C. Clancy "Over-the-Air Deep Learning Based Radio Signal Classification," IEEE Journal of Selected Topics in Signal Processing, vol. 12, no. 1, Feb. 2018.
* [13] Deepsig18 data: https://www.deepsig.ai/datasets. Vectorized deepsig18 data https://drive.google.com/open?id=1vrzz1Dbf98E-Q79-3CFjGNS7sw1HJVM6.
* [14] Richard Lau, T. Woodward “See-through Obscurants via Compressive Sensing in Degraded Visual Environment,” SPIE April 2015. doi:10.1117/12.2178039
|
# Managing Clouds in Cloud Platforms
Hassan Gobjuka Verizon
919 Hidden Ridge
Irving, TX 75083
Email<EMAIL_ADDRESS>Kamal A. Ahmat Hassan Gobjuka Department of
Information Technology Verizon City University of New York 919 Hidden Ridge
New York, NY 11101 Irving, TX 75038<EMAIL_ADDRESS><EMAIL_ADDRESS>
## I Motivation
Managing cloud services is a fundamental challenge in today s virtualized
environments. These challenges equally face both providers and consumers of
cloud services. The issue becomes even more challenging in virtualized
environments that support mobile clouds. Cloud computing platforms such as
Amazon EC2 provide customers with flexible, on demand resources at low cost.
However, they fail to provide seamless infrastructure management and
monitoring capabilities that many customers may need. For instance, Amazon EC2
doesn’t fully support cloud services automated discovery and it requires a
private set of authentication credentials. Salesforce.com, on the other hand,
do not provide monitoring access to their underlying systems. Moreover, these
systems fail to provide infrastructure monitoring of heterogenous and legacy
systems that don’t support agents. In this work, we explore how to build a
cloud management system that combines heterogeneous management of virtual
resources with comprehensive management of physical devices. We propose an
initial prototype for automated cloud management and monitoring framework. Our
ultimate goal is to develop a framework that have the capability of
automatically tracking configuration and relationships while providing full
event management, measuring performance and testing thresholds, and measuring
availability consistently. Armed with such a framework, operators can make
better decisions quickly and more efficiently.
## II Challenges
These tasks are achieved through an agentless monitoring of the cloud’s
infrastructure. While traditional network management methods suffer from
inherited difficulties [1, 2], implementing seamless network management and
monitoring framework entails several new challenges:
* •
Discovering the relationship of virtualized resources to underlying physical
infrastructure.
* •
Minimizing the overhead of monitoring and problem determination across a
physical and virtualized infrastructure.
* •
Handling security-related constraints that may affect data collection is
probably one of the most serious issues.
* •
Response action should be taken regarding a particular virtual or physical
device within the hard response deadline time frame. In agentless-based
monitoring systems, this can be insured only by implementing high number of
threads, which in turn increases complexity.
* •
Dealing with infrastructure management issues such as root-cause analysis
becomes more complex.
## III Design and Implementation
We propose an event-based model where events are placed on an in-memory
publish/subscribe bus on the Management Server, enabling a high throughput of
events.
The event bus architecture, depicted in Figure 1 enables any authorized
mediator to create events on the bus, and any authorized consumer to access
events from the bus. Events on the bus show current status of infrastructure
components.
Figure 1: An initial prototype of our cloud management and monitoring system.
The framework will provide a set of APIs to simplify creation of consumer and
mediator applications. A set of language extensions and Web services will be
used to enable Perl, Ruby, or Java scripts to create events on the bus. To
support high level of reliability and scalability, the Distributed Collector
subsystem will be multi-threaded. Furthermore, events will are normalized from
any source into a common format, which will enable consistent processing.
## References
* [1] T. Benson, A. Akella, and D. A. Maltz, Unraveling the complexity of network management, In NSDI, 2009.
* [2] H. Gobjuka, and Y. Breitbart, Ethernet Topology Discovery for Networks with Incomplete Information, IEEE/ACM Transactions on Networking, 2010, In Press.
|
# Deep Learning in Target Space
Michael Fairbank<EMAIL_ADDRESS>
Spyridon Samothrakis<EMAIL_ADDRESS>
Luca Citi<EMAIL_ADDRESS>
Department of Computer Science and Electronic Engineering
University of Essex
Colchester, CO4 3SQ, UK
###### Abstract
Deep learning uses neural networks which are parameterised by their weights.
The neural networks are usually trained by tuning the weights to directly
minimise a given loss function. In this paper we propose to re-parameterise
the weights into targets for the firing strengths of the individual nodes in
the network. Given a set of targets, it is possible to calculate the weights
which make the firing strengths best meet those targets. It is argued that
using targets for training addresses the problem of exploding gradients, by a
process which we call cascade untangling, and makes the loss-function surface
smoother to traverse, and so leads to easier, faster training, and also
potentially better generalisation, of the neural network. It also allows for
easier learning of deeper and recurrent network structures. The necessary
conversion of targets to weights comes at an extra computational expense,
which is in many cases manageable. Learning in target space can be combined
with existing neural-network optimisers, for extra gain. Experimental results
show the speed of using target space, and examples of improved generalisation,
for fully-connected networks and convolutional networks, and the ability to
recall and process long time sequences and perform natural-language processing
with recurrent networks.
Keywords: Deep Learning, Neural Networks, Targets, Exploding Gradients,
Cascade Untangling
## 1 Introduction
A feed-forward artificial neural network (NN) is a function
$f(\vec{x},\vec{w})$, parameterised by a weights vector $\vec{w}$, that maps
an input vector $\vec{x}$ to an output vector $\vec{y}=f(\vec{x},\vec{w})$.
This paper initially considers feed-forward fully-connected layered NNs with
${n_{\mathrm{L}}}$ layers, as illustrated in Figure 1.
Input $\vec{x}$Output $\vec{y}$Hidden LayersLayer:12345 Figure 1: Example
feed-forward NN with structure “3-2-3-2-3”, with five layers
(${n_{\mathrm{L}}}=5$). An input vector $\vec{x}\in\mathbb{R}^{3}$ (in this
example) is fed in from the left. Data propagates along the forward arrows
(weights) causing nodes to fire, layer by layer, eventually producing output
vector $\vec{y}\in\mathbb{R}^{3}$. The precise equations governing a NN are
given in Section 2.1. Bias weights are not shown here, and this NN does not
include shortcut connections.
NNs can be used in many problem domains, including pattern recognition,
classification and function approximation (Bishop, 1995; Goodfellow et al.,
2016). There are also numerous industrial and scientific applications for NNs,
including vision, neurocontrol, language translation, image captioning,
reinforcement learning and game playing (Silver et al., 2017; Mnih et al.,
2015; Karpathy and Fei-Fei, 2015; Fairbank et al., 2014a, b; Sutskever et al.,
2014; Samothrakis et al., 2016).
Training a NN means deciding upon an appropriate value for the weights vector
$\vec{w}$ so that the NN performs the desired task successfully. This training
process is usually an iterative numerical method that works by trying
continually to adjust $\vec{w}$ so as to minimise some real-valued loss
function $L(\vec{x}_{1},\vec{x}_{2},\ldots,\vec{x}_{n_{\mathrm{p}}},\vec{w})$
for a given set of ${n_{\mathrm{p}}}$ example input vectors
$(\vec{x}_{1},\vec{x}_{2},\ldots,\vec{x}_{n_{\mathrm{p}}})$. In a supervised-
learning task, the loss function is designed so that when minimised, each
output vector $\vec{y}_{i}=f(\vec{x}_{i},\vec{w})$ matches as possible closely
some given data label or desired value $\vec{y}^{*}_{i}$, for each input
vector $\vec{x}_{i}$ for $i\in\\{1,\ldots,{n_{\mathrm{p}}}\\}$. In
unsupervised tasks, the loss function would represent some other objective,
for example a penalty in a reinforcement-leaning problem, or an ability to
reconstruct or group the input data.
The loss function measures how well the NN is achieving its desired task and
its value at each point in weight space creates a surface, which the training
process attempts to traverse to find a suitably low point. Most training
algorithms use the gradient of the error function with respect to the weights,
$\frac{\partial{L}}{\partial{\vec{w}}}$, which is calculated by the celebrated
backpropagation algorithm (Werbos, 1974; Rumelhart et al., 1986). Two major
difficulties for training are that the loss surface can be very crinkly in
places, making the algorithms very slow, and also that the surface may be
riddled with sub-optimal local minima and saddle points. It is these problems
that the various training algorithms in existence, including novel activation
functions and weight-initialisation schemes, are designed to overcome, to
varying extents.
When a NN processes an input vector $\vec{x}$, as illustrated in Figure 1, the
internal (hidden) neurons and output neurons in it will fire at different
strengths, or activations. Hence there is a real number, the activation
strength, associated with each node. These activation values can be gathered
together for all hidden layers and the output layer to form a single vector,
$\vec{a}$.
Hence for each input vector $\vec{x}_{i}$, and given set of weights $\vec{w}$,
there will be an associated activation vector $\vec{a}_{i}$. Given the NN
weights $\vec{w}$ and several input vectors
$\\{\vec{x}_{1},\vec{x}_{2},\ldots,\vec{x}_{n_{\mathrm{p}}}\\}$, the set of
vectors $\\{\vec{a}_{1},\vec{a}_{2},\ldots,\vec{a}_{n_{\mathrm{p}}}\\}$ is
uniquely determined by the equations that govern the NN’s operation.
Conversely, given an arbitrary set of target activation vectors,
$\\{\vec{a}_{1},\vec{a}_{2},\ldots,\vec{a}_{n_{\mathrm{p}}}\\}$, and
corresponding input vectors,
$\\{\vec{x}_{1},\vec{x}_{2},\ldots,\vec{x}_{n_{\mathrm{p}}}\\}$, a relatively
cheap calculation using linear algebra could take place to uniquely determine
the weight vector $\vec{w}$ that most closely achieves the set of target-
activation vectors. Therefore the training process could work by iteratively
improving the targets, instead of the weights. That is the central idea of
this paper: to do NN training in target space (the space of all possible sets
$\\{\vec{a}_{1},\vec{a}_{2},\ldots,\vec{a}_{n_{\mathrm{p}}}\\}$) instead of
the usual weight space (the space of all possible $\vec{w}$).
The motivation for switching from weight-space learning to target space is now
discussed. With weight-space learning, any small adjustment to a weight in an
early layer shown in Fig. 1 will make the activations coming out of that layer
change by a correspondingly small amount. However these changed activations
will have a knock-on effect in changing the activations in the next layer, and
so on with each subsequent layer, often forming a cascade of changes which
reverberate through the later layers.
If the subsequent layers’ neurons are all close to their firing thresholds, or
are on a particularly steep part of the activation function, then the small
change in the early layer could have a catastrophic scrambling effect on the
NN output. This is why the error surface in weight space is so crinkly, or
even chaotic (Skorokhodov and Burtsev, 2019; Phan and Hagan, 2013). This is
not a desirable property for any learning strategy to have to cope with.
Another way of stating that a small change to a weight causes a catastrophic
scrambling of behaviour, is to say that the sensitivity of the loss function
with respect to that weight is very large. This is referred to as the
exploding-gradients problem (Hochreiter and Schmidhuber, 1997a), and we
hypothesise that this is the main reason why NNs with many layers are
difficult to train using standard backpropagation.
With target space, any small change to the targets for one layer will still
cause a correspondingly small change to the activations of that layer. But
then the algorithm that tries to match the node activations to their targets
in the subsequent layers will try to choose the weights intelligently so the
disturbance to later layers is minimised, an effect which we call cascade
untangling (see Fig. 2). If successful, this should minimise the disturbance
caused by the initial small change, and hence make the error surface in target
space much smoother than that of weight space, directly addressing the
exploding-gradients problem. Increased smoothness of the surface will also
reduce the number of local minima in it, and make the crevices in it wider and
easier to follow by gradient descent. This should be increasingly beneficial
for NNs with many layers, and even more so for recurrent neural networks
(RNNs) where the output of a neural network is looped back to be combined with
subsequent inputs, causing data to cycle around the network many times. We
discuss target-space techniques for RNNs in Section 4.1, but initally focus on
feed-forward networks.
If the cascade untangling of target-space learning works as intended, then the
path explored by gradient-descent should be more direct and hence reach lower
minima. This could contribute to better generalisation by the neural network
(Nakkiran et al., 2020). Furthermore, the resulting loss-function surface in
target space should be smoother in general, and in particular it may be
flatter at the final resting place of the optimisation process. This could
also contribute to better generalisation, since flat minima are hypothesised
by Hochreiter and Schmidhuber (1997b) to produce better generalisation than a
sharper minimum (although this is an area for further research because it
might not be straightforward to directly compare the flatness between two
different parameterisations of a loss-surface (Dinh et al., 2017)).
The experimental results given in this paper show that using target space does
indeed allow for gaining better performance in the training of deeper networks
than occurs with weight space, and includes examples of improved
generalisation and improved number of training iterations required for feed-
forward networks, recurrent networks and convolutional layered networks; but
with a higher computational cost per training iteration (due to the linear
algebra process which converts from target space to weight space). We argue
that this extra cost motivates choosing deeper but narrow network
architectures, when training a network in target space.
Input Batch345678654Layer 1Layer 2Layer 3Layer 4Output BatchLoss Figure 2: In
this analogy, a ball bounces deterministically down through a grid of pins,
like in the game bagatelle. This represents a neural network processing a
batch of input vectors and producing a batch of output vectors. The objective
of training the neural network is to arrange the pins to make the ball bounce
into a region of minimum loss at the bottom. Each row of pins represents a
layer of weights in the neural network (but the number of pins in each row is
unrelated to the number of nodes in that network layer). The $x$-coordinate of
the ball’s launch position represents a whole training batch of input vectors,
compressed down to a single $x$-coordinate for this visualisation. Likewise,
the ball’s horizontal position at each layer of pins represents all of the
hidden-state activation vectors at that layer, for each training pattern,
compressed down to a single $x$-coordinate. With weight-space learning, we
consider what effect sliding Layer 1 of pins sideways will have on the final
destination of the ball – clearly it will often catastrophically scramble the
ball’s trajectory (exploding gradients). In contrast, with target-space
learning, whenever Layer 1’s pins are moved, the positions of the lower rows
of pins automatically adjust themselves to try to stabilise the ball’s
trajectory as much as possible. This represents “cascade untangling”. In
target space, learning takes place by actively bending segments of the ball’s
zig-zag trajectory, while causing only minimal disturbances to the other
trajectory segments.
The rest of the paper is structured as follows. In the rest of this section,
we discuss related published work. In Section 2 we define the main target-
space algorithm for feed-forward layered neural networks, and then discuss
background technical information about the method in Section 3. In Section 4
we show how the method can be extended to convolutional and recurrent neural
networks. In Section 5, we give experimental results for feed-forward,
convolutional and recurrent neural networks. Finally, in Section 6, we give
conclusions.
### 1.1 Related work
Target-space techniques were originally proposed by Rohwer (1990) under the
name of “moving targets”, and then re-proposed under different names by Atiya
and Parlos (2000); Enrique Castillo and Alonso-Betanzos (2006). There are some
technical difficulties with these early works, which were later identified and
improved upon by Carreira-Perpinan and Wang (2014), and follow-up work. These
prior target-space methods, and their modern variants, are described in more
detail in Section 3.4.
Other modern deep-learning methods enable the training of deep networks in a
different way from target space. Some of these are described here.
Exploding gradients in deep neural networks were first analysed by Bengio et
al. (1994) and Hochreiter and Schmidhuber (1997a). They also identified and
defined the opposite problem, vanishing gradients, which also occurs in deep
and recurrent networks. The solution proposed by Hochreiter and Schmidhuber
(1997a), Long Short-Term Memory (LSTM) networks, focuses on solving vanishing
gradients in recurrent networks, and is very effective, especially at spotting
and exploiting patterns in long time sequences. The target-space solution we
propose focuses only on addressing exploding gradients, but when combined with
a powerful optimiser like Adam, can also learn and exploit long time sequences
(even compared to LSTM networks); as shown in Sections 5.3-5.4.
Glorot et al. (2011) identified that vanishing and exploding gradients could
largely be controlled by changing the non-linear functions used which affect
the node’s firing activation. They proposed to replace the conventional
logistic-sigmoid and hyperbolic-tangent function by a rectified linear
function, $\operatorname{ReLU}(x)$. Since their proposed activation function
has a maximum gradient of 1, it limits the scale of a cascade of changes
arising from any perturbed weight, and hence eases training of deep networks.
It does not entirely prevent the gradients from decaying/exploding though,
since the magnitude of the gradients are also amplified proportional to the
magnitude of the weights in each layer (Hochreiter and Schmidhuber, 1997a).
Furthermore, the rectified linear function produces some problems of its own,
with its unbound magnitude of its output; which can lead to infinities
appearing, particularly in recurrent networks. These infinities make the
proposed $\operatorname{ReLU}$ activation function inappropriate for recurrent
networks. We compare and include our method with a variant of this activation
function in Section 5.
Another significant recent breakthrough in training deep networks has been
through the careful choice of the magnitude by which weights are randomised
before training commences. The magnitudes derived by Glorot and Bengio (2010)
and He et al. (2015) are carefully chosen so that the mean and variance in
activations of each node remain 0 and 1 respectively, regardless of the depth
of the network. This prevents the activations at each layer growing without
bound, or saturating on the flat parts of the $\tanh$ activation function, and
thus prevent gradients from decaying or exploding.
Batch Normalisation (BN) (Ioffe and Szegedy, 2015) is a powerful method for
helping with the training of deep networks. This method can be viewed as a
simplification and close relative of target space, and also similar in aim as
the above weight-initilisation methods, in that BN prevents the activations of
nodes at subsequent layers from growing or saturating without bound. BN works
by setting an individual “target” for the mean $\mu$ and standard-deviation
$\sigma$ for every node in a layer. These are applied to normalise the entire
training batch passing through the given node. This normalisation can help by
performing some limited form of cascade untangling, but to a lesser extent
than target space does, since with BN the targets are just summary statistics
for a whole node. BN is proven to work well in practice, and there has been
some discussion on how it works so well (Santurkar et al., 2018). BN also has
a relatively low computational cost compared to target space. However target
space can do a better job of cascade untangling and training deep networks. We
describe empirical comparisons of BN to target space in Section 5.
## 2 Target-Space Algorithm for Layered Feed-Forward Networks
In the first two subsections we describe the notation for ordinary weight-
space learning for neural networks. The target-space algorithm is then defined
in the subsequent subsections.
### 2.1 Terminology, feed-forward and training mechanisms for a Neural
Network
We extend the basic NN architecture described in Figure 1 to act on a batch of
size ${n_{\mathrm{b}}}$ patterns simultaneously. Concatenate the batch of
input column vectors
$\\{\vec{x}_{b_{1}},\vec{x}_{b_{2}},\ldots,\vec{x}_{b_{n_{\mathrm{b}}}}\\}$
side by side into a single matrix $X$ with ${n_{\mathrm{b}}}$ columns. Then we
can define a feed-forward neural network (FFNN) as a function that maps this
matrix, $X$, to an output matrix, $Y$. The network is split into
${n_{\mathrm{L}}}$ layers of nodes, each node having an activation function,
$g:\mathbb{R}\rightarrow\mathbb{R}$, and there being a matrix of weights
between each pair of layers. The activation function $g$ is usually smooth,
monotonic and non-linear. Common choices are $g(x)=\tanh(x)$ or the
$\operatorname{ReLU}$ function (Glorot et al., 2011).
The layers, respectively, consist of $d_{1}$, $d_{2}$, $\ldots$,
$d_{{n_{\mathrm{L}}}}$ nodes, as shown in Figure 1. Thus
$X\in\mathbb{R}^{d_{1}\times{n_{\mathrm{b}}}}$ and
$Y\in\mathbb{R}^{d_{n_{\mathrm{L}}}\times{n_{\mathrm{b}}}}$. In the most
general case, each layer $j$ is connected to each later layer $k>j$, via a
matrix of weights $W_{j,k}\in\mathbb{R}^{d_{k}\times d_{j}}$. The network is
then said to have “all shortcut connections”. However in the more common case,
shortcut connections are not included and the only non-zero weight matrices
are between consecutive layers.
Each node has a bias which can be implemented by having an extra “layer 0”
which contains just one node that always has activation of unity. Thus for
each layer $j$, $W_{0,j}\in\mathbb{R}^{d_{j}\times 1}$ is a column vector of
weights coming from layer 0, which represent bias values for layer $j$.
The activations are calculated layer-by-layer, according to Algorithm 1. We
allow the function $g$ to be applied to a vector or matrix in an elementwise
manner, i.e. $(g(A))^{ij}:=g(A^{ij})$, for all $i$ and $j$. In line 4 of the
algorithm, $\mathbb{I}\left({j}\right)$ denotes the set of integer layer-
numbers of all layers that feed forwards into layer $j$. So for example, for a
fully-connected layered network with all shortcut connections,
$\mathbb{I}\left({3}\right)=\\{0,1,2\\}$.
Algorithm 1 Feed-Forward Dynamics
1: $A_{0}\leftarrow[1\ 1\ \ldots\ 1]$ {Bias nodes
$\in\mathbb{R}^{1\times{n_{\mathrm{b}}}}$; a row vector of 1s}
2: $A_{1}\leftarrow X$ {Input matrix.
$X\in\mathbb{R}^{d_{1}\times{n_{\mathrm{b}}}}$.}
3: for $j=2$ to ${n_{\mathrm{L}}}$ do
4: $S_{j}\leftarrow\sum_{k\in\mathbb{I}\left({j}\right)}W_{k,j}A_{k}$ {Sums
received by each node. $S_{j}\in\mathbb{R}^{d_{j}\times{n_{\mathrm{b}}}}$.}
5: $A_{j}\leftarrow g(S_{j})$ {Apply activation function.
$A_{j}\in\mathbb{R}^{d_{j}\times{n_{\mathrm{b}}}}.$}
6: end for
7: $Y\leftarrow A_{{n_{\mathrm{L}}}}$ {Output Matrix.
$Y\in\mathbb{R}^{d_{{n_{\mathrm{L}}}}\times{n_{\mathrm{b}}}}$.}
Running the feed-forward algorithm with an input matrix $X$ generates a
sequence of intermediate work-space matrices, $A_{j}$ and $S_{j}$ for all
layers $j$, whose elements hold the activations and sums, respectively, of
each layer’s nodes. These matrices and the output matrix $Y$ are to be
retained for later use. The $p^{\mathrm{th}}$ column of each matrix $X$,
$A_{j}$, $S_{j}$ and $Y$ all correspond to the same pattern $p$.
To train the neural-network, we first define the loss function, or error
function, $L:(X,\vec{w})\rightarrow\mathbb{R}$, where $\vec{w}$ is a vector of
all of the weights in the network. For supervised learning, the most common
loss functions are the mean-squared error and cross-entropy loss. Then, we
seek to minimise $L$ with respect to $\vec{w}$ using gradient descent:
$\displaystyle\Delta\vec{w}=-\eta\frac{\partial{L}}{\partial{\vec{w}}}.$ (1)
This weight update is applied iteratively, with a small positive learning rate
$\eta$. The learning rate $\eta$ can be changed over training time, or a more
advanced optimiser could be used to try to accelerate learning (e.g. RPROP
(Riedmiller and Braun, 1993), conjugate gradients (Møller, 1993), Levenberg-
Marquardt (Bishop, 1995), RMSProp (Tieleman and Hinton, 2012), or Adam (Kingma
and Ba, 2014)).
To compute the gradients in the right-hand side of (1) efficiently, we can use
the back-propagation algorithm (Werbos, 1974; Rumelhart et al., 1986), or
equivalently automatic differentiation packages provided by a neural-network
software library (Rall, 1981; Werbos, 2005; Abadi et al., 2016).
### 2.2 Stacked Layer Input-Matrix and Weight-Matrix Notation
For layer $j$, define ${A}_{{[0:{j})}}$ to be shorthand form for a vertically
stacked block matrix of all the $A_{k}$ matrices that provide an input to
layer $j$, i.e. for all the $k\in\mathbb{I}\left({j}\right)$. For example, for
a simple layered feed-forward network we would have,
$\displaystyle{A}_{{[0:{j})}}:=\begin{pmatrix}A_{0}\cr A_{j-1}\end{pmatrix},$
(2a) (where $A_{0}$ is the layer of bias nodes), and if all shortcut
connections were present, then this would become,
$\displaystyle{A}_{{[0:{j})}}$ $\displaystyle:=\begin{pmatrix}A_{0}\cr
A_{1}\cr\vdots\cr A_{j-1}\end{pmatrix}.$ (2b)
Also define ${W}_{{[0:{j}]}}$ as a side-by-side block concatenation of all the
weight matrices that input to layer $j$. For example, with for a simple
layered feed-forward network, we would get:
$\displaystyle{W}_{{[0:{j}]}}:=\begin{pmatrix}W_{0,j}&W_{(j-1),j}\end{pmatrix},$
(3a) and, if all shortcut connections were present, we would get,
$\displaystyle{W}_{{[0:{j}]}}$
$\displaystyle:=\begin{pmatrix}W_{0,j}&W_{1,j}&\ldots&W_{(j-1),j}\end{pmatrix}.$
(3b)
This simplifies the formula for the NN feed-forward equations; line 4 of
Algorithm 1 becomes,
$S_{j}\leftarrow{W}_{{[0:{j}]}}{A}_{{[0:{j})}}.$ (4)
### 2.3 Using Targets to Parameterise a Neural Network Instead of Weights
So far the neural-network parameters have been the weights $\vec{w}$. We now
describe how we can switch the representation to “targets”.
Define the matrices $T_{2}$, $T_{3}$, …, $T_{n_{\mathrm{L}}}$, to be the
“target matrices” for each layer. These have the same dimensions as the
corresponding $S_{j}$ matrices. In the target-space approach, the set of
$T_{j}$ matrices will be the learnable parameters, replacing the role of the
weight matrices. The weight matrices are relegated into calculated quantities
that are dependent on the $T_{j}$ matrices.
The target matrix for each layer $T_{j}$ holds the “targets” for the $S_{j}$
matrix at that layer; hence we want to choose the weights which make the
$S_{j}$ matrices get as close as possible to the $T_{j}$ matrices, or to
minimise $\left|\left|{S_{j}-T_{j}}\right|\right|$, where
$\left|\left|{\cdot}\right|\right|$ denotes the Frobenius norm. To simplify
computational complexity, we do this in a greedy layer-by-layer manner.
Substituting (4) shows that we therefore need to find
$\displaystyle{W}_{{[0:{j}]}}=\arg\min_{W}\left[\left|\left|{W{A}_{{[0:{j})}}-T_{j}}\right|\right|^{2}+\lambda\left|\left|{W}\right|\right|^{2}\right],$
(5)
where the $\lambda\left|\left|{W}\right|\right|^{2}$ term is included to
provide Tikhonov regularisation, which ensures that the solution in $W$ is
unique and kept reasonably small. The minimisation in (5) is a standard least-
squares problem from linear algebra, with solution
${W}_{{[0:{j}]}}=T_{j}{\left({A}_{{[0:{j})}}\right)^{\dagger}},$ (6)
where the $\dagger$ indicates a regularised pseudoinverse matrix, defined by
$A^{\dagger}:={A}^{T}(A{A}^{T}+\lambda I)^{-1}.$ (7)
Here $A{A}^{T}$ is referred to as the Gramian matrix, $\lambda\geq 0$
specifies the amount of Tikhonov regularisation, and $I$ is the identity
matrix. The presence of $\lambda I$ in (7) prevents the occurrence of non-
invertible matrices.111An alternative to Tikhonov regularisation would be to
use the Truncated Singlular Value Decomposition pseudoinverse, but this was
avoided because the truncation means the derivatives are not as smooth.
However the SVD (or similar decompositions) may be used to implement (7) in
practice, to obtain improved numerical stability.
Hence the layer weights and activations can be calculated layer by layer. The
full method by which the weights are calculated from the target matrices is
given in Algorithm 2.
Algorithm 2 Converting Targets to Weights, in a FFNN, with Sequential Cascade
Untangling (SCU)
1: $A_{0}\leftarrow[1\ 1\ \ldots\ 1]$ {Bias nodes.
$A_{0}\in\mathbb{R}^{1\times{\overline{n}_{\mathrm{b}}}}$.}
2: $A_{1}\leftarrow\overline{X}$ {Input matrix.
$\overline{X}\in\mathbb{R}^{d_{1}\times{\overline{n}_{\mathrm{b}}}}$.}
3: for $j=2$ to ${n_{\mathrm{L}}}$ do
4: ${W}_{{[0:{j}]}}\leftarrow T_{j}{\left({A}_{{[0:{j})}}\right)^{\dagger}}$
{Calculates weights to layer $j$.
$T_{j}\in\mathbb{R}^{d_{j}\times{\overline{n}_{\mathrm{b}}}}$.}
5: $S_{j}\leftarrow{W}_{{[0:{j}]}}{A}_{{[0:{j})}}$
{$S_{j}\in\mathbb{R}^{d_{j}\times{\overline{n}_{\mathrm{b}}}}$.}
6: $A_{j}\leftarrow g(S_{j})$
{$A_{j}\in\mathbb{R}^{d_{j}\times{\overline{n}_{\mathrm{b}}}}$.}
7: end for
The main inputs to this algorithm are an input matrix $\overline{X}$ with
batch size ${\overline{n}_{\mathrm{b}}}$, and a list of target matrices
$T_{j}$. The main outputs of this algorithm are the realised weight matrices,
$W_{j}$. The quantities $S_{j}$ and $A_{j}$ are work-space matrices.
Because ${A}_{{[0:{j})}}$ is a shorthand for a stack of activation matrices
$A_{j}$, as defined in (2), it is intended that the changes to $A_{j}$ in line
6 will immediately affect the ${A}_{{[0:{j})}}$ matrices referenced in line 4
for higher values of $j$. This is what carries forwards the changes of an
earlier layer, so that they can be corrected for by a later layer.
Once these weight matrices are obtained, they are then used in Alg. 1 to
calculate the actual NN output. Note that Alg. 2 followed by Alg. 1 are run
back-to-back, in that order, and can therefore be viewed as one continuous
computational graph. (This is in contrast with some prior published work on
target space, e.g. Carreira-Perpinan and Wang, 2014, where there are
alternating phases of updating the $W_{j}$ matrices followed by updating
$T_{j}$ matrices. In our method, the $W_{j}$ matrices are defined as functions
of the $T_{j}$ matrices, and there are no alternating phases.)
Alg. 2 is designed to work with a potentially different input batch
$\overline{X}$ from the input matrix $X$ used to evaluate the output of the
main network via Alg. 1. This separation aids using mini-batches when training
the network, which is discussed further in Sec. 3.1. Note that because Alg. 2
uses a different input matrix ($\overline{X}$) compared the input matrix $X$
used by Alg. 1, therefore the work-space matrices $S_{j}$ and $A_{j}$ are a
different set of variables in Alg. 2, compared to Alg. 1.
Note that the aim of matching the $S_{j}$ matrices to their targets by (5)
will not be achieved exactly. In general where the number of patterns
${\overline{n}_{\mathrm{b}}}$ is larger than the rank of the weight matrix,
matching the targets exactly will be impossible. Hence we carry forward the
disturbances actually achieved to the $S_{j}$ matrices, as opposed to the
disturbances intended by $T_{j}$ matrices, in Line 6 of Alg. 2. Then the
subsequent layers’ targets will act to continue to try to dampen down this
disturbance, taking into account the fact that the previous layer’s targets
will not have been met exactly, so that the subsequent cascade of changes is
always minimised as much as possible. Hence we refer to the algorithm as
having Sequential cascade untangling (SCU).
We found SCU to be much more effective when training neural networks than an
alternative of assuming targets are met exactly, which would be implemented by
replacing line 6 of Alg. 2 by
$\displaystyle A_{j}\leftarrow g(T_{j}).$ (8)
Since this approach does not carry forwards the actual cascade of changes
beyond just one layer, we call this alternative approach “optimistic cascade
untangling” (OCU), and this is what prior published research (for example,
Rohwer, 1990; Atiya and Parlos, 2000; Enrique Castillo and Alonso-Betanzos,
2006) has always done. Experiments in Sec. 5.1 (Fig. 6) show a significant
improvement in performance from using SCU over OCU on the Two-Spirals
classification problem, and experiments in Sections 5.3 and 5.4 show the
advantage it gives in recurrent neural networks.
### 2.4 Calculating the Learning Gradient in Target Space
The previous subsection described an algorithm which converts targets to
weights. The next objective is to be able to do gradient descent in target
space, i.e. with respect to the targets themselves.
Algorithm 2 can be viewed as a mapping function $m$ from targets to weights,
such that
$\displaystyle\vec{w}=m(\overline{X},\vec{\tau}),$ (9)
where $\vec{\tau}$ is a shorthand for the vector of all target matrices
flattened and concatenated together. Given such a differentiable mapping
function, $m$, we can define the loss function $L$ in terms of the targets
(which we will denote as $L^{\prime}$), as follows:
$\displaystyle L^{\prime}(X,\vec{\tau}):=L(X,m(\overline{X},\vec{\tau}))$ (10)
Consequently, using the chain rule we can convert gradient descent in weight
space to gradient descent in target space:
$\displaystyle\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\right)^{T}\frac{\partial{L}}{\partial{\vec{w}}},$
(11)
where $\frac{\partial{m}}{\partial{\vec{\tau}}}$ uses Jacobian matrix
notation, and $\frac{\partial{L}}{\partial{\vec{w}}}$ and
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$ are treated as column
vectors.
This gradient $\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$ allows us to
perform gradient descent in target space, directly on the main neural-network
objective function, via
$\displaystyle\Delta\vec{\tau}=-\eta\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}.$
(12)
Algorithm 3 applies (11) to calculate the
$\frac{\partial{L^{\prime}}}{\partial{T_{j}}}$ matrices, for Algorithm 2’s
mapping method. In this code, $A\odot B$ means the Hadamard or elementwise
product. The algorithm uses workspace matrices $\delta A_{j}$ and $\delta
S_{j}$, which are identically dimensioned to their non-prefixed counterparts,
for each layer $j$. The matrix $\delta{A}_{{[0:{j})}}$ is built up of $\delta
A_{k}$ matrices, in the same way as Equation (2), and similarly
$\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}$ is composed of
$\frac{\partial{L}}{\partial{W_{k,j}}}$ matrices like (3). It is assumed that
these matrices point to the same underlying data, so for example, changing
$\delta{A}_{{[0:{3})}}$ will immediately affect $\delta A_{2}$, and vice
versa.
Algorithm 3 Calculation of Learning Gradient in Target Space
0: $S_{j}$, $A_{j}$ and $W_{j}$ matrices calculated by Alg. 2 for input matrix
$\overline{X}$, and $\frac{\partial{L}}{\partial{W_{k,j}}}$ matrices
calculated by back-propagation applied to Alg. 1 for input matrix $X$.
1: $\forall j,\delta A_{j}\leftarrow 0$
2: for $j={n_{\mathrm{L}}}$ to $2$ step $-1$ do
3: $\delta S_{j}\leftarrow(\delta A_{j})\odot g^{\prime}(S_{j})$
4:
$\frac{\partial{L^{\prime}}}{\partial{T_{j}}}\leftarrow\left(\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}+(\delta
S_{j}){{A}_{{[0:{j})}}}^{T}\right)({A}_{{[0:{j})}}^{\dagger})^{T}$
5:
$\delta{A}_{{[0:{j})}}\leftarrow\delta{A}_{{[0:{j})}}+{{W}_{{[0:{j}]}}}^{T}(\delta
S_{j}-\frac{\partial{L^{\prime}}}{\partial{T_{j}}})+({A}_{{[0:{j})}}{A}_{{[0:{j})}}^{T}+\lambda
I)^{-1}\left(\left(\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}\right)^{T}+{{A}_{{[0:{j})}}}(\delta
S_{j})^{T}\right)(T_{j}-S_{j})$
6: end for
The useful outputs of the algorithm are the quantities
$\frac{\partial{L^{\prime}}}{\partial{T_{j}}}$, for all layers $j$, which can
be written collectively as
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$. Hence the algorithm gives
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$, which can be used to
perform gradient descent in target space (12). As with weight-space gradient
descent, a more advanced optimiser might be applied to achieve a speed up.
The target-gradient computation algorithm (Alg. 3) is derived in Appendix B.
The most interesting part of the derivation is the differentiation under the
matrix inverse operation. This was omitted by prior research (Rohwer, 1990;
Enrique Castillo and Alonso-Betanzos, 2006), which indicates that their
learning gradients were incorrect. Our informal experiments (not recorded
here) showed that this severely reduced performance of those prior algorithms.
Modern automatic-differentiation (Abadi et al., 2016) libraries correctly
handle differentiation under a matrix inverse, but as this step is non-obvious
to derive manually, we have included the explicit algorithm here.
Alternatively, if Alg. 2 followed by Alg. 1 followed by the calculation of $L$
is passed through an automatic-differentiation library, then
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$ will be calculated
correctly, automatically.
The algorithmic complexity to implement one iteration of target-space learning
is derived (under various assumptions) in Appendix A.1 to be approximately
$4{\overline{n}_{\mathrm{b}}}/{n_{\mathrm{b}}}$ times larger than time taken
to implement one iteration of weight-space learning. Note that in this ratio,
${\overline{n}_{\mathrm{b}}}$ is the batch-size used for the target space
matrix $\overline{X}$, and ${n_{\mathrm{b}}}$ is the batch size for the
weight-space input matrix $X$. Hence if smaller mini-batches are used to
acquire the weight-space gradient than are used in the target-space
algorithms, then the time per iteration of the target-space algorithm (which
cannot use tiny mini-batches) would become increasingly large in comparison to
the weight-space calculations. Hence in the extreme case of pattern-by-pattern
learning (${n_{\mathrm{b}}}=1$), the target-space algorithm would be slower by
a very significant factor of approximately $4{\overline{n}_{\mathrm{b}}}$. In
the experiments of Section 5.1, we use
${\overline{n}_{\mathrm{b}}}={n_{\mathrm{b}}}$, and the resulting theoretical
ratio of 4 holds out well empirically.
## 3 Technical Aspects for Target-Space Implementations
The previous section has defined the main target-space method. We now consider
some technical aspects, including how to use mini-batching, the effects of
choice of $\lambda$, how to initialise the target variables at the start of
training, detail of differences between this method and previous published
target-space work, and convergence properties of our method.
### 3.1 Mini-batching and the Choice of $\overline{X}$
For very large datasets, it becomes prohibitively expensive to compute
$\frac{\partial{L}}{\partial{\vec{w}}}$ for the whole dataset. Hence with very
large datasets, it is standard practice in deep-learning to use mini-batches;
that is to operate on a smaller, randomly chosen, subset of the training data
in any one training iteration, with ${n_{\mathrm{b}}}\ll{n_{\mathrm{p}}}$. The
mini-batch chosen would be used to build the input matrix $X$ inputted to Alg.
1. Using mini-batching also introduces a stochastic element to the
optimisation process, which is also beneficial in finding flatter final minima
in the loss-function surface, and thus improving generalisation (Bottou, 2010;
Masters and Luschi, 2018).
As noted in Section 2.3, it is possible to use a different $X$ for the
computation of $\frac{\partial{L}}{\partial{\vec{w}}}$ by backpropagation
through Alg. 1 from the $\overline{X}$ used in the target-space calculations
of Algs. 2-3. But unlike the random mini-batches which may be used for
calculating $\frac{\partial{L}}{\partial{\vec{w}}}$, the $\overline{X}$ used
for target space must be fixed; because every time we shuffle the mini-batches
in $\overline{X}$, the corresponding learnable quantities $T_{j}$ would have
their meaning scrambled, which would disrupt learning.
For computational efficiency, it is possible for the patterns in
$\overline{X}$ to be a mini-batch, i.e. a subset of the entire training set,
or even a fixed random matrix222See Sec. 5.4 for an example of this.. But it
must be a fixed matrix.
The larger ${\overline{n}_{\mathrm{b}}}$ is (where
${\overline{n}_{\mathrm{b}}}$ is the number of columns in $\overline{X}$), the
more computationally expensive things will become. So how large should
${\overline{n}_{\mathrm{b}}}$ be? Ideally, ${\overline{n}_{\mathrm{b}}}$
should be sufficiently large so that the Gramian matrix in (7) would not have
any zero eigenvalues. The more non-zero eigenvalues this product has, i.e. the
more linearly independent columns in each $A_{j}$, the more useful the
pseudoinverses calculated will be in performing cascade untangling (defined in
Section 1). If there are too few patterns in $\overline{X}$ then it will mean
that target-space learning will not be able to generate usefully full-rank
weight matrices in any layer where the number of layer inputs exceeds
${\overline{n}_{\mathrm{b}}}$, which can limit the representation capabilities
of the neural network (see section 5.1, Fig. 7, for an example.)
Since the side-dimension of $A_{j}A_{j}^{T}$ is equal to the number of inputs
to layer $j$, as a rule of thumb, we recommend to set
${\overline{n}_{\mathrm{b}}}$ to be preferably as large as the widest layer in
the network, and more so if the computational expense can be spared; as this
will usually ensure the Gramian matrix is full rank. Achieving this while also
maintaining computational efficiency motivates the use of network
architectures which are deep and narrow, as opposed to architectures with a
large number of nodes to each hidden layer.
### 3.2 Choice of $\lambda$
For choosing $\lambda$ in equation (7): if it is too large then the effect of
the pseudoinverses in (7) will be dulled in their ability to perform cascade
untangling. Hence for large $\lambda$, the benefits of target-space learning
start to disappear.
If $\lambda$ is too small, then the inverse might become close-to-singular.
This would mean small changes in $A_{j}$ make large changes to the generated
weight matrices, and hence the learning gradients in target space would become
too steep.
If instability in learning is observed, then $\lambda$ could be increased, to
try to remove any particularly steep gradients in target space caused by the
matrix inversion process. We used either $\lambda=0.001$ or $\lambda=0.1$ in
all experiments in this paper.
Note that the $\lambda$ in equation (5) is performing L2 regularisation only
on the mapping between targets and weights. It does not limit the final
magnitude of the weights in the neural network, since there is no restriction
of the magnitude of $T$ in equation (6), and there is no cost on the magnitude
of $T$ appearing in the main training objective function $L$. Hence, this L2
regularisation should not be confused with a desire to apply L2 regularisation
on the weights of the neural network (weight decay), which would have the
intention of regularising the neural network into having smaller magnitude
weights. If that was required, then explicit weight decay terms (on the
magnitudes of $W$) should be added into $L$.
### 3.3 Target initialisation
At the start of training, the layer target matrices $T_{j}$ need to be
randomised. We used a truncated normal distribution, with mean 0 and a fixed
variance to randomise each element of each $T_{j}$ matrix.
Since these initial layer targets have the same fixed variance at every layer,
the variance of the magnitudes of the layer activations should be the same at
every layer of the initially-randomised network. This is in contrast to
weight-space initialisation, where unless the initial randomised weight
magnitudes are chosen very carefully (such as by using the methods proposed by
He et al. (2015); Glorot and Bengio (2010)), then the activations at
subsequent layers can grow exponentially, eventually either saturating or
becoming zero.
We have empirically found that it may be beneficial to run Alg. 2 once
immediately after the initial targets are randomised, to compute the weight
matrices and $S_{j}$, and then to apply
$T_{j}\leftarrow S_{j},\ \forall j,$ (13)
exactly once before training commences. This simply projects the newly-
randomised targets on to the hypersurface through target space which
represents the subset of targets which are exactly achievable. This step is
done in all of the target space experiments presented in this paper. It
remains to be seen how much value this step adds, although our informal
experiments seemed to show some benefit in our recurrent neural-network
experiments.
### 3.4 Relationship to Prior Target-Space Research
The work by Rohwer (1990) is a stand-out early work on target space which we
discuss here, along with more recent notable work, particularly those
following on from Carreira-Perpinan and Wang (2012). Some of the prior work is
dedicated to recurrent networks (e.g. Atiya and Parlos, 2000), some is
dedicated to feed-forward networks with one hidden layer (Enrique Castillo and
Alonso-Betanzos, 2006), and some (especially more recent publications) is
dedicated to general deep architectures (e.g. Rohwer, 1990; Carreira-Perpinan
and Wang, 2012; Lee et al., 2015a, b; Taylor et al., 2016; Zhang et al., 2016;
Frerix et al., 2017).
In some of the prior works, the process which converts targets into weights
seeks to minimise $\left|\left|{g(S_{j})-T_{j}}\right|\right|$ or
$\left|\left|{S_{j}-g^{-1}(T_{j})}\right|\right|$ instead of
$\left|\left|{S_{j}-T_{j}}\right|\right|$. Unfortunately there is no closed-
form solution to minimise $\left|\left|{g(S_{j})-T_{j}}\right|\right|$ with
respect to the weights, and the second option
$\left|\left|{S_{j}-g^{-1}(T_{j})}\right|\right|$ requires the function $g$ to
be invertible and the domain of $T_{j}$ to be restricted to the range of $g$.
Early prior published work (Rohwer, 1990; Atiya and Parlos, 2000; Enrique
Castillo and Alonso-Betanzos, 2006) is only applicable to the sum-of-squared
loss function, and hence only to supervised regression problems. A significant
defect of these early target-space methods, which probably held back their
greater adoption, is that instead of optimising the main objective function
$L$, they instead optimise an intermediate loss function, similar in concept
(ignoring bias and shortcut connections) to
$E(X,\vec{\tau})=\sum_{j}\left|\left|{W_{j}g(T_{j-1})-T_{j}}\right|\right|^{2},$
(14)
instead of the true sum-of-squares cost function,
$E(X,\vec{\tau})=\sum_{j}\left|\left|{W_{j}g(S_{j-1})-T_{j}}\right|\right|^{2}.$
(15)
They aim to minimise (14) with respect to the variables $T_{j}$, subject to
each $W_{j}$ satisfying (6), and subject to the final layer’s targets
satisfying $T_{n_{\mathrm{L}}}=Y^{*}$, where $Y^{*}$ is the target data in the
supervised regression problem. If (14) is successfully minimised down to zero
then it will follow that $T_{j}=S_{j}$ for all $j$, and (14) will match (15),
and so the supervised learning problem will be solved. However seeing as it is
in general impossible to achieve a zero error in (14), it means that the first
network layer will fail to achieve $S_{1}=T_{1}$ exactly, and hence the
“input” to the second layer in (14), namely $g(T_{1})$, will be wrong. This
misalignment between $S_{j}$ and $T_{j}$ will grow more and more as the layer
number $j$ increases. The end result is that local minima in (14) do not align
with local minima in (15), and so gradient descent on (14) does not actually
minimise the intended loss function. This was a crucial error limiting the
applicability of the methods by Rohwer (1990) and Enrique Castillo and Alonso-
Betanzos (2006). Additionally the work by Rohwer (1990) and Enrique Castillo
and Alonso-Betanzos (2006) make an incorrect derivative calculation in
computing the learning gradient, by omitting to differentiate through the
matrix inverse operation of equation (7). A related error of following the
wrong gradient descent direction appears in the work of Atiya and Parlos
(2000). They approximate $\frac{\partial{L^{\prime}}}{\partial{T_{j}}}=0$ for
all $j<{n_{\mathrm{L}}}$, which is incorrect since cascade untangling can
never occur perfectly.
Later work rectifies these problems. The work by Carreira-Perpinan and Wang
(2012) refers to the target variables as auxiliary coordinates. They solve the
problems associated with (14) by instead using a bespoke objective function
that is something like a weighted sum between (14) and (15), and where the
weighting towards (15) is gradually increased during learning. This ensures
that it is (15) that is finally optimised, while benefitting from the easier
learning of (14) in earlier training. However their method requires
alternating phases of minimisation with respect to $W_{j}$ followed by
minimising with respect to $T_{j}$; and then both of these phases need
interlacing with increasing the weighting of (15) versus (14). Our method
streamlines this process by having a single optimisation to do, which avoids
zig-zagging through the search space, and allows for acceleration methods to
be applied. But in comparison, their method increases the decoupling of the
layers by successfully using an equation based on (14) for the majority of the
learning process.
Frerix et al. (2017) extend upon the work of Carreira-Perpinan and Wang (2012)
but they modify the cost function so that the targets within it are anchored
to the forward-propagated activations (by an equation similar to (13); so that
the targets are no-longer free variables to be learned). This modification
creates an implicit quadratic cost function attached to each layer (similar to
(14)) which enables the use of a semi-implicit optimisation algorithm based on
proximal updates. The proximal updates can converge under much higher learning
rates than would be possible with ordinary gradient descent.
In “Difference Target Propagation”, Lee et al. (2015b) define a method which
uses learnable targets for each hidden layer. In this method, the target at
one layer $T_{j}$ is iteratively set to $L_{P}^{-1}(T_{j+1})$, where
$L_{P}^{-1}$ is an inverse function of the layer’s forward-propagation
function, and where this inverse (being generally an unknown function) is
learned by an auto-associative network which learns to model $L_{P}$ for each
given network layer. This method potentially allows training of networks with
discrete activation functions. In “Deeply-Supervised Nets”, Lee et al. (2015a)
add an extra support-vector machine classifier for the output of each layer.
This provides extra training information; a kind of target for each hidden
layer, which proves very effective in training deep classification networks.
Taylor et al. (2016) use learnable targets for both the $A_{j}$ and $S_{j}$
matrices, and update these learnable variables with iterative application of a
closed-form Bregman method, which trains the network to solve the objective
function, without needing to use any form of gradient descent. Zhang et al.
(2016) use a similar iterative scheme to train neural networks to generate
supervised hash codes.
In summary, much of the prior work shows the potential and power of target
space, and the recent prior work addresses the problems appearing earlier in
novel ways.
Our work provides several notable further enhancements and alternatives to the
prior work, particularly regarding the introduction of the SCU method, which
we show in our experiments is beneficial to performance. Furthermore, none of
the prior work shows how to separate the input matrix $\overline{X}$ (which is
used for calculating the weights from targets) from the input matrix $X$
(which is used to run the neural network in Alg. 1). Our work also introduces
the correction of gradient calculations through the pseudoinverse operation
(which is necessary to apply (11) correctly); the separation of the main
objective’s loss function from the intermediate closed-form least-squares
minimisation; and the introduction of mini-batches. The simplicity of the
method, and the view of searching in “target space”, gives a single, simple,
gradient-descent objective, i.e. (12), which can easily be combined with
existing acceleration schemes such as Adam.
### 3.5 Convergence Properties and Representation Capabilities of Target
Space
The target-space gradient descent update (12) is derived to be true gradient
descent on the loss function $L(\vec{x},m(\overline{X},\vec{\tau}))$ with
respect to $\vec{\tau}$. The loss function $L$ is the main learning objective
function, as chosen by the practitioner. For example, for a regression
problem, $L$ could be the mean-squared error, or for classification problems,
it could be cross-entropy loss.
A potential source of confusion is that there is a second loss function
appearing in the least-squares sub-problem given by (5), and also that the
targets in each layer will not usually be matched exactly; but this least-
squares sub-problem is completely separate from the neural-network’s main
objective function, $L$. To see this more clearly, the mapping from targets to
weights, $\vec{w}=m(\overline{X},\vec{\tau})$, given by Alg. 2, could be
replaced by any other well-defined differentiable mapping function. Regardless
of what the differentiable function $m$ is, and regardless of how well any
targets are matched or not matched, gradient descent is still performed on the
main neural-network objective function $L$ by (11) and (12).
Any sufficiently small step size in target space by (12) will yield a decrease
in $L$, since, to first order:
$\displaystyle\Delta L$
$\displaystyle\approx\left(\Delta\vec{\tau}\right)^{T}\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$
(ignoring higher order terms)
$\displaystyle=-\eta\left(\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}\right)^{T}\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$
(by (12)) $\displaystyle\leq 0$ (16)
Since the function $L$ has a lower bound, $L$ will decrease monotonically but
not beyond that bound. Hence convergence of
$L(\vec{x},m(\overline{X},\vec{\tau}))$ to some limit is guaranteed.
Similarly, the standard convergence proofs for gradient descent with
appropriately chosen step sizes apply here (Bertsekas, 1999, Section 1.2.2).
Since the differentiable mapping function $m$ is arbitrary, the convergence
guarantees work just as well as for the OCU and SCU variants described in
Section 2.3. The difference is that we hope that the target-space loss-surface
is smoother in one variant than the other (and that both variants are smoother
than in weight space), and therefore they will produce faster convergence and
better generalisation (which can only be justified empirically; see discussion
in Section 1 and empirical results in Section 5).
For any given set of weights we can run the neural network forwards, and can
capture the sums $S_{j}$ at each layer $j$, and assign these to the targets at
each layer, by (13). Ignoring the Tikhonov regularisation in (5), this will
mean the targets will generate weights approximately equal to the given set of
weights that we started with. This shows that any point in weight space has at
least one equivalent representation in target space, such that $m$ is a many-
to-one function, and hence any local minimum in weight space could be reached
by gradient descent from an appropriate random start point in target space.
An important question is the relationships between “solutions” (i.e.
stationary points) of the target-space problem,
$L(\vec{x},m(\overline{X},\vec{\tau}))$, and those of the original weight-
space one, $L(\vec{x},\vec{w})$. Equation (11) shows that whenever
$\frac{\partial{L}}{\partial{\vec{w}}}=0$, we must also have
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$. Furthermore, when Alg.
2 is used to define the mapping function $m$, Appendix C shows that whenever
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$, we must also have
$\frac{\partial{L}}{\partial{\vec{w}}}=0$. Hence any stationary point in
target space is also a stationary point in weight space, and also the reverse
is true.
For a step in target space $\Delta\vec{\tau}$, applying a first-order Taylor-
Series Expansion of (9) gives:
$\displaystyle\Delta\vec{w}$
$\displaystyle\approx\frac{\partial{m}}{\partial{\vec{\tau}}}\left(\Delta\vec{\tau}\right)$
(first-order Taylor Series)
$\displaystyle=-\eta\frac{\partial{m}}{\partial{\vec{\tau}}}\left(\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}\right)$
(by (12))
$\displaystyle=-\eta\frac{\partial{m}}{\partial{\vec{\tau}}}\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\right)^{T}\frac{\partial{L}}{\partial{\vec{w}}}$
(by (11)) (17)
Comparing (17) to (1) shows that a first-order approximation to gradient
descent in target space via (12) is equivalent to descent in weight space, but
where each weight-space direction is multiplied by a positive semi-definite
preconditioner matrix
$\frac{\partial{m}}{\partial{\vec{\tau}}}\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\right)^{T}$.
However by explicitly working in target space, we get the benefit of being
able to apply an acceleration procedure to the descent steps in target space,
such as Adam, and still retain the convergence guarantees proven for that
acceleration method. We would lose these guarantees if we applied the semi-
definite preconditioner matrix in weight space, and then applied Adam
afterwards. Also, rather than viewing target space simply as weight space with
this particular preconditioner, we have found empirically that issuing (17)
directly can be more unstable than using the exact function
$\vec{w}=m(\overline{X},\vec{\tau})$, presumably due to the first-order
approximation used in (17); although this is an area for further research.
If mini-batching is used to generate samples of $X$, then the expectation of
the gradient descent direction in target space can be derived as follows.
Denote the sampled mini-batch as $\hat{X}$, and $\hat{L}:=L(\hat{X},\vec{w})$,
and $\mathbb{E}_{\hat{X}}$ to be the expectation operator with respect to
$\hat{X}$. Then,
$\displaystyle\mathbb{E}_{\hat{X}}\left(\Delta\vec{w}\right)$
$\displaystyle=-\eta\mathbb{E}_{\hat{X}}\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\frac{\partial{m}}{\partial{\vec{\tau}}}^{T}\frac{\partial{\hat{L}}}{\partial{\vec{w}}}\right)$
(by (17))
$\displaystyle=-\eta\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\frac{\partial{m}}{\partial{\vec{\tau}}}^{T}\right)\mathbb{E}_{\hat{X}}\left(\frac{\partial{\hat{L}}}{\partial{\vec{w}}}\right)$
$\displaystyle=-\eta\left(\frac{\partial{m}}{\partial{\vec{\tau}}}\frac{\partial{m}}{\partial{\vec{\tau}}}^{T}\right)\frac{\partial{L}}{\partial{\vec{w}}}.$
(18)
The second line above follows because the mapping function $m$ is independent
of the sample chosen $\hat{X}$, for a given
$\vec{w}=m(\overline{X},\vec{\tau})$. The final line concludes that even
though mini-batching may be used, with $\overline{X}$ independent of
$\hat{X}$, the expectation of the learning gradient in target space will still
produce a preconditioned descent step on the loss function on the whole
dataset $L$.
## 4 Specific Deep Architectures
The target-space method can be extended to different neural architectures and
layer types. Here we show specifically how the method can be extended to
convolutional neural networks and recurrent neural networks.
### 4.1 Application to RNNs
Recurrent neural networks (RNNs) are a powerful architecture of neural
networks, which extend the feed-forward network by having one or more
recurrent (backward pointing) weights. These feedback connections allow
information from previous inputs be retained and to contribute extra
information to subsequent inputs to the network. This creates short-term
memory, which allows the network to remember and act on past inputs, enabling
a RNN to potentially have much greater functionality than a FFNN, potentially
allowing it to act like an agent interacting with an environment. Successful
RNN applications are in areas such as neurocontrol, time-series analysis,
image captioning, language translation, and question answering (Karpathy and
Fei-Fei, 2015; Fairbank et al., 2014a, b; Sutskever et al., 2014; Samothrakis
et al., 2016). However RNNs are generally more difficult to train than feed-
forward networks, with major challenges being vanishing or exploding learning
gradients, making it difficult for a RNN to remember information over long
time sequences.
This section describes how a RNN can be trained in target-space. Target-space
methods potentially allow RNNs to tackle more complex time sequences and data-
processing tasks which previously have been very challenging for RNNs to
solve.
A simplified recurrent architecture is shown in Fig. 3. This architecture
consumes ${n_{\mathrm{t}}}$ input matrices $X^{({t})}$, one at each time step
${t}\in\\{1,...,{n_{\mathrm{t}}}\\}$, and produces ${n_{\mathrm{t}}}$ output
matrices $Y^{({t})}$. At each time step, data from an input matrix $X^{({t})}$
enters the RNN from the left and propagates forwards in the usual manner. When
data reaches the “context layer”, layer ${c_{\mathrm{L}}}$, it loops back to
the start of the RNN, and is combined with the next input matrix to go through
the RNN again. Data loops around the recurrent layers many times, each time
also passing through the exit layers which perform some final post-processing
on the data to deliver the output matrices $Y^{({t})}$.
$A_{1}^{(t)}$Next Input$X^{(t)}$$A_{2}^{(t)}$ContextInput
Nodes:$A_{c_{L}}^{(t-1)}$$A_{3}^{(t)}$$\ldots$$A_{c_{L}}^{(t)}$ContextLayer$A_{c_{L}}^{(t)}$$\ldots$Exit
Layers$A_{n_{L}}^{(t)}$Next Output$Y^{(t)}$Recurrent feedback tonext “loop”:
$t\leftarrow t+1$
Figure 3: Diagram showing dataflow in a Recurrent Neural Network (RNN). Arrows
show dataflow. Each rectangle shows a layer of nodes in the neural network;
the layers with only a single rectangle are those that make no transformation
to the incoming data. The data cycles around the network multiple times in
“loops”, each loop indexed by ${t}$. Algorithm 4 describes the process in
greater detail. The “exit layers” do any necessary post-processing on the
data. Extra shortcut connections, or repeated recurrent structure, may be
present to obtain different RNN architectures.
Pseudocode is given in Alg. 4. In this notation, layer 0 is reserved for the
bias nodes; layer 1 is for the input matrices $X^{({t})}$, and layer 2 is for
feedback received from the later context layer ${c_{\mathrm{L}}}$. Superscript
numbers in brackets indicate the time step, ${t}$.
Algorithm 4 Recurrent NN Dynamics
0: On entry, require ${n_{\mathrm{t}}}$ input matrices
${X}^{({t})}\in\mathbb{R}^{d_{1}\times{n_{\mathrm{p}}}}.$
1: $A_{c_{\mathrm{L}}}^{(0)}\leftarrow 0$ {Initial context units are zero}
2: for ${t}=1$ to ${n_{\mathrm{t}}}$ do
3: $A_{0}^{({t})}\leftarrow[1\ 1\ \ldots\ 1]$ {Bias nodes}
4: $A_{1}^{({t})}\leftarrow X^{({t})}$ {${t}^{\mathrm{th}}$ input matrix.}
5: $A_{2}^{({t})}\leftarrow A_{c_{\mathrm{L}}}^{({t}-1)}$ {Feedback from
context layer}
6: for $j=3$ to ${n_{\mathrm{L}}}$ do
7: $S_{j}^{({t})}\leftarrow{W}_{{[0:{j}]}}{A}_{{[0:{j})}}^{({t})}$
8: $A_{j}^{({t})}\leftarrow g(S_{j}^{({t})})$
9: end for
10: $Y^{({t})}\leftarrow A_{{n_{\mathrm{L}}}}^{({t})}$ {${t}^{\mathrm{th}}$
output matrix.
$Y^{({t})}\in\mathbb{R}^{d_{n_{\mathrm{L}}}\times{n_{\mathrm{p}}}}$.}
11: end for
Each input matrix $X^{({t})}$ may itself contain a batch of several patterns
(one in each column). Hence the matrices $A^{({t})}_{j}$ and $S^{({t})}_{j}$
have dimension $d_{j}\times{n_{\mathrm{b}}}$.
An appropriate loss function $L$ would be chosen that is a function of some or
all of the $Y^{({t})}$ matrices, and then the gradient of this loss function
with respect to the weights of the network,
$\frac{\partial{L}}{\partial{\vec{w}}}$, can be found by automatic
differentiation, using for example, backpropagation through time (Werbos,
1990), in execution time
$O({n_{\mathrm{b}}}{n_{\mathrm{t}}}{n_{\mathrm{w}}})$, where
${n_{\mathrm{w}}}$ is the number of weights in the network. Then, assuming
weight-space is being used, an iterative optimizer would use this gradient
information to tune $\vec{w}$, and train the network.
To incorporate target-space learning for a RNN, the intermediate objective is
to make all the $S_{j}^{({t})}$ coming from Alg. 4 match as closely as
possible some given target matrices $T_{j}^{({t})}$, for all time steps ${t}$.
Hence, considering line 7 of Alg. 4, the objective is to choose a weight
matrix ${W}_{{[0:{j}]}}$ so as to achieve,
${W}_{{[0:{j}]}}{A}_{{[0:{j})}}^{({t})}\approx T_{j}^{({t})}\ \ \text{for all
$1\leq{t}\leq{n_{\mathrm{t}}}$}\text{,}$
or equivalently to achieve, as closely as possible,
${W}_{{[0:{j}]}}{A}_{{[0:{j})}}^{(:)}\approx T_{j}^{(:)},$
where we have defined
${A}_{{[0:{j})}}^{(:)}:=\begin{pmatrix}{A}_{{[0:{j})}}^{(1)}&{A}_{{[0:{j})}}^{(2)}&\ldots&{A}_{{[0:{j})}}^{({n_{\mathrm{t}}})}\end{pmatrix},\
3\leq j\leq{n_{\mathrm{L}}}$ (19a) and
$T_{j}^{(:)}:=\begin{pmatrix}T_{j}^{(1)}&T_{j}^{(2)}&\ldots&T_{j}^{({n_{\mathrm{t}}})}\end{pmatrix},\
3\leq j\leq{n_{\mathrm{L}}}$ (19b)
The least squares solution to this is the same as in (6) and (7):
$\displaystyle{W}_{{[0:{j}]}}=T_{j}^{(:)}{\left({A}_{{[0:{j})}}^{(:)}\right)^{\dagger}},$
(20)
however since this is a RNN, we now have the problem in that it is not
possible to know the values of ${A}_{{[0:{j})}}^{(:)}$ until the network can
by run by Alg. 4; but that algorithm cannot be run until equation (20) is
solved.
To break out of this cyclic dependency, we can approximate using the
“optimistic” cascade untangling (OCU), given by (8), and therefore just set:
$\displaystyle A_{j}^{({t})}\leftarrow g\left(T_{j}^{({t})}\right)\
\forall{t}.$ (21)
This OCU step only needs doing on the context layer which feeds backward
connections to the input layers. For the rest of the layers, it is preferable
to use the SCU method. Alg. 5 shows how to do this in detail. This algorithm
calculates the weights of a RNN from a given list of target matrices
$T_{j}^{({t})}$, using the SCU method wherever possible, and the OCU method
for the recurrent layer. The algorithm includes in line 10 an attempt to
correct the error introduced by the OCU step once the exit layers (shown in
Fig. 3) are reached.
To modify the algorithm to a fully OCU method, then we would replace line 8 by
equation (21), and delete lines 7 and 10.
Algorithm 5 Conversion of Targets to Weights for a RNN (using SCU)
0: On entry, require ${\overline{n}_{\mathrm{t}}}$ input matrices
$\overline{X}^{({t})}\in\mathbb{R}^{d_{1}\times{\overline{n}_{\mathrm{b}}}}.$
1: $A_{0}^{({t})}\leftarrow[1\ 1\ \ldots\ 1]\ \forall{t}$ {Bias nodes}
2: $A_{1}^{({t})}\leftarrow\overline{X}^{({t})}\ \forall{t}$
3: $A_{c_{\mathrm{L}}}^{({t})}\leftarrow
g\left(T_{c_{\mathrm{L}}}^{({t})}\right)\ \forall{t}$ {Estimates
$A_{c_{\mathrm{L}}}^{({t})}$ matrices by OCU method.}
4:
$A_{2}^{(:)}\leftarrow\begin{pmatrix}0&A_{c_{\mathrm{L}}}^{(1)}&A_{c_{\mathrm{L}}}^{(2)}&\ldots&A_{c_{\mathrm{L}}}^{({n_{\mathrm{t}}}-1)}\end{pmatrix}$
{Applies recurrent feedback from layer ${c_{\mathrm{L}}}$ to layer 2. Hence
$A_{2}^{(:)}$ is a block shifted-right version of $A_{c_{\mathrm{L}}}^{(:)}$.}
5: for $j=3$ to ${n_{\mathrm{L}}}$ do
6: ${W}_{{[0:{j}]}}\leftarrow
T_{j}^{(:)}{\left({A}_{{[0:{j})}}^{(:)}\right)^{\dagger}}$ {Calculates weights
to layer $j$}
7: $S_{j}^{(:)}\leftarrow{W}_{{[0:{j}]}}{A}_{{[0:{j})}}^{(:)}.$
8: $A_{j}^{(:)}\leftarrow g\left(S_{j}^{(:)}\right)$ {SCU method}
9: if $j={c_{\mathrm{L}}}$ then
10: Use the newly calculated ${W}_{{[0:{j}]}}$ matrices (for $3\leq
j\leq{c_{\mathrm{L}}}$) to run Alg. 4 (using $\overline{X}^{({t})}$ as the
input matrices), up to layer ${c_{\mathrm{L}}}$, to obtain the true
${A}_{{[0:{{c_{\mathrm{L}}}+1})}}^{(:)}$ matrices. {This is an attempt to
correct for the OCU estimation made in line 3.}
11: end if
12: end for
For the reasons discussed in Sec. 3.1, the content and length of the target-
space input matrices, $\overline{X}^{(t)}$ for
$t=1,\ldots,{\overline{n}_{\mathrm{t}}}$, may differ from the content and
length of the weight-space input matrices ($X^{(t)}$ for
$t=1,\ldots,{n_{\mathrm{t}}}$).
This algorithm merely outputs a set of weights of the RNN. The RNN would then
have to be run separately, using Alg. 4, to obtain the set of output matrices
$Y^{({t})}$.
Since Alg. 5 defines the mapping from targets to weights, it is possible to
calculate the learning gradient with respect to the targets (first going via
$\frac{\partial{L}}{\partial{\vec{w}}}$) using automatic differentiation, and
hence train the RNN in target space. For example, if Alg. 5 followed by Alg. 4
is passed to an auto-differentiation toolbox, then the toolbox will be able to
correctly calculate $\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$ by
differentiation through both algorithms sequentially. Section 5 shows
experiments which do this, with successful results.
The bottleneck in algorithmic complexity for Alg. 5 is in forming the Gramian
matrix $AA^{T}$, which will take
${n_{\mathrm{i}}}^{2}{\overline{n}_{\mathrm{b}}}{\overline{n}_{\mathrm{t}}}$
flops by direct multiplication. This is similar to a full forward-unroll of
the RNN with the input matrix $\overline{X}$. Hence the relative complexity of
running Alg. 5 using $\overline{X}$, compared to Alg. 4 using input matrix
$X$, is approximately
${\overline{n}_{\mathrm{b}}}{\overline{n}_{\mathrm{t}}}/{n_{\mathrm{b}}}{n_{\mathrm{t}}}$.
This motivates a choice of using a small value of
${\overline{n}_{\mathrm{t}}}$ where possible. See Section 5.4 for an example.
### 4.2 Application to Convolutional Neural Networks
Convolutional Neural Networks (CNNs) represent one of the most powerful modern
deep-learning architectures and are particularly applicable to vision
problems. The key innovation of the convolutional neural network is the
2D-convolution operation: a smaller weight matrix is “convolved” (i.e. a
sliding dot product is performed) with the source image to calculate the
activations in the next layer. The convolutional operation means the weight
matrix connecting one layer to the next can be much smaller than that of a
fully connected network; and also that this smaller group of weights, the
convolutional “kernel”, will be applied to multiple patches of the image. This
reuse helps in generalisation, and helps preserve spatial relationships in the
image from one layer to the next.
A CNN network structure is usually comprised of a mixture of layer types -
including one or more convolutional layers, one or more down-sampling (max-
pooling) layers, flattening operations that reduce a tensor from rank 4 down
to rank 2, and one or more regular fully-connected layers (as described in
Section 2). Further details of how these layers all work and are arranged with
each other are given by LeCun et al. (1998).
In generating a target-space method for training a CNN, it is only the
convolutional layers and fully-connected layers that have any weights, and so
only those two layer types that need modifying.
Each convolutional layer takes as input a 2D image, of size
width$\times$height, with a third depth dimension representing a number of
input channels. Together with the batch size, ${n_{\mathrm{b}}}$, this input
image is a rank-4 tensor, of shape [${n_{\mathrm{b}}}$, input_height,
input_width, input_channels]. The convolutional kernel that acts on it is a
rank-4 tensor of shape [kernel_height, kernel_width, input_channels,
output_channels], and the layer’s final output is a rank-4 tensor of shape
[${n_{\mathrm{b}}}$, output_height, output_width, output_channels].
The entire convolutional layer’s operation can be split into 6 steps:
1. 1.
Flatten the kernel to a 2-D matrix with shape
[kernel_height$\times$kernel_width$\times$input_channels, output_channels].
Call this matrix $W$.
2. 2.
Extract image patches from the input tensor, and reshape them, to form a
patches matrix $A$ of shape [num_patches,
kernel_height$\times$kernel_width$\times$input_channels], where
num_patches$={n_{\mathrm{b}}}\times$output_height$\times$output_width.
3. 3.
Multiply the kernel matrix $W$ by the patches matrix $A$, obtaining $S=WA$.
4. 4.
Add in the bias to $S$.
5. 5.
Reshape the result back into rank-4 tensor of shape [${n_{\mathrm{b}}}$,
output_height, output_width, output_channels]
6. 6.
Apply the activation function $g$.
To optimise this process, so as to be able to easily modify it for target-
space training, we first combine the bias addition of step 4 with the matrix
multiplication of step 3. This can be achieved by adding an extra row of 1s
into $A$, as was done in equation (2a), and an extra column of weights to $W$,
as was done in equation (3a).
Then we need a target matrix $T$ of the same dimension as $S$ in line 3. Given
this target matrix and the matrix $A$, we can derive the weights which best
achieve the targets using the same least-squares process as with equation (6),
i.e. $W=TA^{\dagger}$.
This derived weight matrix $W$ is then used to calculate the actual product
$S=WA$, and steps 5 and 6 (the reshape and activation function) are applied,
completing the convolutional layer’s behaviour.
The fully-connected layers are handled with their own target matrices and
least-squares solution, as in Alg. 2. The rest of the layer-types in the CNN
are unchanged - down-sampling does not use any targets (or weights), and nor
does the reshape operation.
Automatic differentiation can be used to compute the necessary learning
gradients.
The algorithmic complexity for the target-space CNN layer is derived in
Appendix A.2, and is shown to be slower than the corresponding weight-space
CNN layer by a factor which is bounded above by approximately
$(3(k_{\mathrm{h}}k_{\mathrm{w}})+1){\overline{n}_{\mathrm{b}}}/{n_{\mathrm{b}}}$,
where $k_{\mathrm{h}}$ and $k_{\mathrm{w}}$ are the kernel height and width,
respectively. This is not a constant bound, even when
${\overline{n}_{\mathrm{b}}}={n_{\mathrm{b}}}$, unlike that found for the
fully-connected network.333In future work, it is possible to remove this
numerator factor of $k_{\mathrm{h}}k_{\mathrm{w}}$, since with a stride-length
of 1 there is significant overlap between patches in the matrix $A$, and
therefore optimisations can be made when forming $AA^{T}$. Hence there is an
incentive in target space to choose CNN architectures with smaller kernel
matrices, or to only use a subset of patches when forming the pseudoinverse
matrix. In the CNN architectures used in the experiments of Section 5.2, the
ratio is empirically found to be around 7 (with a 3-by-3 kernel), which is
considerably better than the theoretical upper-bound. Part of this improvement
might be down to the fact that the backward pass of automatic differentiation
can reuse the expensive matrix products and inverses computed in the forward
pass.
This completes the description of how to use target space with a conventional
CNN architecture.
## 5 Experiments
In this section we show the performance of the target-space method on the Two-
Spirals benchmark problem, and on four classic small-image vision benchmark
problems for convolutional neural networks, and then we demonstrate the
target-space method on some bit-stream manipulation tasks and a sentiment-
analysis task for recurrent neural networks.
The experiments show the effectiveness of the target-space method, in ability
to train deep networks and produce improved generalisation. There are improved
generalisation results on the CNN vision benchmarks compared to the equivalent
weight-space method applied to the same CNN architecture. In the recurrent
network tasks, it shows the target-space method being able to solve problems
with long time-sequences, which appear to be intractable in weight space.
All experiments were implemented using Python and Tensorflow v1.14 on a Tesla
K80 GPU.444Source code for experiments is available at
https://github.com/mikefairbank/dlts_paper_code Shading in graphs indicates
95% confidence intervals as calculated by the Python Seaborn package.
### 5.1 Two-Spirals Experiments
The Two-Spirals classification problem consists of 194 two-dimensional
training points, arranged in two interleaving spiral shapes, corresponding to
the two output classes, each spiral revolving through three complete
revolutions. The training and test sets are shown in Fig. 4. The test set was
created as the angular midpoints between consecutive training points.
A layered network architecture was used, with dimensions 2-5-5-5-2, and with
all shortcut connections, following Riedmiller and Braun (1993). The cross-
entropy loss function was used for training, and the $\tanh$ activation
function used on all hidden layers, with softmax on the output layer.
Fig. 4 shows the output function of two trained networks, mapped to a single
scalar output, and visually indicates that the solutions attained in target
space are smoother and capture the essence of the problem better than in
weight space.555Although it should be noted that Levenberg Marquardt and
conjugate gradient training can produce similarly nice solutions as the left
figure.
Figure 4: Typical results for the two-spirals trained network, after 4,000
Adam iterations; target space versus weight space. Red/blue crosses denote
test set; circles denote the training set. Grey-scale background indicates
network output for the given $(x,y)$-coordinate input. Smoothness of the
target-space result shows how successful generalisation is more likely.
Fig. 5-left shows the problem being solved using gradient-descent with optimal
learning rates empirically determined as $\eta=10$ for target space and
$\eta=0.1$ for weight space. The results show that with optimal learning
rates, the target-space algorithm can fully learn the two-spirals problem’s
training set, and generalise well to the test set, in around 1,000 epochs;
compared to around 40,000 epochs for weight space to mostly learn the training
set only. It does not seem possible to generalise as well to the test set in
weight space, likely due to the unevenness appearing in Fig. 4-right.
Figure 5: Results for Two-Spirals learning, using Batch Gradient Descent (on
left) and Adam optimiser (on right).
Fig. 5-right shows results when the Adam optimiser was used, and shows a
similar outcome. The learning rate used was 0.01, which was found to be
beneficial to both target space and weight space on this problem. In this
problem, the target-space gradient descent converges to a solution in fewer
epochs than Adam in weight space.
These results all seem consistent with the target-space motivation for making
the loss-function surface smoother, and the minima commonly found lead to
better generalisation.
In our implementation the processing time was on average 3.5 times longer for
each target-space training iteration compared to each weight-space iteration.
In all experiments, the full data-set was used in all training batches
(${n_{\mathrm{b}}}={\overline{n}_{\mathrm{b}}}=194$). With target space,
$\lambda=0.001$ was used for equation (7), and initial targets were randomised
using a truncated normal distribution with $\sigma=1$, followed by the
projection given by (13). For weight-space learning, the weights were
randomised using the method of Glorot and Bengio (2010).
Fig. 6 shows the effectiveness of the Sequential Cascade untangling (SCU)
variant against the Optimistic Cascade untangling (OCU) target-space algorithm
(described in Section 2.3), and indicates that the SCU method is more stable
and effective than the OCU method.
The same graph also shows that Batch Normalisation does not seem to help on
this problem and network size, and in fact performs worse in weight space than
without batch normalisation. Batch normalisation does significantly help
though in the CNN experiments described in the next subsection.
Figure 6: Results for Two-Spirals learning, using Adam Optimiser, comparing
two forms of target space: Optimistic Cascade untangling (OCU) versus
Sequential Cascade untangling (SCU), and against Batch Normalisation in weight
space.
Fig. 7 demonstrates the sensitivity of the target-space algorithm to two of
its key hyper-parameters. The left diagram shows that reducing
${\overline{n}_{\mathrm{b}}}$, the number of patterns appearing in
$\overline{X}$ (see Section 3.1), reduces the representation capability of the
weights generated by equation (6). In each experimental trial, a random subset
of ${\overline{n}_{\mathrm{b}}}$ columns of $X$ was chosen to form
$\overline{X}$. The results show that as ${\overline{n}_{\mathrm{b}}}$ reduces
below the size of the narrowest network layer (which is 17 in this network),
the weight matrices generated from the targets become low-rank, and it is no
longer possible to fully learn the training set. Fig. 7-right shows that as
the $\lambda$ used in (7) increases, the ability of the algorithm also reduces
(see Section 3.2). Furthermore, with $\lambda\ll 10^{-4}$ the algorithm
stopped due to numerical errors causing non-invertible matrices to appear.
Figure 7: Sensitivity of the Target Space algorithm to algorithm hyper-
parameters ${\overline{n}_{\mathrm{b}}}$ and $\lambda$.
### 5.2 CNN Experiments
In this set of experiments we train convolutional neural networks on the
following four classic small-image classification problems:
* •
The MNIST digit dataset: 60,000 training samples of 28-by-28 grey-scale
pixellated hand-written numeric digits, each labelled from 0-9, and a test set
of 10,000 samples (LeCun et al., 2010).
* •
MNIST-Fashion dataset: 60,000 28x28 grayscale images of 10 labelled fashion
categories, along with a test set of 10,000 images (Xiao et al., 2017).
* •
CIFAR10 dataset: 50,000 32x32 colour training images, labelled over 10
categories, and 10,000 test images (Krizhevsky et al., 2009).
* •
CIFAR100 dataset: 50,000 32x32 colour training images, labelled over 100
categories, and 10,000 test images (Krizhevsky et al., 2009).
All of these datasets were used as training data without any modification to
the training images. For example, we did not use any data-augmentation
techniques, such as image rescaling and distortion, which are known to help
improve neural-network performance (and to be necessary to achieve state-of-
the art classification performance).
The networks used here all had six compound convolutional/pooling layers, each
of which consisted of a convolutional operation (with a square kernel of size
$m\times m$, applied with stride-length 1 with “same” padding, and $c$ output
channels) followed by an application of the activation function, followed by
(possibly) an application of max-pooling (with a square kernel of size
$k\times k$, and applied with stride-length $k$). Each max pooling operation
of side length $k$ reduces the side-length of the image by factor $k$. Hence
each compound convolutional layer can be summarised by a 3-tuple $(m,c,k)$,
with $k=1$ if no max-pooling is used. Using this 3-tuple notation, the network
architectures considered are listed in Table 1.
After the convolutional layers, the layer output is flattened, and then passed
through a number of fully-connected (dense) layers, as described in Table 1.
Benchmark | Convolutional Layers |
---|---|---
Problem | (Convolution size - Number of channels - Max Pool size) | Dense Layers
MNIST | (3-16-1)-(3-16-2)-(3-32-1)-(3-32-2)-(3-64-1)-(3-64-2) | 128-10
MNIST-Fashion | (3-16-1)-(3-16-2)-(3-32-1)-(3-32-2)-(3-64-1)-(3-64-2) | 128-10
CIFAR-10 | (3-32-1)-(3-32-2)-(3-64-1)-(3-64-2)-(3-128-1)-(3-128-2) | 128-10
CIFAR-100 | (3-32-1)-(3-32-2)-(3-64-1)-(3-64-2)-(3-128-1)-(3-128-2) | 512-128-100
Table 1: Convolutional Network Architectures considered for MNIST Problem
All non-final layers used the “leaky-relu” activation function (Maas et al.,
2013) defined by,
$\operatorname{LReL}(x)=\max(x,0.2x),$ (22)
and the final layer used softmax activation. Leaky-relu was found to slightly
be better than the $\operatorname{ReLU}$ function, since it leaves fewer zeros
in the activations which can potentially stall learning after the weights are
initially randomised; and also can potentially make the Gramian matrix in (7)
low rank.
The networks were trained with the cross-entropy loss function and the Adam
optimizer, with learning rate $0.001$ for weight-space learning, and $0.01$
for target-space learning. Mini-batches of size ${n_{\mathrm{b}}}=100$ were
randomly generated at each iteration, for computing the
$\frac{\partial{L}}{\partial{\vec{w}}}$ gradient. A fixed mini-batch of size
${\overline{n}_{\mathrm{b}}}=100$ was used for the targets’ input matrix
$\overline{X}$.
In weight space, the weight initialisation used magnitudes defined by He et
al. (2015), which are derived to work well with $\operatorname{LReL}$. In
target space, the targets values were all initially randomised with a
truncated normal distribution with standard deviation 0.1, followed by the
projection operation given by (13). $\lambda=0.1$ was used in equation (7).
Results are shown in Table 2. The results show the target-space method helping
generalisation performance, both with and without dropout (Srivastava et al.,
2014), and when comparing against weight space both with and without batch-
normalisation; and with ensemble architectures. The benefit of target space is
noticeable in the latter 3 benchmark problems; mostly so in the most
challenging benchmark problem, i.e. CIFAR100.
The two CIFAR problems were given a time budget of 24 GPU hours to train each
network. This allowed approximately 640 epochs in target space, and 5300
epochs in weight space (lowering to 4000 epochs when batch-norm was used). The
two MNIST problems received a 8 GPU-hour time budget, resulting in
approximately 480/3000/2400 epochs for target-space/weight space/BN,
respectively. Hence roughly seven times more processing time was required per
epoch for the target-space algorithms compared to the weight-space algorithms.
Algorithm (no dropout) | MNIST | MNIST-Fashion | CIFAR-10 | CIFAR-100
---|---|---|---|---
Weight Space | $99.26(\pm 0.01)\%$ | $91.6(\pm 0.1)\%$ | $77.9(\pm 0.2)\%$ | $40.3(\pm 0.7)\%$
Weight Space + Batch Normalisation | $\textbf{99.41}(\pm 0.04)\%$ | $91.6(\pm 0.2)\%$ | $80.7(\pm 0.2)\%$ | $46.5(\pm 0.6)\%$
Target Space | $99.29(\pm 0.04)\%$ | $\textbf{92.2}(\pm 0.2)\%$ | $\textbf{82.6}(\pm 0.3)\%$ | $\textbf{50.5}(\pm 0.2)\%$
Algorithm (with dropout) | MNIST | MNIST-Fashion | CIFAR-10 | CIFAR-100
Weight Space | $99.50(\pm 0.06)\%$ | $93.12(\pm 0.01)\%$ | $82.90(\pm 0.09)\%$ | $52.22(\pm 0.04)\%$
Weight Space + Batch Normalisation | $\textbf{99.58}(\pm 0.01)\%$ | $\textbf{94.07}(\pm 0.09)\%$ | $86.70(\pm 0.01)\%$ | $59.7(\pm 0.2)\%$
Target Space | $99.55(\pm 0.03)\%$ | $93.7(\pm 0.1)\%$ | $\textbf{87.4}(\pm 0.1)\%$ | $\textbf{60.4}(\pm 0.1)\%$
Algorithm (with dropout + ensemble) | MNIST | MNIST-Fashion | CIFAR-10 | CIFAR-100
Weight Space | 99.49 % | 93.99 % | 85.43 % | 56.85 %
Weight Space + Batch Normalisation | 99.6 % | 94.5 % | 88.19 % | 62.51 %
Target Space | 99.62 % | 94.34 % | 88.81 % | 63.24 %
Table 2: Test-Set Accuracies for CNN Experiments, on Standard Datasets
When dropout was used, it was applied with a dropout probability of 0.2 to all
non-final dense layers, and all even-numbered convolutional layers. The
results show that dropout provides useful benefit to both weight-space
learning and target-space learning.
When dropout was used in target space, dropout was independently applied
during both the feed-forward algorithm used to calculate
$\frac{\partial{L}}{\partial{\vec{w}}}$ using the mini-batch input matrix $X$,
and the feed-forward algorithm to map from target space to weight space using
the fixed input matrix $\overline{X}$.666Generalisation results were
noticeably worse if dropout in target space was applied to either one of these
two stages without the other.
When batch normalisation was used, it was applied to every convolutional layer
and to every non-final dense layer. Batch normalisation is only applicable to
weight-space learning. In target space learning, the targets for each layer
already define the batch mean and standard-deviation which batch normalisation
hopes to specify; making the combination of batch normalisation with target
space redundant.
The error margins in Table 2 are calculated as the standard-deviations of just
two trials; but are sufficiently small to convey the trend adequately.
When the ensemble of networks were used, the outputs of the two networks
created in the two trials were averaged after softmax. Ensemble networks can
usually generalise better than any of their constituent networks individually,
assuming the outputs of the constituent networks are somewhat independent of
each other. In this scenario the independence comes from different initial
randomisation, different shuffling of mini-batches, and different choices of
the $\overline{X}$ matrix used by the target-space algorithm. The results show
that target space and weight space are assisted by using such an ensemble;
even one comprised of only two networks.
### 5.3 Bit-Stream Recurrent Neural-Network Experiments
In this section we describe two recurrent neural-network experiments regarding
remembering and manipulating streams of bits.
The first experiment is to memorise and recall a random stream of bits. The
RNN receives a new random bit at every time step $t$, and must output the bit
it saw at the time step $t-N$, where $N$ is the delay length. As the delay
length is increased, the problem gets harder, since more bits must be
memorised.
For example if the delay length is $N=2$, and the RNN receives a bit stream
such as “1,1,1,1,0,1” (with most recent bits appearing at the right) then the
RNN is expected to produce an output stream “-,-,1,1,1,1”. (The first two
outputs in the sequence, each indicated by here “-”, are ignored, since the
delay length in this example is 2.)
The neural network has architecture $1-(N+3)-2$, with the hidden layer being
fully connected to itself with recurrent connections (corresponding to setting
${c_{\mathrm{L}}}=3$ in Fig. 3). The hidden layer used $\tanh$ activation
functions, and the final layer used softmax with cross-entropy loss function.
The loss function was made to ignore the first $N$ outputs in the stream
(since these are undefined).
The $N+3$ recurrent hidden nodes are enough to allow the network to remember
the most recent $N$ bits (with 3 spare nodes to add a little flexibility in
solution), as required; for example the RNN could learn manipulate the
remembered bits with a rotate-right bit-wise operation, so as to successfully
queue and recall the bits, and forget about bits older than $N$.
A batch size of 8,000 random bit streams of length ${n_{\mathrm{t}}}=N+50$ was
used to train the network. Random mini-batches of size ${n_{\mathrm{b}}}=100$
were used during each training iteration. A fixed mini-batch of size
${\overline{n}_{\mathrm{b}}}=100$ with
${\overline{n}_{\mathrm{t}}}={n_{\mathrm{t}}}$ was used for the target-space
matrices $\overline{X}^{(t)}$.
In weight space, the weight initialisation used magnitudes defined by Glorot
and Bengio (2010). In target space, the targets values were randomised with a
truncated normal distribution with standard deviation 1, followed by a
projection by equation (13). This projection step seemed to improve results
for the target-space experiments.
The networks were trained with 50,000 iterations of Adam optimiser, with
learning rate 0.001 for both weight-space and target space, and with
$\lambda=0.1$ for target space.
A result was considered a success if a classification accuracy $\geq 99\%$ was
achieved on the test set at any training iteration; otherwise it was a
failure.
Results are shown in Fig. 8 for various delay lengths. They show that the
target-space method is able to learn sequences with a delay length of around
two to three times as long as the weight-space methods are capable of, with a
significantly less steep rise in the number of training iterations required
for success; and that the target-space SCU method is significantly stronger
than the target-space OCU method.
Figure 8: Memorisation of a delayed binary stream of bits using a RNN. The
left graph shows the ratio of trials which were successful in correctly
learning $>99\%$ of the output bits correctly (in a test set). The right-hand
graph shows, for those successful trials, the average iteration number at
which success was first achieved.
For comparison, an extra experiment was made using an LSTM network. Here the
$N+3$ hidden nodes were replaced by $N+3$ LSTM memory cells. The LSTM network
was trained in weight space, again using Adam for 50,000 iterations. Results
are shown in the same Fig. 8. This trial shows that the LSTM network does not
seem to help in solving this problem in weight-space.
In a second RNN experiment, we modify the task from pure mermorization into
one of binary addition. In this experiment, the target output is the binary
sum of the stream of bits with the $N$-step delayed stream. To ease binary
addition, the stream is assumed to arrive in bit-wise little-endian form.
For example, if $N=2$, and the bit stream received is “1,0,1,1,0,1”, then the
target output stream that the RNN must learn is “-,-,0,0,0,1”, which is
calculated by binary addition: 1101+1011=00011. Here the target output stream
terminated before the final carry bit could be delivered, so only the 0001
remained.
As this problem was slightly harder than the previous one, since the
relationship between the target-bit sequence and the past sequence is quite
well disguised (the relationship has similarities to a delayed XOR problem but
there is also a hidden carry-bit process to discover), we gave the recurrent
network $N+5$ hidden recurrent nodes, i.e. two more than previously.
Results are shown in Fig. 9. The experimental conditions are otherwise
unchanged from the previous RNN experiment.
In this experiment the strength of the target-space methods are again shown,
with the SCU method again being capable of coping with delay lengths two to
three times as long as the weight-space methods, and with better scaling of
the number of iterations required. The strength of the SCU method’s results
confirms the value of lines 8 and 10 in Alg. 5, when compared to the OCU
method.
Figure 9: Addition of a delayed binary stream of bits using a RNN.
In both of these RNN experiments, the SCU method significantly beats the LSTM
network. It therefore seems that the exploding-gradients problem (which
target-space networks are designed to address) is more significant in this
problem than the vanishing-gradients problem (which LSTM networks are designed
to address). A complication in making this comparison is that Adam was used.
Adam might have been picking up and aggressively accelerating tiny components
of the gradients in target space, thus counteracting the vanishing gradients
and helping the target-space methods compete with LSTM. Possibly in a more
noisy problem environment, it will not be possible to accelerate such tiny
gradients due to the low signal-to-noise ratio. In that case a combination of
LSTM plus target space could be attempted.
### 5.4 RNN Movie-Review Sentiment Analysis
In this final experiment we trained a RNN to solve the natural-language
processing task of sentiment analysis for 50,000 movies reviews from the
Internet Movie Database (IMDB) website. In this binary classification task,
each review is labelled as either positive or negative. The dataset was
obtained from the Tensorflow/Keras packages, with a 50-50 training/test-set
split, using options of only including the top 5000 most frequent words, and
padding/truncating all reviews to a length of 500 words each.
A word-embedding vector of length 32 was used to encode each word from the
vocabulary of size 5000 (Bengio et al., 2003; Mikolov et al., 2013). Once each
word is converted into an embedded vector, the neural-network architecture is
the same as in the previous experiment, but with 32 inputs, 100 nodes in the
recurrent layer, and two output nodes. Each embedded word of a review is fed
to the RNN one-by-one, making the sequence length ${n_{\mathrm{t}}}=500$. Only
the final output matrix of the neural network, $Y^{(500)}$, is observed.
Results are shown in Fig. 10 and are summarised in Table 3, and show that the
target-space method’s performance slightly exceeds that of the LSTM network,
and significantly exceeds ordinary neural networks trained in weight space.
Figure 10: Results for Movie Sentiment Analysis RNN Problem. Algorithm / | Best Test | Average GPU time
---|---|---
Network Type | Accuracy | per Epoch (s)
Weight Space | $79.0(\pm 5.2)\%$ | 51.3
Weight Space + LSTM | $87.3(\pm 0.6)\%$ | 111.3
Target Space | $87.7(\pm 0.2)\%$ | 64.9
Table 3: Results for Movie Sentiment Analysis RNN Problem
All neural networks were trained using Adam with learning rate 0.001, and
mini-batch sizes of ${n_{\mathrm{b}}}=40$. The target-space algorithm used
$\lambda=0.001$. Weights and targets were initially randomised as in the
previous subsection. Word embeddings were also initially randomised (using a
normal distribution with $\mu=0$ and $\sigma=0.1$). Hence all weight and
target matrices, and the embedding vectors, were learned in an end-to-end
training process.
To customise the target-space method to handle word embeddings efficiently, a
fixed sequence of target-space input matrices $\overline{X}^{(t)}$ was chosen,
for a sequence length of just ${\overline{n}_{\mathrm{t}}}=60$, and mini-batch
size ${\overline{n}_{\mathrm{b}}}=40$. For efficiency, it was chosen that
these matrices would represent some already-embedded word sequences. Hence
each matrix $\overline{X}^{(t)}\in\mathbb{R}^{32\times 40}$, for
$t=1,\ldots,60$. Each of the $\overline{X}^{(t)}$ matrices was generated using
a uniform random distribution in the range [-1,1], and then held constant
throughout training. The lower sequence length
${\overline{n}_{\mathrm{t}}}=60$ improves the algorithmic complexity factor
(given at the end of Section 4.1), and results in a more competitive target-
space training time in Table 3. Even though this sequence length
(${\overline{n}_{\mathrm{t}}}=60$) was less than the true sequence length
(${n_{\mathrm{t}}}=500$), the combination of fixed matrices
$\overline{X}^{(t)}$ and target matrices $T_{j}^{(t)}$ provide enough
information to define the weight matrices ${W}_{{[0:{j}]}}$ unambiguously
using Alg. 5; even though $\overline{X}^{(t)}$ are fixed random matrices, and
therefore do not conform to any valid movie-review style of writing. Hence the
learning gradient $\frac{\partial{L}}{\partial{\vec{w}}}$ (which now includes
the gradient of the learnable embedding matrix,
$W_{embed}\in\mathbb{R}^{5000\times 32}$), can be calculated in weight space,
using the full sequence lengths (500), and then converted to a target-space
gradient $\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}$, using Alg. 5,
followed by automatic differentiation. To compute the gradient
$\frac{\partial{L}}{\partial{W_{embed}}}$ in target space, in order to
optimise those learnable variables too, we just used its value in weight
space, without any modification.
## 6 Conclusions
The target-space method provides an alternative search space in which to train
deep and recurrent neural networks. The theory and experiments indicate that
the loss-function surfaces being optimised are indeed smoother and easier to
optimise in target-space than in weight space. This increased smoothness
potentially leads to easier solution of problems and potentially leads to
better generalisation capabilities in the final neural networks produced.
Using target space comes at an added computational expense. In fully connected
networks, where the batch sizes for $X$ and $\overline{X}$ roughly match, this
is usually a modest constant cost of approximately 3 or 4 times as much
computation per training iteration. With CNNs it can be more, being around 7
times in the CNN architectures considered in this paper, and more so if wider
convolutional kernels are used. With the RNN experiments, which can be
considered as extremely deep and narrow networks, the timings were of similar
order of magnitude between weight space and target space. It is hoped that by
careful choice of architecture, focusing on deeper networks with narrower
hidden layers (possibly with several narrower layers running in parallel,
which has already been proven as a powerful design by Xie et al., 2017, in
their “ResNeXt” CNN design), and avoiding pattern-by-pattern learning, these
costs can be minimised.
It has been shown how to combine mini-batching with target space. The lack of
mini-batching has historically been a major Achilles heel in the adoption of
some previous sophisticated optimisers (for example conjugate gradients or
Levenberg-Marquardt), with very large datasets.
Target-space methods are particularly promising in recurrent neural-network
environments. In the examples given, problems with sequence lengths that were
previously intractable have been solved, and the LSTM results were surpassed
in a natural-language problem. This is despite the fact that LSTM networks
have extra features, such as memory gates, which make the learning task
easier, yet the target-space learning has still managed to make ordinary RNNs
outperform them. In the feed-forward problems given, target space has
consistently produced better generalisation in deeper neural networks.
The theoretical motivation for target space, in that using targets should be
able to untangle the cascades of changes caused during training, with a
beneficial outcome, appears to be feasible. Hence target space aims to
directly address the recognised “exploding-gradients” problem which exists in
deep learning.
Regarding a hypothetical future of neural networks being able to produce
simple programs similar to those formed by human programmers, we hypothesise
that whenever a neural network gets to a really interesting point of training,
then the neural activations will often all be very close to their firing
thresholds, and the exploding-gradients problem becomes really significant in
blocking further learning. For example if the neural network training process
had somehow successfully managed to build a series of interlocking XOR gates,
which were almost all working well together so as to implement a conventional
computer program out of those logic gates, then the scrambling of behaviour
from any potential infinitesimal weight change will always make learning
destabilise in weight space. The target-space approach is designed to be
helpful in these circumstances, and would seem to have more chance of making
further progress than a simple weight-space search would.
Our experimental results with recurrent neural networks over long time
sequences combined with data-processing outperform the equivalent LSTM
networks. Hence it seems that in those problems at least, the exploding-
gradients problem is more significant than vanishing gradients; at least when
Adam is allowed to accelerate the small gradients in target space. This is
particularly paradoxical when it is noted that the objective of the target-
space cascade untangling is to dampen down learning gradients even more, thus
amplifying the vanishing-gradients problem.
Many significant deep-learning innovations exist in prior published work.
These include the closely-related method of batch normalisation, plus modern
activation functions, optimisers, and weight-initialisation techniques. Many
of these are more computationally efficient than target space, but are maybe
slightly less effective; and some can be combined with target space.
Sophisticated neural architectures, such as LSTM, CNNs, and more recently,
attention models, Differentiable Neural Computers and Neural Turing Machines
(Graves et al., 2014, 2016), exist, which all add to neural-network
functionality, and which could all in-principle be trained in target space. So
in final conclusion, the target-space method seems to be a powerful additional
tool which has tremendous potential for the enhancement of deep learning.
## Appendix A Target-Space Algorithmic Complexity Calculations
In this appendix we derive the algorithmic complexity for the main target-
space algorithms. In these derivations, we ignore the computation of
activation functions, and matrix additions, assuming these are dwarfed by
matrix-multiplication operations.
### A.1 Computational Complexity for Fully-Connected Feed-forward networks
First we consider the main target-space algorithm for feed-forward neural
networks (i.e. Algorithms 2-3).
For a given layer $j$, the input matrix to that layer is ${A}_{{[0:{j})}}$,
the weight matrix is ${W}_{{[0:{j}]}}$ and the target matrix is $T_{j}$. For
brevity, we will denote these three matrices without subscripts, as $A$, $W$
and $T$. Let $n_{\mathrm{i}}$ be an initialism for the number of inputs to the
layer (i.e. the number of rows in $A$) and let $n_{\mathrm{o}}$ be the number
of outputs from the layer (i.e. the number of rows in $T$).
Since $A\in\mathbb{R}^{n_{\mathrm{i}}\times{\overline{n}_{\mathrm{b}}}}$, and
if ${\overline{n}_{\mathrm{b}}}>n_{\mathrm{i}}$, then direct multiplication to
form the Gramian $AA^{T}$ will take
${n_{\mathrm{i}}}^{2}{\overline{n}_{\mathrm{b}}}$ floating-point operations
(flops). Assuming matrix inversion takes roughly $n^{3}$ flops, and since the
Gramian is of shape $n_{\mathrm{i}}\times n_{\mathrm{i}}$, the formation of
$(AA^{T}+\lambda I)^{-1}$ will take a further $(n_{\mathrm{i}})^{3}$ flops.
The formation of the product with $A^{T}$ in equation (7) will take a further
$(n_{\mathrm{i}})^{2}{\overline{n}_{\mathrm{b}}}$ flops. Since
$T\in\mathbb{R}^{n_{\mathrm{o}}\times{\overline{n}_{\mathrm{b}}}}$, the
multiplication by $T$ in equation (6) will take a further
$n_{\mathrm{i}}n_{\mathrm{o}}{\overline{n}_{\mathrm{b}}}$ flops. Hence summing
these four terms gives the total time to form the pseudoinverse and calculate
the weight matrix in (6), as
$(n_{\mathrm{i}})^{3}+2(n_{\mathrm{i}})^{2}{\overline{n}_{\mathrm{b}}}+n_{\mathrm{i}}n_{\mathrm{o}}{\overline{n}_{\mathrm{b}}}$.
If however ${\overline{n}_{\mathrm{b}}}<n_{\mathrm{i}}$, then the matrix $A$
is taller than it is wide, and (7) can be rearranged using the Woodbury matrix
identity into an equivalent but more efficient form:
$A^{\dagger}:=(A^{T}{A}+\lambda I)^{-1}{A}^{T}.$ (23)
If this version is used, then the computational complexity is identically
derived, resulting in the same flop-count expression but with all occurrences
of $n_{\mathrm{i}}$ and ${\overline{n}_{\mathrm{b}}}$ swapped.
Hence the resulting overall flop count for calculating $W$ by a pseudoinverse,
assuming the faster of the two equations (7) and (23) is used, is
$\text{Flop count for $W$
calculation}=\begin{cases}(n_{\mathrm{i}})^{3}+2(n_{\mathrm{i}})^{2}{\overline{n}_{\mathrm{b}}}+n_{\mathrm{i}}n_{\mathrm{o}}{\overline{n}_{\mathrm{b}}}&\text{if
$n_{\mathrm{i}}<{\overline{n}_{\mathrm{b}}}$}\\\
({\overline{n}_{\mathrm{b}}})^{3}+2({\overline{n}_{\mathrm{b}}})^{2}n_{\mathrm{i}}+n_{\mathrm{i}}n_{\mathrm{o}}{\overline{n}_{\mathrm{b}}}&\text{otherwise}\end{cases}$
(24)
Once the $W$ matrix for the layer is formed, the feed-forward calculation of
the product $S_{j}=WA$ takes place, which is the same computational complexity
as is required in ordinary weight space, i.e. requiring
$\text{Flop count for $S_{j}$
calculation}=n_{\mathrm{i}}n_{\mathrm{o}}{\overline{n}_{\mathrm{b}}}$ (25)
If it can be assumed that the number of nodes in each layer of the neural
network is approximately the same, so that $d_{j}=\overline{d}$ for all $j$,
and no shortcut connections are present, then we can assume that
$n_{\mathrm{i}}\approx n_{\mathrm{o}}\approx\overline{d}$ (ignoring the single
input from the bias node). If, as advocated in Section 3.1, we further assume
that the size of the batch ${\overline{n}_{\mathrm{b}}}$ is larger than
$\overline{d}$ (so that also ${\overline{n}_{\mathrm{b}}}>n_{\mathrm{i}}$),
then summing the expressions in (24) and (25) and simplifying shows that the
flop count for each layer of the target space Alg. 2 is bounded above by
$4\overline{d}^{2}{\overline{n}_{\mathrm{b}}}$. In comparison, the weight-
space forward-pass algorithm for a single layer is just given by (25), i.e.
$\overline{d}^{2}{n_{\mathrm{b}}}$ flops. Hence the ratio of computation
between target space and weight space is approximately upper-bounded by
$(4{\overline{n}_{\mathrm{b}}}/{n_{\mathrm{b}}})$. Since automatic
differentiation produces backward computations of similar algorithmic
complexity as to the forward pass, the overall computation ratios for forward-
and-backward passes between target space and weight space, when summed over
all layers, is still approximately
$(4{\overline{n}_{\mathrm{b}}}/{n_{\mathrm{b}}})$.
### A.2 Computational Complexity for a CNN layer in Target Space
We now derive the computational complexity of the CNN target-space layer.
Notate the convolutional kernel width and heights by $k_{\mathrm{w}}$ and
$k_{\mathrm{h}}$ respectively, and the number of input and output channels by
$n_{\mathrm{ic}}$ and $n_{\mathrm{oc}}$ respectively. Let $n_{\mathrm{patch}}$
be the number of image patches to be taken from each image. Since the number
of inputs operated on by the flattened $W$ matrix is
$n_{\mathrm{i}}=k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}$, and the number
of outputs is $n_{\mathrm{o}}=n_{\mathrm{oc}}$, and the number of columns in
the patches matrix $A$ is
$n_{\mathrm{b^{\prime}}}={\overline{n}_{\mathrm{b}}}n_{\mathrm{patch}}$, then
substituting these factors into (24) gives a total flop count for the
formation of $W$ as:
CNN Flop count for $W$ formation
$\displaystyle=\begin{cases}(k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}})^{3}+2(k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}})^{2}n_{\mathrm{b^{\prime}}}+k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}n_{\mathrm{oc}}n_{\mathrm{b^{\prime}}}&\text{if
$k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}<n_{\mathrm{b^{\prime}}}$}\\\
(n_{\mathrm{b^{\prime}}})^{3}+2(n_{\mathrm{b^{\prime}}})^{2}k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}+k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}n_{\mathrm{oc}}n_{\mathrm{b^{\prime}}}&\text{otherwise}\end{cases}$
(26)
In contrast, the weight-space CNN forward pass only requires the formation of
$S$, where the flop count is given by (25), which equates to only
$k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}}n_{\mathrm{oc}}{n_{\mathrm{b}}}n_{\mathrm{patch}}$
flops.
If we argue like in Section A.1 that $n_{\mathrm{b^{\prime}}}>n_{\mathrm{i}}$
(which is quite probable with the large number of image patches being
processed by a CNN), and $n_{\mathrm{oc}}\approx n_{\mathrm{ic}}$, then the
flop count in target space is bounded above by
CNN Flop count for $W$ formation $\displaystyle\lessapprox
3(k_{\mathrm{h}}k_{\mathrm{w}}n_{\mathrm{ic}})^{2}n_{\mathrm{b^{\prime}}}+k_{\mathrm{h}}k_{\mathrm{w}}(n_{\mathrm{ic}})^{2}n_{\mathrm{b^{\prime}}}$
$\displaystyle=(3(k_{\mathrm{h}}k_{\mathrm{w}})+1)k_{\mathrm{h}}k_{\mathrm{w}}(n_{\mathrm{ic}})^{2}{\overline{n}_{\mathrm{b}}}n_{\mathrm{patch}}$
(27)
and hence the ratio of the flop count in target space to that in weight space
is bounded above by approximately
$(3(k_{\mathrm{h}}k_{\mathrm{w}})+1){\overline{n}_{\mathrm{b}}}/{n_{\mathrm{b}}}$.
## Appendix B Derivation of Algorithm 3
### B.1 Preliminary Definitions
Single-Entry Matrix:
Define $[J^{ij}]$ to be the single-entry matrix with element at row $i$ and
column $j$ equal to $\begin{cases}1&\text{if $m=i$ and $n=j$}\cr
0&\text{otherwise}\end{cases}$ (Petersen and Pedersen, 2012). This is useful
when differentiating a matrix with respect to one of its elements, since
$\frac{\partial{A}}{\partial{A^{ij}}}=[J^{ij}]$, with $[J^{ij}]$ having the
same dimensions as $A$.
Raised Indices Notation:
Define upper indices (without parentheses) after a matrix variable to indicate
the matrix element, so that for example $A^{ij}$ is the element of $A$ with
row index $i$ and column index $j$.
Define raised indices $[ij]$ in square brackets after a scalar function
$f(i,j)$ to mean the whole matrix whose element at row $i$ and column $j$ is
$f(i,j)$. For example, $\left({A^{ij}}\right)^{[ij]}\equiv A$, and
$\left({A^{ji}}\right)^{[ij]}\equiv A^{T}$.
Frobenius Inner Product, $\left<A,B\right>_{\\!F}$
For two $m\times n$ matrices $A$ and $B$, define
$\left<A,B\right>_{\\!F}:=\sum_{\forall i,j}A^{ij}B^{ij}$. This inner product
is useful when using the chain rule; for example, if $X$, $Y$ and $Z$ are
matrices with $X=X(Y)$ and $Y=Y(Z)$ then
$\frac{\partial{X^{mn}}}{\partial{Z^{ij}}}=\left<\frac{\partial{X^{mn}}}{\partial{Y}},\frac{\partial{Y}}{\partial{Z^{ij}}}\right>_{\\!F}$.
Furthermore, if $L(X)$ is a scalar function, then
$\frac{\partial{L}}{\partial{Y}}=\left({\left<\frac{\partial{X}}{\partial{Y^{ij}}},\frac{\partial{L}}{\partial{X}}\right>_{\\!F}}\right)^{[ij]}$.
### B.2 Basic Lemma for Combining Frobenius Inner Product with Single-entry
Matrix
A useful result for combining the inner product with $[J^{ij}]$ is
$\left({\left<A[J^{ij}]B,C\right>_{\\!F}}\right)^{[ij]}=A^{T}CB^{T}$ (28)
since
$\left<A[J^{ij}]B,C\right>_{\\!F}=\sum_{mn}(A[J^{ij}]B)^{mn}C^{mn}=\sum_{mn}\left(\sum_{pq}A^{mp}[J^{ij}]^{pq}B^{qn}\right)C^{mn}$
$=\sum_{mn}(A^{mi}B^{jn}C^{mn})=(A^{T}CB^{T})^{ij}$.
Similarly,
$\left({\left<A[J^{ij}]^{T}B,C\right>_{\\!F}}\right)^{[ij]}=BC^{T}A$ (29)
### B.3 Matrix Differentiation
Differentiating a scalar by a matrix gives an identically dimensioned matrix,
e.g.
$\left(\frac{\partial{L}}{\partial{X}}\right)^{ij}:=\frac{\partial{L}}{\partial{X^{ij}}}$.
Similarly for differentiating a matrix by a scalar:
$\left(\frac{\partial{X(a)}}{\partial{a}}\right)^{ij}:=\frac{\partial{X^{ij}(a)}}{\partial{a}}$.
Matrix differentiation follows the usual product rule:
$\displaystyle\frac{\partial{AB}}{\partial{X^{mn}}}$
$\displaystyle=\frac{\partial{A}}{\partial{X^{mn}}}B+A\frac{\partial{B}}{\partial{X^{mn}}}.$
(30)
For example, if $A$,$B$ and $C$ are constant matrices, then
$\displaystyle\frac{\partial{AXBXC}}{\partial{X^{mn}}}$
$\displaystyle=A[J^{mn}]BXC+AXB[J^{mn}]C.$
The derivative of an inverse matrix $A^{-1}$ is
$\frac{\partial{A^{-1}}}{\partial{A^{ij}}}=-A^{-1}[J^{ij}]A^{-1}$ (Brookes,
2011). Combining this with the product rule gives
$\displaystyle\frac{\partial{(BB^{T}+\lambda I)^{-1}}}{\partial{B^{ij}}}$
$\displaystyle=-(BB^{T}+\lambda
I)^{-1}\left([J^{ij}]B^{T}+B[J^{ij}]^{T}\right)(BB^{T}+\lambda I)^{-1}$
$\displaystyle=-(BB^{T}+\lambda
I)^{-1}[J^{ij}]B^{\dagger}-(B^{\dagger})^{T}[J^{ij}]^{T}(BB^{T}+\lambda
I)^{-1}$ (31)
And so,
$\displaystyle\frac{\partial{B^{\dagger}}}{\partial{B^{ij}}}=$
$\displaystyle\frac{\partial{B^{T}(BB^{T}+\lambda I)^{-1}}}{\partial{B^{ij}}}$
(by (7)) $\displaystyle=$ $\displaystyle[J^{ij}]^{T}(BB^{T}+\lambda
I)^{-1}-B^{T}\frac{\partial{(BB^{T}+\lambda I)^{-1}}}{\partial{B^{ij}}}$ (by
product rule) $\displaystyle=$ $\displaystyle[J^{ij}]^{T}(BB^{T}+\lambda
I)^{-1}-\big{(}B^{\dagger}[J^{ij}]B^{\dagger}+B^{\dagger}B[J^{ij}]^{T}(BB^{T}+\lambda
I)^{-1}\big{)}$ (by (31)) $\displaystyle=$
$\displaystyle(I-B^{\dagger}B)[J^{ij}]^{T}(BB^{T}+\lambda
I)^{-1}-B^{\dagger}[J^{ij}]B^{\dagger}$ (32)
### B.4 Ordered Partial Derivatives
Define the notation $\frac{\partial{}}{\partial{{}^{*}}}$ to be the ordered
partial derivatives (Werbos, 1974), which take into account cascading changes
to all later layers’ weights and activations by Algorithm 2. For example
$\frac{\partial{\vec{w}}}{\partial{{}^{*}A_{j}^{mn}}}$ describes how all the
layers’ weights would change according to Algorithm 2 if a small perturbation
was forced to occur to $A_{j}^{mn}$.
For a layer $j$, define
$\delta{A_{j}}:=\left({\frac{\partial{\vec{w}}}{\partial{{}^{*}A_{j}^{mn}}}\frac{\partial{L}}{\partial{\vec{w}}}}\right)^{[mn]}$.
This matrix accounts for what effect a small change to $A_{j}$ will have on
$L$, solely through the effect of cascading changes to later layers’ weights
via alg. 2. Note that $\delta{A}$ is subtly different from
$\frac{\partial{L}}{\partial{{}^{*}A}}$ since at the final layer
$\delta{A_{n_{\mathrm{L}}}}=0$ (since there are no later layers whose weights
can change), but
$\frac{\partial{L}}{\partial{{}^{*}A_{n_{\mathrm{L}}}}}=\frac{\partial{L}}{\partial{Y}}\neq
0$. Similarly, define
$\delta{S_{j}}:=\left({\frac{\partial{\vec{w}}}{\partial{{}^{*}S_{j}^{mn}}}\frac{\partial{L}}{\partial{\vec{w}}}}\right)^{[mn]}$
and
$\delta{{W}_{{[0:{j}]}}}:=\left({\frac{\partial{\vec{w}}}{\partial{{}^{*}{W}_{{[0:{j}]}}^{mn}}}\frac{\partial{L}}{\partial{\vec{w}}}}\right)^{[mn]}$.
### B.5 Derivation of Algorithm 3
Define $\delta{{A}_{{[0:{j})}}}$ to be the composite of $\delta{A_{j}}$
matrices in the same way that the ${A}_{{[0:{j})}}$ matrices are composed of
$A_{j}$ matrices, analogous to Eq. (2).
The matrices ${W}_{{[0:{j}]}}$, ${A}_{{[0:{j})}}$, $T_{j}$, $Y_{j}$, $A_{j}$
and $S_{j}$ are for an arbitrary layer $j$. Throughout the following, all
these matrices refer to the same subscripted value of $j$, therefore we omit
this subscript to ease presentation. To avoid the clash of variable names
between $A_{j}$ and ${A}_{{[0:{j})}}$, we define $B\equiv{A}_{{[0:{j})}}$ and
$A\equiv A_{j}$ as shorthand.
First we give useful results for $\frac{\partial{W}}{\partial{B^{mn}}}$ and
$\delta{W}$:
$\displaystyle\frac{\partial{W}}{\partial{B^{mn}}}=$ $\displaystyle
T\big{[}(I-B^{\dagger}B)[J^{mn}]^{T}(BB^{T}+\lambda
I)^{-1}-B^{\dagger}[J^{mn}]B^{\dagger}\big{]}$ (by (6) and (32))
$\displaystyle=$ $\displaystyle(T-S)[J^{mn}]^{T}(BB^{T}+\lambda
I)^{-1}-W[J^{mn}]B^{\dagger}$ (by (6) and (4)) (33)
The derivation for
$\delta{W}=\left({\frac{\partial{\vec{w}}}{\partial{{}^{*}{W}_{{[0:{j}]}}^{mn}}}\frac{\partial{L}}{\partial{\vec{w}}}}\right)^{[mn]}$
in Equation (34) starts by adding two terms. The first term,
$\frac{\partial{L}}{\partial{W}}$, accounts for the contribution from the
changing weights in that particular layer. The second term,
$\left({\left<\delta{S},\frac{\partial{S}}{\partial{W^{pq}}}\right>_{\\!F}}\right)^{[pq]}$,
accounts for the cascading changes to all later layers’ weights (by the
definition of $\delta{S}$).
$\displaystyle\delta{W}=$
$\displaystyle\frac{\partial{L}}{\partial{W}}+\left({\left<\delta{S},\frac{\partial{S}}{\partial{W^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
$\displaystyle=$
$\displaystyle\frac{\partial{L}}{\partial{W}}+\left({\left<\delta{S},[J^{mn}]B\right>_{\\!F}}\right)^{[mn]}$
(by (4) and (30)) $\displaystyle=$
$\displaystyle\frac{\partial{L}}{\partial{W}}+(\delta{S})B^{T}$ (by (28)) (34)
To derive a formula that calculates $\frac{\partial{L^{\prime}}}{\partial{T}}$
for a particular layer given $\frac{\partial{L}}{\partial{\vec{w}}}$, we first
note that changing $T^{mn}$ for one layer will initially just change the
weights of that layer, according to $\frac{\partial{W}}{\partial{T^{mn}}}$.
Then cascading changes to the later layers’ weights will occur via Algorithm
2, as a consequence of this initial single layer’s change of weights, and
therefore all these cascading effects are represented by $\delta{W}$.
Combining these two factors with the Frobenius inner-product gives:
$\displaystyle\frac{\partial{L^{\prime}}}{\partial{T}}=$
$\displaystyle\left({\left<\delta{W},\frac{\partial{W}}{\partial{T^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
$\displaystyle=$
$\displaystyle\left({\left<\delta{W},[J^{mn}]B^{\dagger}\right>_{\\!F}}\right)^{[mn]}$
(by (6) and (30)) $\displaystyle=$
$\displaystyle\left(\frac{\partial{L}}{\partial{W}}+(\delta{S})B^{T}\right)(B^{\dagger})^{T}$
(by (28) and (34)) (35)
This requires calculation of the $\delta{S}$ matrices for each layer. Since
$A^{mn}=g(S^{mn})$, the chain rule gives
$\delta{S}=\delta{A}\odot g^{\prime}(S)$ (36)
The derivation for $\delta{B}$ is given in Equation
(LABEL:eqn:proofLDFR:dEdB). The first line of this derivation consists of two
terms which are present, respectively, because changing $B$ will change the
weights for that layer directly (via the equation $W=TB^{\dagger}$), and will
also change the sums for that layer directly (via the equation $S=WB$). The
effects of these two changes are what the terms $\delta{W}$ and $\delta{S}$,
respectively, are defined to represent.
$\displaystyle\delta{B}=$
$\displaystyle\left({\left<\delta{W},\frac{\partial{W}}{\partial{B^{mn}}}\right>_{\\!F}+\left<\delta{S},\frac{\partial{S}}{\partial{B^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
$\displaystyle=$
$\displaystyle\left({\left<\delta{W},\left((T-S)[J^{mn}]^{T}(BB^{T}+\lambda
I)^{-1}-W[J^{mn}]B^{\dagger}\right)\right>_{\\!F}+\left<\delta{S},W[J^{mn}]\right>_{\\!F}}\right)^{[mn]}$
(by (33), (4) and (30)) $\displaystyle=$ $\displaystyle(BB^{T}+\lambda
I)^{-1}(\delta{W})^{T}(T-S)-W^{T}(\delta{W})(B^{\dagger})^{T}+W^{T}\delta{S}$
(by (28) and (29)) $\displaystyle=$ $\displaystyle
W^{T}\left(\delta{S}-\frac{\partial{L^{\prime}}}{\partial{T}}\right)+\bigg{[}(BB^{T}+\lambda
I)^{-1}\left(\frac{\partial{L}}{\partial{W}}+(\delta{S})B^{T}\right)^{T}(T-S)\bigg{]}$
(by (35) and (34))
This enables us to find $\delta{B}$ from $\delta{S}$ for a particular layer.
Since $\delta{{A}_{{[0:{j})}}}\equiv\delta{B}$ is composed of
$\delta{A_{j-1}}$, and $\delta{A_{n_{\mathrm{L}}}}=0$ we can calculate the
$\delta{A}$ matrices backwards, layer by layer. Thus equations (35), (36) and
(LABEL:eqn:proofLDFR:dEdB) give lines 4, 3, and 5 of Alg. 3 respectively.
## Appendix C Proof that a stationary point in target space corresponds to a
stationary point in weight space
In this appendix we show that if ${\vec{\tau}}^{*}$ is a stationary point for
the target-space problem, i.e.
$\left.\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}\right|_{\vec{\tau}={\vec{\tau}}^{*}}=0$,
then the corresponding vector of weights ${\vec{w}}^{*}$ obtained from
${\vec{\tau}}^{*}$ through Algorithm 2 is a stationary point for the resulting
weight-space problem, i.e.
$\left.\frac{\partial{L}}{\partial{\vec{w}}}\right|_{\vec{w}={\vec{w}}^{*}}=0$.
After a preliminary definition and two lemmas, the main theorem and proof
follows.
Definition: Let
$\displaystyle\qquad A^{\ddagger}:=A+\lambda\,(A^{+})^{T}.$ (38)
where $A^{+}$ denotes the (non-regularised) Moore-Penrose pseudoinverse (Golub
and Van Loan, 2013, Section 5.5.2).
###### Lemma 1
For any real-valued matrix $A$, the following identity holds:
$A\,A^{\dagger}A^{\ddagger}=A$.
Proof We use the singular value decomposition (SVD) to prove this. Let the
shape of $A$ be $m\times n$, and the rank of $A$ be $r$. Let the full SVD of
$A$ be given by
$\displaystyle A=USV^{T},$ (39)
where the matrices $U\in\mathbb{R}^{m\times m}$, $S\in\mathbb{R}^{m\times n}$
and $V\in\mathbb{R}^{n\times n}$, the only non-zero elements of $S$ are on its
leading diagonal, and where $U$ and $V$ are orthogonal. Since $A$ is rank $r$,
the matrix $S$ will have its first $r$ diagonal elements as non-zero and the
remaining elements all zero. Hence we can partition $S$ into block-matrix form
as follows:
$\displaystyle S=\begin{pmatrix}\Sigma&0\\\ 0&0\end{pmatrix},$ (40)
where $\Sigma$ is a diagonal matrix of shape $r\times r$, and the zeros are
rectangular matrices of appropriate shape so as to make
$S\in\mathbb{R}^{m\times n}$. Since $\Sigma$ is square and full rank, its
Moore-Penrose pseudoinverse simplifies into an ordinary inverse:
$\displaystyle\Sigma^{+}=\Sigma^{-1}.$ (41)
Using the SVD, and repeatedly cancelling orthogonal self-products such as
$U^{T}U$ and $V^{T}V$, we can write:
$\displaystyle A^{\,\dagger\,}$ $\displaystyle=VS^{T}(SS^{T}+\lambda
I)^{-1}U^{T}$ (by (7)) (42) $\displaystyle A^{+}$ $\displaystyle=VS^{+}U^{T}$
(Moore-Penrose SVD) (43) $\displaystyle A^{\,\ddagger\,}$
$\displaystyle=U(S+\lambda S^{+})V^{T}$ (by (38) and (43)) (44)
Substituting these into the left-hand side of the lemma’s identity, and
cancelling orthogonal self-products, gives,
$\displaystyle A\,A^{\dagger}A^{\ddagger}$
$\displaystyle=USS^{T}(SS^{T}+\lambda I)^{-1}(S+\lambda S^{+})V^{T}$
$\displaystyle=U\begin{pmatrix}\Sigma^{2}&0\\\
0&0\end{pmatrix}\begin{pmatrix}(\Sigma^{2}+\lambda I)^{-1}&0\\\
0&\lambda^{-1}I\end{pmatrix}\begin{pmatrix}\Sigma+\lambda\Sigma^{-1}&0\\\
0&0\end{pmatrix}V^{T}$ (by (40) and (41))
$\displaystyle=U\begin{pmatrix}\Sigma^{2}(\Sigma^{2}+\lambda
I)^{-1}(\Sigma+\lambda\Sigma^{-1})&0\\\ 0&0\end{pmatrix}V^{T}$ (block
multiplication) $\displaystyle=U\begin{pmatrix}\Sigma(\Sigma^{2}+\lambda
I)^{-1}(\Sigma^{2}+\lambda\Sigma\Sigma^{-1})&0\\\ 0&0\end{pmatrix}V^{T}$
(commute diagonal matrices) $\displaystyle=A$ (by (39) and (40))
This proves the lemma.
Remark: Lemma 1 is analogous to the identity for the non-regularised Moore-
Penrose pseudoinverse given by $A\,A^{+}A=A$ (Golub and Van Loan, 2013,
Section 5.5.2), which also holds for any real-valued matrix $A$.
###### Lemma 2
When the weights are calculated from the targets by Algorithm 2, given any
layer $j$, such that $j={n_{\mathrm{L}}}$ or
$\frac{\partial{L}}{\partial{{W}_{{[0:{k}]}}}}=0$ for all $k>j$, we have
$\frac{\partial{L^{\prime}}}{\partial{T_{j}}}=0\Rightarrow\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}=0$.
Proof First, let us write explicitly the derivative of the loss function $L$
with respect to the weights of layer $j$:
$\displaystyle\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}$
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}},\frac{\partial{S_{j}}}{\partial{{W}_{{[0:{j}]}}^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}},[J^{mn}]{A}_{{[0:{j})}}\right>_{\\!F}}\right)^{[mn]}$
(by (4) and (30))
$\displaystyle=\frac{\partial{L}}{\partial{S_{j}}}\,{A}_{{[0:{j})}}^{T},$ (by
(28)) (45)
And similarly, explicitly state its derivative with respect to the targets of
the same layer:
$\displaystyle\frac{\partial{L^{\prime}}}{\partial{T_{j}}}$
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}}+\sum_{k>j}\left({\left<\frac{\partial{L}}{\partial{{W}_{{[0:{k}]}}}},\frac{\partial{{W}_{{[0:{k}]}}}}{\partial{S_{j}^{pq}}}\right>_{\\!F}}\right)^{[pq]},\frac{\partial{S_{j}}}{\partial{T_{j}^{mn}}}\right>_{\\!F}}\right)^{[mn]}.$
(46)
In this equation, instead of following (35), we have used the chain rule to
produce an expression that explicitly connects
$\frac{\partial{L^{\prime}}}{\partial{T}}$ to
$\frac{\partial{L}}{\partial{S}}$. The summation in (46) evaluates to
$\delta{S}_{j}$ defined in Section B.4, which accounts for the effects of the
weights in all later layers $k>j$ which will change by Alg. 2 as a result of a
change to $T_{j}$. Following on from the initial assumptions of this lemma,
which were that either $j={n_{\mathrm{L}}}$ or the condition
$\frac{\partial{L}}{\partial{{W}_{{[0:{k}]}}}}=0$ holds for all $k>j$,
therefore the summation vanishes and (46) reduces to:
$\displaystyle\frac{\partial{L^{\prime}}}{\partial{T_{j}}}$
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}},\frac{\partial{S_{j}}}{\partial{T_{j}^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}},\frac{\partial{\left(T_{j}{\left({A}_{{[0:{j})}}\right)^{\dagger}}{A}_{{[0:{j})}}\right)}}{\partial{T_{j}^{mn}}}\right>_{\\!F}}\right)^{[mn]}$
(by (4) and (6))
$\displaystyle=\left({\left<\frac{\partial{L}}{\partial{S_{j}}},[J^{mn}]{\left({A}_{{[0:{j})}}\right)^{\dagger}}{A}_{{[0:{j})}}\right>_{\\!F}}\right)^{[mn]}$
(by (30))
$\displaystyle=\frac{\partial{L}}{\partial{S_{j}}}\,{A}_{{[0:{j})}}^{T}\,\left({A}_{{[0:{j})}}^{T}\right)^{\dagger}.$
(by (28)) (47)
Aiming from a contradiction, let us assume that
$\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}\neq 0$. Then one can choose an
appropriately sized column vector $u$ such that
$\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}\,u\neq 0$. We now consider a
second vector $v=\big{(}{A}_{{[0:{j})}}^{T}\big{)}^{\ddagger}u$, using (38),
and write:
$\displaystyle\frac{\partial{L^{\prime}}}{\partial{T_{j}}}\,v$
$\displaystyle=\frac{\partial{L}}{\partial{S_{j}}}\,{A}_{{[0:{j})}}^{T}\,\big{(}{A}_{{[0:{j})}}^{T}\big{)}^{\dagger}\,v$
(by (47)) (48)
$\displaystyle=\frac{\partial{L}}{\partial{S_{j}}}\,{A}_{{[0:{j})}}^{T}\,\big{(}{A}_{{[0:{j})}}^{T}\big{)}^{\dagger}\,\big{(}{A}_{{[0:{j})}}^{T}\big{)}^{\ddagger}\,u$
(by the definition of $v$)
$\displaystyle=\frac{\partial{L}}{\partial{S_{j}}}\,{A}_{{[0:{j})}}^{T}\,u$
(by Lemma 1) $\displaystyle=\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}\,u.$
(by (45)) (49)
While we initially assumed that the last line (49) is non-zero, the first line
(48) must be zero, due to
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$. This contradiction
proves the lemma.
###### Theorem 3
When the weights are calculated from the targets by Algorithm 2, we have
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0\implies\frac{\partial{L}}{\partial{\vec{w}}}=0$.
Proof We shall prove this result by induction, by first showing it holds for
the last layer (used as the base case), and then showing that if it holds for
all subsequent layers then it must also hold for the current layer (the
inductive step).
Lemma 2 explicitly handles the case where $j={n_{\mathrm{L}}}$, thus the base-
case claim, that $\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$ implies
$\frac{\partial{L}}{\partial{{W}_{{[0:{{n_{\mathrm{L}}}}]}}}}=0$, is true.
Next we consider the inductive step, i.e. that if
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$ and
$\frac{\partial{L}}{\partial{{W}_{{[0:{k}]}}}}=0$ for all $k>j$, then
$\frac{\partial{L}}{\partial{{W}_{{[0:{j}]}}}}$ must be zero. Again, Lemma 2
applies here, since it explicitly applies to
$\frac{\partial{L}}{\partial{{W}_{{[0:{k}]}}}}=0$ for all $k>j$, and therefore
the inductive step is also true. This completes the proof by induction.
This final theorem concludes the proof that
$\frac{\partial{L^{\prime}}}{\partial{\vec{\tau}}}=0$ implies
$\frac{\partial{L}}{\partial{\vec{w}}}=0$, i.e. that a stationary point for
the target-space problem is also a stationary point for the corresponding
weight-space problem obtained through Algorithm 2.
## References
|
# A study of the hydrostatic mass bias dependence and evolution within The
Three Hundred clusters
Giulia Gianfagna,1,2 Elena Rasia,3,4 Weiguang Cui,5,6 Marco De Petris,2
Gustavo Yepes,6,7 Ana Contreras-Santos,6 and Alexander Knebe,6,7,8
1INAF, Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Rome, Italy
2Dipartimento di Fisica, Sapienza Universitá di Roma, Piazzale Aldo Moro,
5-00185 Roma, Italy
3IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
4INAF Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34131, Trieste,
Italy
5Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, UK
6Departamento deFísica Teórica, Módulo 15, Facultad de Ciencias, Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
7Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de
Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
8International Centre for Radio Astronomy Research, University of Western
Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
E-mail:
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We use a set of about 300 simulated clusters from The Three Hundred Project to
calculate their hydrostatic masses and evaluate the associated bias by
comparing them with the true cluster mass. Over a redshift range from 0.07 to
1.3, we study the dependence of the hydrostatic bias on redshift,
concentration, mass growth, dynamical state, mass, and halo shapes. We find
almost no correlation between the bias and any of these parameters. However,
there is a clear evidence that the scatter of the mass-bias distribution is
larger for low-concentrated objects, high mass growth, and more generically
for disturbed systems. Moreover, we carefully study the evolution of the bias
of twelve clusters throughout a major-merger event. We find that the
hydrostatic-mass bias follows a particular evolution track along the merger
process: to an initial significant increase of the bias recorded at the begin
of merger, a constant plateaus follows until the end of merge, when there is a
dramatic decrease in the bias before the cluster finally become relaxed again.
This large variation of the bias is in agreement with the large scatter of the
hydrostatic bias for dynamical disturbed clusters. These objects should be
avoided in cosmological studies because their exact relaxation phase is
difficult to predict, hence their mass bias cannot be trivially accounted for.
###### keywords:
methods: numerical – galaxies: clusters: general – galaxies: clusters:
intracluster medium – large-scale structure of Universe.
††pubyear: 2021††pagerange: A study of the hydrostatic mass bias dependence
and evolution within The Three Hundred clusters–A study of the hydrostatic
mass bias dependence and evolution within The Three Hundred clusters
## 1 Introduction
Galaxy clusters are the most massive gravitationally-bound structures in the
Universe and, as such, their mass plays a key role in the estimation of the
cosmological parameters. The majority of their mass is composed by Dark Matter
(DM, almost 80%), the remaining part is composed by stars (galaxies) and hot
gas which is diffused between the galaxies and it is called Intra-Cluster
Medium, ICM (see Kravtsov & Borgani, 2012, for a review).
The mass of these objects in observations can be estimated in several ways:
from galaxy kinematic, applying the virial theorem to the distribution of
cluster galaxies (for example Li et al., 2021), or directly from the velocity
distribution of cluster galaxies (Zwicky, 1937; Pratt et al., 2019; Tian et
al., 2021; Herná ndez-Lang et al., 2022); from scaling laws, linking their
total mass to several observable quantities, like X-rays luminosity or SZ
(Sunyaev-Zeldovich) brightness (Nagarajan et al., 2018; Bulbul et al., 2019);
from weak lensing (Von der Linden et al., 2014; Hoekstra et al., 2015; Okabe &
Smith, 2016; Artis et al., 2022) and, recently, also combining the mentioned
methods and using machine learning (Ntampaka et al., 2015; De Andres et al.,
2022). Another frequently used approach is to derive the cluster mass under
the assumption of the hydrostatic equilibrium (HE). This method uses
temperature, density and pressure profiles of hot gas extracted from X-rays
alone or combined with SZ effect observations (see Pratt et al. 2019 for a
recent review). Three assumptions are the basis of the HE: the ICM must trace
the cluster potential well, it must be spherically symmetric, and its pressure
must be purely thermal. As shown in a recent review by Gianfagna et al.
(2021), the HE mass evaluated in simulated samples is on average
underestimating the true mass. In this paper we investigate more this subject
and in particular the dependence of the bias by several other parameters that
characterise the cluster dynamical state.
Numerical simulation studies find that the bias has a negligible dependence
with the redshift (Piffaretti & Valdarnini, 2008; Le Brun et al., 2017; Henson
et al., 2017; Gianfagna et al., 2021). On the contrary, observational data are
consistent with a decreasing mass bias along the redshift, see for instance
Salvati et al. (2019) and Wicker et al. (2022) on Planck clusters. This
dependence could be linked to a mass dependence of the bias, as clusters
observed at higher redshifts tend to have a higher mass. However, the mass
dependence is not detected in simulations, either using the spectroscopic-like
weighted temperature111Note that spectroscopic-like weighted temperature is
more similar to the X-ray temperature obtained from X-ray spectroscopic
analysis of observed data. (Mazzotta et al., 2004) to estimate the HE mass
(Piffaretti & Valdarnini, 2008; Le Brun et al., 2017; Henson et al., 2017;
Pearce et al., 2019; Ansarifard et al., 2020; Barnes et al., 2021) or the
mass-weighted temperature. Several authors also study the dependence of the
bias on the dynamical state, the general findings also pointed towards no
correlation, even if there is a hint that disturbed objects have a large
scatter with respect to the relaxed ones (Piffaretti & Valdarnini, 2008; Rasia
et al., 2012; Nelson et al., 2014; Henson et al., 2017; Ansarifard et al.,
2020). By comparing results in literature it appears that the hydrostatic mass
bias does not depend on various cosmological-simulation ingredients, such as
the baryon physics included in the simulations, the mass particles resolution
or the number of objects in the samples (Gianfagna et al., 2021). Due to this
consistency for simulated results, the discrepancy on the bias dependence on
redshift between simulation predictions and observation results is still
puzzling. This could be due to less numerous observational samples, due to a
different mass sampling at high redshift and due to the uncertainty in the
total physical mass estimation in observation.
The selection may also play a key role, for example the inclusion of disturbed
objects. Indeed, all of the HE assumptions are violated during major mergers
events which can induce strong dynamical perturbations, adding non-thermal
support to the equilibrium budget (Nelson et al., 2012; Rasia et al., 2012,
2013; Angelinelli et al., 2020; Sereno et al., 2021). Therefore, the
hydrostatic bias can significantly deviate from the expectation. In post-
merger situations, then the merging object might have already been destroyed
and thus it cannot be distinguished as a separate structure. Applying the
hydrostatic-equilibrium technique to these cases (assuming they are in a
relaxed state) will lead to a misinterpretation. Studying the extreme cases in
simulations, where the dynamical state is clearly identified, will allow us to
better understand the situations when the HE assumption is strongly violated
and how the HE mass bias is impacted by mergers.
The organisation of this paper is as follow: in Section 2 we introduce The
Three Hundred set of galaxy clusters, in Section 3 we give an overview on the
hydrostatic equilibrium method. In Section 4 the ICM radial profiles are
introduced, along with their analysis, and we introduce the parameters which
will be used to characterise the dynamical state, the accretion and the
geometry of clusters. In Section 5 we provide our the results, and we conclude
in Section 6.
## 2 The Three Hundred Project
This work is based on clusters simulated for The Three Hundred Project,
introduced in Cui et al. (2018). This project is based on the resimulations of
a set of 324 Lagrangian regions extracted from the MultiDark Planck (MDPL2)
dark-matter-only (DM) cosmological simulation (Klypin et al., 2016) with
different simulation codes. MDPL2 consists of a periodic cube of comoving
length $1h^{-1}$ Gpc and $3840^{3}$ N-body particles. The Lagrangian regions
are identified at $z=0$ around the most massive objects found in the parent
simulation with a comoving radius of $15h^{-1}$ Mpc. The particles in those
massive clusters and in their surrounding were traced back to $z=120$ to
generate the initial conditions for the zoomed-in re-simulations. The original
mass resolution was kept in the central region, and those particles were split
into dark matter and gas particles based on the cosmological baryon fraction
$\Omega_{b}$. The high-resolution dark-matter-particle mass is $1.87\times
10^{9}\rm M_{\odot}$, while the initial gas mass is $3.48\times 10^{8}\rm
M_{\odot}$. In the outer region, the dark matter particles were degraded with
multiple levels of mass refinements in several concentric shells. The 324
zoomed-in regions have been hydro-dynamically re-simulated with three
different baryon models: GADGET-X (Rasia et al., 2015), GADGET-MUSIC
(Sembolini et al., 2013) and GIZMO-Simba (Davé et al., 2019; Cui et al.,
2022). In this work we use the GADGET-X simulated clusters. This code includes
several radiative processes, like gas cooling, star formation, thermal
feedback from supernovae, chemical evolution and enrichment, super massive
black holes with AGN feedback (see Cui et al., 2018, for details).
The simulations assume a standard cosmological model according to the Planck
Collaboration et al. (2016) results: $h=0.6777$ for the reduced Hubble
parameter, $n=0.96$ for the primordial spectral index, $\sigma_{8}=0.8228$ for
the amplitude of the mass density fluctuations in a sphere of $8h^{-1}$ Mpc
comoving radius, $\Omega_{\Lambda}=0.692885$, $\Omega_{m}=0.307115$, and
$\Omega_{b}=0.048206$ respectively for the density parameters of dark energy,
dark matter, and baryonic matter.
### 2.1 Sample
For the analysis presented here, we select the most massive cluster for each
region that does not include any low-resolution particles within its virial
radius. The evolution of the bias is tested at 9 redshifts, covering the
redshift range between $z=0.07$ and $z=1.32$. The mean mass and the number of
objects considered at each redshift are reported in Table 1 for the three
studied overdensities: $\Delta=$ 2500, 500 and 200. These indicate the radius
$R_{\rm\Delta}$ of a sphere whose density is either 2500, 500 or 200 times the
critical density of the Universe at the consider cosmic time:
$\rho_{\rm crit}=(3/8\pi G)H_{0}^{2}[\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}],$
(1)
where $H_{0}$ is the Hubble constant and G the gravitational constant.
Table 1: The redshifts analysed in this work are written in the first column. The number of objects are in the second column. In the third, fourth and fifth columns are the mean masses at the three nominal overdensities 2500, 500, 200 with units of $10^{14}M_{\odot}$. $z$ | $N$ | $<M_{2500}>$ | $<M_{500}>$ | $<M_{200}>$
---|---|---|---|---
| | $\rm[10^{14}M_{\odot}]$ | $\rm[10^{14}M_{\odot}]$ | $\rm[10^{14}M_{\odot}]$
1.32 | 277 | 0.47 | 1.36 | 1.99
1.22 | 281 | 0.55 | 1.58 | 2.31
0.99 | 304 | 0.78 | 2.26 | 3.29
0.78 | 305 | 1.07 | 3.04 | 4.45
0.59 | 305 | 1.48 | 4.05 | 5.91
0.46 | 300 | 1.73 | 4.89 | 7.22
0.33 | 297 | 2.09 | 5.73 | 8.46
0.22 | 298 | 2.43 | 6.73 | 9.97
0.07 | 290 | 2.98 | 8.19 | 12.20
## 3 The hydrostatic equilibrium mass
As said in the introduction, the hydrostatic equilibrium hypothesis is often
at the basis of the procedure that lead to estimate the mass of galaxy
clusters from X-ray observations (Kravtsov & Borgani, 2012; Ettori et al.,
2013). This assumption foresees that the gas thermal pressure, which naturally
leads to expansion of the gas, is balanced by the gravitational forces, which,
on the other side, causes the systems to collapse. The equilibrium is
expressed as the equality between the gradient of the thermal pressure and
that of the gravitational potential and it is supposed to hold at every
concentric distance from the cluster centre. Connecting the gravitational
potential to the mass, under the spherical assumption, leads to the following
formula for the total mass inside a sphere of radius $r$:
$M_{\rm HE,SZ}(<r)=-\frac{r^{2}}{G\rho_{\rm g}(r)}\frac{{\rm d}P_{\rm
th}(r)}{{\rm d}\ r}$ (2)
where $\rho_{\rm g}$ and $P_{\rm th}$ are the density and the thermal pressure
of the gas, respectively. We will refer to this mass as $M_{\rm HE,SZ}$, since
the radial pressure profile can be derived from Compton-$y$ parameter maps
provided by clusters SZ observations at millimetre wavelengths.
Assuming the equation of state of an ideal gas, the cluster mass can also be
estimated with gas density and temperature separately:
$M_{\rm HE,X}(<r)=-\frac{rk_{\rm B}T(r)}{G\mu m_{\rm p}}\left[\frac{{\rm
d}\ln\rho_{\rm g}(r)}{{\rm d}\ln r}+\frac{{\rm d}\ln T(r)}{{\rm d}\ln
r}\right].$ (3)
We refer to this HE mass formulation as $M_{\rm HE,X}$ because historically
this expression was used in the analysis of X-rays observations (Ansarifard et
al., 2020). The detailed computation of the HE mass in our simulated sample
will be introduced in next section.
This mass is compared with the true total mass of the systems, obtained by
summing over all dark matter, stars and gas particle masses inside a fixed
aperture radius. The mass bias $b_{\rm SZ}$ or $b_{\rm X}$is defined as
$b=\frac{M_{\rm true}-M_{\rm HE}}{M_{\rm true}}.$ (4)
This bias can be either positive, when the HE mass is underestimating the true
cluster mass, or negative, when there is an overestimation of the true mass. A
null bias results in a perfect estimate of the true mass through HE.
## 4 Methods
In this work, we aim to estimate the two hydrostatic masses, as presented in
Eqs. (2) and (3). The ICM temperature, gas density and pressure radial
profiles play a fundamental role in the calculation of these masses. In this
section, we describe how we compute them in The Three Hundred simulations
together with other relevant quantities which we use to study the correlations
with the bias. The quantities linked to the study of the mass bias during a
major merger event will be described in Section 5.2.
### 4.1 The ICM profiles
The cluster profiles are extracted in logarithmically equal-spaced radial
bins, centred at the maximum density (mostly coinciding with the minimum of
the potential well, see also Cui et al. 2016; Ansarifard et al. 2020). Only
gas particles with density below the star-formation density threshold and with
temperature above 0.3 keV are considered as ICM for calculation, this is to
insure that we are selecting the hot particles which can generate X-ray or SZ
signals, in order to match the observations.
Once the profiles are extracted, we fit them with analytic models which will
be described below. We then use the analytical best-fitting curves to estimate
the hydrostatic masses. Following this strategy will allow to avoid the small
fluctuations that are presented in the 3D numerical profiles which strongly
impact the calculation of the derivatives in the hydrostatic-equilibrium mass
equations (see Gianfagna et al., 2021). On average, this still allow to well
fit the large fluctuations (bumps or deeps in the profiles), caused by
substructures, which should instead taken into account. This is valid
especially for the temperature and density models, that have the largest
number of parameters. The pressure profile model could instead cancel some
large fluctuation, due to the smaller number of parameters, leading to a non-
reliable SZ bias, but the two biases are compatible between each other (see
Section 5) and also the scatter is not significantly different. The best-fit
procedure of each profile is performed in the radial range [0.2-3]$\rm
R_{500}$. This range was chosen to include the three studied overdensities:
$\Delta=$ 2500, 500 and 200. We study the goodness of the fits using the
$\chi^{2}$ test and found that all the fits can be classified as good fits.
#### 4.1.1 Gas Density
The 3D gas density profiles are estimated as the total gas mass in a spherical
shell, divided by the shell volume: $\rho=\sum m_{\rm i}/V_{\rm shell}$. The
model chosen to fit this profile is the one proposed by Vikhlinin et al.
(2006):
$\rho_{g}(r)=\rho_{0}^{2}\frac{(r/r_{\rm d})^{-a}}{(1+(r/r_{\rm
d})^{2})^{3b-a/2}}\frac{1}{(1+(r/r_{\rm s})^{c})^{e/c}}+$ (5) $\hskip
28.45274pt+\frac{\rho_{02}^{2}}{(1+(r/r_{\rm d2})^{2})^{3b_{2}}},$
where we fix the parameter $c$ equal to 3 and we impose the following
limitation: $e<5$. The other 8 parameters are left free. Often in literature
this model is simplified by discarding the second beta model, here we keep it
to have a reliable fit also in the cluster region near $\rm R_{2500}$. For
reference, we report in Table 2 the medians of all best-fit parameters
computed from the sample at $z=0.07$.
Table 2: The medians and $16^{\rm th}-84^{\rm th}$ percentiles of best-fit free parameters in Eq.5 for gas density profiles and the reduced $\chi^{2}$. The results only for all $z=0.07$ clusters are exhibited here. The $c$ parameter is fixed to 3. $\rho_{0}$[$10^{13}\rm M_{\odot}/Mpc^{3}$] | $r_{d}$[Mpc] | $r_{s}$[Mpc] | $a$
---|---|---|---
$3.5^{+1.5}_{-1.3}$ | $0.71^{+0.20}_{-0.17}$ | $0.73^{+0.65}_{-0.29}$ | $1.2^{+1.2}_{-0.9}$
$b$ | $e$ | $\rho_{0,2}$[$10^{13}\rm M_{\odot}/Mpc^{3}$] | $r_{d,2}$[Mpc]
$2.6^{+0.4}_{-0.7}$ | $2.5^{+0.4}_{-0.6}$ | $2.4^{+0.6}_{-0.8}$ | $0.95^{+0.36}_{-0.28}$
$b_{2}$ | $\chi^{2}$ | |
$1.2^{+0.3}_{-0.2}$ | $1.1^{+0.8}_{-0.4}$ | |
#### 4.1.2 Gas Temperature
The mass-weighted temperature profile is estimated as a weighted average over
the gas particles in the same spherical shells used for the gas density:
$T=\frac{\sum_{i}(m_{i}T_{i})}{\sum_{i}m_{i}}.$ (6)
where $m_{i}$ and $T_{i}$ are the mass and temperature of $i^{th}$ gas
particle. Using the mass-weighted temperature is the optimal choice for
hydrostatic equilibrium studies which theoretically connect the temperature
with the gravitational mass (Biffi et al., 2014) and it is favorable with
respect to the spectroscopic-like temperature (Mazzotta et al., 2004). The
analytical model for the mass-weighted temperature used in this work was
introduced again by Vikhlinin et al. (2006):
$T(r)=T_{0}\ \frac{x+\tau}{x+1}\ \frac{(r/r_{\rm t})^{-\alpha}}{(1+(r/r_{\rm
t})^{\beta})^{\gamma/\beta}},$ (7) $x=(r/r_{\rm cool})^{\alpha_{\rm cool}}.$
All the 8 parameters are free to vary. Their medians for the $z=0.07$ clusters
are reported in Table 3.
Table 3: The medians and $16^{\rm th}$ and $84^{\rm th}$ percentiles of best-fit free parameters of Eq.7 and the reduced $\chi^{2}$ computed for all $z=0.07$ clusters. $T_{0}$[keV] | $\tau$ | $r_{t}$[Mpc] | $\alpha$ | $\beta$ | $\gamma$
---|---|---|---|---|---
$15.3^{+56.4}_{-10.3}$ | $0.37^{+0.32}_{-0.33}$ | $2.3^{+22.6}_{-1.29}$ | $0.29^{+0.71}_{-0.29}$ | $4.6^{+7.4}_{-4.6}$ | $1.3^{+2.2}_{-1.3}$
$r_{cool}$[Mpc] | $\alpha_{cool}$ | $\chi^{2}$ | | |
$1.4^{+1.5}_{-0.7}$ | $5.4^{+4.6}_{-2.8}$ | $1.4^{+0.4}_{-0.1}$ | | |
#### 4.1.3 Gas Pressure
The radial thermal pressure profile is measured from the gas density and
temperature of each gas particle. It is modelled by the generalised Navarro-
Frenk-White (gNFW) model (Nagai et al., 2007):
$P(r)=\frac{P_{0}}{x^{i}(1+x^{g})^{\frac{h-i}{g}}},$ (8)
where $x=r/r_{s}$ is a dimensionless radial distance normalised to the scale
radius, $r_{s}$. The parameters $h$ and $i$ are the slopes for outer and inner
region, respectively, and $g$ is the steepness of the transition between the
two regimes. This model has 4 free parameters, since $i=0.31$ is fixed (Arnaud
et al., 2010). The medians of the best-fit parameters for the $z=0.07$ sample
are listed in Table 4.
Table 4: The medians and $16^{\rm th}$ and $84^{\rm th}$ percentiles of best-fit free parameters of Eq.8 and the reduced $\chi^{2}$ computed for all $z=0.07$ clusters. The $i$ parameter is fixed equal to $0.31$. $P_{0}[10^{-2}\rm keV/cm^{3}]$ | $r_{s}$ [Mpc] | $g$ | $h$ | $\chi^{2}$
---|---|---|---|---
$2.6^{+5.5}_{-1.5}$ | $3.1^{+6.9}_{-2.5}$ | $1.1^{+1.3}_{-0.4}$ | $8.2^{+6.8}_{-4.0}$ | $1.1^{+0.8}_{-0.3}$
### 4.2 The NFW concentration
Similarly to the thermodynamical profiles, we compute the total density
profiles which include the contribution from all particles in the simulations
(dark matter, gas, stars). The total mass profile is fitted with a NFW model
(Navarro et al., 1997):
$M(<r)=M_{0}\left[\log(1+x)-\frac{x}{1+x}\right].$ (9)
As before the radial coordinate is related to the scale radius, $x=r/r_{s}$.
This parameter, together with the normalisation, is derived from a best-
fitting procedure applied in the radial range between 100 kpc and $R_{200}$,
which roughly reproduce a typical radial range used in weak-lensing analysis.
The concentration within R500 is defined from the scale radius:
$c_{500}=R_{500}/r_{s}$.
### 4.3 Relative mass growth
We straightforwardly define the cluster mass growth as the relative mass
difference between an initial, $t_{1}$, and a final time, $t_{2}$:
$\frac{\Delta M/M}{{\rm
d}t}=\frac{(M(t_{2})-M(t_{1}))/M(t_{1})}{[t_{2}-t_{1}]}.$ (10)
Specifically, we estimated the mass growth from the merger tree between
$z_{2}=0.33$ and $z_{1}=0.46$, so that $t_{2}-t_{1}$ corresponds to about 1
Gyr (precisely it is equal to $1.046$ Gyr). This interval corresponds to 5
simulation snapshots.
### 4.4 Dynamical state parameter
The dynamical state of the clusters has been inferred by the combination of
two indicators computed in 3D (Neto et al., 2007; Cui et al., 2017; Cialone et
al., 2018; De Luca et al., 2021):
* •
$f_{s}=M_{\rm sub}/M_{\Delta}$, which is the fraction of cluster mass included
in substructures inside a fixed overdensity;
* •
$\Delta_{\rm r}=|\textbf{r}_{\delta}-\textbf{r}_{\rm cm}|/R_{\Delta}$, which
is the offset between the positions of the maximum density peak and the centre
of mass of the cluster within $R_{\Delta}$ and normalised to that aperture
radius.
When correlating the dynamical-state and the HE mass bias, we evaluate all
quantities of interest at the same overdensity value. In order to have a
relaxed cluster, both indicators should be smaller than 0.1 (Cialone et al.,
2018) while they should be both greater than 0.1 to have a disturbed one. The
other cases are classified as intermediate or hybrid. The percentage of
relaxed clusters is almost 50% at low redshifts, and decreases to 30% at high
redshifts, while the percentage of disturbed clusters is 20% at low redshifts
and increases to 30% at high redshifts (De Luca et al., 2021). As previously
said, in our study we consider one combined parameter as in Haggar et al.
(2020) but derived exclusively from the two mentioned indicators as in De Luca
et al. (2021):
$\chi_{\rm DS}=\sqrt{\frac{2}{\left(\frac{\Delta_{\rm
r}}{0.1}\right)^{2}+\left(\frac{f_{s}}{0.1}\right)^{2}}}.$ (11)
Using the $\chi_{\rm DS}$ parameter is a continuous way to classify the state
of the cluster where the relaxed ones satisfy $\chi_{\rm DS}>1$. We use this
parameter to study the dependence of the dynamical state on the HE bias (see
Section 5.1.4). An extensive study of the relaxation state of The Three
Hundred clusters including its dependence from redshift is presented in De
Luca et al. (2021).
### 4.5 Triaxiality
In order to test how much the lack of spherical symmetry is affecting the HE
mass estimate, we calculate, using all particles within $R_{500}$, the three
axes of the inertia tensor as in (Vega-Ferrero et al., 2017). We then use the
ratio $c/a$ and the triaxiality of the halos:
$t=\frac{1-(b/a)^{2}}{1-(c/a)^{2}},$ (12)
where $a,b$ and $c$ indicate the major, intermediate, and minor axes,
respectively. Depending on the value of the parameter $t$, the halo is
considered as oblate if $0<t<1/3$; triaxial if $1/3<t<2/3$ and prolate if
$2/3<t<1$. More simply, when $c/a$ is equal to $1$ the cluster is spherical.
## 5 Results
In this section we present the results of our work. First, in Section 5.1, we
study the HE mass bias dependencies on the redshift, mass, and on all other
parameters: the concentration, the mass growth, the relaxation parameter and
the triaxiality. We performed all these analyses at the three overdensities
($\Delta=2500,500,200$) and at all considered redshifts. However, most of the
figures related to this part of the work (with the exception of the first) are
only presented with $\Delta=500$ and at one specific redshift because we do
not find a strong dependence of the results on either of these two quantities.
Secondly, in Section 5.2, we analyse how the HE mass bias varies throughout
strong merger events. In this case, we only analyse the behaviour of the bias
at $\rm R_{200}$, since we take as reference the mergers analysis in
Contreras-Santos et al. (2022), which is done considering the entire cluster
volume and thus at overdensity 200.
### 5.1 Bias correlations
#### 5.1.1 Redshift dependence
The variation of $b_{SZ}$ and $b_{X}$ along the redshift range is presented in
Fig. 1 at the 3 overdensities and for the different cluster dynamical states
defined with the dynamical-state parameter, $\chi_{\rm DS}$. For all
overdensities, the un-relaxed clusters (purple shaded region) have the largest
scatter. Interestingly, this large scatter is mostly caused by the presence of
low-value bias. We will present the reason in Section 5.2. No dependence of
the bias on the redshift up to $z=1.25$ is detected in agreement with Le Brun
et al. 2017; Henson et al. 2017; Salvati et al. 2018; Koukoufilippas et al.
2020. Furthermore, we notice that the median $b_{2500}$ is very similar to the
median $b_{500}$ and that both are close to 0.1 (10% of bias) and as such
systematically lower than $b_{200}$ (close to $\sim 0.2$). This is in
agreement with Gianfagna et al. (2021), which showed a declining $b$ from
outer to inner radius. This is occurring despite the differences both in the
code (GADGET2 versus GADGET-X) and in the baryon models. Note, however, that
the SZ and X median $b_{500}$ in Gianfagna et al. (2021) is slightly larger
than this work. The biases dispersion at $R_{2500}$ seems to be slightly
larger with respect to the other overdensities, this is probably due to the
larger deviations in the cluster core properties, which could marginally
affect the profile at $R_{2500}$. In The Three Hundred, the simulation produce
a more diverse variety of simulated cores with respect to the GADGET2 code,
and in some situations, the simulated clusters are extremely peaked at the
centre (see Campitiello et al., 2022).
Figure 1: The redshift evolution of the biases, $b_{\rm SZ}$ (left panels) and
$b_{\rm X}$ (right panels). The median values of the bias for all clusters,
relaxed and un-relaxed clusters are represented with dark cyan crosses, green
diamonds and purple stars respectively. The shaded regions represent the
$16^{th}$ and $84^{th}$ percentiles. The bias estimated at $\rm R_{200}$, $\rm
R_{500}$ and $\rm R_{2500}$ are represented in the top, middle and bottom
panels respectively. The dashed lines show the 0 and 0.2 bias for reference.
#### 5.1.2 Concentration
The concentration parameter $c_{500}$ of a halo is representative of the
halo’s central density. In presence of a disturbed system the concentration is
typically lower since the X-ray peak might have been destroyed by a merger
event which also could have brought more mass in the external regions. For
this it might be interesting to see if there is any correlation between the HE
mass bias and the NFW concentration parameter.
The bias as a function of the concentration is represented in Fig. 2 where the
relaxed, disturbed and hybrid clusters are introduced with different symbols
and colours. Hybrid clusters are these with either $f_{s}<0.1$ or
$\Delta_{r}<0.1$. The two quantities do not show any dependence, as clear from
the trend of the median value shown with a black line. We notice however that
the scatter in the bias substantially decreases going towards larger
concentration values. This is quantified by the half difference between the
$16^{\rm th}$ and the $84^{\rm th}$ percentiles, $d2=0.5\times(p_{84}-p_{16})$
reported in Table 5 together with the bias percentiles. The value of $d2$
decreases from low (third row) to high concentrated clusters (first row) by
70-80%. This trend – reduced scatter with increasing concentration – is
expected because most of the clusters with higher concentration are relaxed.
Figure 2: The bias dependence on the concentration at $\rm R_{500}$ and at $z=0.07$, $b_{\rm SZ}$ is represented in the left panel, $b_{\rm X}$ in the right one. The black line represents the median bias value, with the shaded regions as 16th and 84th percentiles. The relaxed clusters are shown in green diamonds, the disturbed in purple stars and the hybrids in blue dots. Table 5: Table of the biases for the 50 clusters with the highest (first row) and lowest (third row) NFW concentration and the remaining clusters (second row) for the $z=0.59$ 0 sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at $R_{500}$. $c_{500}$ | $b_{SZ}$ | | | | $b_{X}$ | | |
---|---|---|---|---|---|---|---|---
| $m$ | $p_{16}$ | $p_{84}$ | $d_{2}$ | m | $p_{16}$ | $p_{84}$ | $d_{2}$
high | 0.11 | 0.07 | 0.19 | 0.06 | 0.12 | 0.05 | 0.24 | 0.10
med | 0.11 | 0.00 | 0.18 | 0.09 | 0.13 | 0.00 | 0.24 | 0.12
low | 0.06 | -0.03 | 0.22 | 0.12 | 0.11 | -0.07 | 0.24 | 0.15
#### 5.1.3 Relative Mass growth
The dependence of the bias at $z=0.333$ on the mass growth recorded in the
last Gyr (and specifically since $z=0.46$) is shown in the top panel of Fig.
3, where both quantities are estimated at $\rm R_{500}$. The relative median
values for high, low, and intermediate accreting clusters are listed in Table
6. The correlation is again very weak, the Pearson correlation coefficient is
-0.20 for $b_{\rm SZ}$ and -0.36 for $b_{\rm X}$. However, the systems with
the largest mass growth have also the largest dispersion. This is expected
because the systems with the largest mass growth are also the most disturbed
(purple stars in Fig. 3). This relation will be discussed more afterwards,
however, we point out that clusters with large mass accretion will also have
abundant non-thermal gas motions and, thus, a large non-thermal pressure
component. The same trend is seen for the quantities estimated at $\rm
R_{200}$, with a Pearson correlation coefficient of 0.16 for $b_{\rm SZ}$ and
0.21 for $b_{\rm X}$.
In the bottom panel of Fig. 3 the difference between the bias at $z_{2}=0.33$
and at $z_{1}=0.46$ is shown ($\Delta b=b(z_{2})-b(z_{1})$). The variation in
the bias for disturbed clusters (at large accretion rates) is negative,
implying that the HE mass bias overall decreases. This phenomenon which might
seem surprising will be better studied in Section 5.2, but we can already see
from the top panel the origin of this trend: after a major merger (with a mass
variation of at least 50% $\Delta M/M>0.5$) the mass computed by Eqs. 2 or 3
can be significantly larger than the true mass with a large scatter as well
(see also Nelson et al., 2012).
Table 6: Table of the biases values for the 50 clusters with the largest mass growth (first row), the 50 with the smallest (last row) and the remaining clusters (second row) for the $z=0.33$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500. Mass | $b_{SZ}$ | | | | $b_{X}$ | | |
---|---|---|---|---|---|---|---|---
growth | m | $p_{16}$ | $p_{84}$ | d2 | m | $p_{16}$ | $p_{84}$ | d2
high | 0.09 | -0.05 | 0.17 | 0.11 | 0.09 | -0.12 | 0.20 | 0.16
med | 0.10 | -0.02 | 0.17 | 0.08 | 0.12 | 0.00 | 0.22 | 0.11
low | 0.14 | 0.06 | 0.22 | 0.08 | 0.24 | 0.10 | 0.35 | 0.13
Overall Fig.3 shows that the mass growth is a parameter that is capable to
distinguish between relaxed and disturbed clusters, as seen also in Fig. 4.
We, indeed, find a correlation betwee the mass growth parameter and $\chi_{\rm
DS}$ equal to -0.46 both at $\rm R_{500}$ (shown in the Figure) and at $\rm
R_{200}$. The good correlation is actually expected since one of the
indicators defining $\chi_{\rm DS}$ is the mass in substructures. That said
there is a certain level of contamination with the presence of disturbed
objects with reduced mass growth or viceversa relaxed objects with significant
mass growth. Notice, however, that both these peculiar cases have a HE bias
value which is quite consistent with the average expectation of the samples,
while extreme bias values such as those with $b<-0.2$ are only present for the
largest mass growth.
Figure 3: Top panel: the bias dependence on the mass growth $\frac{\Delta
M/M}{dt}$ is represented. $\frac{\Delta M/M}{dt}$ is estimated from $z=0.46$
to $z=0.33$, which corresponds to 1 Gyr, while the biases are estimated at
$z=0.33$. Bottom panel: the bias variation from $z=0.46$ to $z=0.33$ as a
function of $\frac{\Delta M/M}{dt}$. The relaxed and disturbed clusters are
represented with green diamonds and purple stars respectively, while the
hybrids with blue dots. The cluster dynamical states shown in this plot are
estimated at $z=0.33$. Figure 4: The relaxation parameter at $z=0.33$ is
represented as a function of the mass growth from $z=0.46$ to $z=0.33$, all
estimated at $\rm R_{500}$. The dot-dashed line represents the threshold under
which a cluster is considered disturbed.
#### 5.1.4 Relaxation parameter
The bias as a function of the relaxation parameter is represented in Fig. 5,
where, per definition, the coloured points of relaxed and disturbed clusters
are completely separated. Weak or no correlations is found, indeed the Pearson
correlation coefficients with the highest values are only equal to 0.13 for
$b_{\rm SZ}$ and 0.23 for $b_{\rm X}$ at $z=1.32$; and 0.01 for $b_{\rm SZ}$
and 0.01 for $b_{\rm X}$ at $z=0.07$. Also in this case, the disturbed
clusters show a wider dispersion with respect to the relaxed ones, see Table
7, where the difference between the bias percentiles grows by almost a factor
of 2 from the lowest to the largest $\chi_{\rm DS}$ clusters. This is in
agreement with Piffaretti & Valdarnini (2008); Rasia et al. (2012); Nelson et
al. (2014); Henson et al. (2017); Ansarifard et al. (2020).
Figure 5: The biases are represented as a function of the relaxation parameter $\chi_{\rm DS}$. The hybrids, relaxed and disturbed clusters are represented with blue dots, green diamonds and purple stars respectively. In the top panel we show the quantities at $z=0.07$ and in the bottom one at $z=1.3$. The lines are the results of a simple linear fitting result. The black line and the shaded region represent the median bias and the percentiles (16th and 84th). Table 7: Table of the bias values for the 50 highest (first row) and 50 lowest (third row) $\chi_{\rm DS}$ clusters and for the remaining clusters (second row) for the $z=0.59$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500. $\chi_{\rm DS}$ | $b_{SZ}$ | | | | $b_{X}$ | | |
---|---|---|---|---|---|---|---|---
| m | $p_{16}$ | $p_{84}$ | d2 | m | $p_{16}$ | $p_{84}$ | d2
high | 0.10 | 0.05 | 0.16 | 0.06 | 0.12 | 0.05 | 0.23 | 0.09
med | 0.11 | -0.02 | 0.19 | 0.09 | 0.14 | 0.01 | 0.25 | 0.12
low | 0.11 | -0.04 | 0.21 | 0.13 | 0.07 | -0.12 | 0.20 | 0.16
#### 5.1.5 Total mass
We continue to investigate the correlation between the HE bias and the cluster
mass at $\rm R_{500}$ in Fig.6. Since we detect no variation on cosmic time
(see Fig. 1), in this plot, we simultaneously show the biases at four
redshifts (1.32, 0.78, 0.33, 0.07) to increase the halo mass range. We find no
dependence of the bias on the total mass of the clusters in agreement with
Piffaretti & Valdarnini (2008); Le Brun et al. (2017); Henson et al. (2017);
Pearce et al. (2019); Ansarifard et al. (2020); Barnes et al. (2021).
Figure 6: The biases are represented as a function of the cluster total mass,
both at $\rm R_{500}$. The redshifts 1.32, 0.78, 0.33, 0.07 are represented in
red, blue, magenta and green, respectively. The black line and the shaded
region represent the binned median and the 16th-84th percentiles.
#### 5.1.6 Triaxiality
We study the dependence of the bias at $R_{500}$ on the sphericity of the
halos, one of the HE main assumptions, quantified with the parameter $t$ and
on $c/a$, the ratio of the minor and major axis. All the particles within
$R_{500}$ are used to estimate the halo shape (Vega-Ferrero et al., 2017).
The relation between the biases and $c/a$ is represented in Fig. 7, the
results for $t$ are not shown, being very similar. From that Figure we notice
that the overall correlation and the amplitude of the scatter seem to be
uncorrelated with the axial ratio. Although, if we restrict to the most
aspherical cases (third line in Table 8) we see that the both biases increases
by almost 50% with respect to the objects with the highest $c/a$ values. Note
that the halo shape estimated based on total mass is similar to the one based
on hot gas (see Velliscig et al., 2015, for details). Therefore, we believe a
similar result will be in place when the halo shape is estimated only from the
gas particles.
Figure 7: The biases at $\rm R_{500}$ are represented as a function of the axis ratio $c/a$. The hybrids, relaxed and disturbed are represented with blue dots, green diamonds and purple stars respectively. Both $b_{SZ}$ and $b_{X}$ are shown for $z=0.33$. The median value of the bias is represented with the black line, the 16th and 84th percentiles with the shaded region. Table 8: Table of the biases values for the 50 clusters with the highest (first row) and lowest (last row) axial ratio and for the remaining clusters (second row) for the $z=0.46$ sample. We report the median value ($m$), the $16^{\rm th}$ and $84^{\rm th}$ percentiles ($p_{16}$ and $p_{84}$), and their half difference ($d_{2}$) at R500. $c/a$ | $b_{SZ}$ | | | | $b_{X}$ | | |
---|---|---|---|---|---|---|---|---
| m | $p_{16}$ | $p_{84}$ | $d_{2}$ | m | $p_{16}$ | $p_{84}$ | $d_{2}$
high | 0.09 | 0.01 | 0.18 | 0.07 | 0.11 | -0.00 | 0.18 | 0.09
med | 0.09 | 0.00 | 0.17 | 0.08 | 0.14 | -0.02 | 0.23 | 0.13
low | 0.12 | 0.02 | 0.24 | 0.11 | 0.20 | 0.07 | 0.32 | 0.12
### 5.2 HE mass bias and the merger history of the clusters
Unlike observations, hydrodynamical simulations allow for tracking the whole
history of a cluster, and thus to follow a merging event during all its
phases. Specifically in this subsection, we study major merger events that are
identified when a halo experiences a very rapid mass increase, primarily
caused by the accretion of a single massive object (Contreras-Santos et al.,
2022). Our main goal is to understand the evolution of the bias throughout
these violent merging processes.
To define and compute the relevant times to the merger event, we closely
follow the definitions in Contreras-Santos et al. (2022) to which we refer for
a more detailed explanations. In summary, instead of using absolute time as
the variable in the merger history, which can vary with redshift, we normalise
it to the halo dynamical time:
$t_{dyn}=\sqrt{\frac{3}{4\pi}\frac{1}{200G\rho_{\rm crit}}}.$ (13)
Since the critical density depends only on the cosmology and on the redshift,
see Eq. (1), the dynamical time does not depend on the cluster features, but
evolves only with redshift. As described before, we only consider the major
merger events defined as an event that produces a mass increase of 100% within
half of the dynamical time
$\frac{\Delta M}{M}=\frac{M_{f}-M_{i}}{M_{i}}\geq 1,$ (14)
where $M_{i}$ is the initial mass and $M_{f}$ is the final mass. Note that the
masses in this case refer to the overdensity of 200. We use this larger radius
for better capturing the evolution process inside the virial region. The
merger event can easily be characterised by 4 particular redshifts (see Figure
1 of Contreras-Santos et al. 2022):
* •
$z_{\rm before}$: the last time before the merger begins. It is characterised
by a relaxation parameter with high values (typically larger than 1). Soon
after the two systems start to influence each other and the relaxation
parameter decreases;
* •
$z_{\rm start}$: the merging halo enters in the main cluster $R_{200}$ and as
a consequence the mass of the latter starts to grow;
* •
$z_{\rm end}$: the end point of the merger identified as the moment when the
mass growth stops and the $\chi_{\rm DS}$ starts to grow again implying that
the relaxation process is beginning.
* •
$z_{\rm after}$: the end of the whole merger phase when the cluster approaches
a relaxed state by showing $\chi_{DS}$ closer or above $1$.
In this work, we analyse how the bias evolves with the major merger event and
how much time is needed for the bias to return to the average value in the
relaxed population. In our sample, we select 12 clusters which experience a
major merger, from $z_{\rm before}$ to $z_{\rm after}$, in the investigated
redshift range [0.07,1.32]. In the top panel of Fig. 8 the two stacked biases,
presented by mean bias values with $16^{th}-84^{th}$ percentiles as error
bars, are shown as a function of $\Delta t/t_{\rm dyn}$, which is defined as
$\frac{\Delta t}{t_{\rm dyn}}=\frac{t_{0+\textit{i}}-t_{0}}{t_{\rm dyn}}$ (15)
where $t_{0}$ corresponds to the analysed redshift right before $z_{\rm
before}$, and $t_{0+i}$ corresponds to the analysed redshifts following
$t_{0}$. The vertical grey shaded areas represent the $\pm 1\sigma$ regions of
the averaged time before, after, start and end of the merger. The yellow
shaded region shows the $\pm 1\sigma$ region of the SZ bias for the relaxed
clusters at $\rm R_{200}$, averaged over all the redshifts.
At $\Delta t/t_{\rm dyn}=0$ both biases are in agreement with the typical
relaxed clusters values. Right at the beginning of the merger phase ($z_{\rm
before}$), the biases increase to $\sim 0.25-0.3$ due to the incoming
substructure, that causes the true mass of the cluster to increase faster than
the HE mass, leading to a slight increase of the bias. The latter stays almost
constant until the end of the accretion of the secondary object. In this
situation the HE mass is always underestimating the true mass of the object. A
small increase of the bias is detected at around $z_{\rm end}$ followed by a
quick decrease that lasts until $z_{\rm after}$, where a minimum is reached.
Right at this time, around the end of the merger phase, the HE bias can even
reach negative values. The in-fall of the substructure generates shocks
propagating outward. These shocks cause a steep increase of the derivatives of
the pressure, gas density and temperature profiles (see the bottom panel
described later), leading to an increase of the HE mass, and a negative bias.
This trend is in agreement with Bennett & Sijacki (2022) and Nelson et al.
(2012), who conducted a similar study on simulated clusters. When the cluster
approaches a relaxation phase at $\sim z_{\rm after}$, the bias returns to a
value that is closer to the mean $b$ of relaxed clusters. This trend is again
in agreement with what seen in the FABLE simulated clusters (Bennett &
Sijacki, 2022).
In the middle panel of Fig. 8 we show the evolution of the relaxation
parameter during the merger process. The clusters are classified as relaxed at
$t_{0}$ by definition, and then as the secondary cluster approaches (before
$z_{\rm start}$) the $\chi_{\rm DS}$ drops. The relaxation parameter shows
some fluctuation between $z_{\rm start}$ and $z_{\rm end}$ to finally grow
again after the secondary objects is completely incorporated into the main
cluster. At the end of the merger phase ($z_{\rm after}$), per definition
$\chi_{\rm DS}$ is approaching 1. Notice that even after 4 dynamical times
from the beginning of the merger the main cluster does not reach the original
relaxation state, consistently with Contreras-Santos et al. (2022).
In the bottom panel, we shown the relative evolution of all quantities
entering the HE mass equations (both thermodynamic quantities and their
derivatives). To be consistent with the analysis shown in the upper panels,
all the values are computed at $R_{200}$. The pressure derivative is
represented with blue triangles, the temperature and its derivative are
represented as olive green dots and triangles, the gas density and its
derivative are represented with orange dots and triangles. The temperature
derivative does not show any particular trend and fluctuates around the
original values with large error bars. The temperature values instead grows
immediately as the secondary object approaches and keeps growing until the end
of the process, this is expected as it is a consequence of the shock heating
produced by the merger.
The density value does not show any particular change until the secondary
structure merge into the main halo after $z_{\rm start}$. The slope of the gas
density at $R_{200}$ also increases for the presence of the massive companion.
In both cases, the variation continues up to $z_{\rm end}$ before reaching a
plateau. Finally, the pressure derivative is on the mid way between the
temperature and the gas density derivatives. The increase in the derivative of
the gas density, pressure and temperature due to the shocks generated in the
merger or to the sloshing of the subclusters moving around the cluster core
drive the HE masses of Eqs. 2 and 3 to be closer or even above the true mass.
At $\rm R_{500}$ these considerations are still valid, the behaviour of all
the quantities is the same. To conclude, the bias shows a stable evolution
along the whole major merger event, albeit slightly difference between
$b_{SZ}$ and $b_{X}$. However, the dramatic change of the bias value,
especially in the period between $z_{\rm end}$ to $z_{\rm after}$ when the
cluster is identified as un-relaxed, is the origin of the large scatter of the
bias for the disturbed systems shown in all previous figures. For this reason,
it is not suggested to estimate the HE masses for these dynamical un-relaxed
clusters, unless we can correctly identify their states in the merging events
which can be very difficult in observation.
Figure 8: Top panel: the stacked biases at $\rm R_{200}$ (means and standard
deviations) are represented as a function of the difference in times from
$t_{0}$, divided by the dynamical time. The SZ bias is represented in green,
while the X one in orange. The yellow shaded region represents the biases
range of the relaxed clusters. The null bias is represented with a dashed
line. Central panel: the stacked relaxation parameter estimated inside $\rm
R_{200}$, the means and the standard deviations are represented as a function
of $\Delta t/t_{dyn}$. The dashed line represents the threshold between
relaxed and disturbed clusters. Bottom panel: each thermodynamic quantity
which takes part in the estimation of the HE mass (and bias) divided by their
value before the merger (indicated with a subscript 0) is shown as a function
of $\Delta t/t_{dyn}$. The dashed line shows the value of 1 (the quantity does
not change with respect to the age prior to the merger). The temperature and
its derivative are represented in olive green dots and triangles respectively.
Instead, the density and its derivative are represented as orange dots and
triangles, and the pressure derivative is represented with blue triangles. In
the plot the means and the standard deviations are represented. In all the
panels, the vertical grey shaded areas represent the $\pm 1\sigma$ regions of
the averaged time before, after, start and end of the merger.
## 6 Conclusions
Galaxy cluster mass plays a major role in cluster cosmology. Several different
methods are used to estimate it. In this paper we focus on the hydrostatic
equilibrium approximation, which makes use of the temperature, pressure and
density radial profiles of the ICM. These 3D profiles are computed from The
Three Hundred simulations, which include a set of almost 300 simulated galaxy
clusters that we studied at 10 different redshifts. From the profiles, we
recover the masses under the HE approximation and we estimate the bias against
the true cluster total mass. In this work the focus is specifically on the
connection between the hydrostatic mass bias and different cluster properties,
such as dynamical state, cluster total mass, mass-growth rate, NFW
concentration and axial ratio. Moreover, we follow the bias during 12 major-
merger events to understand the bias evolution along these events. The main
findings are as follows:
* •
We do not find correlation between the biases, estimated at different radii,
and the redshift, in agreement with other simulations.
* •
The bias and its scatter are influenced by the radii within which the bias is
estimated, i.e. $R_{2500},R_{500}$ and $R_{200}$ in this work. The largest
bias is at $R_{200}$ (almost 20%), while at $R_{500}$ and $R_{2500}$, the
median value is around 10%.
* •
There is no correlation between the hydrostatic mass bias and the dynamical
state of the clusters as measured with different indicators: the dynamical-
state parameter $\chi_{\rm DS}$, the NFW concentration, the relative mass
growth, and the triaxiality. However, whenever one of these parameters is
typically associated to a disturbed objects (e.g. low $\chi_{\rm DS}$, low
concentration, high mass growth) we detect a bias scatter that can be almost
twice as that of the relaxed systems. The scatter has, instead, little
dependence on mass, redshift, and sphericity.
* •
A moderate correlation is found between the relaxation parameter and the mass
growth. This is expected because the larger is the mass growth, the more
likely the cluster is disturbed.
* •
By stacking clusters while experiencing major merger events we find that the
main object becomes more disturbed along the merger with a slightly higher
$b$, when the secondary structure collides. After $z_{\rm end}$ – the halo
mass growth stops and $b$ drops significantly because of the shocks
propagating outward generated by the in-falling substructure. These shocks
cause a steep increase of the derivatives of the pressure, gas density and
temperature profiles, even leading to an overestimation of the HE mass. At the
end of the merger phase when clusters get closer to a relaxed dynamical state
with $\chi\sim 1$, the bias assumes the typical values of relaxed clusters.
Although no correlation is found between the HE bias and the dynamical state
of the cluster at a given redshift, we find that selecting a sample with the
typical characteristics of regular objects, such as high NFW concentration,
low mass accretion rate, high $\chi_{\rm DS}$, and high $c/a$ ratio, leads to
a reduced cluster-to-cluster scatter. We also found a correlation between
various merger phases and the HE bias which can be close to 0 when the cluster
is at the end of merger process. This, however, occurs when the cluster is not
yet relaxed confirming that including yet-no relaxed clusters could largely
increase the bias scatter.
Our results are in agreement with other simulations, where a large bias is
found mostly generated by deviations from spherical symmetry (which is at the
base of HE), temperature dishomogeneities, but also the presence of
substructures or gas motions, generating non-thermal pressure component. In
the first part of the paper, we have proved that the HE bias is not dependent
on the latter, while in the last part of the paper we explore how the bias
changes during a merger. Here we show that the presence of a substructure, and
consequently a merger, can indeed deeply affect the cluster dynamical state
and the HE mass bias, leading even to an overestimation of the true mass.
During the merger events, see Fig. 8, the bias is almost always positive, i.e.
the HE mass is underestimating the true cluster mass. However, even with the
highest bias value during the merger events, we can only come close to the
bias suggested by Planck. Therefore, we need to look for other reasons for
explaining this bias tension. The next-generation satellites, like XRISM
(X-Ray Imaging and Spectroscopy Mission), hopefully Athena and the proposed
probe LEM (Line Emission Mapper), but also eROSITA-data analysis, are going to
give precise indication on the gas velocities and dispersion and, consequently
on cluster dynamical state.
## Acknowledgements
The authors would like to thank the Referee for the constructive and helpful
comments. The simulations used in this work have been performed in the
MareNostrum Supercomputer at the Barcelona Supercomputing Center, thanks to
CPU time granted by the Red Española de Supercomputación. WC is supported by
the STFC AGP Grant ST/V000594/1 and the Atracción de Talento Contract no.
2020-T1/TIC-19882 granted by the Comunidad de Madrid in Spain. He further
acknowledges the science research grants from the China Manned Space Project
with NO. CMS-CSST-2021-A01 and CMS-CSST-2021-B01. GY acknowledges financial
support from the MICIU/FEDER (Spain) under project grant PGC2018-094975-C21.
MDP acknowledges support from Sapienza Università di Roma thanks to Progetti
di Ricerca Medi 2019, RM11916B7540DD8D. AK is supported by the Ministerio de
Ciencia, Innovación y Universidades (MICIU/FEDER) under research grant
PGC2018-094975-C21 and further thanks Piero Umiliani for ‘svezia, inferno e
paradiso’.
## Data Availability
The data underlying this article were produced as part of The Three Hundred
Project (Cui et al., 2018). They will be shared on request to The Three
Hundred Collaboration, at https://www.the300-project.org.
## References
* Angelinelli et al. (2020) Angelinelli M., Vazza F., Giocoli C., Ettori S., Jones T. W., Brunetti G., Brüggen M., Eckert D., 2020, MNRAS, 495, 864–885
* Ansarifard et al. (2020) Ansarifard S., et al., 2020, A&A, 634, A113
* Arnaud et al. (2010) Arnaud M., Pratt G. W., Piffaretti R., Böhringer H., Croston J. H., Pointecouteau E., 2010, A&A, 517, 1
* Artis et al. (2022) Artis E., Melin J.-B., Bartlett J., Murray C., 2022, EPJ Web of Conferences, 257, 00004
* Barnes et al. (2021) Barnes D. J., Vogelsberger M., Pearce F. A., Pop A.-R., Kannan R., Cao K., Kay S. T., Hernquist L., 2021, MNRAS, 506
* Bennett & Sijacki (2022) Bennett J. S., Sijacki D., 2022, MNRAS, 514, 313
* Biffi et al. (2014) Biffi V., Sembolini F., De Petris M., Valdarnini R., Yepes G., Gottlöber S., 2014, MNRAS, 439, 588
* Bulbul et al. (2019) Bulbul E., et al., 2019, ApJ, 871, 50
* Campitiello et al. (2022) Campitiello M. G., et al., 2022, A&A, 665, A117
* Cialone et al. (2018) Cialone G., De Petris M., Sembolini F., Yepes G., Baldi A. S., Rasia E., 2018, MNRAS, 477, 139
* Contreras-Santos et al. (2022) Contreras-Santos A., et al., 2022, MNRAS, 511, 2897
* Cui et al. (2016) Cui W., et al., 2016, MNRAS, 456, 2566
* Cui et al. (2017) Cui W., Power C., Borgani S., Knebe A., Lewis G. F., Murante G., Poole G. B., 2017, MNRAS, 464, 2502
* Cui et al. (2018) Cui W., et al., 2018, MNRAS, 480, 2898
* Cui et al. (2022) Cui W., et al., 2022, MNRAS, 514, 977
* Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
* De Andres et al. (2022) De Andres D., et al., 2022, EPJ Web of Conferences, 257, 00013
* De Luca et al. (2021) De Luca F., De Petris M., Yepes G., Cui W., Knebe A., Rasia E., 2021, MNRAS, 504, 5383
* Ettori et al. (2013) Ettori S., Donnarumma A., Pointecouteau E., Reiprich H. T., Giodini S., Lovisari L., Schmidt R. W., 2013, Space Sci Rev, 177, 119–154
* Gianfagna et al. (2021) Gianfagna G., et al., 2021, MNRAS, 502, 5115
* Haggar et al. (2020) Haggar R., Gray M. E., Pearce F. R., Knebe A., Cui W., Mostoghiu R., Yepes G., 2020, MNRAS, 492, 6074
* Henson et al. (2017) Henson M. A., Barnes D. J., Kay S. T., McCarthy I. G., Schaye J., 2017, MNRAS, 465, 3361
* Herná ndez-Lang et al. (2022) Herná ndez-Lang D., et al., 2022, MNRAS, 517, 4355
* Hoekstra et al. (2015) Hoekstra H., Herbonnet R., Muzzin A., Babul A., Mahdavi A., Viola M., Cacciato M., 2015, MNRAS, 449, 685
* Klypin et al. (2016) Klypin A., Yepes G., Gottlöber S., Prada F., Heß S., 2016, MNRAS, 457, 4340
* Koukoufilippas et al. (2020) Koukoufilippas N., Alonso D., Bilicki M., Peacock J. A., 2020, MNRAS, 491, 5464
* Kravtsov & Borgani (2012) Kravtsov A. V., Borgani S., 2012, ARA&A, 50, 353
* Le Brun et al. (2017) Le Brun A. M. C., McCarthy I. G., Schaye J., Ponman T. J., 2017, MNRAS, 466, 4442
* Li et al. (2021) Li Q., Han J., Wang W., Cui W., Li Z., Yang X., 2021, MNRAS, 505, 3907
* Mazzotta et al. (2004) Mazzotta P., Rasia E., Moscardini L., Tormen G., 2004, MNRAS, 354, 10
* Nagai et al. (2007) Nagai D., Kravtsov A. V., Vikhlinin A., 2007, ApJ, 668, 1
* Nagarajan et al. (2018) Nagarajan A., et al., 2018, MNRAS, 488, 1728
* Navarro et al. (1997) Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493
* Nelson et al. (2012) Nelson K., Rudd D. H., Shaw L., Nagai D., 2012, ApJ, 751, 121
* Nelson et al. (2014) Nelson K., Lau E. T., Nagai D., Rudd D. H., Yu L., 2014, ApJ, 782, 107
* Neto et al. (2007) Neto A. F., et al., 2007, MNRAS, 381, 1450
* Ntampaka et al. (2015) Ntampaka M., Trac H., Sutherland D. J., Battaglia N., Póczos B., Schneider J., 2015, ApJ, 803, 50
* Okabe & Smith (2016) Okabe N., Smith G. P., 2016, MNRAS, 461, 3794
* Pearce et al. (2019) Pearce F. A., Kay S. T., Barnes D. J., Bower R. G., Schaller M., 2019, MNRAS, 491, 1622
* Piffaretti & Valdarnini (2008) Piffaretti R., Valdarnini R., 2008, A&A, 491, 71
* Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
* Pratt et al. (2019) Pratt G. W., Arnaud M., Biviano A., Eckert D., Ettori S., Nagai D., Okabe N., Reiprich T. H., 2019, Space Sci Rev, 215
* Rasia et al. (2012) Rasia E., et al., 2012, New Journal of Physics, 14, 055018
* Rasia et al. (2013) Rasia E., Borgani S., Ettori S., Mazzotta P., Meneghetti M., 2013, ApJ, 776, 39
* Rasia et al. (2015) Rasia E., et al., 2015, ApJ, 813, L17
* Salvati et al. (2018) Salvati L., Douspis M., Aghanim N., 2018, A&A, 614, A13
* Salvati et al. (2019) Salvati L., Douspis M., Ritz A., Aghanim N., Babul A., 2019, A&A, 626, A27
* Sembolini et al. (2013) Sembolini F., Yepes G., De Petris M., Gottlöber S., Lamagna L., Comis B., 2013, MNRAS, 429, 323
* Sereno et al. (2021) Sereno M., Lovisari L., Cui W., Schellenberger G., 2021, MNRAS, 507, 5214–5223
* Tian et al. (2021) Tian Y., Cheng H., McGaugh S. S., Ko C.-M., Hsu Y.-H., 2021, ApJ Letters, 917, L24
* Vega-Ferrero et al. (2017) Vega-Ferrero J., Yepes G., Gottlöber S., 2017, MNRAS, 467, 3226
* Velliscig et al. (2015) Velliscig M., et al., 2015, MNRAS, 453, 721
* Vikhlinin et al. (2006) Vikhlinin A., Kravtsov A., Forman W., Jones C., Markevitch M., Murray S. S., Speybroeck L. V., 2006, ApJ, 640, 691
* Von der Linden et al. (2014) Von der Linden A., et al., 2014, MNRAS, 443, 1973–1978
* Wicker et al. (2022) Wicker R., Douspis M., Salvati L., Aghanim N., 2022, EPJ Web of Conferences, 257, 00046
* Zwicky (1937) Zwicky F., 1937, ApJ, 86, 217
|
# A criterion for hypersymmetry on discrete groupoids
F. Flores F. Flores 111 2010 Mathematics Subject Classification: Primary
43A20, Secondary 47L65, 47L30.
Key Words: Groupoid, symmetric Banach algebra, Fell Bundle, C* algebra.
###### Abstract
Given a Fell bundle $\mathscr{C}\overset{q}{\to}\Xi$ over the discrete
groupoid $\Xi$, we study the symmetry of the associated Hahn algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ in terms of the isotropy subgroups
of $\Xi$. We prove that $\Xi$ is symmetric (resp. hypersymmetric) if and only
if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also
characterize hypersymmetry using Fell bundles with constant fibers, showing
that for discrete groupoids, ’hypersymmetry’ equals ’rigid symmetry’.
## 1 Introduction
This article treats the symmetry of certain Banach ∗-algebras connected with
Fell bundles over discrete groupoids. The study of symmetry for groupoid
algebras started not long ago, with Austad and Ortega [1], and found a
continuation in [2], where Jauré, Măntoiu and myself first treated the problem
of symmetry for Fell bundles over discrete groupoids.
###### Definition 1.1.
A Banach ∗-algebra $\mathfrak{B}$ is called symmetric if the spectrum of
$b^{*}b$ is positive for every $b\in\mathfrak{B}$ (this happens if and only if
the spectrum of any self-adjoint element is real).
In this regard, the main result of the paper is the following.
###### Theorem 1.2.
Let $\Xi$ be a discrete groupoid. Then:
1. 1.
The algebra $\ell^{\infty,1}(\Xi)$ is symmetric if and only if every isotropy
group is symmetric.
2. 2.
The algebra $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is symmetric for every
Fell bundle $\mathscr{C}$ over $\Xi$ if and only if every isotropy group is
hypersymmetric.
This result is interesting as it reduces the question of the (hyper)symmetry
of a given groupoid to the (hyper)symmetry of its isotropy subgroups and the
study of group $\ell^{1}$-algebras is far more developed. For example, see [1,
2, 4, 10, 13, 9, 5, 8] and the reference therein.
The article is divided in 3 parts: Section 2 deals with preliminares; there we
introduce Fell bundles, the Hahn algebra of a Fell bundle and define what
(hyper)symmetry for a groupoid means. In Section 3 we introduce a
characterization for hypersymmetry using only Fell bundles with constant
fibers. Its analogous to the characterization made for groups by Jauré and
Măntoiu in [4]. Finally, Section 4 deals with the proof of Theorem 1.2. This
is achieved by writing a general discrete groupoid as a disjoint union of
transitive groupoids and proving that transitive groupoids are isomorphic to
transitive transformation groups. Using the available theory for groups yields
the desired result.
## 2 Preliminaries
Let $\Xi$ be a groupoid, with unit space $\mathcal{U}:=\Xi^{(0)}$ , source map
${\rm d}$ and range map ${\rm r}$ . The ${\rm d}$ and ${\rm r}$-fibers are
$\Xi_{u}=\\{\xi\in\Xi\mid{\rm d}(\xi)=u\\}$ and $\Xi^{u}=\\{\xi\in\Xi\mid{\rm
r}(\xi)=u\\}$. The set of composable pairs is
$\Xi^{(2)}\\!:=\\{(x,y)\\!\mid\\!{\rm r}(y)={\rm d}(x)\\}\,.$
The isotropy group associated with the unit $u\in\mathcal{U}$ is
$\Xi_{u}^{u}=\Xi_{u}\cap\Xi^{u}$, it is a subgroupoid of $\Xi$, which happens
to be a group. We endow the groupoid $\Xi$ with the discrete topology. Let us
introduce an important class of groupoids which will be useful.
###### Definition 2.1.
A groupoid $\Xi$ is called transitive if for any pair $u,v\in\mathcal{U}$,
there exists $x\in\Xi$ such that ${\rm r}(x)=u$ and ${\rm d}(x)=v$.
The concept of a transitivity is borrowed from the theory of dynamical
systems. And it comes from a natural but hidden action of the groupoid on its
unit space. So the groupoid is transitive if and only if this action is
transitive, cf [3, Example 2.2, Corollary 7.8].
In this article we are going to work with Fell bundles
$\mathscr{C}\overset{q}{\to}\Xi$ over the groupoid ${\Xi}$ (see [6, 11]). A
Fell bundle is composed of fibers and it satisfies that each fiber
$\mathfrak{C}_{x}\\!:=q^{-1}(\\{x\\})$ is a Banach space with norm
$\parallel\\!\cdot\\!\parallel_{\mathfrak{C}_{x}}$ , the topology of
$\mathscr{C}$ coincides with the norm topology on each fiber, there are
antilinear continuous involutions
$\mathfrak{C}_{x}\ni\\!a\to a^{\bullet}\\!\in\mathfrak{C}_{x^{-1}}$
and for all $(x,y)\in\Xi^{(2)}$ there are continuous multiplications
$\mathfrak{C}_{x}\\!\times\\!\mathfrak{C}_{y}\ni(a,b)\to a\bullet
b\in\mathfrak{C}_{xy}$
satisfying the following axioms valid for
$a\in\mathfrak{C}_{x}\,,b\in\mathfrak{C}_{y}\,$ and $(x,y)\in\Xi^{(2)}$ :
* •
$\parallel\\!ab\\!\parallel_{\mathfrak{C}_{xy}}\,\leq\,\parallel\\!a\\!\parallel_{\mathfrak{C}_{x}}\parallel\\!b\\!\parallel_{\mathfrak{C}_{y}}$
,
* •
$(ab)^{\bullet}=b^{\bullet}a^{\bullet}$,
* •
$\parallel\\!a^{\bullet}a\\!\parallel_{\mathfrak{C}_{{\rm
d}(x)}}=\,\parallel\\!a\\!\parallel_{\mathfrak{C}_{x}}^{2}$ ,
* •
$a^{\bullet}a$ is positive in $\mathfrak{C}_{{\rm d}(x)}$ .
From these axioms it follows that $\mathfrak{C}_{x}$ is a $C^{*}$-algebra for
every unit $x\in\mathcal{U}$ . Sometimes we simply write
$\mathscr{C}=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ for the Fell bundle.
Our object of study is the Hahn algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ adapted to Fell bundles [11],
which in our case it is formed by the sections $\Phi:\Xi\to\mathfrak{C}$ (thus
satisfying $\Phi(x)\in\mathfrak{C}_{x}$ for every $x\in\Xi$) that can be
obtained as a limit of finitely-supported sections in the Hahn-type norm
$\lVert\Phi\rVert_{\infty,1}\,:=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\\!\Big{\\}}.$
(2.1)
It is a Banach ∗-algebra under the multiplication
$(\Phi*\Psi)(x):=\sum_{yz=x}\Phi(y)\bullet\Psi\big{(}z)$ (2.2)
and the involution
$\Phi^{*}(x):=\Phi(x^{-1})^{\bullet}.$ (2.3)
###### Remark 2.2.
Let us point out some of the nature of the functions in
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$. If
$\Phi_{n}\in\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is a sequence of
sections with finite support and $\Phi_{n}\to\Phi$, then the convergence is
uniform. Indeed, let $x\in\Xi$ and observe that
$\displaystyle\lVert\Phi_{n}(x)-\Phi(x)\rVert_{\mathfrak{C}_{x}}$
$\displaystyle\leq\sum_{y\in\Xi^{r(x)}}\lVert\Phi_{n}(y)-\Phi(y)\rVert_{\mathfrak{C}_{y}}$
$\displaystyle\leq\sup_{u\in\mathcal{U}}\sum_{y\in\Xi^{u}}\lVert\Phi_{n}(y)-\Phi(y)\rVert_{\mathfrak{C}_{y}}$
$\displaystyle\leq\lVert\Phi_{n}-\Phi\rVert_{\ell^{\infty,1}(\Xi\,\mid\,\mathscr{C})}.$
So
$\lim_{n}\sup_{x\in\Xi}\lVert\Phi_{n}(x)-\Phi(x)\rVert_{\mathfrak{C}_{x}}=0$.
This implies that the function $\Phi$ has countable support and vanishes at
$\infty$.
We denote by $C^{*}(\Xi\,|\,\mathscr{C})$ the enveloping $C^{*}$-algebra of
the Hahn algebra $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$. It is a known
fact that $\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is a dense ∗-subalgebra
of $C^{*}(\Xi\\!\mid\\!\mathscr{C})$.
###### Definition 2.3.
1. (i)
The discrete groupoid $\Xi$ is called symmetric if the convolution Banach
∗-algebra $\ell^{\infty,1}(\Xi)$ is symmetric.
2. (ii)
The discrete groupoid $\Xi$ is called hypersymmetric if given any Fell bundle
$\mathscr{C}\\!=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ , the Banach ∗-algebra
$\ell^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$ is symmetric.
###### Example 2.4.
If $\Pi\subset X\\!\times\\!X$ is an equivalence relation on $X$, one can make
$\Xi=\Pi$ a discrete groupoid by defining the operations
$\begin{split}{\rm d}(x,y)=(y,y)\,,\quad&{\rm
r}(x,y)=(x,x)\,,\quad(x,y)(y,z)=(x,z)\,,\quad(x,y)^{-1}\\!=(y,x)\,.\end{split}$
The unit space is $\,\mathcal{U}={\sf Diag}(X)$ and it gets canonically
identified with $X$, via the homeomorphism $(x,x)\mapsto x$ . In this case all
of the isotropy groups correspond to the trivial group
$\Pi_{u}^{u}=\\{(u,u)\\}$, so Theorem 4.9 will guarantee that $\Pi$ is
hypersymmetric. A particular examples is the so called pair groupoid,
$\Pi=X\times X$.
## 3 A characterization of hypersymmetry for discrete groupoids
In [2] we introduced some special Fell bundles arising from Hilbert bundles to
characterize the hypersymmetry of a discrete groupoid. However, in this paper
we will improve this characterization: One can actually verify hypersymmetry
by looking at much simpler algebras, associated to Fell bundles with constant
fibers. Let us make precise the Fell bundles of interest.
###### Definition 3.1.
By a (left) groupoid action of a discrete groupoid $\Xi$ on the $C^{*}$-bundle
$\mathscr{A}\\!:=\bigsqcup_{u\in\mathcal{U}}\mathfrak{A}_{u}\overset{p}{\to}\mathcal{U}$
over its unit space we understand a continuous map
$\mathscr{A}\rtimes\Xi:=\\{(\alpha,x)\in\mathscr{A}\\!\times\\!\Xi\\!\mid\\!p(\alpha)={\rm
d}(x)\\}\ni(\alpha,x)\to\mathcal{T}(\alpha,x)\equiv\mathcal{T}_{x}(\alpha)\in\mathscr{A},$
satifying the axioms
1. (a)
$p\big{[}\mathcal{T}_{x}(\alpha)\big{]}={\rm r}(x)$ ,
$\forall\,x\in\Xi\,,\,\alpha\in\mathfrak{A}_{{\rm d}(x)}$ ,
2. (b)
each $\mathcal{T}_{x}$ is a ∗-isomorphism $:\mathfrak{A}_{{\rm
d}(x)}\\!\to\mathfrak{A}_{{\rm r}(x)}$ ,
3. (c)
$\mathcal{T}_{u}={\rm id}_{\mathfrak{A}_{u}}$ , $\forall\,u\in\mathcal{U}$ ,
4. (d)
if $(x,y)\in\Xi^{(2)}$ and $(\alpha,y)\in\mathscr{A}\rtimes\Xi$ , then
$\big{(}\mathcal{T}_{y}(\alpha),x\big{)}\in\mathscr{A}\rtimes\Xi$ and
$\mathcal{T}_{xy}(\alpha)=\mathcal{T}_{x}\big{[}\mathcal{T}_{y}(\alpha)\big{]}$
.
###### Definition 3.2.
Let $\mathcal{T}$ be a groupoid action of $\Xi$ on the $C^{*}$-bundle
$\mathscr{A}\\!:=\bigsqcup_{u\in\mathcal{U}}\mathfrak{A}_{u}\overset{p}{\to}\mathcal{U}$.
We define its associated Fell bundle as follows. The underlying space is
$\mathscr{C}_{\mathcal{T}}:=\mathscr{A}\rtimes\Xi$ with the topology inherited
from the product topology and the obvious projection $q$. We endow it with the
operations
$(\alpha,x)\bullet(\beta,y):=(\alpha\mathcal{T}_{x}(\beta),xy)\,,\textup{when
}(x,y)\in\Xi^{(2)}$
and
$(\alpha,x)^{\bullet}:=(\mathcal{T}_{x^{-1}}(\alpha^{*}),x^{-1})$
to get a Fell bundle over $\Xi$ . A section is now a map
$\Phi:\Xi\to\mathscr{C}_{\mathcal{T}}$ such that
$\Phi(x)\equiv\big{(}\varphi(x),x\big{)}\in\mathfrak{C}_{x}=\mathfrak{A}_{{{\rm
r}}(x)}\\!\times\\!\\{x\\}\,,\quad\forall\,x\in\Xi\,.$
So we may identify every section
$\Phi\in\ell^{\infty,1}(\Xi\,|\,\mathscr{C}_{\mathcal{T}})$ with a function
$\Phi:\Xi\to\mathscr{A}$ (note the abuse of notation), such that
$\Phi(x)\in\mathfrak{A}_{{{\rm r}}(x)}$.
We will also denote the algebra
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C}_{\mathcal{T}})$ by
$\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})$ to recall its particular
nature. If the action $\mathcal{T}$ is trivial, meaning that
$\mathfrak{A}_{u}\equiv\mathfrak{A}$ for all $u\in\mathcal{U}$ and
$\mathcal{T}_{x}\equiv{\rm id}_{\mathfrak{A}}$ for all $x\in\Xi$, then we
denote the resulting algebra simply by $\ell^{\infty,1}(\Xi,\mathfrak{A})$.
###### Remark 3.3.
Let $\Phi,\Psi\in\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})$. In this
case, one may write the algebraic laws as
$\big{[}\Phi*\Psi\big{]}(x)=\\!\sum_{y\in\Xi^{{\rm
r}(x)}}\\!\Phi(y)\mathcal{T}_{y}\big{[}\Psi(y^{-1}x)\big{]}\,,$ (3.1)
$\Phi^{*}(x)=\mathcal{T}_{x}\big{[}\Phi(x^{-1})\big{]}^{*}.$ (3.2)
And the Hahn-type norm as
$\parallel\\!\Phi\\!\parallel_{\ell^{\infty,1}_{\mathcal{T}}(\Xi,\mathscr{A})}=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{A}_{{\rm
r}(x)}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{A}_{{\rm
r}(x)}}\\!\Big{\\}}\,.$ (3.3)
###### Lemma 3.4.
Let $\mathscr{C}\\!=\bigsqcup_{x\in\Xi}\mathfrak{C}_{x}$ be a Fell bundle over
the discrete groupoid $\Xi$. Then there exists and an isometric ∗-monomorphism
$\varphi:\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\to\ell^{\infty,1}\big{(}\Xi,C^{*}(\Xi\,|\,\mathscr{C})\big{)}.$
(3.4)
###### Proof.
Set $\mathfrak{A}:=C^{*}(\Xi\,|\,\mathscr{C})$ and, for every $x\in\Xi$ we
embed $\mathfrak{C}_{x}$ into
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\subset\mathfrak{A}$, by setting for each
$a\in\mathfrak{C}_{x}$
$(\theta_{x}a)(y):=a\ \ {\rm if}\ \
y=x\,,\quad(\theta_{x}a)(y):=0_{\mathfrak{C}_{x}}\ \ {\rm if}\ \ y\neq x\,.$
It is not hard to prove that if $(x,y)\in\Xi^{(2)}$, then
$\theta_{x}a*\theta_{y}b=\theta_{xy}(a\bullet b)\quad\textup{ and
}\quad(\theta_{x}a)^{*}=\theta_{x^{-1}}a^{\bullet}$
hold. On the other hand, one also has
$\lVert\theta_{x}a\rVert_{\mathfrak{A}}=\lVert a\rVert_{\mathfrak{C}_{x}}$: If
$x\in\mathcal{U}$, this equality holds because
$\theta_{x}:\mathfrak{C}_{x}\to\mathfrak{A}$ is a ∗-monomorphism of
$C^{*}$-algebras, hence isometric. If $x\in\Xi$ is not an unit, then one may
apply the -now standard- trick
$\lVert\theta_{x}a\rVert^{2}_{\mathfrak{A}}=\lVert(\theta_{x}a)^{*}*(\theta_{x}a)\rVert_{\mathfrak{A}}=\lVert\theta_{x^{-1}x}(a^{\bullet}\bullet
a)\rVert_{\mathfrak{A}}=\lVert a^{\bullet}\bullet
a\rVert_{\mathfrak{C}_{x^{-1}x}}=\lVert a\rVert^{2}_{\mathfrak{C}_{x}}$
to conclude that $\theta_{x}$ preserves the mentioned norms. This allows us to
successfully define
$\varphi(\Phi)(x)=\theta_{x}\Phi(x),\textup{ for
}\Phi\in\ell^{\infty,1}(\Xi\,|\,\mathscr{C})$
and have an isometry:
$\displaystyle\lVert\varphi(\Phi)\rVert_{\ell^{\infty,1}(\Xi,{\mathfrak{A}})}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\theta_{x}\Phi(x)\\!\parallel_{\mathfrak{A}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\theta_{x}\Phi(x)\\!\parallel_{\mathfrak{A}}\\!\Big{\\}}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{{\rm
r}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\,,\,\sup_{u\in\mathcal{U}}\sum_{{\rm
d}(x)=u}\\!\parallel\\!\Phi(x)\\!\parallel_{\mathfrak{C}_{x}}\\!\Big{\\}}$
$\displaystyle=\lVert\Phi\rVert_{\ell^{\infty,1}(\Xi\,|\,\mathscr{C})}.$
Now we check that $\varphi$ is an ∗-homomorphism,
$\displaystyle\big{[}\varphi(\Phi)*\varphi(\Psi)\big{]}(x)$
$\displaystyle=\sum_{yz=x}\theta_{y}\Phi(y)*\theta_{z}\Psi(z)$
$\displaystyle=\theta_{x}\sum_{yz=x}\Phi(y)\bullet\Psi(z)$
$\displaystyle=\varphi(\Phi*\Psi)(x)$
and
$\varphi(\Phi^{*})(x)=\theta_{x}\Phi^{*}(x)=\theta_{x}\Phi(x^{-1})^{\bullet}=\big{[}\theta_{x^{-1}}\Phi(x^{-1})\big{]}^{*}=\varphi(\Phi)(x^{-1})^{*}=\varphi(\Phi)^{*}(x).$
This finishes the proof.∎
###### Remark 3.5.
Observe that
$\ell^{\infty,1}(\Xi,\mathfrak{A})\cong\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}$,
where $\hat{\otimes}$ denotes the projective tensor product. Indeed, given
$(\varphi,a)\in\ell^{\infty,1}(\Xi)\times\mathfrak{A}$, define the function
$\varphi\otimes a$ by
$\big{[}\varphi\otimes a\big{]}(x):=a\varphi(x),\quad\forall x\in\Xi.$
The map $(\varphi,a)\mapsto\varphi\otimes a$ defined in
$\ell^{\infty,1}(\Xi)\times\mathfrak{A}\to\ell^{\infty,1}(\Xi,\mathfrak{A})$
is bilinear, has norm $1$ (it satisfies $\lVert\varphi\otimes
a\rVert_{\ell^{\infty,1}(\Xi,\mathfrak{A})}\leq\lVert
a\rVert_{\mathfrak{A}}\lVert\varphi\rVert_{\ell^{\infty,1}(\Xi)}$) and it
extends to ∗-isomorphism.
$\iota:\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}\to\ell^{\infty,1}(\Xi,\mathfrak{A})$
The following corollary effectively reduces our concerns to the study of
tensor products, just as in the group case (cf. [4, Theorem 2.4]).
###### Corollary 3.6.
A discrete groupoid $\Xi$ is hypersymmetric if and only if the Banach
∗-algebra
$\ell^{\infty,1}(\Xi,\mathfrak{A})\cong\ell^{\infty,1}(\Xi)\,\hat{\otimes}\,\mathfrak{A}$
is symmetric, for every $C^{*}$-algebra $\mathfrak{A}$.
###### Proof.
Every algebra of the form $\ell^{\infty,1}(\Xi,\mathfrak{A})$ comes from a
Fell bundle (recall Definition 3.2), so it is symmetric if $\Xi$ is
hypersymmetric. On the other hand, because of Lemma 3.4, given any Fell bundle
$\mathscr{C}$, $\ell^{\infty,1}(\Xi\,|\,\mathscr{C})$ may be identified as a
closed ∗-subalgebra of some algebra of the form
$\ell^{\infty,1}(\Xi,\mathfrak{A})$. So it will be symmetric if the latter is
symmetric, by [12, Theorem 11.4.2]. ∎
###### Remark 3.7.
In the group case, this condition has been called ’rigid symmetry’ by many
authors (including myself) and it was introduced by Leptin and Poguntke in
[9]. It can also be seen in [1, 2, 4, 10, 13].
Before going into the following sections, let us simplify some notation. Let
$\mathfrak{A},\mathfrak{B}$ be Banach ∗-algebras. We will denote by
$\mathfrak{A}\hookrightarrow\mathfrak{B}$ the fact that there exists an
isometric ∗-monomorphism $\iota:\mathfrak{A}\to\mathfrak{B}$. In this
language, the conclusion of Lemma 3.4 can be written as
$\ell^{\infty,1}(\Xi\,|\,\mathscr{C})\hookrightarrow\ell^{\infty,1}\big{(}\Xi,C^{*}(\Xi\,|\,\mathscr{C})\big{)}$.
On the other hand, the dual notation
$\mathfrak{A}\twoheadrightarrow\mathfrak{B}$ means that there exists some
contractive ∗-epimorphism $\pi:\mathfrak{A}\to\mathfrak{B}$.
## 4 The result for discrete groupoids
The rest of the paper is devoted to prove Theorem 4.9. We will follow the
strategy detailed in the introduction, starting with a complete (well-known)
characterization of discrete transitive groupoids as group transformation
groupoids.
###### Definition 4.1.
Let $\gamma$ be a continuous action of the (discrete) group ${\sf G}$ on the
topological space $X$, we define the transformation groupoid $\Xi:={\sf
G}\ltimes_{\gamma}\\!X$ associated to it as follows. As a topological space
${\sf G}\ltimes_{\gamma}\\!X$, it is just ${\sf G}\times X$, the maps ${\rm
r},{\rm d}$ are given by ${\rm d}(a,x)=x$ and ${\rm r}(a,x)=\gamma_{a}(x)$.
The composition is $\big{(}b,\gamma_{a}(x)\big{)}(a,x):=(ba,x)$ and inversion
reads $(a,x)^{-1}:=\big{(}a^{-1}\\!,\gamma_{a}(x)\big{)}$ . The unit space is
$\mathcal{U}=\\{{{\sf e}}\\}\times X\equiv X$.
###### Proposition 4.2.
Let $\Xi$ be a discrete transitive groupoid with isotropy group ${\sf G}$.
There is an abelian group structure on $\mathcal{U}$ and an action $\gamma$ of
${\sf G}^{\prime}={\sf G}\times\mathcal{U}$ on $\mathcal{U}$ such that
$\Xi\cong{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}$.
###### Proof.
Let us give $\mathcal{U}$ some abelian group structure with additive notation
and define the action of ${\sf G}^{\prime}$ by $\gamma_{(\alpha,w)}(v)=w+v$.
In order to construct an isomorphism $\varphi$, fix some $u\in\mathcal{U}$ and
realize ${\sf G}$ as $\Xi_{u}^{u}$. Since $\Xi$ is transitive, for every
$v\in\mathcal{U}$, there exists an arrow $z_{v}\in\Xi$ such that ${\rm
d}(z_{v})=v$ and ${\rm r}(z_{v})=u$. Then
$\varphi:{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}\to\Xi\textup{ defined by
}\varphi\big{(}(x,w),v\big{)}=z^{-1}_{w+v}xz_{v},\textup{ is the required
isomorphism.}$
Let us verify that $\varphi$ is indeed a groupoid isomorphism:
* (i)
If
$\big{(}(x_{1},w_{1}),w_{2}+v\big{)}\big{(}(x_{2},w_{2}),v\big{)}=\big{(}(x_{1}x_{2},w_{1}+w_{2}),v\big{)}$,
then
$\displaystyle\varphi\big{(}(x_{1}x_{2},w_{1}+w_{2}),v\big{)}$
$\displaystyle=z^{-1}_{w_{1}+w_{2}+v}x_{1}x_{2}z_{v}$
$\displaystyle=z^{-1}_{w_{1}+w_{2}+v}x_{1}z_{w_{2}+v}z_{w_{2}+v}^{-1}x_{2}z_{v}$
$\displaystyle=\varphi\big{(}(x_{1},w_{1}),w_{2}+v\big{)}\varphi\big{(}(x_{2},w_{2}),v\big{)}$
* (ii)
If $\big{(}(x,w),v\big{)}\in{\sf G}^{\prime}\ltimes_{\gamma}\mathcal{U}$, then
$\big{(}(x,w),v\big{)}^{-1}=\big{(}(x^{-1},-w),w+v\big{)}$ and
$\displaystyle\varphi\big{(}(x^{-1},-w),w+v\big{)}$
$\displaystyle=z^{-1}_{v}x^{-1}z_{w+v}$
$\displaystyle=(z^{-1}_{w+v}xz_{v})^{-1}=\varphi\big{(}(x,w),v\big{)}^{-1}.$
* (iii)
The inverse function $\varphi^{-1}$ is given by
$\varphi^{-1}(\xi)=\big{(}(z_{{\rm r}(\xi)}\xi z_{{\rm d}(\xi)}^{-1},{\rm
r}(\xi)-{\rm d}(\xi)),{\rm d}(\xi)\big{)}$.
∎
###### Remark 4.3.
It follows from Proposition 4.2 that the pair groupoid
$\mathcal{U}\times\mathcal{U}$ is isomorphic to
$\mathcal{U}\ltimes_{\gamma|_{\mathcal{U}}}\mathcal{U}$. (Here ${\sf
G}=\\{{\sf e}\\}$ is the trivial group)
While the commutativity of $\mathcal{U}$ was not essential for the previous
proof -any group structure would have worked out-, it will be of vital
importance in the future (cf. the proof of Theorem 4.7). That is why it will
be occasionally remarked in the following propositions.
The following lemma requires some notation. Given a $C^{*}$-algebra
$\mathfrak{A}$ and a group action $\gamma:{\sf G}\to{\rm Bij}(\mathcal{U})$,
denote by $\Gamma$ the ${\sf G}$-action on
$\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})$ satisfying
$\Gamma_{x}(f)(u)=f\big{(}\gamma_{x^{-1}}(u)\big{)}$.
###### Lemma 4.4.
Suppose that the discrete group ${\sf G}$ acts on $\mathcal{U}$ via $\gamma$
and $\mathfrak{A}$ is a $C^{*}$-algebra. Then
$\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\twoheadrightarrow\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A}).$ (4.1)
###### Proof.
Define $\pi:\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\to\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$ by the formula
$\pi(\Phi)(x,u)=\Phi(x)\big{(}\gamma_{x}(u)\big{)}$. This map is well-defined
and clearly surjective, while it also satisfies
$\displaystyle\lVert\pi(\Phi)\rVert_{\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})}$
$\displaystyle=\max\Big{\\{}\sup_{u\in\mathcal{U}}\sum_{x\in{\sf
G}}\\!\parallel\\!\Phi(x)\big{(}\gamma_{x}(u)\big{)}\\!\parallel_{\mathfrak{A}}\,,\,\sup_{u\in\mathcal{U}}\sum_{x\in{\sf
G}}\\!\parallel\\!\Phi(x)(u)\\!\parallel_{\mathfrak{A}}\\!\Big{\\}}$
$\displaystyle\leq\sum_{x\in{\sf
G}}\sup_{u\in\mathcal{U}}\\!\parallel\\!\Phi(x)(u)\\!\parallel_{\mathfrak{A}}=\sum_{x\in{\sf
G}}\lVert\Phi(x)\rVert_{\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})}=\lVert\Phi\rVert_{\ell^{1}_{\Gamma}({\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A}))}$
and
$\displaystyle\pi(\Phi*\Psi)(x,u)$
$\displaystyle=\big{[}\Phi*\Psi\big{]}(x)\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\Big{[}\sum_{y\in{\sf
G}}\Phi(y)\Gamma_{y}\big{[}\Psi(y^{-1}x)\big{]}\Big{]}\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\sum_{y\in{\sf
G}}\Phi(y)\big{(}\gamma_{x}(u)\big{)}\Psi(y^{-1}x)\big{(}\gamma_{y^{-1}x}(u)\big{)}$
$\displaystyle=\sum_{y\in{\sf
G}}\pi(\Phi)(y,\gamma_{y^{-1}x}(u))\pi(\Psi)(y^{-1}x,u)$
$\displaystyle=\big{[}\pi(\Phi)*\pi(\Psi)\big{]}(x,u).$
Finally, we see that
$\displaystyle\pi(\Phi^{*})(x,u)$
$\displaystyle=\Phi^{*}(x)\big{(}\gamma_{x}(u)\big{)}$
$\displaystyle=\Gamma_{x}\big{[}\Phi(x^{-1})^{*}]\big{(}\gamma_{x}(u)\big{)}=\Phi(x^{-1})(u)^{*}=\big{[}\pi(\Phi)(x^{-1},\gamma_{x}(u))\big{]}^{*}=\pi(\Phi)^{*}(x,u).$
Hence $\pi$ is a contractive ∗-epimorphism. ∎
Lemma 4.4 will be used in the following form, which allows us to focus on the
study of $\ell^{1}$-algebras arising from group $C^{*}$-dynamical systems.
###### Corollary 4.5.
If $\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}$ is a symmetric Banach
∗-algebra, then $\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$ is also symmetric.
###### Proof.
In Lemma 4.4, we showed that $\ell^{1}_{\Gamma}\big{(}{\sf
G},\mathcal{C}_{0}(\mathcal{U},\mathfrak{A})\big{)}\twoheadrightarrow\ell^{\infty,1}({\sf
G}\ltimes_{\gamma}\mathcal{U},\mathfrak{A})$, so the conclusion follows from
[12, Theorem 11.4.2]. ∎
###### Proposition 4.6.
Let $({\sf G}\times{\sf H},\Gamma,\mathfrak{A})$ be a $C^{*}$-dynamical system
and assume that ${\sf G}$ acts trivially on $\mathfrak{A}$. Then one has
$\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\cong\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\Gamma}({\sf H},\mathfrak{A}),$ (4.2)
where ${\sf G}^{\prime}:={\sf G}\times{\sf H}$.
###### Proof.
Observe that
$\iota:\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\to\ell^{1}\big{(}{\sf
G},\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})\big{)}\quad\textup{ defined by
}\quad\iota(\Phi)(x)(y)=\Phi(x,y)$
is an isometric ∗-isomorphism. Indeed, bijectivity is direct, while
$\displaystyle\lVert\iota(\Phi)\rVert_{\ell^{1}({\sf G},\ell^{1}_{\Gamma}({\sf
H},\mathfrak{A}))}$ $\displaystyle=\sum_{x\in{\sf
G}}\lVert\iota(\Phi)(x)\rVert_{\ell^{1}_{\Gamma}({\sf
H},\mathfrak{A})}=\sum_{x\in{\sf G}}\sum_{y\in{\sf
H}}\lVert\Phi(x,y)\rVert_{\mathfrak{A}}=\lVert\Phi\rVert_{\ell^{1}_{\Gamma}({\sf
G}^{\prime},\mathfrak{A})}$
and, identifying $(1_{\sf G},b)\in{\sf G}^{\prime}$ with $b\in{\sf H}$,
$\displaystyle\big{[}\iota(\Phi)*\iota(\Psi)\big{]}(x)(y)$
$\displaystyle=\Big{[}\sum_{a\in{\sf
G}}\iota(\Phi)(a)*\iota(\Psi)(a^{-1}x)\Big{]}(y)$
$\displaystyle=\sum_{a\in{\sf G}}\sum_{b\in{\sf
H}}\iota(\Phi)(a)(b)\Gamma_{b}\big{[}\iota(\Psi)(a^{-1}x)(b^{-1}y)\big{]}$
$\displaystyle=\sum_{(a,b)\in{\sf G}\times{\sf
H}}\Phi(a,b){\Gamma}_{(a,b)}\big{[}\Psi(a^{-1}x,b^{-1}y)\big{]}$
$\displaystyle=\iota(\Phi*\Psi)(x)(y).$
Finally,
$\displaystyle\iota(\Phi^{*})(x)(y)=\Phi^{*}(x,y)=\Gamma_{y}\big{[}\Phi(x^{-1},y^{-1})^{*}\big{]}=\iota(\Phi)(x^{-1})^{*}(y)=\iota(\Phi)^{*}(x)(y).$
So $\ell^{1}_{\Gamma}({\sf G}^{\prime},\mathfrak{A})\cong\ell^{1}\big{(}{\sf
G},\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})\big{)}\cong\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\Gamma}({\sf H},\mathfrak{A})$. ∎
We can put together the previous lemmas to obtain the following result, which
is Theorem 1.2 for transitive groupoids.
###### Theorem 4.7.
Let $\Xi$ be a discrete transitive groupoid with isotropy subgroup ${\sf G}$.
If ${\sf G}$ is symmetric (resp. hypersymmetric), then $\Xi$ is symmetric
(resp. hypersymmetric).
###### Proof.
Let us divide the proof in two cases, the first one being about hypersymmetry.
Because of Proposition 4.2, we may assume that $\Xi={\sf
G}^{\prime}\ltimes_{\gamma}{\sf H}$, with ${\sf G}^{\prime}={\sf G}\times{\sf
H}$ and $\gamma_{(g,h)}(k)=h+k$.
First suppose that ${\sf G}$ is hypersymmetric. Because of corollaries 3.6 and
4.5, it is enough to show that ${\sf G}\times{\sf H}$ is hypersymmetric (or
’rigidly symmetric’, see Remark 3.7). But this follows from the fact that
${\sf H}$ is abelian and [9, Theorem 7].
Now suppose that ${\sf G}$ is only symmetric. Note that $\gamma|_{\sf H}$
coincides with the action of ${\sf H}$ on itself by left translation, so we
will denote it by ${\rm lt}$. Then
$\displaystyle\ell^{1}_{\Gamma}\big{(}{\sf G}^{\prime},\mathcal{C}_{0}({\sf
H})\big{)}$ $\displaystyle\overset{\eqref{eq2}}{\cong}\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}_{\rm lt}\big{(}{\sf H},\mathcal{C}_{0}({\sf
H})\big{)}$ $\displaystyle\overset{\eqref{eq}}{\hookrightarrow}\ell^{1}({\sf
G})\,\hat{\otimes}\,\ell^{1}({\sf H})\,\hat{\otimes}\,\big{(}{\sf
H}\ltimes_{\rm lt}\mathcal{C}_{0}({\sf H})\big{)}$
$\displaystyle\cong\ell^{1}({\sf H})\,\hat{\otimes}\,\ell^{1}\big{(}{\sf
G},\mathcal{K}(\ell^{2}({\sf H})\big{)}).$
The final isomorphism holds because of the Stone-von Neumann theorem: ${\sf
H}\ltimes_{\rm lt}\mathcal{C}_{0}({\sf H})\cong\mathcal{K}(\ell^{2}({\sf
H})\big{)}$. $\ell^{1}\big{(}{\sf G},\mathcal{K}(\ell^{2}({\sf H})\big{)})$ is
symmetric because of [5, Theorem 1], hence $\ell^{1}({\sf
H})\,\hat{\otimes}\,\ell^{1}\big{(}{\sf G},\mathcal{K}(\ell^{2}({\sf
H})\big{)})$ is symmetric because of [8, Theorem 5]. We conclude that
$\ell^{1}_{\Gamma}\big{(}{\sf G}^{\prime},\mathcal{C}_{0}({\sf H})\big{)}$ and
thus $\ell^{\infty,1}(\Xi)$ are symmetric. ∎
###### Corollary 4.8.
The pair groupoid over a discrete set $\mathcal{U}$ is hypersymmetric.
Now we will proceed to upgrade Theorem 4.7 to the case of general discrete
groupoids.
###### Theorem 4.9.
A discrete groupoid $\Xi$ is symmetric (resp. hypersymmetric) if and only if
its isotropy subgroups are symmetric (resp. hypersymmetric).
###### Proof.
Is obvious that symmetry (resp. hypersymmetry) of $\Xi$ implies the symmetry
(resp. hypersymmetry) of its isotropy groups
($\ell^{1}(\Xi_{u}^{u},\mathfrak{A})\hookrightarrow\ell^{\infty,1}(\Xi,\mathfrak{A})$
in a natural way). As the ’only if’ part is clear, let us prove the ’if’ part.
Let $\mathfrak{A}$ be a $C^{*}$-algebra and $\Phi$ be an arbitrary section in
$\ell^{\infty,1}(\Xi,\mathfrak{A})$. $\Phi$ has a countable support (see
Remark 2.2), and since every discrete groupoid can be decomposed as a disjoint
union of discrete transitive subgroupoids, we can find a countable number of
disjoint transitive subgroupoids $\\{\Xi(i)\\}_{i\in\mathbb{N}}$, such that
${\rm supp}(\Phi)\subset\bigcup_{i=1}^{\infty}\Xi(i)$. Define
$\Phi_{n}\in\ell^{\infty,1}(\Xi,\mathfrak{A})$ as
$\Phi_{n}(x):=\left\\{\begin{array}[]{ll}\Phi(x)&\textup{if\
}x\in\bigcup_{i=1}^{n}\Xi(i),\\\ 0_{\mathfrak{A}}&\textup{if\
}x\not\in\bigcup_{i=1}^{n}\Xi(i).\\\ \end{array}\right.$
We have that $\lim_{n}\Phi_{n}=\Phi$ in $\ell^{\infty,1}(\Xi,\mathfrak{A})$,
but more importantly,
${\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi)=\bigcup_{i=1}^{\infty}{\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n}).$ (4.3)
So, to conclude, its enough to prove that ${\rm
Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n})\subset\mathbb{R}$,
whenever $\Phi^{*}=\Phi$. Indeed, if $\Phi^{*}=\Phi$, after fixing $n$ we see
that $\Phi_{n}^{*}=\Phi_{n}$ and $\ell^{\infty,1}(\Xi,\mathfrak{A})$ contains
an isometric ∗-isomorphic copy of
$\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})$, obtained by extending
the sections in the latter algebra with zeros outside its original domain. So
by definition, one has that
$\Phi_{n}\in\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})\subset\ell^{\infty,1}(\Xi,\mathfrak{A})$,
which implies that
${\rm Spec}_{\ell^{\infty,1}(\Xi,\mathfrak{A})}(\Phi_{n})\subset{\rm
Spec}_{\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})}(\Phi_{n}).$
But
$\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})\cong\bigoplus_{i=1}^{n}\ell^{\infty,1}(\Xi(i),\mathfrak{A})$
and since the (finite) direct sum of symmetric Banach ∗-algebras is symmetric,
${\rm
Spec}_{\ell^{\infty,1}(\bigcup_{i=1}^{n}\Xi(i),\mathfrak{A})}(\Phi_{n})\subset\mathbb{R}$
and the result follows. ∎
###### Example 4.10.
Let $\Xi={\sf G}\ltimes_{\gamma}\\!X$ be a transformation groupoid (see
Definition 4.1), where $X$ is discrete. In this case, $\mathcal{U}=X$ and
$\Xi_{u}^{u}=\\{g\in G\mid\gamma_{g}(u)=u\\}\times\\{u\\}$, which is
identificable with the stabilizer of $u$, namely ${\rm
Stab}_{\gamma}(u)\leq{\sf G}$.
###### Remark 4.11.
We will finish this paper with a bit of wishful thinking: Is reasonable to
believe that the groupoid analog of [2, Theorem 3.3] could be true. For a
general locally compact groupoid one can define symmetry and hypersymmetry for
$\Xi$ as the symmetry of the Hahn algebras $L^{\infty,1}(\Xi)$ and
$L^{\infty,1}(\Xi\\!\mid\\!\mathscr{C})$, respectively. So we may pose two
problems:
1. $(i)$
Is the (hyper)symmetry of a Hausdorff locally compact groupoid $\Xi$ implied
by the (hyper)symmetry of its discretization $\Xi^{\rm dis}$?
2. $(ii)$
Is Corollary 3.6 still valid for étale groupoids?
In view of Theorem 1.2, $(i)$ is equivalent to
1. $(i^{\prime})$
Is the (hyper)symmetry of a Hausdorff locally compact groupoid $\Xi$ implied
by the (hyper)symmetry of its discretized isotropy groups $(\Xi_{u}^{u})^{\rm
dis}$?
## References
* [1] A. Austad and E. Ortega: _Groupoids and Hermitian Banach ∗-Algebras_, arXiv:2105.14793 [math.FA].
* [2] F. Flores, D. Jauré and M. Măntoiu: _Symmetry for algebras associated to Fell bundles over groups and groupoids_ , to appear in Journal of Operator Theory.
* [3] F. Flores and M. Măntoiu: _Topological dynamics for groupoid actions_ , to appear in Groups, Geometry and Dynamics.
* [4] D. Jauré and M. Măntoiu: _Symmetry and Spectral Invariance for Topologically Graded $C^{*}$-Algebras and Partial Action Systems_, Bull. London Math. Soc. 54(4), 1448–1469, (2022).
* [5] W. Kugler: _On the Symmetry of Generalized $L^{1}$-Algebras_, Math. Z. 168(3), 241–262, (1979).
* [6] A. Kumjian:_Fell Bundles over Groupoids_ , Proc. Amer. Math. Soc. 126(4), 1115–1125, (1998).
* [7] H. Leptin: _Symmetrie in Banachschen Algebren_ , Arch. Math. (Basel), 27(4), 394–400, (1976).
* [8] H. Leptin: _Ideal Theory in Group Algebras of Locally Compact Groups_. Inventiones mathematicae 31, 259-278, (1976).
* [9] H. Leptin and D. Poguntke: _Symmetry and Nonsymmetry for Locally Compact Groups_ , J. Funct. Anal. 33(2), 119–134, (1979).
* [10] M. Măntoiu: _Symmetry and Inverse Closedness for Banach $C^{*}$-Algebras Associated to Discrete Groups_, Banach Journal of Mathematical Analysis, 9(2), 289–310, (2015).
* [11] P. Muhly and D. Williams: _Equivalence and Disintegration Theorems for Fell Bundles and Their $C^{*}$-Algebras_, Dissertationes Math. (Rozprawy Mat.), 456, 1–57, (2008).
* [12] T.W. Palmer: _Banach Algebras and the General Theory of ∗-Algebras_, Vol. II. ∗-Algebras, Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, Cambridge, 2001.
* [13] D. Poguntke: _Rigidly Symmetric $L^{1}$-Group Algebras_, Seminar Sophus Lie, 2, 189–197(1992).
ADDRESS
F. Flores
Department of Mathematics, University of Virginia,
114 Kerchof Hall. 141 Cabell Dr,
Charlottesville, Virginia, United States
E-mail<EMAIL_ADDRESS>
|
# Separating Subversion Forcing Principles
Hiroshi Sakai Graduate School of System Informatics, Kobe University, Rokko-
dai 1-1, Nada, Kobe, 657-8501, Japan<EMAIL_ADDRESS>and Corey
Bacal Switzer Institute of Discrete Mathematics and Geometry, Vienna
University of Technology. Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract.
We study a family of variants of Jensen’s _subcomplete forcing axiom_ ,
$\mathsf{SCFA}$ and _subproper forcing axiom_ , $\mathsf{SubPFA}$. Using these
we develop a general technique for proving non-implications of
$\mathsf{SCFA}$, $\mathsf{SubPFA}$ and their relatives and give several
applications. For instance we show that $\mathsf{SCFA}$ does not imply
$\mathsf{MA}^{+}(\sigma$-closed$)$ and $\mathsf{SubPFA}$ does not imply
Martin’s Maximum.
###### Key words and phrases:
###### 2010 Mathematics Subject Classification:
03E17, 03E35, 03E50
_Acknowledgments:_ The first author would like to thank JSPS for the support
through grant numbers 18K03397 and 21K03338. The second author would like to
thank the Austrian Science Fund (FWF) for the generous support through grant
number Y1012-N35.
## 1\. Introduction
In this paper we study variants of subcomplete and subproper forcing classes
with an eye towards investigating and distinguishing their forcing principles.
Subcomplete and subproper forcing are two classes of forcing notions
introduced by Jensen in [13] in connection with the extended Namba problem,
see [14, Section 6.4] 111See Definition 1.4 below for precise definitions..
Both are iterable with revised countable support and generalize significantly
$\sigma$-closed and proper forcing notions respectively while allowing, under
some circumstances, new $\omega$-sequences of ordinals to be added to
uncountably cofinal cardinals. As such each comes with a forcing axiom
(consistent relative to a supercompact cardinal). The forcing axiom for
subcomplete forcing in particular, dubbed $\mathsf{SCFA}$ by Jensen in [11,
14] is especially interesting as it is consistent with $\diamondsuit$ while
implying some of the strong, structural consequences of $\mathsf{MM}$ such as
$\mathrm{SCH}$ and strong reflection principles including the failure of
square principles, see [14, Section 4]. Since their initial introduction
subcomplete and subproper forcing have been tied to several applications and
received further treatment, see for instance, [6, 8, 9, 10].
Already in [10] the second author and Fuchs found (seemingly) more general
classes, dubbed “$\infty$-subcomplete” and “$\infty$-subproper” each
containing their non “$\infty$” version respectively, and proved a variety of
iteration and preservation theorems. The main theorem in that work was that
the forcing axiom for $\infty$-subcomplete forcing notions,
$\infty$-$\mathsf{SCFA}$, is compatible with a large variety of behavior on
$\aleph_{1}$ when $\mathsf{CH}$ fails. For instance
$\aleph_{1}=\mathfrak{d}<\mathfrak{c}=\aleph_{2}$ and the existence of Souslin
trees are both consistent with $\infty$-$\mathsf{SCFA}$ $+\neg\mathsf{CH}$.
In this paper we combine the $\infty$-versions of these forcing classes with
further parametrization “above $\mu$” for cardinals $\mu$, initially
investigated, somewhat sparingly, by Jensen in [12, Section 3]. This leads to
a large family of forcing axioms $\infty$-$\mathsf{SubPFA}\upharpoonright\mu$
and $\infty$-$\mathsf{SCFA}\upharpoonright\mu$, where
$\infty$-$\mathsf{SubPFA}$ and $\infty$-$\mathsf{SCFA}$ coincide with
$\infty$-$\mathsf{SubPFA}\upharpoonright 2^{\aleph_{0}}$ and
$\infty$-$\mathsf{SCFA}\upharpoonright 2^{\aleph_{0}}$ respectively. The main
motivation of this work is to investigate how these axioms relate to one
another and to other, more well known axioms such as $\mathsf{MM}$ and
$\mathsf{MA}^{+}(\sigma\mbox{-closed})$. Formal definitions will be given in
the second part of this introduction and Section 2 but the definitions of
these axioms alongside well known results provides almost immediately that the
following diagram of implications holds with $2^{\aleph_{0}}\leq\nu<\mu$
cardinals.
$\mathsf{MM}$$\mathsf{MA}^{+}(\sigma{\rm-
closed})$$\infty$-$\mathsf{SubPFA}$$\infty$-$\mathsf{SubPFA}\upharpoonright\nu$$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$$\mathsf{PFA}$$\forall\kappa\neg\square_{\kappa}$$\infty$-$\mathsf{SCFA}$$\infty$-$\mathsf{SCFA}\upharpoonright\nu$$\infty$-$\mathsf{SCFA}\upharpoonright\mu$$\forall\kappa\geq
2^{\aleph_{0}}\neg\square_{\kappa}$$\forall\kappa\geq\nu^{\aleph_{0}}\neg\square_{\kappa}$$\forall\kappa\geq\mu^{\aleph_{0}}\neg\square_{\kappa}$
Figure 1. Subversion forcing principles and then some
The main result of this work is that essentially no arrows are missing from
Figure 1 above.
###### Main Theorem 1.1.
Let $2^{\aleph_{0}}\leq\nu\leq\lambda<\mu=\lambda^{+}$ be cardinals with
$\nu^{\omega}<\mu$. Assuming the consistency of a supercompact cardinal, the
implications given in Figure 1 are complete in the sense that if no
composition of arrows exists from one axiom to another then there is a model
of $\mathsf{ZFC}$ in which the implication fails222Except for the trivial
$\forall\kappa\neg\square_{\kappa}\to\forall\kappa\geq
2^{\aleph_{0}}\neg\square_{\kappa}$ which did not fit aesthetically into the
picture..
As a corollary of this theorem and its proof we obtain separations of several
“subversion” forcing principles from other, more well-studied reflection
principles and forcing axioms. For instance we show the following.
###### Theorem 1.1 (See Theorem 3.1).
Assuming the consistency of a supercompact cardinal, $\mathsf{SCFA}$ does not
imply the failure of $\square_{\aleph_{1}}$ when $\mathsf{CH}$ fails.
###### Corollary 1.2.
Assuming the consistency of a supercompact cardinal, $\mathsf{SCFA}$ does not
imply $\mathsf{MA}^{+}(\sigma{\mbox{\rm-closed}})$.
The rest of this paper is organized as follows. In the next subsection of this
introduction we give relevant background and terminology. In the next Section
we introduce the variants $\infty$-subcompleteness and $\infty$-subproperness
above $\mu$ and discuss some of their properties. In Section 3 we study the
forcing axioms associated to these classes and show, amongst other things,
that they are distinct as well as the fact $\infty$-$\mathsf{SCFA}$ implies
neither $\mathsf{MA}^{+}(\sigma{\rm-closed})$ nor $\neg\square_{\kappa}$ for
any $\kappa<2^{\omega}$. In Section 4 we continue this investigation and show
that $\infty$-$\mathsf{SubPFA}$ does not imply $\mathsf{MM}$. Section 5
concludes with some final remarks and open problems.
### 1.1. Preliminaries
We conclude this introduction with the key definitions we will use throughout,
beginning with that of subproperness and subcompleteness. These are these two
classes of forcing notions defined by Jensen in [14] which have found several
applications, see e.g [17, 13, 6, 9]. More discussion of these concepts can be
found in [14] or [10]. Before beginning with the definition we will need one
preliminary definition. Below we denote by $\mathsf{ZFC}^{-}$ the axioms of
$\mathsf{ZFC}$ without the power set axiom.
###### Definition 1.3.
A transitive set $N$ (usually a model of $\mathsf{ZFC}^{-}$) is _full_ if
there is an ordinal $\gamma$ so that $L_{\gamma}(N)\models\mathsf{ZFC}^{-}$
and $N$ is regular in $L_{\gamma}(N)$ i.e. for all $x\in N$ and $f\in
L_{\gamma}(N)$ if $f:x\to N$ then ${\rm ran}(f)\in N$.
###### Definition 1.4.
Let $\mathbb{P}$ be a forcing notion and let $\delta(\mathbb{P})$ be the least
size of a dense subset of $\mathbb{P}$.
1. (1)
We say that $\mathbb{P}$ is _subcomplete_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta})=\mathbb{P},s,\theta$ , if
$\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$
2\. $\sigma^{\prime}``\bar{G}\subseteq G$
3\. ${\rm Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma))={\rm
Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma^{\prime}))$
2. (2)
We say that $\mathbb{P}$ is _subproper_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$
2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$
3\. ${\rm Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma))={\rm
Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma^{\prime}))$
Note that the special case where $\sigma=\sigma^{\prime}$ is properness (for
subproperness) and (up to forcing equivalence) $\sigma$-closedness (for
subcomplete). It was pointed out in [10] that the “Hulls” condition 3) in both
definitions is somewhat unnatural. Indeed it is never used in applications and
appears solely for the purpose of proving the iteration theorem, [14, Theorem
3]. In [10] Fuchs and the second author showed that by iterating with
Miyamoto’s _nice iterations_ this condition could be avoided. As such it makes
sense to define the following.
###### Definition 1.5.
Let $\mathbb{P}$ be a forcing notion.
1. (1)
We say that $\mathbb{P}$ is $\infty$-_subcomplete_ if for all sufficiently
large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta})=\mathbb{P},s,\theta$ , if
$\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$
2\. $\sigma^{\prime}``\bar{G}\subseteq G$
2. (2)
We say that $\mathbb{P}$ is $\infty$-_subproper_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$
2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$
To be clear this is just the same as the definitions of the “non-$\infty$”
versions, simply with the additional “Hulls” condition removed. As mentioned
these classes come with an iteration theorem.
###### Theorem 1.6 (Theorem 3.19 (for Subcomplete) and Theorem 3.20 (for
Subproper) of [10]).
Let $\gamma$ be an ordinal and
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\;|\;\alpha<\gamma\rangle$
be a nice iteration in the sense of Miyamoto so that for all $\alpha<\gamma$
we have $\Vdash_{\mathbb{P}_{\alpha}}$“$\dot{\mathbb{Q}}_{\alpha}$ is
$\infty$-subproper (respectively $\infty$-subcomplete). Then
$\mathbb{P}_{\gamma}$ is $\infty$-subproper (respectively
$\infty$-subcomplete).
We note that the above theorem in the case of $\infty$-subproper forcing was
originally proved first independently by Miyamoto in [16]. A consequence of
this theorem (initially observed for the non $\infty$-versions by Jensen) is
that, modulo a supercompact cardinal, these classes have a consistent forcing
axiom.
###### Definition 1.7.
Let $\Gamma$ be a class of forcing notions. The _forcing axiom for_ $\Gamma$,
denoted $\mathsf{FA}(\Gamma)$ is the statement that for all $\mathbb{P}$ in
$\Gamma$ and any $\omega_{1}$-sequence of dense subsets of $\mathbb{P}$, say
$\\{D_{i}\;|\;i<\omega_{1}\\}$ there is a filter $G\subseteq\mathbb{P}$ which
intersects every $D_{i}$.
If $\Gamma$ is the class of ($\infty$-)subproper forcing notions we denote
$\mathsf{FA}(\Gamma)$ by ($\infty$-)$\mathsf{SubPFA}$. Similarly if $\Gamma$
is the class of ($\infty$-)subcomplete forcing notions we denote
$\mathsf{FA}(\Gamma)$ by ($\infty$-)$\mathsf{SCFA}$.
It is not known whether up to forcing equivalence each class is simply equal
to its “$\infty$”-version or if their corresponding forcing axioms are
equivalent. However, since the “$\infty$” versions are more general (or appear
to be) and avoid the unnecessary technicality of computing hulls we will work
with them in this paper. Nearly everything written here however could be
formulated for the “non-$\infty$” versions equally well, though we leave the
translation to the oddly interested reader.
If $\Gamma\subseteq\Delta$ then $\mathsf{FA}(\Delta)$ implies
$\mathsf{FA}(\Gamma)$ so we get the following collection of implications,
which are part of Figure 1.
###### Proposition 1.8.
$\mathsf{MM}\to\infty\mbox{-}\mathsf{SubPFA}\to\mathsf{PFA}$ and
$\mathsf{MM}\to\infty\mbox{-}\mathsf{SubPFA}\to\infty\mbox{-}\mathsf{SCFA}$
Here $\mathsf{MM}$, known as _Martin’s Maximum_ and introduced in [5], is the
forcing axiom for forcing notions which preserve stationary subsets of
$\omega_{1}$ (all $\infty$-subproper forcing notions have this property) and
$\mathsf{PFA}$ is the forcing axiom for proper forcing notions. It is known
from the work of Jensen, see also [10] that none of the above implications can
be reversed with the exception of whether $\mathsf{SubPFA}$ implies
$\mathsf{MM}$. In this paper we will show the consistency of
$\mathsf{SubPFA}+\neg\mathsf{MM}$, see Theorem 4.1 below.
On that note we move to our last preliminary. Many of the theorems in this
paper involve showing that we can preserve some fragment of
$\infty$-$\mathsf{SCFA}$ (or $\infty$-$\mathsf{SubPFA}$) via a forcing killing
another fragment of it. Towards this end we will need an extremely useful
theorem due to Cox. Below recall that a class of forcing notions $\Gamma$ is
_closed under restrictions_ (see Definition 39 of [2]) if for all
$\mathbb{P}\in\Gamma$ and all $p\in\mathbb{P}$ the lower cone
$\mathbb{P}\upharpoonright p:=\\{q\in\mathbb{P}\;|\;q\leq p\\}\in\Gamma$. One
can check that both the classes of $\infty$-subcomplete and $\infty$-subproper
forcing notions (as well as the restrictions “above $\mu$” defined in Section
2) have this property.
###### Theorem 1.9 (Cox, see Theorem 20 of [2]).
Let $\Gamma$ be a class of forcing notions closed under restrictions and
assume $\mathsf{FA}(\Gamma)$ holds. Let $\mathbb{P}$ be a forcing notion.
Suppose that for every $\mathbb{P}$-name $\dot{\mathbb{Q}}$ for a forcing
notion in $\Gamma$ there is a $\mathbb{P}*\dot{\mathbb{Q}}$-name
$\dot{\mathbb{R}}$ for a forcing notion so that the following hold:
1. (1)
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is in $\Gamma$
2. (2)
If $j:V\to N$ is a generic elementary embedding,
$\theta\geq|\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}|^{+}$ is regular in
$V$ and
a) $H_{\theta}^{V}$ is in the wellfounded part of $N$;
b) $j``H_{\theta}^{V}\in N$ has size $\omega_{1}$ in $N$;
c) ${\rm crit}(j)=\omega_{2}^{V}$
d) There exists a $G*H*K$ in $N$ that is
$(H_{\theta}^{V},\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}})$-generic
Then $N$ believes that $j``G$ has a lower bound in $j(\mathbb{P})$
Then $\Vdash_{\mathbb{P}}\mathsf{FA}(\Gamma)$ i.e. $\mathbb{P}$ preserves the
forcing axiom for $\Gamma$.
See [2] for more on strengthenings and generalizations of this wide ranging
theorem.
## 2\. $\infty$-Subcompleteness and $\infty$-Subproperness above $\mu$
Most theorems in this paper filter through the notions of
$\infty$-_subcompleteness (respectively $\infty$-subproperness) above_ $\mu$
for a cardinal $\mu$. These are technical strengthenings of
$\infty$-subcompleteness (respective $\infty$-subproperness). In this section
we define these strengthenings as well as make some elementary observations
which will be used in rest of the paper.
###### Definition 2.1.
Let $\mu$ be a cardinal and $\mathbb{P}$ a forcing notion.
1. (1)
We say that $\mathbb{P}$ is $\infty$-_subcomplete above_ $\mu$ if for all
sufficiently large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$,
if $\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$
2\. $\sigma^{\prime}``\bar{G}\subseteq G$
3\. $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$
2. (2)
We say that $\mathbb{P}$ is $\infty$-_subproper above_ $\mu$ if for all
sufficiently large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=p,\mathbb{P},s,\theta,\mu$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that
1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=p,\mathbb{P},s,\theta,\mu$
2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$
3\. $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$
Concretely being $\infty$-subcomplete above $\mu$ simply means that
$\mathbb{P}$ is subcomplete and, moreover, for any $\sigma:\bar{N}\prec N$ the
corresponding $\sigma^{\prime}$ (in $V[G]$) witnessing the subcompleteness can
be arranged to agree with $\sigma$ “up to $\mu$” i.e. on the ordinals below
$\sigma^{-1}``\mu$ (and idem for $\infty$-subproperness). The “non-$\infty$”
versions of these classes was first introduced by Jensen in [13] under the
name “$\mu$-subcompleteness” and “$\mu$-subproperness”. They were investigated
further by Fuchs in [7] who introduced the terminology “above $\mu$” and made
several of the elementary observations we repeat below. Following Fuchs (as
opposed to Jensen) have moved the parameter $\mu$ to the end to avoid the
awkwardness of “$\mu$-$\infty$-subcomplete/$\mu$-$\infty$-subproper”. The
following is immediate from the definitions.
###### Observation 2.2.
Let $\mu<\nu$ be cardinals. If $\mathbb{P}$ is $\infty$-subcomplete
(respectively $\infty$-subproper) above $\nu$ then it is $\infty$-subcomplete
(respectively subproper) above $\mu$ and it is $\infty$-subcomplete
(respectively $\infty$-subproper) (without any restriction).
It is easy to see that being $\infty$-subcomplete (respectively,
$\infty$-subproper) is equivalent to being $\infty$-subcomplete (respectively
$\infty$-subproper) above $\omega_{1}$, however more is true, an observation
due independently to the first author and Fuchs (see [7, Observation 4.2]).
###### Proposition 2.3.
Let $\mathbb{P}$ be a forcing notion. $\mathbb{P}$ is $\infty$-subcomplete
(respectively $\infty$-subproper) if and only if $\mathbb{P}$ is
$\infty$-subcomplete above $2^{\aleph_{0}}$ (respectively $\infty$-subproper
above $2^{\aleph_{0}}$).
As noted above, this proposition (in the case of subcompleteness) is proved as
Observation 4.2 of [7] but we give a detailed proof in order to help the
reader get accustomed to $\infty$-subversion forcing as well as to include the
mild difference of subproperness. However, let us note that essentially the
point is that, using the definable well order in $L_{\tau}[A]$, the reals of
$\bar{N}$ code the cardinality of the continuum.
###### Proof.
We prove the case of $\infty$-subcomplete and leave the reader to check the
case of $\infty$-subproper. Let $\mathbb{P}$ be a forcing notion. It is
immediate as noted above that if $\mathbb{P}$ is $\infty$-subcomplete above
$2^{\omega}$ then it is $\infty$-subcomplete so we need to check just the
reverse direction. Thus assume that $\mathbb{P}$ is $\infty$-subcomplete and
let $\tau>\theta$ be cardinals so that $\sigma:\bar{N}\prec N:=L_{\tau}[A]$
with $H_{\theta}\subseteq N$ be as in the definition of
$\infty$-subcompleteness. Let $\bar{G}$ be a $\bar{\mathbb{P}}$-generic filter
over $\bar{N}$ with $\bar{\mathbb{P}}$ the preimage of $\mathbb{P}$ under
$\sigma$. Finally let $p\in\mathbb{P}$ force that there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that
$\sigma^{\prime}(\bar{\mathbb{P}})=\mathbb{P}$ and
$\sigma^{\prime}``\bar{G}\subseteq G$ for any generic $G\ni p$ (the existence
of such a condition is the heart of the definition of $\infty$-subcompleteness
of course). We need to show that $p$ forces that
$\sigma^{\prime}\upharpoonright 2^{\aleph_{0}}=\sigma\upharpoonright
2^{\aleph_{0}}$, where, to be clear, $2^{\aleph_{0}}$ denotes cardinal (as
computed in $\bar{N}$) which bijects onto the continuum (as defined in
$\bar{N}$). To avoid confusion let us denote the cardinal
$2^{\aleph_{0}}=\kappa$ (in $V$ and hence $N$) and the preimage of $\kappa$ in
$\bar{N}$ under $\sigma$ as $\bar{\kappa}$. First note that by the
absoluteness of $\omega$ we have that for all reals $x\in\bar{N}$ it must be
the case that $\sigma(x)=\sigma^{\prime}(x)=x$ (and being a real is absolute
between $\bar{N}$ and $V$). Moreover, since $N=L_{\tau}[A]$ there is a
definable well order of the universe, and in particular there is a definable
bijection of the reals onto $\kappa$, say $f:2^{\omega}\to\kappa$. By
elementarity in $\bar{N}$ there is a definable bijection
$\bar{f}:2^{\omega}\cap\bar{N}\to\bar{\kappa}$. But since $f$ is definable we
have $\sigma(\bar{f})=\sigma^{\prime}(\bar{f})=f$ and hence for all
$\alpha\in\bar{\kappa}$ we get
$\sigma(\alpha)=\sigma(\bar{f}(\bar{f}^{-1}(\alpha)))=\sigma^{\prime}(\bar{f}(\bar{f}^{-1}(\alpha)))=\sigma^{\prime}(\alpha)$,
as needed. ∎
In fact $2^{\aleph_{0}}$ is the best possible in $\mathsf{ZFC}$. Jensen showed
that Namba forcing is $\infty$-subcomplete above $\omega_{1}$ assuming
$\mathsf{CH}$ while it is not even $\infty$-subproper above $\omega_{2}$ in
$\mathsf{ZFC}$, a consequence of the next observation.
###### Lemma 2.4.
Let $\mu$ be a cardinal.
1. (1)
If $\mathbb{P}$ is $\infty$-subproper above $\mu$ then any new countable set
of ordinals less than $\mu$ added by $\mathbb{P}$ is covered by an old
countable set of ordinals (less than $\mu$). In particular if
$\Vdash_{\mathbb{P}}$“${\rm cf}(\mu)=\omega$” then ${\rm cf}(\mu)=\omega$ (in
$V$).
2. (2)
If $\mathbb{P}$ is $\infty$-subcomplete above $\mu$ then $\mathbb{P}$ adds no
new countable sets of ordinals below $\mu$.
###### Proof.
The proofs of both are similar to the corresponding proofs that every new
countable set of ordinals added by a proper forcing notion is contained in an
old countable set of ordinals and $\sigma$-closed forcing notions do not add
new countable sets of ordinals at all respectively. The point is that to show
the corresponding fact “below $\mu$” one only needs $\infty$-subproperness
(respectively $\infty$-subcompleteness) above $\mu$.
Let us begin with the first item. Assume that $\mu$ is a cardinal,
$\mathbb{P}$ is $\infty$-subproper above $\mu$, $p\in\mathbb{P}$ and $\dot{x}$
is a $\mathbb{P}$-name so that $p\Vdash\dot{x}:\omega\to\mu$. We need to find
a $q\leq p$ and a countable $X\subseteq\mu$ so that $q\Vdash{\rm
im}(\dot{x})\subseteq\check{X}$. To this end, let $\sigma:\bar{N}\prec N$ be
as in the definition of $\infty$-subproperness with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$.
We claim that $X:=\sigma``\bar{\mu}$ is as needed. Indeed, by the definition
of $\infty$-subproperness above $\mu$ there is a $q\leq p$ forcing that there
is an embedding $\sigma^{\prime}:\bar{N}\prec N$ so that
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$
and $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$.
Fix such a $q$ and let $q\in G$ be $\mathbb{P}$-generic over $V$. Note that by
elementarity we have that $\bar{p}\Vdash\dot{\bar{x}}(\check{n})\in\bar{\mu}$
for all $n<\omega$. Also since $\bar{G}:=(\sigma^{\prime})^{-1}``G$ is
$\bar{\mathbb{P}}$-generic over $\bar{N}$ we can find in $\bar{N}$ ordinals
$\bar{\mu}_{n}<\bar{\mu}$ so that
$\bar{N}[\bar{G}]\models\dot{\bar{x}}^{\bar{G}}(n)=\bar{\mu}_{n}$ for each
$n<\omega$. Finally we have that and
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$
and hence, by elementarity again alongside the fact that
$\sigma\upharpoonright\bar{\mu}=\sigma^{\prime}\upharpoonright\bar{\mu}$ we
get
$N[G]\models\dot{x}^{G}(n)=\sigma^{\prime}(\bar{\mu}_{n})=\sigma(\bar{\mu}_{n})\in
X$ for all $n<\omega$. This completes the proof of the first item.
The second case is nearly verbatim noting that since for $\infty$-subcomplete
forcing we can choose $\bar{G}$ in the ground model we can actually define
$\sigma``\dot{\bar{x}}^{\bar{G}}=(\sigma^{\prime})``\dot{\bar{x}}^{\bar{G}}$
in $V$. ∎
As mentioned before Lemma 2.4 an immediate consequence is the following.
###### Lemma 2.5.
Namba forcing is not $\infty$-subproper above $\omega_{2}$. In particular
Namba forcing is not $\infty$-subproper if $\mathsf{CH}$ fails.
Finally we end this section with some observations about the associated
forcing axioms for the classes we have been discussing.
###### Definition 2.6.
Let $\mu$ be a cardinal. Denote by
$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$ the forcing axiom for forcing
notions $\mathbb{P}$ which are $\infty$-subproper above $\mu$ and
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ the same for $\mathbb{P}$ which are
$\infty$-subcomplete above $\mu$.
The following is immediate by Observation 2.2.
###### Proposition 2.7.
Let $\mu<\nu$ be cardinals. We have that $\infty$-$\mathsf{SCFA}$ implies
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ implies
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$. Similarly for the variants of
$\infty$-$\mathsf{SubPFA}$.
In the next section we will show that (in many cases) the reverse implications
do not hold.
## 3\. Separating the $\infty$-$\mathsf{SCFA}\upharpoonright\mu$ Principles
In this section we show that under certain cardinal arithmetic assumptions
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$ does not imply
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ for $\mu<\nu$. Before proving this
general theorem we introduce our technique with the simple example of
separating $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{1}$ from
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. This involves showing that
adding a $\square_{\omega_{1}}$-sequence to a model of
$\infty$-$\mathsf{SCFA}$ preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. However, it follows from a
result of Jensen, see [14] that $\mathsf{SCFA}+\mathsf{CH}$ implies the
failure of $\square_{\omega_{1}}$.
### 3.1. The Case of $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$:
Adding Non-Reflecting Structures of Size $\aleph_{2}$
Recall that for an uncountable cardinal $\lambda$ a
$\square_{\lambda}$-sequence is a sequence $\langle
C_{\alpha}\;|\;\alpha\in\lambda^{+}\cap{\rm Lim}\rangle$ so that for all
$\alpha$ the following hold:
1. (1)
$C_{\alpha}$ is club in $\alpha$
2. (2)
${\rm ot}(\alpha)\leq\lambda$
3. (3)
For each $\beta\in{\rm lim}(C_{\alpha})$ we have that
$C_{\alpha}\cap\beta=C_{\beta}$
We recall the poset $\mathbb{P}_{0}$ from [3, Example 6.6] for adding a square
sequence. Conditions $p\in\mathbb{P}_{0}$ are functions so that the domain of
$p$ is $\beta+1\cap{\rm Lim}$ for some $\beta\in\lambda^{+}\cap{\rm Lim}$ and
1. (1)
For all $\alpha\in{\rm dom}(p)$ we have that $p(\alpha)$ is club in $\alpha$
with order type $\leq\lambda$; and
2. (2)
If $\alpha\in{\rm dom}(p)$ then for each $\beta\in{\rm lim}(p(\alpha))$ we
have $p(\alpha)\cap\beta=p(\beta)$
The order is end extension. We remark that a moment’s reflection confirms that
this poset is $\sigma$-closed. Moreover it is ${<}\lambda^{+}$-strategically
closed (see [3]). In particular it preserves cardinals up to $\lambda^{+}$.
###### Theorem 3.1.
Assume $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$ and let
$\mathbb{P}_{0}$ be the forcing notion defined above for adding a
$\square_{\omega_{1}}$-sequence. Then $\Vdash_{\mathbb{P}_{0}}$
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. In particular if the
existence of a supercompact cardinal is consistent with $\mathsf{ZFC}$ then
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}+\square_{\omega_{1}}$ is
consistent as well.
Before proving this theorem we need to define one more poset. Recall that if
$G\subseteq\mathbb{P}_{0}$ is generic and $\vec{\mathcal{C}}_{G}=\langle
C_{\alpha}\;|\;\alpha\in\lambda^{+}\cap{\rm Lim}\rangle$ is the generic
$\square_{\lambda}$-sequence added by $G$ then for any cardinal
$\gamma<\lambda$ we can _thread the square sequence_ via the following poset,
$\mathbb{T}_{G,\gamma}$. Conditions are closed, bounded subsets
$c\subseteq\lambda^{+}$ so that $c$ has order type $<\gamma$, and for all
limit points $\beta\in c$ we have that $\beta\cap c=C_{\beta}$. See [4, §6]
and [15, p.7] for more on this threading poset. The point is the following.
###### Fact 3.2 (Lemma 6.9 of [4]).
Let $\gamma<\lambda$ be cardinals, $\mathbb{P}_{0}$ the forcing notion
described above for adding a $\square_{\lambda}$-sequence and
$\dot{\mathbb{T}}_{\dot{G},\gamma}$ be the $\mathbb{P}_{0}$-name for the
forcing to thread the generic square sequence with conditions of size
$<\gamma$. Then $\mathbb{P}_{0}*\dot{\mathbb{T}}_{\dot{G},\gamma}$ has a dense
$<\gamma$-closed subset.
We can now prove Theorem 3.1.
###### Proof.
We let $\mathbb{P}_{0}$ be the forcing described above for adding a
$\square_{\omega_{1}}$-sequence (so $\lambda=\omega_{1}$). Let
$\gamma=\aleph_{1}$ so in $V^{\mathbb{P}_{0}}$ the threading poset
$\dot{\mathbb{T}}:=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$ consists of
countable closed subsets of $\omega_{2}$. We want to apply Theorem 1.9 to
$\mathbb{P}_{0}$. Note that if $\dot{\mathbb{Q}}$ is a $\mathbb{P}_{0}$-name
for an $\infty$-subcomplete above $\omega_{2}$ forcing notion, then
$\dot{\mathbb{T}}=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$ is absolute between
$V^{\mathbb{P}_{0}}$ and $V^{\mathbb{P}_{0}*\dot{\mathbb{Q}}}$ by Lemma 2.4
(2).
###### Claim 3.3.
It is enough to show that for any $\mathbb{P}_{0}$-name $\dot{\mathbb{Q}}$ for
a forcing notion which is $\infty$-subcomplete above $\omega_{2}$, the three
step $\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is
$\infty$-subcomplete above $\omega_{2}$.
###### Proof of Claim.
This is because $\mathbb{T}$ adds a lower bound to $j``G$ as described in the
statement of Theorem 1.9. In more detail, let $\dot{\mathbb{Q}}$ be a
$\mathbb{P}_{0}$-name for a forcing notion which is $\infty$-subcomplete above
$\omega_{2}$, we want to show that for $\dot{\mathbb{R}}=\dot{\mathbb{T}}$ the
hypotheses of Theorem 1.9 are satisfied assuming that
$\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is $\infty$-subcomplete
above $\omega_{2}$. Since this is exactly the first clause we only need to
concern ourselves with the second one. Recall that, relativized to this
situation, this says that if $j:V\to N$ is a generic elementary embedding,
$\theta\geq|\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}|^{+}$ is regular
in $V$ and
a) $H_{\theta}^{V}$ is in the wellfounded part of $N$;
b) $j``H_{\theta}^{V}\in N$ has size $\omega_{1}$ in $N$;
c) ${\rm crit}(j)=\omega_{2}^{V}$
d) There exists a $G*H*K$ in $N$ that is
$(H_{\theta}^{V},\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}})$-generic
Then $N$ believes that $j``G$ has a lower bound in $j(\mathbb{P}_{0})$.
So fix some $\theta$ and $j:V\to N$ as described in a) to d). Note that
$j``G=G$ by c) and the fact that $G$ is coded as a subset of $\omega_{2}^{V}$.
Thus it suffices to find a lower bound of $G$ in $j(\mathbb{P}_{0})$. The
point is now though that since $G*H*K\in N$ we can in particular form $\bigcup
K\in N$ which is a club subset of $\omega_{2}^{V}=\sup_{p\in G}{\rm dom}(p)$
and coheres with all of the elements of $G$, and hence $(\bigcup
G)\cup\langle\omega_{2}^{V},\bigcup K\rangle$ is as needed. ∎
Let us now show that $\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is
$\infty$-subcomplete above $\omega_{2}$. Let $\tau>\theta$ be sufficiently
large cardinals and $\sigma:\bar{N}\prec N=L_{\tau}[A]\supseteq H_{\theta}$ be
as in the definition of $\infty$-subcompleteness above $\omega_{2}$. Let
$\sigma(\bar{\mathbb{P}}_{0},\dot{\bar{\mathbb{Q}}},\dot{\bar{\mathbb{T}}})=\mathbb{P}_{0},\dot{\mathbb{Q}},\dot{\mathbb{T}}$.
Let $\bar{G}*\bar{H}*\bar{K}$ be
$\bar{\mathbb{P}}_{0}*\dot{\bar{\mathbb{Q}}}*\dot{\bar{\mathbb{T}}}$-generic
over $\bar{N}$. There are few things to note. First let us point out that
$\bar{G}$ and $\bar{K}$ are (coded as) subsets of $\bar{\omega}_{2}$, the
second uncountable cardinal from the point of view of $\bar{N}$ (so
$\sigma(\bar{\omega}_{2})=\omega_{2}$). Next note that
$\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is forcing equivalent to
$\mathbb{P}_{0}*\dot{\mathbb{T}}*\dot{{\mathbb{Q}}}$ since both
$\dot{\mathbb{Q}}$ and $\dot{\mathbb{T}}$ are in $V^{\mathbb{P}_{0}}$, and the
same for the “bar” versions in $\bar{N}$. Now note that since
$\mathbb{P}_{0}*\dot{\mathbb{T}}$ has a $\sigma$-closed dense subset,
$\sigma``\bar{G}*\bar{K}$ has a lower bound (in $N$), say $(p,t)$ ($t$ is in
the ground model and the $\sigma$-closed dense subset is simply the collection
of conditions whose second coordinate is a check name decided by $p$). By
$\sigma$-closed-ness $(p,t)$ forces that there is a unique lift of
$\sigma:\bar{N}\prec N$ to some $\sigma_{0}:\bar{N}[\bar{G}]\prec N[G]$ with
$\sigma_{0}(\bar{G})=G$ for any $\mathbb{P}_{0}$-generic $G\ni p$ (technically
we need to work in the extension by $\mathbb{P}_{0}*\dot{\mathbb{Q}}$, but we
only want to specify the embedding of the $\bar{\mathbb{P}}_{0}$ extension).
Fix such a $G$ (from which $\sigma_{0}$ is defined) and work in $V[G]$. Note
that $\sigma_{0}``\bar{K}=\sigma``\bar{K}$ has $t\in N$ as a lower bound.
Since $\mathbb{Q}:=\dot{\mathbb{Q}}^{G}$ is $\infty$-subcomplete above
$\omega_{2}$ here we can apply the definition of $\infty$-subcompleteness to
$\sigma_{0}:\bar{N}[\bar{G}]\prec N[G]$ to obtain a condition
$\dot{q}^{G}:=q\in\mathbb{Q}$ so that if $H\ni q$ is $\mathbb{Q}$-generic over
$V[G]$ then in $V[G][H]$ there is a $\sigma_{1}:\bar{N}[\bar{G}]\prec N[G]$ so
that
$\sigma_{1}(\bar{G},\bar{\mathbb{P}}_{0},\dot{\bar{\mathbb{Q}}}^{\bar{G}},\dot{\bar{\mathbb{T}}}^{\bar{G}})=G,\mathbb{P}_{0},\mathbb{Q},\mathbb{T}$
where $\mathbb{T}\in V[G]$ is $\dot{\mathbb{T}}^{G}$,
$\sigma_{1}``\bar{H}\subseteq H$ and
$\sigma_{1}\upharpoonright\bar{\omega}_{2}=\sigma\upharpoonright\bar{\omega}_{2}$.
Note also that by condensation we have that $\bar{N}=L_{\bar{\tau}}[\bar{A}]$
and hence we can ensure that $\sigma_{1}\upharpoonright\bar{N}:\bar{N}\prec
N$.
Now by the first observation above we know that since $\bar{G}$ and $\bar{K}$
are coded as subsets of $\bar{\omega}_{2}$ it must be the case that in fact
$\sigma_{1}\upharpoonright\bar{G}=\sigma\upharpoonright\bar{G}$ and idem for
$\bar{K}$. In particular $(p,t)$ is still a lower bound in
$\sigma_{1}``\bar{G}*\bar{K}$. But putting all of these observations together
now ensures that the triple
$(p,\dot{q},t)\in\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ forces that
$\sigma_{2}:=\sigma_{1}\upharpoonright\bar{N}$ is as needed to witness that
the three step is $\infty$-subcomplete above $\omega_{2}$ as needed. ∎
Note the following corollary of Theorem 3.1.
###### Corollary 3.4.
$\infty$-$\mathsf{SCFA}$ does not imply $\mathsf{MA}^{+}(\sigma{\mbox{\rm-
closed}})$ assuming the consistency of a supercompact cardinal. In particular
$\infty$-$\mathsf{SCFA}$ does not imply $\mathsf{SCFA}^{+}$.
###### Proof.
Begin with a model of
$\infty$-$\mathsf{SCFA}+2^{\aleph_{0}}=2^{\aleph_{1}}=\aleph_{2}$ (for
instance a model of $\mathsf{MM}$). Force with $\mathbb{P}_{0}$ to preserve
these axioms and add a $\square_{\omega_{1}}$-sequence. Then
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$ and $\square_{\omega_{1}}$
hold in the extension by Theorem 3.1. But, since $\mathbb{P}_{0}$ does not
collapse cardinals (by $2^{\aleph_{1}}=\aleph_{2}$) or add reals, the
continuum is still $\aleph_{2}$ hence $\infty$-$\mathsf{SCFA}$ holds yet
$\mathsf{MA}^{+}(\sigma\mbox{-closed})$ fails since this axiom implies that
$\square_{\kappa}$ fails for all $\kappa$, see [5]. ∎
The proof of Theorem 3.1 can be generalized in many ways. Observe that very
little about $\mathbb{P}_{0}$ is used. For instance almost an analogous proof
gives the following.
###### Theorem 3.5.
Assume $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. The forcing
$\mathbb{S}_{\omega_{2}}$ to add an $\omega_{2}$-Souslin tree preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$.
###### Sketch.
Let $\mathbb{S}_{\omega_{2}}$ be the standard forcing to add an
$\omega_{2}$-Souslin tree: conditions are binary trees $p\subseteq
2^{<\omega_{2}}$ of size $<\aleph_{2}$ ordered by end extension. This adds an
$\omega_{2}$-Souslin tree and is $\sigma$-closed. Let $\dot{T}_{\dot{G}}$ be
the canonical name for the tree added i.e. if
$G\subseteq\mathbb{S}_{\omega_{2}}$ is generic over $V$ then
$(\dot{T}_{\dot{G}})^{G}=\bigcup G$. Let $\dot{\mathbb{Q}}$ be a
$\mathbb{S}_{\omega_{2}}$-name for a forcing notion which is
$\infty$-subcomplete above $\omega_{2}$. As before it is enough to show that
$\mathbb{S}_{\omega_{2}}*\dot{\mathbb{Q}}*\dot{T}_{\dot{G}}$ is
$\infty$-subcomplete above $\omega_{2}$ where $\dot{T}_{\dot{G}}$ is the name
for the tree as a forcing notion. The proof is now though exactly as before
noting that, since $\dot{T}_{\dot{G}}\in V^{\mathbb{S}_{\omega_{2}}}$ we have
again that $\mathbb{S}_{\omega_{2}}*\dot{\mathbb{Q}}*\dot{T}_{\dot{G}}$ is
forcing equivalent to
$\mathbb{S}_{\omega_{2}}*\dot{T}_{\dot{G}}*\dot{\mathbb{Q}}$ and the generic
is coded by a subset of $\omega_{2}$ hence not moved by the new embedding
added by $\dot{\mathbb{Q}}$. ∎
We have the following corollary similar to Corollary 3.4 above by invoking a
model of $\infty$-$\mathsf{SCFA}+2^{\aleph_{0}}=\aleph_{2}$.
###### Corollary 3.6.
Assuming the consistency of a supercompact cardinal we have the consistency of
$\mathsf{SCFA}+\neg\mathsf{CH}+\neg{\rm TP}(\omega_{2})$.
Here ${\rm TP}(\omega_{2})$ is the tree property at $\omega_{2}$ i.e. no
$\omega_{2}$-Aronszajn trees. This result contrasts with [18, Corollary 4.1]
which shows that under Rado’s Conjecture, another forcing axiom-like statement
compatible with $\mathsf{CH}$, ${\rm TP}(\omega_{2})$ is equivalent to
$\neg\mathsf{CH}$.
### 3.2. The General Case
The proof of Theorem 3.1 can be easily generalized to establishe that for any
cardinal $\mu$ adding a $\square_{\mu}$ sequence via $\mathbb{P}_{0}$
preserves $\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$.
###### Theorem 3.7.
Let $\mu$ be an uncountable cardinal and assume
$\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$ holds. If $\mathbb{P}_{0}$ is
the forcing from the previous subsection to add a $\square_{\mu}$-sequence
then $\mathbb{P}_{0}$ preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$.
###### Proof.
In $V^{\mathbb{P}_{0}}$ let
$\dot{\mathbb{T}}:=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$. We only give the
proof of the claim obtained from Claim 3.3 by replacing $\omega_{2}$ with
$\mu$. The rest of the proof is exactly the same as Theorem 3.1 (just replace
$\omega_{2}$ with $(\mu^{+})^{V}$).
Suppose $j:V\to N$, $\theta$ and $G*H*K$ are as in the proof of Claim 3.3. Let
$\beta:=(\mu^{+})^{V}=\sup_{p\in G}{\rm dom}(p)$. Then $\bigcup K\in N$ is a
club subset of $\beta$ and coheres with all of the elements of $G$. Note that
all initial segments of $\bigcup K$ are countable sets in $V$. So
$K^{*}:=j``\bigcup K$ is club in $\beta^{*}:=\sup(j``\beta)$ and coheres with
all of the elements of $G^{*}:=j``G$. Hence $(\bigcup
G^{*})\cup\langle\beta^{*},K^{*}\rangle$ is a lower bound of $j``G$ in
$j(\mathbb{P}_{0})$. ∎
Note that in particular $\mathsf{SCFA}$ does not imply $\square_{\mu}$ for any
$\mu<2^{\aleph_{0}}$. On the other hand, as in Figure 1 at the introduction,
we have the following theorem, which is essentially known.
###### Theorem 3.8.
Let $2^{\aleph_{0}}\leq\nu\leq\kappa<\mu=\kappa^{+}$ be cardinals with
$\nu^{\omega}<\mu$. Modulo the existence of a supercompact cardinal
$\infty\mbox{-}\mathsf{SCFA}\upharpoonright\mu+\neg\infty\mbox{-}\mathsf{SCFA}\upharpoonright\nu$
is consistent.
###### Proof.
By Theorem 3.7 we know that $\infty$-$\mathsf{SCFA}\upharpoonright\mu$ is
consistent with $\square_{\kappa}$ hence it suffices to show that
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$ implies the failure of
$\square_{\kappa}$. This is essentially known though it needs to be pieced
together from a few sources. First, to get the failure of $\square_{\kappa}$
Jensen uses the forcing notion (at $\kappa$) from [14, Lemma 6.3]. Second, [8,
Lemma 3.5] implies that this forcing notion is indeed $\infty$-subcomplete
above $\nu$ under the cardinal arithmetic assumptions mentioned in the theorem
statement. See the proof of [8, Lemma 3.5] and the discussion therein for more
details. ∎
## 4\. Separating $\mathsf{MM}$ from $\mathsf{SubPFA}$
In this section we prove the following result.
###### Theorem 4.1.
Assume there is a supercompact cardinal. Then there is a forcing extension in
which $\infty$-$\mathsf{SubPFA}$ holds but $\mathsf{MM}$ fails. In particular,
modulo the large cardinal assumption, $\infty$-$\mathsf{SubPFA}$ does not
imply $\mathsf{MM}$.
The idea behind this theorem is a combination of the proof technique from [1,
Theorem 2.6] and the proof of Theorem 3.1. Starting from a model of
$\mathsf{MM}$ we will force to add a non-reflecting stationary set to
$2^{\aleph_{0}}$ ($=\aleph_{2}$ since $\mathsf{MM}$ holds). This kills
$\mathsf{MM}$ by the results of [5] but will preserve
$\infty$-$\mathsf{SubPFA}$ by an argument similar to that of [1, Theorem 2.6].
The interesting difference is that $\infty$-$\mathsf{SubPFA}$ (in fact
$\mathsf{SCFA}$) implies that there are no non-reflecting stationary sets
above the continuum (and more) so here $\aleph_{2}$ matters which is not true
for $\mathsf{PFA}$ (though the proof for $\aleph_{2}$ is the same for
$\mathsf{PFA}$ and its “subversion”). We begin by recalling the relevant
definitions.
###### Definition 4.2.
Let $\kappa$ be a cardinal of uncountable cofinality and $S\subseteq\kappa$.
For a limit ordinal $\alpha<\kappa$ of uncountable cofinality we say that $S$
reflects to $\alpha$ if $S\cap\alpha$ is stationary in $\alpha$. We say that
$S$ is non-reflecting if it does not reflect to any $\alpha<\kappa$ of
uncountable cofinality.
###### Fact 4.3 (See Theorem 9 [5]).
$\mathsf{MM}$ implies that for every regular $\kappa>\aleph_{1}$ every
stationary subset of $\kappa\cap{\rm Cof}(\omega)$ reflects.
Compare this with the following.
###### Fact 4.4 (See Lemma 6[14]).
$\mathsf{SCFA}$ implies that for every regular $\kappa>2^{\aleph_{0}}$ every
stationary subset of $\kappa\cap{\rm Cof}(\omega)$ reflects.
###### Remark 1.
Note that in [14] it is claimed that $\mathsf{SCFA}$ implies that the above
holds for all $\kappa>\aleph_{1}$, regardless of the size of the continuum.
However, Sean Cox observed333Private communication, see also the discussion in
[8] preceding Lemma 3.5. that the proof only works for
$\kappa>2^{\aleph_{0}}$. In light of Theorem 3.1 this bound is optimal.
There is a natural forcing notion to add a non-reflecting stationary subset
$S\subseteq\kappa\cap{\rm Cof}(\omega)$ for a fixed regular cardinal $\kappa$.
The definition and basic properties are given in Example 6.5 of [3]. We record
the basics here for reference.
###### Definition 4.5.
Fix a regular cardinal $\kappa>\aleph_{1}$. The forcing notion
$\mathbb{NR}_{\kappa}$ is defined as follows. Conditions are functions $p$
with domain the set of countably cofinal ordinals below some ordinal
$\alpha<\kappa$ mapping into $2$ with the property that if $\beta\leq{\rm
sup}({\rm dom}(p))$ has uncountable cofinality then there is a set
$c\subseteq\beta$ club in $\beta$ which is disjoint from
$p^{-1}(1)=\\{\alpha\in{\rm dom}(p)\mid p(\alpha)=1\\}$. The extension
relation is simply $q\leq_{\mathbb{NR}_{\kappa}}p$ if and only if $q\supseteq
p$.
Proofs of the following can be found in [3].
###### Proposition 4.6.
For any regular $\kappa>\aleph_{1}$ the forcing $\mathbb{NR}_{\kappa}$ has the
following properties.
1. (1)
$\mathbb{NR}_{\kappa}$ is $\sigma$-closed.
2. (2)
$\mathbb{NR}_{\kappa}$ is $\kappa$-strategically closed and in particular
preserves cardinals.
3. (3)
If $G\subseteq\mathbb{NR}_{\kappa}$ is generic then $S_{G}:=\bigcup_{p\in
G}p^{-1}(1)$ is a non-reflecting stationary subset of $\kappa$.
We neglect to give the definition of strategic closure since we will not need
it beyond the fact stated above, see [4] or [3] for a definition.
Let $\kappa$ be as above, $G\subseteq\mathbb{NR}_{\kappa}$ be generic over $V$
and let $S_{G}:=\bigcup_{p\in G}p^{-1}(1)$ be the generic non-reflecting
stationary set. We want to define a forcing to kill $S_{G}$ (this will be the
“$\dot{\mathbb{R}}$” in our application of Theorem 1.9). Specifically we will
define a forcing notion $\mathbb{Q}_{S_{G}}$ so that forcing with
$\mathbb{Q}_{S_{G}}$ will add a club to $\kappa\setminus S_{G}$ and hence kill
the stationarity of $S_{G}$. Note that since $S_{G}$ is non-reflecting its
complement must also be stationary and indeed has to be fat i.e. contain
continuous sequences of arbitrary length $\alpha<\kappa$ cofinally high.
###### Definition 4.7.
Borrowing the notation from the previous paragraph define the forcing notion
$\mathbb{Q}_{S_{G}}$ as the set of closed, bounded subsets of $\kappa\setminus
S_{G}$ ordered by end extension.
Clearly the above forcing generically adds a club to the complement of $S_{G}$
thus killing its stationarity, see [3] Definition 6.10. It is also
$\omega$-distributive.
We are now ready to prove Theorem 4.1.
###### Proof of Theorem 4.1.
Assume $\infty$-$\mathsf{SubPFA}$ holds (the consistency of this is the only
application of the supercompact). Note that the continuum is $\aleph_{2}$ and
will remain so in any cardinal preserving forcing extension which adds no
reals. Let $\mathbb{P}=\mathbb{NR}_{\aleph_{2}}$, $G\subseteq\mathbb{P}$ be
generic over $V$ and work in $V[G]$. Obviously in this model we have “there is
a non-reflecting stationary subset of $\aleph_{2}$” and thus $\mathsf{MM}$
fails by Fact 4.3. We need to show that $\infty$-$\mathsf{SubPFA}$ holds.
We will apply Theorem 1.9 much as in the proof of Theorem 3.1. Let
$\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\infty$-subproper forcing
notion and let $\dot{\mathbb{R}}$ name $\mathbb{Q}_{S_{\dot{G}}}$ in
$V^{\mathbb{P}*\dot{\mathbb{Q}}}$ (NOT just in $V^{\mathbb{P}}$ \- this is
different than the proof of Theorem 3.1 and crucial). By exactly the same
argument as in the proof of Theorem 3.1 it suffices to show that
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is $\infty$-subproper (in $V$).
This is because (2) from Theorem 1.9 follows from the fact that, borrowing the
notation from that Theorem applied to our situation $\dot{\mathbb{R}}$ shoots
a club through the complement of $S_{G}$ hence $j``S_{G}=S_{G}$ is non-
stationary in its supremum and so has a lower bound in $N$.
So we show that $\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is
$\infty$-subproper. This is very similar to the proof of Theorem 3.1 but
enough details are different to warrant repeating everything for completeness.
Let $\tau>\theta$ be sufficiently large cardinals and $\sigma:\bar{N}\prec
N=L_{\tau}[A]\supseteq H_{\theta}$ be as in the definition of
$\infty$-subproperness. Let
$\sigma(\bar{\mathbb{P}},\dot{\bar{\mathbb{Q}}},\dot{\bar{\mathbb{R}}},\bar{\omega_{2}})=\mathbb{P},\dot{\mathbb{Q}},\dot{\mathbb{R}},\omega_{2}$.
Let $(p_{0},\dot{q}_{0},\dot{r}_{0})$ be a condition in
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ with
$\sigma(\bar{p}_{0},\dot{\bar{q}}_{0},\dot{\bar{r}}_{0})=(p_{0},\dot{q}_{0},\dot{r}_{0})$.
Applying the $\sigma$-closure of $\mathbb{P}$ we can find a
$\bar{\mathbb{P}}$-generic $\bar{G}$ over $\bar{N}$ and a condition $p\leq
p_{0}$ so that $p$ is a lower bound on $\sigma``\bar{G}$ and, letting
$\alpha={\rm sup}(\sigma``\bar{\omega}_{2})$, we have $p(\alpha)=0$ (i.e. $p$
forces $\alpha$ to not be in the generic stationary set). Let us assume $p\in
G$ and note that this condition forces $\sigma``\bar{G}\subseteq G$ and hence
$\sigma$ lifts uniquely to a $\tilde{\sigma}:\bar{N}[\bar{G}]\prec N[G]$ that
$\tilde{\sigma}(\bar{G})=G$ and $\alpha:={\rm
sup}(\sigma``\bar{\omega}_{2})\notin S_{G}$. Let
$\bar{\mathbb{Q}}=\dot{\bar{\mathbb{Q}}}^{\bar{G}}$ as computed in
$\bar{N}[\bar{G}]$ and let
$\bar{q}_{0}=\dot{\bar{q}}_{0}^{\bar{G}}\in\bar{N}[\bar{G}]$. Applying the
fact that $\dot{\mathbb{Q}}$ is forced to be $\infty$-subproper let $q\leq
q_{0}=\tilde{\sigma}(\bar{q}_{0})$ be a condition forcing that if
$H\subseteq\mathbb{Q}$ is $V$-generic with $q\in H$ then there is a
$\sigma^{\prime}\in V[G][H]$ so that $\sigma^{\prime}:\bar{N}[\bar{G}]\prec
N[G]$ as in the definition of $\infty$-subproperness (with respect to
$\tilde{\sigma}$). Note that as in the proof of Theorem 3.1
$\sigma^{\prime}\upharpoonright\bar{N}:\bar{N}\prec N$ and
$\sigma^{\prime}\upharpoonright\bar{\omega}_{2}=\sigma\upharpoonright\bar{\omega}_{2}$.
Let $\tilde{\sigma}^{\prime}:\bar{N}[\bar{G}][\bar{H}]\to N[G][H]$ be the lift
of $\sigma^{\prime}$, where $\bar{H}=(\sigma^{\prime})^{-1}``H$.
###### Claim 4.8.
In $V[G][H]$ the set $S_{G}$ does not contain a club.
###### Proof of Claim.
Since $\aleph_{2}$ is the continuum in $V[G]$ note that $\omega_{2}^{V[G]}$
remains uncountably cofinal in $V[G][H]$ (though of course it can be collapsed
to $\omega_{1}$). Suppose towards a contradiction that $S_{G}$ contains a club
and note that since we chose $\theta$ etc sufficiently large we have
$N[G][H]\models$ “$\exists C$ which is club and $C\subseteq S_{G}$”. By
elementarity there is a $\bar{C}\in\bar{N}[\bar{G}][\bar{H}]$ so that
$\bar{N}[\bar{G}][\bar{H}]\models\bar{C}\subseteq\bar{S}_{G}\;{\rm is\,club}$
where $\bar{H}:=\sigma^{\prime}{}^{-1}H$ is $\bar{\mathbb{Q}}$-generic over
$\bar{N}[\bar{G}]$ by the definition of $\infty$-subcompleteness and the
choice of $q$. But now note that if $C=\tilde{\sigma}^{\prime}(\bar{C})$ then
$C\cap\alpha$ is cofinal in $\alpha$ by elementarity so $\alpha\in C$ but
$\alpha\notin S_{G}$ which is a contradiction. ∎
Given the claim we know that $\omega_{2}^{V}\setminus S_{G}$ is a stationary
set in $V[G][H]$ and hence $\mathbb{R}:=\dot{\mathbb{R}}^{G*H}$ is the forcing
to shoot a club through a stationary set. Let
$\bar{\mathbb{R}}\in\bar{N}[\bar{G}][\bar{H}]$ be
$\dot{\bar{\mathbb{R}}}^{\bar{G}*\bar{H}}$. Note that for each
$\beta\in\bar{N}\cap\bar{\omega}_{2}$ it is dense (in
$\bar{N}[\bar{G}][\bar{H}]$) that there is a condition
$\bar{r}\in\bar{\mathbb{R}}$ with $\beta\in{\rm dom}(\bar{r})$. It follows
that if $\bar{K}$ is generic for $\bar{\mathbb{R}}$ over
$\bar{N}[\bar{G}][\bar{H}]$ with
$\bar{K}\ni\bar{r}_{0}:=\dot{\bar{r}}^{\bar{G}*\bar{H}}$ then
$\tilde{\sigma^{\prime}}``\bar{K}$ unions to a club in $\alpha\setminus
S_{G}$. Since $\alpha\notin S_{G}$ we have that
$r:=\bigcup\tilde{\sigma^{\prime}}``\bar{K}\cup\\{\alpha\\}$ is a condition in
$\mathbb{R}$ which is a lower bound on $\tilde{\sigma^{\prime}}``\bar{K}$ and
hence $r\leq\dot{r}_{0}^{G*H}$. Finally let $K\ni r$ be $\mathbb{R}$-generic
over $V[G][H]$. It is now easy to check that the condition
$(p,\dot{q},\dot{r})$ and $\sigma^{\prime}\upharpoonright\bar{N}$ collectively
witness the $\infty$-subproperness of
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ so we are done. ∎
We note that by the same proof adding a nonreflecting stationary set of
$\mu\cap{\rm Cof}(\omega)$ for larger cardinals $\mu$ we can preserve
$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$. The following therefore holds.
###### Theorem 4.9.
Let $2^{\aleph_{0}}\leq\mu\leq\lambda<\nu=\lambda^{+}$ be cardinals with
$\mu^{\omega}<\nu$. Modulo the existence of a supercompact cardinal
$\infty$-$\mathsf{SubPFA}\upharpoonright\nu+\neg\infty$-$\mathsf{SubPFA}\upharpoonright\mu$
is consistent.
The proof of this Theorem finishes the proof of all nonimplications involved
in Main Theorem 2.
## 5\. Conclusion and Open Questions
We view this paper, alongside its predecessor [10] as showing, amongst other
things, that the continuum forms an interesting dividing line for subversion
forcing: below the continuum the “sub” plays no roles as witnessed, as the
same non-implications can hold as those that hold for the non-sub versions.
Above, it adds considerable strength to the associated forcing axioms.
However, as of now we only know how to produce models of $\mathsf{SCFA}$ in
which the continuum is either $\aleph_{1}$ or $\aleph_{2}$. The most pressing
question in this area is therefore whether consistently $\mathsf{SCFA}$ can
co-exist with a larger continuum.
###### Question 1.
Is $\mathsf{SCFA}$ consistent with the continuum $\aleph_{3}$ or greater?
We note here that the most obvious attempt to address this question i.e.
starting with a model of $\mathsf{SCFA}$ and adding $\aleph_{3}$-many reals
with e.g. ccc forcing, does not work, an observation due to the first author.
###### Lemma 5.1.
Suppose $\mathbb{P}$ is a proper forcing notion adding a real. Then
$\mathsf{SCFA}$ fails in $V^{\mathbb{P}}$.
All that is needed about “properness” here is that being proper implies that
stationary subsets of $\kappa\cap{\rm Cof}(\omega)$ are preserved. The proof
of this is standard and generalizes the proof of Lemma 2.4 above (swapping
subproper for proper and removing the bound by the continuum).
###### Proof.
Assume $\mathbb{P}$ is proper. Let $G$ be a $\mathbb{P}$-generic filter over
$V$. For a contradiction, assume $\mathsf{SCFA}$ holds in $V[G]$.
Take a regular cardinal $\nu>2^{\omega}$ in $V[G]$. In $V$, take stationary
partitions $\langle A_{k}:k<\omega\rangle$ of $\nu\cap{\rm Cof}(\omega)$ and
$\langle D_{i}:i<\omega\rangle$ of $\omega_{1}$. In $V[G]$, take a subset $r$
of $\omega$ which is not in $V$. Let $\\{k(i)\\}_{i<\omega}$ be the increasing
enumeration of $r$.
By [14, Lemma 7.1], in $V[G]$, there is an increasing continuous function
$f:\omega_{1}\to\nu$ such that $f[D_{i}]\subseteq A_{k(i)}$ for all
$i<\omega$. Let $\alpha:={\rm sup}({\rm range}(f))$. Then, in $V[G]$ , we have
that $r=\\{k\in\omega:A_{k}\cap\alpha$ is stationary in $\alpha\\}$.
But the set $\\{k\in\omega:A_{k}\cap\alpha$ is stationary in $\alpha\\}$ is
absolute between $V$ and $V[G]$ since $\mathbb{P}$ is proper and hence
preserves stationary subsets of ${\rm Cof}(\omega)$ points. But then $r$ is in
$V$, which is a contradiction. ∎
This shows that either $\mathsf{SCFA}$ implies the continuum is at most
$\aleph_{2}$ \- though given the results of this paper this seems difficult to
prove by methods currently available - or else new techniques for obtaining
$2^{\aleph_{0}}\geq\aleph_{3}$ are needed, which is well known to be in
general an open and difficult area on the frontiers of set theory.
## References
* [1] Robert E. Beaudoin. The proper forcing axiom and stationary set reflection. Pacific J. Math., 149(1):13–24, 1991.
* [2] Sean Cox. Forcing axioms, approchability and stationary set reflection. Journal of Symbolic Logic, 86(2), 2021.
* [3] James Cummings. Iterated forcing and elementary embeddings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages 775–883. Springer, Dordrect, 2010.
* [4] James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection. Journal of Mathematical Logic, 1(01):35–98, 2001.
* [5] Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Annals of Math, 127(1):1 – 47, 1988.
* [6] Gunter Fuchs. Diagonal reflection on squares. Archive for Mathematical Logic, 58(1-2):1–26, 2019.
* [7] Gunter Fuchs. Aronszajn tree preservation and bounded forcing axioms. Journal of Symbolic Logic, 86(1):293 – 315, 2021.
* [8] Gunter Fuchs. Canonical fragments of strong reflection principles. Journal of Mathematical Logic, 21(3):2150023, 2021.
* [9] Gunter Fuchs and Kaethe Minden. Subcomplete forcing, trees and generic absoluteness. Journal of Symbolic Logic, 83(3):920–938, 2018.
* [10] Gunter Fuchs and Corey Bacal Switzer. Iteration theorems for subversions of forcing classes, 2020.
* [11] Ronald B. Jensen. Forcing axioms compatible with CH. Handwritten Notes available at http://www-irm.mathematik.hu-berlin.de/ raesch/org/jensen.htm.
* [12] Ronald B. Jensen. Iteration theorems for subcomplete and related forcings. Handwritten Notes available at http://www-irm.mathematik.hu-berlin.de/ raesch/org/jensen.html.
* [13] Ronald B. Jensen. Subproper and subcomplete forcing. Handwritten Notes available at http://www-irm.mathematik.hu-berlin.de/ raesch/org/jensen.html.
* [14] Ronald B. Jensen. Subcomplete forcing and L-forcing. In C. Chong, Qi Feng, Ted A. Slaman, and W. Hugh Woodin, editors, E-Recursion, Forcing and $C^{*}$-Algebras, volume 27 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, pages 83–182. World Scientific, Singapore, 2014.
* [15] Chris Lambie-Hanson and Menachem Magidor. On the strengths and weaknessess of weak squares. In James Cummings and Ernst Schimmerling, editors, Appalachian Set Theory: 2006-2012, pages 301–330. Cambridge University Press, London, 2012\.
* [16] Tadatoshi Miyamoto. a class of preorders iterated under a type of rcs. RIMS Kokyuroku, (1754):81–90, 2011.
* [17] Corey Bacal Switzer. Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH. PhD thesis, The Graduate Center, The City University of New York, 2020\.
* [18] Víctor Torres-Pérez and Liuzhen Wu. Strong Chang’s conjecture and the tree property at $\omega_{2}$. Topology Appl., 196(part B):999–1004, 2015.
|
# ToolTango: Common sense Generalization in Predicting Sequential Tool
Interactions for Robot Plan Synthesis
Shreshth Tuli<EMAIL_ADDRESS>
Department of Computing
Imperial College London, UK Rajas Bansal∗<EMAIL_ADDRESS>
Department of Computer Science
Stanford University, USA Rohan Paul<EMAIL_ADDRESS>
Department of Computer Science and Engineering / Yardi School of Artificial
Intelligence
Indian Institute of Technology Delhi, India Mausam<EMAIL_ADDRESS>
Department of Computer Science and Engineering / Yardi School of Artificial
Intelligence
Indian Institute of Technology Delhi, India Most work done when the authors
were undergraduate students at Indian Institute of Technology Delhi, India.
###### Abstract
Robots assisting us in environments such as factories or homes must learn to
make use of objects as tools to perform tasks, for instance using a tray to
carry objects. We consider the problem of learning commonsense knowledge of
when a tool may be useful and how its use may be composed with other tools to
accomplish a high-level task instructed by a human. Specifically, we introduce
a novel neural model, termed ToolTango, that first predicts the next tool to
be used, and then uses this information to predict the next action. We show
that this joint model can inform learning of a fine-grained policy enabling
the robot to use a particular tool in sequence and adds a significant value in
making the model more accurate. ToolTango encodes the world state, comprising
objects and symbolic relationships between them, using a graph neural network
and is trained using demonstrations from human teachers instructing a virtual
robot in a physics simulator. The model learns to attend over the scene using
knowledge of the goal and the action history, finally decoding the symbolic
action to execute. Crucially, we address generalization to unseen environments
where some known tools are missing, but alternative unseen tools are present.
We show that by augmenting the representation of the environment with pre-
trained embeddings derived from a knowledge-base, the model can generalize
effectively to novel environments. Experimental results show at least
48.8-58.1% absolute improvement over the baselines in predicting successful
symbolic plans for a simulated mobile manipulator in novel environments with
unseen objects. This work takes a step in the direction of enabling robots to
rapidly synthesize robust plans for complex tasks, particularly in novel
settings.
## 1 Introduction
Advances in autonomy have enabled robots to enter human-centric domains such
as homes and factories where we envision them performing general purpose tasks
such as transport, assembly, and clearing. Such tasks require a robot to
interact with objects, often using them as _tools_. For example, a robot asked
to “take fruits to the kitchen”, can use a _tray_ for carrying items, a
_stick_ to fetch objects beyond physical reach and may use a _ramp_ to reach
elevated platforms. Previous work has shown that the ability to predict the
possible use of tools for a given task is often useful in guiding a robot’s
task planner towards plans likely to be feasible (?, ?, ?, ?, ?). In this work
we consider the problem of predicting _which_ objects could be used as tools
and _how_ their use can be composed for a task. In essence, we focus on the
ability to predict appropriate tools that can guide the robot towards feasible
and efficient plans and delegate the issue of dexterous tool manipulation to
prior work (?, ?).
Learning to predict task-directed tool interactions poses several challenges.
First, real environments (a household or factory-like domain) are typically
large where an expansive number of tool interactions may be possible (e.g.,
objects supporting others while transporting). Acquiring data for all feasible
tool objects or exploring the space of tool interactions is challenging for
any learner. Second, the usefulness of a tool varies with context. For
example, placing milk in the cupboard may require the robot to elevate itself
vertically using a ramp if the milk is placed at a height unreachable by the
robot, but if the milk is kept on a table, a simple tray might suffice. Third,
the robot may encounter new environments populated with novel objects not
encountered during training. Hence, the agent’s model must be able to
_generalize_ by reasoning about interactions with novel objects unseen during
training. Humans possess innate commonsense knowledge about contextual use of
tools for an intended goal (?). For example, a human actor when asked to move
objects is likely to use trays, boxes, or even improvise with a new object
with a flat surface. We aim at providing this commonsense to a robotic agent,
so that it can generalize its knowledge to unseen tools, based on shared
context and attributes of seen tools (see Figure 1).
Figure 1: ToolTango acquires commonsense knowledge from human demonstrations
leveraging graph-structured world representation, knowledge-based corpora and
goal-conditioned attention to perform semantic tasks. Our aim is to acquire
commonsense knowledge to develop a generalized goal-conditioned policy for a
robot.
This paper takes a step in the direction of enabling robots to learn how to
perform high-level tasks such as compositional tool use in semantic tasks,
particularly in novel environments. This paper makes four main contributions.
As the first contribution, we present a crowd-sourced dataset of human-
instructed plans where a human teacher guides a simulated mobile manipulator
to leverage objects as tools to perform multi-step actions such as assembly,
transport and fetch tasks. The process results in a corpus of
$\sim\\!\\!1,500$ human demonstrated robot plans involving diverse goal
settings and environment scenes.
Second, the dataset mentioned above is first used to supervise a (1-step)
neural imitation learner that predicts tool applicability given the knowledge
of the world state and the intended goal. We show how learning a dense
embedding for the environment and that of a background knowledge base can
enable the model to generalize to novel scenes with new object instances that
may share semantic attributes with objects seen during training; a common
problem in state of the art task and policy learning approaches. We introduce
a graph neural architecture, ToolNet, that encodes both the metric and
relational attributes of the world state as well as available taxonomic
resources such as $\mathrm{ConceptNet}$ (?). The ToolNet model predicts tool
use by learning an attention over entities that can potentially serve as
tools. Implicitly, the model acquires knowledge about primitive spatial
characteristics (typically an output of a mapping system) and semantic
attributes (typically contained in taxonomic resources) enabling
generalization to novel contexts with previously unseen objects.
Third, we present an imitation learner that uses the same dataset to make
action predictions towards the intended goal. We term the model, Tool
Interaction Prediction Network for Generalized Object environments (Tango).
Similar to ToolNet, Tango also encodes the world state using a graph neural
network and learns to attend over the scene using knowledge of the goal and
the action history, finally decoding the symbolic action to execute. The
action predictions are interleaved with physics simulation (or execution)
steps, which ameliorates the need for modeling the complex effects of actions
inherent in tool interactions.
As a final contribution, we combine the two models ToolNet and Tango in a
single architecture. Specifically, the joint model first makes the predictions
for the next tool, and then uses this information to make a better action
prediction. We show that this can inform learning of a fine-grained policy
enabling the robot to use a particular tool in sequence and adds a significant
value in making the model more accurate. We term this joint model ToolTango.
Experimental evaluation with a simulated mobile manipulator demonstrates (a)
accurate prediction of tool interaction sequences with high
executability/goal-attainment likelihood, (b) common sense generalization to
novel scenes with unseen object instances, and (c) robustness to unexpected
errors during execution. Additionally, compared to Tango, we demonstrate the
benefit of using ToolNet predictions for specific complex settings requiring
multiple tools to reach the goal state. Experiments show that in previously
seen settings, ToolTango gives an absolute improvement of 3.38-5.59% in
reaching a goal state for a simulated mobile manipulator compared to the
state-of-the-art Tango model. In unseen settings, the unified model gives
2.48-3.58% improvement compared to Tango. In comparison to a simple affordance
prediction approach, the proposed model performs better in learning the pre-
conditions for tool use. Further, the use of a neural model enables higher
goal-reaching performance and faster training compared to a vanilla
reinforcement learning approach.
A preliminary version of ToolNet that performs single tool prediction using
only the initial environment state was presented as part of the Workshop on
Advances & Challenges in Imitation Learning in Robotics at Robotics Science
and Systems (RSS) 2020 conference (?). Our Tango model was presented as part
of the International Joint Conference in Artificial Intelligence (IJCAI) 2021
(?). This journal submission presents a substantially detailed exposition of
the ToolNet and Tango models and includes additional supporting background
material. Moreover, it also extends ToolNet to predict _next_ tool at each
step of a multi-step action execution, instead of a single tool for the whole
sequence, as in the original paper. This paper also combines the two models
into ToolTango, and establishes its superior performance.
The remainder of the paper is organized as follows. Section 2 overviews
related work. Section 3 formulates the problem of predicting tool interaction
sequences. Section 4 details the technical approach and presents the ToolTango
model. The data collection platform and the dataset in detailed in Section 5.
Section 6 provides the experimental details and results. Finally, Sections 7
and 8 summarizes the work and lays out avenues for future work. The associated
code, data set and videos are available at https://github.com/reail-
iitd/tango. Links to all supplementary material are given in Appendix A.
## 2 Related Work
### 2.1 Classical Planning
Reaching goal states from a given world state through symbolic actions is
closely tied to the domain of task and motion planning (TAMP) (?). Literature
presents a large volume of work in this domain, ranging from constraint
satisfaction to search based methods. For instance, ? (?) a planner to compose
physics tool interactions using a logic-based symbolic planner. Similarly, ?
(?) provide an implementation agnostic interface between a task and motion
planner to combine both for goal-directed planning. ? (?) extend PDDL
descriptions to include generic and declarative specifications enabling the
planning to be domain independent. These methods only utilize the task
properties through action constraints. For instance, the knowledge that a tray
can carry multiple objects is only encoded through the constraint that an
object can be placed on it. This allows such methods to create any feasible
plan and not leverage commonsense knowledge to ensure that a goal state is
reached in a few steps. Some recent works such as by ? (?) aim to learn object
importance through human demonstrations. However, such methods can only work
on objects seen previously in training and cannot generalize to unseen
objects. We consider a part of the broad TAMP framework that focuses on
determining the set of actions that are likely to take an agent to the goal
state while taking the motion planning as a behavioural routine. We build our
work on the observation that in domain with a large number of states and
possible interactions, the task planning itself becomes challenging. We
consider home and factory-like domains inspired from VirtualHome (?) and
similar related works. We ensure that in our domains the number of objects is
large and they can be contained within/supported or transported by other
objects as tools (details in Section LABEL:sec:data_collection). Similar to
the work by ? (?), our model learns to prune-away irrelevant objects,
additionally considering a domain with richer inter-object interactions.
Consequently, our learner makes additional use of semantic properties and
exploits correlations between actions and outputs interactions that are likely
to lead to successful plans.
### 2.2 Learning tool manipulation skills
Learning control policies for manipulating tools has received recent attention
in robotics. ? (?) and ? (?) learn tool manipulation policies from human
demonstrations. ? (?) and ? (?) learn physics models and _effects_ enabling
goal-directed compositional use. ? (?) address the problem of learning
primitive physical decomposition of tool like object through its physical and
geometric attributes enabling their human-like use. ? (?) learn physical
properties of objects from unlabeled videos. ? (?) and ? (?) learn to interact
with objects in a self-supervised setup. Efforts such as ? (?), and ? (?) plan
tool interactions modeling contact and force interactions. Another set of
efforts focuses on incorporating the ability to discover the use of objects as
tools and using them for plan completion. Rich symbolic architectures such as
ICARUS (?), KDAP (?) and ROAR (?, ?) attempt to model tool use via PDDL-like
descriptions expressing the applicability and post effects of tool use. In
particular, ICARUS (?) models bridge or staircase construction via lifted
symbolic concepts which can be grounded to real or imagined objects in the
environment. The framework presented in this paper takes inspiration from such
classic works and builds on the use of learned representations for
generalization to new scenes and objects. Further, instead of encoding such
knowledge via a symbolic representation for each tool, the framework acquires
such knowledge in a data-driven way from human demonstration. Thus, the
aforementioned works focus on learning _how_ to manipulate a tool. Our paper
considers the complementary problem of predicting _which_ objects may serve as
tools for a given task while delegating the issue of tool manipulation to the
works as mentioned earlier.
The works of ? (?) and ? (?) considers the specific problem of learning the
physical motion of tools (such as spatial, hammer, mug, etc.) from kinesthetic
demonstration trajectories. Further, the authors present a framework to learn
corrections or replacements enabling improvisation when the actual task
execution scenario differs from the taught demonstration. The focus of the
work is in learning physical motion of tools for short range tasks. This work
lifts the problem of dexterous tool manipulation and focuses on utilizing
tools to reach goal states (e.g., using a box to transfer multiple objects).
This work also extends planning to leverage multiple tools to attain goals
(using boxes and later using ramps). Rather than use fine-grained kinesthetic
teaching, we instead use demonstrations of.long range plan executions. The two
efforts are highly complementary. The tool-use trajectories learned by prior
work could potentially be used in our framework to accomplish long-range
tasks. Similarly, such works can use the framework presented here to compose
skills and generalize to unseen tools.
### 2.3 Learning symbolic action sequences
Others address the problem of acquiring knowledge for completing high-level
task specifications. ? (?, ?) create a knowledge base of task decompositions
as _action sketches_ and learn to translate sketches to executable plans.
These efforts rely on the causal knowledge of sequences on sub-steps required
to achieve an activity which are then contextually grounded. Instead, this
work learns compositional tool use required to achieve the task without any
causal sequence as input. ? (?) learn task decompositions from human
demonstration videos. However, their work does not explicitly model the
physical constraints of the robot and does not generalize to new environments.
? (?) and ? (?) learn trajectory-aware manipulation of deformable objects
using a non-rigid registration method and human demonstrations. These methods
can effectively handle visual variation in manipulating objects; however, they
cannot be extended to generate action sequences to reach goal constraints
given an unseen environment. ? (?) take a similar approach by collecting
natural language corpora describing high-level tasks and learn to associate
instructions to spatial attention over the scene.
Other works study _relational_ planning domains, where model is trained using
RL on small problems, but tested zero-shot on large problems (?, ?). ? (?)
extend this to imitation learning framework, but these works cannot perform
tool generalizations. ? (?) present a symbolic system where a robot imitates
demonstrations from a single teacher. In new environments, it adapts the plan
by performing object replacements using ConceptNet relation edges. A rich set
of approaches learn affordances by mapping objects to preferred locations or
learning the co-use of objects to accomplish a task. For instance, ? (?)
provide a method of acquiring and transferring such knowledge to new tasks by
learning to map between objects in the old and new environments in a task-
dependent manner. This work does not explicitly attempt to learn such
associations and presents a restricted generalization capacity. Instead, such
learning is implicit in the learned policy that predicts a sequence of robot
actions to achieve the goal while using tools (e.g., fetching the tool, using
the tools and attaining its post effects).
Our approach draws inspiration from the above-mentioned works in that, we
learn to predict tools that can be considered as sub-goals to guide planning
for a high-level task. In comparison to these approaches, our method provides
two key contributions. First, we explicitly model the physical constraints
arising from a mobile manipulator interacting in the work space. Second,
instead of learning actions predicated on specific object instances, we
address generalization to new object instances using primitive spatial and
semantic characteristics. We propose a neural model trained using a corpus of
multiple and varied demonstrations provided by several teachers. Our model
uses a dense embedding of semantic concepts, enabling generalization beyond
relationships explicitly stored in ConceptNet.
### 2.4 Commonsense knowledge in instruction following
Acquisition of common sense knowledge has been previously explored for the
task of robot instruction following (?). ? (?) present a symbolic knowledge
base for procedural knowledge of tasks that is utilized for interpreting
underspecified task instructions. Efforts such as ? (?) propose a similar
database encoding common sense knowledge about object affordances (objects and
their common locations). Others such as ? (?) learn motion preferences
implicit in commands. ? (?) ground instructions for recipe preparation tasks.
Their model can generalize to new recipes, but only in environments with
previously _seen_ objects. In contrast, our model generalizes to worlds with
previously _unseen_ tools. Others, such as ? (?) and (?) explore the problem
of robot tool construction, i.e., creating tools from parts available in the
environment. However, the limited set of commonsense concepts, such as
attachment in (?) restricts the space for generalization in such works. A
rule-based approach typically does not scale well to develop a robust
generalization model. We thus utilize a commonsense embedding based approach
that utilizes ConceptNet vectors to encapsulate the concepts and generalize to
unseen tools and object settings.
? (?) present an instruction grounding model that leverages common sense
taxonomic and affordance knowledge learned from linguistic co-associations. ?
(?) consider the problem of learning physical common sense associated with
objects and interactions required to achieve tasks from language only data
sets. They study this problem in the context of question-answering to enable
synthesis of textual responses that capture such physical knowledge. The
aforementioned approaches predict latent constraints or affordances for a
specified task. This work, additionally predicts the _sequence_ of tool
interactions implicitly learning the causal relationships between tools use
and effects. Specifically, this paper focuses on a learning common sense tool
use in the context of following instructions that require multiple object
interactions to attain the intended goal.
### 2.5 Synthetic Interaction Datasets
Virtual environments have been used to collect human demonstrations for high-
level tasks. ? (?) introduce a knowledge base of actions required to perform
activities in a virtual home environment. ? (?) provide a vision-language
dataset translating symbolic actions for a high-level activity to attention
masks in ego-centric images. ? (?) curated data sets that provide a sequence
of _How-To_ instructions for tasks such as preparing recipes. ? (?) present an
affordance detection dataset for tool parts with geometric features. Others
such as ? (?), ? (?) and ? (?) present simulation environments and data sets
for tasks such as learning spatial affordances, situated interaction or
learning low-level motor skills. The present data sets possess two limitations
that make them less usable for the learning task addressed in this work.
First, the data sets are collected using human actors or avatars but do not
explicitly model a robot in their environment. Though virtual agents serve as
a proxy for the robot, they preclude modeling of the physical constraints and
the range of tasks an robot can perform. Second, a majority of the data sets
aim at visual navigation and limited physical interaction with objects. They
are less amenable to interactions (e.g., containment, pushing and attachment
etc.) inherent in tool use. Data sets utilized in robotic tool use literature,
including the UMD Part Affordance Dataset (?), are confined mainly to local
use, such as finding the appropriate tool part for tool use and manipulation.
Such data sets are less amenable to multi-stage plans in large workspaces.
## 3 Problem Formulation
### 3.1 Robot and Environment Model
We consider a mobile manipulator operating in a known environment populated
with objects. An object is associated with a pose, a geometric model and
symbolic states such as $\mathrm{Open/Closed}$, $\mathrm{On/Off}$ etc. We
consider object relations such as (i) _support_ e.g., a block supported on a
tray, (ii) _containment_ : items placed inside a box/carton and (iii)
_attachment_ : a nail attached to a wall, and (iv) _contact_ : a robot
grasping an object. Let $s$ denote the world state that maintains (i) metric
information: object poses, and (ii) symbolic information: object states, class
type and object relations as $\mathrm{OnTop}$, $\mathrm{Near}$,
$\mathrm{Inside}$ and $\mathrm{ConnectedTo}$. Let $s_{0}$ denote the initial
world state and $\mathcal{O}(\cdot)$ denote a map from world state $s$ to the
set of object instances $O=\mathcal{O}(s)$ populating state $s$. Let $\tau$
denote the set of tool objects that the robot can use in its plan. Note that
only movable objects in the scene are considered as potential tools. Hence,
$\tau\subseteq\mathcal{O}(s)$. Online, the robot may encounter _unseen_
objects in its environment.
Let $A$ denote the robot’s symbolic action space. An action $a\in A$ is
abstracted as $I(o^{1},o^{2})$, with an action type predicate
$I\in\mathcal{I}$ that affects the states of objects $o^{1}\in O$ and
$o^{2}\in O$, for instance, $\mathrm{Move(fruit_{0},tray_{0})}$. Here the
arity of the interaction can be 2 or 1 depending on the interaction type. In
case of arity of 1, we can drop the second object. Each action in our
formulation is realized as a set of object relations in the environment that
belongs to
$\\{\mathrm{OnTop},\mathrm{Near},\mathrm{Inside},\mathrm{ConnectedTo}\\}$. The
precondition and postconditions of actions are taken directly from prior work
(?). Realization of these conditions using geometric properties is described
in Appendix D. We shall also use the notion of a timestamp as a subscript to
indicate prediction for each state in the execution sequence. The space of
robot interactions include grasping, releasing, pushing, moving an object to
another location or inducing discrete state changes (e.g.. opening/closing an
object, operating a switch or using a mop). We assume the presence of an
underlying low-level metric planner, encapsulated as a robot _skill_ , which
realizes each symbolic action or returns if the action is infeasible. Robot
actions are stochastic, modeling execution errors (unexpected collisions) and
unanticipated outcomes (objects falling, changing the symbolic state). Let
$\mathcal{T}(\cdot)$ denote the transition function. The successor state
$s_{t+1}$ upon taking the action $a_{t}$ in state $s_{t}$ is generated from a
physics simulator. Let $\eta_{t}=\\{a_{0},a_{1},\dots,a_{t-1}\\}$ denote the
_action history_ till time $t$.
### 3.2 Semantic Goals and Interactions
The robot’s goal is to perform tasks such as transporting or delivering
objects to appropriate destinations, making an assembly, clearing or packing
items or performing abstract tasks such as illuminating or cleaning the room.
The robot is instructed by providing a _declarative_ goal $g$ expressing the
symbolic constraint between world objects (?). For example, the _declarative_
goal, “place milk in fridge” can be expressed as a constraint
$\mathrm{Inside(milk_{0},fridge_{0})}$ between specific object instances.
There may be multiple instance of the same object, for eg. milk carton, in the
environment. The sub-index is used to specify which instance is being referred
to. Another example is of the task of moving all fruits onto the kitchen table
that can be expressed as a list of constraints
$\mathrm{OnTop(apple_{0},table_{0})}$, $\mathrm{OnTop(orange_{0},table_{0})}$
and $\mathrm{OnTop(banana_{0},table_{0})}$. Finally, the robot must synthesize
a plan to satisfy the goal constraints. Goal-reaching plans may require using
some objects as tools, for instance, using a container for moving items, or a
ramp negotiate an elevation.
### 3.3 Predicting Tool Interactions
We assume that the robot is primed with a set of primitive symbolic actions
but lacks knowledge about how object characteristics can facilitate their use
as in attaining high-level goals. Hence, the robot cannot predict the use of
tray-like objects in transportation tasks, or the use of a stick to fetch an
object at a distance. An exception is by discovering such characteristics via
explicit simulation, which may be infeasible or intractable in large planning
domains. Our goal is to learn common sense knowledge about _when_ an object
can be used as a tool and _how_ their use can be sequenced for goal-reaching
plans. We aim at learning a policy $\pi$ that estimates the next action
$a_{t}$ conditioned on the the goal $g$ and the initial state $s$ (including
the action history $\eta_{t}$, such that the robot’s _goal-reaching_
likelihood is maximized. We adopt the MAXPROB-MDP (?) formalism and estimate a
policy that maximizes the goal-reaching likelihood from the given state.
MAXPROB-MDP can be equivalently viewed as an infinite horizon, un-discounted
MDP with a zero reward for non-goal states and a positive reward for goal
states (?). Formally, let $P^{\pi}\left(s,g\right)$ denote the _goal-
probability_ function that represents the likelihood of reaching the goal $g$
from a state $s$ on following $\pi$. Let $S^{\pi_{s}}_{t}$ be a random
variable denoting the state resulting from executing the policy $\pi$ from
state $s$ for $t$ time steps. Let $\mathcal{G}(s,g)$ denote the Boolean _goal
check_ function that determines if the intended goal $g$ is entailed by a
world state $s$ as
$\mathcal{G}(s,g)\in\\{\mathrm{True(T)},\mathrm{False(F)}\\}$. The policy
learning objective is formulated as maximizing the likelihood of reaching a
goal-satisfying state $g$ from an initial state $s_{0}$, denoted as
$\displaystyle\max_{\pi}P^{\pi}(s_{0},g)=\max_{\pi}\sum_{t=1}^{\infty}P\bigg{(}\mathcal{G}(S^{\pi_{s_{0}}}_{t},g)=\mathrm{T}:\mathcal{G}(S^{\pi_{s_{0}}}_{t^{\prime}},g)=\mathrm{F},~{}\forall
t^{\prime}\in[1,t)\bigg{)}.$ (1)
The policy is modeled as a function $f_{\theta}(.)$ parameterized by
parameters $\theta$ that determines the next action for a given world state,
the robot’s action history and the goal as
$a_{t}=f_{\theta}\left(s_{t},g,\eta_{t}\right)$. We adopt imitation learning
approach and learn the function $f_{\theta}(.)$ from demonstrations by human
teachers. Let $\mathcal{D}_{\mathrm{Train}}$ denote the corpus of $N$ goal-
reaching plans,
$\mathcal{D_{\mathrm{Train}}}=\\{(s_{0}^{i},g^{i},\\{s_{j}^{i},a_{j}^{i}\\})\mid
i\in\\{1,\ldots,N\\},j\in\\{0,t_{i}-1\\}\\},$ (2)
where the $i^{th}$ datum consists of the initial state $s^{i}_{0}$, the goal
$g^{i}$ and a state-action sequence
$\\{(s^{i}_{0},a^{i}_{0}),\dots,(s^{i}_{t-1},a^{i}_{t-1})\\}$
of length $t_{i}$. The set of human demonstrations elucidate common sense
knowledge about _when_ and _how_ tools can be used for attaining provided
goals. The data set $\mathcal{D}_{\mathrm{Train}}$ supervises an imitation
loss between the human demonstrations and the model predictions, resulting in
learned parameters $\theta^{*}$. Online, the robot uses the learned model to
sequentially predict actions and execute in the simulation environment till
the goal state is attained. We also consider the _open-world_ case where the
robot may encounter instances of _novel_ object categories _unseen_ in
training, necessitating a _zero-shot_ generalization.
## 4 Technical Approach
We aim at predicting the next robot action $a_{t}$, given the world state
$s_{t}$, the intended goal $g$ and the history $\eta_{t}$ of past actions
taken the robot. We consider learning to predict multi-step plans requiring
complex interaction of objects as tools in possibly novel environments unseen
during training. Our technical approach is built on three insights. First, we
factor the overall learning task of predicting the action for each environment
state by first predicting the tool required to perform the task and then
predicting required interaction. This tool conditioned action prediction
enables us to decouple the learning problem and make the model training
tractable. Second, we incorporate learning dense embeddings of the robot’s
environment as well as semantic representations for words trained on existing
symbolic knowledge corpora. The learned representation allows the robot to
generalize its predictions to novel environments populated with novel objects
that may share semantic attributes with those encountered during training.
Finally, we turn to a corpus of human demonstrated plans to train our learner.
The corpus elucidates the commmon sense knowledge of sequencing tools
interactions for a task. Formally, we introduce an imitation learner, realized
as a hyper-parametric function (a neural network model) denoted as
$f_{\theta}$ as follows:
$a_{t}=f_{\theta}\left(s_{t},g,\eta_{t}\right)=f^{act}_{\theta}\left(f^{goal}_{\theta}\left(f^{state}_{\theta}\left(s_{t}\right),g,f^{hist}_{\theta}\left(\eta_{t}\right)\right),f^{tool}_{\theta}(s_{t},g,\eta_{t})\right).$
(3)
It adopts an object-centric graph representation, learning a state encoding
that fuses metric-semantic information about objects in the environment via
function $f^{state}_{\theta}\left(\cdot\right)$. The function
$f^{hist}_{\theta}\left(\cdot\right)$ encodes the action history. The model
learns to attend over the world state conditioned on the declarative goal and
the history of past actions through $f^{goal}_{\theta}\left(\cdot\right)$. We
leverage an adapted version of our tool likelihood prediction model ToolNet,
denoted as $f^{tool}_{\theta}\left(\cdot\right)$. Finally, the learned
encodings are decoded as the next action for the robot to execute via
$f^{act}_{\theta}\left(\cdot\right)$. Crucially, the predicted action is
grounded over an _a-priori_ unknown state and type of objects in the
environment. The predicted action is executed in the environment and the
updated state action history is used for estimation at the next time step.
Using a neural network as a hyper-parametric function for action prediction
enables us to leverage ConceptNet embeddings to generalize and scale with the
size of object sets, goal and interaction types. We utilize an imitation
learning model with the dataset described in Equation 2 to learn the $\theta$
parameters of the neural network. The constituent model components are
detailed next.
### 4.1 Graph Structured World Representation
Figure 2: Updated ToolNet model encodes the metric-semantic world state using
graph convolution (GGCN) and fully connected (FCN) layers. The model uses goal
information and the robot’s action history to attend over a task-specific
context, finally predicting the likelihood scores for each tool in the
environment. A graph structured representation and inclusion of pre-trained
word embeddings (from a knowledge base) facilitate generalization in
predicting interactions in novel contexts with new objects unseen in training.
#### 4.1.1 Semantic Reasoning
We first describe the $f^{state}_{\theta}\left(\cdot\right)$ function in
equation (3). We denote the robot’s current world state $s_{t}$ as an object-
centric graph $G_{t}=(O,R)$. Each node in the graph represents an object
instance $o\in O=\mathcal{O}(s_{t})$. The edge set consists of binary
relationships $\mathrm{OnTop}$, $\mathrm{ConnectedTo}$, $\mathrm{Near}$ and
$\mathrm{Inside}$ between objects $R\subseteq O\times O$. Let
$l_{o}\in\\{0,1\\}^{p}$ represents discrete object states for the object $o$
(e.g. $\mathrm{Open/Closed}$, $\mathrm{On/Off}$). Here, $p$ represents the
number of possible states of all objects. Next, it incorporates a pre-trained
function $\mathcal{C}(\cdot)$ that embeds a word (like token of an object
class or a relation) to a dense distributed representation, such that
semantically close tokens appear close in the learned space (?). The use of
such embeddings enables generalization, which we discuss subsequently.
Consider the example goal of “Place the box in the cupboard”. In this case we
want to satisfy the relationship $\mathrm{Inside}$ for cupboard and box. The
entire environment can be considered as the graph where relations like
$\mathrm{OnTop}(box,table)$ may be true. The pretrained function
$\mathcal{C}(\cdot)$ can be any pretrained language model.
Let $e_{o}=\mathcal{C}(o)\in\mathcal{R}^{q}$ denote the $q$-dimensional
embedding for an object instance $o$. The embeddings $l_{o}$ and $e_{o}$ model
object attributes that initialize the state of each object node in the graph.
The local context for each $o$ is incorporated via a Gated Graph Convolution
Network (GGCN) (?), which performs message passing between 1-hop vertices on
the graph. Following (?), the gating stage is realized as a Gated Recurrent
Unit (GRU) resulting in _graph-to-graph_ updates as:
$\displaystyle\begin{split}r_{o}^{0}&=\mathrm{tanh}\left(W_{r}\left[l_{o};e_{o}\right]+b^{r}\right),\\\\[-3.0pt]
x^{k}_{o}&=\sum_{j\in R}\sum_{o^{\prime}\in
N^{j}(o)}W_{j}^{k}r_{o^{\prime}}^{k-1},\\\\[-3.0pt]
r^{k}_{o}&=\mathrm{GRU}\left(r^{k-1}_{o},x^{k}_{o}\right).\\\\[-3.0pt]
\end{split}$ (4)
Here, the messages for object $o$ are aggregated over neighbors $N^{j}(o)$
connected by relation $j$ ($\forall j\in R$) during $n$ convolutions,
resulting in an embedding $r^{n}_{o}$ for each object instance in the
environment. Unlike a fully-connected neural network, a GGCN model facilitates
inference over the relations across objects for an input object centric graph.
These relations are crucial to predict the relevant actions and achieve the
intended goal. GRUs in the convolution iterations facilitate stable training
of the model (?).
#### 4.1.2 Fusing Metric Information.
Next, we incorporate the metric information associated with objects. Let
$pose_{o}$ and $size_{o}$ represent the pose and size/extent (along xyz axes)
for each object instance. The pose includes the spatial position and the
orientation of the object. We do not explicitly provide the point-cloud
geometric model and abstract out the surface information that may be required
to infer the utility of objects as tools and instead rely on ConceptNet
embeddings for the same. However, in our evaluation (Section 6), we utilize
CAD models that closely represent the surface properties to model the physics
of tool use. Another point to note here is that for the specific task of
utilizing an object as a tool the pose information has little contribution.
However, the pose information is required to accurately identify the system
state. For instance, the pose information of the door of a fridge indicates
whether the fridge is open or closed. This information is given as a
categorical value i.e. the state vector $l_{o}$ to the GGCN model. To
explicitly pass this information to the FCN model, we use the pose vector of
objects as inputs. The properties are encoded using a $d$-layer Fully
Connected Network (FCN) with a Parameterized ReLU (PReLU) (?) activation as:
$\displaystyle\begin{split}m^{0}_{o}&=\mathrm{PReLU}\left(W_{mtr}^{0}[pose_{o};size_{o}]+b_{mtr}^{0}\right)\\\\[-3.0pt]
m^{k}_{o}&=\mathrm{PReLU}\left(W_{mtr}^{k}m^{k-1}_{o}+b_{mtr}^{k}\right),\end{split}$
(5)
resulting in the metric encoding $m^{d}_{o}$ for each object in the scene. A
world state encoding (for $s_{t}$) is obtained by fusing the semantic and
metric embeddings as
$f^{state}_{\theta}(s_{t})=\\{\tilde{s}_{t}^{o}\\!=\\![r^{n}_{o};m^{d}_{o}]|\
\forall o\in{\cal O}(s_{t})\\}.$ (6)
_Late fusion_ of the two encodings allows downstream predictors to exploit
them independently.
### 4.2 Encoding the Action History
Next, we define the $f^{hist}_{\theta}\left(\cdot\right)$ function in equation
(3). The task of deciding the next action is informed by the agent’s action
history in two ways. First, sequential actions are often temporally
correlated. For example, a placing task often involves moving close to the
box, opening it and then placing an item inside it. If the previous action
involved moving close to a box, this information facilitates the model to
leverage localized contextual information and not jump to manipulate another
object in the goal specification. This, in tandem with the goal-conditioned
attention facilitates seamless action sequence generation. Hence, maintaining
the action history can help in prediction of the next action. Second, the set
of actions the robot executed in the past provides a local context indicating
the objects the the robot may utilize in the future. Formally, we encode the
temporal action history $\eta_{t}$ using an $\mathrm{LSTM}$. We define action
encoding $\mathcal{A}(a_{t-1})$ of $a_{t-1}=I_{t}(o^{1}_{t-1},o^{2}_{t-1})$
independent of the object set, as
$\mathcal{A}(a_{t-1})=[\vec{I}_{t-1};\mathcal{C}(o^{1}_{t-1});\mathcal{C}(o^{2}_{t-1})]$,
where $\vec{I}_{t-1}$ is a one-hot vector over possible interaction types
$\mathcal{I}$, and $\mathcal{C}(o^{1}_{t-1})$ and $\mathcal{C}(o^{2}_{t-1})$
represent the word embeddings of the object instances $o^{1}_{t-1}$ and
$o^{2}_{t-1}$. At each time step $t$, the $\mathrm{LSTM}$ encoder takes in the
encoding of previous action, $\mathcal{A}(a_{t-1})$ and outputs the updated
encoding $\tilde{\eta}_{t}$, given as
$\tilde{\eta}_{t}=\mathrm{LSTM}(\mathcal{A}(a_{t-1}),\tilde{\eta}_{t-1})$.
This results in the embedding vector
$f^{hist}_{\theta}(\eta_{t})=\tilde{\eta}_{t}.$ (7)
### 4.3 Goal-conditioned Attention
We now define the $f^{goal}_{\theta}\left(\cdot\right)$ function in equation
(3). The goal $g$ consists of symbolic relations (e.g. $\mathrm{Inside}$,
$\mathrm{OnTop}$ etc.) between object instances (e.g., carton and cupboard)
that must be true at the end of the robot’s plan execution. The declarative
goal input to the model is partitioned as relations $g_{rel}$ and the object
instances specified in the goal $g_{obj}$. The resulting encondings are
denoted as $\tilde{g}_{rel}$ and $\tilde{g}_{obj}$:
$\tilde{g}_{rel}=\frac{1}{|g_{rel}|}\sum_{j\in g_{rel}}\mathcal{C}(j)\hskip
5.69054pt\mathrm{and}\hskip
5.69054pt\tilde{g}_{obj}=\frac{1}{|g_{obj}|}\sum_{o\in
g_{obj}}\mathcal{C}(o).$
Next, the goal encoding and the action history encoding $\tilde{\eta}_{t}$ is
used to learn attention weights over objects in the environment (?) such that
$\displaystyle\begin{split}\epsilon_{o}=\mathrm{softmax}\left(W_{g}[\tilde{s}_{t}^{o};\tilde{g}_{obj};\tilde{\eta}_{t}]+b_{g}\right),\end{split}$
(8)
where $\tilde{s}_{t}^{o}$ is obtained from $f^{state}_{\theta}(s_{t})$ in (6)
and $\tilde{\eta}_{t}$ is obtained from $f^{hist}_{\theta}(\eta_{t})$ in (7).
This results in the attended scene encoding $\Omega_{t}$ as:
$\Omega_{t}=f^{goal}_{\theta}(\tilde{s}_{t}^{o},\tilde{g}_{obj},\tilde{\eta}_{t})=\sum_{o\in
O}\mathrm{\epsilon_{o}}\tilde{s}_{t}^{o}.$ (9)
The attention mechanism aligns the goal information with the scene learning a
task-relevant context, relieving the model from reasoning about objects in the
environment unrelated to the task, which may be numerous in realistic
environments.
### 4.4 Tool Likelihood Prediction
Now we define the function that predicts the likelihood scores for each tool
in the environment state, i.e., the $f^{tool}_{\theta}\left(\cdot\right)$
function in equation (3). This likelihood score corresponds to the probability
that a particular tool could be used to reach a goal state for a given
environment state and declarative goal specification. In order to allow the
model to generalize to unseen tools, instead of prediction over the pre-
defined tool set $\hat{\tau}$, we allow the model to predict a likelihood
score of a tool $t$ (which may not be present in any of the scenes in the
training set) to be used as a tool using the object embedding
($e_{t}=\mathcal{C}(t)$ as described in Section 4.1.1). This recurrence is
shown in the factored tool likelihood module in Figure 2. The prediction is
made using the encoding of the state, i.e the attended scene embedding
$\Omega_{t}$, the relational description of the goal $\tilde{g}_{rel}$ and the
action history encoding $\tilde{\eta}_{t}$. Likelihood of each tool is
computed for each $t$ using,
$p_{t}=\mathrm{sigmoid}(W[\Omega_{t};\tilde{g}_{rel};\tilde{\eta}_{t};e_{t}])\
\forall\ t\in\hat{\tau},$ (10)
where $\Omega_{t}$ is obtained using (9) and $\tilde{\eta}_{t}$ is obtained
using (7). We now predict over a tool set $\tau$, which may be different from
the fixed toolset seen during training set. We include the possibility of
having tools present at inference time that are unseen during training. The
factored style likelihood prediction enables our model to be flexible to make
likelihood predictions for unseen tools. We now use this to define the
likelihood score for each object as $p_{t}$ for tools and $0$ otherwise. Here
$p_{t}$ is the probability that tool t can be used to complete the goal.
Specifically,
$f^{tool}_{\theta}(s_{t},g_{t},\eta_{t})=\\{p_{o}\text{ if }o\in\tau\text{
else }0\ |\ \forall o\in{\cal O}(s_{t})\\}.$ (11)
Unlike the original ToolNet model (?), $f^{tool}_{\theta}\left(\cdot\right)$
function in this work can predict the tool likelihood scores for each time-
step $t$ and utilizes action history encoding $\tilde{\eta}_{t}$ to capture
temporal context.
Figure 3: ToolTango neural model is architecturally similar to ToolNet but
also incorporates tool likelihood scores predicted from the ToolNet model,
finally decoding the next symbolic action for the robot to execute. The
interaction type and objects are predicted auto-regressively and in a factored
style.
### 4.5 Robot Action Prediction
We now take the encoded information about the world state, goal and action
history to decode the next symbolic action $a_{t}=I_{t}(o^{1}_{t},o^{2}_{t})$.
The three components $I_{t}$, $o^{1}_{t}$ and $o^{2}_{t}$ are predicted auto-
regressively, where the prediction of the interaction, $I_{t}$ is used for the
prediction of the first object, $o^{1}_{t}$ and both their predictions are
used for the second object prediction, $o^{2}_{t}$. For the object predictors
$o^{1}_{t}$ and $o^{2}_{t}$, instead of predicting over a predefined set of
objects, our model predicts a likelihood score of each object $o\in O$ based
on its object embedding $\tilde{s}_{t}^{o}$ and tool likelihood score $p_{o}$
from (11). It then selects the object with highest likelihood score. The
resulting factored likelihood allows the model to generalize to an _a-priori_
unknown number and types of object instances:
$\displaystyle
I_{t}=\mathrm{argmax}_{I\in\mathcal{I}}\left(\mathrm{softmax}(W_{I}[\Omega_{t};\tilde{g}_{rel};\tilde{\eta}_{t}]+b_{I})\right),$
(12)
$\displaystyle\begin{split}o^{1}_{t}&=\mathrm{argmax}_{o\in{O}}\alpha_{t}^{o}\\\
&=\mathrm{argmax}_{o\in{O}}\
(\sigma(W_{\alpha}[\Omega_{t};\tilde{g}_{rel};\tilde{\eta}_{t};e_{o};p_{o};\vec{I}_{t}]+b_{\alpha}))\mathrm{,}\end{split}$
(13) $\displaystyle o^{2}_{t}=\mathrm{argmax}_{o\in{O}}\
(\sigma(W_{\beta}[\Omega_{t};\tilde{g}_{rel};\tilde{\eta}_{t};e_{o};p_{o};\vec{I}_{t};\alpha_{t}^{o}]+b_{\beta})).$
(14)
Here $\alpha^{o}_{t}$ denotes the likelihood prediction of the first object.
Finally, we impose grammar constraints (denoted as $\Lambda$) at inference
time based on the number of arguments that the predicted interaction $I_{t}$
accepts. If $I_{t}$ accepts only one argument, only $o^{1}_{t}$ is selected,
otherwise both are used. Thus, predicted action for time-step $t$ is denoted
as
$a_{t}=f^{act}_{\theta}(\Omega_{t},\tilde{g}_{rel},\tilde{\eta}_{t})=\Lambda[I_{t}(o^{1}_{t},o^{2}_{t})].$
(15)
This action is then executed by the robot in simulation. Our simulation model
follows closely to the one developed by ? (?) wherein symbolic actions such as
$\mathrm{moveTo}$, $\mathrm{pick}$ and $\mathrm{place}$ are realized as
relations in the environment state using a PyBullet simulator. For instance,
the $\mathrm{moveTo}$ action changes the xyz coordinates of the robot such
that it is $\mathrm{Near}$ the target object. Similarly, $\mathrm{pick}$
establishes a $\mathrm{connectedTo}$ relation between the target object and
robot arm. For implementation specific details, visit
https://github.com/reail-iitd/tango/wiki. The executed action and resulting
world state is provided as input to the model for predicting the action at the
next time step. The $f^{act}_{\theta}\left(\cdot\right)$ function denotes the
ToolTango model. In our original Tango model presented in (?), the $p_{o}$
tool likelihood score was not part of equations (13) and (14).
### 4.6 Word Embeddings Informed by a Knowledge Base
ToolTango uses word embedding function $\mathcal{C}(\cdot)$ that provides a
dense vector representation for word tokens associated with object class and
relation types. Contemporary models use word embeddings acquired from language
modeling tasks (?). We adopt embeddings that are additionally informed by an
existing knowledge graph $\mathrm{ConceptNet}$ (?) that provides a
sufficiently large knowledge graph connecting words with edges expressing
relationships such as $\mathrm{SimilarTo}$, $\mathrm{IsA}$,
$\mathrm{UsedFor}$, $\mathrm{PartOf}$ and $\mathrm{CapableOf}$. Word
embeddings (?) can be _retro-fitted_ such that words related using knowledge
graph embeddings are also close in the embedding space (?). Using such (pre-
trained) embeddings incorporates _general purpose_ relational knowledge to
facilitate richer generalization for downstream policy learning. Thus, given
an object, say $apple$, the word embedding function $\mathcal{C}(.)$ returns a
dense $\mathrm{ConceptNet}$ vector that corresponds to this token. The
complete sequence of steps is summarized in Figure 3.
### 4.7 Model Training
We decompose the action prediction task into first predicting tool, i.e., the
$f^{tool}_{\theta}\left(\cdot\right)$ function and then evaluating the
$f^{act}_{\theta}\left(\cdot\right)$ function. We also decompose model
training. First, we train the tool prediction model. The loss used to train
this model is Binary Cross-Entropy with each $t\in\hat{\tau}$ acting as a
class. The class label, $y^{i}_{t}$ is assigned 1 if it is used in a given
demonstration, $i$ and 0 otherwise. We also use categorical weights based on
plan execution time to encourage shorter plans (?). However, the knowledge of
the time taken for different plans has not been injected into the model. In
order to make this notion explicit to the model we use loss weighting such
that for the dataset $\mathcal{D_{\mathrm{Train}}}$ defined in (2),
$\mathcal{L}=-\sum_{i}\alpha_{i}\sum_{j}\sum_{t\in\tau}y^{i}_{j,t}\log(p^{i}_{j,t})+(1-y^{i}_{j,t})\log(1-p^{i}_{j,t}),$
(16)
where, $p^{i}_{j,t}$ is obtained from
$f^{tool}_{\theta}(s^{i}_{j},g^{i},\eta^{i}_{j})$ for each datapoint in
$\mathcal{D_{\mathrm{Train}}}$ and $\alpha_{i}$ is a multiplier that is high
for optimal plans (shortest among human demonstrations) and low otherwise.
For a trained tool prediction model, we then train the
$f^{act}_{\theta}\left(\cdot\right)$ ToolTango model, fine-tuning the weights
of our updated ToolNet function $f^{tool}_{\theta}\left(\cdot\right)$. Here
too we use the Binary Cross-Entropy loss, with the loss for the three
predictors (action and the two objects) being added independently. Additional
model training and hyperparameter details are given in Appendix B.
This decomposed training style has direct implications on the training
stability. First training ToolNet to predict the likelihood scores for tools
enables action prediction informed by the likelihood scores of each tool in
the environment.
## 5 Data Collection Platform and Annotation
Domain | Plan lengths | Objects interact- | Tools used | Sample objects | Sample goal specifications
---|---|---|---|---|---
ed with in a plan | in a plan
Home | 23.25$\pm$12.65 | 4.12$\pm$1.97 | 0.93$\pm$0.70 | floor1, wall, fridge123, cupboard123, tables1, couch1, big-tray1, tray1, book1, paper, cubes, light switch4, bottle, box2, fruits, chair15, stick, dumpster2, milk carton, shelf1, glue6, tape6, stool15, mop8, sponge8, vacuum8, dirt7, door2 | 1\. Place milk in fridge, 2. Place fruits in cupboard, 3. Remove dirt from floor, 4. Stick paper to wall, 5. Put cubes in box, 6. Place bottles in dumpster, 7. Place a weight on paper, 8. Illuminate the room.
Factory | 38.77$\pm$23.17 | 4.38$\pm$1.85 | 1.44$\pm$0.97 | floor1, wall, ramp, worktable1, tray1, box2, crates1, stick, long-shelf1, lift1, cupboard123, drill4, hammer49, ladder5, trolley2, brick, blow dryer48, spraypaint4, welder4, generator4, gasoline, coal, toolbox2, wood cutter4, 3D printer4, screw9, nail9, screwdriver49, wood, platform1, oil7, water7, board, mop8, glue6, tape6, stool15 | 1\. Stack crated on platform, 2. Stick paper to wall, 3. Fix board on wall, 4. Turn on the generator, 5. Assemble and paint parts, 6. Move tools to workbench, 7. Clean spilled water, 8. Clean spilled oil.
Table 1: Dataset characteristics. The average plan length measured as the
length of action sequence), number of objects interacted in plan and number of
tools used in plans with object and goal sets for Home and Factory domains.
Object positions were sampled using Gaussian distribution. Objects in bold can
be used as tools. Legend:- 1: surface, 2: can open/close, 3: container, 4: can
operate, 5: can climb, 6: can apply, 7: can be cleaned, 8: cleaning agent, 9:
can 3D print. Objects in bold can be used as tools. Object affordances derived
from properties in the ConceptNet graph (?). Stool/ladder are objects used to
represent a tool for raising the height of the robot.
### 5.1 Data Collection Environment
We develop a low-fidelity simulation environment where the robot can take
actions and interact with objects. The low-level physics of motion is less
important as we are primarily concerned with the symbolic effects on the world
state. To do this, we use PyBullet, a physics simulator (?), to generate home
and factory-like environments populated with a virtual mobile manipulator (a
Universal Robotics (UR5) arm mounted on a Clearpath Husky mobile base). The
world state is object-centric including the metric locations of objects and
the discrete states of symbolic attributes and relations. The objects in the
domains were derived from real-world home and factory scenes that span
Facebook Replica Dataset (?). These scenes were made to look as photo
realistic as possible. The objects on other hand were chosen to span the YCB
dataset (?). The CAD model for each object obtained from open-source
repositories such as the Google 3D Warehouse111
https://3dwarehouse.sketchup.com/. This was done in order to keep the
simulated physics as real as possible. The set of objects in the two domains
are listed in Table 1.
Each object in the environment is associated with a metric location and
physical extent and may optionally possess discrete states such as
$\mathrm{Open/Closed}$, $\mathrm{On/Off}$, etc. The world model is assumed to
possess spatial notions such as $\mathrm{Near}$ or $\mathrm{Far}$. Objects in
the world model can be supported by, contained within or connected with other
objects (or the agent). Hence, we include semantic relations such as
$\mathrm{OnTop}$, $\mathrm{Inside}$, $\mathrm{ConnectedTo}$ etc. More details
in Appendix D.
Robot Actions
---
Push, Climb up/down, Open/Close, Switch on/off, Drop, Pick, Move to, Operate
device, Clean, Release material on surface, Push until force
Object Attributes
Grabbed/Free, Outside/Inside, On/Off, Open/Close, Sticky/Not Sticky,
Dirty/Clean, Welded/Not Welded, Drilled/Not Drilled, Driven/Not Driven,
Cut/Not Cut, Painted/Not Painted
Semantic Relations
On top, Inside, Connected to, Near
Metric Properties
Position, Orientation, Size
Table 2: Domain Representation. Robot symbolic actions, semantic attributes,
relations to describe the world state and objects populating the scene in Home
and Factory Domains.
The robot possesses a set of behaviours or symbolic actions such as
$\mathrm{Moving}$ towards an object, $\mathrm{Grasping}$,
$\mathrm{Releasing/Dropping}$ or $\mathrm{Pushing}$ an object or
$\mathrm{Operating}$ an entity to imply actions that induce discrete state
changes such as opening the door before exiting, turning on a switch etc. We
assume that the robot’s actions can be realized by the presence of an
underlying controller. We encode the geometric requirements for actions as
symbolic pre-conditions. Examples include releasing an object from the gripper
before grasping another, opening the door before trying to exit the room. The
set of possible actions and the range of interactions are listed in Table 2.
The robot is tasked with goals that involve multiple interactions with objects
derived from standardized data sets (?). These goals include: (a)
_transporting_ objects from one region to another (including space on top of
or inside other objects), (b) _fetching_ objects, which the robot must reach,
grasp and return with, and (c) _inducing state changes_ such as illuminating
the room or removing dirt from floor. We assume that the robot is instructed
by providing _declarative_ goals. For example, the task of moving all fruits
on top of the kitchen table can be modeled as a set intended constraints among
the interacting objects. The effects of actions such as pushing or moving are
simulated via a motion planner and propagated to the next time step. Abstract
actions such as attachment, operating a tool or grasping/releasing objects are
encoded symbolically as the establishment or release constraints. The
simulation for these actions is coarse and considers their symbolic effects
forgoing the exact motion/skills needed to implement them. We assume that the
robot can realize abstract actions through low-level routines.
We present the human agents with a scene and goal conditions we want to attain
and receive actions in a custom grammar. The instructions are grounded in the
world and the agent and the virtual environment show its effects to input the
next action. This is repeated till a goal state is reached. Figure 4
illustrates the interactive platform used for data collection. Using a curated
dataset, the robot must synthesize a plan of executable actions to satisfy the
goal constraints. The presence of a rich space of interactions gives rise to
plans with multiple interactions between objects. For example, “packing items
into a basket and carrying the basket to the goal region”, “using a stick to
fetch and drop an object beyond reach into a box”, “using a ramp/stool to
elevate itself to fetch an object”.
### 5.2 Annotated Corpus
To curate an imitation learning dataset, we recruit human instructors and
provide them with goals. They instruct the robot by specifying a sequence of
symbolic actions (one at a time) to achieve each goal. Each action is
simulated so that they can observe its effects and the new world state. We
encourage the instructors to to complete the task as quickly as possible,
making use of available tools in the environment. To familiarize them with the
simulation platform, we conduct tutorial sessions before data collection. Our
resulting dataset consists of diverse plans with different action sets and
object interactions. We collected plan traces from $\mathrm{12}$ human
subjects using domain randomization with $\mathrm{10}$ scenes and $8$ semantic
goals resulting in a corpus of $\mathrm{708}$ and $\mathrm{784}$ plans for the
home and factory domains. Figure 5(a) and 5(b) show number of interactions
with 10 most interacted objects and frequency of 10 most frequent actions
respectively. The complete set of objects and goals is given in Table 1. We
also perform data augmentation by perturbing the metric states in the human
plan trace, performing random replacements of scene objects and validating
plan feasibility in simulation. The process results in $\mathrm{3540}$ and
$\mathrm{3920}$ plans, respectively. Variation was observed in tool use for an
instructor for different goals, and within similar goals based on context. The
annotated corpus was split as $(75\%:25\%)$ forming the Training data set and
the Test data set to evaluate model accuracy. A $10\%$ fraction of the
training data was used as the Validation set for hyper-parameter search. No
data augmentation was performed on the Test set.
Figure 4: Data Collection Interface. The human teacher instructs (left) a
virtual mobile manipulator robot by specifying symbolic actions. The human-
instructed plan is simulated and visualized (right). The user is shown the
goal needed to be completed in text form. They select the user interaction,
first object and second object to instruct the robot. The interface can be
seen in action in the https://www.youtube.com/watch?v=lUWU3rK1Gno
.
(a) Object interactions
(b) Symbolic actions
Figure 5: Data set Characteristics. Distribution of plans with plan length for
home and factory domains. Frequency of interaction of top 10 objects and
frequency of actions for top 10 actions. The collected data set contains
diverse interactions in complex spaces.
### 5.3 Generalization Test Set
In order to asses the model’s capacity to generalize to unseen worlds, we
curate a second test environment populated by instances of novel object types
placed at randomized locations. The following sampling strategies were used:
(i) _Position_ : perturbing and exchanging object positions in a scene. (ii)
_Alternate_ : removal of the most frequently used tool in demonstrated plans
evaluating the ability to predict the alternative, next best tool to use.
(iii) _Unseen_ : replacing an object with a similar object, which is not
present in training. (iv) _Random_ : replacing a tool with a randomly picked
object which is _unrelated_ to the task. (v) _Goal_ : replacing the goal
objects (objects included as part of the goal specifications). For example:
milk and fridge in the goal “put milk inside fridge”. Here milk may be
replaced with another similar object, such as apple and fridge by a container,
such as cupboard. This process resulted in a Generalization Test set with
$7460$ (goal, plan) pairs.
## 6 Experiments
We first present the results corresponding to high-level tool prediction that
leverages an updated ToolNet model. We then present the goal-reaching
performance of Tango and the unified ToolTango model. Finally, we present a
detailed analysis of the resulting plans in terms of generalization,
robustness to execution errors and plan efficiency.
### 6.1 Sequential Tool Prediction
#### 6.1.1 Baseline
We compare against the basic GGCN model of Section 4.1.1 as our baseline
model. This model is similar to the ResActGraph model proposed by ? (?) and
its encoder incorporates technical ideas from recent imitation learning works
(?) on action prediction. In the subsequent discussion, we consider similar
size of the hyperparameter set across all models performing the same task.
#### 6.1.2 Evaluation Metrics
We first compare the accuracy of the updated ToolNet model on the test and
generalization test sets. We test ToolNet’s tool prediction capabilities in
two settings. In the first setting, we use the dataset as described in Section
5 and split it according to the scene instance, i.e., home and factory. We use
accuracy as our performance measure. Here, a tool prediction is deemed correct
if the predicted tool is used in at least one of the various annotated plans
for the $(\rm{goal,scene})$ pair and incorrect otherwise.
Model | Test Set | Generalization | Generalization Test cases
---|---|---|---
| Home | Factory | Home | Factory | Position | Alternate | Unseen | Random | Goal
Baseline (GGCN) | 32.01 | 20.89 | 21.22 | 19.50 | 8.73 | 6.29 | 12.22 | 9.81 | 46.31
Updated ToolNet | 79.87 | 75.12 | 80.16 | 78.01 | 75.55 | 58.83 | 60.74 | 49.97 | 94.10
Table 3: A comparison of the updated ToolNet model with its previous version
and GGCN baseline for the accuracy of prediction of tool sequences instead of
single tool. Highest scores are shown in bold.
#### 6.1.3 Results
Table 3 shows the final accuracies on the test-set and generalization test-set
with individual accuracies for each test-type. On the regular test set,
ToolNet outperforms the baseline by 47.87 (149% higher) and 54.23 (259%
higher) accuracy points on Home and Factory domains, respectively.
A similar pattern is found in the generalization test set, where each
improvement is 58.94 (278% higher) and 58.51 (300% higher) accuracy points for
Home and Factory domains. The reasoning behind the improvements is as follows.
The goal-conditioned attention gives major improvement in the Goal
generalization test type, since goal objects get replaced in those examples.
Explicitly biasing the model to use the features of those objects (through
conditioned attention) increases their importance, and likely reduces
overfitting. An example for Alternate case is when _generator_ is specified in
the goal, and wood (the fuel for generator, and the most likely tool) is made
absent from the scene. The model could err in giving attention to the _wood-
cutter_ tool, which is often correlated with wood. However, conditioned
attention gives low attention to _wood-cutter_ and predicts _gasoline_ ,
instead.
Moreover, factored likelihood predictions helps the most in Unseen cases,
since without this component, the model cannot predict any unseen tool. A
decent performance of earlier models in this case is attributed to alternative
possible correct answers (any alternative seen tool or no-tool) due to
multiple annotations per scenario. The $\mathrm{ConceptNet}$ embeddings likely
contain commonsense knowledge about unseen tools and objects, for example,
whether a new tool is flat or not (which should help in ascertaining whether
it can be used for transport or not). Using these embeddings makes huge
improvement in Unseen cases where entirely new objects are to be predicted as
tools. Finally, giving higher weight to optimal plans allows the model to
differentiate tools by plan execution time and not human usage frequency. This
helps in improved metric generalization, predicting nearby tools in the
Position test case. Overall, the complete architecture provides the maximum
generalization accuracy among all models.
### 6.2 Action Prediction
#### 6.2.1 Baselines
We compare to the following three baseline models. (1) _ResActGraph_ model
(?), augmented with $\mathrm{FastText}$ embeddings (?). (2) _Affordance-only_
baseline inspired from (?) that learns a co-association between tasks and
tools, implemented by excluding the graph convolutions and attention from
Tango. (3) _Vanilla Deep Q-Learning (DQN)_ approach (?) that learns purely by
interactions with a simulator, receiving positive reward for reaching a goal
state.
#### 6.2.2 Tool Prediction Accuracy
Our experiments use the following accuracy metrics for model evaluation: (i)
_Action prediction accuracy_ : the fraction of tool interactions predicted by
the model that matched the human demonstrated action $a_{t}$ for a given state
$s_{t}$, and (ii) _Plan execution accuracy_ : the fraction of estimated plans
that are successful, i.e., can be executed by the robot in simulation and
attain the intended goal (in max. $50$ steps).
We first present the results for the vanilla Tango model and later present
improvements with the ToolTango.
#### 6.2.3 Comparing Tango with Baselines
Table 4 (top half) compares the Tango model performance with the baseline
models. The Tango model shows a $14-23$ point increase in _Action prediction
accuracy_ and a $66-71$ points increase in the _Plan execution accuracy_ when
compared to the _ResActGraph_ baseline. Note that the ResActGraph model learns
a scene representation assuming a fixed and known set of object types and
hence can only generalize to new randomized scenes of known objects. In
contrast, the Tango model can not only generalize to randomized scenes with
known object types (sharing the GGCN backbone with ResActGraph) but can to
novel scenes new object types (relying on dense semantic embeddings) and an
a-priori unknown number of instances (enabled by a factored likelihood).
The _Affordance-only_ baseline model is confined to learning the possible
association between a tool object type and the task specified by the human
(largely ignoring the environment context). This approach addresses only a
part of our problem as it ignores the sequential decision making aspect, where
tools may need to be used in sequence to attain a goal. Finally, the vanilla
DQN baseline achieves less than $20\%$ policy accuracy (even after a week of
training). In contrast, the Tango model shows accurate results after training
on imitation data for $12-18$ hours. The challenges in scaling can be
attributed to the problem size ($\approx$1000 actions), long plans, sparse and
delayed rewards (no reward until goal attainment).
Model | Action Prediction | Plan Execution | Generalization Plan Execution Accuracy
---|---|---|---
| Home | Factory | Home | Factory | Home | Factory | Position | Alternate | Unseen | Random | Goal
Baseline (ResActGraph) | 27.67 | 45.81 | 26.15 | 0.00 | 12.38 | 0.00 | 0.00 | 0.00 | 0.00 | 25.10 | 9.12
Affordance Only | 46.22 | 52.71 | 52.12 | 20.39 | 44.10 | 4.82 | 17.84 | 47.33 | 29.31 | 29.57 | 34.85
DQN | - | \- | 24.82 | 17.77 | 15.26 | 2.23 | 0.00 | 0.00 | 12.75 | 9.67 | 4.21
Tango | 59.43 | 60.22 | 92.31 | 71.42 | 91.30 | 60.49 | 93.44 | 77.47 | 81.60 | 59.68 | 59.41
Model Ablations
\- GGCN (World Representation) | 59.43 | 60.59 | 84.61 | 27.27 | 78.02 | 38.70 | 70.42 | 58.79 | 60.00 | 56.35 | 38.64
\- Metric (World Representation) | 58.8 | 60.84 | 84.61 | 62.34 | 72.42 | 51.83 | 59.68 | 67.19 | 60.79 | 84.47 | 21.70
\- Goal-Conditioned Attn | 53.14 | 60.35 | 53.85 | 11.69 | 37.02 | 8.80 | 35.33 | 15.05 | 32.14 | 41.67 | 6.51
\- Temporal Action History | 45.91 | 49.94 | 24.61 | 0.00 | 8.55 | 0.00 | 0.00 | 0.00 | 0.00 | 30.56 | 1.15
\- Factored Likelihood | 61.32 | 61.34 | 95.38 | 85.71 | 34.22 | 43.44 | 90.50 | 14.82 | 30.65 | 64.67 | 53.26
\- ConceptNet | 63.52 | 60.35 | 89.23 | 57.14 | 81.86 | 56.97 | 82.33 | 68.61 | 74.57 | 65.73 | 47.92
\- Constraints | 57.23 | 57.74 | 64.62 | 37.66 | 62.98 | 41.95 | 84.95 | 45.39 | 39.99 | 36.11 | 84.85
\- Auto-regression | 56.60 | 60.22 | 69.23 | 50.65 | 73.75 | 53.32 | 71.24 | 66.43 | 61.27 | 70.51 | 34.66
Table 4: A comparison of _Action prediction_ and _Plan execution_ accuracies
for the baseline, the proposed Tango model, and ablations. Results are
presented for test and generalization data sets (under five sampling
strategies) derived from the home and factory domains. Accuracy Prediction is
the percentage of predicted actions matching the human input on the Test set.
Plan Execution is the percentage of plans successfully executed in the Test
set. Generalization Plan Accuracy is the percentage of plans successfully
executed in the set. Highest scores shown in bold.
Next, we assess the zero-shot transfer setting, i.e., whether the model can
perform common sense generalization in worlds with new objects unseen in
training. The same table shows that the plans predicted by Tango lead to an
increase of up to $56$ points in plan execution accuracy on Generalization
Test set over the best-performing baseline model. This demonstrates accurate
prediction and use of unseen tool objects for a given goal. Specifically, in
the home domain, if the _stool_ is not present in the scene, the model is able
to use a _stick_ instead to fetch far-away objects. Similarly, if the robot
can predict the use of a box for transporting objects even if it has only seen
the use of a tray for moving objects during training. The ResActGraph model is
unable to adapt to novel worlds and obtains zero points in several
generalization tests.
The poorer performance of the _Affordance-only_ model can again be attributed
to the fact that planning tool interactions involves sequential decision-
making. Even if the robot can use affordance similarity to replace a _tray_
object with a _box_ , it still needs to predict the opening of the _box_
before placing an item in its plan for a successful execution. This is
corroborated by the drop in performance for the Unseen generalization tests
for this model by $52.3$ points. Finally, the vanilla DQN model lacks a clear
mechanism for transferring to novel settings, hence shows poor generalization
in our experiments.
#### 6.2.4 Ablation Analysis of Tango Components
We analyze the importance of each component of the Tango model by performing
an ablation study. Table 4 (lower half) presents the results. For a fair
comparison, the model capacities remain the same during the ablation
experiments.
The model builds on the _GGCN_ environment representation encoding the inter-
object and agent-object relational properties. The ablation of the GGCN
component results in a reduction of 22% in the generalization accuracy in the
factory domain (where tools may be placed at multiple levels in the factory).
The inclusion of this component allows the robot to leverage relational
properties such as OnTop to predict the use of tools such as a ramp to
negotiate an elevation or a stick to fetch an object immediately beyond the
manipulator’s reach.
The _Metric_ component encodes the metric properties of objects in the scene
such as positions, size etc. Experiments demonstrate its effectiveness in
prediction tool interactions based on relative sizes of interacting objects.
E.g., the model successfully predicts that _fruits_ can be transported using a
_tray_ but larger _cartons_ require a _box_ for the same task. The ablation of
this component leads to a reduction of $10.2$ points in the Alternate
generalization tests as the ablated model unable to adapt the tool when there
are unseen objects with different sizes than those seen during training.
Next, we assess the impact of removing the _Goal-Conditioned Attention
component_. This experiment shows a a significant reduction ($\approx 50$
points) in the _Plan execution accuracy_ on the Generalization Test set,
particularly in scenes with a large number of objects. The attention mechanism
allows learning of a restricted context of tool objects that may be useful for
attaining the provided goal, in essence, filtering away goal-irrelevant
objects populating the scene. Additionally, note that the inclusion of this
component allows tool predictions to be _goal-aware_. Consequently, we observe
ablating this component leads to a reduction of $53$ points in the Goal
generalization test set where the goal objects are perturbed.
The _Action History_ component utilizes the agent’s past interactions for the
purpose of predicting the next tool interaction. The inclusion of this
component allows learning of correlated and commonly repeated action
sequences. For instance, the task of exiting from a room typically involves a
plan fragment that includes moving to a door, opening it and exiting from the
door and are commonly observed in a number of longer plans. The ablation of
this component leads to erroneous predictions where a particular action in a
common plan fragment is missing or incorrectly predicted. E.g., a robot
attempting to pick an object inside an enclosure without opening the lid. In
our experiments, we observe that ablating the model leads to a significant
decrease in goal reach-ability, causing a $70$ point decrease in the _Plan
Execution accuracy_ and $72$ point drop in the _Generalization accuracy_.
The need for generalization to novel scenes implies that our model cannot
assume a fixed and a-priori known set of objects that the robot can interact
with. Generalization to an arbitrary number of objects in the scene is
accomplished by factoring model predictions over individual objects in a
recurrent manner. Ablating the factored likelihood components results in a
simpler model that performs predictions over a known fixed-size object set.
The simplified model displays a higher action-prediction and plan-execution
accuracies in known world. Crucially, we observe that ablating this component
results in a significant decrease of $51$ and $63$ points in the Unseen and
the Alternate generalization test sets.
Finally, the _ConceptNet embeddings_ are important for semantic generalization
to unseen tool types. We replace ConceptNet embeddings with
$\mathrm{FastText}$ (?) embeddings in the _-ConceptNet_ model to show their
importance. The _-ConceptNet_ model shows poorer generalization (6.5%
decrease) as it models word affinity as expressed in language only.
$\mathrm{ConceptNet}$ embedding space models relational affinity between
objects as present in the knowledge-base.
Model | Action Prediction | Plan Execution | Generalization Plan Execution Accuracy
---|---|---|---
| Home | Factory | Home | Factory | Home | Factory | Position | Alternate | Unseen | Random | Goal
TANGO | 59.43 | 60.22 | 92.31 | 71.42 | 91.30 | 60.49 | 93.44 | 77.47 | 81.60 | 59.68 | 59.41
ToolTANGO | 64.12 | 63.72 | 95.69 | 77.01 | 92.88 | 62.97 | 94.38 | 88.12 | 92.71 | 78.01 | 64.43
Table 5: A comparison of TANGO with ToolTANGO model.
#### 6.2.5 Comparing Tango with ToolTango
Table 5 compares the performance of ToolTango with the Tango model. ToolTango
gives improved scores in both domains and for every generalization test case.
The Action prediction accuracy of ToolTango is higher than Tango by 4.69 and
3.50 points for Home and Factory domains, respectively. Similarly, we see an
increase in the Plan execution accuracy both for Test and Generalization Test
sets. We get 3.38- 5.59 higher points on Test and 1.58-2.48 points in
Generalization Test set for Home and Factory domains. The improvement is
higher in Factory domain where the probability of using multiple tools in the
same plan to reach a goal state is higher. This is particularly due to the
independent tool prediction in ToolTango that aids action prediction in
complex settings requiring longer action sequences (more details in Section
6.3). The performance improvement is highest for the Alternate and Random
cases where the tools are replaced by an alternate tool or another non-tool
object. This shows the importance of independent tool-likelihood score
prediction in the ToolTango model.
Figure 6: A simulated robot manipulator uses ToolTango to synthesize tool
interactions in novel contexts with unseen objects. (top) ToolTango predicts
the use of a box when tray is unavailable. (bottom) ToolTango predicts the use
of hammer and nails when screws are unavailable. Figure 7: The model predicts
the instance of tray (on the left) which is closer to the fruits (goal
objects) other instance (one on the right).
Figure 8: Interleaved action prediction and execution enables adaptation in
case of unexpected errors during action execution. (top) robot recovers after
the milk carton falls. (bottom) recovers after a fruit falls from the tray by
picking it up again.
### 6.3 Analysis of Resulting Plans
#### 6.3.1 Evidence of Generalization
Figure 6 shows the robot using the learned model to synthesize a plan for a
declarative goal. Here, if the goal is to transport fruits and human
demonstrates usage of _tray_ and the model never sees _box_ while training,
ToolTango uses _box_ in a scene where tray is absent, showing that it is able
to predict semantically similar tools for task completion. Similarly, for the
goal of fixing a board on the wall, if humans use _screws_ the agent uses
_nails_ and _hammer_ when screws are absent from the scene. Figure 7 shows how
the model uses the position information of tool objects to predict the tool
closer to the goal object or the agent. The world representation encodes the
metric properties of objects (position and orientation) that allows the robot
to interact with nearer tool objects.
Figure 9: Scatter plot comparing the lengths of plans obtained from model
predictions and those from human demonstrations. The plans generated from
ToolTango are of similar length as that of human demonstrators.
#### 6.3.2 Robustness to Errors
Figure 8 shows the robustness to unexpected errors and stochasticity in action
execution. Consider the task of “fetching a carton”, where the milk carton is
on an elevated platform, the model predicts the uses a stool to elevate
itself. The carton falls due to errors during action execution. Following
which, the robot infers that the stool is no longer required and directly
fetches the carton. Similarly, for the task of “storing away the fruits in the
cupboard”, the robot predicts the use of tray for the transport task. During
execution the apple falls off the tray. The robot correctly re-collects the
apple.
#### 6.3.3 Plan Efficiency Comparison
Figure 9 compares the length of robot plans predicted by the learned model
against human demonstrated plans. We observe that, on average, the predicted
plan lengths are close to the human demonstrated ones. In 12% cases, the plans
predicted by ToolTango utilize tools satisfying the goal condition in fewer
steps compared to the human demonstrated plan.
Figure 10: Test cases where TANGO fails but ToolTANGO reaches the goal state.
Typically in cases with complex and long plans with multiple tools being used.
Tango predicts stool in the first case and coal in the second case, both of
which are unreachable. ToolTango predicts reachable tools. (Top) Here the
robot needs to place the milk carton which is at a higher location which is
unreachable without a stool. In this test case, the stool in unreachable so
the model uses a stick to reach the milk carton. (Bottom) Switching on the
generator requires a fuel. In this test case coal is unreachable/unavailable
and so the model instead uses gasoline to turn on the generator.
Figure 11: Execution accuracy of inferred plans with plan length for Tango and
ToolTango. The latter gives a higher goal-reaching performance than the former
for plans with longer action sequences.
Figure 12: An analysis of fractional errors during plan execution using the
learned ToolTango model. The horizontal axis denotes the total errors for the
ablated model. The absolute value of the total errors are shown in Table 4
#### 6.3.4 Analysis of Independent Tool Prediction
Figure 10 demonstrates the advantage of independent tool prediction and how it
augments the ToolTango to improve goal-reaching performance. The figure shows
two cases where the Tango model is unable to reach the goal state, but
ToolTango reaches a goal. The first example shows a setting where a milk
carton is placed on top of the fridge and stool is kept directly in front of
the fridge. In this case, Tango predicts the use of stool to reach the carton;
however, it first opens the fridge door, making stool inaccessible. On the
other hand, ToolTango predicts using a stick to drop the carton on top of the
table and is able to place it inside the fridge. The second example shows a
case where the generator needs to be powered on after adding a fuel source.
Here, ToolTango predicts the picking coal that is on top of the shelf.
However, it does not use any tool to elevate itself and the execution returns
an error of unreachable object. Instead, ToolTango predicts the use of
gasoline that is placed on the ground and is successfully able to execute the
plan.
Independent tool prediction also allows the model to perform better in complex
settings requiring longer action sequences and using multiple tools. Figure 11
assesses the model accuracy with the lengths of the inferred plans. We observe
that ToolTango has a higher plan execution accuracy in case of (goal, scene)
pairs where the average plan length to attain a goal state is high. For
instance, if the goal is to place fruits in side the cupboard and all the
fruits are on top of the fridge, the robot needs to use a tool to first reach
the fruits and then use another tool to carry them to the cupboard. In this
case the Tango model directly tries to pick unreachable fruits or predicts an
incorrect tool to reach the fruits. ToolTango is able to execute complex task
by subsequently using a stool and tray.
## 7 Limitations and Future Work
### 7.1 Scaling to Longer Plans
As we observed in Figure 11, the plan execution accuracy decreases by 20% on
the Test sets and 30% on Generalization Test sets. This merits investigation
into planning abstractions (?) for scaling to longer plan lengths in realistic
domains. Figure 12 analyzes the errors encountered during plan execution using
actions predicted by the proposed model. In 27% of the cases, the model misses
a pre-requisite actions required for a pre-condition for initiating the
subsequent action. For example, missing the need to open the door before
exiting the room (object unreachable $19\%$) or missing opening the cupboard
before picking an object inside it (object inside enclosure $8\%$). There is
scope for improvement here by incorporating explicit causal structure (?).
### 7.2 Partially observable environments
Figure 13: Likelihood estimates of exploration for different objects where
“screw” might be found.
In realistic scenarios, the robot’s environment may only be partially-known
due to the limited view and scope of the robot’s sensors. For example, tools
such as screws may be stored away in a tool box and may not be easily observed
by the robot. In such a scenario, we expect the robot to learn to predict
possible locations for exploration based on common sense knowledge. For
example, the robot should explore the tool box. Further, the subgeometry
information may be fed into the model to determine the set of objects in the
environment. In order to address partially-observed worlds, we can extend the
prediction model as follows. Instead of learning attention only over candidate
objects, we can learn a _generalized_ attention over spatial relations modeled
in the graph network. Such an extension allows the model to predict a
preference order for locations that the robot should explore to find the
required tool. Figure 13 illustrates an example, where the robot shows an
example, where the robot predicts that the screws may be found in toolbox or
workbench. Please note that this result is indicative of the possibility of
using the model in partially-known world and will be investigated in detail as
part of future work.
### 7.3 Extension to realistic robot control settings
The action representations adopted in this work are inspired from task and
motion planning communities (?, ?). We simulate a mobile manipulator
(Clearpath Robotics Husky with a mounted Universal Robotics UR5 arm). The
robot’s action space is modeled as high-level skills (or behaviors), each in-
turn realized using a low-level motion plan or controller that is
parameterized at run time. For navigation actions, a standard state-space
planner (A* with Manhattan distance as a heuristic) for a coarse collision
free path and realize point to point motion through a velocity-based
controller. For the manipulation, we consider a 6 degree of freedom model of
the manipulator. For object manipulation a crane-grasp was implemented where
the arm would move to a pre-grasp position to the intended object of interest.
The arm motion was realized using a joint state controller moving the arm to
the end-effector pose. The precise manipulation of tools is abstracted as
follows: once the robot end-effector makes contact a rigid joint is
established for subsequent motions till it is released. As mentioned in
Section 1, learning the precise manipulation of tools is not the focus of this
work and can be delegated to a method such as the one proposed by ? (?). All
robot actions are executed in the PyBullet physics engine with a mesh-based
model of objects and a URDF-model for the robot manipulation system. The
technical details appear in Appendix D. The entire implementation of the robot
behaviors and the simulation environment is available at
https://github.com/reail-iitd/tango for the use of the research community.
## 8 Conclusions
This paper proposes ToolTango, a novel neural architecture that learns a
policy to attain intended goals as tool interaction sequences leveraging
fusion of semantic and metric representations, goal-conditioned attention,
knowledge-base corpora. ToolTango is trained using a data set of human
instructed robot plans with simulated world states in home and factory like
environments. It uses an independent model that predicts likelihood scores for
each tool at each time-step. The imitation learner demonstrates accurate
commonsense generalization to environments with novel object instances using
the learned knowledge of shared spatial and semantic characteristics. It also
shows the ability to adapt to erroneous situations and stochasticity in action
execution. Finally, ToolTango synthesizes a sequence of tool interactions with
a high accuracy of goal-attainment.
Acknowledgments
Mausam is supported by an IBM SUR award, grants by Google, Bloomberg and 1MG,
Jai Gupta chair fellowship, and a Visvesvaraya faculty award by Govt. of
India. Rohan Paul acknowledges support from Pankaj Gupta Faculty Fellowship
and DST’s Technology Innovation Hub (TIH) for Cobotics. Shreshth Tuli is
supported by the President’s PhD Scholarship at Imperial College London. We
thank the IIT Delhi HPC facility and Prof. Prem Kalra and Mr. Anil Sharma at
the CSE VR Lab for compute resources. We thank Mr. Pulkit Sapra and Prof. P.
V. M. Rao for assistance with CAD model creation. We thank ? (?) for sharing
the implementation for baseline comparison. We are grateful to anonymous
turkers and student volunteers for assisting in the data collection.
## References
* Allen et al. Allen, K. R., Smith, K. A., and Tenenbaum, J. (2019). Rapid trial-and-error learning in physical problem solving.. In CogSci, p. 90.
* Antunes et al. Antunes, A., Saponaro, G., Dehban, A., Jamone, L., Ventura, R., Bernardino, A., and Santos-Victor, J. (2015). Robotic tool use and problem solving based on probabilistic planning and learned affordances. In Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. Workshop Learn. Object Affordances Fundamental Step to Allow Prediction Plan. Tool use.
* Bae et al. Bae, H., Kim, G., Kim, J., Qian, D., and Lee, S. (2019). Multi-robot path planning method using reinforcement learning. Applied Sciences, 9(15), 3057.
* Bahdanau et al. Bahdanau, D., Cho, K. H., and Bengio, Y. (2015). Neural machine translation by jointly learning to align and translate. In 3rd International Conference on Learning Representations, ICLR 2015.
* Bansal et al. Bansal, R., Tuli, S., Paul, R., and Mausam (2020). Toolnet: Using commonsense generalization for predicting tool use for robot plan synthesis. In Workshop on Advances & Challenges in Imitation Learning for Robotics at Robotics Science and Systems (RSS).
* Bisk et al. Bisk, Y., Zellers, R., Bras, R. L., Gao, J., and Choi, Y. (2020). Piqa: Reasoning about physical commonsense in natural language. In AAAI.
* Boteanu et al. Boteanu, A., Kent, D., Mohseni-Kabir, A., Rich, C., and Chernova, S. (2015). Towards robot adaptability in new situations. In 2015 AAAI Fall Symposium Series. Citeseer.
* Calli et al. Calli, B., Singh, A., Bruce, J., Walsman, A., Konolige, K., Srinavasa, S. S., Abbeel, P., and Dollar, A. M. (2017). YCB Benchmarking Project: Object Set, Data Set and Their Applications. Journal of The Society of Instrument and Control Engineers, 56(10), 792–797.
* Chen et al. Chen, H., Tan, H., Kuntz, A., Bansal, M., and Alterovitz, R. (2020). Enabling robots to understand incomplete natural language instructions using commonsense reasoning. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 1963–1969. IEEE.
* Choi et al. Choi, D., Langley, P., and To, S. T. (2018). Creating and using tools in a hybrid cognitive architecture.. In AAAI Spring Symposia.
* Coumans and Bai Coumans, E., and Bai, Y. (2016). Pybullet, a python module for physics simulation for games, robotics and machine learning. In GitHub repository.
* Driess et al. Driess, D., Oguz, O., Ha, J.-S., and Toussaint, M. (2020). Deep visual heuristics: Learning feasibility of mixed-integer programs for manipulation planning. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 9563–9569. IEEE.
* Finn et al. Finn, C., Yu, T., Zhang, T., Abbeel, P., and Levine, S. (2017). One-shot visual imitation learning via meta-learning. In Conference on Robot Learning, pp. 357–368. PMLR.
* Fitzgerald et al. Fitzgerald, T., Goel, A., and Thomaz, A. (2018). Human-guided object mapping for task transfer. ACM Transactions on Human-Robot Interaction (THRI), 7(2), 1–24.
* Fitzgerald et al. Fitzgerald, T., Goel, A., and Thomaz, A. (2021). Modeling and learning constraints for creative tool use. Frontiers in Robotics and AI, 8.
* Gajewski et al. Gajewski, P., et al. (2019). Adapting everyday manipulation skills to varied scenarios. In 2019 International Conference on Robotics and Automation (ICRA), pp. 1345–1351. IEEE.
* Garg et al. Garg, S., Bajpai, A., and Mausam (2019). Size independent neural transfer for RDDL planning. In Benton, J., Lipovetzky, N., Onaindia, E., Smith, D. E., and Srivastava, S. (Eds.), Proceedings of the Twenty-Ninth International Conference on Automated Planning and Scheduling, ICAPS 2018, Berkeley, CA, USA, July 11-15, 2019, pp. 631–636. AAAI Press.
* Garg et al. Garg, S., Bajpai, A., and Mausam (2020). Symbolic network: Generalized neural policies for relational mdps. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, Vol. 119 of Proceedings of Machine Learning Research, pp. 3397–3407. PMLR.
* Garrett et al. Garrett, C. R., et al. (2021). Integrated task and motion planning. Annu. Rev. Control Robot. Auton. Syst, 2021(4), 1–30.
* Garrett et al. Garrett, C. R., Lozano-Pérez, T., and Kaelbling, L. P. (2020). Pddlstream: Integrating symbolic planners and blackbox samplers via optimistic adaptive planning. In Proceedings of the International Conference on Automated Planning and Scheduling, Vol. 30, pp. 440–448.
* He et al. He, K., Zhang, X., Ren, S., and Sun, J. (2015). Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026–1034.
* Hermans et al. Hermans, T., Rehg, J. M., and Bobick, A. (2011). Affordance prediction via learned object attributes. In ICRA: Workshop on Semantic Perception, Mapping, and Exploration, pp. 181–184.
* Holladay et al. Holladay, R., Lozano-Pérez, T., and Rodriguez, A. (2019). Force-and-motion constrained planning for tool use. In IROS.
* Huang et al. Huang, D.-A., Nair, S., Xu, D., Zhu, Y., Garg, A., Fei-Fei, L., Savarese, S., and Niebles, J. C. (2019). Neural task graphs: Generalizing to unseen tasks from a single video demonstration. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8565–8574.
* Huang et al. Huang, S. H., Pan, J., Mulcaire, G., and Abbeel, P. (2015). Leveraging appearance priors in non-rigid registration, with application to manipulation of deformable objects. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 878–885. IEEE.
* Hübner et al. Hübner, J. F., Bordini, R. H., and Wooldridge, M. (2006). Programming declarative goals using plan patterns. In International Workshop on Declarative Agent Languages and Technologies, pp. 123–140. Springer.
* Jain et al. Jain, A., Das, D., Gupta, J. K., and Saxena, A. (2015). Planit: A crowdsourcing approach for learning to plan paths from large scale preference feedback. In ICRA, pp. 877–884.
* Kho et al. Kho, G., Hung, C., and Cunningham, H. (2014). Robo brain: Massive knowledge base for robots. In Cornell Univ., USA, Tech. Rep.
* Kingma and Ba Kingma, D. P., and Ba, J. (2014). Adam: A method for stochastic optimization. In CoRR arXiv:1412.6980.
* Kolobov et al. Kolobov, A., Mausam, Weld, D. S., and Geffner, H. (2011). Heuristic search for generalized stochastic shortest path mdps. In ICAPS.
* Kroemer et al. Kroemer, O., Ugur, E., Oztop, E., and Peters, J. (2012). A kernel-based approach to direct action perception. In 2012 IEEE international Conference on Robotics and Automation, pp. 2605–2610. IEEE.
* Lee et al. Lee, A. X., Gupta, A., Lu, H., Levine, S., and Abbeel, P. (2015). Learning from multiple demonstrations using trajectory-aware non-rigid registration with applications to deformable object manipulation. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5265–5272. IEEE.
* Levihn and Stilman Levihn, M., and Stilman, M. (2014). Using environment objects as tools: Unconventional door opening. In 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2502–2508. IEEE.
* Li et al. Li, Y., Tarlow, D., Brockschmidt, M., and Zemel, R. (2015). Gated graph sequence neural networks. In arXiv preprint arXiv:1511.05493.
* Liao et al. Liao, Y.-H., Puig, X., Boben, M., Torralba, A., and Fidler, S. (2019). Synthesizing environment-aware activities via activity sketches. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6291–6299.
* Liu et al. Liu, Z., Freeman, W. T., Tenenbaum, J. B., and Wu, J. (2018). Physical primitive decomposition. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 3–19.
* Lynch et al. Lynch, C., Khansari, M., Xiao, T., Kumar, V., Tompson, J., Levine, S., and Sermanet, P. (2020). Learning latent plans from play. In Conference on Robot Learning, pp. 1113–1132. PMLR.
* Mandlekar et al. Mandlekar, A., Zhu, Y., Garg, A., Booher, J., Spero, M., Tung, A., Gao, J., Emmons, J., Gupta, A., Orbay, E., et al. (2018). Roboturk: A crowdsourcing platform for robotic skill learning through imitation. In Conference on Robot Learning, pp. 879–893. PMLR.
* Mausam and Kolobov Mausam, and Kolobov, A. (2012). Planning with Markov decision processes: An AI perspective. Synthesis Lectures on Artificial Intelligence and Machine Learning, 6(1), 1–210.
* Migimatsu and Bohg Migimatsu, T., and Bohg, J. (2020). Object-centric task and motion planning in dynamic environments. IEEE Robotics and Automation Letters, 5(2), 844–851.
* Mikolov et al. Mikolov, T., Grave, E., Bojanowski, P., Puhrsch, C., and Joulin, A. (2018). Advances in pre-training distributed word representations. In LREC.
* Misra et al. Misra, D. K., Sung, J., Lee, K., and Saxena, A. (2016). Tell me dave: Context-sensitive grounding of natural language to manipulation instructions. IJRR, 35(1-3), 281–300.
* Myers et al. Myers, A., Teo, C. L., Fermüller, C., and Aloimonos, Y. (2015). Affordance detection of tool parts from geometric features. In 2015 IEEE International Conference on Robotics and Automation (ICRA), pp. 1374–1381. IEEE.
* Nair et al. Nair, A., Chen, D., Agrawal, P., Isola, P., Abbeel, P., Malik, J., and Levine, S. (2017). Combining self-supervised learning and imitation for vision-based rope manipulation. In 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 2146–2153. IEEE.
* Nair and Chernova Nair, L., and Chernova, S. (2020). Feature guided search for creative problem solving through tool construction. In Frontiers in Robotics and AI, p. 205. Frontiers.
* Nair et al. Nair, L., Srikanth, N. S., Erickson, Z. M., and Chernova, S. (2019a). Autonomous tool construction using part shape and attachment prediction.. In Robotics: Science and Systems.
* Nair et al. Nair, S., Zhu, Y., Savarese, S., and Fei-Fei, L. (2019b). Causal induction from visual observations for goal directed tasks. In CoRR arXiv:1910.01751.
* Nyga and Beetz Nyga, D., and Beetz, M. (2018). Cloud-based probabilistic knowledge services for instruction interpretation. In Robotics Research, pp. 649–664. Springer.
* Nyga et al. Nyga, D., Roy, S., Paul, R., Park, D., Pomarlan, M., Beetz, M., and Roy, N. (2018). Grounding robot plans from natural language instructions with incomplete world knowledge. In Conference on Robot Learning, pp. 714–723.
* Park et al. Park, D., Noseworthy, M., Paul, R., Roy, S., and Roy, N. (2019). Inferring task goals and constraints using bayesian nonparametric inverse reinforcement learning. In Proceedings of the 3rd Conference on Robot Learning (CoRL).
* Puig et al. Puig, X., Ra, K., Boben, M., Li, J., Wang, T., Fidler, S., and Torralba, A. (2018). Virtualhome: Simulating household activities via programs. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 8494–8502.
* Sarathy and Scheutz Sarathy, V., and Scheutz, M. (2018). Macgyver problems: Ai challenges for testing resourcefulness and creativity. Advances in Cognitive Systems, 6, 31–44.
* Scalise et al. Scalise, R., Li, S., Admoni, H., Rosenthal, S., and Srinivasa, S. S. (2018). Natural language instructions for human–robot collaborative manipulation. The International Journal of Robotics Research, 37(6), 558–565.
* Sharma et al. Sharma, S., Gupta, J., Tuli, S., Paul, R., and Mausam (2022a). Goalnet: Inferring conjunctive goal predicates from human plan demonstrations for robot instruction following. In International Conference on Automated Planning and Scheduling (ICAPS) - Planning and Reinforcement Learning Workshop.
* Sharma et al. Sharma, V., Arora, D., Geißer, F., Mausam, and Singla, P. (2022b). Symnet 2.0: Effectively handling non-fluents and actions in generalized neural policies for RDDL relational MDPs. In Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence (UAI).
* Shridhar et al. Shridhar, M., Thomason, J., Gordon, D., Bisk, Y., Han, W., Mottaghi, R., Zettlemoyer, L., and Fox, D. (2020). Alfred: A benchmark for interpreting grounded instructions for everyday tasks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10740–10749.
* Silver et al. Silver, T., et al. (2021). Planning with learned object importance in large problem instances using graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 35, pp. 11962–11971.
* Speer et al. Speer, R., Chin, J., and Havasi, C. (2017). ConceptNet 5.5: An open multilingual graph of general knowledge. In AAAI.
* Speer et al. Speer, R., Chin, J., and Havasi, C. (2019). ConceptNet Numberbatch, the best pre-computed word embeddings you can use. In GitHub repository.
* Srivastava et al. Srivastava, S., Fang, E., Riano, L., Chitnis, R., Russell, S., and Abbeel, P. (2014). Combined task and motion planning through an extensible planner-independent interface layer. In 2014 IEEE international conference on robotics and automation (ICRA), pp. 639–646. IEEE.
* Straub et al. Straub, J., Whelan, T., Ma, L., Chen, Y., Wijmans, E., Green, S., Engel, J. J., Mur-Artal, R., Ren, C., Verma, S., et al. (2019). The replica dataset: A digital replica of indoor spaces. In CoRR arXiv:1906.05797.
* Toussaint et al. Toussaint, M. A., Allen, K. R., Smith, K. A., and Tenenbaum, J. B. (2018). Differentiable physics and stable modes for tool-use and manipulation planning. In Robotics: Science and Systems Foundation.
* Tuli et al. Tuli, S., Bansal, R., Paul, R., and Mausam (2021). Tango: Commonsense generalization in predicting tool interactions for mobile manipulators. In International Joint Conference on Artificial Intelligence (IJCAI).
* Vega-Brown and Roy Vega-Brown, W., and Roy, N. (2020). Asymptotically optimal planning under piecewise-analytic constraints. In Algorithmic Foundations of Robotics XII, pp. 528–543. Springer.
* Wu et al. Wu, J., Lim, J. J., Zhang, H., Tenenbaum, J. B., and Freeman, W. T. (2016). Physics 101: Learning physical object properties from unlabeled videos. In British Machine Vision Conference.
* Wu et al. Wu, J., Yildirim, I., Lim, J. J., Freeman, B., and Tenenbaum, J. (2015). Galileo: Perceiving physical object properties by integrating a physics engine with deep learning. In Cortes, C., Lawrence, N. D., Lee, D. D., Sugiyama, M., and Garnett, R. (Eds.), Advances in Neural Information Processing Systems 28, pp. 127–135. Curran Associates, Inc.
* Xie et al. Xie, A., Ebert, F., Levine, S., and Finn, C. (2019). Improvisation through physical understanding: Using novel objects as tools with visual foresight. In Robotics at Robotics Science and Systems (RSS).
* Zhu et al. Zhu, Y., Tremblay, J., Birchfield, S., and Zhu, Y. (2021). Hierarchical planning for long-horizon manipulation with geometric and symbolic scene graphs. In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 6541–6548. IEEE.
## A Supplementary Material
All our code is available as a GitHub repository under BSD-2 License
https://github.com/reail-iitd/tango. Instructions for reproducing the results
are given at https://github.com/reail-iitd/tango/wiki. A supplementary video
demonstrating our data collection platform and model’s generalization
capability is available at https://www.youtube.com/watch?v=lUWU3rK1Gno.
## B Hyperparameter Details
We detail the hyper-parameters for the ToolTango architecture introduced in
this paper.
* •
_Graph Structured World Representation._ The Gated Graph Convolution Network
(GGCN) was implemented with $4$-hidden layers, each of size $128$, with
convolutions across $2$ time steps for every relation passing through a layer
normalized GRU cell. The Parameterized ReLU activation function with a $0.25$
negative input slope was used in all hidden layers.
* •
_Word Embeddings._ The word embeddings (derived from $\mathrm{ConceptNet}$)
were of size $300$. Additionally, the semantic state of each object was
encoded as a one-hot vector of size $29$. Typically, there were $35$ and $45$
objects in the home and factory domains respectively.
* •
_Fusing Metric Information._ The metric encodings were generated from the
metric information associated with objects using a $2$-layer Fully Connected
Network (FCN) with $128$-sized layers.
* •
_Encoding Action History._ A Long Short Term Memory (LSTM) layer of size $128$
was used to encode the action history using the generalized action encoding
$\mathcal{A}(I_{t}(o^{1}_{t},o^{2}_{t}))$.
* •
_Goal-conditioned Attention._ The attention network was realized as a
$1$-layer FCN of layer size $128$ with a $\mathrm{softmax}$ layer at the end.
* •
_Tool Prediction._ To predict the tool-likelihood score $p_{t}$, a $3$-layer
FCN was used, each hidden layer with size $128$ and output layer with size $1$
with sigmoid activation function.
* •
_Action Prediction._ To predict the action $I_{t}$, a $3$-layer FCN was used,
each hidden layer with size $128$ and output layer with size $|\mathcal{I}|$.
$I_{t}$ was converted to a one-hot encoding $\vec{I}_{t}$. This, with the
object embedding $e_{o}$ was passed to the $o^{1}_{t}$ predictor via an FCN.
This FCN consists of 3-hidden layers of size $128$ and a final layer of size
$1$ with a sigmoid activation (for likelihood). The $\vec{I}_{t}$ and
$o^{1}_{t}$ likelihoods were sent to the $o^{2}_{t}$ predictor to predict
likelihoods for all object embeddings $e_{o}$. This part was realized as a
$3$-layer FCN with hidden layer size $128$ and final layer of size $1$ with a
sigmoid activation function.
* •
_Training parameters._ Model training used a learning rate of $5\times
10^{-4}$. The Adam optimizer (?) with a weight decay parameter of $10^{-5}$
and a batch size of $1$ was used. An early stopping criterion was applied for
convergence. The _action prediction accuracy_ was used as the comparison
metric on the validation set or up to a maximum of $200$ epochs.
## C World Scenes
The figure 14 illustrates the object-centric graph representation of the 10
world scenes we used for Home and Factory domains. The agent node is shown in
green, tools in black and objects with states in red. The relations of Close
are shown in green (only populated for agent), On in black, Inside in red and
Stuck to in blue.
The legend for node IDs to objects are given below:
Home: 0: floor, 1: walls, 2: door, 3: fridge, 4: cupboard, 5: husky, 6: table,
7: table2, 8: couch, 9: big-tray, 10: book, 11: paper, 12: paper2, 13: cube
gray, 14: cube green, 15: cube red, 16: tray, 17: tray2, 18: light, 19: bottle
blue, 20: bottle gray, 21: bottle red, 22: box, 23: apple, 24: orange, 25:
banana, 26: chair, 27: ball, 28: stick, 29: dumpster, 30: milk, 31: shelf, 32:
glue, 33: tape, 34: stool, 35: mop, 36: sponge, 37: vacuum, 38: dirt.
Factory: 0: floor warehouse, 1: 3D printer, 2: assembly station, 3: blow
dryer, 4: board, 5: box, 6: brick, 7: coal, 8: crate green, 9: crate peach,
10: crate red, 11: cupboard, 12: drill, 13: gasoline, 14: generator, 15: glue,
16: hammer, 17: ladder, 18: lift, 19: long shelf, 20: mop, 21: nail, 22: oil,
23: paper, 24: part1, 25: part2, 26: part3, 27: platform, 28: screw, 29:
screwdriver, 30: spraypaint, 31: stick, 32: stool, 33: table, 34: tape, 35:
toolbox, 36: trolley, 37: wall warehouse, 38: water, 39: welder, 40: wood, 41:
wood cutter, 42: worktable, 43: ramp, 44: husky, 45: tray.
(a) A sample home scene.
(b) A sample factory scene.
Figure 14: Sample World Scenes in Home and Factory domains
## D Realization of Object Relations
Figure 15: Manipulation region of an object
To implement various relations among objects PyBullet allows to define
_manipulation regions_ around objects. These regions are virtual regions
inside which a robot can manipulate that object. A visual description of this
region is shown in Figure 15.
We now define two object types.
1. 1.
Standalone object: Such an object is defined as a collection of stereo-
lithographic tetrahedrons with visual and collision properties. Each such
object has a single manipulation region, denoted as $MR(object)$, which is the
region around the object where it is considered to be in “vicinity” to the
object and hence can be manipulated by the robot. Examples: cubes, fruits,
milk, etc.
2. 2.
Containing object: Such an object is a standalone object with an additional
containment region, denoted as $CR(object)$, which is a proper subset of the
manipulation region where if another object’s center lies is called to be
“contained inside” this object. Examples: box, cupboard and fridge.
Every object has a center defined arithmetic mean of the centers of the
tetrahedrons, denoted by $C(object)$. We now define the two relation
constraints of which others are different versions of combinations of:
1. 1.
Inside: An object $A$ is said to be contained inside a containing object $B$
if the $C(A)\in CR(B)$.
2. 2.
On Top: An object $A$ is said to be on top on another object $B$ if
$C(A)_{z}>C(B)_{z}$ and $MR(A)\cap MR(B)\neq\phi$, where $C(O)_{z}$ denotes
the z component of the center of object $O$.
|
expł{i ( k_y y + k_z z ) }̊
Stability of solar atmospheric structures harboring standing slow waves
An analytical model in a compressible plasma
M. Geeraerts 1
T. Van Doorsselaere 1
Centre for mathematical Plasma Astrophysics (CmPA), Mathematics Department, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium
In the context of the solar coronal heating problem, one possible explanation for the high coronal temperature is the release of energy by magnetohydrodynamic (MHD) waves. The energy transfer is believed to be possible, among others, by the development of the Kelvin-Helmholtz instability (KHI) in coronal loops.
Our aim is to determine if standing slow waves in solar atmospheric structures such as coronal loops, and also prominence threads, sunspots, and pores, can trigger the KHI due to the oscillating shear flow at the structure's boundary.
We used linearized nonstationary MHD to work out an analytical model in a cartesian reference frame. The model describes a compressible plasma near a discontinuous interface separating two regions of homogeneous plasma, each harboring an oscillating velocity field with a constant amplitude which is parallel to the background magnetic field and aligned with the interface. The obtained analytical results were then used to determine the stability of said interface, both in coronal and photospheric conditions.
We find that the stability of the interface is determined by a Mathieu equation. In function of the parameters of this equation, the interface can either be stable or unstable. For coronal as well as photospheric conditions, we find that the interface is stable with respect to the KHI. Theoretically, it can, however, be unstable with respect to a parametric resonance instability, although it seems physically unlikely. We conclude that, in this simplified setup, a standing slow wave does not trigger the KHI without the involvement of additional physical processes.
§ INTRODUCTION
Although it has been known for a long time that the corona is three orders of magnitude hotter than the photosphere, a definite explanation for this unexpected feature has continued to elude researchers. One possible explanation that has been advanced over the years, and which is supported by both observations and theoretical models, is the deposition of energy of magnetohydrodynamic (MHD) waves into the corona [Parnell & De Moortel, 2012, Arregui, 2015, Van Doorsselaere et al., 2020]. There are several mechanisms that allow for the conversion of wave energy into heating of the surrounding plasma. Possibilities include dissipation by Ohmic resistivity, values of which were calculated by Kovitya & Cram, 1983 for the photosphere and used by Chen et al., 2018, Geeraerts et al., 2020, and Chen et al., 2020 to infer its importance in damping slow waves in a photospheric pore. Mode coupling [Pascoe et al., 2010, Pascoe et al., 2012, Hollweg et al., 2013, De Moortel et al., 2016] and resonant absorption, both in the Alfvén [Hollweg & Yang, 1988, Hollweg et al., 1990, Goossens et al., 1992, Goossens et al., 2002, Soler et al., 2013] and cusp [Cadez et al., 1997, Erdélyi et al., 2001, Soler et al., 2009, Yu et al., 2017] continua, have also been studied for their potential to damp waves by transferring their energy to local oscillations. Another mechanism for damping waves in atmospheric structures and which has received considerable attention recently is the Kelvin-Helmholtz instability (KHI). Indeed, the transition of wave energy to turbulence and plasma heating on smaller scales in the coronal plasma is known to be facilitated by this instability [Heyvaerts & Priest, 1983, Ofman et al., 1994, Karpen et al., 1994, Karampelas et al., 2017, Afanasyev et al., 2019, Hillier et al., 2020, Shi et al., 2021]. The KHI has been observed in the solar atmosphere, for example on coronal mass ejection flanks [Ofman & Thompson, 2011, Foullon et al., 2011], and, more recently, on coronal loops [Samanta et al., 2019].
Slow magnetosonic waves were observed in solar coronal loops more than a decade ago, both as propagating waves and standing waves. Upwardly propagating disturbances along coronal loops have been observed, for example, by Berghmans & Clette, 1999, Nightingale et al., 1999, and De Moortel et al., 2000. These have been interpreted as propagating slow modes through a theoretical model by Nakariakov et al., 2000. Other observations of disturbances in coronal loops by the SUMER spectrometer have been analyzed and interpreted by Wang et al., 2002, Wang et al., 2003, Wang et al., 2003, Wang et al., 2007 as standing slow modes, whereas Kumar et al., 2013 and Mandal et al., 2016 reported the observation of a reflecting, also referred to as sloshing, slow wave in a coronal loop. Wang, 2011 provides a review of observations and modeling of standing slow waves in coronal loops. These modes have also been studied through numerical simulations, for example, by De Moortel & Hood, 2003, who found that thermal conduction is an important damping mechanism for propagating slow waves in coronal conditions, and by Mandal et al., 2016, who reported about the frequency dependent damping of propagating slow waves in a coronal loop. Structures in the lower solar atmosphere, such as sunspots and pores in the photosphere, have also been observed to harbor slow magnetosonic waves [Dorotovič et al., 2008, Dorotovič et al., 2014, Morton et al., 2011, Grant et al., 2015, Moreels et al., 2015, Freij et al., 2016], which were then classified as either surface or body waves [Moreels et al., 2013, Keys et al., 2018].
The KHI and its growth rate at the interface between two aligned sheared stationary flows have been known for a long time [Chandrasekhar, 1961]. Although in the purely hydrodynamics (HD) model the instability always develops in the presence of a shear flow, in the MHD model an aligned magnetic field can prevent its triggering. A natural question that then arises in the context of wave heating is under which conditions the KHI would develop in the presence of an oscillating shear flow. Indeed, it is possible that waves propagating or standing in solar coronal structures, such as loops, can trigger the KHI on the boundary and hereby convey some of their energy to the plasma background. Zaqarashvili et al., 2015, for example, studied the KHI in rotating and twisted magnetized jets in the solar atmosphere in the presence of a stationary shear flow and used their derivations to discuss the stability of standing kink and torsional Alfvén waves at the velocity antinodes. They found that the standing waves are always unstable whereas the propagating waves are stable. Transverse oscillations are known to be unstable to the KHI in coronal loops and numerical studies on this topic include Terradas et al., 2008, Antolin et al., 2014, Antolin et al., 2015, Antolin et al., 2017, Magyar et al., 2015, Guo et al., 2019, Karampelas et al., 2019 and Pascoe et al., 2020.
There have also been several analytical studies regarding the stability of the interface between oscillating sheared flows. Kelly, 1965 looked at the HD case of two parallel sheared flows aligned with the interface, whereas Roberts, 1973 studied the same setup in the MHD model. More recently, Barbulescu et al., 2019 and Hillier et al., 2019 investigated the stability of transverse oscillations in coronal loops by modeling them locally at the loop boundary as a cartesian interface between sheared background flows, with the background velocity perpendicular to the background magnetic field. Each of these studies revealed that the interface between oscillating sheared flows is always unstable, in contrast to the constant sheared flows case. All of these studies relying on the simplifying assumption that the fluid is incompressible, it is worth asking whether their conclusions remain unchanged when compression is included.
The goal of this paper is therefore twofold. Firstly, it aims at extending the known incompressible model of Roberts, 1973 for a plasma with an oscillating velocity field aligned with both the magnetic field and the interface to a compressible version. The focus lies on finding expressions for the eigenfunctions and, in particular, to derive their evolution over time. The interest in doing this lies in identifying the shortcomings made by the approximation of the incompressible model, at the cost of considerably more involved analytical derivations. This is the subject of Sections 2 and 3. Secondly, it aims at expressing more general instability conditions as compared to the incompressible model, by taking into account the subtleties arising due to the inclusion of compression. In particular, it will allow us to assess the stability of certain solar atmospheric structures harboring a standing slow wave. We will do so by using this model as a local approximation of the structure's boundary, at the position where the velocity shear is the greatest (for instance in a cusp resonance). We will also compare our findings for the local stability of slow waves to those of Barbulescu et al., 2019 and Hillier et al., 2019 for the local stability of fast kink waves. This is the subject of Section 4.
§ MODEL
We derived an analytical model for a compressible plasma at the boundary of solar atmospheric structures which can, in a first rough approximation, be modeled as a cylinder with a discontinuous boundary separating two regions of homogeneous plasma with different properties. Such structures include coronal loops, prominence threads, sunspots and photospheric pores.
The physical setup here is the same as in Roberts, 1973, except we included compression. We point out that, in a more realistic setup, a smooth transition layer would have to be included at the interface. This would result in the possibility of resonance occuring between the main oscillation of the structure and local slow mode oscillations in the boundary layer, as studied theoretically by Yu et al., 2017. The analytical derivations of Goossens et al., 2021 show that, when slow waves are resonantly absorbed in the cusp continuum, both the azimuthal component of vorticity and the parallel component of the plasma displacement are large. The huge amount of vorticity could indicate the possibility of the KHI developing in those conditions. However, the sharp spatial variation of the parallel displacement and the truly discontinuous interface separating two plasma regions are absent from the present model. This should be kept in mind when drawing conclusions regarding shear flows in resonances.
In order to be able to make progress in the analytical derivations, the model uses a Cartesian coordinate system $(x, y, z)$, where $x=0$ is the interface and the $z$-direction is the longitudinal direction (i.e., along the cylinder's axis). The region $x<0$ represents the interior of the structure, whereas the region $x>0$ represents the surrounding plasma. This is a model for the local stability at the boundary, in the region of the structure where the shear in longitudinal velocity is the greatest, that is to say, at an antinode of the longitudinal component of the velocity eigenfunction. For the longitudinally fundamental slow mode, this would be at the middle of the structure (see Figure <ref>), whereas for the first longitudinal overtone, for example, this would be at a quarter of the structure's length measured from either end. The time variable is denoted by $t$.
Sketch of a longitudinal cut along the axis of a coronal loop harboring a fundamental standing slow body sausage mode, at the velocity's maximal amplitude (left) and the local cartesian model at the boundary (right). The arrows on the left represent the velocity field and their lengths are to scale with the relative local magnitude of the field for a slow body sausage mode at a given time. The lengths and directions of the magnetic field arrows on the right figure are consistent for that same slow mode, whereas for the velocity arrows only the direction is consistent (the length of the exterior velocity arrow having been increased for visual clarification).
After linearizing the ideal MHD equations, the equilibrium quantities are denoted by the subscript $0$ and the perturbed quantities are denoted by the subscript $1$. The regions on each side of the interface are two homogeneous but different plasmas. This means each region has its own values for the background quantities, which are assumed spatially constant. The background magnetic field is assumed to be a straight and constant axial field along the $z$-coordinate: $\pmb{B}_0 = B_{0z} \pmb{1}_z$. Furthermore, we assume that the background flow is oscillating with a certain frequency $\omega_0$, which represents the frequency of the standing slow wave: $\pmb{v}_0 = V_0 \cos \left( \omega_0 t \right) \pmb{1}_z$. It would, of course, be more accurate to also include a background oscillation for the other quantities, such as magnetic field, pressure, and density. In the context of solar atmospheric structures, the magnetic field oscillations that occur because of this external forcing in the background can, however, be neglected in a first approximation with respect to the strong longitudinal magnetic field. As for the density and pressure background oscillations, they can be neglected in this model because they are in antiphase with respect to the velocity in their longitudinal profile and thus have a node where the logitudinal component of velocity has an antinode. For slow modes, the longitudinal component of the velocity is typically much larger than its other components (see left of Figure <ref>), which can thus be neglected at the former's antinode as well.
The perturbed density, thermal pressure, velocity and magnetic field are denoted with $\rho_1$, $p_1$, $\pmb{v}_1$ and $\pmb{B}_1$, respectively. In what follows the perturbed eulerian total pressure will be used as well, which is given by $P_1 = p_1 + \frac{\pmb{B}_0 \cdot \pmb{B}_1}{\mu_0}$. Although gravity certainly has a role in solar atmospheric wave dynamics, it is neglected in this model in order to make some analytical progress.
The linearized compressible ideal MHD equations can, under these assumptions, be written as follows:
\begin{align}
&\d\frac{D \rho_1}{D t} +\rho_0 \l( \nabla \cdot \pmb{v}_1 \r) = 0\text{,} \label{eq1}\\
& \rho_1 \frac{\partial \pmb{v}_0}{\partial t} + \rho_0 \frac{D \pmb{v}_1}{D t} = -\nabla P_1 + \frac{1}{\mu_0} \left( \pmb{B}_0 \cdot \nabla \right) \pmb{B}_1 \text{,} \label{eq2}\\
&\frac{D \pmb{B}_1}{D t} = - \pmb{B}_0 \left( \nabla \cdot \pmb{v}_1 \right) + \left( \pmb{B}_0 \cdot \nabla \right) \pmb{v}_1 \text{,} \label{eq3}\\
&\frac{D p_1}{D t} + \rho_0 v_{\text{s}}^2 \l( \nabla \cdot \pmb{v}_1 \r) = 0\text{,} \label{eq4}
\end{align}
where $\frac{D f}{Dt} = \frac{\partial f}{\partial t} + \left(\pmb{v}_0 \cdot \nabla\right) f$ is the Lagrangian derivative of a quantity $f$, $v_{\text{s}} = \sqrt{\frac{\gamma p_0}{\rho_0}}$ is the speed of sound and $\mu_0$ the magnetic permeability of free space. The first equation, Eq. (<ref>), has this simpler form because $\nabla \rho_0 = 0$ in each region (i.e., both inside and outside the cylinder). Since the background quantities depend only on $x$ and $t$, the perturbed quantities can be Fourier-analyzed in the $y$ and $z$ coordinates and are thus assumed to have the following form: $f_1 = \overline{f}_1(x,t) \e$, for each of the perturbed quantities $p_1$, $\rho_1$, $v_{1x}$, $v_{1 y}$, $v_{1z}$, $B_{1x}$, $B_{1 y}$ and $B_{1z}$.
§ EXPRESSIONS FOR THE PERTURBED QUANTITIES
In this section, we derive the governing equations for the evolution of linear MHD perturbations in a compressible plasma for the model described in the previous section. In the next section, we then try to use the obtained information to describe the stability of the interface in compressible plasma conditions.
§.§ The central quantity $\c$}
In what follows, an expression is derived for the compression term $$̧. This term is central to finding expressions for all other physical quantities, in the sense that they can all be derived solely from it.
By using the Lagrangian displacement $\pmb{\xi}$ defined by $\frac{D \pmb{\xi}}{D t} = \pmb{v}_1$, it can be shown (see Appendix <ref>) that the compression term $\c$ must satisfy the following partial differential equation:
\begin{align}
&\d\frac{D^4 \l( \c \r)}{Dt^4} \; + \; i k_z \o_0 V_0 \frac{D^2 \l( \sin \l(\o_0 t \r) \l( \c \r) \r)}{Dt^2} \notag\\
& \qquad \; - \; \l( v_A^2 + v_s^2 \r) \frac{D^2}{Dt^2} \l( \frac{\pa^2 \l( \c \r)}{\pa x^2} \r) \; + \; k^2 \l( v_A^2 + v_s^2 \r) \frac{D^2 \l( \c \r)}{Dt^2} \notag\\
& \qquad \; - \; i k_z V_0 \o_0 \sin \l( \o_0 t \r) v_A^2 \frac{\pa^2 \l( \c \r)}{\pa x^2} \notag\\
& \qquad \; + \; i k_z k^2 V_0 \o_0 \sin \l(\o_0 t \r) v_A^2 \l( \c \r) \; - \; k_z^2 v_A^2 v_s^2 \frac{\pa^2 \l( \c \r)}{\pa x^2} \notag \\
& \qquad \; + \; k_z^2 k^2 v_A^2 v_s^2 \l( \c \r) = 0 \text{,} \label{Eqdivxi}
\end{align}
where $k = √(k_y^2 + k_z^2)$ and $v_A = B_0z / √(μ_0 ρ_0)$ is the Alfv\'en speed. This partial differential equation is of fourth order in $t$ and second order in $x$. Note that if $V_0 = 0$, such that the time dependence of the background quantities disappears, we retrieve an ordinary differential equation (ODE) of second order in $x$ which is the governing equation for the compression term $$̧ in a plasma without a background flow. This is the equation used for example in Edwin
& Roberts, 1983 to study wave modes in a magnetic cylinder in the framework of ideal MHD in a plasma without background flow.
By drawing inspiration from the derivations of Barbulescu et al., 2019, one finds that Eq. (<ref>) can take a simpler form if $\c$ is written as
\begin{equation}\label{f-g}
\c \; = \; f(x,t) g(t) \e \text{,}
\end{equation}
where $g(t) = expł{ -i k_z V_0 sinł( ø_0 t )̊/ø_0 }̊$. Since $g(t)$ has modulus equal to $1$ for all $t ∈ℝ$, the stability of $$̧ over time is entirely determined by $f$. Inserting expression (<ref>) into Eq. (<ref>) simplifies this equation considerably, now taking the following form:
\begin{align}
& {\frac {\partial ^{4} f(x,t)}{\partial {t}^{4}}} \; - \; \left( {v_{A}}^{2}+{v_{s}}^{2} \right) {\frac {\partial ^{4} f(x,t)}{
\partial {x}^{2}\partial {t}^{2}}} \notag\\
& \qquad \; + \; \left[ i \omega_{0} k_{z} V_{0} \sin \left( \omega_{0} t \right) + k^2 \left( {v_{A}}^{2}+{v_{s}}^{2} \right) \right] {\frac {\partial ^{2} f(x,t)}{\partial {t}^{2}}} \notag \\
& \qquad \; - \; \left[ i \omega_{0} k_{z} V_{0} {v_{A}}^{2} \sin \left( \omega_{0} t \right)+{k_{z}}^{2}{v_{A}}^{2}{v_{s}}^{2}
\right] {\frac {\partial ^{2} f(x,t)}{\partial {x}^{2}}} \notag \\
& \qquad \; + \; 2 i \omega_{0}^2 k_{z} V_{0} \cos \left( \omega_{0} t \right) {\frac {\partial f(x,t)}{\partial t}} \notag \\
& \qquad \; + \; \l[ i \omega_0 k_z V_0 \sin \l( \omega_0 t \r) \l( k^2 v_A^2 - \o_0^2 \r) \r. \notag\\
& \l. \hspace{4.5cm} + k^2 {k_{z}}^{2}{v_{A}}^{2}{v_{s}}^2 \r] f(x,t) = 0 \text{.} \label{Eqf(x,t)}
\end{align}
Equation (<ref>) has a solution in the form of $F(x)G(t)$, where $F$ and $G$ satisfy the following ODEs (with $m$ a constant):
\begin{equation} \label{EqF}
\d\frac{d^2 F(x)}{dx^2} \; + \; m^2 F(x) \; = \; 0
\end{equation}
\begin{align}
&\d\frac{d^4 G(\t)}{d \t^4} \; + \; \l( a_1 + q_1 \sin \l( \t \r) \r) \frac{d^2 G(\t)}{d \t^2} \; + \; q_3 \cos \l( \t \r) \frac{d G(\t)}{d \t} \notag \\
& \hspace{4.0cm} + \; \l( a_2 + q_2 \sin \l( \t \r) \r) G(\t) \; = \; 0 \text{,} \label{EqG}
\end{align}
with $\t = \o_0 t$, and where
\begin{align*}
&\qquad a_1 = \frac{\l(v_A^2 + v_s^2 \r) \l(m^2 + k^2 \r)}{\o_0^2} \text{, } \;\; q_1 = \frac{i k_z V_0}{\o_0} \text{, } \\
&\qquad q_3 = \d\frac{2 i k_z V_0}{\o_0} \text{,} \\
&\qquad a_2 = \frac{k_z^2 v_A^2 v_s^2 \l( m^2 + k^2 \r)}{\o_0^4} \text{, } \;\;\;\;\; q_2 = \frac{i k_z V_0 \l[ \l( m^2 + k^2 \r) v_A^2 - \o_0^2 \r] }{\o_0^3} \text{.}
\end{align*}
These two equations are related by the constant $m$, which occurs in both of them.
§.§.§ The spatial function $F$
Equation (<ref>) has a simple analytical solution, namely
\begin{equation}\label{SolF}
F(x) \; = \; C_1 \exp \l( i m x \r) \; + \; C_2 \exp \l( -i m x \r) \text{,}
\end{equation}
with $C_1$ and $C_2$ arbitrary constants. From this it can be inferred that $m$ plays the role of the wavenumber along $x$, which is the direction normal to the interface. The focus of this paper goes to standing slow waves in solar atmospheric structures that can be modeled as a straight cylinder with a circular base and a discontinuous boundary separating two homogeneous but different plasma regions. Therefore, each region has its own value for $m$ (namely $m_i$ for the interior and $m_e$ for the exterior), each being able to only take on specific values depending on the boundary conditions at the interface. Finding expressions for $m_i$ and $m_e$ is, however, not straightforward. This will be discussed in Section <ref>. We note that for the purpose of studying the local stability of the interface, $m$ will be taken purely imaginary in order to have evanescent spatial profiles in the normal direction.
§.§.§ The temporal function $G$
Equation (<ref>) does not have a simple analytical solution. This equation governs the stability of the compression term $\c$ over time, its parameters depending on the background quantities $v_A$, $v_s$, $V_0$ and $ø_0$. The function $h(t) = G(t) g(t)$, where $G$ is the solution to equation \eqref{EqG}, actually describes the general time evolution of $$̧ in a compressible homogeneous plasma of inifinite extent with an oscillating background velocity which is parallel to the straight and constant equilibrium magnetic field. Although there is no closed-form analytical solution, the boundedness of $\c$ over time can be derived from the properties of Eq. \eqref{EqG}. Indeed, this linear fourth-order ODE has periodic coefficients with the same period and hence it falls in the category of equations which obey Floquet theory [Chicone, 2008].
Defining $τ= ø_0 t$, it can be rewritten as a $4 ×4$ system of first-order ODEs of the form $x'() = A() x()$, where the coefficient matrix $A()$ is periodic with a period of $2 π$. One can find four linearly independent fundamental solution vectors $x_1()$, $x_2()$, $x_3()$ and $x_4()$, which depend on their respective initial conditions. The matrix obtained by putting these four vectors into its columns is called a fundamental solution matrix. If we take as initial condition matrix the identity matrix, Floquet theory states that the corresponding fundamental solution matrix (which we denote by $X()$) evaluated at one period (i.e., at $τ= 2 π$ in our case) is intimately linked to the boundedness of the solutions of the ODE. Indeed, denoting the eigenvalues of $X(2π)$ by $λ_1$, $λ_2$, $λ_3$ and $λ_4$, Floquet's theorem states that for each distinct eigenvalue $λ$, if we write $λ= e^2 πμ$ where $μ∈ℂ$ is called the Floquet exponent, there exists an independent solution of the system which has the form
\begin{equation} \label{FloquetSol}
\pmb{x}(\t) = e^{\mu \t} \pmb{\Phi}(\t) \text{,}
\end{equation}
where $Φ$ contains $2 π$-periodic functions. This means that a solution to Eq. \eqref{EqG} is unstable if and only if $λ > 1$ for at least one eigenvalue $λ$ of $X(2 π)$. If there are less than four distinct eigenvalues of $X(2 π)$, it is possible there are less than four independent Floquet solutions of the form \eqref{FloquetSol}. If the eigenspace of $X(2 π)$ has dimension $4$, there are still four independent eigenvectors $Φ_i$ and thus four independent Floquet solutions. If the eigenspace of $X(2 π)$ has dimension less than $4$, then there are less than four independent Floquet solutions. In that case, extra independent solutions have to be found by using the Jordan normal form of $X(2 π)$ (see e.g., Cesari, 1963 for more details). These extra independent Jordan solutions are always unstable. The stability of $$̧ is thus entirely determined by the Floquet exponents $\mu$.
A solution to Eq. (<ref>) can be written in the form of a series. Knowing that each independent Floquet solution has the form $e^{\mu \t} \Phi(\t)$ with $\Phi$ a $2 \pi$-periodic function, one can assume that such a basic independent solution can be written as
\begin{equation}\label{Gseries}
G(\t) \; = \; e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \varphi_j \; e^{i j \t} \text{,}
\end{equation}
where the $\varphi_j$ are unknown coefficients. Writing $\mu=i \nu$ for convenience, the following recurence relation between the coefficients $\varphi_j$ can then be derived by inserting expression (<ref>) into Eq. (<ref>):
\begin{equation} \label{recursion}
- \varepsilon_j \; \varphi_{j-1} \;\; + \;\; \varphi_j \;\; + \;\; \varepsilon_j \; \varphi_{j+1} \;\; = \;\; 0\text{,}
\end{equation}
\begin{equation} \label{epsj}
\varepsilon_j \; = \; \d\frac{\frac{1}{2} k_z V_0 \o_0 \l[ \l( j + \nu \r)^2 \o_0^2 - K^2 v_A^2 \r]}{\l( j + \nu \r)^4 \o_0^4 - \l( j + \nu \r)^2 \o_0^2 K^2 \l( v_A^2 + v_s^2 \r) + k_z^2 K^2 v_A^2 v_s^2}
\end{equation}
with $K = \sqrt{k^2 + m^2}$. Equations (<ref>) are nontrivially satisfied if and only if the following infinite determinant vanishes:
\begin{equation} \label{Delta}
\Delta = \begin{vmatrix}
\ddots & \ddots & & & 0\\ \ddots & 1 & \varepsilon_{-1} & & \\ & -\varepsilon_0 & 1 & \varepsilon_0 & \\ & & -\varepsilon_1 & 1 & \ddots \\ 0 & & & \ddots & \ddots
\end{vmatrix} \text{.}
\end{equation}
This is a Fredholm determinant. Denoting with $A$ the operator defined by the corresponding infinite matrix, this Fredholm determinant is well-defined if the operator $A-I$ (where $I$ is the identity operator) is a trace class operator on $\ell^2(\mathbb{Z})$, the Hilbert space of square-summable sequences of complex numbers with entire index. It can be shown that this is the case here (see for example Sträng, 2005 for the method to follow), except if the denominator of one of the $\varepsilon_j$ vanishes. This happens if the perturbation, which is a normal mode with wave vector $\pmb{K} = (m, k_y, k_z)$, satisfies the equation
\begin{equation} \label{res}
(j+\nu)^2 \o_0^2 \; = \; \d\frac{K^2 \l(v_A^2 + v_s^2 \r)}{2} \l\{ 1 \pm \l[ 1 - \d\frac{4 k_z^2 v_A^2 v_s^2}{K^2 \l( v_A^2 + v_s^2 \r)^2} \r]^{1/2} \r\} \text{,}
\end{equation}
for a $j \in \mathbb{Z}$. This represents a resonance between the background oscillator with frequency $\o_0$ and a magnetosonic wave, in a homogeneous plasma of infinite extent. In this case $m$ is a free parameter (like $k_y$ and $k_z$) and can potentially take any real value. In the model we describe in this paper, with an interface separating two such homogeneous plasmas, only certain specific values of $m$ (different in both regions) are physically possible. Because surface waves on the interface are actually of interest in this case, $m$ should be imaginary. In Section <ref> we study the resonance of these surface waves with the background oscillator.
Considering $\Delta$ as a function of the two unknowns $\nu$ and $m$, it is thus well-defined for every $(\nu,m)$ except in the poles of the $\varepsilon_j$. It can also easily be checked with Eqs. (<ref>) and (<ref>) that $\Delta(\nu, m) = \Delta(\nu +1, m)$ and $\Delta(\nu, m) = \Delta(- \nu, m)$. As previously stated, there are in general four independent Floquet solutions of the form (<ref>) to Eq. (<ref>). Hence, all solutions for $\nu$ of $\Delta(\nu,m) = 0$ (as functions of $m$) which correspond to distinct solutions of the differential equation lie on the strip $-0.5 < \text{Re}\l[ \nu \r] \leq 0.5$. These distinct solutions relate as follows: $\nu_2 = -\nu_1$ and $\nu_4 = -\nu_3$. The four independent Floquet solutions in the most general case are thus of the form $e^{\mu_1 \t} \Phi_1(\t)$, $e^{-\mu_1 \t} \Phi_2(\t)$, $e^{\mu_3 \t} \Phi_3(\t)$ and $e^{-\mu_3 \t} \Phi_4(\t)$, where $\Phi_1, \Phi_2, \Phi_3$ and $\Phi_4$ are $2 \pi$-periodic functions. Recalling the earlier discussion in this section, we note that it is possible that there are less than four independent Floquet solutions if at least two eigenvalues $\lambda_j = e^{2 \pi i \nu_j}$ of $X(2 \pi)$ are equal. From the properties of $\Delta$, we can see that this happens in two situations. One possibility is if we have $\text{Re}[\nu_j] = n/2$ for some $n \in \mathbb{Z}$ and $\text{Im}[\nu_j] = 0$, for one of the solutions $\nu_j$. Indeed, the eigenvalues $e^{2 \pi i \nu_j}$ and $e^{-2 \pi i \nu_j}$ are equal in that case. Another possibility is if, for two solutions $\nu_j$ and $\nu_l$, we have $\text{Re}[\nu_j] = \text{Re}[\nu_l] + n$ for some $n \in \mathbb{Z}$ and $\text{Im}[\nu_j] = \text{Im}[\nu_l]$. In this case $e^{2 \pi i \nu_j}$ and $e^{2 \pi i \nu_l}$ will also be equal.
We find the following formula for $\Delta$, derived in Appendix <ref>:
\begin{equation} \label{DeltaFormula}
\Delta \; = \; 1 + \d\sum_{n=1}^{\infty} \l[ \d\sum_{j_1 = - \infty}^{\infty} \d\sum_{j_2 = j_1 + 2}^{\infty} \ldots \d\sum_{j_n = j_{n-1} + 2}^{\infty} \l( \d\prod_{l=1}^n \varepsilon_{j_l} \varepsilon_{j_l +1} \r) \r] \text{.}
\end{equation}
It is clear from Eq. (<ref>) that $\varepsilon_j \approx \frac{k_z V_0}{2 j^2 \o_0}$ as $\abs{j} \to \infty$. We can then try to use the fact that $\varepsilon_j$ drops off as $1/j^2$ to approximate the cumbersome formula (<ref>). If we can assume that $\abs{\text{Im}[\nu]} \ll \abs{\text{Re}[\nu]}$, then since $-0.5 < \text{Re}\l[ \nu \r] \leq 0.5$ the quantity $\nu$ becomes negligible with respect to $j$ already for quite small $\abs{j}$ in that case. In Section <ref>, we see that this is a good assumption in both coronal and photospheric conditions, if the Alfvén Mach number $M_A = (V_{0i}-V_{0e})/v_{Ai}$ is small enough.
One could then try considering, as a first approximation for $\Delta$, only the first few terms on the right-hand side of Eq. (<ref>). Taking $n=1$ and $j_1 \in \{-1,0,1\}$, this would yield:
\begin{equation} \label{DeltaApprox}
\Delta \;\; \approx \;\; 1 \;\; + \;\; \varepsilon_{-1} \; \varepsilon_0 \;\; + \;\; \varepsilon_0 \; \varepsilon_1 \text{.}
\end{equation}
Equation (<ref>) corresponds to Eq. (<ref>) with the infinite determinant in the right-hand side truncated to a $3 \times 3$ determinant centered on the row with $\varepsilon_0$. If one starts from the incompressible versions of Eqs. (<ref>)-(<ref>), one can derive that $m=ik$ in an incompressible plasma. The approximation of truncating the series in Eq. (<ref>) is only valid for large enough $\abs{j}$ and away from the poles of the $\varepsilon_j$. Under the assumption that $\abs{K} v_A \ll \o_0$ and $\abs{K} v_s \ll \o_0$, this condition is fulfilled. We note that in the solar atmosphere, $v_A$ and $v_s$ are rougly of the same order of magnitude. Since $K = k^2 - \abs{m}^2$ for surface waves, we have $K \to 0$ in the incompressible limit. Eq. (<ref>) thus gives us an approximation for $\Delta$ at least in a weakly compressible plasma, but could maybe even be correct more generally.
Analytical solutions for $m_i$ and $m_e$ in function of $\nu$ can then be derived from Eq. (<ref>). The obtained expressions for $m_i(\nu)$ and $m_e(\nu)$ are very complicated but, introducing numerical values relevant for the physical conditions of the solar structure being considered, we can use them together with a third relation involving $\nu$, $m_i$ and $m_e$. This other relation can theoretically be the one derived in the next subsection, although in practice one of those we derive in Section <ref> under approximating circumstances will probably be preferred. We note that $\nu$ is the same on both sides of the interface, as will be explained in the next section. In contrast to this, $m$ is in general not identical on both sides. It can also be seen that $\Delta(\nu, m) = \Delta(\nu, -m)$. Hence, if $m$ is a solution then so is $-m$. This is reflected in the form of the spatial function $F$ in Eq. (<ref>).
§.§ Other perturbed physical quantities
With $h(t) = G(t) g(t)$, the compression term can be written as $\c = F(x) h(t) \exp \{ i (k_y y + k_z z) \}$. From this and Eqs. (<ref>)-(<ref>), the following expressions for the perturbed quantities can then easily be derived:
\begin{align}
\xi_x &= - \d\frac{i m}{k^2 + m^2} \; \tilde{F}(x)\; h_1(t) \; \e \text{,} \label{xix}\\
\xi_y &= - \d\frac{i k_y}{k^2 + m^2} \; F(x)\; h_2(t) \; \e \text{,} \label{xiy}\\
\xi_z &= - \d\frac{i k_z}{k^2 + m^2} \; F(x)\; h_3(t) \; \e \text{,} \label{xiz}\\
B_{1x} &= B_{0z} \; \d\frac{k_z m}{k^2 + m^2} \; \tilde{F}(x) \; h_1(t) \; \e \text{,} \\
B_{1y} &= B_{0z} \; \d\frac{k_z k_y}{k^2 + m^2} \; F(x) \; h_2(t) \; \e \text{,} \\
B_{1z} &= B_{0z} \; F(x) \l( \d\frac{k_z^2}{k^2 + m^2} \; h_3(t) \; - \; h(t) \r) \e \text{,}\\
p_1 &= -\rho_0 \; v_s^2 \; F(x) \; h(t) \; \e \text{,} \\
\rho_1 &= -\rho_0 \; F(x) \; h(t) \; \e \text{,} \\
P_1 &= \rho_0 \; \l( \d\frac{k_z^2 v_A^2}{k^2 + m^2} h_3(t) \; - \; \l( v_A^2 + v_s^2 \r) h(t) \r) \notag\\
& \hspace{4cm} F(x) \; \e \text{,} \label{P1}
\end{align}
with $\tilde{F}(x) = C_1 e^{imx} - C_2 e^{-imx}$ and where $h_1$, $h_2$, $h_3$ are still unknown but have to satisfy
\begin{equation}\label{h1h2h3h}
\d\frac{m^2}{k^2 + m^2} \; h_1(t) \; + \; \d\frac{k_y^2}{k^2 + m^2} \; h_2(t) \; + \; \d\frac{k_z^2}{k^2 + m^2} \; h_3(t) \; = \; h(t) \text{,}
\end{equation}
by the definition $\c = \partial \xi_x / \partial x + \partial \xi_y / \partial y + \partial \xi_z / \partial z$. Each of the expressions (<ref>)-(<ref>) is different for each region on both sides of the interface. The functions $\tilde{F}$, $F$, $h$, $h_1$, $h_2$ and $h_3$ in particular will have a different version in both regions. For the spatial functions $\tilde{F}$ and $F$, only one of the two coefficients $C_1$ and $C_2$ must be retained in each region. We make the choice that $C_1 = 0$ in the $x<0$ region, and $C_2 = 0$ in the $x>0$ region.
Writing $h_1(t) = G_1(t) g(t)$, $h_2(t) = G_2(t) g(t)$ and $h_3(t) = G_3(t) g(t)$, the following expressions for $G_1$, $G_2$ and $G_3$ can be derived from Eqs. (<ref>)-(<ref>), Eqs. (<ref>)-(<ref>) and Eq. (<ref>) (see Appendix <ref>):
\begin{align}
G_1(\t) \; &= \; G_2(\t) \notag \\
&= \; K^2 v_s^2 \; e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \d\frac{1}{\l(j + \nu \r)^2 \o_0^2 - K^2 v_A^2} \; \varphi_j \; e^{i j \t} \text{,} \\
G_3(\t) \; &= \; \d\frac{K^2}{k_z^2} \; e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \l(1 - \frac{\l( k_y^2+ m^2 \r) v_s^2}{\l(j + \nu \r)^2 \o_0^2 - K^2 v_A^2} \r) \; \varphi_j \; e^{i j \t} \text{.}
\end{align}
It can also be checked that Eq. (<ref>) is indeed fulfilled with these expressions. Now, the following two boundary conditions have to be fulfilled at the interface between the two regions (i.e., at $x=0$) for physical reasons:
\begin{align}
[ \xi_x ] = 0 \text{,} \label{BCxi}\\
[ P_1 ] = 0 \text{,} \label{BCP1}
\end{align}
where $[f] = \lim_{x \downarrow 0} f(x) - \lim_{x \uparrow 0} f(x)$ denotes the jump in a quantity $f$ across the interface. We note that, similarly as for the frequency of a normal mode in the case without background flow, $\nu$ is the same inside and outside. It has proven to be too difficult to show mathematically, but it can be explained as follows. When a perturbation is unstable, its growth rate is determined solely by $\nu$: the growth of every perturbed quantity is namely expressed by the factor $e^{-\text{Im}[\nu]t}$. Therefore, since quantities such as $\xi_x$ and $P_1$ have to be continuous at the interface $x=0$, $\nu$ must be the same on both sides of the interface.
From the two equations (<ref>) and (<ref>) linking the interior and exterior solutions, a relation can be derived which has to be satisfied in order for the system determined by these equations to have nontrivial solutions:
\begin{equation} \label{disp}
\d\sum_{j=- \infty}^{\infty} \l( \rho_{0i} m_e \zeta_{A,j} \; + \; \rho_{0e} m_i \zeta_{B,j} \r) \; e^{ij \t} = 0 \text{,}
\end{equation}
\begin{align*}
\zeta_{A,j} &= \d\sum_{l=- \infty}^{\infty} \frac{2 \o_0^2 v_{si}^2 v_{se}^2 \l[ \l( l+\nu \r)^2 \o_0^2 - k_z^2 v_{Ai}^2 \r] \varphi_{l,i} \varphi_{j-l,e}}{\l[ \l( l+\nu \r)^2 \o_0^2 - K_i^2 v_{Ai}^2 \r] \l[ \l(j- l+\nu \r)^2 \o_0^2 - K_e^2 v_{Ae}^2 \r] } \text{,} \\
\zeta_{B,j} &= \d\sum_{l=- \infty}^{\infty} \frac{2 \o_0^2 v_{si}^2 v_{se}^2 \l[ \l( l+\nu \r)^2 \o_0^2 - k_z^2 v_{Ae}^2 \r] \varphi_{l,e} \varphi_{j-l,i}}{\l[ \l( l+\nu \r)^2 \o_0^2 - K_e^2 v_{Ae}^2 \r] \l[ \l(j- l+\nu \r)^2 \o_0^2 - K_i^2 v_{Ai}^2 \r] } \text{.}
\end{align*}
Equation (<ref>), arising from the boundary conditions at the interface, is usually called the dispersion relation when there is no oscillating background flow. Both $m_i$ and $m_e$ being determined by Eq. (<ref>) (by inserting respectively interior and exterior values for the background quantities), Eq. (<ref>) determines the only remaining unknown, $\nu$.
From Eq. (<ref>), the following has to hold for every $j \in \mathbb{Z}$:
\begin{equation} \label{mRatios}
\d\frac{m_e}{m_i} = -\d\frac{\rho_{0e} \zeta_{B,j}}{\rho_{0i} \zeta_{A,j}} \text{.}
\end{equation}
For $j = 0$ this gives us the relation
\begin{equation} \label{mRatiosj0}
\d\frac{m_e}{m_i} = - \d\frac{\rho_{0e}}{\rho_{0i} } \d\frac{\d\sum_{l=- \infty}^{\infty} \frac{ \l[ \l( l+\nu \r)^2 \o_0^2 - k_z^2 v_{Ae}^2 \r] \varphi_{l,e} \varphi_{-l,i}}{\l[ \l( l+\nu \r)^2 \o_0^2 - K_e^2 v_{Ae}^2 \r] \l[ \l(l-\nu \r)^2 \o_0^2 - K_i^2 v_{Ai}^2 \r] } }{ \d\sum_{l=- \infty}^{\infty} \frac{\l[ \l(l -\nu \r)^2 \o_0^2 - k_z^2 v_{Ai}^2 \r] \varphi_{l,e} \varphi_{-l,i}}{\l[ \l( l+\nu \r)^2 \o_0^2 - K_e^2 v_{Ae}^2 \r] \l[ \l(l-\nu \r)^2 \o_0^2 - K_i^2 v_{Ai}^2 \r] } } \text{.}
\end{equation}
We note that this is not an explicit solution for $m_e/m_i$, because both $m_i$ and $m_e$ appear on the right-hand side, through $K_i$ and $K_e$ respectively. While Eq. (<ref>) seems difficult to use directly, it gives us some information. Indeed, we learn from this equation that, whereas in the incompressible case we have $m_e/m_i = 1$, in the compressible case $m_e/m_i$ seems to depend on $k_z$ as well as $k_y$, the latter appearing only in $K_i$ and $K_e$. In the next section, we see that the only perturbation quantities the stability of the interface depends on are $k_z$ and $m_e/m_i$. This means that, while the abstraction is made that only the longitudinal wavenumber $k_z$ is important for the stability of the interface in an incompressible plasma, the perpendicular wavenumber $k_y$ seems to have some influence as well in the more realistic case of a compressible plasma.
§ STABILITY OF THE INTERFACE
§.§ Governing equation
The results derived in the preceding sections permit us to say something about the stability of the interface at the structure's boundary. This stability is determined by the temporal evolution of the normal displacement $\xi_x$, which is governed by the following equation:
\begin{equation} \label{Eqxix}
\d\frac{D^2 \xi_x}{Dt^2} \; + \; k_z^2 v_A^2 \xi_x \; = \; \frac{-1}{\rho_0} \frac{\pa P_1}{\pa x} \text{.}
\end{equation}
There is again a different version of Eq. (<ref>) for each region, namely for $x<0$ and for $x>0$. For each of the two regions, the respective versions of expression (<ref>) for $P_1$ can then be inserted in the respective versions of Eq. (<ref>). These can then be put together by expressing $C_1$ in function of $C_2$ thanks to Eq. (<ref>), and yield the following equation governing the displacement of the interface over time:
\begin{align}
&\l( \rho_{0e} m_i \d\frac{D_e^2 \l(\xi_x\bigr\rvert_{x=0} \r)}{Dt^2} \; + \; \rho_{0i} m_e \frac{D_i^2 \l(\xi_x\bigr\rvert_{x=0} \r)}{Dt^2} \r) \notag\\
& \hspace{1.5cm} + \; k_z^2 \l( \rho_{0e} v_{Ae}^2 m_i \xi_x\bigr\rvert_{x=0} \; + \; \rho_{0i} v_{Ai}^2 m_e \xi_x\bigr\rvert_{x=0} \r) = 0 \text{,} \label{xixie}
\end{align}
where $D_i/Dt = \pa / \pa t + i k_z V_{0i} \sin (\o_0 t)$, $D_e/Dt = \pa / \pa t + i k_z V_{0e} \sin (\o_0 t)$, and $\xi_x\bigr\rvert_{x=0}$ is $\xi_x$ evaluated at $x=0$.
Using a similar trick as was used for $\c$ before and which was also used by Barbulescu} {et~al.}, 2019 in a different setup, we write $ξ_x|_x=0$ as
\begin{equation}\label{trick}
\xi_x\bigr\rvert_{x=0}(t) = \eta(t) \exp \l\{\frac{- i A \sin \l( \o_0 t \r)}{\o_0} \r\} \text{,}
\end{equation}
with $A = k_z V_0e m_i ρ_0e + V_0i m_e ρ_0i/m_i ρ_0e + m_e ρ_0i$. This assumption changes nothing to the stability of $ξ_x|_x=0(t)$ since the moduli of $ξ_x|_x=0(t)$ and $η(t)$ are the same for every $t ∈ℝ^+$, and therefore $ξ_x|_x=0$ is unbounded if and only if $η$ is unbounded. Inserting Eq. \eqref{trick} in Eq. \eqref{xixie} greatly simplifies the equation, which can be worked out to yield the following ODE for $η$:
\begin{equation}\label{Mathieu}
\d\frac{d^2 \eta(\tau)}{d \tau^2} + \l( a - 2 q \cos \l( 2 \tau \r) \r) \eta(\tau) = 0 \text{,}
\end{equation}
with $τ= ø_0 t$ and
%a &= \d \frac{k_z^2}{ \o_0^2 \l( \rho_{0i} m_e + \rho_{0e} m_i \r)^2} \l\{ m_i^2 \rho_{0e}^2 v_{Ae}^2 + m_e^2 \rho_{0i}^2 v_{Ai}^2 \r. \notag \\
%& \hspace{1.5cm} \l. - m_i m_e \rho_{0i} \rho_{0e} \l[ \frac{1}{2} \l(V_{0i} - V_{0e} \r)^2 - \l( v_{Ai}^2 + v_{Ae}^2 \r) \r] \r\} \text{,} \label{a1} \\
%q &= \d\frac{k_z^2 \rho_{0i} \rho_{0e} m_i m_e \l( V_{0e} - V_{0i} \r)^2}{4 \o_0^2 \l( \rho_{0i} m_e + \rho_{0e} m_i \r)^2} \text{.} \label{q1}
\begin{align}
a &= \d\frac{k_z^2}{\o_0^2} \d\frac{\rho_{0i} v_{Ai}^2 m_e + \rho_{0e} v_{Ae}^2 m_i}{\rho_{0i} m_e + \rho_{0e} m_i} - \frac{k_z^2 \rho_{0i} \rho_{0e} m_i m_e \l(V_{0i} - V_{0e} \r)^2}{2 \o_0^2 \l( \rho_{0i} m_e + \rho_{0e} m_i \r)^2} \text{,} \label{a1} \\
q &= \d\frac{k_z^2 \rho_{0i} \rho_{0e} m_i m_e \l( V_{0i} - V_{0e} \r)^2}{4 \o_0^2 \l( \rho_{0i} m_e + \rho_{0e} m_i \r)^2} \text{.} \label{q1}
\end{align}
The parameters $a$ and $q$ can be rewritten in normalized form as follows:
%a &= \d\frac{\kappa_z^2 \l( r^2 \overline{v}_A^2 + \overline{m}^2 + \overline{m} r \l( 1 + \overline{v}_A^2 - \frac{1}{2} M_A^2 \r) \r) }{\l( \overline{m} + r \r)^2} \text{,} \label{a2}\\
%q &= \d\frac{\kappa_z^2 r \overline{m} M_A^2}{4 \l( \overline{m} + r \r)^2} \label{q2} \text{,}
\begin{align}
a &= \kappa_z^2 \l[ \d\frac{\ov{m} + r \ov{v}_A^2}{\ov{m} + r} - \frac{r \ov{m} M_A^2}{2 \l( \ov{m} + r \r)^2} \r] \text{,} \label{a2}\\
q &= \kappa_z^2 \d\frac{ r \overline{m} M_A^2}{4 \l( \overline{m} + r \r)^2} \label{q2} \text{,}
\end{align}
where we introduced the normalized quantities $κ_z = k_z v_Ai / ø_0$, $v_A = v_Ae / v_Ai$, $r = ρ_0e / ρ_0i$, $m = m_e/m_i$ and $M_A = (V_0i-V_0e) / v_Ai$.
We notice that the first term of $a$ consists of an expression that resembles the squared kink frequency and which we call the pseudo squared kink expression. The second term of $a$ resembles a Doppler-shifting correction and equals $-2q$. In an incompressible plasma, the first term would be the squared kink frequency (since then $m_i, m_e →k$). We also note that if there is no background flow (i.e., $V_0i = V_0e = 0$), we are left with only the pseudo squared kink expression in the factor in front of $η()$ in Eq. \eqref{Mathieu}. The solution is then a surface wave with frequency $ν$ determined by the dispersion relation
\begin{equation} \label{nuConstFlow}
\nu^2 = \kappa_z^2 \d\frac{\ov{m} + r \ov{v}_A^2}{\ov{m} + r} \text{,}
\end{equation}
where $m_i$ and $m_e$ are determined in function of $ν$ through their respective version of Eq. \eqref{DeltaFormula}. Eq. \eqref{nuConstFlow} is a transcendental equation in $ν$ with more than one solution, which means that there are several different modes in a compressible plasma. This is in contrast with an incompressible plasma, where there is only one mode, for which the expression of the frequency is readily readable from Eq. \eqref{nuConstFlow} and is equal to the kink frequency. These results about modes at an interface in a compressible plasma without background flow are already known (see for example Priest, 2014).
\subsection{Instability conditions}
Equation \eqref{Mathieu} is the Mathieu equation, which also pertains to the class of equations obeying Floquet theory and which has been extensively studied for example by McLachlan, 1947. It does not have an analytical solution but the stability of its solution in function of its parameters is well-known. The stability of the solutions of the Mathieu equation \eqref{Mathieu} in function of the parameters $a$ and $q$ is often represented in the so-called stability diagram (see Figure \ref{StabDiag}). The white zones in the figure are regions where the solutions are stable, whereas the gray zones are region where the solutions are unstable.
\begin{figure}
\centering
\includegraphics[scale=0.165]{MathDiag3.png}
\caption{Stability diagram of the Mathieu equation. The white zones represent regions of the parameter space where the solution is stable, whereas the gray zones represent regions of the parameter space where the solution is unstable.}
\label{StabDiag}
\end{figure}
\subsubsection{Constant shear flow}
In order to gain some insight on the nature of the potential instabilities, we follow Hillier} {et~al.}, 2019, who studied the stability of an interface in a similar model. We start by looking at the case with a constant background shear flow (i.e., we assume $ø_0 = 0$). We have to modify the procedure above a little bit in order to be mathematically correct: instead of Eq. \eqref{trick}, we assume $ξ_x|_x=0(t) = η(t) expł{- i A t }̊$ in this particular case. The obtained equation is then
\begin{equation}\label{PseudoMathieu}
\d\frac{d^2 \eta(\tau)}{d \tau^2} + \kappa_z^2 \l[ \d\frac{\ov{m} + r \ov{v}_A^2}{\ov{m} + r} - \frac{r \ov{m} M_A^2}{\l( \ov{m} + r \r)^2} \r]\; \eta(\tau) = 0 \text{.}
\end{equation}
It has normal mode solutions with normalized frequency $ν$ defined by the dispersion relation
\begin{equation} \label{DispMat1}
\nu^2 = \kappa_z^2 \l[ \d\frac{\ov{m} + r \ov{v}_A^2}{\ov{m} + r} - \frac{r \ov{m} M_A^2}{\l( \ov{m} + r \r)^2} \r] \text{.}
\end{equation}
This transcendental equation in $ν$ has multiple solutions as well, and thus we see that there are also different modes in the case of a constant shear flow in a compressible plasma. A particular mode will be unstable to the KHI if
\begin{equation} \label{instCondConst}
M_A^2 > \d\frac{\l( \ov{m} + r \r) \l( \ov{m} + r \ov{v}_A^2 \r)}{r \ov{m}}
\end{equation}
is satisfied. Different modes will thus have different instability conditions in a compressible plasma. If we take the incompressible limit, the above derivations match the already well-known results of a constant shear flow in an incompressible plasma (see for example Chandrasekhar}, 1961).
\subsubsection{Oscillating shear flow}
In the case of an oscillatory background flow ($ø_0 ≠0$), Eq. \eqref{Mathieu} governs the evolution of the normal displacement at the interface over time. In the same way as for Eq. \eqref{EqG}, Floquet theory allows us to write an independent solution to the Mathieu equation as follows [McLachlan, 1947]:
\begin{equation}\label{seriesMathieu}
\eta(\t) \; = \; e^{i \nu \t} \d\sum_{j = -\infty}^{\infty} \phi_j \; e^{2 i j \t} \text{.}
\end{equation}
This resembles a normal mode with frequency $ν$, but with a periodic function replacing the constant. One can then find the following recursion relation by inserting Eq. \eqref{seriesMathieu} into Eq. \eqref{Mathieu}:
\begin{equation} \label{recursionMathieu}
\epsilon_j \; \phi_{j-1} \;\; + \;\; \phi_j \;\; + \;\; \epsilon_j \; \phi_{j+1} \;\; = \;\; 0\text{,}
\end{equation}
\begin{equation}
\epsilon_j = \d\frac{q}{\l( 2j + \nu \r)^2-a} \text{.}
\end{equation}
If the denominator of one of the $ϵ_j$ vanishes, there can be a resonance between the background oscillator and the induced surface waves.
Now some analytical progress can be made in the limiting case $q →0$, corresponding to a small squared shear rate $k_z^2 (V_0i-V_0e)^2$ with respect to the squared frequency $ø_0^2$. Indeed, for $q ≪1$, we see from Eq. \eqref{recursionMathieu} that in order to have a nontrivial solution \eqref{seriesMathieu} we need the denominator of at least one of the $ϵ_j$ to vanish. For that, we need to have
\begin{equation} \label{res}
\l( 2j + \nu \r)^2-a \approx 0
\end{equation}
for a $j ∈ℤ$. In case $ν^2 ≈a$, that is to say, if
\begin{equation} \label{j0}
\nu^2 \approx \kappa_z^2 \l[ \d\frac{\ov{m} + r \ov{v}_A^2}{\ov{m} + r} - \frac{r \ov{m} M_A^2}{2 \l( \ov{m} + r \r)^2} \r] \text{,}
\end{equation}
then Eq. \eqref{res} is fulfilled for $j = 0$. Since Eq. \eqref{j0} is again a transcendental equation in $ν$ with multiple solutions, there are multiple modes possible. These modes are surface waves with frequency $ν$. Such a mode will be unstable if
\begin{equation} \label{instCondOsc}
M_A^2 > 2 \d\frac{\l( \ov{m} + r \r) \l( \ov{m} + r \ov{v}_A^2 \r)}{r \ov{m}}
\end{equation}
is satisfied. This renders the right-hand side in Eq. \eqref{j0}, which is $a$, negative. Hence, for $q ≪1$, the region of the parameters space which corresponds to the KHI is $a<0$. It can be seen on Figure \ref{StabDiag} that this region is indeed uniformly unstable. We note that the right-hand side in Eq. \eqref{instCondOsc} is a factor of $2$ larger than in Eq. \eqref{instCondConst}. In a first approximation, we could thus evaluate the minimum value of $M_A$ for the KHI to develop in the case of an oscillatory shear flow with $q ≪1$ to be $√(2)$ times higher than in the case of a constant shear flow with $q ≪1$. This is assuming that the corresponding values of $m_i$ and $m_e$ are equal in both cases. In an incompressible plasma, this is obviously correct. In a compressible plasma, however, there is no reason to suggest a priori that this is the case in general. Nevertheless, the values might still be close, such as in a weakly compressible plasma for example.
If on top of Eq. \eqref{res} being satistfied for $j=0$ it is also satisfied for another $j ∈ℤ$, then there is a resonance between a mode satisfying Eq. \eqref{j0} and the background oscillator. From $ν^2 = a$ and Eq. \eqref{res} with $j≠0$, this means that we must have
\begin{equation}
a = j^2 \text{.}
\end{equation}
In Figure \ref{StabDiag} we see the regions of resonance instability around $a=j^2$, with $j ∈ℤ_0$. Hence, for $q ≪1$, the KHI and the resonance instability are clearly distinct features and are mutually exclusive.
As was also mentionned by Hillier} {et~al.}, 2019 (themselves based on Bender \& Orszag, 1999), the dominant resonance in this limit is the one with $j = 1$. Its instability reagion is borded by the curves
\begin{equation}
a = j^2 \pm q + O(q^2) \hspace{1cm} \text{ as } q \to 0
\end{equation}
and its maximum growth rate is
\begin{equation}
\text{Im}[\nu] = \d\frac{q}{2} + O(q^2) \hspace{1cm} \text{ as } q \to 0 \text{.}
\end{equation}
We could thus express this growth rate through the equation $Im[ν] = q/2$ as an approximation. This is an implicit equation in $ν$, because $q$ depends on it through $m$.
\subsection{Solar applications: Relevant region of the parameter space}
From Eqs. \eqref{a2} and \eqref{q2}, we see the parameters $a$ and $q$ are related as follows:
\begin{equation} \label{a=f(q)}
a = \l\{ \d\frac{4 \l( \ov{m} + r \r) \l( \ov{m} + r \ov{v}_A^2 \r)}{r \ov{m} M_A^2} - 2 \r\} \; q \text{.}
\end{equation}
In the incompressible limit, $m →1$ and we find the same relation as Barbulescu} {et~al.}, 2019 for their model. For fixed values of the background quantities and for a fixed value of $m$, Eq. \eqref{a=f(q)} represents a line in the $aq$-plane. We note, however, that changing the value of $k_z$ and/or $k_y$ a priori changes the value of $m$ as well and that hence the modes do not lie all on the same line. This is unlike the case of an incompressible plasma, where one stays on the same line when changing $k_z$, as was already found for example by Barbulescu} {et~al.}, 2019 in a similar model.
We find that the slope in Eq. \eqref{a=f(q)} is minimized for $m = r v_A$, in which case the line's equation is
\begin{equation} \label{minSlope}
a= \l\{ \frac{ 4 \l( 1+ \overline{v}_A \r)^2 }{M_A^2} - 2 \r\} \; q \text{.}
\end{equation}
For each set of fixed background quantities, the only physically possible values of $a$ and $q$ thus lie in the region of the stability diagram between the positive $a$-axis and the line defined by Eq. \eqref{minSlope}. We see that, in this model, it is not physically possible to have values of $(q,a)$ in the region below the line through the origin and with slope $-2$.
\subsubsection{Coronal loops}
A first application to a solar atmospheric structure would be a coronal loop, where slow waves occur near footpoints [De Moortel} {et~al.}, 2002] or during flares [Wang} {et~al.}, 2002, Kumar} {et~al.}, 2013]. For a standing slow wave under realistic coronal loop conditions, for which we took $(v_Ae, v_si, v_se) = (5/2,1/2,1/4) v_Ai$, the line of minimum slope, Eq. \eqref{minSlope}, lies almost on the positive $a$-axis. Indeed, with $v_A = 5/2$ and for $M_A = 0.066$ for example, we find the line of minimum slope to be given by $a = 11415q$. If we assume incompressibility and fix $m=1$, Eq. \eqref{a=f(q)} defines a line as well and we find $a=12486q$. For the less realistic value of $M_A = 1$, we find that the line of minimum slope is given by $a=47q$ whereas in the incompressible limit the line \eqref{a=f(q)} becomes $a=52q$. As we can see from Figure \ref{StabDiag}, the region of the $aq$-diagram relevant for coronal loop conditions is thus stable with respect to the KHI. It could be unstable with respect to resonance; however, since in reality a finite cylindrical structure is closed both azimuthally and longitudinally, the respective wavenumbers will be entire and thus take on discrete values. In the $aq$-diagram, the possible values will thus be a set of discrete points $(a_n, q_n)$. Seeing how the vast majority of the area is a stable region in the relevant region between the positive $a$-axis and the minimum-slope line \eqref{minSlope}, it is likely that this parametric resonance instability is avoided as well.
\begin{table}[!htb]
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{conditions} & \textbf{$M_A$} & \textbf{minimum slope} & \textbf{incomp. slope} \\ \hline
coronal & 0.066 & 11415 & 12486 \\ \hline
photospheric & 0.066 & 1450 & 1615 \\ \hline
photospheric & 0.36 & 46 & 52 \\ \hline
coronal & 1 & 47 & 52 \\ \hline
photospheric & 1 & 4.25 & 5 \\ \hline
\end{tabular}
\caption{Table summarizing the minimum slope in Eq. \eqref{minSlope} and the slope of Eq. \eqref{a=f(q)} in the incompressible limit, for different atmospheric conditions. The first two rows are for realistic values of the longitudinal velocity for a slow surface wave on a discontinuous boundary of a cylindrical structure. The third row is for realistic values of the longitudinal velocity around the resonant position for a resonant slow wave in a photospheric pore. The last two lines are examples where $M_A$ begins to be unrealistically high and which show that even then the slopes are not low enough to be in an unstable regime. For coronal conditions we took $(v_{Ae}, v_{si}, v_{se}) = (5/2,1/2,1/4) v_{Ai}$, whereas for photospheric conditions we took $(v_{Ae}, v_{si}, v_{se}) = (1/4,1/2,3/4) v_{Ai}$.}
\label{tab:my-table}
\end{table}
We conclude that the oscillating shear velocity of a standing slow wave in a coronal loop would not trigger the KHI on its own, without the involvement of other physical processes. This is in clear contrast with the fast kink waves. Indeed, according to the incompressible models of Barbulescu} {et~al.}, 2019 and Hillier} {et~al.}, 2019, which are similar to this model but where the velocity and magnetic fields are perpendicular instead of parallel, fast kink waves are predicted to excite the KHI on the loop boundary. This has also been confirmed by numerical simulations.
\subsubsection{Photospheric pores}
As was mentionend in the introduction, propagating slow waves have also been observed in photospheric pores. One would be curious whether standing versions of these modes could trigger the KHI under these conditions. The model can be used to examine the stability of slow waves in these structures as well. However, the qualitative conclusion is the same as in the case of coronal loops: with photospheric pore conditions set to $(v_Ae, v_si, v_se) = (1/4,1/2,3/4) v_Ai$, we have $v_A = 1/4$, and, taking $M_A = 0.066$, the minimum slope is $1450$ whereas for the slope of the line defined by Eq. \eqref{a=f(q)} is $1615$ in the incompressible limit. As a comparison, even for the unrealistic value of $M_A=1$, the minimum slope is $4.25$ and the slope in Eq. \eqref{a=f(q)} in the incompressible limit is $5$. These values are an order of magnitude lower then for their coronal counterpart, but looking at Figure \ref{StabDiag} we see that the same conclusion remains: every possible pair of parameters $(a,q)$ lie in a region that does not permit the development of the KHI. Similarly as for a coronal loop in the previous subsection, it is likely that the parametric instability arising from resonance with the driver is also avoided and that the pore therefore remains stable.
Additionally, one could investigate the stability of resonant slow waves, which can occur in the presence of a transition layer at the pore's boundary. Indeed, slow surface waves are known to be resonantly absorbed in photospheric conditions. It is worth wondering whether the great longitudinal velocity shear that arises around the resonant point could be enough to trigger the KHI. In ideal MHD, the longitudinal velocity of resonant slow modes displays a hyperbolic singularity [Sakurai} {et~al.}, 1991], but in the presence of finite resistivity this singularity disappears to make place for a sharp but continuous profile. Kovitya} \& {Cram}, 1983 reported values of electrical resistivity in sunspots, with a minimum resistivity corresponding to a magnetic Reynolds number of $10^7$. Erdelyi}, 1997 derived an expression for the longitudinal Lagrangian displacement in the dissipative layer around the cusp resonance in the presence of finite electrical resistivity (see Eq. (30) therein). Assuming a sinusoidal transition profile with width $l=0.1R$ (where $R$ is the radius of the pore) in the squared cusp and sound speeds, a ratio of external to internal magnetic field of $B_0ze/B_0zi=0.33$, a cusp resonant position at $r_C=0.955R$ (based on the numerical computations of Goossens} {et~al.}, 2021), a value for the perturbed total pressure at the resonant position equal to its value on the interface for the corresponding surface mode in the absence of the transition layer, a real part of the frequency equal to the frequency of the corresponding surface mode in the absence of the transition layer, and a magnetic Reynolds number of $10^7$ to have a lowerbound on resistivity and thus an upperbound on realistic values for the longitudinal velocity at the resonance, we found a value of $M_A=0.36$ around the cusp resonance point for a sausage mode with $k_z R = 2$. This value of $k_z R$ is within the range of validity for the longitudinal wavenumber of the observed slow surface sausage mode in a photospheric pore by Grant} {et~al.}, 2015. The value found for $M_A$ implies a minimum slope of $46$ in Eq. \eqref{minSlope} and a slope of $52$ for Eq. \eqref{a=f(q)} in the incompressible limit. These values are still well above the values for the slopes found for $M_A=1$ in photospheric conditions. Through the model derived in this paper, we conclude from this that even resonant slow waves do not display a large enough velocity shear for an instability to develop.
\subsection{Limitations of the model}
We point out that the local model discussed in this paper is only valid under certain conditions. Firstly, the azimuthal wavenumber of the perturbation must be large, such that the azimuthal direction can be approximated by the cartesian direction of $y$. This means that we must have $k_y R ≫1$, with $R$ the structure's radius. Secondly, for the amplitude of the oscillating background velocity to be assumed constant, the longitudinal wavelength of the perturbation must be small with respect to the structure's length $L$. We must thus have $k_z L ≫1$. Lastly, we note that our model might not be suitable in the presence of a smooth transition layer between the interior and the exterior of the structure. Indeed, resonant absorption of the standing slow wave in the background could occur in that case [Yu} {et~al.}, 2017], which leads to a steep and continuous variation in the longitudinal component of the velocity along the $x$ direction. Although one can use the model to estimate the effect of the logitudinal velocity shear in resonant absorption on the stability of the interface, as we did in last subsection, one must keep in mind that certain assumptions on which the model is based are not fulfilled in the presence of a transition layer. Firstly, there is no true discontinuous interface and, secondly, the assumption of a constant amplitude profile along $x$ for the oscillating background velocity is violated.
\section{Conclusion}
In this paper, we developed an analytical model for the local stability of a cylindrical solar atmospheric structure harboring a standing slow wave. We assumed that the structure is a straight cylinder with a circular base, of which the boundary is an interface discontinuously separating two homogeneous and compressible plasmas. The magnetic fields on both sides were assumed to be constant, straight and aligned with the interface. The velocity fields were assumed to be oscillating in time but spatially constant, in order to model the standing slow wave in the background.
We used linearized MHD to derive an equation for the compression term, $$̧. The expressions for all the other perturbed quantities could then be expressed in terms of the solution of $\c$. The spatial part of the solution along the direction normal to the interface could be analytically found to be a normal mode, with wavenumber $m$. In contrast to the incompressible approximation, the value of this wavenumber $m$ in a compressible plasma is different on both sides of the interface. We saw that $m_e/m_i$ is dependent on both $k_z$ and $k_y$, which entails that the stability of the interface depends on $k_y$ as well as $k_z$. This is in contrast with the incompressible version of this model, in which only $k_z$ has an influence on said stability.
We then found that the governing equation for the displacement component of the perturbation normal to the interface is a Mathieu equation. The stability of the solution of this equation in function of the different involved parameters being known from the litterature, we were able to describe two kinds of instabilities that could theoretically arise in the presence of an oscillating shear background flow. As an application to the solar atmosphere, we found that the physically relevant region of the parameter space corresponds to an interface that is locally stable with respect to the KHI, both in a coronal loop and in a photospheric pore. Although the interface can be unstable to a parametric resonance between the background slow wave and the induced perturbations on the interface, we concluded that it is unlikely to happen in reality. Even in the case of resonance in the cusp continuum, we concluded from our model that the longitudinal velocity shear of the resonant slow waves is not enough to trigger an instability.
We ended by noting that this model is only applicable under specific conditions. It is indeed a local model, in the sense that the spatial scale of azimuthal variations must be small compared to the structure's radius and the spatial scale of longitudinal variations must be small with respect to the loop's length. Furthermore, for a structure with a smooth transition layer the assumptions of a discontinuous transition at the boundary and a constant amplitude for the background oscillating velocity fields would be violated. This needs to be kept in mind when using this model to infer stability conditions around resonances.
\begin{acknowledgements}
This research was supported by the European Research Council (ERC) under
the European Union's Horizon 2020
research and innovation program (TVD via grant agreement No 724326),
which MG gratefully acknowledges for his Ph.D. studentship.
\end{acknowledgements}
% WARNING
% Please note that we have included the references to the file aa.dem in
% order to compile it, but we ask you to:
% - use BibTeX with the regular commands:
% \bibliographystyle{aa} % style aa.bst
% \begin{thebibliography}{77}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
[Afanasyev} {et~al.}, 2019]
{Afanasyev}, A., {Karampelas}, K., \& {Van Doorsselaere}, T. 2019, \apj, 876,
[Antolin} {et~al.}, 2017]
{Antolin}, P., {De Moortel}, I., {Van Doorsselaere}, T., \& {Yokoyama}, T.
2017, \apj, 836, 219
[Antolin} {et~al.}, 2015]
{Antolin}, P., {Okamoto}, T.~J., {De Pontieu}, B., {et~al.} 2015, \apj, 809, 72
[Antolin} {et~al.}, 2014]
{Antolin}, P., {Yokoyama}, T., \& {Van Doorsselaere}, T. 2014, \apjl, 787, L22
[Arregui}, 2015]
{Arregui}, I. 2015, Philosophical Transactions of the Royal Society of London
Series A, 373, 20140261
[Barbulescu} {et~al.}, 2019]
{Barbulescu}, M., {Ruderman}, M.~S., {Van Doorsselaere}, T., \& {Erd{\'e}lyi},
R. 2019, \apj, 870, 108
[Bender \& Orszag, 1999]
Bender, C. \& Orszag, S. 1999, Advanced Mathematical Methods for Scientists and
Engineers I: Asymptotic Methods and Perturbation Theory, Advanced
Mathematical Methods for Scientists and Engineers (Springer)
[Berghmans} \& {Clette}, 1999]
{Berghmans}, D. \& {Clette}, F. 1999, \solphys, 186, 207
[Cadez} {et~al.}, 1997]
{Cadez}, V.~M., {Csik}, A., {Erdelyi}, R., \& {Goossens}, M. 1997, \aap, 326,
[Cesari, 1963]
Cesari, L. 1963, Asymptotic Behavior and Stability Problems in Ordinary
Differential Equations (Springer-Verlag)
[Chandrasekhar}, 1961]
{Chandrasekhar}, S. 1961, {Hydrodynamic and hydromagnetic stability}
[Chen} {et~al.}, 2018]
{Chen}, S.-X., {Li}, B., {Shi}, M., \& {Yu}, H. 2018, \apj, 868, 5
[Chen} {et~al.}, 2020]
{Chen}, S.~X., Li, B., Van~Doorsselaere, T., Goossens, M., \& Geeraerts, M.
2020, \apj, submitted
[Chicone, 2008]
Chicone, C. 2008, Ordinary Differential Equations with Applications, Texts in
Applied Mathematics (Springer New York)
[Conway, 1990]
Conway, J. 1990, A Course in Functional Analysis, Graduate texts in mathematics
[De Moortel} \& {Hood}, 2003]
{De Moortel}, I. \& {Hood}, A.~W. 2003, \aap, 408, 755
[De Moortel} {et~al.}, 2000]
{De Moortel}, I., {Ireland}, J., \& {Walsh}, R.~W. 2000, \aap, 355, L23
[De Moortel} {et~al.}, 2002]
{De Moortel}, I., {Ireland}, J., {Walsh}, R.~W., \& {Hood}, A.~W. 2002,
\solphys, 209, 61
[De Moortel} {et~al.}, 2016]
{De Moortel}, I., {Pascoe}, D.~J., {Wright}, A.~N., \& {Hood}, A.~W. 2016,
Plasma Physics and Controlled Fusion, 58, 014001
[Dorotovi{\v{c}}} {et~al.}, 2014]
{Dorotovi{\v{c}}}, I., {Erd{\'e}lyi}, R., {Freij}, N., {Karlovsk{\'y}}, V., \&
{M{\'a}rquez}, I. 2014, \aap, 563, A12
[Dorotovi{\v{c}}} {et~al.}, 2008]
{Dorotovi{\v{c}}}, I., {Erd{\'e}lyi}, R., \& {Karlovsk{\'y}}, V. 2008, in IAU
Symposium, Vol. 247, Waves \& Oscillations in the Solar Atmosphere: Heating
and Magneto-Seismology, ed. R.~{Erd{\'e}lyi} \& C.~A. {Mendoza-Briceno},
[Edwin} \& {Roberts}, 1983]
{Edwin}, P.~M. \& {Roberts}, B. 1983, \solphys, 88, 179
[Erdelyi}, 1997]
{Erdelyi}, R. 1997, \solphys, 171, 49
[Erd{\'e}lyi} {et~al.}, 2001]
{Erd{\'e}lyi}, R., {Ballai}, I., \& {Goossens}, M. 2001, \aap, 368, 662
[Foullon} {et~al.}, 2011]
{Foullon}, C., {Verwichte}, E., {Nakariakov}, V.~M., {Nykyri}, K., \&
{Farrugia}, C.~J. 2011, \apjl, 729, L8
[Freij} {et~al.}, 2016]
{Freij}, N., {Dorotovi{\v{c}}}, I., {Morton}, R.~J., {et~al.} 2016, \apj, 817,
[Geeraerts} {et~al.}, 2020]
{Geeraerts}, M., {Van Doorsselaere}, T., {Chen}, S.-X., \& {Li}, B. 2020, \apj,
897, 120
[Goossens} {et~al.}, 2002]
{Goossens}, M., {Andries}, J., \& {Aschwanden}, M.~J. 2002, \aap, 394, L39
[Goossens} {et~al.}, 2021]
{Goossens}, M., {Chen}, S.~X., {Geeraerts}, M., {Li}, B., \& {Van
Doorsselaere}, T. 2021, \aap, 646, A86
[Goossens} {et~al.}, 1992]
{Goossens}, M., {Hollweg}, J.~V., \& {Sakurai}, T. 1992, \solphys, 138, 233
[Grant} {et~al.}, 2015]
{Grant}, S.~D.~T., {Jess}, D.~B., {Moreels}, M.~G., {et~al.} 2015, \apj, 806,
[Guo} {et~al.}, 2019]
{Guo}, M., {Van Doorsselaere}, T., {Karampelas}, K., {et~al.} 2019, \apj, 870,
[Heyvaerts} \& {Priest}, 1983]
{Heyvaerts}, J. \& {Priest}, E.~R. 1983, \aap, 117, 220
[Hillier} {et~al.}, 2019]
{Hillier}, A., {Barker}, A., {Arregui}, I., \& {Latter}, H. 2019, \mnras, 482,
[Hillier} {et~al.}, 2020]
{Hillier}, A., {Van Doorsselaere}, T., \& {Karampelas}, K. 2020, \apjl, 897,
[Hollweg} {et~al.}, 2013]
{Hollweg}, J.~V., {Kaghashvili}, E.~K., \& {Chandran}, B. D.~G. 2013, \apj,
769, 142
[Hollweg} \& {Yang}, 1988]
{Hollweg}, J.~V. \& {Yang}, G. 1988, \jgr, 93, 5423
[Hollweg} {et~al.}, 1990]
{Hollweg}, J.~V., {Yang}, G., {Cadez}, V.~M., \& {Gakovic}, B. 1990, \apj, 349,
[Karampelas} {et~al.}, 2017]
{Karampelas}, K., {Van Doorsselaere}, T., \& {Antolin}, P. 2017, \aap, 604,
[Karampelas} {et~al.}, 2019]
{Karampelas}, K., {Van Doorsselaere}, T., {Pascoe}, D.~J., {Guo}, M., \&
{Antolin}, P. 2019, Frontiers in Astronomy and Space Sciences, 6, 38
[Karpen} {et~al.}, 1994]
{Karpen}, J.~T., {Dahlburg}, R.~B., \& {Davila}, J.~M. 1994, \apj, 421, 372
[Kelly}, 1965]
{Kelly}, R.~E. 1965, Journal of Fluid Mechanics, 22, 547
[Keys {et~al.}, 2018]
Keys, P.~H., Morton, R.~J., Jess, D.~B., {et~al.} 2018, The Astrophysical
Journal, 857, 28
[Kovitya} \& {Cram}, 1983]
{Kovitya}, P. \& {Cram}, L. 1983, \solphys, 84, 45
[Kumar} {et~al.}, 2013]
{Kumar}, P., {Innes}, D.~E., \& {Inhester}, B. 2013, \apjl, 779, L7
[Magyar} {et~al.}, 2015]
{Magyar}, N., {Van Doorsselaere}, T., \& {Marcu}, A. 2015, \aap, 582, A117
[Mandal} {et~al.}, 2016]
{Mandal}, S., {Magyar}, N., {Yuan}, D., {Van Doorsselaere}, T., \& {Banerjee},
D. 2016, \apj, 820, 13
[McLachlan, 1947]
McLachlan, N. 1947, Theory and Application of Mathieu Functions (Clarendon
[Moreels} {et~al.}, 2015]
{Moreels}, M.~G., {Freij}, N., {Erd{\'e}lyi}, R., {Van Doorsselaere}, T., \&
{Verth}, G. 2015, \aap, 579, A73
[Moreels} {et~al.}, 2013]
{Moreels}, M.~G., {Goossens}, M., \& {Van Doorsselaere}, T. 2013, \aap, 555,
[Morton} {et~al.}, 2011]
{Morton}, R.~J., {Erd{\'e}lyi}, R., {Jess}, D.~B., \& {Mathioudakis}, M. 2011,
\apjl, 729, L18
[Nakariakov} {et~al.}, 2000]
{Nakariakov}, V.~M., {Verwichte}, E., {Berghmans}, D., \& {Robbrecht}, E. 2000,
\aap, 362, 1151
[Nightingale} {et~al.}, 1999]
{Nightingale}, R.~W., {Aschwanden}, M.~J., \& {Hurlburt}, N.~E. 1999, \solphys,
190, 249
[Ofman} {et~al.}, 1994]
{Ofman}, L., {Davila}, J.~M., \& {Steinolfson}, R.~S. 1994, \grl, 21, 2259
[Ofman} \& {Thompson}, 2011]
{Ofman}, L. \& {Thompson}, B.~J. 2011, \apjl, 734, L11
[Parnell} \& {De Moortel}, 2012]
{Parnell}, C.~E. \& {De Moortel}, I. 2012, Philosophical Transactions of the
Royal Society of London Series A, 370, 3217
[Pascoe} {et~al.}, 2020]
{Pascoe}, D.~J., {Goddard}, C.~R., \& {Van Doorsselaere}, T. 2020, Frontiers in
Astronomy and Space Sciences, 7, 61
[Pascoe} {et~al.}, 2012]
{Pascoe}, D.~J., {Hood}, A.~W., {de Moortel}, I., \& {Wright}, A.~N. 2012,
\aap, 539, A37
[Pascoe} {et~al.}, 2010]
{Pascoe}, D.~J., {Wright}, A.~N., \& {De Moortel}, I. 2010, \apj, 711, 990
[Priest, 2014]
Priest, E. 2014, Magnetohydrodynamics of the Sun (Cambridge University Press)
[Roberts}, 1973]
{Roberts}, B. 1973, Journal of Fluid Mechanics, 59, 65
[Sakurai} {et~al.}, 1991]
{Sakurai}, T., {Goossens}, M., \& {Hollweg}, J.~V. 1991, \solphys, 133, 227
[Samanta} {et~al.}, 2019]
{Samanta}, T., {Tian}, H., \& {Nakariakov}, V.~M. 2019, \prl, 123, 035102
[Shi} {et~al.}, 2021]
{Shi}, M., {Van Doorsselaere}, T., {Guo}, M., {et~al.} 2021, arXiv e-prints,
[Simon, 2005]
Simon, B. 2005, Trace Ideals and Their Applications, Mathematical surveys and
monographs (American Mathematical Society)
[Soler} {et~al.}, 2013]
{Soler}, R., {Goossens}, M., {Terradas}, J., \& {Oliver}, R. 2013, \apj, 777,
[Soler} {et~al.}, 2009]
{Soler}, R., {Oliver}, R., {Ballester}, J.~L., \& {Goossens}, M. 2009, \apjl,
695, L166
[Sträng, 2005]
Sträng, J.-E. 2005
[Terradas} {et~al.}, 2008]
{Terradas}, J., {Andries}, J., {Goossens}, M., {et~al.} 2008, \apjl, 687, L115
[Van Doorsselaere} {et~al.}, 2020]
{Van Doorsselaere}, T., {Srivastava}, A.~K., {Antolin}, P., {et~al.} 2020,
\ssr, 216, 140
[Wang}, 2011]
{Wang}, T. 2011, \ssr, 158, 397
[Wang} {et~al.}, 2007]
{Wang}, T., {Innes}, D.~E., \& {Qiu}, J. 2007, \apj, 656, 598
[Wang} {et~al.}, 2002]
{Wang}, T., {Solanki}, S.~K., {Curdt}, W., {Innes}, D.~E., \& {Dammasch}, I.~E.
2002, \apjl, 574, L101
[Wang} {et~al.}, 2003]
{Wang}, T.~J., {Solanki}, S.~K., {Curdt}, W., {et~al.} 2003{\natexlab{a}},
\aap, 406, 1105
[Wang} {et~al.}, 2003]
{Wang}, T.~J., {Solanki}, S.~K., {Innes}, D.~E., {Curdt}, W., \& {Marsch}, E.
2003{\natexlab{b}}, \aap, 402, L17
[Yu} {et~al.}, 2017]
{Yu}, D.~J., {Van Doorsselaere}, T., \& {Goossens}, M. 2017, \aap, 602, A108
[Zaqarashvili} {et~al.}, 2015]
{Zaqarashvili}, T.~V., {Zhelyazkov}, I., \& {Ofman}, L. 2015, \apj, 813, 123
\end{thebibliography}
% your references Yourfile.bib
% - join the .bib files when you upload your source files
\bibliographystyle{aa}
\begin{thebibliography}{77}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
[Afanasyev} {et~al.}, 2019]
{Afanasyev}, A., {Karampelas}, K., \& {Van Doorsselaere}, T. 2019, \apj, 876,
[Antolin} {et~al.}, 2017]
{Antolin}, P., {De Moortel}, I., {Van Doorsselaere}, T., \& {Yokoyama}, T.
2017, \apj, 836, 219
[Antolin} {et~al.}, 2015]
{Antolin}, P., {Okamoto}, T.~J., {De Pontieu}, B., {et~al.} 2015, \apj, 809, 72
[Antolin} {et~al.}, 2014]
{Antolin}, P., {Yokoyama}, T., \& {Van Doorsselaere}, T. 2014, \apjl, 787, L22
[Arregui}, 2015]
{Arregui}, I. 2015, Philosophical Transactions of the Royal Society of London
Series A, 373, 20140261
[Barbulescu} {et~al.}, 2019]
{Barbulescu}, M., {Ruderman}, M.~S., {Van Doorsselaere}, T., \& {Erd{\'e}lyi},
R. 2019, \apj, 870, 108
[Bender \& Orszag, 1999]
Bender, C. \& Orszag, S. 1999, Advanced Mathematical Methods for Scientists and
Engineers I: Asymptotic Methods and Perturbation Theory, Advanced
Mathematical Methods for Scientists and Engineers (Springer)
[Berghmans} \& {Clette}, 1999]
{Berghmans}, D. \& {Clette}, F. 1999, \solphys, 186, 207
[Cadez} {et~al.}, 1997]
{Cadez}, V.~M., {Csik}, A., {Erdelyi}, R., \& {Goossens}, M. 1997, \aap, 326,
[Cesari, 1963]
Cesari, L. 1963, Asymptotic Behavior and Stability Problems in Ordinary
Differential Equations (Springer-Verlag)
[Chandrasekhar}, 1961]
{Chandrasekhar}, S. 1961, {Hydrodynamic and hydromagnetic stability}
[Chen} {et~al.}, 2018]
{Chen}, S.-X., {Li}, B., {Shi}, M., \& {Yu}, H. 2018, \apj, 868, 5
[Chen} {et~al.}, 2020]
{Chen}, S.~X., Li, B., Van~Doorsselaere, T., Goossens, M., \& Geeraerts, M.
2020, \apj, submitted
[Chicone, 2008]
Chicone, C. 2008, Ordinary Differential Equations with Applications, Texts in
Applied Mathematics (Springer New York)
[Conway, 1990]
Conway, J. 1990, A Course in Functional Analysis, Graduate texts in mathematics
[De Moortel} \& {Hood}, 2003]
{De Moortel}, I. \& {Hood}, A.~W. 2003, \aap, 408, 755
[De Moortel} {et~al.}, 2000]
{De Moortel}, I., {Ireland}, J., \& {Walsh}, R.~W. 2000, \aap, 355, L23
[De Moortel} {et~al.}, 2002]
{De Moortel}, I., {Ireland}, J., {Walsh}, R.~W., \& {Hood}, A.~W. 2002,
\solphys, 209, 61
[De Moortel} {et~al.}, 2016]
{De Moortel}, I., {Pascoe}, D.~J., {Wright}, A.~N., \& {Hood}, A.~W. 2016,
Plasma Physics and Controlled Fusion, 58, 014001
[Dorotovi{\v{c}}} {et~al.}, 2014]
{Dorotovi{\v{c}}}, I., {Erd{\'e}lyi}, R., {Freij}, N., {Karlovsk{\'y}}, V., \&
{M{\'a}rquez}, I. 2014, \aap, 563, A12
[Dorotovi{\v{c}}} {et~al.}, 2008]
{Dorotovi{\v{c}}}, I., {Erd{\'e}lyi}, R., \& {Karlovsk{\'y}}, V. 2008, in IAU
Symposium, Vol. 247, Waves \& Oscillations in the Solar Atmosphere: Heating
and Magneto-Seismology, ed. R.~{Erd{\'e}lyi} \& C.~A. {Mendoza-Briceno},
[Edwin} \& {Roberts}, 1983]
{Edwin}, P.~M. \& {Roberts}, B. 1983, \solphys, 88, 179
[Erdelyi}, 1997]
{Erdelyi}, R. 1997, \solphys, 171, 49
[Erd{\'e}lyi} {et~al.}, 2001]
{Erd{\'e}lyi}, R., {Ballai}, I., \& {Goossens}, M. 2001, \aap, 368, 662
[Foullon} {et~al.}, 2011]
{Foullon}, C., {Verwichte}, E., {Nakariakov}, V.~M., {Nykyri}, K., \&
{Farrugia}, C.~J. 2011, \apjl, 729, L8
[Freij} {et~al.}, 2016]
{Freij}, N., {Dorotovi{\v{c}}}, I., {Morton}, R.~J., {et~al.} 2016, \apj, 817,
[Geeraerts} {et~al.}, 2020]
{Geeraerts}, M., {Van Doorsselaere}, T., {Chen}, S.-X., \& {Li}, B. 2020, \apj,
897, 120
[Goossens} {et~al.}, 2002]
{Goossens}, M., {Andries}, J., \& {Aschwanden}, M.~J. 2002, \aap, 394, L39
[Goossens} {et~al.}, 2021]
{Goossens}, M., {Chen}, S.~X., {Geeraerts}, M., {Li}, B., \& {Van
Doorsselaere}, T. 2021, \aap, 646, A86
[Goossens} {et~al.}, 1992]
{Goossens}, M., {Hollweg}, J.~V., \& {Sakurai}, T. 1992, \solphys, 138, 233
[Grant} {et~al.}, 2015]
{Grant}, S.~D.~T., {Jess}, D.~B., {Moreels}, M.~G., {et~al.} 2015, \apj, 806,
[Guo} {et~al.}, 2019]
{Guo}, M., {Van Doorsselaere}, T., {Karampelas}, K., {et~al.} 2019, \apj, 870,
[Heyvaerts} \& {Priest}, 1983]
{Heyvaerts}, J. \& {Priest}, E.~R. 1983, \aap, 117, 220
[Hillier} {et~al.}, 2019]
{Hillier}, A., {Barker}, A., {Arregui}, I., \& {Latter}, H. 2019, \mnras, 482,
[Hillier} {et~al.}, 2020]
{Hillier}, A., {Van Doorsselaere}, T., \& {Karampelas}, K. 2020, \apjl, 897,
[Hollweg} {et~al.}, 2013]
{Hollweg}, J.~V., {Kaghashvili}, E.~K., \& {Chandran}, B. D.~G. 2013, \apj,
769, 142
[Hollweg} \& {Yang}, 1988]
{Hollweg}, J.~V. \& {Yang}, G. 1988, \jgr, 93, 5423
[Hollweg} {et~al.}, 1990]
{Hollweg}, J.~V., {Yang}, G., {Cadez}, V.~M., \& {Gakovic}, B. 1990, \apj, 349,
[Karampelas} {et~al.}, 2017]
{Karampelas}, K., {Van Doorsselaere}, T., \& {Antolin}, P. 2017, \aap, 604,
[Karampelas} {et~al.}, 2019]
{Karampelas}, K., {Van Doorsselaere}, T., {Pascoe}, D.~J., {Guo}, M., \&
{Antolin}, P. 2019, Frontiers in Astronomy and Space Sciences, 6, 38
[Karpen} {et~al.}, 1994]
{Karpen}, J.~T., {Dahlburg}, R.~B., \& {Davila}, J.~M. 1994, \apj, 421, 372
[Kelly}, 1965]
{Kelly}, R.~E. 1965, Journal of Fluid Mechanics, 22, 547
[Keys {et~al.}, 2018]
Keys, P.~H., Morton, R.~J., Jess, D.~B., {et~al.} 2018, The Astrophysical
Journal, 857, 28
[Kovitya} \& {Cram}, 1983]
{Kovitya}, P. \& {Cram}, L. 1983, \solphys, 84, 45
[Kumar} {et~al.}, 2013]
{Kumar}, P., {Innes}, D.~E., \& {Inhester}, B. 2013, \apjl, 779, L7
[Magyar} {et~al.}, 2015]
{Magyar}, N., {Van Doorsselaere}, T., \& {Marcu}, A. 2015, \aap, 582, A117
[Mandal} {et~al.}, 2016]
{Mandal}, S., {Magyar}, N., {Yuan}, D., {Van Doorsselaere}, T., \& {Banerjee},
D. 2016, \apj, 820, 13
[McLachlan, 1947]
McLachlan, N. 1947, Theory and Application of Mathieu Functions (Clarendon
[Moreels} {et~al.}, 2015]
{Moreels}, M.~G., {Freij}, N., {Erd{\'e}lyi}, R., {Van Doorsselaere}, T., \&
{Verth}, G. 2015, \aap, 579, A73
[Moreels} {et~al.}, 2013]
{Moreels}, M.~G., {Goossens}, M., \& {Van Doorsselaere}, T. 2013, \aap, 555,
[Morton} {et~al.}, 2011]
{Morton}, R.~J., {Erd{\'e}lyi}, R., {Jess}, D.~B., \& {Mathioudakis}, M. 2011,
\apjl, 729, L18
[Nakariakov} {et~al.}, 2000]
{Nakariakov}, V.~M., {Verwichte}, E., {Berghmans}, D., \& {Robbrecht}, E. 2000,
\aap, 362, 1151
[Nightingale} {et~al.}, 1999]
{Nightingale}, R.~W., {Aschwanden}, M.~J., \& {Hurlburt}, N.~E. 1999, \solphys,
190, 249
[Ofman} {et~al.}, 1994]
{Ofman}, L., {Davila}, J.~M., \& {Steinolfson}, R.~S. 1994, \grl, 21, 2259
[Ofman} \& {Thompson}, 2011]
{Ofman}, L. \& {Thompson}, B.~J. 2011, \apjl, 734, L11
[Parnell} \& {De Moortel}, 2012]
{Parnell}, C.~E. \& {De Moortel}, I. 2012, Philosophical Transactions of the
Royal Society of London Series A, 370, 3217
[Pascoe} {et~al.}, 2020]
{Pascoe}, D.~J., {Goddard}, C.~R., \& {Van Doorsselaere}, T. 2020, Frontiers in
Astronomy and Space Sciences, 7, 61
[Pascoe} {et~al.}, 2012]
{Pascoe}, D.~J., {Hood}, A.~W., {de Moortel}, I., \& {Wright}, A.~N. 2012,
\aap, 539, A37
[Pascoe} {et~al.}, 2010]
{Pascoe}, D.~J., {Wright}, A.~N., \& {De Moortel}, I. 2010, \apj, 711, 990
[Priest, 2014]
Priest, E. 2014, Magnetohydrodynamics of the Sun (Cambridge University Press)
[Roberts}, 1973]
{Roberts}, B. 1973, Journal of Fluid Mechanics, 59, 65
[Sakurai} {et~al.}, 1991]
{Sakurai}, T., {Goossens}, M., \& {Hollweg}, J.~V. 1991, \solphys, 133, 227
[Samanta} {et~al.}, 2019]
{Samanta}, T., {Tian}, H., \& {Nakariakov}, V.~M. 2019, \prl, 123, 035102
[Shi} {et~al.}, 2021]
{Shi}, M., {Van Doorsselaere}, T., {Guo}, M., {et~al.} 2021, arXiv e-prints,
[Simon, 2005]
Simon, B. 2005, Trace Ideals and Their Applications, Mathematical surveys and
monographs (American Mathematical Society)
[Soler} {et~al.}, 2013]
{Soler}, R., {Goossens}, M., {Terradas}, J., \& {Oliver}, R. 2013, \apj, 777,
[Soler} {et~al.}, 2009]
{Soler}, R., {Oliver}, R., {Ballester}, J.~L., \& {Goossens}, M. 2009, \apjl,
695, L166
[Sträng, 2005]
Sträng, J.-E. 2005
[Terradas} {et~al.}, 2008]
{Terradas}, J., {Andries}, J., {Goossens}, M., {et~al.} 2008, \apjl, 687, L115
[Van Doorsselaere} {et~al.}, 2020]
{Van Doorsselaere}, T., {Srivastava}, A.~K., {Antolin}, P., {et~al.} 2020,
\ssr, 216, 140
[Wang}, 2011]
{Wang}, T. 2011, \ssr, 158, 397
[Wang} {et~al.}, 2007]
{Wang}, T., {Innes}, D.~E., \& {Qiu}, J. 2007, \apj, 656, 598
[Wang} {et~al.}, 2002]
{Wang}, T., {Solanki}, S.~K., {Curdt}, W., {Innes}, D.~E., \& {Dammasch}, I.~E.
2002, \apjl, 574, L101
[Wang} {et~al.}, 2003]
{Wang}, T.~J., {Solanki}, S.~K., {Curdt}, W., {et~al.} 2003{\natexlab{a}},
\aap, 406, 1105
[Wang} {et~al.}, 2003]
{Wang}, T.~J., {Solanki}, S.~K., {Innes}, D.~E., {Curdt}, W., \& {Marsch}, E.
2003{\natexlab{b}}, \aap, 402, L17
[Yu} {et~al.}, 2017]
{Yu}, D.~J., {Van Doorsselaere}, T., \& {Goossens}, M. 2017, \aap, 602, A108
[Zaqarashvili} {et~al.}, 2015]
{Zaqarashvili}, T.~V., {Zhelyazkov}, I., \& {Ofman}, L. 2015, \apj, 813, 123
\end{thebibliography}
\appendix
\section{Deriving the governing equation for $\c$} \label{A1}
In this appendix, we derive Eq. \eqref{Eqdivxi} from Eqs. \eqref{eq1}-\eqref{eq4}. Equations \eqref{eq1}, \eqref{eq3} and \eqref{eq4} can be rewritten using $\pmb{v} = \frac{D \pmb{\xi}}{Dt}$ to yield the following expressions for $\rho_1$, $\pmb{B}_1$ and $p_1$:
\begin{align}
\rho_1 &= -\rho_0 \c \label{rho_1}\\
\pmb{B}_1 &= -\pmb{B}_0 \l( \c \r) + \l( \pmb{B}_0 \cdot \nabla \r) \pmb{\xi} \label{B_1}\\
p_1 &= -\rho_0 v_s^2 \c \text{.} \label{p_1}
\end{align}
Using Eq. \eqref{rho_1} and Eq. \eqref{p_1}, Eq. \eqref{eq2} can be rewritten as follows:
\begin{align}
&\o_0 V_0 \sin \l( \o_0 t \r) \l( \c \r) \pmb{1}_z + \d\frac{D^2 \pmb{\xi}}{Dt^2} = \notag\\
& \hspace{3cm} \frac{-1}{\rho_0} \nabla P_1 + i k_z v_A^2 \l[- \pmb{1}_z \l( \c \r) + i k_z \pmb{\xi} \r] \text{,} \label{6bis}
\end{align}
where $v_A = B_{0z} / \sqrt{\mu_0 \rho_0}$ is the Alfv\'en speed. By now taking the divergence on both sides of Eq. \eqref{6bis} we get the following:
\begin{equation}
i k_z \o_0 V_0 \sin \l( \o_0 t \r) \l( \c \r) + \d\frac{D^2 \l( \c \r)}{Dt^2} = \frac{-1}{\rho_0} \nabla^2 P_1 \text{,} \label{7}
\end{equation}
where $\nabla^2$ denotes the Laplace operator.
Next, from the definition of total pressure $P_1 = p_1 + \pmb{B}_0 \cdot \pmb{B}_1 / \mu_0$ we obtain the following equation with the use of Eqs. \eqref{B_1} and \eqref{p_1}:
\begin{equation}
\d\frac{1}{\rho_0} P_1 + (v_A^2 + v_s^2) \l( \c \r) - i k_z v_A^2 \xi_z = 0 \text{.} \label{8}
\end{equation}
Taking the $z$-component of Eq. \eqref{6bis} and using Eq. \eqref{8} yields
\begin{equation}
\d\frac{D^2 \xi_z}{Dt^2} = - \o_0 V_0 \sin \l( \o_0 t \r) \l( \c \r) + i k_z v_s^2 \l( \c \r) \text{,} \label{9}
\end{equation}
while taking twice the Lagrangian derivative on both sides of Eq. \eqref{8} and rearanging the terms yields
\begin{equation}
\d\frac{D^2 \xi_z}{Dt^2} = \frac{1}{i k_z v_A^2 \rho_0} \frac{D^2 P_1}{Dt^2} +\frac{v_A^2 + v_s^2}{i k_z v_A^2} \frac{D^2 \l( \c \r)}{Dt^2} \text{.} \label{10}
\end{equation}
We can then combine Eq. \eqref{9} and Eq. \eqref{10} into the following equation:
\begin{align}
&\d\frac{1}{i k_z v_A^2 \rho_0} \frac{D^2 P_1}{Dt^2} +\frac{v_A^2 + v_s^2}{i k_z v_A^2} \frac{D^2 \l( \c \r)}{Dt^2} = \notag\\
& \hspace{2.5cm} - \o_0 V_0 \sin \l( \o_0 t \r) \l( \c \r) + i k_z v_s^2 \l( \c \r) \text{.} \label{11}
\end{align}
If we now take twice the Lagrangian derivative on both sides of Eq. \eqref{7} and we take the Laplacian on both sides of Eq. \eqref{11}, we can combine the obtained equations, yielding Eq. \eqref{Eqdivxi}:
\begin{align}
&\d\frac{D^4 \l( \c \r)}{Dt^4} + i k_z \o_0 V_0 \frac{D^2 \l( \sin \l(\o_0 t \r) \l( \c \r) \r)}{Dt^2} \notag\\
& \qquad - \l( v_A^2 + v_s^2 \r) \frac{D^2}{Dt^2} \l( \frac{\pa^2 \l( \c \r)}{\pa x^2} \r) + k^2 \l( v_A^2 + v_s^2 \r) \frac{D^2 \l( \c \r)}{Dt^2} \notag\\
& \qquad - i k_z V_0 \o_0 \sin \l( \o_0 t \r) v_A^2 \frac{\pa^2 \l( \c \r)}{\pa x^2} \notag\\
& \qquad + i k_z k^2 V_0 \o_0 \sin \l(\o_0 t \r) v_A^2 \l( \c \r) - k_z^2 v_A^2 v_s^2 \frac{\pa^2 \l( \c \r)}{\pa x^2} \notag \\
& \qquad + k_z^2 k^2 v_A^2 v_s^2 \l( \c \r) = 0 \text{,}
\end{align}
where $k = \sqrt{k_y^2 + k_z^2}$.
\section{Deriving the expression for $\Delta$} \label{ExtPow}
In this appendix, we show formula \eqref{DeltaFormula} for $\Delta$. We recall the definition of $\Delta$, Eq. \eqref{Delta}, for the reader's convenience:
\begin{equation} \label{Delta2}
\Delta = \begin{vmatrix}
\ddots & \ddots & & & 0\\ \ddots & 1 & \varepsilon_{-1} & & \\ & -\varepsilon_0 & 1 & \varepsilon_0 & \\ & & -\varepsilon_1 & 1 & \ddots \\ 0 & & & \ddots & \ddots
\end{vmatrix} \text{.}
\end{equation}
Denoting with $A$ the operator defined by the infinite matrix corresponding to the determinant in the right-hand side of Eq. \eqref{Delta2}, we explained in Section \ref{Gfunction} that $A-I$ (with $I$ the identity operator) is a trace class operator on $\ell^2(\mathbb{Z})$ (except in the poles of the $\varepsilon_j$). This is the Hilbert space of square-summable sequences of complex numbers with entire index, and with inner product $\langle \cdot, \cdot \rangle$ defined by
\begin{equation}
\langle x, y \rangle = \d\sum_{l=-\infty}^{\infty} x(l) \overline{y(l)}
\end{equation}
for $x,y \in \ell^2(\mathbb{Z})$. To work out the right-hand side of Eq. \eqref{Delta2}, we use a result from Fredholm theory. It states that, for a trace class operator $M$, the following holds:
\begin{equation} \label{FredholmDet}
\det(I+M) \; = \; \d\sum_{n=0}^{\infty} \text{Tr} \l( \Lambda^n \l( M \r) \r) \text{,}
\end{equation}
where $\text{Tr} \l( \Lambda^n \l( M \r) \r)$ is the trace of the $n$th exterior power of $M$ [Simon, 2005].
The $n$th exterior power $\Lambda^n (\ell^2(\mathbb{Z}))$ (with $n \in \mathbb{N}_0$) of $\ell^2(\mathbb{Z})$ is the Hilbert space whose elements are of the form $v_1 \wedge ... \wedge v_n$, that is to say, the exterior product of $v_1$, ..., $v_n$ (in that order) where $v_i \in \ell^2(\mathbb{Z})$ for every $i \in \{1, ..., n \}$, and with a uniquely defined inner product $\langle \cdot , \cdot \rangle^{\otimes n}$ that is determined by the inner product on $\ell^2(\mathbb{Z})$. The vector space $\Lambda^n (\ell^2(\mathbb{Z}))$ is the subspace of the tensor product $\ell^2(\mathbb{Z}) \otimes ... \otimes \ell^2(\mathbb{Z})$ (n times) whose elements are totally antisymmetric tensors of rank $n$. Furthermore, the 0th exterior power $\Lambda^0 (\ell^2(\mathbb{Z}))$ of $\ell^2(\mathbb{Z})$ is $\mathbb{C}$.
For a trace class operator $M$ on $\ell^2(\mathbb{Z})$, the $n$th exterior power of $M$ is defined by
\begin{align}
\Lambda^n (M): \; &\Lambda^n ( \ell^2 (\mathbb{Z})) \to \Lambda^n ( \ell^2 (\mathbb{Z})): \notag\\
&\hspace{2cm} v_1 \wedge ... \wedge v_n \mapsto M v_1 \wedge ... \wedge M v_n \label{Lambda^n(M)}
\end{align}
and is a trace class operator on $\Lambda^n (\ell^2(\mathbb{Z}))$. Now, if $\{e_j \}_{j \in \mathbb{Z}}$ is an orthonormal basis for a Hilbert space $H$, then the trace of a trace class operator $M$ on $H$ is defined as [Conway, 1990]:
\begin{equation} \label{trace}
\text{Tr}(M) = \d\sum_{j=-\infty}^{\infty} \langle M e_j, e_j \rangle \text{.}
\end{equation}
We also have that, if $\{ e_j \}_{j \in \mathbb{Z}}$ is an orthonormal basis of $\ell^2(\mathbb{Z})$, then $\{e_{j_1} \wedge ... \wedge e_{j_n} \}_{j_1 < \cdots < j_n}$ is an orthonormal basis of $\Lambda^n ( \ell^2 (\mathbb{Z}))$ and
\begin{equation} \label{innprodextprod}
\d \langle e_{j_1} \wedge ... \wedge e_{j_n}, f_{1} \wedge ... \wedge f_{n} \rangle^{\otimes n} = \det \l(\langle e_{j_k}, f_m \rangle_{1 \leq k,m \leq n} \r) \text{,}
\end{equation}
for any $f_{1} \wedge ... \wedge f_{n} \in \Lambda^n ( \ell^2 (\mathbb{Z}))$ [Simon, 2005]. Hence, from Eqs. \eqref{Lambda^n(M)}, \eqref{trace} and \eqref{innprodextprod}, we find for the trace class operator $\Lambda^n \l( M \r)$ on $\Lambda^n (\ell^2(\mathbb{Z}))$ that
\begin{align}
&\text{Tr} \l(\Lambda^n \l( M \r) \r) \notag\\
&= \d\sum_{j_1 = -\infty}^{\infty} \sum_{j_2 = j_1 + 1}^{\infty} ... \sum_{j_n = j_{n-1} + 1}^{\infty} \ip{ \Lambda^n \l( M \r) \l( e_{j_1} \wedge ... \wedge e_{j_n} \r) , e_{j_1} \wedge ... \wedge e_{j_n} }^{\otimes n} \notag \\
&= \d\sum_{j_1 = -\infty}^{\infty} \sum_{j_2 = j_1 + 1}^{\infty} ... \sum_{j_n = j_{n-1} + 1}^{\infty} \ip{ M e_{j_1} \wedge ... \wedge M e_{j_n} , e_{j_1} \wedge ... \wedge e_{j_n} }^{\otimes n} \notag \\
&= \d\sum_{j_1 = -\infty}^{\infty} \sum_{j_2 = j_1 + 1}^{\infty} ... \sum_{j_n = j_{n-1} + 1}^{\infty} \begin{vmatrix} a_{1,1} & \cdots & a_{1,n} \\ \vdots & & \vdots\\ a_{n,1} & \cdots & a_{n,n} \end{vmatrix} \text{,} \label{TrExpr1}
\end{align}
where $a_{k,m} = \langle M e_{j_k}, e_{j_m} \rangle$ for $k,m \in \mathbb{N}_0$.
Looking at Eq. \eqref{Delta2}, we see that the trace class operator $M=A-I$ is defined by
\begin{equation}
M : \ell^2\l(\mathbb{Z} \r) \to \ell^2\l(\mathbb{Z} \r) : \begin{pmatrix} \vdots \\ \varphi(l) \\ \vdots \end{pmatrix} \mapsto \begin{pmatrix} \vdots \\ \varepsilon_l \l[ \varphi \l( l+1 \r) - \varphi \l( l-1 \r) \r] \\ \vdots \end{pmatrix} \text{.}
\end{equation}
For the orthonormal basis $\{ e_j \}_{j \in \mathbb{Z}}$ of $\ell^2(\mathbb{Z})$ defined by $e_j(l) = \delta_{jl}$ for all $j,l \in \mathbb{Z}$, we then have the following:
\begin{equation}
M e_{j_k} = M \begin{pmatrix} \vdots \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ \vdots \end{pmatrix} = \begin{pmatrix} \vdots \\ 0 \\ \varepsilon_{j_k-1} \\ 0 \\ -\varepsilon_{j_k+1} \\ 0 \\ \vdots \end{pmatrix} \text{,}
\end{equation}
for every $k \in \mathbb{N}_0$. This means that
\begin{align}
\ip{M e_{j_k},e_{j_m}} &= \d\sum_{l=-\infty}^{\infty} \l(M e_{j_k} \r)(l) \; \overline{e_{j_m}(l)} \notag \\
&= \begin{cases} \varepsilon_{j_k-1} & \text{if } j_m = j_k -1 \\ - \varepsilon_{j_k+1} & \text{if } j_m = j_k+1 \\ 0 & \text{else}\end{cases} \text{.} \label{innprod}
\end{align}
We thus find that $\langle M e_{j_k}, e_{j_m} \rangle = 0$ if $\abs{k-m} \ge 2$ (since $j_1 < ... < j_n$) or $k=m$. Hence, from Eq. \eqref{TrExpr1},
\begin{equation} \label{traceExpr2}
\text{Tr} \l(\Lambda^n \l( M \r) \r) = \d\sum_{j_1 = -\infty}^{\infty} \sum_{j_2 = j_1 + 1}^{\infty} \ldots \sum_{j_n = j_{n-1} + 1}^{\infty} \Delta_{j_1,...,j_n} \text{,}
\end{equation}
where we define for any $q \in \mathbb{N}_0$ and $s_1,..., s_q \in \mathbb{N}_0$
\begin{equation} \label{Delta_js}
\Delta_{j_{s_1},...,j_{s_q}} = \begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes,nodes in empty cells,right delimiter={|},left delimiter={|} ]{
0 & a_{s_1,s_2} & 0 & & 0 \\
a_{s_2,s_1} & & & & \\
0 & & & & 0 \\
& & & &a_{s_{q-1},s_q} \\
0 & & 0 & a_{s_q,s_{q-1}} & 0 \\
} ;
\draw[loosely dotted][thick] (m-1-1)-- (m-5-5);
\draw[loosely dotted][thick] (m-1-2)-- (m-4-5);
\draw[loosely dotted][thick] (m-2-1)-- (m-5-4);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-1);
\draw[loosely dotted][thick] (m-5-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-1-3)-- (m-1-5);
\draw[loosely dotted][thick] (m-1-5)-- (m-3-5);
\draw[loosely dotted][thick] (m-1-3)-- (m-3-5);
\end{tikzpicture} \text{.}
\end{equation}
We can now rewrite the determinants $\Delta_{j_1,...,j_n}$ composing the terms of the sum in Eq. \eqref{traceExpr2}:
\begin{eqnarray}
\Delta_{j_1,...,j_n} &=&\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes,nodes in empty cells,right delimiter={|},left delimiter={|} ]{
0 & a_{1,2} & 0 & & 0 \\
a_{2,1} & & & & \\
0 & & & & 0 \\
& & & &a_{n-1,n} \\
0 & & 0 & a_{n,n-1} & 0 \\
} ;
\draw[loosely dotted][thick] (m-1-1)-- (m-5-5);
\draw[loosely dotted][thick] (m-1-2)-- (m-4-5);
\draw[loosely dotted][thick] (m-2-1)-- (m-5-4);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-1);
\draw[loosely dotted][thick] (m-5-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-1-3)-- (m-1-5);
\draw[loosely dotted][thick] (m-1-5)-- (m-3-5);
\draw[loosely dotted][thick] (m-1-3)-- (m-3-5);
\end{tikzpicture} \label{TrPowerprev} \\
&=& - a_{1,2} \begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes,nodes in empty cells,right delimiter={|},left delimiter={|} ]{
a_{2,1} & a_{2,3} & 0 & & & 0 \\
0 & 0 & a_{3,4} & 0 & & 0 \\
& a_{4,3} & & & & \\
& 0 & & & & 0 \\
& & & & &a_{n-1,n} \\
0 & 0 & & 0 & a_{n,n-1} & 0 \\
} ;
\draw[loosely dotted][thick] (m-2-2)-- (m-6-6);
\draw[loosely dotted][thick] (m-2-3)-- (m-5-6);
\draw[loosely dotted][thick] (m-3-2)-- (m-6-5);
\draw[loosely dotted][thick] (m-4-2)-- (m-6-2);
\draw[loosely dotted][thick] (m-6-2)-- (m-6-4);
\draw[loosely dotted][thick] (m-4-2)-- (m-6-4);
\draw[loosely dotted][thick] (m-2-4)-- (m-2-6);
\draw[loosely dotted][thick] (m-2-6)-- (m-4-6);
\draw[loosely dotted][thick] (m-2-4)-- (m-4-6);
\draw[loosely dotted][thick] (m-2-1)-- (m-6-1);
\draw[loosely dotted][thick] (m-1-3)-- (m-1-6);
\end{tikzpicture} \notag \\
&=& - a_{1,2} a_{2,1}\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes,nodes in empty cells,right delimiter={|},left delimiter={|} ]{
0 & a_{3,4} & 0 & & 0 \\
a_{4,3} & & & & \\
0 & & & & 0 \\
& & & &a_{n-1,n} \\
0 & & 0 & a_{n,n-1} & 0 \\
} ;
\draw[loosely dotted][thick] (m-1-1)-- (m-5-5);
\draw[loosely dotted][thick] (m-1-2)-- (m-4-5);
\draw[loosely dotted][thick] (m-2-1)-- (m-5-4);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-1);
\draw[loosely dotted][thick] (m-5-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-3-1)-- (m-5-3);
\draw[loosely dotted][thick] (m-1-3)-- (m-1-5);
\draw[loosely dotted][thick] (m-1-5)-- (m-3-5);
\draw[loosely dotted][thick] (m-1-3)-- (m-3-5);
\end{tikzpicture} \notag \\
&=& - a_{1,2} a_{2,1} \Delta_{j_3,...,j_n} \label{TrPower} \text{,}
\end{eqnarray}
where we expanded along the first row to find the second equality and along the first column to find the third equality. Clearly, we can omit terms that are $0$ from the sum in the right-hand side of Eq. \eqref{traceExpr2}. Now, the factors $a_{1,2}$ and $a_{2,1}$ are different from $0$ only if $j_2 = j_1+1$, in which case
\begin{align*}
&a_{1,2} = \ip{ M e_{j_1}, e_{j_1+1}} = - \varepsilon_{j_1 + 1} \text{, and}\\
& a_{2,1} = \ip{ M e_{j_1+1}, e_{j_1}} = \varepsilon_{j_1} \text{.}
\end{align*}
This means that, from Eqs. \eqref{traceExpr2}, \eqref{Delta_js} and \eqref{TrPower}, we have
\begin{equation} \label{traceExpr3}
\text{Tr} \l(\Lambda^n \l( M \r) \r) = \d\sum_{j_1 = -\infty}^{\infty} \varepsilon_{j_1} \varepsilon_{j_1 + 1} \sum_{j_3 = j_1 + 2}^{\infty} \sum_{j_4 = j_3 + 1}^{\infty} \ldots \sum_{j_n = j_{n-1} + 1}^{\infty} \Delta_{j_3,...,j_n} \text{.}
\end{equation}
We can now make a distinction between even and odd $n$. Indeed, if $n = 2p$ for a $p \in \mathbb{N}_0$, we find the following by continuing to expand every determinant of the form \eqref{Delta_js} in the sum on the right-hand side of Eq. \eqref{traceExpr3} in the same way as we went from Eq. \eqref{TrPowerprev} to \eqref{TrPower}:
\begin{equation} \label{TraceEven}
\text{Tr} \l(\Lambda^{2p} \l( M \r) \r) = \d\sum_{j_1 = - \infty}^{\infty} \d\sum_{j_2 = j_1 + 2}^{\infty} \ldots \d\sum_{j_p = j_{p-1} + 2}^{\infty} \l( \d\prod_{l=1}^{p} \varepsilon_{j_l} \varepsilon_{j_l +1} \r) \text{.}
\end{equation}
In contrast to this, if $n=2p+1$ for a $p \in \mathbb{N}$, we find
\begin{align}
\text{Tr} \l(\Lambda^{2p+1} \l( M \r) \r) &= \d\sum_{j_1 = - \infty}^{\infty} \d\sum_{j_2 = j_1 + 2}^{\infty} \ldots \d\sum_{j_p = j_{p-1} + 2}^{\infty} \l( \d\prod_{l=1}^{p} \varepsilon_{j_l} \varepsilon_{j_l +1} \r) 0 \notag\\
&= 0 \text{.} \label{TrUneven}
\end{align}
Hence, from Eqs. \eqref{FredholmDet}, \eqref{TraceEven} and \eqref{TrUneven}, and from the fact that $\text{Tr} \l(\Lambda^0 \l( M \r) \r) = 1$ [Simon, 2005], we now find Eq. \eqref{DeltaFormula}:
\begin{equation}
\Delta \; = \; 1 + \d\sum_{n=1}^{\infty} \l[ \d\sum_{j_1 = - \infty}^{\infty} \d\sum_{j_2 = j_1 + 2}^{\infty} \ldots \d\sum_{j_n = j_{n-1} + 2}^{\infty} \l( \d\prod_{l=1}^n \varepsilon_{j_l} \varepsilon_{j_l +1} \r) \r] \text{.}
\end{equation}
\section{Deriving solutions for $G_1$, $G_2$ and $G_3$} \label{AppendixG1G2G3}
In this appendix, we show how to derive the solutions to the temporal functions $G_1$, $G_2$ and $G_3$. We recall the equations of interest to this matter, Eqs. \eqref{eq1}-\eqref{eq3}, Eqs. \eqref{xix}-\eqref{xiz} and Eq. \eqref{P1} for the reader's convenience:
\begin{align}
&\d\frac{D \rho_1}{D t} +\rho_0 \l( \nabla \cdot \pmb{v}_1 \r) = 0\text{,} \label{B1}\\
& \rho_1 \frac{\partial \pmb{v}_0}{\partial t} + \rho_0 \frac{D \pmb{v}_1}{D t} = -\nabla P_1 + \frac{1}{\mu_0} \left( \pmb{B}_0 \cdot \nabla \right) \pmb{B}_1 \text{,} \label{B2}\\
&\frac{D \pmb{B}_1}{D t} = - \pmb{B}_0 \left( \nabla \cdot \pmb{v}_1 \right) + \left( \pmb{B}_0 \cdot \nabla \right) \pmb{v}_1 \text{,} \label{B3}\\
&\xi_x = - \d\frac{i m}{k^2 + m^2} \; \tilde{F}(x)\; h_1(t) \; \e \text{,} \label{B5}\\
&\xi_y = - \d\frac{i k_y}{k^2 + m^2} \; F(x)\; h_2(t) \; \e \text{,} \label{B6}\\
&\xi_z = - \d\frac{i k_z}{k^2 + m^2} \; F(x)\; h_3(t) \; \e \text{,} \label{B7}\\
&P_1 = \rho_0 \; \l( \d\frac{k_z^2 v_A^2}{k^2 + m^2} h_3(t) \; - \; \l( v_A^2 + v_s^2 \r) h(t) \r) \notag\\
& \hspace{4cm} F(x) \; \e \text{.} \label{B8}
\end{align}
We recall Eq. \eqref{6bis} from Appendix \ref{A1}:
\begin{align}
&\o_0 V_0 \sin \l( \o_0 t \r) \l( \c \r) \pmb{1}_z + \d\frac{D^2 \pmb{\xi}}{Dt^2} = \notag\\
& \hspace{3cm} \frac{-1}{\rho_0} \nabla P_1 + i k_z v_A^2 \l[- \pmb{1}_z \l( \c \r) + i k_z \pmb{\xi} \r] \text{.} \label{B9}
\end{align}
Replacing $h_i(t)$ by their definition $G_i(t) g(t)$ for every $i \in \{1,2,3 \}$ (where $g(t) = \exp \l\{ \frac{-i k_z V_0 \sin \l( \o_0 t \r)}{\o_0} \r\}$) and inserting Eqs. \eqref{B5}-\eqref{B8} into the $x$-component, $y$-component and $z$-component of Eq. \eqref{B9}, we find that the following equations have to hold:
\begin{align}
&\d\frac{d^2 G_1(t)}{dt^2} + K^2 \l(v_A^2 + v_s^2 \r) G(t) + k_z^2 v_A^2 \l[ G_1(t) - G_3(t) \r] = 0\text{,} \label{B.G1}\\
&\d\frac{d^2 G_2(t)}{dt^2} + K^2 \l(v_A^2 + v_s^2 \r) G(t) + k_z^2 v_A^2 \l[ G_2(t) - G_3(t) \r] = 0\text{,} \label{B.G2}\\
&\d\frac{d^2 G_3(t)}{dt^2} + K^2 \l(v_s^2 + \d\frac{i \o_0 V_0}{k_z} \sin \l( \o_0 t \r) \r) G(t) = 0 \text{,} \label{B.G3}
\end{align}
with $K = \sqrt{k^2 + m^2}$.
We see that $G_1$ and $G_2$ obey the same differential equation and that they thus have the same general solution. They depend on $G_3$, which by solving Eq. \eqref{B.G3} can be found to be equal to
\begin{equation} \label{B.G3eq}
G_3(t) = - \d\frac{K^2}{k_z} \d\iint \l(i \o_0 V_0 \sin \l( \o_0 t \r) + k_z v_s^2 \r) G(t) \; dt \; dt \text{.}
\end{equation}
We recall that we defined $\t = \o_0 t$. So, since $G(\t) = e^{\mu \t} \sum_{j = -\infty}^{\infty} \varphi_j e^{i j \t}$, we find that
\begin{align}
G_3(\t) &= -\d\frac{K^2}{2 k_z \o_0^2} e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \d\frac{V_0 \o_0 \l( \varphi_{j+1} - \varphi_{j-1} \r) - 2 k_z v_s^2 \varphi_j}{\l( j + \nu \r)^2} \; e^{i j \t} \notag \\
&= \d\frac{K^2}{k_z^2} e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \l(1 - \d\frac{\l( k_y^2+ m^2 \r) v_s^2}{\l(j + \nu \r)^2 \o_0^2 - K^2 v_A^2} \r) \; \varphi_j \; e^{i j \t} \text{,} \label{B.G3exp}
\end{align}
where we used Eq. \eqref{recursion} to find the second equality.
Now, solving Eq. \eqref{B.G1}, we find that
\begin{align}
G_1(t) &= \d\frac{1}{k_z v_A} \Bigg\{ \cos \l( k_z v_A t \r) \notag \\
& \qquad \l[ \d \int \sin \l( k_z v_A t \r) \l( K^2 \l( v_A^2 + v_s^2 \r) G(t) - k_z^2 v_A^2 G_3(t) \r) dt \r] \notag\\
& \quad- \sin \l( k_z v_A t \r) \notag \\
& \qquad \l[ \d \int \cos \l( k_z v_A t \r) \l( K^2 \l( v_A^2 + v_s^2 \r) G(t) - k_z^2 v_A^2 G_3(t) \r) dt \r] \Bigg\} \text{.} \label{B.G1eq}
\end{align}
Having derived expression \eqref{B.G3exp}, we can also work out Eq. \eqref{B.G1eq} and find the following solution for $G_1$:
\begin{equation} \label{B.G1exp}
G_1(\t) = K^2 v_s^2 \; e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \d\frac{1}{\l(j + \nu \r)^2 \o_0^2 - K^2 v_A^2} \; \varphi_j \; e^{i j \t} \text{.}
\end{equation}
\begin{equation} \label{B.G1exp}
G_2(\t) = K^2 v_s^2 \; e^{\mu \t} \d\sum_{j = -\infty}^{\infty} \d\frac{1}{\l(j + \nu \r)^2 \o_0^2 - K^2 v_A^2} \; \varphi_j \; e^{i j \t} \text{.}
\end{equation}
% [Baker, 1966] Baker, N. 1966,
% in Stellar Evolution,
% ed.\ R. F. Stein,\& A. G. W. Cameron
% (Plenum, New York) 333
% [Balluch, 1988] Balluch, M. 1988,
% A\&A, 200, 58
% [Cox, 1980] Cox, J. P. 1980,
% Theory of Stellar Pulsation
% (Princeton University Press, Princeton) 165
% [Cox, 1969] Cox, A. N.,\& Stewart, J. N. 1969,
% Academia Nauk, Scientific Information 15, 1
% [Mizuno, 1980] Mizuno H. 1980,
% Prog. Theor. Phys., 64, 544
% [Tscharnuter, 1987] Tscharnuter W. M. 1987,
% A\&A, 188, 55
% [Terlevich, 1992] Terlevich, R. 1992, in ASP Conf. Ser. 31,
% Relationships between Active Galactic Nuclei and Starburst Galaxies,
% ed. A. V. Filippenko, 13
% [Yorke, 1980] Yorke, H. W. 1980a,
% A\&A, 86, 286
% [Zheng, 1997] Zheng, W., Davidsen, A. F., Tytler, D. \& Kriss, G. A.
% 1997, preprint
\end{document}
|
# Pervasive beyond room-temperature ferromagnetism in a doped van der Waals
magnet
Xiang Chen<EMAIL_ADDRESS>Materials Sciences Division, Lawrence
Berkeley National Lab, Berkeley, California 94720, USA Physics Department,
University of California, Berkeley, California 94720, USA Yu-Tsun Shao
School of Applied and Engineering Physics, Cornell University, Ithaca, New
York 14853, USA Rui Chen Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, USA Materials Sciences
Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
Sandhya Susarla Department of Materials Science and Engineering, University
of California, Berkeley, California 94720, USA Materials Sciences Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA Tom Hogan
Quantum Design, Inc., San Diego, CA 92121, USA Yu He Department of Applied
Physics, Yale University, New Haven, Connecticut, 06511, USA Physics
Department, University of California, Berkeley, California 94720, USA
Materials Sciences Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA Hongrui Zhang Department of Materials Science and
Engineering, University of California, Berkeley, California 94720, USA Siqi
Wang NSF Nanoscale Science and Engineering Center (NSEC), 3112 Etcheverry
Hall, University of California, Berkeley, California 94720, USA Jie Yao
Department of Materials Science and Engineering, University of California,
Berkeley, California 94720, USA Materials Sciences Division, Lawrence
Berkeley National Lab, Berkeley, California 94720, USA Peter Ercius The
Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA David A. Muller School of Applied and Engineering Physics, Cornell
University, Ithaca, New York 14853, USA Kavli Institute at Cornell for
Nanoscale Science, Cornell University, Ithaca, New York 14853, USA
Ramamoorthy Ramesh Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, USA Materials Sciences
Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
Physics Department, University of California, Berkeley, California 94720, USA
Robert J. Birgeneau<EMAIL_ADDRESS>Physics Department, University of
California, Berkeley, California 94720, USA Materials Sciences Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA Department of
Materials Science and Engineering, University of California, Berkeley,
California 94720, USA
###### Abstract
The existence of long range magnetic order in low dimensional magnetic
systems, such as the quasi-two-dimensional (2D) van der Waals (vdW) magnets,
has attracted intensive studies of new physical phenomena. The vdW FeNGeTe2
($N$ = 3, 4, 5; FGT) family is exceptional owing to its vast tunability of
magnetic properties. In particular, a ferromagnetic ordering temperature
($T_{\text{C}}$) above room temperature at $N$ = 5 (F5GT) is observed. Here,
our study shows that, by nickel (Ni) substitution of iron (Fe) in F5GT, a
record high $T_{\text{C}}$ = 478(6) K is achieved. Importantly, pervasive,
beyond-room-temperature ferromagnetism exists in almost the entire doping
range of the phase diagram of Ni-F5GT. We argue that this striking observation
in Ni-F5GT can be possibly due to several contributing factors, including
increased 3D magnetic couplings due to the structural alterations.
## I Introduction
Spontaneous symmetry breaking is forbidden at non-zero temperatures in
isotropic spin systems with dimensions $d$ $\leq$ 2 [1, 2]. Long range
magnetic order in materials with reduced dimensionality can still be
stabilized via both magnetic anisotropy and weak three-dimensional (3D)
magnetic couplings [3, 4, 5]. In quasi-two-dimensional (quasi-2D) Heisenberg
magnets, such as the van der Waals (vdW) bonded materials, the ordering
process typically occurs as follows [6, 7, 8]. At the highest temperature, the
system is expected to exhibit 2D, classical isotropic magnetic correlations.
As the temperature $T$ is lowered, the correlation length grows exponentially
with $1/T$ and at sufficiently large length scales there is inevitably a
crossover from 2D Heisenberg to 3D Ising or XY behavior followed by a phase
transition to the 3D long range magnetic order. The details of the crossover
depend on the strength of the 3D magnetic interactions and the symmetry and
strength of the magnetic anisotropy. Van der Waals materials represent
exciting realizations of these phenomena plus they contain the broad prospect
of important technological applications [9, 10, 11, 12, 13, 14, 15].
Among the prominent bulk vdW materials for studying quasi-2D magnetism, such
as Cr2Ge2Te6 [16, 17], CrI3 [18, 19], Fe3GeTe2 (F3GT) [20, 21, 22, 23, 24, 25]
and CrTe2 [26, 27], the F3GT system is exceptional owing to the coupling
between the electronic and magnetic degrees of freedom and its remarkable
tunability. The bulk form of F3GT has a ferromagnetic (FM) transition
temperature $T_{\text{C}}$ $\sim$ 230 K, which can be readily enhanced up to
room temperature (RT), by either patterning the microstructure or applying
ionic gating [28, 29]. By intercalating more iron (Fe) into F3GT, $i.e.$,
FeNGeTe2 ($N$=4 for F4GT or 5 for F5GT) [30, 31, 32], the marked effects are
the elevated $T_{\text{C}}$ up to RT and enhanced magnetization while only
moderately increasing the magnetic moment size [30, 32, 33]. The F5GT compound
is particularly interesting because of its advantageous characteristics for
potential RT spintronic applications, including a $T_{\text{C}}$ above RT
($\sim$315 K), large magnetization ($\sim$700 kA/m), strong spin lattice
coupling [32, 33] and exotic magnetic textures [34, 35].
To achieve practical applications of the quasi-2D magnetic materials, one
common theme is to strengthen the FM and enhance the $T_{\text{C}}$ of the vdW
magnets. Some proven avenues to dramatically raise the $T_{\text{C}}$ include
gating [28, 37], applying pressure or strain [38], ion intercalation or
carrier doping [39, 40]. By cobalt (Co) substitution of F5GT, $i.e.$
$({\text{Fe}}_{1-x}{\text{Co}}_{x})_{5+\delta}{\text{GeTe}}_{2}$ (Co-F5GT),
the magnetic ordering temperature is further increased to $\sim$ 360 K, along
with the evolution of the magnetic ground state [41, 42, 36,
Zhang_2022_FCGT_sciadv]. More strikingly, at $x$ = 0.5 of Co-F5GT, a novel
wurtzite-type polar magnetic metal was discovered, along with the
N$\acute{\text{e}}$el-type skyrmion lattice at RT [36,
Zhang_2022_FCGT_sciadv]. The aforementioned discoveries of the F5GT system
highlight its immense tunability and capacity for applications in next-
generation spintronics.
In this letter, we report pervasive, well-beyond-room-temperature FM in the
F5GT vdW magnet with nickel (Ni) substitution, $i.e.$,
$({\text{Fe}}_{1-x}{\text{Ni}}_{x})_{5+\delta}{\text{GeTe}}_{2}$ (Ni-F5GT).
Strikingly, a record high $T_{\text{C}}$ = 478(6) K is reached at a Ni doping
level of $x=0.36(2)$. In addition, the FM order persists robustly against Ni
replacement until $x$ $\sim$ 0.86(1), beyond which only weak paramagnetism
exists. Several factors might be relevant for the dramatic enhancement of the
FM in Ni-F5GT.
Figure 1: (Color online) (a) Illustration of the AA-stacking order of both the
Fe-rich and Ni-rich domains in Ni doped Fe5GeTe2 (Ni-F5GT) (Green symbols: Fe;
Orange: Ni; Purple: Te; Blue: Ge) [43]. (b) The (0 0 L) type of peaks of
Ni-F5GT at select Ni doping levels $x$. (c) Effective layer spacing $d$ as a
function of Ni doping (colored symbols) or Co doping (gray symbols, [36]) in
Fe5+δGeTe2. Vertical dashed lines indicate different regions in Ni-F5GT.
The Ni-F5GT single crystals were grown by the chemical vapor transfer method
[32, 33, 43]. The exact Ni doping level $x$ and total cation count per formula
unit ($f.u.$) $5+\delta$ were verified by energy-dispersive X-ray spectroscopy
(EDX/EDS) (Fig. S1). The lattice and atomic structure of Ni-F5GT were
investigated by a combination of techniques, including powder and/or single
crystal x-ray diffraction (XRD) and high-angle annular dark-field scanning
transmission electron microscopy (HAADF-STEM). Magnetization measurements were
performed via a commercial Quantum Design MPMS3.
Figure 2: (Color online) Nanoscale phase separation of Ni-F5GT at $x$ =
0.36(2). (a) Energy-dispersive X-ray spectroscopy map showing both the Fe-rich
(green) and Ni-rich (red) regions. (b)-(d) Atomic resolution HAADF-STEM image
demonstrating the two types of domains. The enlarged (c) Fe-rich and (d) Ni-
rich domains show flat and rumpled atomic planes, respectively. The Te-Te
planes in (b) are color-coded based on the number and the rumpling of the
atomic layers. Scale bars in (c)-(d), 5 Å.
The unit cell of F5GT is composed of three identical layers (stacks) with the
rhombohedral layer stacking (space group R$\bar{3}$m), labelled as ABC-
stacking [32]. Upon introducing Ni into F5GT, such as at $x$ = 0.19(1), the
crystal structure undergoes a transition from ABC-stacking to AA-stacking
(space group P$\bar{3}$m1, illustrated in Fig. 1(a)), as confirmed by single
crystal XRD (Fig. S2). The AA-stacking order is also verified by HAADF-STEM
images at other doping levels (Fig. 2 and Fig. S5). When viewed along the [1 1
0] direction, the identical layers stack exactly on top of each other along
the $c$ direction and are separated by a vdW spatial gap. Within each layer
(stack), the center germanium (Ge, blue symbols) atom is surrounded by three
different sites of iron (Fe1, Fe2 and Fe3, green) atoms which are further
protected by the outer Te (purple) atoms, $i.e.$, forming a Te-Fe1-Fe3-Fe2-Ge-
Fe2-Fe3-Fe1-Te plane-like (labelled as TGT-$plane$) structure.
With increasing Ni doping in Ni-F5GT, the excess cation count $\delta$
gradually increases from $\delta\sim$ 0 at $x$ = 0 to $\delta$ $\sim$ 0.5 at
$x$ $\sim$ 0.3 (labelled as region (I) with 0 $\leq$ $x$ $\leq$ 0.3), and
saturates when $x$ $>$ 0.3 (Fig. S1). Interestingly, in the intermediate
doping range 0.36(2) $\leq$ $x$ $\leq$ 0.86(1) (region (II)), although the
sample still maintains the AA-stacking, two types of domains coexist, as
evidenced from the splitting of the (0 0 $L$) peaks from the XRD measurements
(Fig. 1(b)), the direct atomic visualization from the HAADF-STEM images (Fig.
2 and Fig. S5) and the STEM-EDX maps (Fig. 2 and Fig. S4). As an example at
$x$ = 0.36(2) (Fig. 2 and illustrated in Fig. 1(a)), the two types of domains
correspond to Fe-rich domains (Fig. 2(c)) and Ni-rich domains (Fig. 2(d)),
respectively. In the Fe-rich domains, the TGT-$planes$ are almost flat and
more extended in space; while the TGT-$planes$ are rumpled in the Ni-rich
domains. Because of the contrasting local atomic arrangements of the Fe and Ni
atoms within the TGT-$planes$, the effective layer spacing $d$ of either the
Fe-rich or Ni-rich domain shows a strong deviation from the value at $x$ = 0
(Fig. 1(c)). This strong alteration may have a dramatic impact on the
electronic and/or magnetic properties of Ni-F5GT.
While the Ni-F5GT samples are still metallic (Fig. S6), the magnetic
properties are strongly influenced by Ni substitution (Fig. 3 and Fig. S7).
With moderate Ni replacement, the $T_{C}$ of Ni-F5GT is already considerably
enhanced (Figs. 3(a)-3(b)). For instance, at $x$ = 0.19(1), the experimentally
determined $T_{C}$ = 395(6) K already approaches 400 K (Fig. 3(a)). Meanwhile,
the Ising spin moment switches to the in-plane direction and seems to remain
so until $x$ $\sim$ 0.86(1) (Fig. 3(e) and Fig. S7). Upon further increasing
the Ni content, a record high $T_{C}$ $=$ 478(6) K is achieved at $x$ =
0.36(2), together with the maximum in-plane ($H_{C}^{ab}$ $\approx$ 500 Oe)
and out-of-plane ($H_{C}^{c}$ $\approx$ 1600 Oe) coercive fields (Figs. 3(c),
3(f) and Fig. S7). After reaching the maximum $T_{C}$ in Ni-F5GT, the FM order
is only gradually weakened by further Ni doping. Strikingly, even at $x$ =
0.86(1), the as-grown sample of Ni-F5GT maintains an above-room-temperature
$T_{C}$ $\approx$ 380 K, which is only lowered and stabilized at 220 K or 150
K, depending on how the sample is thermally cycled above its original $T_{C}$
(Fig. S8). Only beyond this point of further Ni replacement (0.86(1) $<$ $x$
$\leq$ 1, region (III)), the FM order of Ni-F5GT is completely suppressed,
rendering the weak paramagnetism, accompanied by the adoption of a different,
layered tetragonal structure (space group I4/mmm) as Ni5.5GeTe2 [44].
The saturation magnetic moment per $f.u.$ of Ni-F5GT at 2 K decreases
approximately linearly with increasing Ni content (Figs. 3(c),(e) and Fig.
S7), from $\sim$10 $\mu_{B}$/$f.u.$ at $x$ = 0 to nearly zero (weakly
paramagnetic) at $x$ = 1. This implies that the Ni dopants are not magnetic
and only dilute the FM in Ni-F5GT. Surprisingly, the magnetic ordering
temperature $T_{C}$ does not follow this expected trend (Fig. 4). Instead,
pervasive, above-room-temperature FM exists almost over the entire range of
the phase diagram (0 $\leq$ $x$ $\leq$ 0.79(1)). Particularly, in region (I)
of the phase diagram of Ni-F5GT, the $T_{C}$ shows a strong, positive
deviation from the Vegard’s Law behavior ($i.e.$, a dilution of the magnetic
component), thus suggesting additional factors might be present for the
unusual evolution of $T_{C}$. Our study demonstrates the remarkable Ni
enhanced FM in Ni-F5GT, as summarized in Fig. 4, where three different regions
are categorized based on the structural and magnetic characterization.
Figure 3: (Color online) Magnetization data of Ni-F5GT at select Ni doping
level $x$. (a)-(b) Temperature dependent magnetization of Ni-F5GT (external
in-plane magnetic field is 50 Oe). For clarity, the magnetization data at $x$
= 0.97(1) and $x$ = 1 are multiplied by 200 and 2000, respectively. (c)-(d)
In-plane isothermal magnetization at $T$ = 2 K (c) and $T$ = 400 K (d),
respectively. (e) Saturation moment per formula unit at 2 K under the magnetic
field of 7 T. (f) Coercive fields extracted from the isothermal magnetization
at $T$ = 2 K. In (e)-(f), the magnetic field is applied along the $ab$-plane
(red symbols) or $c$-direction (green symbols).
The unique feature of Ni-F5GT is the immense impact on the lattice and
magnetism by Ni substitution, including the radical change of the layer
spacing $d$ (Fig. 1(c)) and the local atomic arrangements (Fig. 1(a), Fig. 2,
Figs. S2-S5), as compared to F5GT or Co-F5GT [32, 36]. It is evident that the
change of the layer spacing $d$ in Ni-F5GT is significantly more pronounced
than its evolution in Co-F5GT. Importantly, the layer spacing $d$ of the Fe-
rich domains (green symbols in Fig. 1(c)) closely tracks the evolution of
$T_{\text{C}}$ in Ni-F5GT at 0 $\leq$ $x$ $\leq$ 0.86(1) (Fig. 4). This is
supported by the experimental observation that the Fe-rich domains maintain
the AA-stacking order with the straight-and-flat TGT-$planes$ over a broad Ni
doping range 0 $<$ $x$ $\leq$ 0.86(1) and thus may be mainly responsible for
the robust FM in Ni-F5GT. The Ni dopants are also indispensable for donating
electrons to the system and forming the Ni-rich domains in region (II), which
help preserve the Fe-rich domains. This phase separation in region (II)
naturally explains the domain pinning enhanced coercive fields (Fig. 3(f)).
Meanwhile, the preserved Fe-rich domains maintain the robust $T_{C}$ of
Ni-F5GT in this region.
Our work on Ni-F5GT reveals a complicated yet intriguing system, in which the
lattice, electronic and magnetic degrees of freedom are closely intertwined.
To understand the enhancement of FM in Ni-F5GT, especially the region (I)
(Fig. 4), it is necessary to consider the possible effects of both electron
doping and structural alteration by Ni replacement. Carrier doping is often
seen to be detrimental to the correlation induced long range magnetic order,
as recognized in the unconventional superconductors, such as the cuprates and
iron pnictides [45, 46]. For the itinerant FM, charge doping can, in some
cases, help meet the Stoner criterion and actually promote magnetic order
[47]. The related F3GT system, which may also apply to F5GT, has both
itinerant and localized spin moment contributions to the magnetism [21, 23,
28, 24, 25, 48, 49]. Hence, the ordering temperature of F3GT can be elevated
by electrostatic gating of its thin layer form [28, 37]. In Ni-F5GT, the
electrons provided by the Ni dopants may greatly influence the electronic band
structure and the density of states (DOS) near the Fermi level
($E_{\text{F}}$). Alternatively, the magnetic exchange coupling $J_{i,j}$
between spins $\boldsymbol{S}_{i}$ and $\boldsymbol{S}_{j}$ on sites $i$ and
$j$, might also be altered via the Ruderman–Kittel–Kasuya–Yosida (RKKY)
exchange since the electronic structure is likely altered substantially by Ni-
doping [28, 25, 50, 51]. All together, the itinerant FM might also be enhanced
by electron doping in Ni-F5GT.
Now focusing on the localized spin moments and considering a simple Heisenberg
model with a weak magnetic anisotropy, in which the magnetic contributions
from the itinerant FM can also be effectively mapped into the Hamiltonian [52,
28]:
$H=\sum_{\begin{subarray}{c}i<j\end{subarray}}J_{i,j}\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}-\sum_{\begin{subarray}{c}i\end{subarray}}A(S_{i}^{z})^{2}$
(1)
Here, $A$ is the single-ion anisotropy ($A$ $>$ 0 for Ising spin moment).
Since $A$ is small and close to zero in Ni-F5GT [33], a mean-field treatment
results in the magnetic transition temperature [28, 12]:
$T_{C}=\frac{S(S+1)}{3k_{B}}({z_{nn}}J_{nn}+...)$ (2)
where $z_{nn}$ is the coordination number of the nearest neighbouring sites,
$S$ is the magnetic spin quantum number and $J_{nn}$ the nearest-neighbour
exchange coupling. On the mean-field level, a larger $z_{nn}$ or $J_{nn}$
promises a larger $T_{C}$. Perhaps, this is why the ordering temperature in
FeNGeTe2 is quickly increased from $T_{C}$ $\sim$ 230 K at $N$ = 3 to $T_{C}$
$\sim$ 317 K at $N$ = 5 [20, 21, 30, 32, 33]. In Ni-F5GT, since the average
magnetic moment per Fe is almost unchanged and the in-plane FM indicates a
small yet negative $A$ (Fig. 3(e) and Fig. S7), hence neither $S$ nor $A$ is
responsible for the enhancement of the FM. However, the structural alterations
may affect both $z_{nn}$ and $J_{i,j}$, and therefore are strong candidates
for explaining the further enhanced $T_{C}$.
Firstly, a small increment of $\delta$ is observed for lightly Ni doped F5GT
(region (I) in Fig. 4 and Fig. S1). One direct consequence is the larger site
occupancy of the Fe1 site, which is up to $\sim$75$\%$ at $x$ $\sim$ 0.3 as
compared to a maximum of $\sim$50$\%$ at $x$ = 0. Considering that the Fe1-Fe3
bond length is the shortest among all of the direct Fe-Fe bonds (Fig. S3), the
Fe1 site might be critical for determining the $T_{C}$ in Ni-F5GT. Therefore,
a larger $\delta$, which implies a greater $z_{nn}$ of the Fe1 site, might
promote a higher $T_{C}$ (Eqn. (2)). Secondly, with more Ni substitution in
region (I), the explicit effect is that the TGT-$planes$ of the Fe-rich
domains are becoming more flat and extended in space (Fig. 2(c), Fig. S3,
Figs. S5(c),(e)). Microscopically, other than the small spatial re-
arrangements of the Fe2 and Fe3 sites, it is the Fe1 site that is becoming
increasingly distant from the Fe3 site along the $c$ direction while keeping
the Fe-Fe bond lengths nearly unchanged (Fig. S3). This explains the
enlargement of the layer spacing $d$ (Fig. 1(c)), which positively correlates
with the $T_{C}$ in Ni-F5GT. Understandably, the intralayer and interlayer
exchange coupling, especially these related to the Fe1 site, might be
effectively strengthened, which render the 3D magnetic interactions stronger
and eventually lead to the enhancement of $T_{C}$ in Ni-F5GT.
Figure 4: (Color online) Ni doping level $x$ dependent $T_{C}$ in Ni-F5GT.
Empty symbols: $T_{C}$ for as-grown samples; solid symbols: $T_{C}$ after
thermal cycling. Vertical dashed lines indicate different regions in Ni-F5GT.
In summary, our work reveals a new arena for studying the vdW magnetic metals
with strong room temperature ferromagnetism. A record high ferromagnetic order
with a $T_{C}$ $=$ 478(6) K is realized in Ni-F5GT at $x$ = 0.36(2). Albeit
with the non-magnetic dilution, several factors are speculated to assist the
pervasive, well-above-room-temperature FM in Ni-F5GT. Candidate contributors
include the increased site occupancy, structural modifications altered
magnetic exchange couplings and the electron doping effect. Clearly, as
stated, these ideas are purely speculative and require much more thorough
investigation. Although further research is needed to understand fully the
mechanism of the enhanced magnetism, our study highlights that the Ni-F5GT
system is an extremely rare example of strongly enhanced ferromagnetism, in
spite of the detrimental factors such as the non-magnetic dilution and
electron doping effects introduced by Ni dopants. In addition, Ni-F5GT offers
unique or alternative avenues towards enhanced coercivity, varying length
scale of phase separation, thermal cycling influenced $T_{C}$ and potential
relevance to skyrmionics in F5GT and other related vdW magnets [41, 42, 36,
Zhang_2022_FCGT_sciadv].
X.C. wishes to thank Nicholas S. Settineri and Weiwei Xie for some single
crystal XRD measurement and acknowledge the Applications Group at Quantum
Design for their contribution of high temperature (300-600 K) magnetization
measurements to this work. Work at Lawrence Berkeley National Laboratory was
funded by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division under Contract
No. DE-AC02-05-CH11231 within the Quantum Materials Program (KC2202). Y.T.S.,
H.Z. and D.A.M acknowledge financial support from the Department of Defense,
Air Force Office of Scientific Research under award FA9550-18-1-0480. R.C. and
J.Y. acknowledge the support by Intel Corporation under an award titled
Valleytronics center. The electron microscopy studies were performed at the
Cornell Center for Materials Research, a National Science Foundation (NSF)
Materials Research Science and Engineering Centers program (DMR 1719875). The
microscopy work at Cornell was supported by the NSF PARADIM DMR-2039380, with
additional support from Cornell University, the Weill Institute and the Kavli
Institute at Cornell. S.S. acknowledges the help from Dr. Rohan Dhall and is
supported by the Quantum Materials Program under the Basic Energy Sciences,
Department of Energy. The microscopy work was performed at Molecular Foundry
that is supported by the Office of Science, Office of Basic Energy Sciences,
of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The
devices for transport measurements were fabricated in the UC Berkeley Marvell
Nanofabrication Laboratory.
## References
* Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1133 (1966).
* Hohenberg [1967] P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158, 383 (1967).
* Onsager [1944] L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944).
* J L Lado and J Fernández-Rossier [2017] J L Lado and J Fernández-Rossier, On the origin of magnetic anisotropy in two dimensional ${\mathrm{Cr}}{\mathrm{I}}_{3}$, 2D Materials 4, 035002 (2017).
* Kim _et al._ [2019] D.-H. Kim, K. Kim, K.-T. Ko, J. Seo, J. S. Kim, T.-H. Jang, Y. Kim, J.-Y. Kim, S.-W. Cheong, and J.-H. Park, Giant Magnetic Anisotropy Induced by Ligand $LS$ Coupling in Layered Cr Compounds, Phys. Rev. Lett. 122, 207201 (2019).
* Als-Nielsen _et al._ [1976] J. Als-Nielsen, R. J. Birgeneau, H. J. Guggenheim, and G. Shirane, Critical behaviour of a two-dimensional random antiferromagnet: Rb2Mn${}_{0}.5$Ni${}_{0}.5$F4, Journal of Physics C: Solid State Physics 9, L121 (1976).
* Chakravarty _et al._ [1989] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Two-dimensional quantum Heisenberg antiferromagnet at low temperatures, Phys. Rev. B 39, 2344 (1989).
* Birgeneau _et al._ [1999] R. J. Birgeneau, M. Greven, M. A. Kastner, Y. S. Lee, B. O. Wells, Y. Endoh, K. Yamada, and G. Shirane, Instantaneous spin correlations in ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$, Phys. Rev. B 59, 13788 (1999).
* Kosterlitz and Thouless [1973] J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6, 1181 (1973).
* Park [2016] J.-G. Park, Opportunities and challenges of 2D magnetic van der Waals materials: magnetic graphene?, Journal of Physics: Condensed Matter 28, 301001 (2016).
* Burch _et al._ [2018] K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in two-dimensional van der Waals materials, Nature 563, 47 (2018).
* Gibertini _et al._ [2019] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Magnetic 2D materials and heterostructures, Nature Nanotechnology 14, 408 (2019).
* Gong and Zhang [2019] C. Gong and X. Zhang, Two-dimensional magnetic crystals and emergent heterostructure devices, Science 363, 10.1126/science.aav4450 (2019).
* Wang _et al._ [2020] M.-C. Wang, C.-C. Huang, C.-H. Cheung, C.-Y. Chen, S. G. Tan, T.-W. Huang, Y. Zhao, Y. Zhao, G. Wu, Y.-P. Feng, H.-C. Wu, and C.-R. Chang, Prospects and Opportunities of 2D van der Waals Magnetic Systems, Annalen der Physik 532, 1900452 (2020).
* Du _et al._ [2021] L. Du, T. Hasan, A. Castellanos-Gomez, G.-B. Liu, Y. Yao, C. N. Lau, and Z. Sun, Engineering symmetry breaking in 2D layered materials, Nature Reviews Physics 3, 193 (2021).
* Carteaux _et al._ [1995] V. Carteaux, D. Brunet, G. Ouvrard, and G. Andre, Crystallographic, magnetic and electronic structures of a new layered ferromagnetic compound ${\mathrm{Cr}}_{2}{\mathrm{Ge}}_{2}{\mathrm{Te}}_{6}$, Journal of Physics: Condensed Matter 7, 69 (1995).
* Gong _et al._ [2017] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature 546, 265 (2017).
* McGuire _et al._ [2015] M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Coupling of Crystal Structure and Magnetism in the Layered, Ferromagnetic Insulator ${\mathrm{Cr}}{\mathrm{I}}_{3}$, Chem. Mater. 27, 612 (2015).
* Huang _et al._ [2017] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature 546, 270 (2017).
* Deiseroth _et al._ [2006] H.-J. Deiseroth, K. Aleksandrov, C. Reiner, L. Kienle, and R. K. Kremer, ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ and ${\mathrm{Ni}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ – Two New Layered Transition-Metal Compounds: Crystal Structures, HRTEM Investigations, and Magnetic and Electrical Properties, European Journal of Inorganic Chemistry 2006, 1561 (2006).
* Chen _et al._ [2013] B. Chen, J. Yang, H. Wang, M. Imai, H. Ohta, C. Michioka, K. Yoshimura, and M. Fang, Magnetic Properties of Layered Itinerant Electron Ferromagnet ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$, Journal of the Physical Society of Japan 82, 124711 (2013).
* Fei _et al._ [2018] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J.-H. Chu, and X. Xu, Two-dimensional itinerant ferromagnetism in atomically thin ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$, Nature Materials 17, 778 (2018).
* May _et al._ [2016] A. F. May, S. Calder, C. Cantoni, H. Cao, and M. A. McGuire, Magnetic structure and phase stability of the van der Waals bonded ferromagnet ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_{2}$, Phys. Rev. B 93, 014411 (2016).
* Kim _et al._ [2018] K. Kim, J. Seo, E. Lee, K.-T. Ko, B. S. Kim, B. G. Jang, J. M. Ok, J. Lee, Y. J. Jo, , W. Kang, J. H. Shim, C. Kim, H. W. Yeom, B. Il Min, B.-J. Yang, and J. S. Kim, Large anomalous Hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal, Nature Materials 17, 794 (2018).
* Zhang _et al._ [2018] Y. Zhang, H. Lu, X. Zhu, S. Tan, W. Feng, Q. Liu, W. Zhang, Q. Chen, Y. Liu, X. Luo, D. Xie, L. Luo, Z. Zhang, and X. Lai, Emergence of Kondo lattice behavior in a van der Waals itinerant ferromagnet, ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$, Science Advances 4, 10.1126/sciadv.aao6791 (2018).
* Freitas _et al._ [2015] D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. Strobel, F. Gay, J. Marcus, and M. Núñez-Regueiro, Ferromagnetism in layered metastable 1$T$-${\mathrm{Cr}}{\mathrm{Te}}_{2}$ , Journal of Physics: Condensed Matter 27, 176002 (2015).
* Sun _et al._ [2020] X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang, L. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V. Bouchiat, M. Nunez Regueiro, N. Rougemaille, J. Coraux, A. Purbawati, A. Hadj-Azzem, Z. Wang, B. Dong, X. Wu, T. Yang, G. Yu, B. Wang, Z. Han, X. Han, and Z. Zhang, Room temperature ferromagnetism in ultra-thin van der Waals crystals of 1$T$-${\mathrm{Cr}}{\mathrm{Te}}_{2}$ , Nano Research 13, 3358 (2020).
* Deng _et al._ [2018] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferromagnetism in two-dimensional ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$, Nature 563, 94 (2018).
* Li _et al._ [2018] Q. Li, M. Yang, C. Gong, R. V. Chopdekar, A. T. N’Diaye, J. Turner, G. Chen, A. Scholl, P. Shafer, E. Arenholz, A. K. Schmid, S. Wang, K. Liu, N. Gao, A. S. Admasu, S.-W. Cheong, C. Hwang, J. Li, F. Wang, X. Zhang, and Z. Qiu, Patterning-Induced Ferromagnetism of ${\mathrm{Fe}}_{3}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ van der Waals Materials beyond Room Temperature, Nano Letters 18, 5974 (2018).
* Seo _et al._ [2020] J. Seo, D. Y. Kim, E. S. An, K. Kim, G.-Y. Kim, S.-Y. Hwang, D. W. Kim, B. G. Jang, H. Kim, G. Eom, S. Y. Seo, R. Stania, M. Muntwiler, J. Lee, K. Watanabe, T. Taniguchi, Y. J. Jo, J. Lee, B. I. Min, M. H. Jo, H. W. Yeom, S.-Y. Choi, J. H. Shim, and J. S. Kim, Nearly room temperature ferromagnetism in a magnetic metal-rich van der Waals metal, Science Advances 6, 10.1126/sciadv.aay8912 (2020).
* Stahl _et al._ [2018] J. Stahl, E. Shlaen, and D. Johrendt, The van der Waals Ferromagnets ${\mathrm{Fe}}_{5-\delta}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ and ${\mathrm{Fe}}_{5-\delta-x}{\mathrm{Ni}}_{x}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ – Crystal Structure, Stacking Faults, and Magnetic Properties, Zeitschrift f${\ddot{\text{u}}}$r anorganische und allgemeine Chemie 644, 1923 (2018).
* May _et al._ [2019] A. F. May, D. Ovchinnikov, Q. Zheng, R. Hermann, S. Calder, B. Huang, Z. Fei, Y. Liu, X. Xu, and M. A. McGuire, Ferromagnetism Near Room Temperature in the Cleavable van der Waals Crystal ${\mathrm{Fe}}_{5}{\mathrm{Ge}}{\mathrm{Te}}_{2}$ , ACS Nano 13, 4436 (2019).
* Zhang _et al._ [2020] H. Zhang, R. Chen, K. Zhai, X. Chen, L. Caretta, X. Huang, R. V. Chopdekar, J. Cao, J. Sun, J. Yao, R. Birgeneau, and R. Ramesh, Itinerant ferromagnetism in van der Waals ${\mathrm{Fe}}_{5-x}{\mathrm{Ge}\mathrm{Te}}_{2}$ crystals above room temperature, Phys. Rev. B 102, 064417 (2020).
* Ly _et al._ [2021] T. T. Ly, J. Park, K. Kim, H.-B. Ahn, N. J. Lee, K. Kim, T.-E. Park, G. Duvjir, N. H. Lam, K. Jang, C.-Y. You, Y. Jo, S. K. Kim, C. Lee, S. Kim, and J. Kim, Direct Observation of Fe-Ge Ordering in ${\mathrm{Fe}}_{5-x}{\mathrm{Ge}\mathrm{Te}}_{2}$ Crystals and Resultant Helimagnetism, Advanced Functional Materials 31, 2009758 (2021).
* Gao _et al._ [2020] Y. Gao, Q. Yin, Q. Wang, Z. Li, J. Cai, T. Zhao, H. Lei, S. Wang, Y. Zhang, and B. Shen, Spontaneous (Anti)meron Chains in the Domain Walls of van der Waals Ferromagnetic ${\mathrm{Fe}}_{5-x}{\mathrm{Ge}\mathrm{Te}}_{2}$, Advanced Materials 32, 2005228 (2020).
* Zhang _et al._ [2021] H. Zhang, Y.-T. Shao, R. Chen, X. Chen, S. Susarla, J. T. Reichanadter, L. Caretta, X. Huang, N. S. Settineri, Z. Chen, J. Zhou, E. Bourret-Courchesne, P. Ercius, J. Yao, J. B. Neaton, D. A. Muller, R. J. Birgeneau, and R. Ramesh, A room temperature polar ferromagnetic metal (2021), arXiv:2106.00833 [cond-mat.mtrl-sci] .
* Verzhbitskiy _et al._ [2020] I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y. P. Feng, and G. Eda, Controlling the magnetic anisotropy in ${\mathrm{Cr}}_{2}{\mathrm{Ge}}_{2}{\mathrm{Te}}_{6}$ by electrostatic gating, Nature Electronics 3, 460 (2020).
* Bhoi _et al._ [2021] D. Bhoi, J. Gouchi, N. Hiraoka, Y. Zhang, N. Ogita, T. Hasegawa, K. Kitagawa, H. Takagi, K. H. Kim, and Y. Uwatoko, Nearly room temperature ferromagnetism in pressure-induced correlated metallic state of van der Waals insulator CrGeTe3 (2021), arXiv:2107.10573 [cond-mat.str-el] .
* Weber _et al._ [2019] D. Weber, A. H. Trout, D. W. McComb, and J. E. Goldberger, Decomposition-Induced Room-Temperature Magnetism of the Na-Intercalated Layered Ferromagnet $\mathrm{F}{\mathrm{e}}_{3-x}\mathrm{GeT}{\mathrm{e}}_{2}$, Nano Letters 19, 5031 (2019).
* Wang _et al._ [2019] N. Wang, H. Tang, M. Shi, H. Zhang, W. Zhuo, D. Liu, F. Meng, L. Ma, J. Ying, L. Zou, Z. Sun, and X. Chen, Transition from Ferromagnetic Semiconductor to Ferromagnetic Metal with Enhanced Curie Temperature in ${\mathrm{Cr}}_{2}{\mathrm{Ge}}_{2}{\mathrm{Te}}_{6}$ via Organic Ion Intercalation, Journal of the American Chemical Society 141, 17166 (2019).
* May _et al._ [2020] A. F. May, M.-H. Du, V. R. Cooper, and M. A. McGuire, Tuning magnetic order in the van der Waals metal ${\mathrm{Fe}}_{5}{\mathrm{GeTe}}_{2}$ by cobalt substitution, Phys. Rev. Materials 4, 074008 (2020).
* Tian _et al._ [2020] C. Tian, F. Pan, S. Xu, K. Ai, T. Xia, and P. Cheng, Tunable magnetic properties in van der Waals crystals (Fe1-xCox)5GeTe2, Applied Physics Letters 116, 202402 (2020).
* [43] See Supplemental Material at ? for additional data of sample synthesis, characterizations, transport measurements, magnetization, etc.
* Deiseroth _et al._ [2007] H.-J. Deiseroth, F. Spirovski, C. Reiner, and M. Schlosser, Crystal structures of nickel germanium selenide, Ni5.45GeSe2, and nickel germanium telluride, Ni5.45GeTe2, Zeitschrift f${\ddot{\text{u}}}$r Kristallographie - New Crystal Structures 222, 171 (2007).
* Lee _et al._ [2006] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
* Dai [2015] P. Dai, Antiferromagnetic order and spin dynamics in iron-based superconductors, Rev. Mod. Phys. 87, 855 (2015).
* Huang _et al._ [2021] J. Huang, Z. Wang, H. Pang, H. Wu, H. Cao, S.-K. Mo, A. Rustagi, A. F. Kemper, M. Wang, M. Yi, and R. J. Birgeneau, Flat-band-induced itinerant ferromagnetism in ${\mathrm{RbCo}}_{2}{\mathrm{Se}}_{2}$, Phys. Rev. B 103, 165105 (2021).
* Calder _et al._ [2019] S. Calder, A. I. Kolesnikov, and A. F. May, Magnetic excitations in the quasi-two-dimensional ferromagnet ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_{2}$ measured with inelastic neutron scattering, Phys. Rev. B 99, 094423 (2019).
* Xu _et al._ [2020] X. Xu, Y. W. Li, S. R. Duan, S. L. Zhang, Y. J. Chen, L. Kang, A. J. Liang, C. Chen, W. Xia, Y. Xu, P. Malinowski, X. D. Xu, J.-H. Chu, G. Li, Y. F. Guo, Z. K. Liu, L. X. Yang, and Y. L. Chen, Signature for non-Stoner ferromagnetism in the van der Waals ferromagnet $\mathrm{F}{\mathrm{e}}_{3}\mathrm{GeT}{\mathrm{e}}_{2}$, Phys. Rev. B 101, 201104 (2020).
* Seo _et al._ [2021] J. Seo, E. S. An, T. Park, S.-Y. Hwang, G.-Y. Kim, K. Song, W.-s. Noh, J. Y. Kim, G. S. Choi, M. Choi, E. Oh, K. Watanabe, T. Taniguchi, J. H. Park, Y. J. Jo, H. W. Yeom, S.-Y. Choi, J. H. Shim, and J. S. Kim, Tunable high-temperature itinerant antiferromagnetism in a van der Waals magnet, Nature Communications 12, 2844 (2021).
* Zhao _et al._ [2021] M. Zhao, B.-B. Chen, Y. Xi, Y. Zhao, H. Xu, H. Zhang, N. Cheng, H. Feng, J. Zhuang, F. Pan, X. Xu, W. Hao, W. Li, S. Zhou, S. X. Dou, and Y. Du, Kondo Holes in the Two-Dimensional Itinerant Ising Ferromagnet Fe3GeTe2, Nano Letters 21, 6117 (2021).
* Prange and Korenman [1979] R. E. Prange and V. Korenman, Local-band theory of itinerant ferromagnetism. IV. Equivalent Heisenberg model, Phys. Rev. B 19, 4691 (1979).
|
# A three-wave mixing kinetic inductance traveling-wave amplifier with near-
quantum-limited noise performance
M. Malnou<EMAIL_ADDRESS>National Institute of Standards and
Technology, Boulder, Colorado 80305, USA Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA M. R. Vissers National Institute of
Standards and Technology, Boulder, Colorado 80305, USA J. D. Wheeler
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
J. Aumentado National Institute of Standards and Technology, Boulder,
Colorado 80305, USA J. Hubmayr National Institute of Standards and
Technology, Boulder, Colorado 80305, USA J. N. Ullom National Institute of
Standards and Technology, Boulder, Colorado 80305, USA Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA J. Gao National
Institute of Standards and Technology, Boulder, Colorado 80305, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
###### Abstract
We present a theoretical model and experimental characterization of a
microwave kinetic inductance traveling-wave amplifier (KIT), whose noise
performance, measured by a shot-noise tunnel junction (SNTJ), approaches the
quantum limit. Biased with a dc current, the KIT operates in a three-wave
mixing fashion, thereby reducing by several orders of magnitude the power of
the microwave pump tone and associated parasitic heating compared to
conventional four-wave mixing KIT devices. It consists of a 50 Ω artificial
transmission line whose dispersion allows for a controlled amplification
bandwidth. We measure $16.5^{+1}_{-1.3}$ dB of gain across a 2 GHz bandwidth
with an input 1 dB compression power of -63 dBm, in qualitative agreement with
theory. Using a theoretical framework that accounts for the SNTJ-generated
noise entering both the signal and idler ports of the KIT, we measure the
system-added noise of an amplification chain that integrates the KIT as the
first amplifier. This system-added noise, $3.1\pm 0.6$ quanta (equivalent to
$0.66\pm 0.15$ K) between 3.5 and 5.5 GHz, is the one that a device replacing
the SNTJ in that chain would see. This KIT is therefore suitable to read large
arrays of microwave kinetic inductance detectors and promising for multiplexed
superconducting qubit readout.
## I INTRODUCTION
Is it possible to build a quantum limited microwave amplifier with enough
gain, bandwidth and power handling to simultaneously read thousands of
frequency-multiplexed superconducting resonators, like those in qubit systems
or microwave kinetic inductance detectors (MKIDs)? When designed with resonant
structures, Josephson junction-based parametric amplifiers have demonstrated
high gain and quantum limited performances Castellanos-Beltran _et al._
(2008); Bergeal _et al._ (2010); Roch _et al._ (2012); Mutus _et al._
(2013); Zhong _et al._ (2013); Lecocq _et al._ (2017); Malnou _et al._
(2018). However, despite efforts to increase the bandwidth up to a few hundred
megahertz via impedance engineering Mutus _et al._ (2014); Roy _et al._
(2015), or to increase the power handling up to a few hundred femtowatts via
Kerr engineering Frattini _et al._ (2017, 2018); Sivak _et al._ (2019), they
still cannot read more than a handful of resonators simultaneously. When
designed with nonresonant structures, i.e. transmission lines, Josephson
traveling-wave parametric amplifiers (JTWPAs) have high gain over gigahertz
bandwidth Macklin _et al._ (2015); White _et al._ (2015); Planat _et al._
(2020), but so far still exhibit similar power handling capabilities as their
resonant counterparts. Recent studies Sivak _et al._ (2020); Zorin (2016,
2019) suggest that a three-wave mixing (3WM) JTWPA with a finely controlled
and canceled Kerr nonlinearity should increase the device’s power handling
tenfold. Compelling experiments have yet to prove the feasibility of this
approach, for which the JTWPA’s design and fabrication increase in complexity.
We propose to tackle this challenge using a different non-linear medium:
starting with the intrinsic broadband and high power handling capabilities of
a kinetic inductance traveling-wave amplifier (KIT) Eom _et al._ (2012), we
build a near-quantum-limited amplifier.
The current limitations on microwave amplification affect many scientific
endeavors. Although proof-of-principle “quantum supremacy” was demonstrated by
a quantum computer containing a few tens of qubits Arute _et al._ (2019),
this number has to scale by at least an order of magnitude to run powerful
quantum algorithms Shor (1994); Grover (1997). In the hunt for exoplanets,
cameras with tens of thousands of MKID pixels are being built Szypryt _et
al._ (2017), and proposals to search for very light warm dark matter also
necessitate the use of a great number of MKID pixels Hochberg _et al._
(2016a, b). All these applications are either already limited by amplifier
noise, or would greatly benefit from wideband, high gain, high power handling,
quantum limited amplifiers.
The KIT we present in this article is a step toward a practical, quantum-
limited amplifier, whose bandwidth and power handling are compatible with high
channel-count applications. Operating in a 3WM fashion, and fabricated out of
a single layer of NbTiN, it consists of a weakly dispersive artificial
transmission line Chaudhuri _et al._ (2017); Zobrist _et al._ (2019), for
which we control the phase matched bandwidth with dispersion engineering. This
limits spurious parametric conversion processes that otherwise degrade the
power handling and noise performance. We measure an average gain of 16.5 dB
over a 2 GHz bandwidth, and a typical 1 dB input compression power of -63 dBm
within that bandwidth. Using a shot-noise tunnel junction (SNTJ) Spietz _et
al._ (2003, 2006) we measure the added noise of a readout chain with the KIT
as the first amplifier. To quote the true system-added noise of the chain,
i.e. the one that a device replacing the SNTJ in that chain would see, we
develop a novel theoretical framework that accounts for the SNTJ-generated
noise illuminating both the signal and idler ports of the KIT. Failure to
account for the idler port’s noise makes the system-added noise look
significantly better than its true value. We demonstrate a true system-added
noise of $3.1\pm 0.6$ quanta between 3.5 and 5.5 GHz, and estimate that the
KIT alone is near-quantum-limited. It is the first time that the broadband
noise properties of a KIT are fully characterized rigorously.
## II THEORY AND DESIGN
KITs exploit the nonlinear kinetic inductance of a superconducting line to
generate parametric interaction between pump, signal, and idler photons. In
3WM, a single pump photon converts into signal and idler photons, whereas
four-wave mixing (4WM) converts two pump photons in this fashion. Operating a
KIT with 3WM offers two key advantages over 4WM. First, as the pump frequency
is far detuned from the amplification band, it is easily filtered, which is
often necessary to avoid saturating the following amplifier. Second, it
reduces the rf pump power because energy is extracted from dc power to convert
pump photons, which avoids undesirable heating effects from the strong rf
pump, including those happening in the packaging. More precisely, when biased
with a dc current $I_{d}$, the KIT inductance per unit length L is Vissers
_et al._ (2016):
$L=L_{d}(1+\epsilon I+\xi I^{2}+\mathcal{O}(I^{3})),$ (1)
where $I$ is the rf current, $L_{d}$ is the KIT inductance under dc bias, at
zero rf current, $\epsilon=2I_{d}/(I_{*}^{2}+I_{d}^{2})$ and
$\xi=1/(I_{*}^{2}+I_{d}^{2})$. The current $I_{*}$ sets the scale of the
nonlinearity; it is typically $\sim 10^{3}$ higher than that of Josephson
devices, thereby conferring KITs $\sim 10^{4}-10^{6}$ higher power handling
capabilities than their Josephson equivalents. The term $\epsilon I$ permits
3WM, while $\xi I^{2}$ permits 4WM.
Solving the coupled mode equations (CME) for a pump at frequency $\omega_{p}$,
signal at $\omega_{s}$, and idler at $\omega_{i}$, such that
$\omega_{p}=\omega_{s}+\omega_{i}$ yields the 3WM phase matching condition for
exponential gain:
$\Delta_{k}=-\frac{\xi I_{p0}^{2}}{8}(k_{p}-2k_{s}-2k_{i}),$ (2)
see appendix A.1. Here, $\Delta_{k}=k_{p}-k_{s}-k_{i}$ with $k_{j}$,
$j\in\\{p,s,i\\}$ the pump, signal and idler wavenumbers, and $I_{p0}$ is the
rf pump amplitude at the KIT’s input. In a non-dispersive transmission line
$\Delta_{k}=0$, and thus equation 2 can naturally be fulfilled over a very
wide frequency range in KITs, where $I_{p0}\ll I_{*}$. Although desirable
within the amplification bandwidth, it is undesirable outside of that
bandwidth, where multiple parametric conversion processes take place Vissers
_et al._ (2016); Erickson and Pappas (2017). These processes deplete the pump,
thereby degrading the amplifier’s power handling, and they also induce
multiple conversion mechanisms at each frequency, thereby increasing the
amplifier-added noise.
Figure 1: Schematic of the KIT artificial transmission line. (a) Three cells
in series (not to scale) are arranged in a CPW topology. The highly inductive
central line (red) is flanked by IDC fingers, which match it to 50 Ω. Each
finger constitutes a low-Q, quarter-wave resonator. (b) In the equivalent
electrical circuit, each cell consists of a series inductance $L_{d}$ and two
resonators with inductance $L_{f}$ and capacitance to ground $C/2$.
While in conventional traveling-wave amplifiers dispersion engineering
prevents only pump harmonic generation, we suppress all unwanted parametric
conversion processes by designing our KIT as a weakly dispersive artificial
transmission line. Originally developed to have the KIT matched to 50 Ω
Chaudhuri _et al._ (2017), this line consists of a series of coplanar
waveguide (CPW) sections, or cells, each with inductance $L_{d}$, flanked by
two interdigitated capacitor (IDC) fingers that form the capacitance to ground
$C$, such that $Z=\sqrt{L_{d}/C}=50$ Ω, see Fig. 1a. Each IDC finger then
constitutes a low-Q quarter-wave resonator, with capacitance $C_{f}=C/2$ and
inductance $L_{f}$ (see Fig. 1b), set by the finger’s length. In practice, we
choose $\omega_{f}=1/\sqrt{L_{f}C_{f}}=2\pi\times 36$ GHz, and
$Q=1/Z\sqrt{L_{f}/C_{f}}=3.3$, to generate a slight dispersion at low
frequencies, where the pump, signal and idler modes lie. The dashed line in
Fig. 2a shows the dispersive part of the wavenumber $k^{*}=k-k_{0}$ with
$k_{0}=\omega\sqrt{L_{d}C}$, as a function of frequency. It is calculated by
cascading the ABCD matrices of the KIT cells, see appendix B. As it slightly
deviates from zero, no triplet $\\{k_{s},k_{i},k_{p}\\}$ can satisfy Eq. 2.
Figure 2: Influence of the phase matching on the gain profile. (a) The
nonlinear wavenumber $k^{*}$ is calculated as a function of frequency for the
transmission line represented in Fig. 1 (dashed), and for a similar line,
periodically loaded at $80$ Ω (black). The nonlinear part of four triplets
$\\{k_{s},k_{i},k_{p}\\}$ that satisfy the phase matching condition (Eq. 2)
are indicated with colored dots: in purple $\omega_{i}-\omega_{s}=4$ GHz,
green $\omega_{i}-\omega_{s}=3$ GHz, red $\omega_{i}-\omega_{s}=2$ GHz, and
blue $\omega_{i}=\omega_{s}$. Additionally, a gray line indicates a pump
frequency for which phase matching is nowhere fulfilled (b) Solving the CME
(Eqs. 11), the signal power gain profile is calculated (Eq. 15) for the
related pump frequencies (the colors match with panel a). The KIT length is
$N_{c}=6.6\times 10^{4}$ cells.
To retrieve phase-matching over a desired bandwidth, we engineer another
dispersion feature by increasing the line impedance periodically on short
sections. This feature creates a resonance in the line’s phase response (and a
stopband in the line’s amplitude response), at a frequency $\omega_{l}$
controlled by the loading periodicity Chaudhuri _et al._ (2017); Pozar
(2011). Figure 2a shows $k^{*}$ in a line periodically loaded at $80$ Ω, with
$\omega_{l}=2\pi\times 8.5$ GHz. Because the nonlinear wavenumber close to
resonance varies sharply, there exists values of $k^{*}_{p}$ (above
$\omega_{l}$) for which we can find triplets $\\{k_{s},k_{i},k_{p}\\}$ that
satisfy Eq. 2 (examples of their nonlinear parts are shown in colored dots). A
slight variation of the pump frequency $\omega_{p}$ significantly affects
which pair of signal and idler frequencies is phase matched.
At these matched frequencies, the 3WM gain grows exponentially with $k_{p}$
(in radians per cell), with the KIT length, and with $\delta_{L}$, the
relative inductance modulation amplitude generated by the rf current and
scaling with $I_{d}$. More precisely, the phase matched, small signal power
gain can be written as:
$G_{s}\simeq\cosh^{2}\left(\frac{1}{8}\delta_{L}k_{p}N_{c}\right),$ (3)
where $N_{c}$ is the total number of cells, see appendix A.2. Typically, when
operating our KIT, $\delta_{L}\sim 7\times 10^{-3}$; thus, with $L_{d}\sim 50$
pH/cell (see Sec. III), we need $N_{c}>5\times 10^{4}$ to get $G_{s}>15$ dB at
$\omega_{p}\sim 2\pi\times 9$ GHz.
Since maximum gain is achieved with phase-matching, $\omega_{p}$ influences
the gain profile. To calculate this profile, we insert the dispersion relation
$k(\omega)$ into the CME, and solve them numerically, see appendix B. Figure
2b shows signal power gain profiles, calculated for the pump frequencies
represented in Fig. 2a. As expected, the gain is maximal at the signal and
idler frequencies for which the triplets $\\{k_{s},k_{i},k_{p}\\}$ satisfy Eq.
2. When these frequencies are far apart, there is a region in between where
phase matching is not sufficient and the gain drops. By reducing $\omega_{p}$,
we can lower the distance between phase-matched signal and idler frequencies,
and therefore obtain a wideband, flat gain region. Further reducing
$\omega_{p}$, we reach the value where phase-matched signal and idler
frequencies are equal, beyond what phase matching is nowhere fulfilled
anymore, and the gain drops. Fundamentally, the wideband nature of the gain
depends on the convexity of the dispersion relation, and therefore on the
fingers’ length and capacitance to ground. As $\omega_{f}$ or Q increase,
$k^{*}$ is less convex, and thus closer to a broadband, phase-matched
situation, but at the cost of allowing extra parametric processes to arise.
## III EXPERIMENTAL REALIZATION
Because the kinetic inductance nonlinearity is weak, in practice a KIT
comprises a transmission line tens of centimeters long. To maximize this
nonlinearity, and to minimize its length, the line as well as IDC fingers are
made 1 $\upmu$m wide, and each unit cell is 5 $\upmu$m long, see Fig. 3a and
b. Fabricated from a 20 nm thick NbTiN layer on 380 $\upmu$m high-resistivity
intrinsic silicon via optical lithography, it yields $I_{*}\sim 7$ mA, and a
sheet inductance $\sim 10$ $\mathrm{pH}/$square. Thus, $L_{d}\sim 50$ pH, and
in order to retrieve $Z=50$ Ω, each finger is made 102 $\upmu$m long. The
loading (Fig.3a) consists of cells with shorter fingers (33.5
$\upmu\mathrm{m}$, $Z=80$ Ω), arranged periodically to generate a resonance at
8.5 GHz, thereby positioning the pump frequency in a way compatible with our
filtering capabilities.
We lay out the line in a spiral topology, on a $2\times 2$ cm chip (Fig. 3c),
which contains $N_{c}=6.6\times 10^{4}$ cells, equivalent to 33 cm. To avoid
spurious chip and microstrip modes, electrical grounding inside the chip is
ensured with spring-loaded pogo pins, descending from the top copper lid of
the packaging, and contacting the chip between the line traces, see appendix
D. The pogo pins also improve the chip-to-packaging thermal link, which
otherwise mostly relies on wire-bonds.
Figure 3: Micrographs of the KIT. (a) The transmission line (false color red)
is periodically loaded with shorter IDC fingers. (b) Line and fingers are 1
$\upmu$m wide, and each cell is 5 $\upmu$m long. (c) The overall KIT is laid
out in a spiral configuration on a $2\times 2$ cm chip, and clamped on a
copper packaging.
In a first experiment, we measure the gain, bandwidth, and power handling of
the KIT when cooled to $\sim 30$ mK. The KIT is mounted as the sole amplifier
in the chain, thereby facilitating comparison of its gain profile to the
theoretical profiles, and revealing the gain ripples of the isolated KIT,
which otherwise also depend on the return loss of external components. Two
mandatory bias tees at the KIT input and output ports combine dc and rf
currents. Figure 4a shows KIT gain profiles, acquired at two different pump
frequencies. The current amplitudes are $I_{d}=1.5$ mA, and $I_{p0}=160$
$\upmu$A ($-29$ dBm in power, calibrated in situ by comparing dc and rf
nonlinear phase shifts, see appendix A.3). For the higher pump frequency, the
gain drops in the middle of the amplification bandwidth. For the lower one,
the gain profile is flatter, with an average value of $16.5^{+1}_{-1.3}$ dB
between 3.5 and 5.5 GHz, where the subscript and superscript denote the
amplitude of the gain ripples. Both profiles agree qualitatively with
behaviors explained in Sec. II. The gain ripples have a $8$ MHz characteristic
frequency (see Fig. 4c), equivalent to $62.5$ cm in wavelength (the phase
velocity being $v_{p}=1/\sqrt{L_{d}C}\sim 1000$ cell per nanosecond), or about
twice the KIT length. We thus attribute their presence to a mismatch in
impedance between KIT and microwave feed structures before and after the KIT.
This mismatch results in a pump standing wave pattern, which influences the
signal amplification, depending on its frequency. Figure 4b shows the gain at
4.5 GHz (obtained at the lower pump power), as a function of $P_{t}$, the
input power of a probe tone. The gain compresses by 1 dB from its small signal
value for $P_{\mathrm{-1dB}}=-63$ dBm, about $7$ dB lower than theoretical
predictions, see appendix C.2. This discrepancy, suggesting substantial room
for improvement, may be due to effects not included in our model, such as
standing wave patterns, or defects in the line, which locally lower $I_{*}$.
Nonetheless, $P_{\mathrm{-1dB}}$ is already about $30$ dB higher than JTWPAs
Macklin _et al._ (2015); White _et al._ (2015); Planat _et al._ (2020), and
about $10$ dB less than 4WM KITs with similar gain (see appendix H). This KIT
is therefore suitable to read thousands of frequency multiplexed MKIDs, that
use drive tone powers typically around $-90$ dBm Zobrist _et al._ (2019);
Vissers _et al._ (2016) or even more qubits, whose readout involves powers
substantially less than $-90$ dBm.
Figure 4: Amplification properties of the KIT. (a) The power gain is measured
with a vector network analyzer (VNA), when $\omega_{p}=2\pi\times 8.895$ GHz
(blue), and $\omega_{p}=2\pi\times 8.931$ GHz (red). (b) The gain of a probe
tone at 4.5 GHz compresses as the tone’s power increases. At
$P_{\mathrm{-1dB}}=-63$ dBm, referred to the KIT’s input, the gain has lowered
by 1 dB from its small signal value. It is measured for $\omega_{p}=2\pi\times
8.895$ GHz. (c) At the same pump frequency, a close-up on the small signal
gain around 4.5 GHz shows ripples with 8 MHz characteristic frequency.
As in any phase-insensitive traveling-wave and resonant parametric amplifier,
the practical, usable bandwidth, is half of the presented amplification
bandwidth. It is the bandwidth in which signals coming from microwave devices
can be phase-insensitively amplified. The other half, barring the idler
frequencies, contains a copy of signals in the first half. That is why the
gain in Fig. 4a is nearly symmetric about the half pump frequency ($\sim 4.5$
GHz). The asymmetry - gain and ripples marginally bigger above the half pump
frequency - originates from a frequency dependent saturation power. In fact,
higher frequencies possess a higher saturation power, see appendix C.1. We see
this effect here because we chose a signal power close to $P_{\mathrm{-1dB}}$
in order to maximize the signal-to-noise ratio in this measurement, where the
KIT remains the sole amplifier.
At higher dc current bias (bounded by the dc critical current of the
transmission line, $\sim 2.4$ mA in our device), lower rf pump power can be
used to obtain equivalent small signal gain, at the cost of a reduced 1 dB
compression power. Conversely, reducing $I_{d}$ and increasing $I_{p0}$
improves power handling capabilities, but the gain is then limited by a
superconductivity breaking phenomenon. We suspect the presence of weak links
Bockstiegel _et al._ (2014), and we are currently investigating the line
breakdown mechanism.
## IV NOISE PERFORMANCE
The combined gain, bandwidth, and power handling performance are promising,
provided that the KIT also presents competitive noise performance. Measuring
this noise is a topic of current interest Eom _et al._ (2012); Ranzani _et
al._ (2018); Zobrist _et al._ (2019), and we execute the task using a self-
calibrated shot-noise tunnel junction (SNTJ) Spietz _et al._ (2003, 2006). We
measure the output spectral density, whose power depends on the chain’s gain
and loss. The SNTJ acts as a dynamic variable noise source, allowing for a
continuous sweep in input noise temperature by sweeping its bias voltage, and
our measurement scans the entire KIT bandwidth.
Figure 5: Schematic of the noise measurement setup. Each component is labeled
with its gain or transmission efficiency, and with its input-referred added
noise. Calibrated noise $N_{\mathrm{in}}^{s}$ ($N_{\mathrm{in}}^{i}$) is
generated by the SNTJ at the signal (idler) frequency. It is routed to the KIT
with transmission efficiency $\eta_{1}^{s}$ ($\eta_{1}^{i}$), i.e. it
undergoes a beamsplitter interaction and part of it is replaced with vacuum
noise whose value $N_{f}=0.5$ quanta. At the KIT’s input, the noise is
$N_{1}^{j}$, $j\in\\{s,i\\}$. On the signal-to-signal path, the KIT then adds
$N_{\mathrm{ex}}^{s}$ quanta of excess noise and has a gain $G$; on the idler-
to-signal path, the KIT adds $N_{\mathrm{ex}}^{i}$ quanta of excess noise and
has a gain $G-1$. Noise $N_{2}^{s}$ at the KIT’s output is then routed with
efficiency $\eta_{2}$ to the HEMT (input noise $N_{3}^{s}$). With gain $G_{H}$
and added noise $N_{H}$, it further directs the noise $N_{4}^{s}$ to room
temperature components. Amplification and loss at room temperature are
excluded from our schematic but not our analysis. The noise reaching the
spectrum analyzer (SA) is $N_{o}^{s}$. The full setup is described in appendix
E. Figure 6: System-added noise measurement of a microwave amplification chain
with a KIT as first amplifier. (a) The gain’s frequency dependence is measured
with a VNA (with a 1 kHz intermediate frequency bandwidth). (b) The output
noise $N_{o}^{s}$ is measured with a SA (5 MHz resolution bandwidth,
comparable to typical resonant amplifier’s bandwidth), while varying the SNTJ
dc voltage bias V. Fitting the whole output noise response, we obtain the
frequency dependent system-added noise $N_{\Sigma}$ and the chain’s signal-to-
signal gain $G_{c}^{ss}$. We divide $N_{o}^{s}$ by $G_{c}^{ss}$ to refer it to
the KIT input, and subtract the zero bias noise value $N_{f}=0.5$, so that
$N_{\Sigma}$ (panel c) visually matches the zero bias value of $N_{o}^{s}$
(see appendix F.3). Three colored curves indicate output noises at 4, 5, and 6
GHz, with fits superimposed in black lines. (c) Data from the output noise
spectra are compiled to form $N_{\Sigma}$. Uncertainties are indicated by the
gray area surrounding the black line. They predominantly come from the fit of
the curves in (b). The quantum limit (QL) on amplifier-added noise is
indicated by the dashed black line.
Because the SNTJ is a wideband noise source, it illuminates the KIT at both
its signal and idler frequency inputs; these input noises are then
parametrically amplified. We thus model the KIT as a three-port network: two
inputs at $\omega_{s}$ and $\omega_{i}$, and one output at $\omega_{s}$.
Figure 5 schematizes the entire amplification chain, where we have labeled the
gain or transmission efficiency, and input-referred added noise of each
component. The KIT power gain on the signal-to-signal path is $G$, while its
power gain on the idler-to-signal path is $G-1$ Caves (1982). The output noise
at the signal frequency (in units of photons) measured on the spectrum
analyzer (SA) can then be written as
$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s})+G_{c}^{si}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i}),$
(4)
where $N_{\mathrm{in}}^{s}$ ($N_{\mathrm{in}}^{i}$) is the SNTJ-generated
noise at the signal (idler) frequency, $G_{c}^{ss}$ ($G_{c}^{si}$) is the
signal-to-signal (idler-to-signal) gain of the entire chain from SNTJ to SA,
and $N_{\mathrm{eff}}^{s}$ ($N_{\mathrm{eff}}^{i}$) is the effective signal-
to-signal (idler-to-signal) path KIT-excess noise, see appendix F.1. When
varying $N_{\mathrm{in}}^{s}$ and $N_{\mathrm{in}}^{i}$ we retrieve
$N_{\mathrm{eff}}^{s}$ and $N_{\mathrm{eff}}^{i}$, equal to zero for a
quantum-limited amplifier.
The system-added noise that a device replacing the SNTJ would see is therefore
$N_{\Sigma}=N_{\mathrm{eff}}^{s}+\frac{G_{c}^{si}}{G_{c}^{ss}}(N_{f}+N_{\mathrm{eff}}^{i}),$
(5)
where $N_{f}=0.5$ quanta is the vacuum noise (provided that the idler port of
this device is cold); for a high gain, phase-insensitive quantum-limited
amplifier $N_{f}$ is the minimum added noise Caves (1982). Failure to account
for the additional change in idler noise at the amplifier’s input leads to an
underestimate of the system-added noise by about a factor two, thereby making
it look significantly better than its true value, given by Eq. 5, see appendix
F.2.
In practice, we measure $N_{o}^{s}$ while varying simultaneously
$N_{\mathrm{in}}^{s}$ and $N_{\mathrm{in}}^{i}$ using the SNTJ voltage bias
(Fig. 6b). We then fit to obtain $G_{c}^{ss}$, $G_{c}^{si}$,
$N_{\mathrm{eff}}^{s}$ and $N_{\mathrm{eff}}^{i}$ (see appendix F.3), and form
$N_{\Sigma}$ with Eq. 5. Figure 6c presents the system-added noise
$N_{\Sigma}$, measured over the KIT’s amplification bandwidth. In this
experimental configuration, the KIT gain is $G=16.6^{+1.8}_{-3.1}$ dB between
3.5 and 5.5 GHz (Fig. 6a), stable over the acquisition time ($\sim 12$ hrs).
In that bandwidth, $N_{\Sigma}=3.1\pm 0.6$ quanta. It is an unprecedented
broadband, true system-added noise performance (see appendix H).
This performance depends on the intrinsic signal (idler) KIT-excess noise
$N_{\mathrm{ex}}^{s}$ ($N_{\mathrm{ex}}^{i}$), but also on the chain’s
transmission efficiencies as well as on the HEMT-added noise $N_{H}$. More
precisely, when $\\{G,N_{H}\\}\gg 1$, Eq. 5 becomes:
$N_{\Sigma}=\frac{N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}}{\eta_{1}^{s}}+\frac{2(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{N_{H}}{\eta_{2}G\eta_{1}^{s}}+N_{f}.$
(6)
From left to right, the first three terms in the right-hand side represent the
contribution to $N_{\Sigma}$ from the KIT alone, from the lossy link between
the SNTJ and the KIT, and from the HEMT-added noise; the last term is the
minimum half quantum of added noise that a quantum-limited amplifier must add
Caves (1982). Measuring the individual loss of the chain’s components, we
estimate the value of the transmission efficiencies to be
$\eta_{1}^{s}=0.57\pm 0.02$ and $\eta_{2}=0.64\pm 0.10$; in addition, by
measuring the system-added noise of the amplification chain with the KIT
turned off, we estimate $N_{H}=8\pm 1$ quanta, see appendix G. With these
additional information, we estimate that the overall KIT-excess noise is
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}=0.77\pm 0.40$ quanta, suggesting that
the KIT alone operates near the quantum limit.
Several strategies can improve the system-added noise. First, increasing the
transmission efficiencies would have a major impact, because all the system-
dependent noise contributions are enlarged by $1/\eta_{1}^{s}$. For example,
$\eta_{1}^{s}=0.8$ would yield $N_{\Sigma}=1.6$ quanta. To that end, we are
currently developing lossless superconducting circuits (bias tees, directional
couplers and low-pass filters) that can be integrated on-chip with the KIT.
Second, increasing the KIT gain $G$ would reduce the HEMT contribution to
$N_{\Sigma}$ (third term in Eq. 6). Here, this contribution is estimated at
$0.5\pm 0.3$ quanta. The solutions to achieve higher gain directly follow from
Eq. 3: higher pump power (i.e. increase $\delta_{L}$), longer line (increase
$N_{c}$), or higher inductance per unit cell (increase $k_{p}$). All represent
non-trivial challenges, starting with better understanding of the line
breakdown mechanism Bockstiegel _et al._ (2014). If it comes from
imperfections in the line (weak links), a higher resolution fabrication
process, like electron beam lithography, may improve the performance of the
device. Also, running the amplifier at higher gain will require better damping
of the gain ripples, whose amplitude grows with gain. Finally, we are
investigating the origin of the remaining excess noise
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}$. It may be due to parasitic chip
heating or two-level system noise Gao _et al._ (2008).
## V CONCLUSION
It is possible to build a microwave amplifier with broadband and near-quantum-
limited performance without sacrificing power handling capability. Engineering
the phase-matched bandwidth is key because it suppresses spurious parametric
conversion processes. We demonstrate this idea on a KIT, whose combined gain,
bandwidth, power handling, and noise performance are fully characterized. In
addition, we develop a theoretical framework adapted to noise measurements
performed with wideband noise sources. Using a SNTJ we evaluate the true
system-added noise of an amplification containing the KIT as the first
amplifier. The KIT itself is estimated to be near-quantum-limited, therefore
it has the potential to initiate a qualitative shift in the way arrays of
superconducting detectors, such as MKIDs, process information, moving them
into a quantum-limited readout regime.
## Acknowledgments
We thank Kim Hagen for his help in the design and fabrication of the KIT
packaging, and Felix Vietmeyer and Terry Brown for their help in the design
and fabrication of room temperature electronics. We acknowledge useful
discussions with John Teufel, Gangqiang Liu and Vidul Joshi. Certain
commercial materials and equipment are identified in this paper to foster
understanding. Such identification does not imply recommendation or
endorsement by the National Institute of Standards and Technology, nor does it
imply that the materials or equipment identified are necessarily the best
available for the purpose. This work was supported by the NIST Innovations in
Measurement Science Program, as well as NASA, under grant NNH18ZDA001N-APRA.
## Appendix A COUPLED-MODE THEORY OF A DC-BIASED KIT
### A.1 Coupled-mode equations
The phase matching condition for exponential gain, Eq. 2 is obtained by
solving the CME while pumping in a 3WM fashion, i.e. such that
$\omega_{p}=\omega_{s}+\omega_{i}$, and in the presence of 3WM and 4WM terms
see, Eq. 1. The CME relate the current amplitudes $I_{j}$, $j\in\\{p,s,i\\}$
at the frequencies $\omega_{j}$, $j\in\\{p,s,i\\}$; they are obtained by
injecting equation 1 into the telegrapher’s equations, and by operating the
harmonic balance (HB) with only these three frequencies.
More precisely, the telegrapher’s equations in a lossless transmission line
relate current I and voltage V as:
$\displaystyle-\frac{\partial I}{\partial x}$ $\displaystyle=C\frac{\partial
V}{\partial t}$ (7) $\displaystyle-\frac{\partial V}{\partial x}$
$\displaystyle=L\frac{\partial I}{\partial t},$
with x a length per unit cell. Injecting Eq. 1 into Eqs. 7, we obtain:
$v_{p}^{2}\frac{\partial^{2}I}{\partial x^{2}}-\frac{\partial^{2}I}{\partial
t^{2}}=\frac{\partial}{\partial t^{2}}\left(\frac{1}{2}\epsilon
I^{2}+\frac{1}{3}\xi I^{3}\right),$ (8)
with $v_{p}=1/\sqrt{CL_{d}}$ the phase velocity. To solve Eq. 8 we perform the
HB: we assume that the current in the transmission line is a sum of three
terms at three different frequencies:
$\displaystyle I=\frac{1}{2}\big{(}$ $\displaystyle
I_{p}(x)e^{i(k_{p}x-\omega_{p}t)}$ (9) $\displaystyle+$ $\displaystyle
I_{s}(x)e^{i(k_{s}x-\omega_{s}t)}$ $\displaystyle+$ $\displaystyle
I_{i}(x)e^{i(k_{i}x-\omega_{i}t)}+c.c\big{)},$
and we then keep only the mixing terms from Eq. 8 that emerge at these
frequencies. This approach is valid in our case, because the phase matching
bandwidth is limited by dispersion engineering (see appendix B), and thus
mostly these three frequencies are able to mix together. Under the slow-
varying envelope approximation, $\lvert d^{2}I_{j}/dx^{2}\rvert\ll\lvert
k_{j}dI_{j}/dx\rvert$ for $j\in\\{p,s,i\\}$, the left hand side of Eq. 8
yields:
$v_{p}^{2}\frac{\partial^{2}I}{\partial x^{2}}-\frac{\partial^{2}I}{\partial
t^{2}}=v_{p}^{2}\sum_{j=p,s,i}ik_{j}\frac{dI_{j}}{dx}e^{i(k_{j}x-\omega_{j}t)}+c.c.$
(10)
Using $\omega_{p}=\omega_{s}+\omega_{i}$, we collect terms at $\omega_{j}$,
$j\in\\{p,s,i\\}$ in the right hand side (rhs) and form the CME:
$\displaystyle\frac{dI_{p}}{dx}$
$\displaystyle=\frac{ik_{p}\epsilon}{4}I_{s}I_{i}e^{-i\Delta_{k}x}+\frac{ik_{p}\xi}{8}I_{p}(\lvert
I_{p}\rvert^{2}+2\lvert I_{s}\rvert^{2}+2\lvert I_{i}\rvert^{2})$ (11)
$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}e^{i\Delta_{k}x}+\frac{ik_{s}\xi}{8}I_{s}(2\lvert
I_{p}\rvert^{2}+\lvert I_{s}\rvert^{2}+2\lvert I_{i}\rvert^{2})$
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}e^{i\Delta_{k}x}+\frac{ik_{i}\xi}{8}I_{i}(2\lvert
I_{p}\rvert^{2}+2\lvert I_{s}\rvert^{2}+\lvert I_{i}\rvert^{2}),$
with $\Delta_{k}=k_{p}-k_{s}-k_{i}$. The phase matching condition, Eq. 2, is
found for a strong pump, where $\\{\lvert I_{s}\rvert,\lvert
I_{i}\rvert\\}\ll\lvert I_{p}\rvert$. Assuming the pump undepleted, $\lvert
I_{p}(x)\rvert=I_{p0}$, Eqs. 11 rewrite:
$\displaystyle\frac{dI_{p}}{dx}$
$\displaystyle=\frac{ik_{p}\xi}{8}I_{p}I_{p0}^{2}$ (12)
$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}e^{i\Delta_{k}x}+\frac{ik_{s}\xi}{4}I_{s}I_{p0}^{2}$
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}e^{i\Delta_{k}x}+\frac{ik_{i}\xi}{4}I_{i}I_{p0}^{2},$
which results in $I_{p}(x)=I_{p0}\exp{(i\xi k_{p}I_{p0}^{2}x/8)}$. Signal and
idler amplitudes are then searched with the form:
$I_{j}(x)=\tilde{I}_{j}(x)\exp{(i\xi I_{p0}^{2}k_{j}x/4)}$, $j\in\\{s,i\\}$.
Equations 12 then yield:
$\displaystyle\frac{d\tilde{I}_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p0}\tilde{I}_{i}^{*}e^{i\Delta_{\beta}x}$
(13) $\displaystyle\frac{d\tilde{I}_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p0}\tilde{I}_{s}^{*}e^{i\Delta_{\beta}x},$
with $\Delta_{\beta}=\Delta_{k}+\frac{\xi I_{p0}^{2}}{8}(k_{p}-2k_{s}-2k_{i})$
and $\Delta_{k}=k_{p}-k_{s}-k_{i}$. The system of equations 13 has known
solutions Boyd (2019). In particular, when phase matching is achieved, i.e.
$\Delta_{\beta}=0$, we obtain:
$\displaystyle\tilde{I}_{s}$ $\displaystyle=\cosh{(g_{3}x)\tilde{I}_{s0}}$
(14) $\displaystyle\tilde{I}_{i}$
$\displaystyle=i\sqrt{\frac{k_{i}}{k_{s}}}\sinh{(g_{3}x)\tilde{I}_{s0}},$
with $g_{3}=\frac{\epsilon I_{p0}}{4}\sqrt{k_{i}k_{s}}$, and with initial
conditions $I_{s}(0)=I_{s0}$ and $I_{i}(0)=0$. The signal power gain
$G_{s}(x)=\left\lvert\frac{I_{s}(x)}{I_{s0}}\right\rvert^{2}$ (15)
is then exponential with $x$: $G_{s}=\cosh^{2}(g_{3}x)$.
### A.2 3WM gain
We can re-write $g_{3}$ as a function of more meaningful quantities. In fact,
the linear inductance $L$ exposed in Eq. 1 also writes:
$\displaystyle L$
$\displaystyle=L_{0}\left(1+\frac{I_{d}^{2}}{I_{*}^{2}}\right)\left(1+\frac{2I_{d}I}{I_{*}^{2}+I_{d}^{2}}+\frac{I^{2}}{I_{*}^{2}+I_{d}^{2}}\right)$
(16) $\displaystyle\simeq
L_{d}\left(1+\frac{2I_{d}I}{I_{*}^{2}+I_{d}^{2}}\right),$
when $I\ll I_{d}$, and with $L_{d}=L_{0}(1+I_{d}^{2}/I_{*}^{2})$. Here,
$L_{0}$ is the bare linear inductance. It is the one directly derived from the
sheet kinetic inductance and the geometry of the line, while $L_{d}$ is the
inductance per unit length under dc bias. Because $I_{d}^{2}\ll I_{*}^{2}$,
for design purposes $L_{d}\sim L_{0}$. In the strong pump regime $I=I_{p0}$,
therefore, $2I_{d}I/(I_{*}^{2}+I_{d}^{2})=\epsilon I_{p0}$; assuming
$k_{s}=k_{i}\simeq k_{p}/2$, we therefore get:
$G_{s}(x)\simeq\cosh^{2}\left(\frac{1}{8}\delta_{L}k_{p}x\right),$ (17)
where
$\delta_{L}=\frac{L-L_{d}}{L_{d}}=\frac{2I_{d}I_{p0}}{I_{*}^{2}+I_{d}^{2}}$
(18)
is the relative inductance variation due to $I_{p0}$. With $I_{*}=7$ mA, and
typical values: $I_{d}=1.5$ mA (limited by the dc critical current, $\sim 2.4$
mA, value specific to our NbTiN film and to the line’s width) and
$I_{p0}=I_{*}/60$, we get $\delta_{L}=6.8\times 10^{-3}$.
### A.3 Pump phase shift
From the phase matching condition, Eq. 2, it is clear that only the 4WM term
$\xi$ creates a dispersive phase shift of pump, signal and idler. In other
words, in a pure 3WM case, $\xi=0$ and the phase matching condition becomes
$\Delta_{k}=0$, naturally fulfilled in a dispersion-less transmission line.
While detrimental for noise properties (see Sec. IV), we can use the continued
presence of 4WM to our advantage, because it allows us to calibrate the pump
power, down to the KIT input.
In fact, in such a situation $I_{d}$ and $I_{p0}$ influence the pump tone
phase shift, which we can measure unambiguously (i.e. not $\mod 2\pi$) with a
VNA. More precisely, although the phase $\phi=\arg(S_{21})$ read by a VNA is
$2\pi$-wrapped, its shift $\delta_{\phi}=\phi-\phi_{0}$ from an initial value
$\phi_{0}$ can be continuously monitored when $I_{d}$ and $I_{p0}$ vary, and
thus unambiguously determined. This phase shift in turn translates into a
wavenumber variation $\delta_{p}=-\delta_{\phi}/N_{c}$. If initially at zero
dc bias and small rf pump amplitude, then $\delta_{p}=\beta_{p}-k_{p}$, with
$\beta_{p}$ the pump wavenumber, dependent on $I_{d}$ and $I_{p0}$, and
$k_{p}$ the linear wavenumber. When a single pump tone travels along the line
(no input signal), we are by default under the strong pump approximation, and
the first equation of Eqs. 12 gives $I_{p}(x)=I_{p0}\exp{(i\xi
k_{p}I_{p0}^{2}x/8)}$. In addition, the current I in the line then writes as
$I=1/2\\{I_{p}(x)\exp[i(k_{p}x-\omega_{p}t)]+c.c\\}$, and thus the pump
wavenumber is $\beta_{p}=\xi k_{p}I_{p0}^{2}/8+k_{p}$, which leads to
$\delta_{p}=\xi k_{p}I_{p0}^{2}/8$. Because $k_{p}=\omega_{p}\sqrt{L_{d}C}$,
we can rewrite:
$\delta_{p}=\frac{1}{8}\frac{I_{p0}^{2}}{I_{*}^{2}}\omega_{p}\sqrt{L_{0}C}\sqrt{1+\frac{I_{d}^{2}}{I_{*}^{2}}},$
(19)
therefore $I_{p0}$ and $I_{d}$ induce similar phase shift in the line. Knowing
$I_{d}$ (room temperature parameter), we thus get $I_{p0}$ at the KIT input.
## Appendix B ABCD TRANSFER MATRICES
The dispersion relations, Fig. 2a are calculated by cascading the ABCD
matrices of the KIT cells, a method suitable for any periodic loading pattern.
We then compute the KIT $S_{21}$ scattering parameter as
$S_{21}=2/(A+B/Z_{0}+CZ_{0}+D)$ Pozar (2011), where $Z_{0}=50$ Ω is the input
and output ports impedance, and finally get
$k=-\mathrm{unwrap}[\arg(S_{21})]/N_{c}$.
In the unloaded case, represented in Fig. 1, the ABCD matrix cell is:
$\bm{T_{c}}=\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}.$
(20)
All the cells being identical, the KIT’s ABCD matrix is simply
$\bm{T_{K}}=\bm{T_{c}}^{N_{c}}$. In Fig. 2a we used $N_{c}=6.6\times 10^{4}$
to match the length of our fabricated KIT, and $L_{d}=45.2$ pH, $C=18.8$ fF,
and $L_{f}=1.02$ nH, values that match our design (fingers are $102$ $\upmu$m
long and $1$ $\upmu$m wide).
In the loaded case, some cells have shorter fingers, see Fig. 3a. In these
cells, the capacitance to ground is $C_{l}=L_{d}/Z_{l}^{2}$, where $Z_{l}$ is
the load impedance, and a finger’s inductance is $L_{l}$. To compute the KIT
scattering parameter, we form the ABCD matrix of the repetition pattern
comprised with unloaded and loaded cells:
$\displaystyle\bm{T_{\mathrm{sc}}}=$
$\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}^{N_{u}/2}$
(21) $\displaystyle\times$ $\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C_{l}\omega}{2-L_{l}C_{l}\omega^{2}}&1-\frac{2L_{d}C_{l}\omega^{2}}{2-L_{l}C_{l}\omega^{2}}\end{bmatrix}^{N_{l}}$
$\displaystyle\times$ $\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}^{N_{u}/2},$
where $N_{u}$ is the number of unloaded cells and $N_{l}$ is the number of
loaded cells in the pattern, which we call a supercell. As before, to get the
KIT’s ABCD matrix, we simply form
$\bm{T_{K}}=\bm{T_{\mathrm{sc}}}^{N_{\mathrm{sc}}}$, where
$N_{\mathrm{sc}}=N_{c}/(N_{u}+N_{l})$ is the number of supercells in the KIT.
Here, $N_{u}=60$ (equivalent to $300\upmu$m), $N_{l}=6$ (equivalent to
$30\upmu$m), $N_{c}=66000$ (equivalent to $33$ cm), therefore
$N_{\mathrm{sc}}=1000$. The plain line in Fig. 2a shows the wavenumber $k^{*}$
thus found, with $Z_{l}=80$ Ω, and $L_{l}=335$ pH, as the finger length in a
loaded cell is $33.5$ $\upmu$m .
To compute the signal power gain, Fig. 2b, we inject the expression of
$k(\omega)$ for a periodically loaded KIT (from $\bm{T_{K}}$) in the CME, Eqs.
11. We solve them numerically for different pump frequencies. For $8.8812$ GHz
(blue curve), $8.8992$ GHz (red), $8.9256$ GHz (green) and $8.9736$ GHz
(purple), phase matched signal and idler are detuned from the half pump
frequency by $0$, $1$, $1.5$ and $2$ GHz respectively. For $8.855$ GHz (gray
curve), phase matching is nowhere achieved. We used $N_{c}=6.6\times 10^{4}$,
$I_{*}=7$ mA, $I_{d}=1.5$ mA, and the initial conditions $I_{p0}=I_{*}/60$,
$I_{s0}=I_{p0}/100$, and $I_{i0}=0$, close to experimental values.
## Appendix C KIT SATURATION
### C.1 Strong signal gain profile asymmetry
When the input signal power amounts to a significant fraction of the pump
power, parametric amplification depletes the pump. It surprisingly generates
asymmetry in the signal gain profile, with respect to the half pump frequency.
Figure 7a shows signal gain profiles, calculated when phase matching is
achieved at exactly half the pump frequency, i.e. for $\omega_{s}=\omega_{i}$
(corresponding to $\omega_{p}=8.8812$ GHz), at various initial signal powers
$P_{s0}$. They are obtained by solving the CME 11, which incorporate pump
depletion effects. As $P_{s0}$ increases, the gain diminishes, and the
originally flat profile tilts, with higher frequencies presenting higher gain.
Figure 7: Theory of KIT saturation, calculated by solving the full CME 11. (a)
Distortion of the gain profile, at four different input signals:
$I_{s0}=I_{p0}/100$ (blue curve), corresponding to a small signal regime,
$I_{s0}=I_{p0}/12$ (red), $I_{s0}=I_{p0}/8$ (green), and $I_{s0}=I_{p0}/6$
(purple). The KIT length, is $N_{c}=6.6\times 10^{4}$ cells. Vertical lines
indicate three frequencies: 3.4406 GHz (long dashed), 4.4406 GHz (plain),
equal to the half pump frequency, and 5.4406 GHz (short dashed). (b) The gain
of a probe tone is calculated at these three frequencies, as a function of the
probe tone power. (c) The 1 dB compression power is shown as a function of
frequency.
Fundamentally, this asymmetry stems from the fact that when solving the CME in
the case where $I_{p}$ varies along the KIT transmission line, the initial
conditions (ICs) vary as a function of frequency. More precisely, the second
and third equations in Eqs. 11, govern the evolution of $I_{s}$ and $I_{i}$
respectively; in a simplified version, they write as
$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}$ (22)
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}.$
We dropped the second terms in the rhs of Eqs. 11, representing the 4WM
conversion processes, as the asymmetry still holds when $\xi=0$, and we
assumed perfect phase matching in 3WM, $\Delta_{k}=0$, i.e. a dispersionless
line.
In the undepleted pump regime $I_{p}(x)=I_{p0}$, and we can decouple the
equations on $I_{s}$ and $I_{i}$. Deriving with respect to $x$ Eqs. 22, we get
$\frac{d^{2}I_{j}}{dx^{2}}=g_{3}^{2}I_{j},$ (23)
with $j\in\\{s,i\\}$, and $g_{3}=\frac{\epsilon I_{p0}}{4}\sqrt{k_{i}k_{s}}$,
as defined in appendix A.1. Using the ICs $I_{s}(0)=I_{s0}$ and
$dI_{s}/dx(0)=0$ (because $I_{i}^{*}(0)=0$), $I_{s}=\cosh(g_{3}x)I_{s0}$, as
seen in Eqs. 14. Signal and idler wavenumbers appear as a product in this
solution, hence for any $x$ the signal amplitude $I_{s}$ is symmetric with
respect to the half pump frequency.
In the soft pump regime, where $I_{p}(x)$ is not constant, Eqs. 22 cannot be
decoupled. We can however write them in a canonical form, deriving with
respect to $x$:
$\frac{d^{2}I_{j}}{dx^{2}}-\frac{1}{I_{p}}\frac{dI_{p}}{dx}\frac{dI_{j}}{dx}-\frac{k_{s}k_{i}\epsilon^{2}}{16}\lvert
I_{p}\rvert^{2}I_{j}=0,$ (24)
with $j\in\\{s,i\\}$. Here, the interplay between $I_{s}$ and $I_{i}$ comes
from $I_{p}$, which contains the product $I_{s}I_{i}$ (see Eqs. 11). Although
signal and idler wavenumbers also appear only as a product in Eqs. 24, these
CME lead to an asymmetric signal amplitude profile, because out of the five
ICs required to solve them, one changes with frequency: $I_{p}(0)=I_{p0}$,
$I_{s}(0)=I_{s0}$, $I_{i}(0)=0$, $dI_{s}/dx(0)=0$, and $dI_{i}/dx(0)=i\epsilon
I_{p0}I_{s0}k_{i}/4$. This last IC depends on $k_{i}$, which depends on the
signal frequency. In the small signal limit, $dI_{i}/dx(0)\xrightarrow{}0$,
and we recover a symmetric gain profile with respect to the half pump
frequency.
### C.2 Compression power calculation
This asymmetry produces higher power handling capabilities at frequencies
above $\omega_{p}/2$, compared to below $\omega_{p}/2$. Thus, considering that
only half the bandwidth is usable for resonator readout, it is more
advantageous to have these lie above $\omega_{p}/2$. Figure 7b shows the gain
as a function of a probe tone power $P_{t}$, calculated from the CME 11 at
three frequencies: one at $\omega_{p}/2$, and two at $\omega_{p}/2\pm 1$ GHz.
Because phase matching is set to be optimal at
$\omega_{s}=\omega_{i}=\omega_{p}/2$, gain is maximal at this frequency. The
small signal gain is identical for $\omega_{p}/2\pm 1$ GHz, however it visibly
compresses at higher tone power for $\omega_{p}/2+1$ GHz. Figure 7c presents
the 1 dB compression power $P_{\mathrm{-1dB}}$, calculated in the interval
$[\omega_{p}/2-1,\omega_{p}/2+1]$ gigahertz. As expected, $P_{\mathrm{-1dB}}$
is a few dB higher when $\omega>\omega_{p}/2$. This phenomenon is reminiscent
of gain distortion, seen in JPAs Malnou _et al._ (2018). Effects not included
in the CME 11, such as standing wave patterns, or defects in the line, which
locally lower $I_{*}$, may cause the discrepancy between these theoretical
calculations and the measurements (see Sec. III)
## Appendix D KIT PACKAGING
There are three main concerns when packaging a KIT: first, the package should
be matched to 50 Ω. Any mismatch will result in reflections, creating gain
ripples (see Sec. III). Second, the package should ensure good electrical
grounding of the transmission line. Otherwise, given the fairly large chip
size, spurious chip modes can appear within the frequency range of interest.
Third, the package should ensure good thermalization inside the chip. Because
the pump power remains high for millikelvin operations, any inhomogeneous rise
in temperature may trigger a hot spot near a weak-link, and possibly break
superconductivity. We implemented a series of technologies to address these
concerns.
Figure 8: KIT packaging. (a) The KIT is clamped on a gold plated copper box
and wire-bonded to PCBs and to the box itself. Although not very sensitive to
magnetic field, the KIT is shielded with aluminum and A4K cryogenic shielding.
Two SMA connectors protrude on both side of the packaging. (b) A close-up on
the central region of the KIT shows the periodically loaded transmission line,
with gold strips deposited between its spiral arms. (c) The top copper lid
contains spring-loaded pogo pins inserted half-way into the box. They are
arranged to contact the chip between the KIT trace. (d) A close-up on the pins
shows their top thinner part, which can retract inside the body of the pin.
They are fixed on the copper lid with dried silver paint.
Figure 8a presents the chip, clamped onto the bottom part of the copper
packaging. The chip is wire-bonded on both sides to printed circuit boards
(PCBs). They convert the on-chip CPW layout to microstrip, and then the
central pin of sub-miniature version A (SMA) connectors are soldered onto the
microstrip. We suspect imperfect PCBs, with measured impedance close to 52 Ω,
play a role in creating gain ripples. When designing the spiral, we carefully
adjusted the radius of the turns, in conjunction with the unit cell length, to
have these turns match the straight sections’ inductance and capacitance per
length.
Electrical grounding inside the chip is ensured with pogo pins inserted in the
top lid of the packaging, see Fig. 8c and d. When closing the box, these pins
contact the chip between the line traces. If absent, we have measured spurious
resonant modes with harmonics at gigahertz frequencies. Pins are $140$
$\upmu$m in diameter, and each applies a 20 g force to the chip.
Figure 9: Full schematic of the noise measurement experimental setup. The KIT
is represented by a spiral, enclosed in a square. A permanent magnet
suppresses the Josephson effect in the SNTJ. A magnetic shield protects other
microwave components from the effect of this magnet. The 4-12 GHz isolator is
from Quinstar technology, model CWJ1015-K13B. The dashed purple line
represents the bypass.
These pins also act as thermal links to the packaging. In addition, we
deposited gold strips onto the NbTiN layer, inside the spiral, between the KIT
line traces, and near the chip edges, see Fig. 8b. These strips contact the
pins. Absent the pogo pins, superconductivity breaks down before high gain can
be reached. Finally, we gold-bond the chip ground plane (from the deposited
gold) to the copper box (instead of using standard aluminum bonding): gold
remains a normal metal at millikelvin temperatures, thereby better
thermalizing the chip.
## Appendix E NOISE MEASUREMENT EXPERIMENTAL SETUP
Figure 9 presents a schematic of the full experimental setup used to measure
the system-added noise of a readout chain using a KIT as a pre-amplifier. In
total, noise generated by the SNTJ travels through three amplifiers: the KIT,
a HEMT at 4K, and a room temperature amplifier, before finally being recorded
with a SA.
The SNTJ is packaged with a permanent magnet suppressing the Josephson effect,
and a magnetic shield protects other elements from this magnetic field. In
addition, a bias tee routes SNTJ-generated rf noise to the microwave readout
chain, while at the same time allowing for dc bias. In fact, the SNTJ is
biased with an arbitrary waveform generator (AWG). It outputs a low frequency
(50 Hz) triangular voltage wave on a $10$ kΩ current limiting resistor,
thereby creating a current $I_{\mathrm{AWG}}$, varying between $\pm
12$$\upmu$A, which sweeps the SNTJ-generated noise value.
An oscilloscope reads the SNTJ voltage in situ while the AWG outputs a known
current, allowing the computation of the SNTJ impedance
$Z_{\mathrm{SNTJ}}=54\pm 4$ Ω, and with it, the SNTJ voltage bias
$V=Z_{\mathrm{SNTJ}}I_{\mathrm{AWG}}$.
Noise from the SNTJ is combined with rf tones (pump, and probe from a VNA to
measure the gain profile) via a 20 dB directional coupler (DC) connected to
the KIT input. Additionally, a 7 GHz low-pass filter placed between the SNTJ
and the DC prevents the rf pump from leaking back to the SNTJ. Because the KIT
requires a fairly high pump power ($-29$ dBm), we only attenuate the pump line
by 10 dB at 4K. Then, an 8 GHz high-pass filter at 30 mK rejects noise at
frequencies within the KIT amplification band, while allowing the pump to
pass. A bias tee at the KIT input port combines rf signals (including noise
from the SNTJ) with the KIT dc current bias, and a second bias tee at the KIT
output separates dc from rf. The rf signal then passes through a $4-12$ GHz
isolator, and a 7 GHz low-pass filter (Pasternack PE87FL1015, $\sim 45$ dB
rejection at 9 GHz), preventing the pump tone from saturating the HEMT.
The SA is operated in a zero-span mode, its acquisition triggered by the AWG.
This measures the output noise at a single frequency, over a 5 MHz resolution
bandwidth (RBW), and directly traces out the curves of Fig. 6b. Varying the SA
center frequency, we obtain the system-added noise over the full 3-6.5 GHz
bandwidth, Fig. 6c.
## Appendix F NOISE THEORY
### F.1 System-added noise
When propagating through the experimental setup presented in Fig. 9, noise
generated by the SNTJ undergoes loss and amplification. Both processes affect
the effective noise at each amplifier input, and therefore also, the noise
reaching the SA. While the overall system-added noise $N_{\Sigma}$ encompasses
microwave loss, we derive its complete expression as a function of signal
($N_{\mathrm{ex}}^{s}$) and idler ($N_{\mathrm{ex}}^{i}$) amplifier-excess
noise, gain, and transmission efficiencies to estimate
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}$.
Figure 5 represents the lossy amplification chain, with the transmission
efficiencies, gain and added noise associated with each interconnection and
amplification stages. We have:
$\displaystyle N_{1}^{j}$
$\displaystyle=\eta_{1}^{j}\left[N_{\mathrm{in}}^{j}+\frac{N_{f}(1-\eta_{1}^{j})}{\eta_{1}^{j}}\right]$
(25) $\displaystyle N_{2}^{s}$
$\displaystyle=G(N_{1}^{s}+N_{\mathrm{ex}}^{s})+(G-1)(N_{1}^{i}+N_{\mathrm{ex}}^{i})$
(26) $\displaystyle N_{3}^{s}$
$\displaystyle=\eta_{2}\left[N_{2}^{s}+\frac{N_{f}(1-\eta_{2})}{\eta_{2}}\right]$
(27) $\displaystyle N_{4}^{s}$ $\displaystyle=G_{H}(N_{3}^{s}+N_{H})$ (28)
$\displaystyle N_{o}^{s}$ $\displaystyle=G_{r}N_{4}^{s}.$ (29)
Here, $N_{1}^{j}$ with $j\in\\{s,i\\}$ is the KIT-input noise at the signal
and idler frequency respectively; then, at the signal frequency, $N_{2}^{s}$
is the KIT-output noise, $N_{3}^{s}$ is the HEMT-input noise, $N_{4}^{s}$ is
the HEMT-output noise, and $N_{o}^{s}$ is the noise measured by the SA. Note
that the beamsplitter interaction between the KIT and the HEMT assumes the
loss to be mostly cold, which is the case in our setup where the lossy
components before the HEMT are at the 30 mK stage. Refer to table 1 for the
other variable definitions. From Eqs. 25-29 we can derive:
$\displaystyle N_{o}^{s}$
$\displaystyle=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s})+G_{c}^{si}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i})$
(30)
$\displaystyle=G_{c}^{ss}\left[N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s}+\frac{G-1}{G}\frac{\eta_{1}^{i}}{\eta_{1}^{s}}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i})\right],$
(31)
where
$\displaystyle G_{c}^{ss}$ $\displaystyle=G_{r}G_{H}\eta_{2}G\eta_{1}^{s}$
(32) $\displaystyle G_{c}^{si}$
$\displaystyle=G_{r}G_{H}\eta_{2}(G-1)\eta_{1}^{i},$ (33)
and where
$\displaystyle N_{\mathrm{eff}}^{s}$
$\displaystyle=\frac{N_{\mathrm{ex}}^{s}+(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{(1-\eta_{2})N_{f}+N_{H}}{\eta_{2}G\eta_{1}^{s}}$
(34) $\displaystyle N_{\mathrm{eff}}^{i}$
$\displaystyle=\frac{N_{\mathrm{ex}}^{i}+(1-\eta_{1}^{i})N_{f}}{\eta_{1}^{i}}.$
(35)
The system-added noise is defined as the part of the output noise in Eq. 31
that is not due to the input $N_{\mathrm{in}}^{s}$, and by assuming a cold
input at the idler port, i.e. assuming $N_{f}=N_{\mathrm{in}}^{i}$. We thus
find
$N_{\Sigma}=N_{\mathrm{eff}}^{s}+\frac{G-1}{G}\frac{\eta_{1}^{i}}{\eta_{1}^{s}}(N_{f}+N_{\mathrm{eff}}^{i}).$
(36)
This equation is equivalent to Eq. 5, where we did not simplify the ratio
$G_{c}^{si}/G_{c}^{ss}$.
Figure 10: System-added noise measurement of a microwave amplification chain
with the HEMT as first amplifier. (a) The transmission through the whole noise
setup (KIT un-pumped), when the SNTJ has been replaced by a transmission line
capacitively coupled to an array of resonators (not used in the present
experiment). It is performed in two situations: without (black curve) and with
(purple curve) the bypass. (b) Examples of
$T_{o}^{s^{\prime}}=N_{o}^{s^{\prime}}\hbar\omega/k_{B}$ are shown for the SA
centered at 4, 5 and 6 GHz, with fits superimposed in black lines. The SA RBW
is 5 MHz. (c) From the fits, we extract the system-added noise temperature
$T_{\Sigma}^{\prime}=N_{\Sigma}^{\prime}\hbar\omega/k_{B}$ as a function of
frequency without (black curve) and with (purple curve) the bypass.
Uncertainties are indicated by the gray area surrounding the lines.
Assuming $\\{G,N_{H}\\}\gg 1$ and inserting Eqs. 34 and 35 into Eq. 36 we get
$N_{\Sigma}=\frac{N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}}{\eta_{1}^{s}}+\frac{2(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{N_{H}}{\eta_{2}G\eta_{1}^{s}}+N_{f}.$
(37)
which is Eq. 6 presented in the main text.
When the KIT is not pumped, we consider it as a lossless, noiseless, passive
element. We therefore have $G=1$, $N_{\mathrm{ex}}^{s}=0$ and
$N_{\mathrm{ex}}^{i}=0$. In that situation
$N_{o}^{s^{\prime}}=G_{c}^{ss^{\prime}}(N_{\mathrm{in}}^{s}+N_{\Sigma}^{\prime}),$
(38)
where $G_{c}^{ss^{\prime}}=G_{r}G_{H}\eta_{2}\eta_{1}^{s}$. From Eq. 36 and 34
we thus get
$N_{\Sigma}^{\prime}=\frac{(1-\eta_{2}\eta_{1}^{s})N_{f}+N_{H}}{\eta_{2}\eta_{1}^{s}}$
(39)
as the chain’s system-added noise with the HEMT as first amplifier.
### F.2 Discarding the idler port input noise
In a simple case where $G-1\simeq G$, and $\eta_{1}^{s}\simeq\eta_{1}^{i}$,
Eq. 30 becomes
$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}).$
(40)
Thus, varying the SNTJ bias i.e. $N_{\mathrm{in}}^{s}$ and
$N_{\mathrm{in}}^{i}$ simultaneously, the y-intercept gives
$N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$, equal to zero for a quantum-
limited amplifier. Also, in that simple case, the system-added noise is
$N_{\Sigma}=N_{f}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$, as shown by Eq.
36.
On the other hand, if the calibrated noise (coming from the SNTJ or any other
wideband noise source, such as a hot/cold load, or a variable temperature
stage) only illuminates the signal port of the amplifier, or if the noise at
the idler port is wrongly discarded from the analysis, we get
$N_{\mathrm{in}}^{i}=N_{f}$ and Eq. 30 becomes
$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{f}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}).$
(41)
Usually, no distinction is made between $N_{\mathrm{eff}}^{s}$ and
$N_{\mathrm{eff}}^{i}$, and the effective excess noise is simply
$N_{\mathrm{eff}}=N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$. The sum
$N_{f}+N_{\mathrm{eff}}$ is then what is commonly defined as the system-added
noise. Varying the SNTJ bias, i.e. $N_{\mathrm{in}}^{s}$, the y-intercept
gives $N_{f}+N_{\mathrm{eff}}$, equal to half a quantum for a quantum-limited
amplifier.
In practice, we fit Eqs. 40 or 41 to find the chain’s gain and the
y-intercept. Fitting a situation correctly described by Eq. 40 with Eq. 41
leads to an underestimate of the system-added noise. In fact, assuming
$N_{\mathrm{in}}^{s}\simeq N_{\mathrm{in}}^{i}$, Eq. 40 can be rewritten into
a form comparable to Eq. 41:
$N_{o}^{s}\simeq
2G_{c}^{ss}\left(N_{\mathrm{in}}^{s}+\frac{N_{\mathrm{eff}}}{2}\right).$ (42)
The interpretation of the y-intercept is crucial. Here, the fit yields
$N_{\mathrm{eff}}/2$ as the y-intercept, and $2G_{c}^{ss}$ as the chain’s
gain. The true system-added noise is then twice the y-intercept value plus a
half quantum, $N_{\Sigma}=N_{\mathrm{eff}}+N_{f}$, and the true chain’s gain
is half of the fitted slope. Conversely, assuming Eq. 41 leads to the
conclusion that the y-intercept value is already the system-added noise, and
that the slope is the chain’s gain. The true system-added noise is thus more
than twice as high, and the chain’s true gain is half as much (i.e. 3 dB
lower, a mistake usually hard to detect).
### F.3 Shot-noise fit
The SNTJ generates a known noise power Spietz _et al._ (2006). The amount of
noise $N_{\mathrm{in}}^{j}$, with $j\in\\{s,i\\}$ delivered to the 50 Ω
transmission line can be written as Lecocq _et al._ (2017)
$\displaystyle N_{\mathrm{in}}^{j}=\frac{k_{B}T}{2\hbar\omega_{j}}\bigg{[}$
$\displaystyle\frac{eV+\hbar\omega_{j}}{2k_{B}T}\coth\left(\frac{eV+\hbar\omega_{j}}{2k_{B}T}\right)$
(43) $\displaystyle+$
$\displaystyle\frac{eV-\hbar\omega_{j}}{2k_{B}T}\coth\left(\frac{eV-\hbar\omega_{j}}{2k_{B}T}\right)\bigg{]},$
where T is the physical temperature of the SNTJ, and V the SNTJ voltage bias.
In practice, the AWG has a slight voltage offset, which we include as a fit
parameter: we write $V-V_{\mathrm{off}}$ instead of V in Eq. 43. In a first
step we fit the asymptotes of the output noise response, for which $\lvert
eV/(2\hbar\omega)\rvert>3$ quanta. In that case, Eq. 43 reduces to
$N_{\mathrm{in}}^{j}=eV/(2\hbar\omega_{j})$, and thus Eq. 30 reduces to
$\displaystyle N_{o}^{s}=$ $\displaystyle G_{c}^{ss}\left(\frac{eV-
V_{\mathrm{off}}}{2\hbar\omega_{s}}+N_{\mathrm{eff}}^{s}\right)$ (44)
$\displaystyle+$ $\displaystyle G_{c}^{si}\left(\frac{eV-
V_{\mathrm{off}}}{2\hbar\omega_{i}}+N_{\mathrm{eff}}^{i}\right).$
We thus get $V_{\mathrm{off}}$, $G_{c}^{ss}$ and $G_{c}^{si}$. In a second
step, we fix $V_{\mathrm{off}}$, $G_{c}^{ss}$ and $G_{c}^{si}$ to their values
derived in the first step and fit the central region (where $\lvert
eV/(2\hbar\omega)\rvert\leq 3$ quanta) to Eq. 30 to get
$N_{\mathrm{eff}}^{s}$, $N_{\mathrm{eff}}^{i}$ and T.
The spectrum analyzer (SA) records a power spectrum $P_{o}^{s}$ (in Watts),
which we convert into a number of photons:
$N_{o}^{s}=P_{o}^{s}/(B\hbar\omega)$, where B is the SA resolution bandwidth.
Dividing by $G_{c}^{ss}$, we then refer $N_{o}^{s}$ to the chain’s input.
Finally, we can subtract $N_{f}$ (in quanta), to visually read $N_{\Sigma}$ at
$V=0$ on the SNTJ curves of Fig. 6b. In fact,
$\frac{N_{o}^{s}(0)}{G_{c}^{ss}}-N_{\mathrm{in}}^{s}(0)-\frac{G_{c}^{si}}{G_{c}^{ss}}N_{\mathrm{in}}^{i}(0)+N_{f}\frac{G_{c}^{si}}{G_{c}^{ss}}=N_{\Sigma}.$
(45)
At $V=0$,
$N_{\mathrm{in}}^{j}(0)=\frac{1}{2}\coth{\left(\frac{\hbar\omega_{j}}{k_{B}T}\right)},$
(46)
where $j\in\\{s,i\\}$, therefore if $\hbar\omega_{j}\gg k_{B}T$,
$N_{\mathrm{in}}^{j}(0)\simeq N_{f}=0.5$. Considering that $G_{c}^{ss}\simeq
G_{c}^{si}$, Eq. 45 yields:
$\frac{N_{o}^{s}(0)}{G_{c}^{ss}}-N_{f}\simeq N_{\Sigma}.$ (47)
### F.4 List of variables
Table 1 lists the variables used throughout Sec. IV and appendix F.1.
variable name | definition
---|---
$N_{\mathrm{in}}^{s}$ | SNTJ-generated noise at the signal frequency
$N_{\mathrm{in}}^{i}$ | SNTJ-generated noise at the idler frequency
$N_{f}$ | vacuum (or thermal) noise, set by the refrigerator temperature
$N_{\mathrm{ex}}^{s}$ | signal-to-signal path KIT-excess noise
$N_{\mathrm{ex}}^{i}$ | idler-to-signal path KIT-excess noise
$N_{\mathrm{eff}}^{s}$ | signal-to-signal path effective KIT-excess noise, Eq. 34
$N_{\mathrm{eff}}^{i}$ | idler-to-signal path effective KIT-excess noise, Eq. 35
$N_{H}$ | HEMT-added noise
$N_{o}^{s}$ | noise measured by the SA at the signal frequency, Eq. 30
$N_{o}^{s^{\prime}}$ | noise measured by the SA when the KIT is off, Eq. 38
$N_{\Sigma}$ | system-added noise, Eq. 36
$N_{\Sigma}^{\prime}$ | system-added noise when the KIT is off, Eq. 39
$\eta_{1}^{s}$ | transmission efficiency between SNTJ and KIT at the signal frequency
$\eta_{1}^{i}$ | transmission efficiency between SNTJ and KIT at the idler frequency
$\eta_{2}$ | transmission efficiency between KIT and HEMT at the signal frequency
$G$ | KIT signal power gain
$G_{H}$ | HEMT signal power gain
$G_{r}$ | room-temperature signal power gain
$G_{c}^{ss}$ | signal-to-signal chain’s gain, Eq. 32
$G_{c}^{si}$ | idler-to-signal chain’s gain, Eq. 33
Table 1: List of the variables used in the noise theory. All the variables
designating a noise quantity are in units of quanta. All the transmission
efficiencies are dimensionless. All the gains are in linear units.
## Appendix G LOSS BUDGET IN THE NOISE MEASUREMENT SETUP
To quote the KIT-excess noise terms $N_{\mathrm{ex}}^{s}$ and
$N_{\mathrm{ex}}^{i}$, it is necessary to account for the transmission
efficiencies (and therefore the loss) in the measurement setup,
$\eta_{1}^{s}$, $\eta_{1}^{i}$ and $\eta_{2}$. Note that $\eta_{1}^{i}$ is
simply the transmission efficiency at the idler frequency
$\omega_{i}=\omega_{p}-\omega_{s}$, symmetric of the signal frequency
$\omega_{s}$ with respect to the half-pump frequency $\omega_{p}/2$. Thus, a
broadband measurement of $\eta_{1}^{s}$ on both sides of $\omega_{p}/2$
provides both $\eta_{1}^{s}$ and $\eta_{1}^{i}$. We estimate these
efficiencies in this appendix and give an overall loss budget per component
between the SNTJ and the HEMT. Knowing where the loss comes from also provides
guidance on how to improve the amplifier because many lossy components could
be optimized, particularly by integrating them on-chip with the KIT.
### G.1 System-added noise temperature with un-pumped KIT
First, we measured the system-added noise temperature of the amplification
chain $T_{\Sigma}^{\prime}=N_{\Sigma}^{\prime}\hbar\omega/k_{B}$ (with $\hbar$
the reduced Planck’s constant and $k_{B}$ the Boltzmann’s constant) when the
KIT is off, i.e. with the HEMT as the first amplifier. In that case,
$N_{\Sigma}^{\prime}$ is given by Eq. 39. We measured $N_{\Sigma}^{\prime}$ in
two situations: one for the measurement setup presented in Fig. 9, and one
where the KIT surrounded by its two bias tees, the directional coupler and the
low pass filter next to it have been bypassed, i.e. replaced by a microwave
cable. In the limit where $N_{f}\ll N_{\Sigma}^{\prime}$, the ratio in
$N_{\Sigma}^{\prime}$ between these two situations gives direct access to the
bypassed components’ insertion loss (IL) $\mathcal{I}_{\mathrm{BP}}$ (the
exact transmission efficiency ratio from which we calculate
$\mathcal{I}_{\mathrm{BP}}$ is equivalent to finding $\eta_{2}\eta_{1}^{s}$ in
Eq. 39). Figure 10c shows $T_{\Sigma}^{\prime}$ as a function of frequency,
obtained from fitting curves like those presented in Fig. 10b, obtained with
and without the bypass. There are ripples with 130 MHz characteristic
frequency, likely due to reflections in coaxial cables between the SNTJ and
the HEMT. Without the bypass $T_{\Sigma}^{\prime}=5.1\pm 1.4$ K between 3.5
and 5.5 GHz, while with the bypass $T_{\Sigma}^{\prime}=2.9\pm 1$ K. Thus, the
ratio gives $\mathcal{I}_{\mathrm{BP}}=2.5$ dB.
### G.2 Component loss
Second, we replaced the SNTJ (including the bias tee) with a transmission
line, to measure the transmission of a probe tone through the setup (with a
VNA). This transmission line is capacitively coupled to an array of
resonators, whose resonant frequencies span between $4$ and $5$ GHz, and is
part of a future experiment. These resonances are not relevant for the current
characterization. We measured the transmission with and without the bypass,
shown in Fig. 10a. Once again, the transmission ratio between these two
situations gives direct access to the bypassed components’ IL. We find
$\mathcal{I}_{BP}=2.4\pm 0.6$ dB, in agreement with the system-added noise
measurements of Fig. 10c.
authors | mixing | | pump
---
power [dBm]
| gain
---
[dBm]
| bandwidth
---
[GHz]
| saturation
---
[dBm]
| noise
---
bandwidth
| system
---
noise [K]
this work | 3WM | $-29$ | $16.5^{+1}_{-1.3}$ | 2 | $-63$ | $3$-$6.5$ GHz | $0.66\pm 0.15$
Eom et.al. Eom _et al._ (2012) | 4WM | $-8$ | $10^{+3}_{-3}$ | $4$ | $-52$ | $1$ Hz | $1.5$111idler noise input not accounted for
Bockstiegel et.al. Bockstiegel _et al._ (2014) | 4WM | $-10$ | $20^{+3}_{-3}$ | $8$ | - | - | -
Vissers et.al. Vissers _et al._ (2016) | 3WM | $-10$ | $20^{+5}_{-5}$ | $4$ | $-45$222at $10$ dB gain | - | -
Chaudhuri et.al. Chaudhuri _et al._ (2017) | 4WM | $-10$ | $15^{+3}_{-3}$ | $3$ | - | - | -
Ranzani et.al. Ranzani _et al._ (2018) | 3WM | $-30$ | $10^{+2}_{-2}$ | $4$ | - | $6$-$10$ GHz | $1.5$-$5^{\mathrm{\ref{ftn:ftn2}}}$
Zobrist et.al. Zobrist _et al._ (2019) | 3WM | $-23$ | $15$333data quoted but not shown in the paper | $5^{\mathrm{\ref{ftn:ftn1}}}$ | $-53^{\mathrm{\ref{ftn:ftn1}}}$ | 1 MHz | ${0.58^{+0.2}_{-0.03}}^{\mathrm{\ref{ftn:ftn2}}}$
Table 2: Comparison of our KIT to other published KIT results. As discussed in
appendix F.2, failure to account for the noise at the idler input results in
an underestimate of the system-added noise temperature by about a factor of
two.
We also measured the individual transmissions of the chain’s components at 4K:
bias tee (BT), filter (LPF), directional coupler (DC) and isolator (ISO).
Figure 11a shows the IL of these four components. Between $3.5$ and $5.5$ GHz,
$\mathcal{I}_{\mathrm{BT}}=0.3\pm 0.04$ dB, $\mathcal{I}_{\mathrm{LPF}}=0.2\pm
0.1$ dB, $\mathcal{I}_{\mathrm{DC}}=0.2\pm 0.04$ dB, and
$\mathcal{I}_{\mathrm{ISO}}=0.7\pm 0.6$ dB. Its IL increases below $4$ GHz as
we leave its operating band, which degrades the system-added noise performance
in the same manner as reducing the KIT gain would, see Eq. 34. The SNTJ
packaging loss, including the bias tee, has been previously reported to be
$\mathcal{I}_{\mathrm{SNTJ}}=1$ dB (transmission efficiency of $0.8$) Chang
_et al._ (2016).
Figure 11: Insertion loss (IL) from the microwave components used in the noise
measurement setup (see Fig. 9). The ILs (a) are measured at 4K: the bias tee,
Anritsu K250 (red curve), the low pass filter, Pasternack PE87FL1015 (green
curve), the directional coupler, Pasternack PE2204-20 (blue curve), and the
isolator, Quinstar CWJ1015-K13B (purple curve). (b) Combining the IL from the
components before and after the KIT, we estimate the transmission efficiencies
$\eta_{1}^{s}$ (black curve) and $\eta_{2}$ (purple curve) respectively.
### G.3 HEMT-added noise, transmission efficiencies, KIT-excess noise
We can estimate the HEMT-added noise temperature
$T_{H}=N_{H}\hbar\omega/k_{B}$ from eq. 39, because we measured
$T_{\Sigma}^{\prime}$ (see Fig. 10c), and because we can have an estimation of
the total IL $\mathcal{I}_{T}$ from the SNTJ to the HEMT. In fact, without the
bypass
$\mathcal{I}_{T}=\mathcal{I}_{\mathrm{SNTJ}}+\mathcal{I}_{\mathrm{BP}}+\mathcal{I}_{\mathrm{ISO}}+\mathcal{I}_{\mathrm{LPF}}$
(48)
(in dB), which gives $\mathcal{I}_{T}=4.3\pm 0.6$ dB. Then,
$\eta_{2}\eta_{1}^{s}=10^{-\mathcal{I}_{T}/10}$, and we get $T_{H}=1.8\pm 0.2$
K (i.e. $N_{H}=8\pm 1$ quanta) between 3.5 and 5.5 GHz, in agreement with the
HEMT data sheet, which gives $T_{H}=1.6\pm 0.3$ K in that frequency range.
Subtracting $\mathcal{I}_{\mathrm{LPF}}$, $\mathcal{I}_{\mathrm{DC}}$ and
$2\times\mathcal{I}_{\mathrm{BT}}$ from $\mathcal{I}_{\mathrm{BP}}$, the KIT’s
packaging (see Fig. 8a) is responsible for about
$\mathcal{I}_{\mathrm{KIT}}=1.4\pm 0.6$ dB of IL. This loss may be decreased
in future optimization, for example by coating the PCBs and the SMA pins with
superconducting material.
Finally, we can separately estimate $\eta_{1}^{s}$ and $\eta_{2}$ by adding
(in dB) the IL of the chain’s components, and then use Eqs. 34 and 35 to
estimate the KIT-excess noise $N_{\mathrm{ex}}^{s}$ and $N_{\mathrm{ex}}^{i}$.
Between the SNTJ and the KIT,
$\mathcal{I}_{\eta_{1}^{s}}=\mathcal{I}_{\mathrm{SNTJ}}+\mathcal{I}_{\mathrm{LPF}}+\mathcal{I}_{\mathrm{DC}}+\mathcal{I}_{\mathrm{BT}}+\frac{\mathcal{I}_{\mathrm{KIT}}}{2},$
(49)
which gives $\mathcal{I}_{\eta_{1}}=2.4\pm 0.1$ dB, and between the KIT and
the HEMT
$\mathcal{I}_{\eta_{2}}=\frac{\mathcal{I}_{\mathrm{KIT}}}{2}+\mathcal{I}_{\mathrm{BT}}+\mathcal{I}_{\mathrm{ISO}}+\mathcal{I}_{\mathrm{LPF}},$
(50)
which gives $\mathcal{I}_{\eta_{2}}=1.9\pm 0.6$ dB. Figure 11b shows
$\eta_{1}^{s}=10^{-\mathcal{I}_{\eta_{1}}/10}$ and
$\eta_{2}=10^{-\mathcal{I}_{\eta_{2}}/10}$ thus obtained. Therefore, between
3.5 and 5.5 GHz, $\eta_{1}^{s}=\eta_{1}^{i}=0.57\pm 0.02$ (both equal since
the frequency band is nearly symmetric with respect to $\omega_{p}/2=4.4$
GHz), and $\eta_{2}=0.64\pm 0.10$.
## Appendix H KIT PERFORMANCE COMPARISON
Table 2 compares the performance of our KIT with previous results on KITs. The
pump power (third column) is the power at the KIT’s input. The gain (fourth
column) is the average gain observed over the KIT amplification bandwidth
(fifth column). The amplitude of gain ripples over that bandwidth is indicated
next to the gain value. The saturation power (sixth column) corresponds to the
input $1$ dB compression point. The noise bandwidth (seventh column) is the
bandwidth over which a noise measurement was performed. Finally, column eight
reports the measured system noise temperature
$T_{\Sigma}=N_{\Sigma}\hbar\omega/k_{B}$ in that bandwidth. In our case,
$N_{\Sigma}=3.1\pm 0.6$ quanta, which translates into $T_{\Sigma}=0.66\pm
0.15$ K. Note that Eom et.al Eom _et al._ (2012), Ranzani et.al. Ranzani _et
al._ (2018) and Zobrist et.al. Zobrist _et al._ (2019) used a wideband noise
source (a hot/cold load) but did not account for the effect of the idler
port’s input noise. Therefore, their true system-added noise is about twice of
what they reported (see appendix F.2). We do not compare the amplifier-added
noise or amplifier-excess noise, because it is subject to approximations in
the amount of loss present in the noise measurement setup. In addition, the
inferred added noise usually does not include the loss of mandatory microwave
components used when operating the amplifier, and as such it is an under-
estimation of the true, useful amplifier-added noise.
## References
* Castellanos-Beltran _et al._ (2008) M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable Josephson metamaterial,” Nature Physics 4, 929–931 (2008).
* Bergeal _et al._ (2010) N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Phase-preserving amplification near the quantum limit with a josephson ring modulator,” Nature 465, 64–68 (2010).
* Roch _et al._ (2012) N. Roch, E. Flurin, F. Nguyen, P. Morfin, P. Campagne-Ibarcq, M. H. Devoret, and B. Huard, “Widely tunable, nondegenerate three-wave mixing microwave device operating near the quantum limit,” Phys. Rev. Lett. 108, 147701 (2012).
* Mutus _et al._ (2013) J. Y. Mutus, T. C. White, E. Jeffrey, D. Sank, R. Barends, J. Bochmann, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, I. Siddiqi, R. Vijay, A. N. Cleland, and John M. Martinis, “Design and characterization of a lumped element single-ended superconducting microwave parametric amplifier with on-chip flux bias line,” Applied Physics Letters 103, 122602 (2013).
* Zhong _et al._ (2013) L. Zhong, E. P. Menzel, R. Di Candia, P. Eder, M. Ihmig, A. Baust, M. Haeberlein, E. Hoffmann, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, F. Deppe, A. Marx, and R. Gross, “Squeezing with a flux-driven josephson parametric amplifier,” New J. Phys. 15, 125013 (2013).
* Lecocq _et al._ (2017) F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable josephson amplifier,” Phys. Rev. Applied 7, 024028 (2017).
* Malnou _et al._ (2018) M. Malnou, D. A. Palken, Leila R. Vale, Gene C. Hilton, and K. W. Lehnert, “Optimal operation of a josephson parametric amplifier for vacuum squeezing,” Phys. Rev. Applied 9, 044023 (2018).
* Mutus _et al._ (2014) J. Y. Mutus, T. C. White, R. Barends, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, K. M. Sundqvist, A. N. Cleland, and John M. Martinis, “Strong environmental coupling in a josephson parametric amplifier,” Applied Physics Letters 104, 263513 (2014).
* Roy _et al._ (2015) T. Roy, S. Kundu, M. Chand, A. M. Vadiraj, A. Ranadive, N. Nehra, M. P. Patankar, J. Aumentado, A. A. Clerk, and R. Vijay, “Broadband parametric amplification with impedance engineering: Beyond the gain-bandwidth product,” Applied Physics Letters 107, 262601 (2015).
* Frattini _et al._ (2017) N. E. Frattini, U. Vool, S. Shankar, A. Narla, K. M. Sliwa, and M. H. Devoret, “3-wave mixing josephson dipole element,” Applied Physics Letters 110, 222603 (2017).
* Frattini _et al._ (2018) N. E. Frattini, V. V. Sivak, A. Lingenfelter, S. Shankar, and M. H. Devoret, “Optimizing the nonlinearity and dissipation of a snail parametric amplifier for dynamic range,” Phys. Rev. Applied 10, 054020 (2018).
* Sivak _et al._ (2019) V. V. Sivak, N. E. Frattini, V. R. Joshi, A. Lingenfelter, S. Shankar, and M. H. Devoret, “Kerr-free three-wave mixing in superconducting quantum circuits,” Phys. Rev. Applied 11, 054060 (2019).
* Macklin _et al._ (2015) C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, “A near–quantum-limited josephson traveling-wave parametric amplifier,” Science 350, 307–310 (2015).
* White _et al._ (2015) T. C. White, J. Y. Mutus, I.-C. Hoi, R. Barends, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, S. Chaudhuri, J. Gao, and J. M. Martinis, “Traveling wave parametric amplifier with josephson junctions using minimal resonator phase matching,” Applied Physics Letters 106, 242601 (2015).
* Planat _et al._ (2020) L. Planat, A. Ranadive, R. Dassonneville, J. Puertas Martínez, S. Léger, C. Naud, O. Buisson, W. Hasch-Guichard, D.M. Basko, and N. Roch, “Photonic-crystal josephson traveling-wave parametric amplifier,” Phys. Rev. X 10, 021021 (2020).
* Sivak _et al._ (2020) V. V. Sivak, S. Shankar, G. Liu, J. Aumentado, and M. H. Devoret, “Josephson array-mode parametric amplifier,” Phys. Rev. Applied 13, 024014 (2020).
* Zorin (2016) A. B. Zorin, “Josephson traveling-wave parametric amplifier with three-wave mixing,” Phys. Rev. Applied 6, 034006 (2016).
* Zorin (2019) A.B. Zorin, “Flux-driven josephson traveling-wave parametric amplifier,” Phys. Rev. Applied 12, 044051 (2019).
* Eom _et al._ (2012) B. H. Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas, “A wideband, low-noise superconducting amplifier with high dynamic range,” Nature Physics 8, 623–627 (2012).
* Arute _et al._ (2019) F. Arute _et al._ , “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505–510 (2019).
* Shor (1994) P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in _Proceedings 35th Annual Symposium on Foundations of Computer Science_ (IEEE, 1994) pp. 124–134.
* Grover (1997) L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325–328 (1997).
* Szypryt _et al._ (2017) P. Szypryt, S. R. Meeker, G. Coiffard, N. Fruitwala, B. Bumble, G. Ulbricht, A. B. Walter, M. Daal, C. Bockstiegel, G. Collura, N. Zobrist, I. Lipartito, and B. A. Mazin, “Large-format platinum silicide microwave kinetic inductance detectors for optical to near-ir astronomy,” Opt. Express 25, 25894–25909 (2017).
* Hochberg _et al._ (2016a) Y. Hochberg, Y. Zhao, and K. M. Zurek, “Superconducting detectors for superlight dark matter,” Phys. Rev. Lett. 116, 011301 (2016a).
* Hochberg _et al._ (2016b) Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek, “Detecting superlight dark matter with fermi-degenerate materials,” Journal of High Energy Physics 2016, 57 (2016b).
* Chaudhuri _et al._ (2017) S. Chaudhuri, D. Li, K. D. Irwin, C. Bockstiegel, J. Hubmayr, J. N. Ullom, M. R. Vissers, and J. Gao, “Broadband parametric amplifiers based on nonlinear kinetic inductance artificial transmission lines,” Applied Physics Letters 110, 152601 (2017).
* Zobrist _et al._ (2019) N. Zobrist, B. H. Eom, P. Day, B. A. Mazin, S. R. Meeker, B. Bumble, H. G. LeDuc, G. Coiffard, P. Szypryt, N. Fruitwala, I. Lipartito, and C. Bockstiegel, “Wide-band parametric amplifier readout and resolution of optical microwave kinetic inductance detectors,” Applied Physics Letters 115, 042601 (2019).
* Spietz _et al._ (2003) L. Spietz, K. W. Lehnert, I. Siddiqi, and R. J. Schoelkopf, “Primary electronic thermometry using the shot noise of a tunnel junction,” Science 300, 1929–1932 (2003).
* Spietz _et al._ (2006) L. Spietz, R. J. Schoelkopf, and P. Pari, “Shot noise thermometry down to 10mk,” Applied Physics Letters 89, 183123 (2006).
* Vissers _et al._ (2016) M. R. Vissers, R. P. Erickson, H.-S. Ku, L. Vale, X. Wu, G. C. Hilton, and D. P. Pappas, “Low-noise kinetic inductance traveling-wave amplifier using three-wave mixing,” Applied Physics Letters 108, 012601 (2016).
* Erickson and Pappas (2017) R. P. Erickson and D. P. Pappas, “Theory of multiwave mixing within the superconducting kinetic-inductance traveling-wave amplifier,” Phys. Rev. B 95, 104506 (2017).
* Pozar (2011) D.M. Pozar, _Microwave Engineering, 4th Edition_ (Wiley, 2011).
* Bockstiegel _et al._ (2014) J. Bockstiegel, C.and Gao, M.R. Vissers, M. Sandberg, S. Chaudhuri, A. Sanders, L.R. Vale, K.D. Irwin, and D.P. Pappas, “Development of a broadband nbtin traveling wave parametric amplifier for mkid readout,” Journal of Low Temperature Physics 176, 476–482 (2014).
* Ranzani _et al._ (2018) L. Ranzani, M. Bal, Kin Chung Fong, G. Ribeill, X. Wu, J. Long, H.-S. Ku, R. P. Erickson, D. Pappas, and T. A. Ohki, “Kinetic inductance traveling-wave amplifiers for multiplexed qubit readout,” Applied Physics Letters 113, 242602 (2018).
* Caves (1982) Carlton M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
* Gao _et al._ (2008) Jiansong Gao, Miguel Daal, Anastasios Vayonakis, Shwetank Kumar, Jonas Zmuidzinas, Bernard Sadoulet, Benjamin A. Mazin, Peter K. Day, and Henry G. Leduc, “Experimental evidence for a surface distribution of two-level systems in superconducting lithographed microwave resonators,” Applied Physics Letters 92, 152505 (2008).
* Boyd (2019) Robert W Boyd, _Nonlinear optics_ (Academic press, 2019).
* Chang _et al._ (2016) S.-W. Chang, J. Aumentado, W.-T. Wong, and J.C. Bardin, “Noise measurement of cryogenic low noise amplifiers using a tunnel-junction shot-noise source,” in _2016 IEEE MTT-S International Microwave Symposium (IMS)_ (IEEE, 2016) pp. 1–4.
|
# Phy-Taylor: Physics-Model-Based Deep Neural Networks
Yanbing Mao Engineering Technology Division, Wayne State University, Detroit,
MI 48201, USA corresponding author: Yanbing Mao (e-mail<EMAIL_ADDRESS>Lui Sha Department of Computer Science, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA Huajie Shao Department of Computer Science,
College of William & Mary, Williamsburg, VA 23185, USA Yuliang Gu Department
of Mechanical Engineering, University of Illinois at Urbana–Champaign, Urbana,
IL 61801, USA Qixin Wang Department of Computing, Hong Kong Polytechnic
University, Hong Kong SAR, China Tarek Abdelzaher Department of Computer
Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
###### Abstract
Purely data-driven deep neural networks (DNNs) applied to physical engineering
systems can infer relations that violate physics laws, thus leading to
unexpected consequences. To address this challenge, we propose a physics-
model-based DNN framework, called Phy-Taylor, that accelerates learning
compliant representations with physical knowledge. The Phy-Taylor framework
makes two key contributions; it introduces a new architectural physics-
compatible neural network (PhN), and features a novel compliance mechanism, we
call Physics-guided Neural Network Editing. The PhN aims to directly capture
nonlinearities inspired by physical quantities, such as kinetic energy,
potential energy, electrical power, and aerodynamic drag force. To do so, the
PhN augments neural network layers with two key components: (i) monomials of
Taylor series expansion of nonlinear functions capturing physical knowledge,
and (ii) a suppressor for mitigating the influence of noise. The neural-
network editing mechanism further modifies network links and activation
functions consistently with physical knowledge. As an extension, we also
propose a self-correcting Phy-Taylor framework that introduces two additional
capabilities: (i) physics-model-based safety relationship learning, and (ii)
automatic output correction when violations of safety occur. Through
experiments, we show that (by expressing hard-to-learn nonlinearities directly
and by constraining dependencies) Phy-Taylor features considerably fewer
parameters, and a remarkably accelerated training process, while offering
enhanced model robustness and accuracy.
## 1 Introduction
The paper proposes a novel physics-model-based deep neural network framework,
called Phy-Taylor, that addresses a critical flaw in purely data-driven neural
networks, when used to model aspects of physical engineering systems. Namely,
it addresses the potential lack of agreement between learned latent neural
network representations and prior physical knowledge – a flaw that sometimes
leads to catastrophic consequences [1]. As shown in Figure 1, the Phy-Taylor
framework introduces two contributions: the deep physics-compatible neural
networks and a physics-guided neural network editing mechanism, aiming at
ensuring compliance with prior physical knowledge.
Figure 1: Architectures of Phy-Taylor and physics-compatible neural network,
having neural network (NN) editing including link editing and activation
editing.
The work contributes to emerging research on physics-enhanced deep neural
networks. Current approaches include physics-informed neural networks [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], physics-guided neural-network
architectures [17, 18, 19, 20, 21, 22] and physics-inspired neural operators
[23, 24]. The physics-informed networks and physics-guided architectures use
compact partial differential equations (PDEs) for formulating loss functions
and/or architectural components. Physics-inspired neural operators, such as
the Koopman neural operator [23] and the Fourier neural operator [24], on the
other hand, map nonlinear functions into alternative domains, where it is
easier to train their parameters from observational data and reason about
convergence. These frameworks improve consistency with prior analytical
knowledge, but remain problematic in several respects. For example, (i) due to
incomplete knowledge, the compact or precise PDEs may not always be available,
and (ii) fully-connected neural networks can introduce spurious correlations
that deviate from strict compliance with available well-validated physical
knowledge. Instead, through the use of a Taylor-series expansion, the Phy-
Taylor is able to leverage partial knowledge. Moreover, thanks to the neural
editing mechanism, the framework removes links and reshapes activation
functions not consistent with physics-based representations.
The Phy-Taylor framework leverages the intuition that most physical relations
live in low-dimensional manifolds, shaped by applicable physical laws. It’s
just that the estimation of key physical variables from high-dimensional
system observations is often challenging. By expressing known knowledge as
relations between yet-to-be-computed latent variables, we force representation
learning to converge to a space, where these variables represent desired
physical quantities, shaped by the applicable (expressed) physical laws. In
effect, by shaping non-linear terms and relations in the latent space, we
arrive at a desired physics-compliant latent representation. More
specifically, Phy-Taylor offers the following two advantages:
* •
Non-linear Physics Term Representation: Classical neural networks can learn
arbitrary non-linear relations by unfolding them into layers of linear
weighting functions and switch-like activations. This mechanism is akin to
constructing nonlinearities by stitching together piecewise linear behaviors.
Instead, by directly exploiting non-linear terms of the Taylor series
expansion, we offer a set of features that express physical nonlinearities
much more succinctly, thereby reducing the number of needed parameters and
improving accuracy of representation. Monomials of the Taylor series can
capture common nonliearities present in physics equations, such as kinetic
energy, potential energy, rolling resistance and aerodynamic drag force. The
(controllable) model error of the series drops significantly as the series
order increases [25]. The approach constructs input features that represent
monomials of the Taylor series and adds a compressor for mitigating influence
of noise on augmented inputs.
* •
Removing Spurious Correlations: The general topology of neural networks allows
for models that capture spurious correlations in training samples
(overfitting) [26, 27]. In contrast, we develop a neural network (topology)
editing mechanism in the latent space that removes links among certain latent
variables, when these links contradict their intended physical behaviors,
thereby forcing the latent representation to converge to variables with the
desired semantic interpretation that obey the desired physical relations.
Through experiments with learning the dynamics of autonomous vehicles and
other non-linear physical systems, we show that Phy-Taylor exhibits a
considerable reduction in learning parameters, a remarkably accelerated
training process, and greatly enhanced model robustness and accuracy (viewed
from the perspective of long-horizon prediction of a trajectory). Experiments
with safe velocity regulation in autonomous vehicles further demonstrate that
the self-correcting Phy-Taylor successfully addresses the dilemma of
prediction horizon and computation time that nonlinear model-predictive
control and control barrier function are facing in safety-critical control.
## 2 Problem Formulation
Table 1: Table of Notation $\mathbb{R}^{n}$: set of n-dimensional real vectors | $\mathbb{R}_{\geq 0}$: set of non-negative real numbers
---|---
$\mathbb{N}$: set of natural numbers | $[\mathbf{x}]_{i}$: $i$-th entry of vector $\mathbf{x}$
$[\mathbf{x}]_{i:j}$: a sub-vector formed by the $i$-th to $j$-th entries of vector $\mathbf{x}$ | $[\mathbf{W}]_{i,j}$: element at row $i$ and column $j$ of matrix $\mathbf{W}$
$[\mathbf{W}]_{i,:}$: $i$-th row of matrix $\mathbf{W}$ | $\left[\mathbf{x}\leavevmode\nobreak\ ;\leavevmode\nobreak\ \mathbf{y}\right]$: stacked (tall column) vector of vectors $\mathbf{x}$ and $\mathbf{y}$
$\mathbf{0}_{n}$: $n$-dimensional vector of all zeros | $\mathbf{1}_{n}$: $n$-dimensional vector of all ones
$\mathbf{O}_{m\times n}$: $m\times n$-dimensional zero matrix | $\mathbf{I}_{n}$: $n\times n$-dimensional identity matrix
$||\cdot||$: Euclidean norm of a vector or absolute value of a number | $\odot$: Hadamard product
$\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{$\scriptscriptstyle\bullet$}}}}}$: multiplication operator | $\mathrm{len}(\mathbf{x})$: length of vector $\mathbf{x}$
act: activation function | sus: suppressor function
$\top$: matrix or vector transposition | ina: a function that is inactive
$\boxplus$: a known model-substructure parameter | $*$: an unknown model-substructure parameter
Consider the problem of computing some output vectors, $\mathbf{y}$, from a
set of observations, $\mathbf{x}$. The relation between $\mathbf{x}$ and
$\mathbf{y}$ is partially determined by physical models of known structure
(but possibly unknown parameter values) and partially unknown, thus calling
for representation learning of the missing substructures using neural network
observables. For example, $\mathbf{y}$ might denote the estimated momentum and
future position of a target as a function of a vector of $\mathbf{x}$, that
include its position, velocity, and type. In this formulation, position and
velocity might be directly related to output quantities via known physical
relations, but type is represented only indirectly by an image that requires
some representation learning in order to translate it into relevant parameters
(such as mass and maneuverability) from which the outputs can be computed. We
express the overall input/output relation by the function:
$\displaystyle\mathbf{y}=\underbrace{\mathbf{A}}_{\text{weight
matrix}}\cdot\underbrace{\mathfrak{m}(\mathbf{x},r)}_{\text{node-
representation vector}}+\underbrace{\mathbf{f}(\mathbf{x})}_{\text{model
mismatch}}\triangleq\underbrace{\mathbf{g}(\mathbf{x})}_{\text{ground truth
model}},$ (1)
where $\mathbf{y}$ and $\mathbf{x}$ are the output and input vectors of
overall system model, respectively, and the parameter $r\in\mathbb{N}$
controls model size. For convenience, Table 1 summarizes the remaining
notations used throughout the paper. Since Equation (1) combines known and
unknown model substructures, we distinguish them according to the definition
below.
###### Definition 1.
For all $i\in\\{1,2,\ldots,\mathrm{len}(\mathbf{y})\\}$,
$j\in\\{1,2,\ldots,\mathrm{len}(\mathfrak{m}(\mathbf{x},r))\\}$, element
$[\mathbf{A}]_{i,j}$ is said to be a known model-substructure parameter in
Equation (1) if and only if
$\frac{{\partial[\mathbf{f}(\mathbf{x})]_{i}}}{{\partial{[\mathfrak{m}}(\mathbf{x},r)]_{j}}}\equiv
0$. Otherwise, it is called an unknown model-substructure parameter.
Definition 1 indicates that the model-substructure knowledge includes:
* •
(i) Known parameter values but completely unknown model formula (see e.g., the
Example 1).
* •
(ii) Partially-known model formula. For example, in the (lateral) force
balance equation of autonomous vehicle [28]:
$\displaystyle
m\left({\ddot{y}+\dot{\psi}{V_{x}}}\right)={F_{\text{yf}}}+{F_{\text{fr}}}+{F_{\text{bank}}},$
(2)
the force formula due to road bank angle $\phi$, i.e.,
${F_{\text{bank}}}=mg\sin(\phi)$, is known while other force formulas are
unknown because of complex and unforeseen driving environments.
* •
(iii) Known model formula but unknown parameter values.
###### Example 1 (Identify Known Model-Substructure Parameters).
DNN Design Goal: Use the current position $p(k)$, velocity $v(k)$ and mass $m$
to estimate a vehicle’s next velocity $v(k+1)$, sensed road friction
coefficient $r(k+1)$ and safety metric $s(k+1)$ of velocity regulation, in the
dynamic driving environments. With the knowledge of vehicle dynamics and
control [28], the problem can be mathematically described by
$\displaystyle v\left({k+1}\right)$
$\displaystyle={g_{1}}\left({p\left(k\right),\leavevmode\nobreak\
v\left(k\right),\leavevmode\nobreak\ m}\right),$ (3a) $\displaystyle
r\left({k+1}\right)$ $\displaystyle={g_{2}}\left({m,\leavevmode\nobreak\
v\left(k\right)}\right),$ (3b) $\displaystyle s\left({k+1}\right)$
$\displaystyle={g_{3}}\left({v\left(k\right)}\right),$ (3c)
We here assume the formulas of ${g_{1}}(\cdot)$–${g_{3}}(\cdot)$ are unknown.
While the Equations (3b) and (3c) indicate that given the inputs, the
available knowledge are (i) the road friction coefficient varies with a
vehicle’s mass and real-time velocity only, and (ii) the considered safety
metric depends on velocity only. We let $\mathbf{x}=[p(k);\leavevmode\nobreak\
v(k);\leavevmode\nobreak\ m]$, $\mathbf{y}=[v(k+1);\leavevmode\nobreak\
r(k+1);\leavevmode\nobreak\ s(k+1)]$,
$\mathbf{g}(\mathbf{x})=[{g_{1}}({p(k),v(k),m});\leavevmode\nobreak\
{g_{2}}({m,v(k)});\leavevmode\nobreak\ {g_{3}}({v(k)})]$, and
$\mathfrak{m}(\mathbf{x},r)=[1;\leavevmode\nobreak\ p(k);\leavevmode\nobreak\
v(k);\leavevmode\nobreak\ m;\leavevmode\nobreak\ p^{2}(k);\leavevmode\nobreak\
p(k)v(k);\leavevmode\nobreak\ mp(k);\leavevmode\nobreak\
v^{2}(k);\leavevmode\nobreak\ mv(k);\leavevmode\nobreak\ m^{2}]$. The ground
truth model (3) is then equivalently rewritten in the form of (1):
$\displaystyle\mathbf{y}=\underbrace{\left[{\begin{array}[]{*{20}{c}}*&*&*&*&*&*&*&*&*&*\\\
&0&*&*&0&0&0&*&*&*\\\
&0&*&0&0&0&0&*&0&0\end{array}}\right]}_{\mathbf{A}}\cdot\mathfrak{m}(\mathbf{x},r)+\underbrace{\mathbf{g}(\mathbf{x})-\mathbf{A}\cdot\mathfrak{m}(\mathbf{x},r)}_{\mathbf{f}(\mathbf{x})}=\mathbf{g}(\mathbf{x}),$
(7)
which thus encodes the available knowledge points (i) and (ii) to the known
model-substructure parameters (i.e., zeros) in system matrix $\mathbf{A}$.
Considering this definition, the problem addressed in this paper is formally
stated below.
###### Problem 1.
Given a time-series of inputs, $\mathbf{x}$, the corresponding outputs,
$\mathbf{y}$, and the known model substructures in Equation (1), it is desired
to develop an end-to-end neural network that directly estimates $\mathbf{y}$
(denoted by $\widehat{\mathbf{y}}$), given $\mathbf{x}$, consistently with all
known model substructures. In other words, the model must satisfy the property
that for each known model-substructure parameter, $[\mathbf{A}]_{i,j}$, the
end-to-end model must ensure that
$\frac{{\partial{{[\widehat{\mathbf{y}}}]_{i}}}}{{\partial{[\mathfrak{m}}\left({\mathbf{x},r}\right)]_{j}}}\equiv[\mathbf{A}]_{i,j}$
for any $\mathfrak{m}(\mathbf{x},r)$.
The above definition allows the system described by Equation (1) to have an
end-to-end model that intertwines well-known substructure properties with
high-order unmodeled correlations of unknown nonlinear structure. In this, our
problem differs from past seminal frameworks of physics-enhanced DNNs [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 29,
30, 31, 32, 33, 34, 35], that use a compact partial differential equations
(PDEs) for formulating the PDEs-regulated loss function and/or DNN
architectures to count the degree mean of consistency with PDEs. The proposed
solution to Problem 1 is the Phy-Taylor framework, which will rely on two
building blocks: a deep physics-compatible neural network (PhN) and a physics-
guided neural network editing mechanism, presented in the next section.
## 3 Phy-Taylor Framework
The proposed Phy-Taylor for addressing Problem 1 is shown in Figure 1, which
is built on the conjunctive deep physics-compatible neural network (PhN) and
physics-guided neural network (NN) editing. In other words, implementing NN
editing according to Taylor’s theorem for embedding available physical
knowledge into deep PhN yields the Phy-Taylor. The PhN is a neural network
layer with a key component: a physics-inspired augmentation (called Phy-
Augmentation) for generating monomials in Equation (1) of Taylor series
expansion of nonlinear functions capturing physical knowledge. The physics-
guided NN editing – including link editing and activation editing – further
modifies network links and activation functions consistently with physical
knowledge. Specifically, the link editing performs removing and preserving
links according to the consistency with physical knowledge. Meanwhile, the
activation editing performs the physics-knowledge-preserving computing in
output channel of each PhN. Collaboratively through link and activation
editing, the input/output of Phy-Taylor strictly complies with the available
physical knowledge, which is a desired solution to Problem 1. Next, we detail
the two components.
### 3.1 The Physics-compatible Neural Network (PhN)
In order to capture non-linear features of physical functions, we introduce a
new type of network layer that is augmented with terms derived from Taylor
series expansion. The Taylor’s Theorem offers a series expansion of arbitrary
nonlinear functions, as shown below. Taylor’s Theorem (Chapter 2.4 [25]): Let
$\mathbf{g}\\!:\leavevmode\nobreak\ \mathbb{R}^{n}\to\mathbb{R}$ be a
$r$-times continuously differentiable function at the point
$\mathbf{o}\in\mathbb{R}^{n}$. Then there exists
$\mathbf{h}_{\alpha}\\!:\leavevmode\nobreak\ \mathbb{R}^{n}\to\mathbb{R}$,
where $\left|\alpha\right|=r$, such that
$\displaystyle\mathbf{g}(\mathbf{x})=\sum\limits_{\left|\alpha\right|\leq
r}{\frac{{{\partial^{\alpha}}\mathbf{g}(\mathbf{o})}}{{\alpha!}}}{\left({\mathbf{x}-\mathbf{o}}\right)^{\alpha}}+\sum\limits_{\left|\alpha\right|=r}{{\mathbf{h}_{\alpha}}(\mathbf{x}){{({\mathbf{x}-\mathbf{o}})^{\alpha}}}},\hskip
5.69046pt\text{and}\leavevmode\nobreak\
\mathop{\lim}\limits_{\mathbf{x}\to\mathbf{o}}{\mathbf{h}_{\alpha}}\left(\mathbf{x}\right)=\mathbf{0},$
(8) where $\alpha=\left[{{\alpha_{1}};\leavevmode\nobreak\
{\alpha_{2}};\leavevmode\nobreak\ \ldots;\leavevmode\nobreak\
{\alpha_{n}}}\right]$,
$\left|\alpha\right|=\sum\limits_{i=1}^{n}{{\alpha_{i}}}$,
$\alpha!=\prod\limits_{i=1}^{n}{{\alpha_{i}}}!$,
${\mathbf{x}^{\alpha}}=\prod\limits_{i=1}^{n}{\mathbf{x}_{i}^{{\alpha_{i}}}}$,
and
${\partial^{\alpha}}\mathbf{g}=\frac{{{\partial^{\left|\alpha\right|}}\mathbf{g}}}{{\partial\mathbf{x}_{1}^{{\alpha_{1}}}\cdot\ldots\cdot\partial\mathbf{x}_{n}^{{\alpha_{n}}}}}$.
The Taylor’s theorem has several desirable properties:
* •
Non-linear Physics Term Representation: The high-order monomials (i.e., the
ones included in $\left({\mathbf{x}-\mathbf{o}}\right)^{\alpha}$ with
$|\alpha|\geq 2$) of the Taylor series (i.e.,
$\sum\limits_{\left|\alpha\right|\leq
r}{\frac{{{D^{\alpha}}\mathbf{g}(\mathbf{o})}}{{\alpha!}}}{\left({\mathbf{x}-\mathbf{o}}\right)^{\alpha}}$)
capture core nonlinearities of physical quantities such as kinetic energy
($\triangleq\frac{1}{2}m{v^{2}}$), potential energy
($\triangleq\frac{1}{2}k{x^{2}}$), electrical power ($\triangleq V\cdot I$)
and aerodynamic drag force ($\triangleq\frac{1}{2}\rho{v^{2}}{C_{D}}A$), that
drive the state dynamics of physical systems.
* •
Controllable Model Accuracy: Given ${\mathbf{h}_{\alpha}}(\mathbf{x})$ is
finite and $\left\|{\mathbf{x}-\mathbf{o}}\right\|<1$, the error
$\sum\limits_{\left|\alpha\right|=r}{{\mathbf{h}_{\alpha}}(\mathbf{x}){{({\mathbf{x}-\mathbf{o}})^{\alpha}}}}$
for approximating the ground truth $\mathbf{g}(\mathbf{x})$ will drop
significantly as the order $r=|\alpha|$ increases and
$\mathop{\lim}\limits_{|\alpha|=r\to\infty}{\mathbf{h}_{\alpha}}(\mathbf{x}){\left({\mathbf{x}-\mathbf{o}}\right)^{\alpha}}=\mathbf{0}$.
This allows for controllable model accuracy via controlling order $r$.
* •
Knowledge Embedding: The Taylor series can directly project the known model
substructure parameters of the ground-truth model (1) into neural network
parameters including the weight matrix
(${\frac{{{D^{\alpha}}\mathbf{g}(\mathbf{o})}}{{\alpha!}}}$ with $|\alpha|>0$)
and bias (${\frac{{{D^{\alpha}}\mathbf{g}(\mathbf{o})}}{{\alpha!}}}$ with
$|\alpha|=0$), thus paving the way to embed the available physical knowledge
in the form of an appropriately weighted neural network layer.
Figure 2: Phy-Augmentation architecture.
We note the Taylor theorem relies on an assumption that the ground truth
$\mathbf{g}(\mathbf{x})$ is a $r$-times continuously differentiable function
at the point $\mathbf{o}$. If the assumption does not hold, what the Taylor
series will approximate is a proximity of ground truth that is $r$-times
continuously differentiable. For continuous functions, this is often a
sufficient approximation. Next, we describe how PhNs embed the Taylor series
expansion into neural network layers. The resulting architecture (of a single
PhN layer) is shown in Figure 1. Compared with a classical neural network
layer, we introduce the Phy-Augmentation, whose architecture is shown in
Figure 2. The Phy-Augmentation has two components: (i) augmented inputs that
represent monomials of a Taylor series expansion, and (ii) a suppressor for
mitigating the influence of noise on such augmented inputs (i.e., high-order
monomials). Next, we detail them.
#### 3.1.1 Phy-Augmentation: Taylor Series Monomials
Input: augmentation order $r$, input $\mathbf{x}$, point $\mathbf{o}$,
suppressor mapping $\chi(\cdot)$.
1 Suppress input:
$[\mathbf{x}]_{i}\leftarrow\begin{cases}\chi([{\mathbf{x}}]_{i}),&\text{if
suppressor is active}\\\ [{\mathbf{x}}]_{i},&\text{otherwise}\end{cases}$,
$i\in\\{1,2,\ldots,\text{len}(\mathbf{x})\\}$;
2 Generate index vector of input entries:
$\mathbf{i}\leftarrow[1;\leavevmode\nobreak\ 2;\leavevmode\nobreak\
\ldots;\leavevmode\nobreak\ \mathrm{len}({\mathbf{x}})]$;
3 Generate augmentations:
${\mathfrak{m}}({\mathbf{x}},r)\leftarrow{\mathbf{x}}$;
4 for _$\\_\ =2$ to $r$_ do
5 for _$i=1$ to $\mathrm{len}({\mathbf{x}})$ _ do
6 Compute temporaries: ${\mathbf{{t}}}_{a}$
$\leftarrow[{\mathbf{x}}]_{i}\cdot[{\mathbf{x}}]_{\left[{[\mathbf{i}]_{i}\leavevmode\nobreak\
:\leavevmode\nobreak\ \mathrm{len}({\mathbf{x}})}\right]}$;
7 if _$i==1$ _ then
8 Generate temporaries:
$\widetilde{\mathbf{{t}}}_{b}\leftarrow\widetilde{\mathbf{{t}}}_{a}$;
9 else
10 Generate temporaries:
$\widetilde{\mathbf{{t}}}_{b}\leftarrow\left[\widetilde{\mathbf{{t}}}_{b};\leavevmode\nobreak\
\widetilde{\mathbf{{t}}}_{a}\right]$;
11 end if
12 Update index entry: $[\mathbf{i}]_{i}\leftarrow\mathrm{len}({\mathbf{x}})$;
13 Update augmentations:
${\mathfrak{m}}({\mathbf{x}},r)\leftarrow\left[{\mathfrak{m}}({\mathbf{x}},r);\leavevmode\nobreak\
{{\mathbf{{t}}}_{b}}\right]$;
14
15 end for
16
17 end for
Output vector of augmented monomials:
${\mathfrak{m}}({\mathbf{x}},r)\leftarrow\left[1;\leavevmode\nobreak\
{\mathfrak{m}}({\mathbf{x}},r)\right]$.
Algorithm 1 Phy-Augmentation Procedure Figure 3: An example of Algorithm 1 in
TensorFlow framework, where input $\tilde{\mathbf{x}}\in\mathbb{R}^{3}$ is
from Line 1 of Algorithm 1.
The function of physical-features augmentation in Figure 2 is to generate the
vector of physical features (i.e., node representations) in form of Taylor
series monomials, which is formally described by Algorithm 1. The Lines 1–1 of
Algorithm 1 guarantee that the generated node-representation vector embraces
all the non-missing and non-redundant monomials of Taylor series. The Line 1
shows that Algorithm 1 finally stacks vector with one. This operation means a
PhN node will be assigned to be one, and the bias (corresponding to
${\frac{{{D^{\alpha}}\mathbf{g}(\mathbf{o})}}{{\alpha!}}}$ with $|\alpha|=0$
in Taylor series) will be thus treated as link weights in PhN layers. As an
example shown in Figure 3, the Phy-Augmentation empowers PhN to well capture
core nonlinearities of physical quantities (see e.g., kinetic energy,
potential energy, electrical power and aerodynamic drag force) that drive the
state dynamics of physical systems, and then represent or approximate physical
knowledge in form of the Taylor series.
We note the Line 1 of Algorithm 1 means the noise suppressor is not applied to
all the input elements. The critical reason is the extra mapping induced by
suppressor on inputs can destroy the compliance with physical knowledge, when
the available model-substructure knowledge do not include the mapping of
suppressor. We next is going to present the function of suppressor.
#### 3.1.2 Phy-Augmentation: Noise Suppressor
The suppressor in Figure 2 is to mitigate the influence of noise on the
augmented high-order monomials. Before proceeding on the working mechanism, we
present a metric pertaining to noise and true data.
###### Definition 2.
Consider the noisy data and define the data-to-noise ratio ($\mathrm{DNR}$):
$\displaystyle[\bar{\mathbf{x}}]_{i}=\underbrace{[\mathbf{h}]_{i}}_{\text{true
data}}+\underbrace{[{\mathbf{w}}]_{i}}_{\text{noise}}\in\mathbb{R},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{DNR}_{i}\triangleq\frac{[\mathbf{h}]_{i}}{[\mathbf{w}]_{i}}.$ (9)
The auxiliary Theorem 4 presented in Supplementary Information 9.1 implies
that the high-order monomials can shrink their DNRs due to nonlinear mapping.
This means the PhN can be vulnerable to the noisy inputs, owning to Phy-
Augmentation for generating high-order monomials. Hence, mitigating the
influence of noise is vital for enhancing the robustness of PhNs, consequently
the Phy-Taylor. As shown in Figure 2, we incorporate a suppressor into PhN to
process the raw input data, such that the high-order monomial from Phy-
Augmentation can enlarge their DNRs. Building on Definition 2, the proposed
noise suppressor mapping is
$\displaystyle\chi([\bar{\mathbf{x}}]_{i})=\chi([\mathbf{h}]_{i}+[\mathbf{w}]_{i})=\begin{cases}0,&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}<0\\\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i},&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}<0\\\
([\mathbf{h}]_{i}+[\mathbf{w}]_{i})\cdot\kappa_{i}+\rho_{i},&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}>0\end{cases},$ (10)
where the parameters $\rho_{i}$ and $\kappa_{i}$ satisfy
$\displaystyle|\rho_{i}|\geq|[\mathbf{h}]_{i}+[\mathbf{w}]_{i}|\cdot|\kappa_{i}|.$
(11)
We next present the suppressor properties in the following theorem, whose
proof appears in Supplementary Information 9.2.
###### Theorem 1.
Consider the noisy data $[\bar{\mathbf{x}}]_{i}$ and the suppressor described
in Equations (9) and (10), respectively. Under the condition (11), the
suppressor output, denoted by
$[\widehat{\mathbf{x}}]_{i}=\chi([\bar{\mathbf{x}}]_{i})$, has the properties:
The DNR magnitude of high-order monomial
$[\widehat{\mathbf{x}}]_{i}^{p}[\widehat{\mathbf{x}}]_{j}^{q}$ ($p+q\geq 2$)
is strictly increasing with respect to DNR magnitudes of
$[\widehat{\mathbf{x}}]_{i}$ and $[\widehat{\mathbf{x}}]_{j}$. (12) The true
data and the noise of suppressor output $[\widehat{\mathbf{x}}]_{i}$ are
$\displaystyle[\widetilde{\mathbf{h}}]_{i}=\begin{cases}\\![\mathbf{h}]_{i}\cdot\kappa_{i}+\rho_{i},\\!\\!&[\mathbf{h}]_{i}\\!+\\![\mathbf{w}]_{i}\\!\geq\\!0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}\\!>\\!0\\\
\\![\mathbf{h}]_{i},\\!\\!&\text{otherwise}\\\
\end{cases},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
[\widetilde{\mathbf{w}}]_{i}=\begin{cases}\\!-[\mathbf{h}]_{i},\\!\\!&[\mathbf{h}]_{i}\\!+\\![\mathbf{w}]_{i}\\!<\\!0\\\
\\![\mathbf{w}]_{i},\\!\\!&[\mathbf{h}]_{i}\\!+\\![\mathbf{w}]_{i}\\!\geq\\!0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}\\!<\\!0\\\
\\![\mathbf{w}]_{i}\cdot\kappa_{i},\\!\\!&[\mathbf{h}]_{i}\\!+\\![\mathbf{w}]_{i}\\!\geq\\!0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}\\!>\\!0\end{cases},$ (13)
such that
$[\widehat{\mathbf{x}}]_{i}=[\widetilde{\mathbf{h}}]_{i}+[\widetilde{\mathbf{w}}]_{i},\leavevmode\nobreak\
i\in\\{1,2,\dots,\mathrm{len}(\widehat{\mathbf{x}})\\}$.
The result (13) implies the parameters $\kappa$ and $\rho$ control the DNRs of
suppressed data, consequently the high-order monomials. Furthermore, the
result (12) suggests that through designing parameters $\kappa_{i}$,
$\rho_{i}$, $\kappa_{j}$ and $\rho_{j}$ for increasing the DNR magnitudes of
data $[\widehat{\mathbf{x}}]_{i}$ and $[\widehat{\mathbf{x}}]_{j}$, the DNR of
high-order monomial
$[\widehat{\mathbf{x}}]_{i}^{p}[\widehat{\mathbf{x}}]_{j}^{q}$ can be enlarged
consequently, such that the influence of noise is mitigated.
### 3.2 Physics-guided Neural Network Editing
Input: Available knowledge included in system matrix $\mathbf{A}$ of ground-
truth model (1), terminal output dimension $\text{len}(\mathbf{y})$, number
$p$ of PhNs, activation functions $\text{act}(\cdot)$,
$\mathbf{y}_{\left\langle 0\right\rangle}=\mathbf{x}$ and $r_{\left\langle
1\right\rangle}=r$.
1 for _$t=1$ to $p$_ do
2 if _$t==1$ _ then
3 Deactivate noise suppressor;
4 Generate node-representation vector $\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle})$ via Algorithm 1;
5 Generate knowledge matrix $\mathbf{K}_{\left\langle t\right\rangle}$:
$[\mathbf{K}_{\left\langle
t\right\rangle}]_{i,j}\leftarrow\begin{cases}[\mathbf{A}_{\left\langle
t\right\rangle}]_{i,j},&[\mathbf{A}_{\left\langle
t\right\rangle}]_{i,j}=\boxplus\\\ 0,&\text{otherwise}\end{cases}$;
6 Generate weight-masking matrix $\mathbf{M}_{\left\langle t\right\rangle}$:
$[\mathbf{M}_{\left\langle
t\right\rangle}]_{i,j}\leftarrow\begin{cases}0,&[\mathbf{A}_{\left\langle
t\right\rangle}]_{i,j}=\boxplus\\\ 1,&\text{otherwise}\end{cases}$;
7 Generate activation-masking vector $\mathbf{a}_{\left\langle
t\right\rangle}$: $[\mathbf{a}_{\left\langle
t\right\rangle}]_{i}\\!\leftarrow\\!\begin{cases}0,\\!\\!\\!&[\mathbf{M}_{\left\langle
t\right\rangle}]_{i,j}\\!=\\!0,\forall
j\\!\in\\!\\{1,\ldots,\text{len}(\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle}))\\}\\\
1,\\!\\!\\!&\text{otherwise}\end{cases}$;
8
9 else
10 Generate node-representation vector $\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle})$ via Algorithm 1;
11 Generate knowledge matrix $\mathbf{K}_{\left\langle t\right\rangle}$:
$\displaystyle\mathbf{K}_{\left\langle
t\right\rangle}\leftarrow\left[\begin{gathered}\leavevmode\hbox
to270.7pt{\vbox to68.29pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.8947pt}{14.4014pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mbox{\small
O}_{(\text{len}(\mathbf{y}_{\left\langle
t\right\rangle})-\text{len}(\mathbf{y}))\times\text{len}(\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle}))}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.22885pt}{47.09738pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathbf{0}_{\text{len}(\mathbf{y})}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{73.00107pt}{47.24461pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mbox{\small
I}_{\text{len}(\mathbf{y})}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{139.64648pt}{48.54462pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mbox{\small
O}_{\text{len}(\mathbf{y})\times(\text{len}(\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle
t\right\rangle}))-\text{len}(\mathbf{y})-1)}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{34.14322pt}\pgfsys@lineto{270.30121pt}{34.14322pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{51.21504pt}{34.14322pt}\pgfsys@lineto{51.21504pt}{68.28644pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{113.81104pt}{34.14322pt}\pgfsys@lineto{113.81104pt}{68.28644pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}\right];$
(15)
12 Generate weight-masking matrix $\mathbf{M}_{\left\langle t\right\rangle}$:
$[\mathbf{M}_{\left\langle
t\right\rangle}]_{i,j}\leftarrow\begin{cases}0,&\frac{{\partial[\mathfrak{m}(\mathbf{y}_{\left\langle
t\right\rangle},r_{\left\langle
t\right\rangle})]_{j}}}{{\partial[\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})]_{v}}}\neq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{M}_{\left\langle
1\right\rangle}]_{i,v}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
v\in\\{1,2,\dots,\text{len}(\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle}))\\}\\\ 1,&\text{otherwise}\end{cases}$;
13 Generate activation-masking vector $\mathbf{a}_{\left\langle
t\right\rangle}\leftarrow\left[\mathbf{a}_{\left\langle
1\right\rangle};\leavevmode\nobreak\
\mathbf{1}_{\text{len}(\mathbf{y}_{\left\langle
t\right\rangle})-\text{len}(\mathbf{y})}\right]$;
14
15 end if
16 Generate weight matrix: ${\mathbf{W}_{\left\langle t\right\rangle}}$;
17 Generate uncertainty matrix $\mathbf{U}_{\left\langle
t\right\rangle}\leftarrow\mathbf{M}_{\left\langle
t\right\rangle}\odot{\mathbf{W}_{\left\langle t\right\rangle}}$;
18 Compute output: $\mathbf{y}_{\left\langle
t\right\rangle}\leftarrow\mathbf{K}_{\left\langle
t\right\rangle}\cdot\mathfrak{m}(\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle})+\mathbf{a}_{\left\langle
t\right\rangle}\odot\text{act}\left({\mathbf{U}_{\left\langle
t\right\rangle}\cdot\mathfrak{m}\left({\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle t\right\rangle}}\right)}\right)$ ;
19 end for
Output : terminal output:
$\widehat{\mathbf{y}}\leftarrow\mathbf{y}_{\left\langle p\right\rangle}$
.
Algorithm 2 Physics-guided NN Editing
Building on the deep PhN, this section presents the neural network (NN)
editing for embedding and preserving the available physical knowledge, through
the physics-guided link editing and activation editing. Specifically, the link
editing centers around removing and preserving the links according to the
consistency with physical knowledge. Meanwhile, the activation editing
performs the physical-knowledge-preserving computing in the output channels of
PhNs. Thanks to the concurrent link and activation editing, the input/output
of Phy-Taylor can strictly comply with the available physical knowledge. Using
the notation ‘$\boxplus$’ defined in the Table 1, the procedure of physics-
guided NN editing is described in Algorithm 2.
For the edited weight matrix $\mathbf{W}_{\left\langle t\right\rangle}$, if
its entries in the same row are all known model-substructure parameters, the
associated activation should be inactivate. Otherwise, the Phy-Taylor cannot
strictly preserve the available physical knowledge due to the extra nonlinear
mappings induced by the activation functions. This thus motivates the physics-
knowledge preserving computing, i.e., the Line 2 of Algorithm 2. Figure 4
summarizes the flowchart of NN editing in a single PhN layer:
* •
(i) Given the node-representation vector from Algorithm 1, the original
(fully-connected) weight matrix is edited via link editing to embed assigned
physical knowledge, resulting in $\mathbf{W}_{\left\langle t\right\rangle}$.
* •
(ii) The edited weight matrix ${\mathbf{W}_{\left\langle t\right\rangle}}$ is
separated into knowledge matrix $\mathbf{K}_{\left\langle t\right\rangle}$ and
uncertainty matrix $\mathbf{U}_{\left\langle t\right\rangle}$, such that
${\mathbf{W}_{\left\langle t\right\rangle}}=\mathbf{K}_{\left\langle
t\right\rangle}+\mathbf{U}_{\left\langle t\right\rangle}$. Specifically, the
$\mathbf{K}_{\left\langle t\right\rangle}$, generated in Lines 2 and 11,
includes all the known model-substructure parameters. While the
$\mathbf{M}_{\left\langle t\right\rangle}$, generated in Lines 2 and 2, is
used to generate uncertainty matrix $\mathbf{U}_{\left\langle t\right\rangle}$
(see Line 2) to include all the unknown model-substructure parameters, through
freezing the known model-substructure parameters of $\mathbf{W}_{\left\langle
t\right\rangle}$ to zeros.
* •
(iii) The $\mathbf{K}_{\left\langle t\right\rangle}$,
$\mathbf{M}_{\left\langle t\right\rangle}$ and activation-masking vector
$\mathbf{a}_{\left\langle t\right\rangle}$ (generated in Lines 2 and 2) are
used by activation editing for the physical-knowledge-preserving computing of
output in each PhN layer. The function of $\mathbf{a}_{\left\langle
t\right\rangle}$ is to avoid the extra mapping (induced by activation) that
prior physical knowledge does not include.
Figure 4: Flowchart of NN editing in single PhN layer.
The flowchart of NN editing operating in cascade PhN is depicted in Figure 5.
The Lines 2-2 of Algorithm 2 means that $\mathbf{A}=\mathbf{K}_{\left\langle
1\right\rangle}+\mathbf{M}_{\left\langle 1\right\rangle}\odot\mathbf{A}$,
leveraging which and the setting $r_{\left\langle 1\right\rangle}=r$, the
ground-truth model (1) is rewritten as
$\displaystyle\mathbf{y}=(\mathbf{K}_{\left\langle
1\right\rangle}+\mathbf{M}_{\left\langle
1\right\rangle}\odot\mathbf{A})\cdot\mathfrak{m}(\mathbf{x},r)+\mathbf{f}(\mathbf{x})=\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+(\mathbf{M}_{\left\langle
1\right\rangle}\odot\mathbf{A})\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+\mathbf{f}(\mathbf{x}).$ (16)
Figure 5: Implementing NN editing, i.e., Algorithm 2, in the Example 1. (i)
Unknown substructures are formed by the grey links, the known substructures
are formed by the red and blue links. (ii) Cutting black links to avoid
spurious correlations, otherwise, the links introduce the dependence of
$s\left({k+1}\right)$ on mass m, thus contradicting physical knowledge.
We obtain from the Line 2 of Algorithm 2 that the output of the first PhN
layer is
$\displaystyle\mathbf{y}_{\left\langle
1\right\rangle}=\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+\mathbf{a}_{\left\langle
1\right\rangle}\odot\text{act}\left({\mathbf{U}_{\left\langle
1\right\rangle}\cdot{\mathfrak{m}}\left({\mathbf{x},r_{\left\langle
1\right\rangle}}\right)}\right).$ (17)
Recalling that $\mathbf{K}_{\left\langle 1\right\rangle}$ includes all the
known model-substructure parameters of $\mathbf{A}$ while the
$\mathbf{U}_{\left\langle 1\right\rangle}$ includes remainders, we conclude
from (16) and (17) that the available physical knowledge pertaining to the
ground-truth model (1) has been embedded to the first PhN layer. As Figure 5
shows the embedded knowledge shall be passed down to the remaining cascade
PhNs and preserved therein, such that the end-to-end Phy-Taylor model can
strictly with the prior physical knowledge. This knowledge passing is achieved
by the block matrix $\mathbf{K}_{\left\langle p\right\rangle}$ generated in
Line 11, due to which, the output of $t$-th PhN layer satisfies
$\displaystyle[\mathbf{y}_{\left\langle
t\right\rangle}]_{1:\text{len}(\mathbf{y})}=\underbrace{\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})}_{\text{knowledge
passing}}+\underbrace{[\mathbf{a}_{\left\langle
t\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
t\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
t-1\right\rangle},r_{\left\langle
t\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}}_{\text{knowledge
preserving}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall t\in\\{2,3,\ldots,p\\}.$ (18)
Meanwhile, the $\mathbf{U}_{\left\langle
t\right\rangle}=\mathbf{M}_{\left\langle
t\right\rangle}\odot{\mathbf{W}_{\left\langle t\right\rangle}}$ means the
masking matrix $\mathbf{M}_{\left\langle t\right\rangle}$ generated in Line 2
is to remove the spurious correlations in the cascade PhN, which is depicted
by the cutting link operation in Figure 5.
### 3.3 The Solution to Problem 1: Phy-Taylor
Described in Figure 1, with the guidance of Taylor series, implementing the
physics-guided NN editing in the deep PhN yields the Phy-Taylor. The Phy-
Taylor embeds the available physical knowledge into each PhN, such that its
input/output strictly complies with the physical knowledge, which is formally
stated in the following theorem. The theorem proof appears in Supplementary
Information 9.3.
###### Theorem 2.
Consider the Phy-Taylor described by Figure 1. The input/output (i.e.,
${\mathbf{x}}$/$\widehat{\mathbf{y}}$) of Phy-Taylor strictly complies with
the available knowledge pertaining to the physics model (1) of ground truth,
i.e., if the $[\mathbf{A}]_{i,j}$ is a known model-substructure parameter,
then
$\frac{{\partial{{[\widehat{\mathbf{y}}}]_{i}}}}{{\partial{[\mathfrak{m}}\left({\mathbf{x},r}\right)]_{j}}}\equiv\frac{{\partial{[\mathbf{y}]_{i}}}}{{\partial{[\mathfrak{m}}\left({\mathbf{x},r}\right)]_{j}}}\equiv[\mathbf{A}]_{i,j}$
always holds.
## 4 Phy-Taylor Properties
Figure 6: (a): Two Fully-connected NN layers. (b): A single PhN layer.
Moving forward, this section focuses on the property analysis of Phy-Taylor.
### 4.1 Parameter Quantity Reduction
The Figures 2 and 3 show that the Phy-Augmentation, i.e., the Algorithm 1,
expands the input without involving weight-matrix multiplication. This trait
can be leveraged to significantly reduce the quantity of learning parameters
including weights and bias. For the demonstration, we consider the two network
models in Figure 6 (a) and (b), where the (a) describes a fully-connected two-
layer network, while the (b) describes a single PhN. Observing them, we obtain
that given the same dimensions of input and terminal output, the number of
learning parameters of Figure 6 (a) is $(m+1)n+(n+1)p$ (including $(m+p)n$
weights and $n+p$ bias), while the number of learning parameters of Figure 6
(b) is $(n+1)p$ (including $n\times p$ weights and $p$ bias). The number
difference of learning parameters is thus
$\displaystyle(m+1)n+(n+1)p-(n+1)p=(m+1)n.$ (19)
We note the quantity of reduced parameters (19) is the lower bound of PhN in
Phy-Taylor framework, since it is obtained without considering physics-guided
NN editing for removing and freezing links and bias according to the available
physical knowledge.
### 4.2 Single PhN v.s. Cascade PhN
We next investigate if the space complexity (i.e., the quantity of augmented
monomials) of Phy-Augmentation of a single PhN with a large augmentation order
can be reduced via cascade PhN with relatively small orders. To simplify the
presentation, a single PhN and cascade PhN are respectively represented in the
following equations.
$\displaystyle\widehat{\mathbf{y}}=\mathrm{PhN}({\left.{\mathbf{x}}\in\mathbb{R}^{n}\right|r})\in\mathbb{R}^{m},$
(20) $\displaystyle{\mathbf{x}}\in{{\mathbb{R}}^{n}}\leavevmode\nobreak\
\leavevmode\nobreak\ \longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathbf{y}_{\left\langle
1\right\rangle}}=\mathrm{PhN}({\left.{\mathbf{x}}\right|{r_{\left\langle
1\right\rangle}}})\in{{\mathbb{R}}^{{n_{\left\langle
1\right\rangle}}}}\leavevmode\nobreak\ \leavevmode\nobreak\
\longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\
\ldots\leavevmode\nobreak\ \leavevmode\nobreak\
\longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathbf{y}_{{\left\langle
d-1\right\rangle}}}=\mathrm{PhN}({\left.{{\mathbf{y}_{{\left\langle
d-2\right\rangle}}}}\right|{r_{{\left\langle
d-1\right\rangle}}}})\in{{\mathbb{R}}^{{n_{{\left\langle
d-1\right\rangle}}}}}$ $\displaystyle\hskip
255.79042pt\longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\
\widehat{\mathbf{y}}=\mathrm{PhN}({\left.{{\mathbf{y}_{{\left\langle
d-1\right\rangle}}}}\right|{r_{\left\langle
d\right\rangle}}})\in\mathbb{R}^{m},$ (21)
where the cascade architecture consists of $d$ PhNs. To guarantee the cascade
PhN (21) and the single PhN (20) have the same monomials, their augmentation
orders are required to satisfy
$\displaystyle\prod\limits_{v=1}^{d}{{r_{{\left\langle
v\right\rangle}}}}=r,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall r_{{\left\langle
v\right\rangle}},\leavevmode\nobreak\ \leavevmode\nobreak\ r\in\mathbb{N}.$
(22)
The space complexity difference of Phy-Augmentation is formally presented in
the following theorem, whose proof is presented in Supplementary Information
9.4.
###### Theorem 3.
Under the condition (22), the space complexity difference between single PhN
(20) and cascade PhN (21), due to Phy-Augmentation, is
$\displaystyle\mathrm{len}(\mathfrak{m}(\mathbf{x},r))-\sum\limits_{p=1}^{d}{\mathrm{len}(\mathfrak{m}(\mathbf{x},r_{{\left\langle
p\right\rangle}}))}=\sum\limits_{s={r_{\left\langle
1\right\rangle}}+1}^{r}{\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)!s!}}}-\sum\limits_{v=1}^{d-1}{\sum\limits_{s=1}^{{r_{{\left\langle
v+1\right\rangle}}}}{\frac{{\left({{n_{\left\langle
v\right\rangle}}+s-1}\right)!}}{{\left({{n_{\left\langle
v\right\rangle}}-1}\right)!s!}}}}+1-d.$ (23)
The Theorem 3 implies that the output dimensions and the augmentation orders
of intermediate PhNs are critical in the significant reduction of space
complexity via cascade PhN. However, an intuitive question arises: Does the
cascade PhN reduce the complexity at the cost of model accuracy? Without loss
of generality, we use the following example to answer the question.
###### Example 2.
For simplicity in explanation, we ignore the bias and consider the scenario
that both the activation and the suppressor are inactive. For the single PhN
(20), we let $\mathbf{x}\in\mathbb{R}^{2}$ and
$\widehat{\mathbf{y}}\in\mathbb{R}$ and $r=4$. Its output is then computed
according to
$\displaystyle\widehat{\mathbf{y}}={w_{1}}{[\mathbf{x}]_{1}}+{w_{2}}{[\mathbf{x}]_{2}}+{w_{3}}[\mathbf{x}]_{1}^{2}+{w_{4}}{[\mathbf{x}]_{1}}{[\mathbf{x}]_{2}}+{w_{5}}[\mathbf{x}]_{2}^{2}+{w_{6}}[\mathbf{x}]_{1}^{3}+{w_{7}}[\mathbf{x}]_{1}^{2}{[\mathbf{x}]_{2}}+{w_{8}}{[\mathbf{x}]_{1}}[\mathbf{x}]_{2}^{2}+{w_{9}}[\mathbf{x}]_{2}^{3}$
$\displaystyle\hskip
137.42685pt+{w_{10}}[\mathbf{x}]_{1}^{4}+{w_{11}}[\mathbf{x}]_{1}^{3}{[\mathbf{x}]_{2}}+{w_{12}}[\mathbf{x}]_{1}^{2}[\mathbf{x}]_{2}^{2}+{w_{13}}{[\mathbf{x}]_{1}}[\mathbf{x}]_{2}^{3}+{w_{14}}[\mathbf{x}]_{2}^{4},$
(24)
For the corresponding cascade PhN (21), we let $r_{{\left\langle
1\right\rangle}}=r_{{\left\langle 2\right\rangle}}=2$, the output dimension of
first PhN is 2 while dimension of terminal output is 1. We thus have
$\displaystyle\widehat{\mathbf{y}}$
$\displaystyle={{\hat{w}}_{6}}{\mathbf{y}_{\left\langle
1\right\rangle}}+{{\hat{w}}_{7}}\mathbf{y}^{2}_{\left\langle
1\right\rangle}={{\hat{w}}_{6}}({{{\tilde{w}}_{1}}{[\mathbf{x}]_{1}}+{{\tilde{w}}_{2}}{[\mathbf{x}]_{2}}+{{\tilde{w}}_{3}}[\mathbf{x}]_{1}^{2}+{{\tilde{w}}_{4}}{[\mathbf{x}]_{1}}{[\mathbf{x}]_{2}}+{{\tilde{w}}_{5}}[\mathbf{x}]_{2}^{2}})$
$\displaystyle\hskip
152.22241pt+{{\hat{w}}_{7}}{\left({{{\tilde{w}}_{1}}{[\mathbf{x}]_{1}}+{{\tilde{w}}_{2}}{[\mathbf{x}]_{2}}+{{\tilde{w}}_{3}}[\mathbf{x}]_{1}^{2}+{{\tilde{w}}_{4}}{[\mathbf{x}]_{1}}{[\mathbf{x}]_{2}}+{{\tilde{w}}_{5}}[\mathbf{x}]_{2}^{2}}\right)^{2}}.$
(25)
Observing (24) and (25), we discover that (i) both the single and cascade
architectures have the same monomials due to the satisfying condition (22),
and (ii) the single PhN layer has 14 weight parameters (i.e., the ${w_{1}}$,
${w_{2}}$, $\dots$, ${w_{14}}$ in Equation (24)), while the cascade layers
have only 7 weight parameters (i.e., the ${\hat{w}_{1}}$, $\ldots$, and
$\hat{w}_{7}$ in Equation (25)) in total. Intuitively, we can conclude that if
the reduced weights are associated with the links that contradict with
physical knowledge, the cascade PhN can further increase model accuracy,
otherwise, it can reduce the space complexity at the cost of model accuracy.
## 5 Extension: Self-Correcting Phy-Taylor
Figure 7: Self-correcting Phy-Taylor Architecture: $\mathbf{u}(k)$ denotes the
vector of real-time decisions, ${\bf{s}}(u(k))$ denotes the vector of real-
time safety metrics, $\bf{c}$ is the vector of safety bounds.
The recent incidents due to deployment of DNNs overshadow the revolutionizing
potential of AI in the physical engineering systems [36, 1], especially the
safety-critical systems, whose unintended behavior results in death or serious
injury to people, severe damage to equipment or environments [37, 38]. The
safe control and planning is a fundamental solution for enhancing safety
assurance of the AI-assisted physical systems often operating in environments
where time and safety are critical, such as the airplanes, medical drones and
autonomous vehicles. To comply with safety constraints in face of potential
conflicts from control objectives, the framework of control barrier function
(CBF) has been proposed for the computation of real-time safety-critical
control commands [39, 40]. The CBFs however use only current state information
without prediction, whose control policy is thus greedy and challenging for
proactive safe control. It is well known that model predictive control (MPC)
yields a less greedy safe control policy, since it takes future state
information into account [41, 42]. Motivated by the observations, MPC with
incorporation of CBF, i.e., MPC-CBF, was proposed [43]. Due to the nonlinear
dynamics, the MPC-CBF however faces a dilemma of prediction horizon and
computation time of safe control commands, which induces considerable feedback
delays and thus leads to failures in the time- and safety-critical operating
environments. To address the dilemma, we propose the self-correcting Phy-
Taylor, whose architecture is shown in Figure 7. Its one mission is learning
the safety relationship between the real-time decisions and the safety
metrics, with consideration of future information:
$\displaystyle\mathbf{s}(\mathbf{x}(k),\mathbf{u}(k),\tau)=\sum\limits_{t=k}^{k+\tau-1}{\tilde{\mathbf{f}}(\mathbf{x}(t),\mathbf{u}(t))},$
(26)
where $\tilde{\mathbf{f}}(\mathbf{x}(t),\mathbf{u}(t))$ is the predefined
vector of safety metrics at time $t$, and $\tau$ denotes the horizon of future
information of safety metrics.
Inside the self-correcting Phy-Taylor, the learned safety relationship for
approximating (26) will first be subject to the off-line verification of
available physical knowledge, based on which, the necessary revisions can be
needed. According to the off-line verified and revised (if needed) safety
relationship, the correcting of real-time decision $\mathbf{u}(k)$ will be
triggered if any safety metric
$[\mathbf{s}(\mathbf{x}(k),\mathbf{u}(k),\tau)]_{i}$,
$i\in\\{1,2,\ldots,h\\}$, exceeds (or leaves) the preset safety bounds (or
safety envelopes). However, the current learned formula corresponding to (26)
is not ready (if not impossible) for delivering the procedure, owning to the
complicated dependence of $[\mathbf{s}(\mathbf{x}(k),\mathbf{u}(k),\tau)]_{i}$
on both system state $\mathbf{x}(k)$ and decision $\mathbf{u}(k)$. To address
this problem, as shown in Figure 7, we decouple the real-time decisions from
the real-time system states. Specifically,
* •
Given the real-time system state $\mathbf{x}(k)$ as the origin input, the
first Phy-Taylor outputs the real-time decision $\mathbf{u}(k)$, which is
motivated by the fact that the state-feedback control is used most commonly in
physical engineering systems [44]. In other words, the computation of raw
$\mathbf{u}(k)$ directly depends on real-time system state $\mathbf{x}(k)$.
* •
Given the real-time decision $\mathbf{u}(k)$ (i.e., the output of the first
Phy-Taylor) as the input of the second Phy-Taylor, the terminal output is the
real-time safety metric $\mathbf{s}(\mathbf{u}(k))$, which is motivated by the
fact that the decision $\mathbf{u}(k)$ manipulates system state. In other
words, the safety metric $\mathbf{s}(\mathbf{u}(k))$ directly depends on
decision $\mathbf{u}(k)$ and indirectly depends on system state
$\mathbf{x}(k)$.
* •
The two Phy-Taylors are trained simultaneously according to the training loss
function:
$\displaystyle\mathcal{L}=\alpha\left\|{{\bf{s}}(\mathbf{u}(k))-{\bf{s}}({\bf{x}}(k),{\bf{u}}(k),\tau)}\right\|+\beta\left\|{\breve{\bf{u}}(k)-\bf{u}(k)}\right\|,$
(27)
where the ${\bf{s}}({\bf{x}}(k),{\bf{u}}(k),\tau)$ given in Equation (26) is
ground truth of safety-metric vector, the $\breve{\bf{u}}(k)$ is ground truth
of decision vector, the $\alpha$ and $\beta$ are hyperparameters. The two
cascade Phy-Taylors thus depend on each other.
* •
To render the learned safety relationship ${\bf{s}}(\mathbf{u}(k))$ tractable,
the activation and compressor inside the second Phy-Taylor are inactive, such
that the ${\bf{s}}(\mathbf{u}(k))$ is expressed in the form of Taylor series.
Given the verified and revised (if needed) relationship, the self-correcting
procedure will be triggered (if a safety metric exceeds the safety bound
$\mathbf{c}$) for correcting decision according to
$\displaystyle\mathbf{u}(k)\leftarrow\mathop{\arg\min}\limits_{\widetilde{\mathbf{u}}(k)\in{\mathbb{R}^{m}}}\left\\{{\left.{\left\|{\widetilde{\mathbf{u}}(k)-\mathbf{u}(k)}\right\|}\right|[\bf{s}(\widetilde{\mathbf{u}}(k)]_{i}<[\mathbf{c}]_{i},\leavevmode\nobreak\
\leavevmode\nobreak\
i\in\\{1,2,\ldots,\text{len}({\bf{s}}(\mathbf{u}(k)))\\}}\right\\}.$ (28)
We note the self-correcting mechanism and the safety revision of relationship
between ${\bf{s}}(\mathbf{u}(k))$ and $\mathbf{u}(k)$ for delivering (28) vary
with safety problems and physical systems. An example in this paper is the
safe control of autonomous vehicles: Algorithm 3 in the Experiments section.
## 6 Experiments
The demonstrations are performed on three different systems with different
degrees of available physical knowledge: autonomous vehicles, coupled
pendulums, and a subsystem of US Illinois climate.
### 6.1 Autonomous Vehicles
Figure 8: Vehicle’s driving environment.
The first experiment performs the demonstration of two functions: (i) the
learning of vehicle’s conjunctive lateral and longitudinal dynamics via Phy-
Taylor, and (ii) the safe velocity regulation via self-correcting Phy-Taylor.
The vehicle’s driving environment is shown in Figure 8, operating in the
AutoRally platform [45].
#### 6.1.1 Vehicle Dynamics Learning
We first leverage our available physical knowledge to identify the known
model-substructure parameters for the physics-guided NN editing. According to
the Newton’s second law for motion along longitudinal and lateral axes [28],
we have the following governing equations:
$\displaystyle\bar{m}\ddot{\mathrm{p}}={F_{\mathrm{p}f}}+{F_{\mathrm{p}r}}-{F_{\text{aero}}}-{R_{\mathrm{p}f}}-{R_{\mathrm{p}r}},\quad\quad\bar{m}(\ddot{\mathrm{y}}+\dot{\psi}{v_{\mathrm{p}}})={F_{\mathrm{y}f}}+{F_{\mathrm{y}r}},$
(29)
where $\mathrm{p}$ is the longitudinal position, $\mathrm{y}$ is the lateral
position, $\psi$ is the vehicle yaw angle, $\bar{m}$ is the vehicle mass,
$v_{\mathrm{p}}\triangleq\dot{\mathrm{p}}$ is the longitudinal velocity,
${F_{\mathrm{p}f}}$ and ${F_{\mathrm{p}r}}$ denote the longitudinal tire force
at the front and rear tires, respectively, ${R_{\mathrm{p}f}}$ and
${R_{\mathrm{p}r}}$ denote the rolling resistance at the front and rear tires,
respectively, ${F_{\text{aero}}}$ represents the longitudinal aerodynamic drag
force, ${F_{\mathrm{y}f}}$ and ${F_{\mathrm{y}r}}$ are the lateral tire forces
of the front and rear wheels, respectively. With the notations of lateral
velocity $v_{\mathrm{y}}\triangleq\dot{\mathrm{y}}$ and yaw velocity
$v_{\psi}\triangleq\dot{\psi}$, the following state space model is derived
from the force balance equation (29) in the literature [28].
$\displaystyle\frac{\mathrm{d}}{{\mathrm{d}t}}\left[\begin{gathered}\mathrm{p}\hfill\\\
\mathrm{y}\hfill\\\ \psi\hfill\\\ {v_{\mathrm{p}}}\hfill\\\
{v_{\mathrm{y}}}\hfill\\\ {v_{\psi}}\hfill\\\
\end{gathered}\right]=\left[{\begin{array}[]{*{20}{c}}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ 0&0&0&*&0&0\\\ 0&0&0&0&*&*\\\
0&0&0&0&*&*\end{array}}\right]\underbrace{\left[\begin{gathered}\mathrm{p}\hfill\\\
\mathrm{y}\hfill\\\ \psi\hfill\\\ {v_{\mathrm{p}}}\hfill\\\
{v_{\mathrm{y}}}\hfill\\\ {v_{\psi}}\hfill\\\
\end{gathered}\right]}_{\triangleq{\mathbf{x}}}+\left[\begin{gathered}0\hfill\\\
0\hfill\\\ 0\hfill\\\ *\hfill\\\ 0\hfill\\\ 0\hfill\\\
\end{gathered}\right]\theta+\left[\begin{gathered}0\hfill\\\ 0\hfill\\\
0\hfill\\\ 0\hfill\\\ *\hfill\\\ *\hfill\\\ \end{gathered}\right]\delta,$ (64)
where ‘*’ can represent a state-dependent or time-dependent function or mixed
of them or just a scalar, but is unknown to us, and $\theta$ and $\delta$
denote throttle and steering, respectively. Given the practical physical
knowledge that the throttle computation depends on the longitudinal velocity
and position only, while the dependencies of steering are unknown, the state
space model (64) updates with
$\displaystyle\dot{{\mathbf{x}}}=\left[{\begin{array}[]{*{20}{c}}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ &0&0&*&0&0\\\ &*&*&*&*&*\\\
&*&*&*&*&*\end{array}}\right]{\mathbf{x}}.$ (71)
The sampling technique, with sampling period denoted by $T$, converts the
continuous-time state-space model above to the following discrete-time one:
$\displaystyle{\mathbf{x}}\left({k+1}\right)=\left[{\begin{array}[]{*{20}{c}}1&0&0&T&0&0\\\
0&1&0&0&T&0\\\ 0&0&1&0&0&T\\\ &0&0&*&0&0\\\ &*&*&*&*&*\\\
&*&*&*&*&*\end{array}}\right]{\mathbf{x}}\left(k\right).$ (78)
Figure 9: (a): Phy-Taylor${}_{\text{large order}}$ has a PhN with large order
$r=4$. (b): Phy-Taylor${}_{\text{small order}}$ has cascading PhNs with
relatively small orders satisfying $r_{\left\langle 1\right\rangle}\cdot
r_{\left\langle 2\right\rangle}=2\cdot 2=4=r$.
We first consider two Phy-Taylor models, named ‘Phy-Taylor${}_{\text{large
order}}$’ and ‘Phy-Taylor${}_{\text{small order}}$’, which can embed the
available knowledge (i.e., the known parameters included in system matrix of
model (78)). Their architectures are shown in Figure 9 (a) and (b). The Phy-
Taylor${}_{\text{large order}}$ has one PhN with a large augmentation order
while the Phy-Taylor${}_{\text{small order}}$ has two cascade PhN layers with
two relatively small augmentation orders. Meanwhile, the three orders satisfy
the condition (22) for having the same monomials of Taylor series. We also
consider the corresponding models without NN editing (i.e., without physical
knowledge embedding), which degrades the Phy-Taylor to the deep PhN (DPhN).
The two DPhN models are named ‘DPhN${}_{\text{large order}}$’ and
‘DPhN${}_{\text{small order}}$’. The final model we considered is the seminal
Deep Koopman [23], following the same model configurations therein. The
configurations of five models are summarized in Table 2.
Table 2: Model Configurations
| Layer 1 | Layer 2 | Layer 3 | |
---|---|---|---|---|---
Model ID | #weights | #bias | #Weights | #bias | #weights | #bias | #parameter sum | prediction error $e$
DPhN${}_{\text{large order}}$ | $3135$ | $15$ | $90$ | $6$ | $-$ | $-$ | $3246$ | nan
DPhN${}_{\text{small order}}$ | $270$ | $10$ | $520$ | $8$ | $48$ | $6$ | $862$ | $45.57277$
Phy-Taylor${}_{\text{large order}}$ | $2313$ | $12$ | $30$ | $3$ | $-$ | $-$ | $2358$ | $0.047758$
Phy-Taylor${}_{\text{small order}}$ | $167$ | $7$ | $265$ | $5$ | $18$ | $3$ | $465$ | $0.003605$
| Encoder | Decoder | Auxiliary Networks | |
---|---|---|---|---|---
Model ID | #weights | #bias | #weights | #bias | #weights | #bias | #parameter sum | prediction error
Deep Koopman | $2040$ | $176$ | $2040$ | $176$ | $19920$ | $486$ | $24838$ | $0.232236$
Figure 10: Training and Testing. (a)-(c): The trajectories of averaged
training loss (5 random seeds) of different models described in Table 2.
(i)-(vi): Ground truth and predicted trajectories via trained models.
The trajectories of training loss are presented in Figure 10 (a)–(c). The
(training loss, validation loss) of trained DPhN${}_{\text{large order}}$,
DPhN${}_{\text{small order}}$, Phy-Taylor${}_{\text{large order}}$ and Phy-
Taylor${}_{\text{small order}}$ are (0.00389, 0.00375), (0.000344, 0.000351),
(0.000222, 0.000238) and (0.000915, 0.000916), respectively. To perform the
testing, we consider the long-horizon prediction of system trajectories, given
the same initial conditions. The prediction error is measured by the mean
squared error:
$e=\frac{1}{\kappa}\sum\limits_{t=k+1}^{k+\kappa}{\frac{1}{6}}\left\|{\underline{\bf{x}}\left(t\right)-{\bf{x}}\left(t\right)}\right\|\leavevmode\nobreak\
\text{with}\leavevmode\nobreak\
\underline{\bf{x}}\left(k\right)={\bf{x}}\left(k\right)$, where
$\underline{\bf{x}}\left(t\right)$ is the prediction of ground truth
${\bf{x}}\left(t\right)$ at time $t$. The prediction errors over the horizon
$\kappa=300$ and initial time $k=950$ are summarized in Table 2. The ground-
truth trajectories and predicted ones from Deep Koopman and the Phy-Taylor
models are presented in Figure 10 (i)–(vi). Observing from Table 2 and Figure
10, we can conclude:
* •
Figure 10 (i)–(vi) and Figure 10 (a)–(b): the physics-guided NN editing can
significantly accelerate model training, reduce validation and training loss
as well as improve model accuracy (viewed from long-horizon prediction of
trajectory).
* •
Figure 10 (i)–(vi) and Figure 10 (c): with physics-guided NN editing, the
cascade PhN with small augmentation orders can further significantly reduce
training loss, and increase model accuracy. This can be due to the further
removed spurious correlations or NN links contradicting with physics law, via
the cascade architecture.
* •
Figure 10 (i)–(vi) and Table 2: compared with the fully-connected DPhN models,
i.e., DPhN${}_{\text{large order}}$ and DPhN${}_{\text{small order}}$, the
seminal Deep Koopman strikingly increases model accuracy, viewed from the
perspective of long-horizon prediction of trajectory. Conversely, compared to
Deep Koopman, the Phy-Taylor models (both Phy-Taylor${}_{\text{large order}}$
and Phy-Taylor${}_{\text{small order}}$) notably reduce the model learning
parameters (weights and bias) and further remarkably increase model accuracy
simultaneously.
#### 6.1.2 Self-Correcting Phy-Taylor
This experiment demonstrates the effectiveness of self-correcting Phy-Taylor
in guaranteeing vehicle’s safe driving in the environment shown in Figure 8.
The architecture of self-correcting Phy-Taylor is presented in Figure 11. Its
real-time input vector is ${\mathbf{x}}(k)=[w(k);\leavevmode\nobreak\
\mathrm{p}(k);\leavevmode\nobreak\ \mathrm{y}(k);\leavevmode\nobreak\
\psi(k);\leavevmode\nobreak\ v_{\mathrm{p}}(k);\leavevmode\nobreak\
v_{\mathrm{y}}(k);v_{\psi}(k)]$, where $w(k)$ is the average of four wheels’
velocities. The mediate output
$\mathbf{u}(k)=\left[\theta(k);\leavevmode\nobreak\ \gamma(k)\right]$ denotes
the vector of control commands, where
$\theta(k)\in[-0.156,\leavevmode\nobreak\ 0.156]$ is the throttle command and
$\gamma(k)\in[-0.6,\leavevmode\nobreak\ 0.6]$ is the steering command. The
considered safety-metric vector in Equation (26) is
$\displaystyle\mathbf{s}({\mathbf{x}}(k),\mathbf{u}(k),\tau)=\sum\limits_{t=k+1}^{k+\tau}{\left[{{{\left({{v_{\mathrm{p}}}(t)-\mathrm{v}}\right)}^{2}};\leavevmode\nobreak\
\leavevmode\nobreak\ {{\left({{v_{\mathrm{p}}}(t)-r\cdot
w(k)}\right)}^{2}}}\right]}\in{\mathbb{R}^{2}},$ (79)
Figure 11: Self-correcting Phy-Taylor for safe control of autonomous vehicle.
where $\mathrm{v}$ and $r$ denote the reference of longitudinal velocity and
the wheel radius, respectively. The safety metric (79) together with the
driving scenario in Figure 8 indicate the objective of safe control command is
to simultaneously steer vehicle’s longitudinal velocity to reference
$\mathrm{v}$, and constrain the slip (i.e., $(v_{\mathrm{x}}(t)-r\cdot
w(k))^{2}$) to prevent slipping and sliding. The hyperparameters in the
training loss function (27) are set to $\alpha=\beta=1$.
The testing results of trained model are presented in Figure 12 (a)–(d) (blue
curves), which in conjunction with the ground truth (orange curves)
demonstrate the trustfulness of trained model. We next output the learned
safety relationships for off-line verification and necessary revision:
$\displaystyle[{\bf{s}}({\bf{u}}(k))]_{1}$
$\displaystyle=0.00111007+{\left[\\!\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\\!\right]^{\top}}\left[{\begin{array}[]{*{20}{c}}{-0.04581441}&{0.00100625}\\\
{0.00100625}&{0.00342825}\end{array}}\right]\left[\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\right],$ (80g) $\displaystyle[{\bf{s}}({\bf{u}}(k))]_{2}$
$\displaystyle=0.14376973-{\left[\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\right]^{\top}}\left[{\begin{array}[]{*{20}{c}}{6.06750536}&{0.02701398}\\\
{0.02701398}&{0.00601609}\end{array}}\right]\left[\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\right].$ (80n)
The safety metrics (79) of ground truth are always non-negative. We thus need
to verify that given the ranges of control commands (i.e.,
$\theta(k)\in[-0.156,0.156]$, $\gamma(k)\in[-0.6,0.6]$, $\forall
k\in\mathbb{N}$), if both $[{\bf{s}}({\bf{u}}(k))]_{1}$ and
$[{\bf{s}}({\bf{u}}(k))]_{2}$ in Equation (80) can always be non-negative. If
a violation occurs, we will make revisions to the relationships. We can verify
from (80) that the non-negativity constraint does not always hold, such as
$\theta(k)=0.156$ and $\gamma(k)=0$. Therefore, the revision of safety
relationship is needed before working on the self-correcting procedure. The
regulated safety relationships (revisions are highlighted in red color) are
presented below.
$\displaystyle[{\bf{s}}({\bf{u}}(k))]_{1}$
$\displaystyle=\underbrace{0.00{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.00,0.00}02}1007}_{{[\mathbf{b}]_{1}}}+{\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right]^{\top}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.00,0.00}0.0018}1441}&{0.00100625}\\\
{0.00100625}&{0.00342825}\end{array}}\right]}_{{\triangleq{\bf{P}}_{1}}}\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right],$ (81g)
$\displaystyle[{\bf{s}}({\bf{u}}(k))]_{2}$
$\displaystyle=\underbrace{0.14376973}_{{[\mathbf{b}]_{2}}}-{\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right]^{\top}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.00,0.00}5.90769724}}&{{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.00,0.00}0.01201398}}\\\
{{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{1.00,0.00,0.00}0.01201398}}&{0.00601609}\end{array}}\right]}_{\triangleq{\bf{P}}_{2}}\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right],$ (81n)
which satisfy $\left[{\bf{s}}({\bf{u}}(k))\right]_{1}\geq 0$ and
$\left[{\bf{s}}({\bf{u}}(k))\right]_{2}\geq 0$, for any
$\theta(k)\in[-0.156,0.156]$ and $\gamma(k)\in[-0.6,0.6]$, and can be
demonstrated by the green curves in Figure 12 (c) and (d).
Input: Real-time control-command vector
${\bf{u}}(k)=[\theta(k);\leavevmode\nobreak\ \gamma(k)]$, safety bounds
$[\mathbf{c}]_{1}$ and $[\mathbf{c}]_{2}$, and learned matrices
$\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ and bias $[\mathbf{b}]_{1}$ and
$[\mathbf{b}]_{2}$ defined in Equation (81).
1 Update original safety relationship with off-line verified and revised one:
${\bf{s}}(\bf{u}(k))\leftarrow\eqref{rchoo}$;
2 if _$[{\bf{s}}({\bf{u}}(k))]_{1} >[\mathbf{c}]_{1}$ or
$[{\bf{s}}({\bf{u}}(k))]_{2}>[\mathbf{c}]_{2}$ _ then
3 if _$[{\bf{s}}({\bf{u}}(k))]_{i}\geq[\mathbf{c}]_{i}$ , $i\in\\{1,2\\}$ _
then
4 Update safety metric:
$[\widehat{\mathbf{c}}]_{i}\leftarrow[\mathbf{c}]_{i},i\in\\{1,2\\}$;
5 else
6 Maintain safety metric:
$[\widehat{\mathbf{c}}]_{i}\leftarrow[{\bf{s}}({\bf{u}}(k))]_{i},i\in\\{1,2\\}$;
7 end if
8 Compute orthogonal matrix $\mathbf{P}_{1}$ and eigenvalues $\lambda_{1}$ and
$\lambda_{1}$ according to (88);
9 Compute matrix:
$\mathbf{S}\leftarrow\mathbf{Q}_{1}\cdot\mathbf{P}_{2}\cdot\mathbf{Q}_{1}$;
10 Compute $\widehat{\theta}(k)$ and $\widehat{\gamma}(k)$ according to (99);
11 Correct real-time control commands:
$\displaystyle\theta(k)\leftarrow\mathop{\arg\min}\limits_{\left\\{{\widehat{\theta}(k),-\widehat{\theta}(k)}\right\\}}\left\\{{|{\theta(k)-\widehat{\theta}(k)}|,|{\theta(k)+\widehat{\theta}(k)}|}\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\
\gamma(k)\leftarrow\mathop{\arg\min}\limits_{\left\\{{\widehat{\gamma}(k),-\widehat{\gamma}(k)}\right\\}}\left\\{{|{\gamma(k)-\widehat{\gamma}(k)}|,|{\gamma(k)+\widehat{\gamma}(k)}|}\right\\}.$
12else
13 Maintain real-time control commands: $\theta(k)\leftarrow\theta(k)$ and
$\gamma(k)\leftarrow\gamma(k)$.
14 end if
Algorithm 3 Self-Correcting Procedure for Safe Control Commands
Figure 12: (a)–(b): Ground truth and inference of control commands. (c)–(d):
Ground truth, original inference and inference (according to the revised
safety relationships) of safety metrics (tracking error
${({{v_{\mathrm{p}}}(t)-\mathrm{v}})}^{2}$ and wheel slip
$(v_{\mathrm{p}}(t)-r\cdot w(k))^{2}$). (e): Safety metric: tracking error.
(f): Safety metric: wheel slip. (g): Average of wheel velocities. (h):
Vehicle’s driving curve (phase plot).
With the revised safety relationships (81) at hand, we are ready to develop
the self-correcting procedure. Considering the two matrices ${\bf{P}}_{1}$ and
${\bf{P}}_{2}$ defined in Equation (81) are symmetric, we have
$\displaystyle{\bf{P}}_{1}$
$\displaystyle=\underbrace{\left[{\begin{array}[]{*{20}{c}}{-0.934}&{-0.3572}\\\
{-0.3572}&{0.934}\end{array}}\right]}_{\triangleq{\bf{Q}}_{1}}\left[{\begin{array}[]{*{20}{c}}{\underbrace{0.0008}_{\triangleq{\lambda}_{1}}}&0\\\
0&{\underbrace{0.0038}_{\triangleq{\lambda}_{2}}}\end{array}}\right]\underbrace{\left[{\begin{array}[]{*{20}{c}}{-0.934}&{-0.3572}\\\
{-0.3572}&{0.934}\end{array}}\right]}_{={\bf{Q}}^{\top}_{1}},$ (88)
$\displaystyle{\bf{P}}_{2}$
$\displaystyle={\bf{Q}}_{1}\cdot{\bf{Q}}_{1}\cdot{\bf{P}}_{2}\cdot{\bf{Q}}_{1}\cdot{\bf{Q}}_{1},$
(89)
based on which, we further define:
$\displaystyle\widehat{\mathbf{u}}(k)\triangleq{\bf{Q}}_{1}\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bf{S}\triangleq{\bf{Q}}_{1}\cdot{\bf{P}}_{2}\cdot{\bf{Q}}_{1}=\left[{\begin{array}[]{*{20}{c}}{{s_{11}}}&{{s_{12}}}\\\
{{s_{12}}}&{{s_{22}}}\end{array}}\right].$ (94)
We let $[\widehat{\mathbf{c}}]_{1}$ and $[\widehat{\mathbf{c}}]_{2}$ denote
the two assigned safety metrics. According to the derivations appearing in
Supplementary Information 9.5, the control commands included in the safety
formulas $[{\bf{s}}({\bf{u}}(k))]_{1}=[\widehat{\mathbf{c}}]_{1}$ and
$\left[{\bf{s}}({\bf{u}}(k))\right]_{2}=[\widehat{\mathbf{c}}]_{2}$ are
obtained as
$\displaystyle\left[\begin{array}[]{l}\pm\widehat{\theta}(k)\\\
\pm\widehat{\gamma}(k)\end{array}\right]\triangleq{\mathbf{Q}_{1}}\left[\begin{array}[]{l}\pm\sqrt{\frac{{{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}}{{{\lambda_{1}}}}-\frac{{{\lambda_{2}}}}{{{\lambda_{1}}}}\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}}\\\
\pm\sqrt{\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}}\end{array}\right],$
(99)
where
$\displaystyle{\varpi_{1}}$
$\displaystyle\triangleq{\left({\frac{{{\lambda_{2}}}}{{{\lambda_{1}}}}}\right)^{2}}s_{11}^{2}+s_{22}^{2}+\frac{{\left({4s_{12}^{2}-2{s_{11}}{s_{22}}}\right){\lambda_{2}}}}{{{\lambda_{1}}}},$
(100) $\displaystyle{\varpi_{2}}$
$\displaystyle\triangleq\frac{{2({{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}){s_{11}}{s_{22}}-4({{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}})s_{12}^{2}+2{\lambda_{2}}({{[\mathbf{b}]_{2}}-{[\widehat{\mathbf{c}}]_{2}}}){s_{11}}}}{{{\lambda_{1}}}}$
$\displaystyle\hskip
213.39566pt-\frac{{2({{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}){\lambda_{2}}s_{11}^{2}}}{{\lambda_{1}^{2}}}-2({{[\mathbf{b}]_{2}}-{[\widehat{\mathbf{c}}]_{2}}}){s_{22}},$
(101) $\displaystyle{\varpi_{3}}$
$\displaystyle\triangleq{\left({{[\mathbf{b}]_{2}}-{[\widehat{\mathbf{c}}]_{2}}}\right)^{2}}+{\left({\frac{{{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}}{{{\lambda_{1}}}}}\right)^{2}}s_{11}^{2}-\frac{{2\left({{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}\right)\left({{[\mathbf{b}]_{2}}-{[\widehat{\mathbf{c}}]_{2}}}\right){s_{11}}}}{{{\lambda_{1}}}}.$
(102)
The solution (99) has paved the way to delivering the self-correcting
procedure: Algorithm 3. The algorithm can be summarized as if the real-time
safety metric $[{\bf{s}}({\bf{u}}(k))]_{1}$ or $[{\bf{s}}({\bf{u}}(k))]_{2}$
is larger than the corresponding safety bound $[\mathbf{c}]_{1}$ or
$[\mathbf{c}]_{2}$, the real-time safety metric will be updated with the
corresponding safety bound (indicated by Line 3 of Algorithm 3). The corrected
control commands are then computed according to (99) (see Lines 3-3). The
solutions however are not unique. To address the problem, the Line 11 of
Algorithm 3 picks up the control commands that are most close to current ones.
Under the control of Phy-Taylor, with and without the self-correcting
procedure, the system performances are presented in Figure 12 (e)–(h). We can
observe from the figure that the self-correcting procedure can significantly
enhance the safe assurance of velocity regulation. The demonstration video of
implementing the self-correcting Phy-Taylor in AutoRally is available at
https://ymao578.github.io/pubs/taylorcontrol2.mp4.
### 6.2 Coupled Pendulums
Figure 13: Mechanical analog of three coupled pendulums.
The second experiment demonstrates the proposed approach to learn the dynamics
of three coupled pendulums, whose mechanical analog is shown in Figure 13.
Each pendulum is characterized by its phase angle $\theta_{i}$ and velocity
$\dot{\theta}_{i}$. The physical topology is described by
$a_{12}=a_{21}=a_{23}=a_{32}=1$ and $a_{13}=a_{31}=0$. We let
${\upsilon_{i}}\triangleq{\dot{\theta}_{i}}$. Referring to Figure 13, the
available physical knowledge for NN editing are summarized in Table 3.
Table 3: Models with Different Degrees of Embedded Physical Knowledge | Available Physical Knowledge | |
---|---|---|---
Model ID | Physics Law: $\upsilon=\dot{\theta}$ | Sampling Period: $T$ | Coupling Topology: $1\leftrightsquigarrow 2\leftrightsquigarrow 3$ | Force Dependency | Training Loss | Out-of-Distribution: Prediction Error $e$
Phy-Taylor-1 | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $1.75146\\!\cdot\\!10^{-6}$ | $0.06486147$
Phy-Taylor-2 | $\surd$ | $\times$ | $\surd$ | $\times$ | $2.42426\\!\cdot\\!10^{-6}$ | $1.46647886$
FDNN | $\times$ | $\times$ | $\times$ | $\times$ | $6.63564\cdot 10^{-7}$ | $3.65772883$
Figure 14: (i): Phy-Taylor architecture. (ii): FDNN architecture. (a)–(f):
Ground truth and predicted trajectories.
We consider the dynamics learning via Phy-Taylor and fully-connected DNN
(FDNN), whose architectures are shown in Figure 14 (a) and (b). The inputs of
both networks are identical as $\mathbf{x}(k)$ $=$ [${\theta_{1}}({k})$;
${\theta_{2}}({k})$; ${\theta_{3}}({k})$; ${\upsilon_{1}}({k})$;
${\upsilon_{2}}({k})$; ${\upsilon_{3}}({k})$]. For a fair comparison, we let
each layer of FDNN has the same number of nodes as the Phy-Taylor (after
augmentation), which suggests the network configuration in Figure 14 (b).
We aim to demonstrate the robustness of Phy-Taylor owning to NN editing. To
achieve this, we consider the testing data that is out-of-distribution.
Specifically, the initial conditions of phase angles for generating training
data are ${\theta_{1}}(0),{\theta_{2}}(0),{\theta_{3}}(0)\in[-1,1]$, while the
initial conditions for generating testing data are outside the range:
${\theta_{1}}(0),{\theta_{2}}(0),{\theta_{3}}(0)\in[-1.5,-1)$. The training
loss (mean square error) of the considered models with different degrees of
embedded physical knowledge are summarized in Table 3. To review the
robustness of trained models, we consider the prediction of long-horizon
trajectory, given the same initial input. The prediction error is measured by
$e=\frac{1}{6}\sum\limits_{i=1}^{3}{\frac{1}{\tau}\left({\sum\limits_{t=k+1}^{k+\tau}{\left({{{\widehat{\theta}}_{i}}(t)-{\theta_{i}}(t)}\right)^{2}}+\sum\limits_{t=k+1}^{k+\tau}{\left({{{\widehat{\upsilon}}_{i}}(t)-{\upsilon_{i}}(t)}\right)^{2}}}\right)}$,
where ${{\widehat{\theta}}_{i}}(t)$ and ${{\widehat{\upsilon}}_{i}}(t)$ denote
the predicted angle and angular speed of $i$-th pendulum at time $t$,
respectively. The prediction errors are summarized in Table 3, which, in
conjunction with ground-truth and predicted trajectories in Figure 14 (c)–(h),
demonstrate that (i) the NN editing can significantly enhance the model
robustness, and (ii) the more embedded physical knowledge can lead to stronger
robustness, viewed from the perspective of long-horizon (out-of-distribution)
predictions of trajectories.
### 6.3 US Illinois Climate
In final example, we consider a climate system without any available physical
knowledge for NN editing, which degrades Phy-Taylor to deep PhN. The dataset
is the hourly climate normals in Illinois state, including five stations111
NOAA: https://www.ncdc.noaa.gov/cdo-
web/datasets/NORMAL_HLY/locations/FIPS:17/detail. The period of record is
01/01/2010–12/31/2010. The locations of considered stations are shown in
Figure 15, where $S_{1}$-$S_{4}$ denote stations GHCND:USW00094846,
GHCND:USW00094822, GHCND:USW00014842 and GHCND:USW00093822, respectively. The
input of deep PhN is
$\displaystyle\mathbf{x}(k)=\left[{{x_{1}}(k);\leavevmode\nobreak\
{x_{2}}(k);\leavevmode\nobreak\ {x_{3}}(k);\leavevmode\nobreak\
{x_{4}}(k);\leavevmode\nobreak\ {{\dot{x}}_{1}}(k);\leavevmode\nobreak\
{{\dot{x}}_{2}}(k);\leavevmode\nobreak\
}\right.{{\dot{x}}_{3}}(k);\leavevmode\nobreak\
{{\dot{x}}_{4}}(k);\leavevmode\nobreak\
{{\ddot{x}}_{1}}(k);\leavevmode\nobreak\
{{\ddot{x}}_{2}}(k);\leavevmode\nobreak\
{{\ddot{x}}_{3}}(k);\leavevmode\nobreak\ \left.{{{\ddot{x}}_{4}}(k)}\right],$
Figure 15: Station locations.
where
${{\dot{x}}_{i}}\left(k\right)={{x}_{i}}\left(k\right)-{{x}_{i}}\left(k-1\right)$
and
${{\ddot{x}}_{i}}\left(k\right)={\dot{x}_{i}}\left(k\right)-{\dot{x}_{i}}\left(k-1\right)=({{x}_{i}}\left(k\right)-{{x}_{i}}\left(k-1\right))-({{x}_{i}}\left(k-1\right)-{{x}_{i}}\left(k-2\right))$,
$i=1,2,3,4$, and the $x_{1}$, $x_{2}$, $x_{3}$ and $x_{4}$ denote the dew
point mean, the heat index mean, the wind chill mean and the average wind
speed, respectively. The training loss function is
$\displaystyle\mathcal{L}={\left\|{{[\widehat{\mathbf{y}}]_{1}}-{x_{2}}\left({k+1}\right)}\right\|}+{\left\|{{[\widehat{\mathbf{y}}]_{2}}-{x_{4}}\left({k+1}\right)}\right\|},$
which indicates the network models are to predict the heat index mean and the
average wind speed for next hour.
#### 6.3.1 Phy-Augmentation
Figure 16: Architectures of deep PhN and classical DNN.
We use the data of station $S_{1}$ to study the influence of augmentation
order $r$ from the perspective of training loss. The considered network model
is shown in Figure 16 (a). The trajectories of training loss under two
different augmentation orders are presented in Figure 17 (a). It is
straightforward to observe from the figure that the input augmentation
significantly speeds up the training process and reduces the training loss.
The intuitive explanation is the input augmentation enlarges node number.
To validate the statement, we perform the comparisons with the classical DNN,
whose structure is given in Figure 16 (b). To guarantee the comparisons are
carried in the fair settings, we let the input dimension (i.e., $n$) of the
final layer of DNN in Figure 16 (b) equates to the output dimension of input
augmentation of PhN in Figure 16 (a). According to (112), we let
$n={\rm{len}}(\mathfrak{m}(\mathbf{x},r=5))=253$. The comparison results are
presented in Figure 17 (b), which indicates that given the same number of
nodes, the deep PhN still has much faster convergence speed and smaller
training loss of mean. The phenomenon can be explained by the fact that Phy-
Augmentation well captures physical features.
Figure 17: Deep PhN v.s. classical DNN: log of training loss (5 random seeds).
#### 6.3.2 Large Noise
To test the robustness of deep PhN faced with large noise, we consider the
network structure in Figure 16 (c). To introduce noisy datasets for training,
we use the inputs $\mathbf{x}(k)$ from stations $S_{2}$–$S_{4}$, while the
dataset of outputs $[\widehat{\mathbf{y}}]_{1}$ and
$[\widehat{\mathbf{y}}]_{2}$ is from the station $S_{1}$. This setting means
we will use station $S_{1}$’s neighbors’ inputs to predict its heat index mean
and average wind speed. For the suppressor of deep PhN in (11), we let
$\beta=90$ and $\alpha=-1$. The trajectories of training loss in Figure 18
together with station locations in Figure 15 show that the training loss
decreases as the distance with station $S_{1}$ increases. Considering the fact
that noise of training (input) datasets can increase as distance with (truth)
station $S_{1}$ increases, the result demonstrates the superiority of
suppressor in mitigating the influence of large noise in augmented input
features.
Figure 18: Training loss with and without suppressor function (5 random
seeds).
## 7 Discussion
In this article, we have proposed a physics-model-based deep neural network
framework, called Phy-Taylor, that addresses the challenge the purely data-
driven deep DNNs are facing: the violation of inferred relations with physics
laws. The Phy-Taylor framework introduces two contributions: the deep physics-
compatible neural networks (PhNs) and a physics-guided neural network (NN)
editing mechanism, aiming at ensuring strict compliance with prior physical
knowledge. The PhN aims to directly capture nonlinearities inspired by
physical quantities, such as kinetic energy and aerodynamic drag force. The NN
editing mechanism further modifies network links and activation functions
consistently with physical knowledge to avoid spurious correlations. As an
extension, we have also proposed a self-correcting Phy-Taylor framework that
introduces a core capability of automatic output correction when violation of
safety occurs. The self-correcting Phy-Taylor successfully addresses the
dilemma of prediction horizon and computation time that nonlinear model-
predictive control and control barrier function are facing in safety- and
time-critical systems.
The experiments show that through physics-guided NN editing and direct capture
of hard-to-learn nonlinearities, the Phy-Taylor exhibits a considerable
reduction in learning parameters, a remarkably accelerated training process,
and greatly enhanced model robustness and accuracy. This suggests that
building on the Phy-Taylor framework, the concurrently energy-efficient,
robustness and high-accurate DNNs are promising for the energy-constraint
physical engineering systems (see e.g., internet of things and battery-powered
drones), which constitutes a part of our future research. The current Phy-
Taylor however suffers from curse of dimensionality, so it can hardly be
applied to high-dimensional data, such as image and text. Overcoming the
dimensionality curse problem will thus make the Phy-Taylor scalable. The
Tucker Decomposition is a potential to address this problem, since it can
decompose the higher-order derivatives in Taylor expansions parameterized by
DNNs into small core tensors and a set of factor matrices.
## 8 Methods
### 8.1 Datasets
All the datasets used in the experiments are publicly available at
https://waynestateprod-
my.sharepoint.com/:f:/g/personal/hm9062_wayne_edu/EoXO99WN8zJEidtGRj-
dISIBQPBWCnL_Ji6QOZ1uJACjug.
The datasets in Experiment 6.1 are collected from the AutoRally platform [45].
The vehicle therein is set to run 6 times, each running time is 30 minutes,
which can generate around 1100 samples of ellipse trajectories. The datasets
in Experiment 6.2 are generated by solving the differential equations of
coupled pendulum in MATLAB using the ode45 solver, given 1000 initial
conditions. The climate datasets in Experiment 6.3 are from the Climate Data
Online–National Oceanic and Atmospheric Administration222 NOAA:
https://www.ncdc.noaa.gov/cdo-
web/datasets/NORMAL_HLY/locations/FIPS:17/detail, whose period of record is
01/01/2010–12/31/2010. The datasets in Experiments 6.1 and 6.2 are evenly
distributed to 6 files: two files are preserved for testing and validating,
the remaining 4 files are used for training. For the Experiment 6.3, the data
over the final 100 hours is preserved for testing, the data over another 864
hours are used for validating, the remaining data are used for training.
### 8.2 Code
For the code, we use the Python API for the TensorFlow framework [46] and the
Adam optimizer [47] for training. The Python version is 2.7.12. The TensorFlow
version is 1.4.0. Our source code is publicly available at GitHub:
https://github.com/ymao578/Phy-Taylor. The source code of implementing self-
correcting Phy-Taylor in the AutoRally platform is publicly available at
https://waynestateprod-
my.sharepoint.com/:f:/g/personal/hm9062_wayne_edu/EnDmRmmbKlJFmlQfC3qLMHwBf4KG9FRGVVo3FFVk1TrZeg?e=TQZSgr.
### 8.3 Training
We set the batch-size to 200 for the Experiments 6.1 and 6.2, while the batch-
size of Experiment 6.3 is 150. The learning rates of Experiments 6.1–6.3 are
set to 0.0005, 0.00005 and 0.00005, respectively. In all the experiments, each
weight matrix is initialized randomly from a (truncated) normal distribution
with zero mean and standard deviation, discarding and re-drawing any samples
that are more than two standard deviations from the mean. We initialize each
bias according to the normal distribution with zero mean and standard
deviation.
## References
* [1] Zachary, A. & Helen, T. AI Accidents: An emerging threat. _Center for Security and Emerging Technology_ DOI: https://doi.org/10.51593/20200072 (2021).
* [2] Wang, R. & Yu, R. Physics-guided deep learning for dynamical systems: A survey. _arXiv preprint_ https://arxiv.org/abs/2107.01272.
* [3] Willard, J., Jia, X., Xu, S., Steinbach, M. & Kumar, V. Integrating scientific knowledge with machine learning for engineering and environmental systems. _ACM Computing Surveys_ (2021).
* [4] Jia, X. _et al._ Physics-guided machine learning for scientific discovery: An application in simulating lake temperature profiles. _ACM/IMS Transactions on Data Science_ 2, 1–26 (2021).
* [5] Jia, X. _et al._ Physics-guided RNNs for modeling dynamical systems: A case study in simulating lake temperature profiles. In _Proceedings of the 2019 SIAM International Conference on Data Mining_ , 558–566 (2019).
* [6] Wang, S. & Perdikaris, P. Deep learning of free boundary and stefan problems. _Journal of Computational Physics_ 428, 109914 (2021).
* [7] Lu, L. _et al._ Physics-informed neural networks with hard constraints for inverse design. _SIAM Journal on Scientific Computing_ 43, B1105–B1132 (2021).
* [8] Chen, Y. _et al._ Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method. _Journal of Computational Physics_ 445, 110624 (2021).
* [9] Wang, N., Zhang, D., Chang, H. & Li, H. Deep learning of subsurface flow via theory-guided neural network. _Journal of Hydrology_ 584, 124700 (2020).
* [10] Xu, K. & Darve, E. Physics constrained learning for data-driven inverse modeling from sparse observations. _Journal of Computational Physics_ 110938 (2022).
* [11] Karniadakis, G. E. _et al._ Physics-informed machine learning. _Nature Reviews Physics_ 3, 422–440 (2021).
* [12] Wang, R., Kashinath, K., Mustafa, M., Albert, A. & Yu, R. Towards physics-informed deep learning for turbulent flow prediction. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 1457–1466 (2020).
* [13] Daw, A., Karpatne, A., Watkins, W., Read, J. & Kumar, V. Physics-guided neural networks (PGNN): An application in lake temperature modeling. _arXiv preprint_ https://arxiv.org/abs/1710.11431.
* [14] Cranmer, M. _et al._ Lagrangian neural networks. In _ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations_ (2020).
* [15] Finzi, M., Wang, K. A. & Wilson, A. G. Simplifying Hamiltonian and Lagrangian neural networks via explicit constraints. _Advances in Neural Information Processing Systems_ 33, 13880–13889 (2020).
* [16] Greydanus, S., Dzamba, M. & Yosinski, J. Hamiltonian neural networks. _Advances in Neural Information Processing Systems_ 32 (2019).
* [17] Muralidhar, N. _et al._ Phynet: Physics guided neural networks for particle drag force prediction in assembly. In _Proceedings of the 2020 SIAM International Conference on Data Mining_ , 559–567 (2020).
* [18] Masci, J., Boscaini, D., Bronstein, M. & Vandergheynst, P. Geodesic convolutional neural networks on riemannian manifolds. In _Proceedings of the IEEE international conference on computer vision workshops_ , 37–45 (2015).
* [19] Monti, F. _et al._ Geometric deep learning on graphs and manifolds using mixture model cnns. In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 5115–5124 (2017).
* [20] Horie, M., Morita, N., Hishinuma, T., Ihara, Y. & Mitsume, N. Isometric transformation invariant and equivariant graph convolutional networks. _arXiv preprint_ https://arxiv.org/abs/2005.06316.
* [21] Wang, R. Incorporating symmetry into deep dynamics models for improved generalization. In _International Conference on Learning Representations_ (2021).
* [22] Li, Y., He, H., Wu, J., Katabi, D. & Torralba, A. Learning compositional Koopman operators for model-based control. _arXiv preprint_ https://arxiv.org/abs/1910.08264.
* [23] Lusch, B., Kutz, J. N. & Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. _Nature communications_ 9, 1–10 (2018).
* [24] Li, Z. _et al._ Fourier neural operator for parametric partial differential equations. _arXiv:2010.08895_ (2020).
* [25] Königsberger, K. _Analysis 2_ (Springer-Verlag, 2013).
* [26] Yang, Y.-Y. & Chaudhuri, K. Understanding rare spurious correlations in neural networks. _arXiv preprint_ https://arxiv.org/abs/2202.05189.
* [27] Sagawa, S., Raghunathan, A., Koh, P. W. & Liang, P. An investigation of why overparameterization exacerbates spurious correlations. In _International Conference on Machine Learning_ , 8346–8356 (2020).
* [28] Rajamani, R. _Vehicle dynamics and control_ (Springer Science & Business Media, 2011).
* [29] Kani, J. N. & Elsheikh, A. H. DR-RNN: A deep residual recurrent neural network for model reduction. _arXiv preprint_ https://arxiv.org/abs/1709.00939.
* [30] Belbute-Peres, F. D. A., Economon, T. & Kolter, Z. Combining differentiable PDE solvers and graph neural networks for fluid flow prediction. In _International Conference on Machine Learning_ , 2402–2411 (2020).
* [31] Wu, D. _et al._ DeepGLEAM: A hybrid mechanistic and deep learning model for COVID-19 forecasting. _arXiv preprint_ https://arxiv.org/abs/2102.06684 (2021).
* [32] Guen, V. L. & Thome, N. Disentangling physical dynamics from unknown factors for unsupervised video prediction. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 11474–11484 (2020).
* [33] Garcia Satorras, V., Akata, Z. & Welling, M. Combining generative and discriminative models for hybrid inference. _Advances in Neural Information Processing Systems_ 32 (2019).
* [34] Long, Y., She, X. & Mukhopadhyay, S. HybridNet: Integrating model-based and data-driven learning to predict evolution of dynamical systems. In _Conference on Robot Learning_ , 551–560 (2018).
* [35] Yin, Y. _et al._ Augmenting physical models with deep networks for complex dynamics forecasting. _Journal of Statistical Mechanics: Theory and Experiment_ 2021, 124012 (2021).
* [36] AI incident database. _AID_ DOI: https://incidentdatabase.ai/summaries/incidents.
* [37] TOM, K. & STEFANIE, D. Felony charges are 1st in a fatal crash involving autopilot. _AP News_ DOI: https://apnews.com/article/tesla-autopilot-fatal-crash-charges-91b4a0341e07244f3f03051b5c2462ae (2022).
* [38] Hawkins, A. J. Tesla didn’t fix an autopilot problem for three years, and now another person is dead. _The Verge_ DOI: https://www.theverge.com/2019/5/17/18629214/tesla-autopilot-crash-death-josh-brown-jeremy-banner (2019).
* [39] Ames, A. D. _et al._ Control barrier functions: Theory and applications. In _2019 18th European Control Conference_ , 3420–3431 (2019).
* [40] Singletary, A., Swann, A., Chen, Y. & Ames, A. Onboard safety guarantees for racing drones: High-speed geofencing with control barrier functions. _IEEE Robotics and Automation Letters_ (2022).
* [41] Williams, G., Drews, P., Goldfain, B., Rehg, J. M. & Theodorou, E. A. Information-theoretic model predictive control: Theory and applications to autonomous driving. _IEEE Transactions on Robotics_ 34, 1603–1622 (2018).
* [42] Falcone, P., Borrelli, F., Asgari, J., Tseng, H. E. & Hrovat, D. Predictive active steering control for autonomous vehicle systems. _IEEE Transactions on Control Systems Technology_ 15, 566–580 (2007).
* [43] Zeng, J., Zhang, B. & Sreenath, K. Safety-critical model predictive control with discrete-time control barrier function. In _2021 American Control Conference_ , 3882–3889 (2021).
* [44] Doyle, J. C., Francis, B. A. & Tannenbaum, A. R. _Feedback control theory_ (Courier Corporation, 2013).
* [45] Goldfain, B. _et al._ AutoRally: An open platform for aggressive autonomous driving. _IEEE Control Systems Magazine_ 39, 26–55 (2019).
* [46] Kingma, D. P. & Ba, J. Adam: A method for stochastic optimization. _arXiv preprint_ https://arxiv.org/abs/1412.6980.
* [47] Abadi, M. _et al._ Tensorflow: Large-scale machine learning on heterogeneous distributed systems. _arXiv preprint_ https://arxiv.org/abs/1603.04467.
* [48] Stanley, R. P. What is enumerative combinatorics? In _Enumerative combinatorics_ , 1–63 (Springer, 1986).
## 9 Supplementary Information
### 9.1 Auxiliary Theorems
###### Theorem 4.
The DNR magnitude of high-order monomial
$[\bar{\mathbf{x}}]_{i}^{p}[\bar{\mathbf{x}}]_{j}^{q}$, $p$, $q\in\mathbb{N}$,
is strictly increasing with respect to $|\mathrm{DNR}_{i}|$ and
$|\mathrm{DNR}_{j}|$, if
$\displaystyle\mathrm{DNR}_{i},\leavevmode\nobreak\
\mathrm{DNR}_{j}\in(-\infty,-1]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{DNR}_{i},\leavevmode\nobreak\
\mathrm{DNR}_{j}\in[-\frac{1}{2},0)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{DNR}_{i},\leavevmode\nobreak\ \mathrm{DNR}_{j}\in(0,\infty).$ (103)
###### Proof.
In view of definition 2, the true data can be equivalently expressed as
$[{\mathbf{h}}]_{i}=\mathrm{DNR}_{i}\cdot[{\mathbf{w}}]_{i}$, according to
which we have
$[\bar{\mathbf{x}}]_{i}=(1+\mathrm{DNR}_{i}){[{\mathbf{w}}]_{i}}$ such that
$\displaystyle[\bar{\mathbf{x}}]_{i}^{p}[\bar{\mathbf{x}}]_{j}^{q}={({1+\mathrm{DNR}_{i}})^{p}}{({1+\mathrm{DNR}_{j}})^{q}}[\mathbf{w}]_{i}^{p}[\mathbf{w}]_{j}^{q},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
[\mathbf{h}]_{i}^{p}[\mathbf{h}]_{j}^{q}=\mathrm{DNR}_{i}^{p}\cdot\mathrm{DNR}_{j}^{q}\cdot[\mathbf{w}]_{i}^{p}[\mathbf{w}]_{j}^{q}.$
(104)
We note the true data of high-order monomial
$[\bar{\mathbf{x}}]_{i}^{p}[\bar{\mathbf{x}}]_{j}^{q}$ is
$[{\mathbf{h}}]_{i}^{p}[{\mathbf{h}}]_{j}^{q}$, the corresponding noise can
thus be derived from the formula (104) as
$\displaystyle[\bar{\mathbf{x}}]_{i}^{p}[\bar{\mathbf{x}}]_{j}^{q}-[\mathbf{h}]_{i}^{p}[\mathbf{h}]_{j}^{q}=\left[{({1+\mathrm{DNR}_{i}})^{p}}{({1+\mathrm{DNR}_{j}})^{q}}-\mathrm{DNR}_{i}^{p}\cdot\mathrm{DNR}_{j}^{q}\right][\mathbf{w}]_{i}^{p}[\mathbf{w}]_{j}^{q},$
which, in conjunction with the second formula in Equation (104), lead to
$\displaystyle\left|\mathrm{DNR}^{p+q}_{ij}\right|\triangleq\left|\frac{{[\mathbf{h}]_{i}^{p}[\mathbf{h}]_{j}^{q}}}{{[\bar{\mathbf{x}}]_{i}^{p}[\bar{\mathbf{x}}]_{j}^{q}-[\mathbf{h}]_{i}^{p}[\mathbf{h}]_{j}^{q}}}\right|=\left|\frac{1}{{{{\left({1+\frac{1}{{\mathrm{DNR}_{i}}}}\right)}^{p}}{{\left({1+\frac{1}{{\mathrm{DNR}_{j}}}}\right)}^{q}}-1}}\right|,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ p,\leavevmode\nobreak\ q\in\mathbb{N}.$ (105)
We can straightforwardly verify from formula (105) that if
$\mathrm{DNR}_{i},\leavevmode\nobreak\ \mathrm{DNR}_{j}\in(0,\infty)$, we have
$\displaystyle\left|\mathrm{DNR}^{p+q}_{ij}\right|=\frac{1}{{{{\left({1+\frac{1}{{|\mathrm{DNR}_{i}|}}}\right)}^{p}}{{\left({1+\frac{1}{{|\mathrm{DNR}_{j}|}}}\right)}^{q}}-1}},$
(106)
which implies $|\mathrm{DNR}^{p+q}_{ij}|$ is strictly increasing with respect
to $|\mathrm{DNR}_{i}|$ and $|\mathrm{DNR}_{j}|$ under this condition.
The condition $\mathrm{DNR}_{i},\leavevmode\nobreak\
\mathrm{DNR}_{j}\in(-\infty,-1]$ means that
$\displaystyle\frac{{1}}{{\mathrm{DNR}_{j}}}\in[-1,0),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\frac{{1}}{{\mathrm{DNR}_{j}}}\in[-1,0),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
1+\frac{{1}}{{\mathrm{DNR}_{i}}}\in[0,1),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ 1+\frac{{1}}{{\mathrm{DNR}_{j}}}\in[0,1),$ (107)
considering which, the formula (105) equivalently transforms to
$\displaystyle\left|\mathrm{DNR}^{p+q}_{ij}\right|=\frac{1}{{{{1-\left({1+\frac{1}{{\mathrm{DNR}_{i}}}}\right)}^{p}}{{\left({1+\frac{1}{{\mathrm{DNR}_{j}}}}\right)}^{q}}}}=\frac{1}{{{{1-\left({1-\frac{1}{{|\mathrm{DNR}_{i}|}}}\right)}^{p}}{{\left({1-\frac{1}{{|\mathrm{DNR}_{j}|}}}\right)}^{q}}}},$
(108)
which reveals that $|\mathrm{DNR}^{p+q}_{ij}|$ is strictly increasing with
respect to $|\mathrm{DNR}_{i}|$ and $|\mathrm{DNR}_{j}|$.
The condition $\mathrm{DNR}_{i},\leavevmode\nobreak\
\mathrm{DNR}_{j}\in[-\frac{1}{2},0)$ means
$\displaystyle\frac{{1}}{{\mathrm{DNR}_{j}}}\in(-\infty,-2],\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{{1}}{{\mathrm{DNR}_{j}}}\in(-\infty,-2],\leavevmode\nobreak\
\leavevmode\nobreak\
1+\frac{{1}}{{\mathrm{DNR}_{i}}}\in(-\infty,-1],\leavevmode\nobreak\
\leavevmode\nobreak\ 1+\frac{{1}}{{\mathrm{DNR}_{j}}}\in(-\infty,-1],$ (109)
in light of which, the formula (105) can equivalently express as
* •
if $p+q$ is even,
$\displaystyle\left|\mathrm{DNR}^{m+n}_{ij}\right|=\frac{1}{{{{\left|{\frac{1}{{|\mathrm{DNR}_{i}|}}-1}\right|}^{p}}{{\left|{\frac{1}{{|\mathrm{DNR}_{j}|}}-1}\right|}^{q}}}-1},$
(110)
* •
if $p+q$ is odd,
$\displaystyle\left|\mathrm{DNR}^{p+q}_{ij}\right|=\frac{1}{{{{1+\left|{\frac{1}{{|\mathrm{DNR}_{i}|}}-1}\right|}^{p}}{{\left|{\frac{1}{{|\mathrm{DNR}_{j}|}}-1}\right|}^{q}}}}.$
(111)
We note both the functions (110) and (111) imply $|\mathrm{DNR}^{m+n}_{ij}|$
is strictly increasing with respect to $|\mathrm{DNR}_{i}|$ and
$|\mathrm{DNR}_{j}|$, which completes the proof. ∎
###### Theorem 5.
[48] For any pair of positive integers $n$ and $k$, the number of $n$-tuples
of non-negative integers whose sum is $r$ is equal to the number of multisets
of cardinality $n-1$ taken from a set of size $n+r-1$, i.e.,
$\left(\begin{array}[]{l}n+r-1\\\
n-1\end{array}\right)=\frac{{\left({n+r-1}\right)!}}{{\left({n-1}\right)!r!}}$.
###### Theorem 6.
The space complexity of Phy-Augmentation, i.e., the dimension of terminal
output generated by Algorithm 1, in PhN layer (20) is
$\displaystyle\mathrm{len}(\mathfrak{m}({\mathbf{x}},r))=\sum\limits_{s=1}^{r}{\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)!s!}}}+1.$
(112)
###### Proof.
We denote the output from Line 1 of Algorithm 1 by $\overline{\mathbf{x}}$.
Let us first consider the case $r=1$. In this case, the Algorithm 1 skips the
Lines 1–1 and arrives at $\mathfrak{m}(\mathbf{x},r)=[1;\leavevmode\nobreak\
\overline{\mathbf{x}}]$ in Line 1. Noticing from Line 1 that
$\overline{\mathbf{x}}\in\mathbb{R}^{n}$, we obtain
$\text{len}(\mathfrak{m}(\mathbf{x},r))=n+1$, which verifies the correctness
of (112) with $r=1$.
We next consider the case $r\geq 2$. Given the input dimension $n$ and an
order $s\in\\{2,\ldots,r-1,r\\}$, the Lines 1–1 of Algorithm 1 are to generate
all the non-missing and non-redundant monomials included in
${\left({\sum\limits_{i=1}^{n}{{[\overline{\mathbf{x}}]_{i}}}}\right)^{s}}$.
The problem of the number of generated monomials via Algorithm 1 is equivalent
to the problem that for any pair of positive integers $n$ and $s$, the number
of $n$-tuples of non-negative integers (whose sum is $s$) is equal to the
number of multisets of cardinality $n-1$ taken from a set of size $n+s-1$.
Additionally, we note that the vector generated in Line 1 of Algorithm 1,
denoted by $\widetilde{\mathfrak{m}}(\mathbf{x},s)$, stacks all the generated
monomials. According to the auxiliary Theorem 5 in Supplementary Information
9.1, we then have
$\text{len}(\widetilde{\mathfrak{m}}(\mathbf{x},s))=\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)!s!}}$.
Finally, we note that the Lines 1, and 1 of Algorithm 1 imply that the
generated vector $\mathfrak{m}(\mathbf{x},r)$ stack the $1$ with
$\widetilde{\mathfrak{m}}(\mathbf{x},s)$ over $s\in\\{1,\ldots,r-1,r\\}$,
respectively. We thus can obtain (112). ∎
### 9.2 Proof of Theorem 1
We note the $[\widetilde{\mathbf{h}}]_{i}$ given in Equation (13) can be
written as
$\displaystyle[\widetilde{\mathbf{h}}]_{i}=\begin{cases}[\mathbf{h}]_{i},&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}<0\\\
[\mathbf{h}]_{i},&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}<0\\\
[\mathbf{h}]_{i}\cdot\kappa_{i}+\rho_{i},&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}>0\end{cases},$ (113)
subtracting $[{\mathbf{h}}]_{i}$ from which yields
$\displaystyle\left|[\widetilde{\mathbf{h}}]_{i}-[{\mathbf{h}}]_{i}\right|$
$\displaystyle=\begin{cases}0,&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}<0\\\ 0,&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}<0\\\
\left|{[\mathbf{h}]_{i}\cdot\left({\kappa_{i}-1}\right)+\rho_{i}}\right|,&\text{if}\leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ [\mathbf{w}]_{i}>0\\\ \end{cases}.$ (114)
Referring to the output $\chi([\bar{\bf{x}}]_{i})$ of suppressor in Equation
(10), we can conclude the $[\widetilde{\mathbf{h}}]_{i}$ given in Equation
(113) is the true data of suppressor output. Subtracting the
$[\widetilde{\mathbf{h}}]_{i}$ from the $\chi([\bar{\bf{x}}]_{i})$ results in
the noise $[\widetilde{\mathbf{w}}]_{i}$ of suppressor output, given in
Equation (13).
To prove the property (12), we consider the following three cases:
* •
If $[\mathbf{h}]_{i}+[\mathbf{w}]_{i}<0$, we obtain from the the first item of
$[\widetilde{\mathbf{h}}]_{i}$ in Equation (113) and
$[\widetilde{\mathbf{w}}]_{i}$ in Equation (13) that
$\mathrm{DNR}_{i}=\frac{[\widetilde{\mathbf{h}}]_{i}}{[\widetilde{\mathbf{w}}]_{i}}=-1$.
* •
If $[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0$ and $[\mathbf{w}]_{i}<0$, we have
$[\mathbf{h}]_{i}>0$ and $[\mathbf{h}]_{i}>-[\mathbf{w}]_{i}|>0$. We then
obtain from the second item of $[\widetilde{\mathbf{h}}]_{i}$ in Equation
(113) and $[\widetilde{\mathbf{w}}]_{i}$ in Equation (13) that
$\mathrm{DNR}_{i}=\frac{[\widetilde{\mathbf{h}}]_{i}}{[\widetilde{\mathbf{w}}]_{i}}=\frac{[{\mathbf{h}}]_{i}}{[{\mathbf{w}}]_{i}}<-1$.
* •
If $[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0$ and $[\mathbf{w}]_{i}>0$, we
obtain from the third item of $[\widetilde{\mathbf{h}}]_{i}$ in Equation (113)
and $[\widetilde{\mathbf{w}}]_{i}$ in Equation (13) that
$\mathrm{DNR}_{i}=\frac{[\widetilde{\mathbf{h}}]_{i}}{[\widetilde{\mathbf{w}}]_{i}}=\frac{{[\mathbf{h}]_{i}\cdot\kappa_{i}+\rho_{i}}}{{[\mathbf{w}]_{i}\cdot\kappa_{i}}}$.
Recalling $[\widetilde{\mathbf{w}}]_{i}>0$, if $\kappa_{i}>0$, the
$\frac{{[\mathbf{h}]_{i}\cdot\kappa_{i}+\rho_{i}}}{{[\mathbf{w}]_{i}\cdot\kappa_{i}}}\leq-1$
is equivalent to
$\displaystyle\rho_{i}\leq-([\mathbf{h}]_{i}+[\mathbf{w}]_{i})\kappa_{i}<0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{with}\leavevmode\nobreak\
\leavevmode\nobreak\ \kappa_{i}>0,\leavevmode\nobreak\ \leavevmode\nobreak\
[\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0.$ (115)
If $\kappa_{i}<0$, the
$\frac{{[\mathbf{h}]_{i}\cdot\kappa_{i}+\rho_{i}}}{{[\mathbf{w}]_{i}\cdot\kappa_{i}}}\leq-1$
is equivalent to
$\displaystyle\rho_{i}\geq-\left({[\mathbf{w}]_{i}+[\mathbf{h}]_{i}}\right)\kappa_{i}\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ \kappa_{i}<0,\leavevmode\nobreak\
\leavevmode\nobreak\ [\mathbf{h}]_{i}+[\mathbf{w}]_{i}\geq 0.$ (116)
We finally conclude from Equations (115) and (116) that
$\mathrm{DNR}_{i}=\frac{[\widetilde{\mathbf{h}}]_{i}}{[\widetilde{\mathbf{w}}]_{i}}\in(-\infty,-1]$
under the condition (11). Then, according to Theorem 4, we arrive in the
property (12), which completes the proof.
### 9.3 Proof of Theorem 2
Let us first consider the first PhN layer, i.e., the case $t=1$. The Line 2 of
Algorithm 2 means that the knowledge matrix $\mathbf{K}_{\left\langle
1\right\rangle}$ includes all the known model-substructure parameters, whose
corresponding entries in the masking matrix $\mathbf{M}_{\left\langle
1\right\rangle}$ (generated in the Line 2 of Algorithm 2) are frozen to be
zeros. Consequently, both $\mathbf{M}_{\left\langle
1\right\rangle}\odot\mathbf{A}$ and $\mathbf{U}_{\left\langle
1\right\rangle}=\mathbf{M}_{\left\langle
1\right\rangle}\odot\mathbf{W}_{\left\langle 1\right\rangle}$ excludes all the
known model-substructure parameters (included in $\mathbf{K}_{\left\langle
1\right\rangle}$). With the consideration of Definition 1, we thus conclude
that $\mathbf{M}_{\left\langle
1\right\rangle}\odot\mathbf{A}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+\mathbf{f}(\mathbf{x})$ in the ground-truth model (16) and
$\mathbf{a}_{\left\langle
1\right\rangle}\odot\text{act}\left(\mathbf{U}_{\left\langle
1\right\rangle}\cdot{\mathfrak{m}}\left(\mathbf{x},r_{\left\langle
1\right\rangle}\right)\right)$ in the output computation (17) are independent
of the term $\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle 1\right\rangle})$.
Moreover, the activation-masking vector (generated in Line 2 of Algorithm 2)
indicates that the activation function corresponding to the output’s $i$-th
entry is inactive, if the all the entries in the $i$-th row of masking matrix
are zeros (implying all the entries in the $i$-th row of weight matrix are
known model-substructure parameters). Finally, we arrive in the conclusion
that the input/output (i.e., $\mathbf{x}$/$\mathbf{y}_{\left\langle
1\right\rangle}$) of the first PhN layer strictly complies with the available
physical knowledge pertaining to the ground truth (16), i.e., if the
$[\mathbf{A}]_{i,j}$ is a known model-substructure parameter, the
$\frac{{\partial{[\mathbf{y}_{\left\langle
1\right\rangle}]_{i}}}}{{\partial[{\mathfrak{m}}\left({\mathbf{x},r}\right)]_{j}}}\equiv\frac{{\partial{[\mathbf{y}]_{i}}}}{{\partial{[\mathfrak{m}}({\mathbf{x},r})]_{j}}}\equiv{[\mathbf{A}]_{i,j}}$
always holds.
We next consider the remaining PhN layers. Considering the Line 2 Algorithm 2,
we have
$\displaystyle[\mathbf{y}_{\left\langle
p\right\rangle}]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=[\mathbf{K}_{\left\langle
p\right\rangle}\cdot\mathfrak{m}(\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle})]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=\mbox{\small
I}_{\text{len}(\mathbf{y})}\cdot[\mathfrak{m}(\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle})]_{2:(\text{len}(\mathbf{y})+1)}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$ (117a)
$\displaystyle=\mbox{\small
I}_{\text{len}(\mathbf{y})}\cdot[\mathbf{y}_{\left\langle
p-1\right\rangle}]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$ (117b)
$\displaystyle=[\mathbf{y}_{\left\langle
p-1\right\rangle}]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=[\mathbf{K}_{\left\langle
p-1\right\rangle}\cdot\mathfrak{m}(\mathbf{y}_{\left\langle
p-2\right\rangle},r_{\left\langle
p-1\right\rangle})]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=\mbox{\small
I}_{\text{len}(\mathbf{y})}\cdot[\mathfrak{m}(\mathbf{y}_{\left\langle
p-2\right\rangle},r_{\left\langle
p-1\right\rangle})]_{2:(\text{len}(\mathbf{y})+1)}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=\mbox{\small
I}_{\text{len}(\mathbf{y})}\cdot[\mathbf{y}_{\left\langle
p-2\right\rangle}]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=[\mathbf{y}_{\left\langle
p-2\right\rangle}]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot\breve{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$ $\displaystyle=\ldots$
$\displaystyle=[\mathbf{y}_{\left\langle
1\right\rangle}]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=[\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})]_{1:\text{len}(\mathbf{y})}+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})},$
$\displaystyle=\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})},$ (117c)
where (117a) and (117b) are obtained from their previous steps via considering
the structure of block matrix $\mathbf{K}_{\left\langle t\right\rangle}$
(generated in Line 11 of Algorithm 2) and the formula of augmented monomials:
$\mathfrak{m}(\mathbf{x},r)=\left[1;\leavevmode\nobreak\
\mathbf{x};\leavevmode\nobreak\
[\mathfrak{m}({\mathbf{x},r})]_{(\text{len}(\mathbf{x})+2):\text{len}(\mathfrak{m}(\mathbf{x},r))}\right]$
(generated via Algorithm 1). The remaining iterative steps follow the same
path.
The training loss function is to push the terminal output of Algorithm 2
(i.e., $\widehat{\mathbf{y}}=\mathbf{y}_{\left\langle p\right\rangle}$) to
approximate the real output $\mathbf{y}$, which in light of (117c) yields
$\displaystyle\widehat{\mathbf{y}}$ $\displaystyle=\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+[\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle
p\right\rangle}})}\right)]_{1:\text{len}(\mathbf{y})}$
$\displaystyle=\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle
1\right\rangle})+\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle p\right\rangle}})}\right),$ (118)
where (118) from its previous step is obtained via considering the fact
$\text{len}(\widehat{\mathbf{y}})=\text{len}(\mathbf{y})=\text{len}(\mathbf{y}_{\left\langle
p\right\rangle})$. Meanwhile, the condition of generating weight-masking
matrix in Line 2 of Algorithm 2 removes all the node-representations’
connections with the known model-substructure parameters included in
$\mathbf{K}_{\left\langle 1\right\rangle}$. Therefore, we can conclude that in
the terminal output computation (118), the term $\mathbf{a}_{\left\langle
p\right\rangle}\odot\text{act}\\!\left({\mathbf{U}_{\left\langle
p\right\rangle}\cdot{\mathfrak{m}}({\mathbf{y}_{\left\langle
p-1\right\rangle},r_{\left\langle p\right\rangle}})}\right)$ does not have
influence on the computing of knowledge term $\mathbf{K}_{\left\langle
1\right\rangle}\cdot\mathfrak{m}(\mathbf{x},r_{\left\langle 1\right\rangle})$.
Thus, the Algorithm 2 strictly embeds and preserves the available knowledge
pertaining to the physics model of ground truth (1), or equivalently the (16).
### 9.4 Proof of Theorem 3
Due to Theorem 6 in Supplementary Information 9.1, the number of augmented
monomials of $d$ cascading PhNs (21) is obtained as
$\displaystyle\sum\limits_{p=1}^{d}{\text{len}(\mathfrak{m}({\mathbf{x}},r_{{\left\langle
p\right\rangle}}))}$
$\displaystyle=\underbrace{\sum\limits_{s=1}^{{r_{\left\langle
1\right\rangle}}}{\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)s!}}}+1}_{\text{the
first
PhN}}+\underbrace{\sum\limits_{v=1}^{d-1}{\sum\limits_{s=1}^{{r_{{\left\langle
v+1\right\rangle}}}}{\frac{{\left({{n_{\left\langle
v\right\rangle}}+s-1}\right)!}}{{\left({{n_{\left\langle
v\right\rangle}}-1}\right)!s!}}}}+d-1}_{\text{the remaining PhNs}}.$ (119)
The condition (22) implies that $r>r_{{\left\langle 1\right\rangle}}$, which
in conjunction with (112), lead to
$\displaystyle\text{len}(\mathfrak{m}({\mathbf{x}},r))=\sum\limits_{s=1}^{{r_{\left\langle
1\right\rangle}}}{\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)!s!}}}+\sum\limits_{s={r_{\left\langle
1\right\rangle}}+1}^{r}{\frac{{\left({n+s-1}\right)!}}{{\left({n-1}\right)!s!}}}+1.$
(120)
Subtracting (119) from (120) yields (23).
### 9.5 Derivations of Solution (99)
With the consideration of (81)–(94), the safety formulas:
$\left[{\bf{s}}({\bf{u}}(k))\right]_{1}=[\widehat{\mathbf{c}}]_{1}$ and
$\left[{\bf{s}}({\bf{u}}(k))\right]_{2}=[\widehat{\mathbf{c}}]_{2}$ can be
rewritten as
$\displaystyle{\lambda_{1}}[\widehat{\mathbf{u}}(k)]_{1}^{2}+{\lambda_{2}}[\widehat{\mathbf{u}}(k)]_{2}^{2}$
$\displaystyle={[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}},$ (121)
$\displaystyle{\left[\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\right]^{\top}}{\mathbf{P}}_{2}\left[\begin{array}[]{l}\theta(k)\\\
\gamma(k)\end{array}\right]$
$\displaystyle={s_{11}}[\widehat{\mathbf{u}}(k)]_{1}^{2}+2{s_{12}}{[\widehat{\mathbf{u}}}(k)]_{1}{[\widehat{\mathbf{u}}}(k)]_{2}+{s_{22}}[\widehat{\mathbf{u}}(k)]_{2}^{2}=[\mathbf{b}]_{2}-{[\widehat{\mathbf{c}}]_{2}},$
(126)
We now define:
$\displaystyle\overline{\mu}_{1}\triangleq\frac{{{[\widehat{\mathbf{c}}]_{1}}-{[\mathbf{b}]_{1}}}}{{{\lambda_{1}}}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\overline{\lambda}\triangleq\frac{{{\lambda_{2}}}}{{{\lambda_{1}}}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\bar{b}\triangleq{[\mathbf{b}]_{2}}-{[\widehat{\mathbf{c}}]_{2}}.$ (127)
leveraging which, the (121) is rewritten as
$\displaystyle[\widehat{\mathbf{u}}(k)]_{1}^{2}=\overline{\mu}_{1}-\overline{\lambda}[\widehat{\mathbf{u}}(k)]_{2}^{2},$
(128)
and we can obtain from (126) that
$\displaystyle
4s_{12}^{2}[\widehat{\mathbf{u}}(k)]_{2}^{2}[\widehat{\mathbf{u}}(k)]_{1}^{2}={{\overline{b}}^{2}}+s_{11}^{2}[\widehat{\mathbf{u}}(k)]_{1}^{4}+s_{22}^{2}[\widehat{\mathbf{u}}(k)]_{2}^{4}-2\overline{b}{s_{11}}[\widehat{\mathbf{u}}(k)]_{1}^{2}-2\overline{b}{s_{22}}[\widehat{\mathbf{u}}(k)]_{2}^{2}+2{s_{11}}{s_{22}}[\widehat{\mathbf{u}}(k)]_{1}^{2}[\widehat{\mathbf{u}}(k)]_{2}^{2},$
substituting (128) into which yields
$\displaystyle{\varpi_{1}}[\widehat{\mathbf{u}}]_{2}^{4}\left(k\right)+{\varpi_{2}}[\widehat{\mathbf{u}}]_{2}^{2}\left(k\right)+{\varpi_{3}}\left(k\right)=0,$
(129)
where
$\displaystyle{\varpi_{1}}$ $\displaystyle\triangleq
s_{11}^{2}{\overline{\lambda}^{2}}+s_{22}^{2}-2{s_{11}}{s_{22}}\overline{\lambda}+4s_{12}^{2}\overline{\lambda},$
(130) $\displaystyle{\varpi_{2}}$ $\displaystyle\triangleq
2\bar{b}{s_{11}}\overline{\lambda}-2s_{11}^{2}{\overline{\mu}_{1}}\overline{\lambda}-2\bar{b}{s_{22}}+2{s_{11}}{s_{22}}{\overline{\mu}_{1}}-4s_{12}^{2}{\overline{\mu}_{1}},$
(131) $\displaystyle{\varpi_{3}}$
$\displaystyle\triangleq{\bar{b}^{2}}+s_{11}^{2}\overline{\mu}_{1}^{2}-2\bar{b}{s_{11}}{\overline{\mu}_{1}}.$
(132)
Considering $\widehat{u}_{2}^{2}(k)\geq 0$, the solution of (129) is
$\displaystyle[\widehat{\mathbf{u}}]_{2}^{2}(k)=\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}},$
(133)
substituting which into (128) yields
$\displaystyle[\widehat{\mathbf{u}}]_{1}^{2}(k)={{\overline{\mu}}_{1}}-\overline{\lambda}\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}.$
(134)
The $\widehat{\mathbf{u}}(k)$ is then straightforwardly obtained from (133)
and (134):
$\displaystyle\widehat{\mathbf{u}}(k)=\left[{\pm\sqrt{{{\overline{\mu}}_{1}}-\overline{\lambda}\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}};\pm\sqrt{\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}}}\right],$
which, in conjunction with (94) and $Q^{-1}=Q=Q^{\top}$, lead to
$\displaystyle\left[\begin{array}[]{l}\theta\left(k\right)\\\
\gamma\left(k\right)\end{array}\right]={\mathbf{Q}_{1}}\left[\begin{array}[]{l}\pm\sqrt{{{\overline{\mu}}_{1}}-\overline{\lambda}\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}}\\\
\pm\sqrt{\frac{{\sqrt{\varpi_{2}^{2}-4{\varpi_{1}}{\varpi_{3}}}-{\varpi_{2}}}}{{2{\varpi_{1}}}}}\end{array}\right].$
(139)
Substituting the notations defined in (127) into (139) and (130)–(132) results
in (99) and (100)–(102), respectively.
## Acknowledgements
This work was supported in part by National Science Foundation (award number:
CPS-1932529), Hong Kong Research Grants Council (RGC) Theme-based Research
Scheme T22-505/19-N (P0031331, RBCR, P0031259, RBCP), and RGC Germany/HK Joint
Research Scheme under Grant G-PolyU503/16.
## Author Contributions
Y.M., L.S., and H.S. designed research; Y.M. performed research; Y.M. led
experiments, H.S. conducted experiment of deep Koopman, and Y. L. collected
vehicle data and implemented self-correcting Phy-Taylor in AutoRally; Y.M.,
L.S., Q.W., and T.A. participated in research discussion and data analysis;
Y.M. and S.H. wrote this manuscript; T.A. edited this manuscript; L.S. and
T.A. led the research team.
|
approach. For my needs this is more than enough, since in any case I
eventually restrict attention to a small interval around $0$.
Corollary 4.11 can be formulated in a slightly more general fashion. Using the
notation
$\delta(W):=\limsup_{r\rightarrow\infty}\frac{\log\big{(}b_{W}(r)\big{)}}{r}$
for a general subset $W$ in a general metric space $X$, the following version
holds:
###### Corollary 4.13.
Let $(X,x_{0})$ be a pointed metric space, $Y,Z\subset X$ two subsets, and
$u(r)\preceq_{\infty}\varepsilon r$ for some $\varepsilon<\frac{1}{2}$. Assume
that $b_{Z}^{u}(r)=b_{Z}(r)$. If $Z\subset\mathcal{N}_{u}(Y)$, then
$\delta(Y)\geq(1-4\varepsilon)\cdot\delta(Z)$. Moreover, if $u$ is sublinear,
then $\delta(Z)=\delta(Y)$.
In particular, Corollary 4.11 holds even when the group $G$ does not have
property $(T)$. Corollary 4.11 follows from Corollary 4.13 because the fact
that $\Gamma$ is a group of isometries implies
$b_{\Gamma}^{u}(r)=b_{\Gamma}(r)$.
###### Proof of Corollary 4.13.
Consider the closest point projection $p_{Y}:Z\rightarrow Y$, denote
$y_{z}:=p_{Y}(z)$ and observe:
1. 1.
$|y_{z}|\leq|z|+u(|z|)$.
2. 2.
$d\big{(}y_{z},y_{z^{\prime}}\big{)}\geq
d(z,z^{\prime})-2u(\max\\{|z|,|z^{\prime}|\\})$.
The first item follows from the fact
$y_{z}\in\overline{B}\big{(}z,u(|z|)\big{)}$ and triangle inequality. The
second item follows from the quadrilateral inequality, i.e., using triangle
inequality twice along the quadrilateral
$[z,z^{\prime},y_{z^{\prime}},y_{z}]$.
The above properties allow me to use Proposition 4.10 with constant $L=1$ and
function $u^{\prime}=2u$ to get
$b_{Y}(r)\geq
b_{Z}\big{(}r-u^{\prime}(r)\big{)}/b_{Z}^{u}\Big{(}u^{\prime}\big{(}r-u^{\prime}(r)\big{)}\Big{)}$
Since I assume $b_{Z}^{u}=b_{Z}$, I can omit the superscript $u$ in the last
expression. Recalling the definition
$\delta(W)=\limsup_{r\rightarrow\infty}\frac{b_{W}(r)}{r}$, it remains to
prove:
$\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}/b_{Z}\Big{(}u^{\prime}\big{(}r-u^{\prime}(r)\big{)}\Big{)}\bigg{)}\geq(1-4\varepsilon)\cdot\delta(Z)$
The proof of this inequality involves nothing more than $\log$ rules and
arithmetic of limits:
$\begin{split}&\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}/b_{Z}\Big{(}u^{\prime}\big{(}r-u^{\prime}(r)\big{)}\Big{)}\bigg{)}\\\
&=\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\Bigg{(}\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\bigg{)}-\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}r-u^{\prime}(r)\big{)}\Big{)}\bigg{)}\Bigg{)}\\\
&\geq\limsup_{r\rightarrow\infty}\Bigg{[}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\bigg{)}-\limsup_{s\rightarrow\infty}\bigg{[}\frac{1}{s}\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}s-u^{\prime}(s)\big{)}\Big{)}\bigg{)}\bigg{]}\Bigg{]}\\\
&=\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\bigg{)}-\limsup_{s\rightarrow\infty}\frac{1}{s}\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}s-u^{\prime}(s)\big{)}\Big{)}\bigg{)}\\\
&\geq\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-2\varepsilon
r\big{)}\bigg{)}-\limsup_{s\rightarrow\infty}\frac{1}{s}\log\bigg{(}b_{Z}\Big{(}2\varepsilon
s\Big{)}\bigg{)}\\\
&=(1-2\varepsilon)\delta(Z)-2\varepsilon\delta(Z)=(1-4\varepsilon)\delta(Z)\end{split}$
(3)
Below I justify the steps in the above inequalities:
1. 1.
First equality is by rules of $\log$.
2. 2.
Second and third inequalities are by arithmetic of limits: let
$(a_{n})_{n},(b_{n})_{n}$ be two sequences of positive numbers, and
$A=\limsup_{n}a_{n},\ B=\limsup_{n}b_{n}$. Then
$\limsup(a_{n}-b_{n})\geq\limsup_{n}(a_{n}-B)=A-B$.
3. 3.
Fourth inequality: $u^{\prime}(r)<2\varepsilon(r)$ for all large enough $r$.
4. 4.
Fifth equality: definition of $\delta$.
This completes the proof in the general case, which is what is needed for the
proof of Theorem 4.3. For the more refined statement in the case $u$ is
sublinear, one has to show a bit more. From inequality 3 (specifically from
the fourth line of the inequality) it is clearly enough to prove:
1. 1.
$\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\bigg{)}=\delta(Z)$.
2. 2.
$\limsup_{s\rightarrow\infty}\frac{1}{s}\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}s-u^{\prime}(s)\big{)}\Big{)}\bigg{)}=0$.
Starting from the second item, indeed it holds that
$\frac{1}{s}\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}s-u^{\prime}(s)\big{)}\Big{)}\bigg{)}=\frac{u^{\prime}\big{(}s-u^{\prime}(s)\big{)}}{s}\cdot\frac{\log\bigg{(}b_{Z}\Big{(}u^{\prime}\big{(}s-u^{\prime}(s)\big{)}\Big{)}\bigg{)}}{u^{\prime}\big{(}s-u^{\prime}(s)\big{)}}$
Clearly $\limsup$ of the right factor in the above product is bounded by
$\delta(Z)$, and in particular it is uniformly bounded. On the other hand
sublinearity of $u^{\prime}$ implies that the left factor tends to $0$. I
conclude that this product tends to $0$ as $s$ tends to $\infty$.
It remains to prove
$\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\bigg{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\bigg{)}=\delta(Z)$.
In a similar fashion,
$\frac{1}{r}\log\Big{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\Big{)}=\frac{r-u^{\prime}(r)}{r}\cdot\frac{\log\Big{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\Big{)}}{r-u^{\prime}(r)}$
The left factor limits to $1$ by sublinearity of $u^{\prime}$. The right
factor is nearly the expression in the definition of $\delta(Z)$, and I want
to prove that indeed taking $\limsup$ of it equals $\delta(Z)$. _A priori_
$\\{r-u^{\prime}(r)\\}_{r\in\mathbb{R}_{>0}}$ is just a subset of
$\mathbb{R}_{>0}$, so changing variable and writing $t:=r-u^{\prime}(r)$
requires a justification. But there is no harm in assuming that $u^{\prime}$
is a non-decreasing continuous function, hence $\mathbb{R}_{\geq
R}\subset\\{r-u^{\prime}(r)\\}_{r\in\mathbb{R}_{>0}}$ for some
$R\in\mathbb{R}_{>0}$. Therefore for any sequence $r_{n}\rightarrow\infty$
there is a sequence $r_{n}^{\prime}$ with
$r_{n}=r_{n}^{\prime}-u^{\prime}(r_{n}^{\prime})$ for all large enough $n$
(note that in particular $r_{n}^{\prime}\rightarrow\infty$). In the other
direction, for every sequence $r_{n}^{\prime}\rightarrow\infty$ there is
clearly a sequence $r_{n}\rightarrow\infty$ for which
$r_{n}=r_{n}^{\prime}-u^{\prime}(r_{n}^{\prime})$. I conclude
$\limsup_{r\rightarrow\infty}\frac{\log\Big{(}b_{Z}\big{(}r-u^{\prime}(r)\big{)}\Big{)}}{r-u^{\prime}(r)}=\limsup_{r\rightarrow\infty}\frac{1}{r}\cdot\log\big{(}b_{Z}(r)\big{)}=\delta(Z)$
This completes the proof.
∎
###### Proof of Theorem 4.3.
Define $\varepsilon(G)=\frac{c^{*}(G)}{4\cdot 2\|\rho\|}$, and assume
$u(r)\preceq_{\infty}\varepsilon(G)\cdot r$. Notice that
$\varepsilon(G)<\frac{1}{2}$, and since $\delta(\Gamma)=2\|\rho\|$ Corollary
4.11 gives
$\delta(\Lambda)\geq\big{(}1-4\varepsilon(G)\big{)}\cdot
2\|\rho\|=2\|\rho\|-4\varepsilon(G)\cdot 2\|\rho\|\geq 2\|\rho\|-c^{*}(G)$
By Theorem 4.6, $\Lambda$ is a lattice. ∎
###### Remark 4.14.
The question of existence of interesting groups that coarsely cover a lattice
is a key question that arises naturally from this paper. The first question
that comes to mind is whether there exist groups that are not commensurable to
a lattice but that sublinearly, or even $\varepsilon$-linearly, cover one.
Perhaps the growth rate point of view could be used to rule out groups that
cover a lattice $\varepsilon$-linearly but not sublinearly.
## 5 Uniform Lattices
In this section I prove:
###### Theorem 5.1.
Let $G$ be a finite-centre semisimple Lie group without compact factors. Let
$\Gamma\leq G$ be a lattice, $\Lambda\leq G$ a discrete subgroup such that
$\Gamma\subset\mathcal{N}_{u}(\Lambda)$ for some sublinear function $u$. If
$\Gamma$ is uniform, then $\Lambda$ is a uniform lattice.
As in the case of lattices with property (T), uniform lattices admit the
stronger version of $\varepsilon$-linear rigidity, for any $\varepsilon<1$:
###### Theorem 5.2.
In the setting of Theorem 5.1, the conclusion holds also under the relaxed
assumption that $u(r)\preceq_{\infty}\varepsilon r$ for any $0<\varepsilon<1$.
Clearly Theorem 5.2 implies Theorem 5.1. Throughout this section, the standing
assumptions are those of Theorem 5.2.
##### Lattice Criterion.
A discrete subgroup is a uniform lattice if and only if it admits a relatively
compact fundamental domain. The criterion I use is the immediate consequence
that if $\Gamma$ is uniform and $u$ is bounded (i.e.
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$), then $\Lambda$ is a
uniform lattice.
##### Line of Proof and Use of $\varepsilon$-Linearity.
The goal is to show that the $\varepsilon$-linearity of $u$ forces
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$, i.e. that $\Gamma$
actually lies inside a bounded neighbourhood of $\Lambda$. The proof is by way
of contradiction. If there is no such $D>0$ then there are arbitrarily large
balls that do not intersect $\Lambda$. The proof goes by finding such large
$\Lambda$-free balls that are all tangent to some fixed arbitrary point $x\in
X$ (see Figure 1). The $\varepsilon$-linearity then gives rise to concentric
$\Gamma$-free balls that are arbitrarily large, contradicting the fact that
$\Gamma$ is a uniform lattice.
###### Remark 5.3.
The main difference from the non-uniform case is that for a non-uniform
lattice $\Gamma$, the space $X$ does admit arbitrarily large $\Gamma$-free
balls. This situation requires different lattice criteria and much extra work.
Still the proof for the uniform case, though essentially no more than a few
lines, lies the foundations for and presents the logic of the much more
involved case of $\mathbb{Q}$-rank $1$ lattices.
### 5.1 Notations and Terminology
The following definitions will be used repeatedly in both this section and in
Section 6. It mainly fixes terminology and notation of the geometric situation
illustrated in Figure 1.
###### Definition 5.4.
Let $H\leq G=\mathrm{Isom}(X)^{\circ}$. A set $U\subset X$ is called _$H$
-free_ if $H\cdot x_{0}\cap\mathrm{Int}(U)=\emptyset$, where $\mathrm{Int}(U)$
is the topological interior of $U$. That is, $U$ is called $H$-free if its
interior does not intersect the $H$-orbit $H\cdot x_{0}$.
###### Definition 5.5.
Denote $P_{\Lambda}(\gamma x_{0})=P_{\Lambda}(\gamma)=\lambda_{\gamma}x_{0}$.
1. 1.
$d_{\gamma}:=d(\gamma x_{0},\lambda_{\gamma}x_{0})$.
2. 2.
$B_{\gamma}:=B(\gamma x_{0},d_{\gamma})$. It is a $\Lambda$-free ball centred
at $\gamma x_{0}$ and tangent to $\lambda_{\gamma}x_{0}$.
3. 3.
$x_{\gamma}^{\prime}:=\lambda_{\gamma}^{-1}\gamma x_{0}$. Notice
$|x_{\gamma}^{\prime}|=d_{\gamma}$.
4. 4.
$B_{\gamma}^{\prime}:=\lambda_{\gamma}^{-1}B_{\gamma}=B(x_{\gamma}^{\prime},d_{\gamma})$.
It is $\Lambda$-free as a $\Lambda$-translate of the $\Lambda$-free ball
$B_{\gamma}$, and is tangent to $x_{0}$.
5. 5.
For $s\in\mathbb{R}_{>0}$ and a ball $B=B(x,r)$, denote $sB:=B(x,sr)$, the
rescaled ball with same centre and radius $sr$.
6. 6.
For a sequence $\gamma_{n}$, denote by
$\lambda_{n},d_{n},B_{n},B_{n}^{\prime},x_{n}^{\prime}$ the respective
$\lambda_{\gamma_{n}},d_{\gamma_{n}}$, etc.
$x_{0}$$x_{\gamma}^{\prime}$$=d_{\gamma}$$=s\cdot d_{\gamma}$$\Gamma-
free$$\Lambda-free$$\gamma x_{0}$$=d_{\gamma}$$\Lambda-
free$$\lambda_{\gamma}x_{0}$$L_{\lambda_{\gamma}^{-1}}$ Figure 1: Basic
Setting and Lemma 5.6. A $\Lambda$-free ball about $\gamma x_{0}$ of radius
$d_{\gamma}$, translated by $\lambda_{\gamma}^{-1}$ to a ball tangent to
$x_{0}$. The linear ratio between $|x_{\gamma}^{\prime}|=d_{\gamma}$ and the
$\Lambda$-free radius forces the red ball to be $\Gamma$-free.
### 5.2 Proof of Theorem 5.2
###### Lemma 5.6.
Let $x\in X$. There exists $S=S(x,u)\in(0,1)$ such that for every $s\in(0,S)$
there is $R=R(s,S)$ such that if $r>R$ and $B=B(y,r)$ is a $\Lambda$-free ball
tangent to $x$, then $sB$ is $\Gamma$-free.
In particular, the existence of arbitrarily large $\Lambda$-free balls that
are all tangent to a fixed point $x\in X$ implies the existence of arbitrarily
large $\Gamma$-free balls.
There is a slightly stronger version of Lemma 5.6 if $u$ is sublinear:
###### Lemma 5.7.
Let $G,\Gamma,\Lambda$ and $u$ be as in Theorem 5.1 (in particular, $u$ is a
sublinear function and $\Gamma\subset\mathcal{N}_{u}(\Gamma)$). For every
$x\in X$ and every $s\in(0,1)$ there exists $R=R(x,s)>0$ such that for every
$r>R$, if $B=B(y,r)$ is a $\Lambda$-free ball tangent to $x$ then $sB$ is
$\Gamma$-free.
I omit the proof of Lemma 5.7, which is a slightly simpler version of the
proof of Lemma 5.6.
###### Proof of Lemma 5.6..
The proof is more easily read if one assumes $x=x_{0}$ and $u(r)=\varepsilon
r$ so I begin with this case. Assume $B=B(y,R)$ is $\Lambda$-free for some
$y\in X,r>0$. The assumption $x=x_{0}$ gives $|y|=d(y,x_{0})=r$. For a fixed
$s\in(0,1)$, assume $sB\cap\Gamma\cdot x_{0}\neq\emptyset$. This gives rise to
an element $\gamma\in\Gamma$ such that:
1. 1.
$|\gamma|=d(\gamma x_{0},x_{0})\leq d(y,x_{0})+d(\gamma x_{0},y)=(1+s)r$. In
particular, $d(\gamma x_{0},\Lambda\cdot x_{0})\leq\varepsilon(1+s)r$.
2. 2.
$B\big{(}\gamma x_{0},(1-s)r\big{)}\subset B(y,r)$, so it is $\Lambda$-free.
I conclude that for such $s$ one has the inequality
$(1-s)r\leq\varepsilon(1+s)r$, i.e. $\frac{1-s}{1+s}\leq\varepsilon$. The
constant $\varepsilon$ is fixed and smaller than $1$, while $\frac{1-s}{1+s}$
limit to $1$ monotonically from below as $s>0$ tend to $0$. I conclude that
there is a segment $(0,S)\subset(0,1)$ such that for all $s\in(0,S)$, $sB$ is
$\Gamma$-free (independently of $r$).
Assume now that $x\neq x_{0}$ and $u(r)\preceq_{\infty}\varepsilon r$. As
above, if $\gamma x_{0}\in B(y,sr)$ then it is the centre of a $\Lambda$-free
ball of radius $(1-s)r$, and so $(1-s)r\leq u(|\gamma|)$. I wish to use the
$\varepsilon$-linear bound on $u$ as I did before, only this time $u$ is only
asymptotically smaller than $\varepsilon r$. To circumvent this I just need to
show that $|\gamma|$ is large enough. Indeed since $B\big{(}\gamma
x_{0},(1-s)r\big{)}$ is $\Lambda$-free it does not contain
$x_{0}\in\Lambda\cdot x_{0}$ and in particular $(1-s)r\leq d(x_{0},\gamma
x_{0})=|\gamma|$. For some $R_{1}(s)=R_{1}(s,u)$ one therefore has for all
$r>R_{1}(s)$
$(1-s)r\leq u(|\gamma|)\leq\varepsilon|\gamma|$
On the other hand $|y|\leq d(x,y)+d(x,x_{0})=r+|x|$. Consequently
$|\gamma|\leq(1+s)r+|x|$ and, for $r>R_{1}(s)$,
$(1-s)r\leq
u(|\gamma|)\leq\varepsilon|\gamma|\leq\varepsilon\big{(}(1+s)r+|x|\big{)}$
This means that $s$ for which $\Gamma\cdot x_{0}\cap B(y,sr)\neq\emptyset$
must admit, for all $r>R_{1}(s)$,
$\frac{1-s}{1+s+\frac{|x|}{r}}=\frac{(1-s)r}{(1+s)r+|x|}\leq\varepsilon<1$ (4)
The rest of the proof is just Calculus 1, and concerns with finding
$S=S(x,\varepsilon,u)\in(0,1)$ so that for any $s\in(0,S)$ there is $R(s)$
such that all $r>R(s)$ satisfy
$\varepsilon<\frac{1-S}{1+S+\frac{|x|}{r}}\leq\frac{1-s}{1+s+\frac{|x|}{r}}$
(5)
The lemma readily follows from inequalities 4,5.
Explicitly, fix $\varepsilon^{\prime}>\varepsilon$. As before, monotonic
approach of $\frac{1-s}{1+s}$ to $1$ allows to fix $S\in(0,1)$ for which
$\varepsilon<\varepsilon^{\prime}<\frac{1-s}{1+s}$ for all $s\in(0,S)$. Next
note that for any fixed $s\in(0,S)$,
$\lim_{r\rightarrow\infty}\frac{1-s}{1+s+\frac{|x|}{r}}=\frac{1-s}{1+s}$, and
that the approach in monotonically increasing with $r$. Since
$\varepsilon<\varepsilon^{\prime}$, this limit implies that for some
$R_{2}>R_{1}(S)$, all $r>R_{2}$ admit
$\varepsilon<\frac{1-S}{1+S+\frac{|x|}{r}}$. Finally notice that for any fixed
$r$ the function $\frac{1-s}{1+s+\frac{|x|}{r}}$ is again monotonically
increasing as $s$ tends to $0$ from above. Therefore inequality 5 holds for
every $s\in(0,S)$ and all $r>R_{2}(S)$ (capital $S$ is intentional and
important).
To conclude the proof, notice that if moreover $r>R_{1}(s)$ (again lowercase
$s$ is intentional and important) then inequalities 4,5 both hold. This means
that for any $s\in(0,S)$ there is $R(s):=\max\\{R_{1}(s),R_{2}(S)\\}$ such
that $r>R(s)\Rightarrow B(y,sr)$ is $\Gamma$-free. The constants
$R_{1}(s),R_{2}(S)$ have the desired dependencies, hence so does $R(s)$,
proving the lemma. ∎
###### Corollary 5.8 (Theorem 5.2).
There is a uniform bound on $\\{d_{\gamma}\\}_{\gamma\in\Gamma}$, i.e.,
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$. In particular,
$\Lambda$ is a uniform lattice.
## 6 $\mathbb{Q}$-rank 1 Lattices
In this section I prove:
###### Theorem 6.1.
Let $G$ be a real finite-centre semisimple Lie group without compact factors,
$\Gamma\leq G$ an irreducible non-uniform $\mathbb{Q}$-rank $1$ lattice,
$\Lambda\leq G$ a discrete irreducible subgroup. If
$\Gamma\subset\mathcal{N}_{u}(\Lambda)$ for some sublinear function $u$, then
$\Lambda$ is a lattice. Moreover, if
$\Gamma\not\subset\mathcal{N}_{D}(\Lambda)$ for any $D>0$, then $\Lambda$ is
also of $\mathbb{Q}$-rank $1$.
If $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$, then $\Lambda$
could be a uniform lattice. An obvious obstacle for that is if
$\Lambda\subset\mathcal{N}_{u^{\prime}}(\Gamma)$ for some sublinear function
$u^{\prime}$. This condition turns out to be sufficient for commensurability.
###### Proposition 6.2 (Proposition 6.27).
Let $G$ be a real finite-centre semisimple Lie group without compact factors,
$\Gamma\leq G$ an irreducible $\mathbb{Q}$-rank $1$ lattice, $\Lambda\leq G$ a
discrete subgroup such that $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some
$D>0$, and $\Lambda\subset\mathcal{N}_{u}(\Gamma)$ for some sublinear function
$u$. Then $\Lambda\subset\mathcal{N}_{D^{\prime}}(\Gamma)$ for some
$D^{\prime}$.
From Eskin’s and Schwartz’s results on groups at finite Hausdorff distance
(see Section 6.3), I conclude:
###### Corollary 6.3.
In the setting of Proposition 6.2 and unless $G$ is locally isomorphic to
$\mathrm{SL}_{2}(\mathbb{R})$, $\Lambda$ is commensurable to $\Gamma$.
###### Proof of Theorem 1.8.
The theorem is an immediate result of Theorem 6.1 and Corollary 6.3 ∎
### 6.1 Strategy
##### Lattice Criteria.
I use three different lattice criteria, depending on the $\mathbb{R}$-rank of
$G$ and on whether or not $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some
$D>0$. My proof is motivated by a conjecture of Margulis, that can be viewed
as an algebraic converse to the geometric structure of the compact core
described in Theorem 2.21. Over the past $30$ years this conjecture was
resolved in many major cases by Oh, Benoist and Miquel, and was recently
proved in full generality by Benoist-Miquel (see Section $1.2$ in [5] for an
overview of the milestones in solving this conjecture).
###### Theorem 6.4 (Theorem $2.16$ in [5]).
Let $G$ be a semisimple real algebraic Lie group of real rank at least $2$ and
$U$ be a non-trivial horospherical subgroup of $G$. Let $\Delta$ be a discrete
Zariski dense subgroup of $G$ that contains an indecomposable lattice
$\Delta_{U}$ of $U$. Then $\Delta$ is a non-uniform irreducible arithmetic
lattice of $G$.
See Definition 6.44 for the precise meaning of an _indecomposable_
horospherical lattice.
For $\mathbb{R}$-rank$\ 1$ groups, one has the following theorem by Kapovich
and Liu, stating that a group is geometrically finite so long as ‘most’ of its
limit points are conical. Recall $\mathcal{L}(\Delta)$ is the limit set of
$\Delta\leq\mathrm{Isom}(X)$, and $\mathcal{L}_{\mathrm{con}}(\Delta)$ is the
set of its conical limit points.
###### Theorem 6.5 (Theorem $1.5$ in [29]).
Let $X$ be a $\mathbb{R}$-rank$\ 1$ symmetric space. A discrete subgroup
$\Delta\leq\mathrm{Isom}(X)$ is geometrically infinite if and only if the set
$\mathcal{L}(\Delta)\setminus\mathcal{L}_{\mathrm{con}}(\Delta)$ of non-
conical limit points has the cardinality of the continuum.
As a direct corollary I obtain the following criterion:
###### Corollary 6.6.
Let $X$ be a $\mathbb{R}$-rank$\ 1$ symmetric space, $\Gamma\leq
G=\mathrm{Isom}(X)$ a non-uniform lattice and $\Lambda\leq G$ a discrete
subgroup. If $\mathcal{L}(\Lambda)=X(\infty)$ and
$\mathcal{L}_{\mathrm{con}}(\Gamma)\subset\mathcal{L}_{\mathrm{con}}(\Lambda)$,
then $\Lambda$ is a lattice.
###### Proof.
Since $\Gamma$ is a lattice, $\mathcal{L}(\Gamma)=X(\infty)$ and it is
geometrically finite. Theorem 6.5 implies the cardinality of
$X(\infty)\setminus\mathcal{L}_{\mathrm{con}}(\Gamma)$ is strictly smaller
than the continuum. The assumption
$\mathcal{L}_{\mathrm{con}}(\Gamma)\subset\mathcal{L}_{\mathrm{con}}(\Lambda)$
implies the same holds for $\Lambda$, and in particular that $\Lambda$ is
geometrically finite. The assumption that $\mathcal{L}(\Lambda)=X(\infty)$
implies that $\Lambda$ is geometrically finite if and only if it is a lattice.
∎
###### Corollary 6.7.
Let $X$ be a $\mathbb{R}$-rank$\ 1$ symmetric space, $\Gamma\leq
G=\mathrm{Isom}(X)$ a non-uniform lattice and $\Lambda\leq G$ a discrete
subgroup. If $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$, then
$\Lambda$ is a lattice.
###### Proof.
By definition of the limit set and of conical limit points, it is clear that
every $\Gamma$-limit point is a $\Lambda$-limit point, and every
$\Gamma$-conical limit point is also $\Lambda$-conical limit point. I conclude
from Corollary 6.6 that $\Lambda$ is a lattice. ∎
Also in higher rank the inclusion $\Gamma\subset\mathcal{N}_{D}(\Lambda)$
implies that $\Lambda$ is a lattice. This result is due to Eskin.
###### Theorem 6.8 (Eskin, see Remark 6.10 below).
Let $G$ be a real finite-centre semisimple Lie group without compact factors
and of higher rank, $\Gamma\leq G$ an irreducible non-uniform lattice,
$\Lambda\leq G$ a discrete subgroup such that
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$. Then $\Lambda$ is a
lattice. If moreover $\Lambda\subset\mathcal{N}_{D}(\Gamma)$ then $\Lambda$
and $\Gamma$ are commensurable.
Theorem 6.8 was used in the proof of quasi-isometric rigidity for higher rank
non-uniform lattices in [15] and [21]. In the (earlier) $\mathbb{R}$-rank$\ 1$
case, Schwartz [53] used an analogous statement that requires one extra
assumption.
###### Theorem 6.9 (Schwartz, see Section $10.4$ in [53] and Remark 6.10
below).
Let $G$ be a real finite-centre simple Lie group of $\mathbb{R}$-rank$\ 1$,
$\Gamma\leq G$ an irreducible non-uniform lattice, $\Lambda\leq G$ a discrete
subgroup such that both $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ and
$\Lambda\subset\mathcal{N}_{D}(\Gamma)$ for some $D>0$. Then $\Lambda$ is a
lattice. If moreover $G$ is not locally isomorphic to
$\mathrm{SL}_{2}(\mathbb{R})$, then $\Lambda$ and $\Gamma$ are commensurable.
###### Remark 6.10.
Theorem 1.6 should be viewed as a generalization of the bounded case depicted
in Theorems 6.8 and 6.9, which were known to experts in the field in the late
1990’s. Complete proofs for these statements were never given in print, and I
take the opportunity to include them here. See Section 6.3, where I also prove
Proposition 6.2. I thank Rich Schwartz and Alex Eskin for supplying me with
their arguments and allowing me to include them in this paper. I also thank my
thesis examiner Emmanuel Breuillard for encouraging me to find and make these
proofs public.
##### Line of Proof and Use of Sublinearity.
Lattices of $\mathbb{Q}$-rank $1$ admit a concrete geometric structure (see
Section 2.2). This structure is manifested in the geometry of an orbit of such
a lattice in the corresponding symmetric space $X=G/K$. One important
geometric property is the existence of a set of horoballs which the orbit of
the lattice intersects only in the bounding horospheres, and in each such
horosphere the orbit forms a (metric) cocompact lattice.
Corollary 6.7 and Theorem 6.8 reduce the proof to the case where
$\Gamma\not\subset\mathcal{N}_{D}(\Lambda)$ for any $D>0$. In that case, the
essence lies in proving the existence of horospheres in $X$ which a
$\Lambda$-orbit intersects in a cocompact lattice. This is proved purely
geometrically, using the geometric structure of $\mathbb{Q}$-rank $1$ lattices
and the sublinear constraint. Together with some control on the location of
these horospheres, I prove two strong properties:
1. 1.
$\Lambda\cdot x_{0}$ intersects a horosphere $\mathcal{H}\subset X$ in a
cocompact lattice (Proposition 6.11).
2. 2.
Every $\Gamma$-conical limit point is also a $\Lambda$-conical limit point
(Corollary 6.26).
The $\mathbb{R}$-rank$\ 1$ case of Theorem 6.1 follows directly from Corollary
6.6 using the second item above. The higher rank case requires a bit more,
namely to deduce from the above items that $\Lambda$ meets the hypotheses of
the Benoist-Miquel Theorem 6.4. To that end I use a well known geometric
criterion (Lemma 6.47) in order show that $\Lambda$ is Zariski dense, and a
lemma of Mostow (Lemma 6.32) to show that $\Lambda$ intersects a horospherical
subgroup in a cocompact lattice.
##### Outline for Section 6.
Section 6.2 is the core of the original mathematics of this paper. It is
devoted to proving that $\Lambda\cdot x_{0}$ intersects some horospheres in a
cocompact lattice. The proof is quite delicate and somewhat involved, and I
include a few figures and a detailed informal overview of the proof. The
figures are detailed and may take a few moments to comprehend, but I believe
they are worth the effort.
Section 6.3 deals with the case where $\Gamma\subset\mathcal{N}_{D}(\Lambda)$,
and elaborates on Schwartz’s and Eskin’s proofs of Theorem 6.8 and Theorem
6.9. Section 6.4 is devoted to the translation of the geometric results of
Section 6.2 to the algebraic language used in Theorem 6.4. Though the work is
indeed mainly one of translation, some of it is non-trivial. Finally, in
Section 6.5 I put everything together for a complete proof of Theorem 6.1.
I highly recommend the reader to have a look at the uniform case in Section 5
before reading this one.
### 6.2 A $\Lambda$-Cocompact Horosphere
Recall that $d_{\gamma}:=d(\gamma x_{0},\lambda_{\gamma}x_{0})$. In this
section I prove:
###### Proposition 6.11 (Proposition 1.9).
If $\\{d_{\gamma}\\}_{\gamma\in\Gamma}$ is unbounded, then there exists a
horosphere $\mathcal{H}$ based at $\mathcal{W}_{\mathbb{Q}}(\Gamma)$ such that
$\big{(}\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H})\big{)}\cdot x_{0}$
intersects $\mathcal{H}$ in a cocompact metric lattice. Moreover, the bounded
horoball $\mathcal{HB}$ is $\Lambda$-free.
Throughout Section 6.2 the standing assumptions are that
$\\{d_{\gamma}\\}_{\gamma}\in\Gamma$ is unbounded, and $\Gamma$ is an
irreducible $\mathbb{Q}$-rank $1$ lattice.
##### The Argument.
The proof is by chasing down the geometric implications of unbounded
$d_{\gamma}$. These implications are delicate, but similar in spirit to the
straight-forward proof for uniform lattices. The proof consists of the
following steps:
1. 1.
Unbounded $d_{\gamma}$ results in $\Lambda$-free horoballs
$\mathcal{HB}^{\Lambda}$ tangent to $\Lambda$-orbit points. Each such horoball
is based at $\mathcal{W}_{\mathbb{Q}}(\Gamma)$, giving rise to corresponding
horoballs of $\Gamma$, denoted $\mathcal{HB}^{\Gamma}$.
2. 2.
If $d_{\gamma}$ is large, then $\gamma x_{0}$ must lie deep inside a unique
$\Lambda$-free horoball tangent to $\lambda_{\gamma}x_{0}$. I use:
1. (a)
A bound on the distance $d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})$.
2. (b)
A bound on the angle
$\angle_{\lambda_{\gamma}x_{0}}([\lambda_{\gamma}x_{0},\gamma
x_{0}],[\lambda_{\gamma}x_{0},\xi))$, where $\xi\in X(\infty)$ is the base
point of a suitable $\Lambda$-free horoball tangent to
$\lambda_{\gamma}x_{0}$.
3. 3.
There exist horospheres of $\Gamma$, say $\mathcal{H}^{\Gamma}$, such that if
$\gamma x_{0}\in\mathcal{H}^{\Gamma}$ then large $d_{\gamma}$ implies large
$\Lambda$-free areas along the bounding horosphere of some
$\mathcal{HB}^{\Lambda}$.
4. 4.
If $\mathcal{HB}^{\Lambda}$ is (uniformly) boundedly close to some
$\Lambda$-orbit point, then I show that $\mathcal{H}^{\Lambda}$ is _almost
$\Lambda$-cocompact_, that is
$\mathcal{H}^{\Lambda}\subset\mathcal{N}_{D}(\Lambda\cdot x_{0})$ for some
universal $D=D(\Lambda)$. Together with the previous step, this yields a
uniform bound on $d_{\gamma}$ along certain horospheres of $\Gamma$.
5. 5.
Finally I elevate the almost cocompactness to actual cocompactness and show
$\mathcal{H}^{\Lambda}\subset\mathcal{N}_{D}(\Lambda\cdot
x_{0}\cap\mathcal{H}^{\Lambda})$ for some $D>0$. This immediately elevates to
$\mathcal{H}^{\Lambda}\subset(\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H}^{\Lambda}))\cdot
x_{0}$, proving the proposition.
##### The Properties of $\Gamma$.
The geometric properties of $\Gamma$ that are used in the proof are:
1. 1.
In higher rank, the characterization of $\mathcal{W}_{\mathbb{Q}}(\Gamma)$
using conical / non-horospherical limit points (Corollary 2.31). In
$\mathbb{R}$-rank$\ 1$, the dichotomy of limit points being either non-
horospherical or conical (Theorem 2.30).
2. 2.
$\Gamma$-cocompactness along the horospheres of $\Gamma$.
3. 3.
For every point $x\in X$ and $C>0$ there is a bound $K(C)$ on the number of
horospheres of $\Gamma$ that intersect $B(x,C)$ (Corollary 2.23).
#### 6.2.1 $\Lambda$-Free Horoballs
I retain the notations and objects defined in Section 5.1.
###### Lemma 6.12.
There exists a $\Lambda$-free horoball tangent to $x_{0}$.
###### Proof.
Since $\\{d_{\gamma}\\}_{\gamma\in\Gamma}$ is unbounded, there are
$\gamma_{n}\in\Gamma$ with
$d_{n}=d_{\gamma_{n}}=d(\gamma_{n},\lambda_{n})\rightarrow\infty$
monotonically, where $\lambda_{n}\in\Lambda$ is a $\Lambda$-orbit point
closest to $\gamma_{n}$. Denote
$x_{n}^{\prime}=\lambda_{n}^{-1}\gamma_{n}x_{0}$,
$\eta_{n}:=[x_{0},x_{n}^{\prime}]$, and $v_{n}\in S_{x_{0}}X$ the initial
velocity vectors $v_{n}:=\dot{\eta_{n}}(0)$. The tangent space $S_{x_{0}}X$ is
compact, so up to a subsequence, $v_{n}$ converge monotonically in angle to a
direction $v\in S_{x_{0}}X$. Let $\eta$ be the unit speed geodesic ray
emanating from $x_{0}$ with initial velocity $\dot{\eta}(0)=v$. Denote
$\xi:=\eta(\infty)$ the limit point of $\eta$ in $X(\infty)$.
I claim that the horoball $\mathcal{HB}:=\cup_{t>0}B\big{(}\eta(t),t\big{)}$,
based at $\xi$ and tangent to $x_{0}$, is $\Lambda$-free. Let $t>0$ and
consider $\eta(t)$. For every $\varepsilon>0$, there is some angle
$\alpha=\alpha(t,\varepsilon)$ such that any geodesic $\eta^{\prime}$ with
$\angle_{x_{0}}(\eta,\eta^{\prime})<\alpha$ admits
$d\big{(}\eta(t),\eta^{\prime}(t)\big{)}<\varepsilon/2$. The convergence
$v_{n}\rightarrow v$ implies
$d\big{(}\eta(t),\eta_{n}(t)\big{)}<\varepsilon/2$ for all but finitely many
$n\in\mathbb{N}$. In particular,
$B\big{(}\eta(t),t\big{)}\subset\mathcal{N}_{\varepsilon}\Big{(}B\big{(}\eta_{n}(t),t\big{)}\Big{)}$
for all such $n\in\mathbb{N}$.
For a fixed $t\leq d_{n}$, it is clear from the definitions that
$B\big{(}\eta_{n}(t),t\big{)}\subset B_{n}^{\prime}=B(x_{n}^{\prime},d_{n})$.
One has $d_{n}\rightarrow\infty$, and so for a fixed $t>0$ it holds that
$t<d_{n}$ for all but finitely many $n\in\mathbb{N}$. I conclude that for any
fixed $t>0$ there is $n$ large enough such that
$B\big{(}\eta(t),t\big{)}\subset\mathcal{N}_{\varepsilon}\Big{(}B\big{(}\eta_{n}(t),t\big{)}\Big{)}\subset\mathcal{N}_{\varepsilon}B_{n}^{\prime}$
I conclude that for every $\varepsilon>0$,
$\mathcal{HB}\subset\bigcup_{n}\mathcal{N}_{\varepsilon}(B_{n}^{\prime})=\mathcal{N}_{\varepsilon}\big{(}\bigcup_{n}B_{n}^{\prime}\big{)}$.
This implies that any point in the interior of $\mathcal{HB}$ is contained in
the interior of one of the $\Lambda$-free balls $B_{n}^{\prime}$, proving
$\mathcal{HB}$ is $\Lambda$-free.
∎
###### Lemma 6.13.
Suppose $\mathcal{HB}$ is a $\Lambda$-free horoball, based at some point
$\xi\in X(\infty)$. Then $\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
###### Proof.
For any geodesic $\eta$ with limit $\xi$, the size $d(x_{0},\gamma x_{0})$ of
the $\Gamma$-orbit points $\gamma x_{0}$ that lie boundedly close to $\eta$
grows linearly in the distance to any fixed horosphere based at $\xi$, and in
particular to $\mathcal{H}=\partial\mathcal{HB}$. The sublinear constraint
$d(\gamma x_{0},\lambda_{\gamma}x_{0})\leq u(|\gamma|)$ together with the fact
that $\mathcal{HB}$ is $\Lambda$-free imply that the size of such $\gamma$ is
bounded. In $\mathbb{R}$-rank$\ 1$ every limit point is either conical or in
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$, proving the lemma in this case. For higher
rank, the above argument actually shows more: it shows that a point
$\xi^{\prime}\in\mathcal{N}_{\frac{\pi}{2}}(\xi)$ is not conical, because
every geodesic with limit $\xi^{\prime}\in\mathcal{N}_{\frac{\pi}{2}}(\xi)$
entres $\mathcal{HB}$ at a linear rate (Lemma 2.32). Hattori’s
characterization of $\mathcal{W}_{\mathbb{Q}}(\Gamma)$ (Corollary 2.31)
implies $\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
∎
###### Definition 6.14.
Given a $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$, Lemma 6.13 gives
rise to a horoball of $\Gamma$ based at the same point at $X(\infty)$. Call
this the _horoball corresponding to $\mathcal{HB}^{\Lambda}$_, and denote it
by $\mathcal{HB}^{\Gamma}$. The corresponding horosphere is denoted
$\mathcal{H}^{\Gamma}$.
###### Remark 6.15.
In the course of my work I had had a few conversations with Omri Solan
regarding the penetration of geodesics into $\Lambda$-free horoballs. Assuming
$\Lambda\subset\mathcal{N}_{u}(\Gamma)$ implies that $\Lambda$ preserves
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$ (see Lemma 6.29). This is the case in the
motivating setting where $\Lambda$ is an abstract finitely generated group
that is SBE to $\Gamma$, see Claim 3.26 in Chapter 3. In the case of
$SL_{2}(\mathbb{R})$ Omri suggested to use the action of $\Lambda$ on the
Bruhat-Tits tree of $SL_{2}(\mathbb{Q}_{p})$ (for all primes $p$) and the
classification of these elements into elliptic and hyperbolic elements
(separately for each $p$) in order to deduce that $\Lambda$ actually lies in
$SL_{2}(\mathbb{Z})$. We did not pursue that path nor its possible
generalization to the $SL_{n}$ case and general Bruhat-Tits buildings.
#### 6.2.2 A $\Gamma$-orbit point Lying Deep Inside a $\Lambda$-Free Horoball
I established the existence of $\Lambda$-free horoballs. It may seem odd that
the first step in proving $\Lambda\cdot x_{0}$ is ‘almost everywhere’ is
proving the existence of $\Lambda$-free regions. But this fits perfectly well
with the algebraic statement that non-uniform lattices must admit unipotent
elements (see Proposition $5.3.1$ in [38]). The goal of this section is to
obtain some control on the location of the $\Lambda$-free horoballs, in order
to conclude that some $\gamma x_{0}$ lies deep inside
$\mathcal{HB}^{\Lambda}$. This results in yet more $\Lambda$-free regions,
found on the bounding horosphere.
I need one property of sublinear functions. I thank Panos Papazoglou for
noticing a mistake in the original formulation.
###### Lemma 6.16.
Let $u$ be a sublinear function, $f,g:\mathbb{R}_{\geq
0}\longrightarrow\mathbb{R}_{>0}$ two positive monotone functions with
$\lim_{x\rightarrow\infty}f(x)+g(x)=\infty$. If for all large enough $x$ it
holds that $f(x)\leq u\big{(}f(x)+g(x)\big{)}$, then for every $1<s$ and all
large enough $x$ it holds that $f(x)\leq u\big{(}s\cdot g(x)\big{)}$. In
particular $f(x)\leq u^{\prime}\big{(}g(x)\big{)}$ for some sublinear function
$u^{\prime}$.
###### Proof.
Assume as one may that $u$ is non-decreasing. By definition of sublinearity
$\lim_{x\rightarrow\infty}\frac{u\big{(}f(x)+g(x)\big{)}}{f(x)+g(x)}=0$, so by
hypothesis $\lim_{x\rightarrow\infty}\frac{f(x)}{f(x)+g(x)}=0$. This means
that for every $\varepsilon>0$ one has $f(x)\leq\varepsilon\cdot g(x)$ for all
large enough $x$, resulting in
$f(x)\leq u\big{(}f(x)+g(x)\big{)}\leq u\big{(}(1+\varepsilon)\cdot
g(x)\big{)}$
Notice that for a fixed $s>0$, the function $u^{\prime}(x)=u(sx)$ is
sublinear, as
$\lim_{x\rightarrow\infty}\frac{u(sx)}{x}=\lim_{x\rightarrow\infty}s\cdot\frac{u(sx)}{sx}=0$
∎
###### Lemma 6.17.
Let $C>0$. There is $L=L(C)$ such that if $\mathcal{HB}^{\Lambda}$ is any
$\Lambda$-free horoball tangent to a point $x\in B(x_{0},C)$ then
$d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})\leq L$. Moreover, there is a
sublinear function $u^{\prime}$ such that:
$L(C)\leq\begin{cases}u^{\prime}(C)&\text{if
}\mathcal{HB}^{\Gamma}\subset\mathcal{HB}^{\Lambda}\\\ C&\text{if
}\mathcal{HB}^{\Lambda}\subset\mathcal{HB}^{\Gamma}\end{cases}$
###### Proof.
If $\mathcal{HB}^{\Lambda}\subset\mathcal{HB}^{\Gamma}$, then clearly
$d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})\leq C$, simply because
$\mathcal{HB}^{\Gamma}$ is $\Gamma$-free and in particular cannot contain
$x_{0}$. Therefore $\mathcal{H}^{\Gamma}$ must separate
$\mathcal{H}^{\Lambda}$ from $x_{0}$ and in particular
$d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})\leq C$.
Assume that $\mathcal{HB}^{\Gamma}\subset\mathcal{HB}^{\Lambda}$, and denote
$l=d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})$. The horoball
$\mathcal{HB}^{\Gamma}$ is a horoball of $\Gamma$, hence $\Gamma\cdot x_{0}$
is $D_{\Gamma}$-cocompact along $\mathcal{H}^{\Gamma}$ and there is an element
$\gamma\in\Gamma$ with $|\gamma|\leq C+l+D_{\Gamma}$ and $\gamma
x_{0}\in\mathcal{H}^{\Gamma}$. Since $\mathcal{HB}^{\Lambda}$ is
$\Lambda$-free one has
$l\leq d(\gamma x_{0},\lambda_{\gamma}x_{0})\leq u(|\gamma|)\leq
u(C+l+D_{\Gamma})$
$C,D_{\Gamma}$ are fixed, so this inequality can only occur for boundedly
small $l$, say $l<L^{\prime}(C)$ ($D_{\Gamma}$ is a universal constant and may
be ignored). Consult figure 2 for a geometric visualization of this situation.
It remains to show that $L^{\prime}(C)$ is indeed sublinear in $C$. First
define $L(C)$ to be the minimal $L$ that bounds the distance
$d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})$ for all possible
$\mathcal{HB}^{\Lambda}$ tangent to a point $x\in B(x_{0},C)$. This is indeed
a minimum, since by Corollary 2.23 there are only finitely many horoballs of
$\Gamma$ intersecting $B\big{(}x_{0},C+L^{\prime}(C)\big{)}$. For every $C$
there is thus a horoball $\mathcal{HB}^{\Gamma}_{C}$ and an element
$\gamma\in\Gamma$ such that $\gamma x_{0}\in\mathcal{H}^{\Gamma}$,
$d(\mathcal{H}^{\Lambda}_{C},\mathcal{H}^{\Gamma}_{C})=L(C)$ and $|\gamma|\leq
C+L(C)+D_{\Gamma}$. The fact that $\mathcal{HB}^{\Lambda}_{C}$ is
$\Lambda$-free implies
$L(C)=d(\mathcal{H}^{\Lambda}_{C},\mathcal{H}^{\Gamma}_{C})\leq
u(|\gamma|)\leq u\big{(}C+L(C)+D_{\Gamma}\big{)}$
Lemma 6.16 implies that $L(C)\leq u^{\prime}(C)$ for some sublinear function
$u^{\prime}$. ∎
$Rad=C$$x_{0}$$\Lambda-free\ \mathcal{HB}^{\Lambda}$$\xi$$\Gamma-free\
\mathcal{HB}^{\Gamma}$$\leq L(C)$$\gamma
x_{0}$$P_{\mathcal{H}^{\Gamma}}(x_{0})$$\leq L(C)+D_{\Gamma}$ Figure 2: Lemma
6.17. A $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$ intersects a ball of
radius $C$ about $x_{0}$. The associated $\Gamma$-free horoball
$\mathcal{HB}^{\Gamma}$ is boundedly close, essentially due to the uniform
cocompactness of $\Gamma\cdot x_{0}$ along the $\Gamma$ horospheres.
The following is an immediate corollary, apparent already in the above proof.
###### Corollary 6.18.
For every $C>0$ there is a bound $K=K(C)$ and a fixed set
$\xi_{1},\xi_{2},\dots\xi_{K}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)\subset
X(\infty)$ so that every $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$
which is tangent to some point $x\in B(x_{0},C)$ is based at $\xi_{i}$ for
some $i\in\\{1,\dots,K\\}$. In particular, for any specific point
$x\in\mathcal{N}_{C}(\Lambda\cdot x_{0})$ there are at most $K$ $\Lambda$-free
horoballs tangent to $x$.
###### Proof.
Let $\mathcal{HB}^{\Lambda}$ be a horoball tangent to a point $x\in
B(x_{0},C)$. Lemma 6.17 bounds
$d(\mathcal{HB}^{\Lambda},\mathcal{HB}^{\Gamma})$ by $L(C)$, hence
$\mathcal{HB}^{\Gamma}$ is tangent to a point $x^{\prime}\in
B\big{(}x_{0},C+L(C)\big{)}$. By Corollary 2.23 there are only finitely many
possibilities for such $\mathcal{HB}^{\Gamma}$. In particular there are
finitely many base-points for these horoballs, say
$\xi_{1},\xi_{2},\dots,\xi_{K(C)}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
Finally, recall that a horoball is determined by a base point and a point
$x\in X$ tangent to it, so the last statement of the corollary holds for any
$x\in B(x_{0},C)$. But the property in question is $\Lambda$-invariant so the
same holds for any point $x\in\Lambda\cdot
B(x_{0},C)=\mathcal{N}_{C}(\Lambda\cdot x_{0})$. ∎
The bound on $d(\mathcal{HB}^{\Lambda},\mathcal{HB}^{\Gamma})$ given by Lemma
6.17 further strengthen the relation between $\mathcal{HB}^{\Lambda}$ and
$\mathcal{HB}^{\Gamma}$. The ultimate goal is to show that the
$\mathcal{HB}^{\Lambda}$-s play the role of the $\Gamma$-horoballs in the
geometric structure of $\mathbb{Q}$-rank $1$ lattices, namely to show that
$\Lambda\cdot x_{0}$ is cocompact on the $\mathcal{H}^{\Lambda}$-s. This
requires to actually find $\Lambda$-orbit points somewhere in $X$, and not
just $\Lambda$-free regions as was done up to now. As one might suspect, these
points arise as $\lambda_{\gamma}x_{0}$ corresponding to points $\gamma
x_{0}\in\mathcal{H}^{\Gamma}$, which exist in abundance since $\Gamma\cdot
x_{0}\cap\mathcal{H}^{\Gamma}$ is a cocompact lattice in
$\mathcal{H}^{\Gamma}$.
The hope is that a $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$ tangent to
$\lambda_{\gamma}x_{0}$ would correspond to a horoball of $\Gamma$ tangent to
$\gamma x_{0}$. This would have forced all the $\lambda_{\gamma}$ to actually
lie on the same bounding horosphere, and $\\{\lambda_{\gamma}x_{0}\mid\gamma
x_{0}\in\mathcal{H}^{\Gamma}\\}$ would then be a cocompact lattice in
$\mathcal{H}^{\Lambda}$. This hope turns out to be more or less true, but it
requires some work. The goal of the rest of this section is to establish a
relation between a $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$ tangent to
$\lambda_{\gamma}x_{0}$ and $\gamma x_{0}$. I start with some notations.
###### Definition 6.19.
In light of Corollary 6.18, there is a finite number $N$ of $\Lambda$-free
horoballs tangent to $x_{0}$. Denote:
1. 1.
$\\{\mathcal{HB}^{\Lambda}_{i}\\}_{1=1}^{N}$ are the $\Lambda$-free horoballs
tangent to $x_{0}$.
2. 2.
$\xi^{i}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$ is the base point of
$\mathcal{HB}^{\Lambda}_{i}$.
3. 3.
$v^{i}\in S_{x_{0}}X$ is the unit tangent vector in the direction $\xi^{i}$.
4. 4.
$\eta^{i}:=[x_{0},\xi_{i})$ is the unit speed geodesic ray emanating from
$x_{0}$ with limit $\xi^{i}$. In particular $v^{i}=\frac{d}{dt}\eta^{i}(0)$.
5. 5.
$\mathcal{HB}^{\Gamma}_{i}$ is the horoball of $\Gamma$ that corresponds to
$\mathcal{HB}^{\Lambda}_{i}$, based at $\xi^{i}$.
6. 6.
$\mathcal{HB}^{\Lambda}_{\lambda,i},\xi^{i}_{\lambda},\eta^{i}_{\lambda}$ are
the respective $\lambda$-translates of the objects above. For example,
$\mathcal{HB}^{\Lambda}_{\lambda,i}:=\lambda\cdot\mathcal{HB}^{\Lambda}_{i}$.
7. 7.
$\mathcal{H}$ decorated by the proper indices denotes the horosphere bounding
$\mathcal{HB}$, the horoball with respective indices, e.g.
$\mathcal{H}^{\Lambda}_{i}:=\partial\mathcal{HB}^{\Lambda}_{i}$.
8. 8.
For an angle $\alpha>0$ and a tangent vector $v_{0}\in S_{x}X$, define
1. (a)
The _$\alpha$ -sector of $v$ in $S_{x}X$_ is the set $\\{v\in S_{x}X\mid
v\in\mathcal{N}_{\alpha}(v_{0})\\}$. Recall that the metric on $S_{x}X$ is the
angular metric.
2. (b)
The _$\alpha$ -sector of $v$ in $X$_ are all points $y\in X$ for which the
tangent vector at $0$ of the unit speed geodesic $[x,y]$ lies in the
$\alpha$-sector of $v$ in $SxX$.
###### Lemma 6.20.
For every angle $\alpha\in(0,\frac{\pi}{2})$ there exists $D=D(\alpha)$ such
that if $d_{\gamma}>D$ then for some $i\in\\{1,\dots,N\\}$, $\gamma x_{0}$
lies inside the $\alpha$-sector of $v^{i}_{\lambda_{\gamma}}$ at
$\lambda_{\gamma}x_{0}$. Furthermore whenever $\alpha$ is uniformly small
enough, there is a unique such $i=i(\gamma)$, independent of $\alpha$.
###### Proof.
Translation by the isometry $\lambda_{\gamma}^{-1}$ preserves angles and
distances, so it is enough to prove that there is an $i$ for which
$x^{\prime}_{\gamma}:=\lambda^{-1}\gamma x_{0}$ lies inside the
$\alpha$-sector of $v^{i}$, and that this $i$ is unique if $\alpha$ is
uniformly small. Assume towards contradiction that there is
$\alpha\in(0,\frac{\pi}{2})$ and a sequence $\gamma_{n}\in\Gamma$,
$\lambda_{n}:=\lambda_{\gamma_{n}}\in\Lambda$ with $d_{\gamma_{n}}$ unbounded,
and $x^{\prime}_{n}:=\lambda_{n}^{-1}\gamma_{n}x_{0}$ not lying in the union
of the $\alpha$-sectors of $v^{i}$. By perhaps taking smaller $\alpha$ I may
assume all the $\alpha$-sectors of the $v^{i}$ in $S_{x_{0}}X$ are pairwise
disjoint. This can be done because there are only finitely many $v^{i}$.
Compactness of $S_{x_{0}}X$ allows me to take a converging subsequence
$v^{\prime}_{n}:=\dot{[x_{0},x^{\prime}_{n}]}$, with limit direction
$v^{\prime}$. Denote by $\eta^{\prime}$ the geodesic ray emanating from
$x_{0}$ with initial velocity $v^{\prime}$. The exact same argument of Lemma
6.12 proves that $\eta^{\prime}(\infty)$ is the base point of a $\Lambda$-free
horoball tangent to $x_{0}$. But this means $v^{\prime}=v^{i}$ for some
$i\in\\{1,\dots,N\\}$, contradicting the fact that all $v^{\prime}_{n}$ lie
outside the $\alpha$-sectors of the $v^{i}$. This proves that there is a bound
$D=D(\alpha)$ such that if $d_{\gamma}>D$ then $x_{\gamma}^{\prime}$ lies
within the $\alpha$-sector of some $v^{i}$.
The proof clearly shows that whenever $\alpha$ is small enough so that the
$\alpha$-sectors of the $v^{i}$ are disjoint, $x_{\gamma}^{\prime}$ lies in
the $\alpha$-sector of a unique $v^{i}$ as soon as $d_{\gamma}>D(\alpha)$.
∎
###### Remark 6.21.
In the proof of Lemma 6.12 I used compactness of $S_{x_{0}}X$ to induce a
converging subsequence of directions. Lemma 6.20 actually shows that the fact
there are finitely many $\Lambda$-free horoballs tangent to $x_{0}$ implies _a
posteriori_ that there was not much choice in the process - all directions
$[x_{0},x_{\gamma}^{\prime}]$ must fall into one of the finitely many
directions $v^{i}$.
Next, I want to control the actual location of certain points with respect to
the horoballs of interest, and not just the angles. This turns out to be a
more difficult of a task than one might suspect, since control on angles does
not immediately give control on distances.
Recall that large $\Lambda$-free balls near $x_{0}$ imply large concentric
$\Gamma$-free balls. The precise quantities and bounds are given by Lemma 5.6
(one can use Lemma 5.7 to obtain a slightly cleaner statement).
###### Proposition 6.22.
Let $S\in(0,1)$ be the constant given by Lemma 5.6, and let $s\in(0,S)$. There
is a bound $D=D(s)$ such that $d_{\gamma}>D$ implies that $\gamma x_{0}$ lies
$sd_{\gamma}$ deep in $\mathcal{HB}^{\Lambda}_{\lambda_{\gamma}}$.
###### Proof.
The proof is a bit delicate but very similar to that of Lemma 5.6. In essence,
I use the $\Gamma$-free balls near $x_{0}$ to produce a $\Gamma$-free
cylinder, which would force a certain geodesic not to cross a horosphere of
$\Gamma$, i.e. force it to stay inside a $\Gamma$-free horoball.
As in Lemma 6.20 it is only required to show that
$x^{\prime}=\lambda_{\gamma}^{-1}\gamma x_{0}$ is $sd_{\gamma}$ deep inside
$\mathcal{HB}^{\Lambda}_{i(\gamma)}$. I start with proving that
$x^{\prime}\in\mathcal{HB}^{\Gamma}_{i(\gamma)}$. I learned the hard way that
even this is not a triviality. Recall the notation $B_{\gamma}=B(\gamma
x_{0},d_{\gamma})$. The ball
$B^{\prime}_{\gamma}=\lambda_{\gamma}^{-1}B_{\gamma}$ is a $\Lambda$-free ball
of radius $d_{\gamma}$ about $x^{\prime}=\lambda_{\gamma}^{-1}\gamma x_{0}$.
Denote by $x^{\prime}_{t}$ the point at time $t$ along the unit speed geodesic
$\eta^{\prime}:=[x_{0},x^{\prime}]$. It holds that $|x_{t}^{\prime}|=t$ and,
for $t\leq d_{\gamma}$, $x_{t}^{\prime}$ is the centre of a $\Lambda$-free
ball of radius $t$ tangent to $x_{0}$. The constant $s$ is fixed and by Lemma
5.6 there is $T^{\prime}=T^{\prime}(s)$ such that if $t>T^{\prime}$, the ball
$sB\big{(}x^{\prime}_{t},t\big{)}$ is $\Gamma$-free.
The next goal is to show that $x_{T}^{\prime}\in\mathcal{HB}^{\Gamma}$ for
some adequate $T$. For any time $T>0$, let $\alpha=\alpha(\varepsilon,T)$ be
the angle for which $d\big{(}\eta(T),\eta^{i(\gamma)}(T)\big{)}<\varepsilon$
for every $\eta$ in the $\alpha$-sector of $v^{i(\gamma)}$. By perhaps taking
smaller $\alpha$ I may assume that $\alpha$ is uniformly small as stated in
Lemma 6.20. Let $D(\alpha)$ be the bound given by Lemma 6.20 guaranteeing
$D(\alpha)<d_{\gamma}\Rightarrow
d\big{(}x^{\prime}_{T},\eta^{i(\gamma)}(T)\big{)}<\varepsilon$
For my needs in this lemma $\varepsilon$ may as well be chosen to be $1$. I
now choose a specific time $T$ for which I want $x_{T}^{\prime}$ and
$\eta^{i(\gamma)(T)}$ to be close. There are only finitely many $\Lambda$-free
horoballs $\\{\mathcal{HB}^{\Lambda}_{i}\\}_{i\in\\{1,\dots,N\\}}$ tangent to
$x_{0}$, giving rise to a uniform bound
$L=\max_{i\in\\{1,\dots,N\\}}\\{d(\mathcal{H}^{\Lambda}_{i},\mathcal{H}^{\Gamma}_{i})\\}$
on the distance
$d(\mathcal{H}^{\Lambda}_{i(\gamma)},\mathcal{H}^{\Gamma}_{i(\gamma)})$. Fix
$T$ to be any time in the open interval
$(T^{\prime}+L+\varepsilon,d_{\gamma})$. The fact that $L+\varepsilon<T$
implies that $\eta^{i(\gamma)}(T)$ lies at least $\varepsilon$-deep inside
$\mathcal{HB}^{\Gamma}_{i(\gamma)}$, and therefore
$\eta^{\prime}(T)\in\mathcal{HB}^{\Gamma}_{i(\gamma)}$. Recall that any point
on $\mathcal{H}^{\Gamma}$ is $D_{\Gamma}$-close to a point $\gamma
x_{0}\in\mathcal{H}^{\Gamma}$. By perhaps enlarging $T$ and shrinking $\alpha$
if necessary, I may assume that $D_{\Gamma}<sT$. Thus for all $T<t\leq
d_{\gamma}$, $x_{t}^{\prime}$ is the centre of a $\Gamma$-free ball of radius
$st>sT>D_{\Gamma}$, hence $\\{x^{\prime}_{t}\\}_{T\leq t\leq d_{\gamma}}$ does
not cross a horosphere of $\Gamma$. Since
$x_{T}^{\prime}\in\mathcal{HB}^{\Gamma}_{i(\gamma)}$, this implies that
$x_{t}^{\prime}$ stays in $\mathcal{HB}^{\Gamma}_{i(\gamma)}$ for all $T<t\leq
d_{\gamma}$. In particular
$x^{\prime}_{d_{\gamma}}=x^{\prime}\in\mathcal{HB}^{\Gamma}_{i(\gamma)}$.
To get the result of the proposition, recall that
$sB_{\gamma}^{\prime}=B(x^{\prime},sd_{\gamma})$ is $\Gamma$-free, so
$x^{\prime}$ must be at distance at least $sd_{\gamma}-D_{\Gamma}$ from any
horosphere of $\Gamma$, and in particular from
$\mathcal{H}^{\Gamma}_{i(\gamma)}$. In terms of Busemann functions, this means
that $b_{\eta^{i(\gamma)}}(x^{\prime})\leq-sd_{\gamma}+D_{\Gamma}$ whenever
one can find such $T^{\prime}+L+\varepsilon<T<d_{\gamma}$. Since
$\mathcal{HB}^{\Lambda}_{i(\gamma)}$ is tangent to $x_{0}$, the corresponding
horoball $\mathcal{HB}^{\Gamma}_{i(\gamma)}$ lies inside it, and so
$x^{\prime}$ lies $(sd_{\gamma}-D_{\Gamma})$-deep inside
$\mathcal{HB}^{\Lambda}_{i(\gamma)}$. A close look at the argument yields the
desired bound $D=D(s)$ such that the above holds whenever $d_{\gamma}>D(s)$.
To help the reader take this closer look, I reiterate the choice of constants
and their dependencies as they appear in the proof:
1. 1.
Fix $\varepsilon=1$.
2. 2.
Let $T^{\prime}=T^{\prime}(s)$ the constant from Lemma 5.6 and
$L=\max_{i\in\\{1,\dots,N\\}}\\{d(\mathcal{HB}^{\Lambda}_{i},\mathcal{HB}^{\Gamma}_{i})\\}$.
3. 3.
Fix $T>T^{\prime}+L+1$.
4. 4.
Fix $\alpha=\alpha(1,T)$.
5. 5.
Fix $D(s)=\max\\{D(\alpha),T+1\\}$.
I remark, for the reader worried about the $D_{\Gamma}$ which appears in the
final bound but not in the statement, that (a) $D_{\Gamma}$ is a fixed
universal constant and may as well be ignored, and (b) the discrepancy can be
formally corrected by taking a slightly larger $s<s^{\prime}$ to begin with
and as a result perhaps enlarging the bound $D$ for $d_{\gamma}$). Also note
that $L=L(\Lambda)$ is a universal constant.
∎
#### 6.2.3 Intersection of $\Lambda$-Free Regions and the Existence of a
$\Lambda$-Cocompact Horosphere
In this section I find $\Lambda$-orbit points that lie close to the bounding
horosphere of a $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$. In order to
find such points I need to make sure $\mathcal{HB}^{\Lambda}$ is not contained
inside a much larger $\Lambda$-free horoball. I introduce the following
definition.
###### Definition 6.23.
A $\Lambda$-free horoball $\mathcal{HB}^{\Lambda}$ is called _maximal_ if it
is tangent to a point $x=\lambda x_{0}\in\Lambda\cdot x_{0}$. It is called
_$\varepsilon$ -almost maximal_ if $d(\Lambda\cdot
x_{0},\mathcal{H}^{\Lambda})<\varepsilon$.
###### Remark 6.24.
It may happen that a discrete group admits free but not _maximally free_
horoballs - see discussion in section $4$ of [56]. In any case it is clear
that any $\Lambda$-free horoball can be ‘blown-up’ to an $\varepsilon$-almost
maximal $\Lambda$-free horoball, for every $\varepsilon>0$. Moreover, every
two $\varepsilon$-almost maximal horoballs based at the same point
$\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$ lie at distance at most $\varepsilon$
of one another. For my needs any fixed $\varepsilon$ would suffice, and I fix
$\varepsilon=1$.
###### Lemma 6.25.
There is $D_{\Lambda}>0$ such that if $\mathcal{HB}^{\Lambda}$ is $1$-almost
maximal $\Lambda$-free horoball then
$\mathcal{H}^{\Lambda}\subset\mathcal{N}_{D_{\Lambda}}(\Lambda\cdot x_{0})$,
i.e. $d(x,\Lambda\cdot x_{0})\leq D_{\Lambda}$ for all
$x\in\mathcal{H}^{\Lambda}$.
Notice that Lemma 6.25 does not state $\Lambda\cdot x_{0}$ even intersects
$\mathcal{H}^{\Lambda}$.
###### Proof.
I start with a short sketch of the proof. Consider a $1$-maximal horoball and
a point $x$ on its bounding horosphere with $d(x,\Lambda\cdot x_{0})=D$. One
may translate this situation to $x_{0}$, which results in a $\Lambda$-free
horoball $\mathcal{HB}^{\Lambda}$ intersecting the (closed) $D$-ball about
$x_{0}$ at a point $w$ with $B(w,D)$ $\Lambda$-free.
The proof differs depending on whether
$\mathcal{HB}^{\Gamma}\subset\mathcal{HB}^{\Lambda}$ or the other way round,
since I use the bounds from 6.17:
1. 1.
If $\mathcal{HB}^{\Gamma}\subset\mathcal{HB}^{\Lambda}$, there is a sublinear
bound on $d(\mathcal{HB}^{\Lambda},\mathcal{HB}^{\Gamma})$, which readily
yields a bound on $D$.
2. 2.
if $\mathcal{HB}^{\Lambda}\subset\mathcal{HB}^{\Gamma}$ there is a bound on
$d(x_{0},\mathcal{HB}^{\Gamma})$ that is independent of $D$. So there are only
finitely many possibilities for $\mathcal{HB}^{\Gamma}$, independent of $D$.
Hence there are only finitely many possible base points for
$\mathcal{HB}^{\Gamma}$. These in turn correspond to possible base points for
such $\mathcal{HB}^{\Lambda}$, and this finiteness yields a bound on the
distance $d(\mathcal{HB}^{\Gamma},\mathcal{HB}^{\Lambda})<L$ that is
independent of $D$. The rest of the proof is quite routine.
Let $\mathcal{HB}^{\Lambda}$ be a $1$-almost maximal $\Lambda$-free horoball.
By definition there is $\lambda\in\Lambda$ and $z\in\mathcal{H}^{\Lambda}$
such that $d(\lambda x_{0},z)<1$. Fix $D>0$. I show that if there is some
$z^{\prime}\in\mathcal{H}^{\Lambda}$ for which $d(z^{\prime},\Lambda\cdot
x_{0})\geq D$, then $D$ must be uniformly small. Exactly how small will be set
in the course of the proof.
Fix $D>1$ and assume that there is $z^{\prime}\in\mathcal{H}^{\Lambda}$ with
$d(z,\Lambda\cdot x_{0})\geq D$. Up to sliding $z^{\prime}$ along
$\mathcal{H}^{\Lambda}$, the continuity of the function $x\mapsto
d(x,\Lambda\cdot x_{0})$ together with Intermediate Value Theorem allows to
assume that $d(z^{\prime},\Lambda\cdot x_{0})=D$. Let
$\lambda^{\prime}\in\Lambda$ be the element for which
$d(z^{\prime},\lambda^{\prime}x_{0})=D$. Translating by $\lambda^{\prime-1}$
yields
1. 1.
A $\Lambda$-free horoball
$\mathcal{HB}^{\Lambda}_{0}:=\lambda^{\prime-1}\mathcal{HB}^{\Lambda}$.
2. 2.
A point $w:=\lambda^{\prime-1}z^{\prime}\in\mathcal{H}^{\Lambda}_{0}$ for
which $|w|=d(w,x_{0})=d(w,\Lambda\cdot x_{0})=D$.
Assume first that
$\mathcal{HB}^{\Gamma}_{0}\subset\mathcal{HB}^{\Lambda}_{0}$. By Lemma 6.17
there is a sublinear function $u^{\prime}$ such that
$d(\mathcal{H}^{\Gamma}_{0},\mathcal{H}^{\Lambda}_{0})\leq u^{\prime}(D)$.
This yields a point $\gamma x_{0}\in\mathcal{H}^{\Gamma}_{0}$ for which
$d(w,\gamma x_{0})\leq u^{\prime}(D)+D_{\Gamma}$. Thus $|\gamma x_{0}|\leq
D+u^{\prime}(D)+D_{\Gamma}$ and the reverse triangle inequality gives
$D-\big{(}u^{\prime}(D)+D_{\Gamma}\big{)}\leq
d(w,\lambda_{\gamma}x_{0})-d(w,\gamma x_{0})<d(\gamma
x_{0},\lambda_{\gamma}x_{0})$
Together with the bound $d(\gamma x_{0},\lambda_{\gamma}x_{0})\leq u(|\gamma
x_{0}|)$ and rearranging, one obtains
$D\leq u\big{(}D+u^{\prime}(D)+D_{\Gamma}\big{)}+u^{\prime}(D)+D_{\Gamma}$
The right hand side is clearly a sublinear function in $D$, hence this
inequality may hold only for boundedly small $D$, say $D<D_{1}$. I conclude
that $\mathcal{HB}^{\Gamma}_{0}\subset\mathcal{HB}^{\Lambda}_{0}$ may occur
only when $D<D_{1}$. Notice that $D_{1}$ depends only on $u$ and $u^{\prime}$,
and not on $\mathcal{HB}^{\Lambda}$.
Assume next that $\mathcal{HB}^{\Lambda}_{0}\subset\mathcal{HB}^{\Gamma}_{0}$,
and that the containment is strict. Since $x_{0}\in\Gamma\cdot x_{0}$, the
geodesic $\tau:=[x_{0},w]$ is of length $D$ and intersects
$\mathcal{H}^{\Gamma}_{0}$. Denote by $t_{0}\in[0,D)$ the time in which $\tau$
intersects $\mathcal{H}^{\Gamma}_{0}$, and let
$w^{\prime}:=\tau(t_{0})\in\mathcal{H}^{\Gamma}_{0}$ be the intersection
point. In particular $|w^{\prime}|=t_{0}$. It is clear that
$B(w^{\prime},t_{0})$ is $\Lambda$-free, as a subset of the ball $B(w,D)$.
Again there is $\gamma x_{0}\in
B(w^{\prime},D_{\Gamma})\cap\mathcal{H}^{\Gamma}_{0}$ and so $|\gamma
x_{0}|\leq t_{0}+D_{\Gamma}$. By reverse triangle inequality
$t_{0}-D_{\Gamma}\leq d(w^{\prime},\lambda_{\gamma}x_{0})-d(w^{\prime},\gamma
x_{0})\leq d(\gamma x_{0},\lambda_{\gamma}x_{0})$
and the sublinear constraint gives $t_{0}-D_{\Gamma}\leq u(t_{0}+D_{\Gamma})$.
This can only happen for boundedly small $t_{0}$, say $t_{0}<T$. I conclude
that if $\mathcal{HB}^{\Lambda}_{0}\subset\mathcal{HB}^{\Gamma}_{0}$, then
$\mathcal{HB}^{\Gamma}_{0}$ is a horoball of $\Gamma$ tangent to some point
$y\in B(x_{0},T)$.
By Corollary 2.23 there are finitely many horoballs of $\Gamma$ tangent to
points in $B(x_{0},T)$. In particular there is a finite set
$\\{\xi_{1}^{\prime},\dots,\xi_{K}^{\prime}\\}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$
of possible base points for $\mathcal{HB}^{\Gamma}_{0}$. This set depends only
on $T$, and since the choice of $T$ was completely independent of $D$, the set
of possible base points is independent of $D$ as well. Let
$\widetilde{\mathcal{HB}^{\Gamma}_{i}}$ be the horoball of $\Gamma$ based at
$\xi_{i}^{\prime}$.
I can now bound the distance
$d(\mathcal{HB}^{\Gamma}_{0},\mathcal{HB}^{\Lambda}_{0})$. Let $1\leq i\leq K$
be an index for which there is a $\Lambda$-free horoball based at
$\xi_{i}^{\prime}$ that is contained in
$\widetilde{\mathcal{HB}^{\Gamma}_{i}}$. There is thus some $1$-almost-maximal
$\Lambda$-free horoball based at $\xi^{\prime}_{i}$. Fix an arbitrary such
$1$-almost-maximal $\Lambda$-free horoball
$\widetilde{\mathcal{HB}^{\Lambda}_{i}}$ for each such $i$, and let
$L_{i}:=d(\widetilde{\mathcal{HB}^{\Lambda}_{i}},\widetilde{\mathcal{HB}^{\Gamma}_{i}})$.
Finally, define $L:=\max\\{L_{i}\\}+1$ among such $i$. As stated in Remark
6.24,
$d(\mathcal{HB}^{\Lambda}_{0},\widetilde{\mathcal{HB}^{\Lambda}_{i}})\leq 1$
for some $i$, therefore
$d(\mathcal{HB}^{\Gamma}_{0},\mathcal{HB}^{\Lambda}_{0})\leq L$.
Recall $|w|=D$ and $B(w,D)$ is $\Lambda$-free. It holds that
$d(w,\mathcal{H}^{\Gamma}_{0})\leq L$, and so there is $\gamma
x_{0}\in\mathcal{H}^{\Gamma}_{0}$ for which $d(w,\gamma x_{0})\leq
L+D_{\Gamma}$. In particular $|\gamma x_{0}|\leq D+L+D_{\Gamma}$ (in fact it
is clear that $|\gamma x_{0}|\leq T+D_{\Gamma}$, but this won’t be necessary).
Reverse triangle inequality gives
$D-(L+D_{\Gamma})\leq d(w,\lambda_{\gamma}x_{0})-d(w,\gamma x_{0})\leq
d(\gamma x_{0},\lambda_{\gamma}x_{0})$
and from the sublinear constraint I conclude $D-(L+D_{\Gamma})\leq
u(D+L+D_{\Gamma})$. Since $L,D_{\Gamma}$ are fixed constants independent of
$D$, this can only hold for boundedly small $D$, say $D<D_{2}$. In particular,
one gets a uniform bound $D_{\Lambda}:=\max\\{D_{1},D_{2}\\}$ such that
$x\in\mathcal{H}^{\Lambda}\Rightarrow d(x,\Lambda\cdot x_{0})<D_{\Lambda}$.
∎
$w$$x_{0}$$\begin{array}[]{l}\Lambda-free\\\ \
B(w,D)\end{array}$$\tau(t_{0})$$\tau$$\Lambda-free\
\mathcal{HB}^{\Lambda}{}$$\xi$$\Gamma-free\ \mathcal{HB}^{\Gamma}{}$$\gamma
x_{0}$ Figure 3: Lemma 6.25, case
$\mathcal{HB}^{\Lambda}\subset\mathcal{HB}^{\Gamma}$. The red horosphere of
$\Gamma$ is trapped between $x_{0}$ and $\mathcal{H}^{\Lambda}$, and is at
distance $t_{0}$ from $x_{0}$. A $\Gamma$-orbit point on the red horosphere
close to $x_{0}$ allows to use sublinearity to get a bound on $t_{0}$.
###### Corollary 6.26.
Every $\Gamma$-conical limit point is a $\Lambda$-conical limit point.
###### Proof.
Let $\xi\in X(\infty)$ be a $\Gamma$-conical limit point. Let
$\eta:\mathbb{R}_{\geq 0}\rightarrow X$ be a geodesic with $\eta(\infty)=\xi$.
By definition there is a bound $D>0$ and sequences
$t_{n}\rightarrow\infty,\gamma_{n}\in\Gamma$ such that
$d\big{(}\gamma_{n}x_{0},\eta(t_{n})\big{)}<D$. Consider the corresponding
$\lambda_{n}:=\lambda_{\gamma_{n}}$ and $\lambda_{n}x_{0}$. If $d_{n}$ is
uniformly bounded, then $\xi$ is $\Lambda$-conical by definition. Otherwise,
assume $d_{n}$ is monotonically increasing to $\infty$. For some fixed
$s\in(0,1)$ it holds that for all but finitely many $n\in\mathbb{N}$,
$\gamma_{n}x_{0}$ is $sd_{n}$ deep inside
$\mathcal{HB}^{\Lambda}_{n}:=\mathcal{HB}^{\Lambda}_{\lambda_{n},i(\gamma_{n})}$.
I assume $d_{n}$ is large enough so that $sd_{n}>D$, and in particular
$\eta(t_{n})\in\mathcal{HB}^{\Lambda}_{n}$. Let
$\xi_{n}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$ be the respective base points of
$\mathcal{HB}^{\Lambda}_{n}$. The point $\xi$ is $\Gamma$-conical, and by
Theorem 2.29 $\frac{\pi}{2}\leq d(\xi,\mathcal{W}_{\mathbb{Q}}(\Gamma))\leq
d(\xi,\xi_{n})$.
The proof differs depending on whether the above inequality is strict or not
for any $n\in\mathbb{N}$. Assume first that for some $m\in\mathbb{N}$,
$d(\xi,\xi_{m})=\frac{\pi}{2}$. By item $2$ of Lemma 2.32,
$d\big{(}\mathcal{H}^{\Lambda}_{m},\eta(t)\big{)}$ is uniformly bounded, i.e.,
there is $C>0$ such that for every $t>0$ there is
$x_{t}\in\mathcal{H}^{\Lambda}_{m}$ for which
$d\big{(}x_{t},\eta(t)\big{)}<C$. By Lemma 6.25, $d(x_{t},\Lambda\cdot
x_{0})<D_{\Lambda}$, hence $d\big{(}\eta(t_{n}),x_{t_{n}}\big{)}\leq
C+D_{\Lambda}$. This means that $\xi$ is $\Lambda$-conical.
Otherwise, for all $n\in\mathbb{N}$ it holds that
$\frac{\pi}{2}<d(\xi,\xi_{n})$. The fact that
$\eta(t_{n})\in\mathcal{HB}^{\Lambda}_{n}$ together with Lemma 2.32 implies
that at some later time the geodesic ray $\eta$ leaves
$\mathcal{HB}^{\Lambda}_{n}$. Thus there is $s_{n}>t_{n}$ for which
$\eta(s_{n})\in\mathcal{H}^{\Lambda}_{n}$. Since $\mathcal{HB}^{\Lambda}_{n}$
are maximal $\Lambda$-free horoballs, Lemma 6.25 gives rise to points
$\lambda_{n}x_{0}$ such that $d\big{(}\lambda_{n}x_{0},\eta(s_{n})\big{)}\leq
D_{\Lambda}$. This renders $\xi$ as a $\Lambda$-conical limit point, as
wanted.
∎
###### Proof of Proposition 6.11.
The strategy is as follows. For
$\mathcal{HB}^{\Lambda}=\mathcal{HB}^{\Lambda}_{{\lambda_{\gamma}},i(\gamma)}$,
one uses Proposition 6.22 to get that
$\mathcal{HB}^{\Gamma}\subset\mathcal{HB}^{\Lambda}$ and that the distance
$d(\mathcal{H}^{\Lambda},\mathcal{H}^{\Gamma})$ is large with $d_{\gamma}$.
The horosphere $\mathcal{H}^{\Gamma}$ admits a $\Gamma$-cocompact metric
lattice, and so the projections of these metric lattice points onto
$\mathcal{H}^{\Lambda}$ form a cocompact metric lattice in
$\mathcal{H}^{\Lambda}$. It remains to show that for each
$\gamma^{\prime}x_{0}\in\Gamma\cdot x_{0}\cap\mathcal{H}^{\Gamma}$, the
corresponding $\lambda^{\prime}=\lambda_{\gamma^{\prime}}$ indeed lies on the
same $\mathcal{H}^{\Lambda}$ and boundedly close to the projection
$P_{\mathcal{H}^{\Lambda}}(\gamma^{\prime}x_{0})$. This is done by putting
together all the geometric facts obtained up to this point, specifically Lemma
6.25. One delicate fact that will be of use is that two maximal $\Lambda$-free
horoballs that are based at the same point must be equal, because none of them
can contain a $\Lambda$-orbit point while on the other hand both bounding
horospheres intersect $\Lambda\cdot x_{0}$.
Fix $s>0$ for which Proposition 6.22 yields a corresponding bound $D(s)$, and
let $\gamma\in\Gamma$ such that
$sd_{\gamma}>2\cdot\big{(}D_{\Lambda}+D(s)\big{)}$. Consider the (maximal)
$\Lambda$-free horoball $\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$
based at $\xi^{i(\gamma)}_{\lambda}$. I show that $\Lambda\cdot
x_{0}\cap\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ is a cocompact
metric lattice in $\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$. I keep
the subscript notation because the proof is a game between
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ and another
$\Lambda$-free horoball.
Let $\mathcal{HB}^{\Gamma}_{\lambda_{\gamma},i(\gamma)}$ be the
$\Gamma$-horoball corresponding to
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$. I can conclude that
$\mathcal{HB}^{\Gamma}_{\lambda_{\gamma},i(\gamma)}\subset\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$,
because the choice of $d_{\gamma}>D(s)$ guarantees $\gamma x_{0}$ is
$sd_{\gamma}$ deep inside
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$. In particular
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ is not $\Gamma$-free.
Moreover, it holds that
$L:=d(\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)},\mathcal{H}^{\Gamma}_{\lambda_{\gamma},i(\gamma)})\geq
sd_{\gamma}$. Let $\gamma^{\prime}\in\Gamma$ be any element in the cocompact
metric lattice $\Gamma\cdot
x_{0}\cap\mathcal{H}^{\Gamma}_{\lambda_{\gamma},i(\gamma)}$, and consider two
associated points: (a) $\lambda^{\prime}x_{0}=\lambda_{\gamma^{\prime}}x_{0}$
and (b) the projection of $\gamma^{\prime}x_{0}$ on
$\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$, denoted
$p_{\gamma}^{\prime}:=P_{\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}}(\gamma^{\prime}x_{0})\in\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$.
The horoball $\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ is a
maximal $\Lambda$-free horoball so it is also $1$-almost maximal, hence
$d(p_{\gamma}^{\prime},\Lambda\cdot x_{0})\leq D_{\Lambda}$ and the following
holds:
$sd_{\gamma}\leq L\leq d_{\gamma^{\prime}}\leq
d(\gamma^{\prime}x_{0},p_{\gamma}^{\prime})+d(p_{\gamma}^{\prime},\Lambda\cdot
x_{0})\leq L+D_{\Lambda}$ (6)
Consider $\xi^{i(\gamma^{\prime})}_{\lambda^{\prime}}$, and assume towards
contradiction that
$\xi^{i(\gamma^{\prime})}_{\lambda_{\gamma^{\prime}}}\neq\xi^{i(\gamma)}_{\lambda_{\gamma}}$.
Both points lie in $\mathcal{W}_{\mathbb{Q}}(\Gamma)$ and therefore must be at
Tits distance $\pi$ of each other. Therefore the fact that
$\gamma^{\prime}x_{0}$ lies in
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ implies that the
geodesic $[\gamma^{\prime}x_{0},\xi^{i(\gamma^{\prime})}]$ leaves
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ at some point
$z\in\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$.
The fact that $D(s)\leq sd_{\gamma}\leq d_{\gamma^{\prime}}$ implies that
$\gamma^{\prime}x_{0}$ lies $s^{2}d_{\gamma}$ deep inside
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}$.
Therefore the point $z$ also lies at least $s^{2}d_{\gamma}$ deep inside
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}$, and
therefore $z$ is the centre of a $\Lambda$-free horoball of radius at least
$s^{2}d_{\gamma}$.
By choice of $d_{\gamma}$ the point $z$ therefore admits a $2D_{\Lambda}$
neighbourhood that is $\Lambda$-free. But $z$ lies on
$\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$, a maximal horosphere of
$\Lambda$, contradicting Lemma 6.25. I conclude that
$\xi^{i(\gamma^{\prime})}_{\lambda_{\gamma^{\prime}}}=\xi^{i(\gamma)}_{\lambda_{\gamma}}$,
so $\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ and
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}$ are
two $\Lambda$-free horoballs that are tangent to a $\Lambda\cdot x_{0}$ point
and based at the same point at $\infty$. This implies
$\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}=\mathcal{HB}^{\Lambda}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}$,
and in particular $\lambda_{\gamma^{\prime}}x_{0}\in\mathcal{H}^{\Lambda}$.
Finally, it is clearly seen from Inequality 6 that
$d(\lambda^{\prime}x_{0},p_{\gamma}^{\prime})\leq
d(\lambda^{\prime}x_{0},\gamma^{\prime}x_{0})+d(\gamma^{\prime}x_{0},p_{\gamma}^{\prime})\leq
d_{\gamma^{\prime}}+L\leq L+D_{\Lambda}+L$
The element $\gamma^{\prime}x_{0}\in\Gamma\cdot
x_{0}\cap\mathcal{H}^{\Gamma}_{\lambda_{\gamma},i(\gamma)}$ was as arbitrary
element, and the above argument shows that the corresponding $\Lambda$-orbit
points satisfy:
1. 1.
$\lambda^{\prime}x_{0}$ all lie on
$\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$.
2. 2.
Each $p_{\gamma}^{\prime}$ is $2L+D_{\Lambda}$ close to the point
$\lambda^{\prime}x_{0}$.
This shows that the cocompact metric lattice
$\\{p_{\gamma}^{\prime}\mid\gamma^{\prime}x_{0}\in\mathcal{H}^{\Gamma}_{\lambda_{\gamma},i(\gamma)}\\}$
lies in a bounded neighbourhood of the set of points $\Lambda\cdot
x_{0}\cap\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$, proving that
$\Lambda\cdot x_{0}\cap\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$ is
a cocompact metric lattice in
$\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$. Lemma 2.25 elevates this
to
$\big{(}\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)})\big{)}\cdot
x_{0}\cap\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$
being a cocompact metric lattice in
$\mathcal{H}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$, completing the proof. ∎
$\gamma
x_{0}$$d_{\gamma}$$\gamma^{\prime}x_{0}$$\lambda_{\gamma^{\prime}}x_{0}$$\mathcal{HB}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}^{\Lambda}$$z^{\prime}:=\pi_{\mathcal{HB}_{\lambda_{\gamma^{\prime}},i(\gamma^{\prime})}^{\Lambda}}(\gamma^{\prime}x_{0})$$\overbrace{}^{D_{1}}_{1}$$\xi_{\lambda_{\gamma^{\prime}}}^{i(\gamma^{\prime})}$$\xi_{\lambda_{\gamma}}^{i(\gamma)}$$\geq\frac{1}{2}d_{\gamma}$$\mathbf{z}$$\geq\frac{1}{2}d_{\gamma^{\prime}}$$\begin{array}[]{l}\Lambda-
free\ ball\\\ \
B\left(z,\frac{1}{2}d_{\gamma}\right)\end{array}$$\mathcal{HB}_{\lambda_{\gamma},i(\gamma)}^{\Lambda}$$\mathcal{HB}_{\lambda_{\gamma},i(\gamma)}^{\Gamma}$$\lambda_{\gamma}x_{0}$
Figure 4: Proposition 6.11. Assuming towards contradiction that
$\xi^{i(\gamma)}_{\lambda_{\gamma}}\neq\xi^{i(\gamma^{\prime})}_{\lambda_{\gamma^{\prime}}}$
results in a point $z\in\mathcal{HB}^{\Lambda}_{\lambda_{\gamma},i(\gamma)}$
(blue coloured and bold faced in the bottom part of the figure) admitting a
large $\Lambda$-free neighbourhood, contradicting almost cocompactness.
### 6.3 The Bounded Case
Proposition 6.11 is enough in order to prove Theorem 6.1 in the case that
$\Gamma$ does not lie in a bounded neighbourhood of $\Lambda$. The case where
$\Gamma$ and $\Lambda$ lie at bounded Hausdorff distance, i.e. where
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ and
$\Lambda\subset\mathcal{N}_{D}(\Gamma)$ for $D>0$, arose naturally in the
context of quasi-isometric completeness of non-uniform lattices. The precise
statements are given in Theorems 6.8 and 6.9 above. I present their proofs in
Section 6.3.2 below.
The notable difference between Theorem 6.8 and Theorem 6.9 is that for higher
rank groups, the inclusion $\Lambda\subset\mathcal{N}_{D}(\Gamma)$ is only
required to prove commensurability. In view of Corollary 6.7, this allows me
to omit that assumption from Theorem 1.6. Notice also that for groups with
property (T) the result easily follows from the (much more recent) result by
Leuzinger in Theorem 4.7.
In the context of commensurability in the sublinear setting, I can only prove
a limited result, Namely that $\Lambda$ is commensurable to $\Gamma$ if
$\Gamma$ is an irreducible $\mathbb{Q}$-rank $1$ lattice and both
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ and
$\Lambda\subset\mathcal{N}_{u}(\Gamma)$ for some constant $D>0$ and a
sublinear function $u$. This is done via a reduction to the bounded case.
###### Proposition 6.27.
Let $G$ be a real finite-centre semisimple Lie group without compact factors,
$\Gamma\leq G$ an irreducible lattice of $\mathbb{Q}$-rank $1$, $\Lambda\leq
G$ a discrete subgroup, and $u$ a sublinear function. If
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ for some $D>0$ and
$\Lambda\subset\mathcal{N}_{u}(\Gamma)$, then actually
$\Lambda\subset\mathcal{N}_{D^{\prime}}(\Gamma)$ for some $D^{\prime}>0$.
Moreover, if $G$ is of $\mathbb{R}$-rank$\ 1$, the conclusion holds under the
relaxed assumption that $u(r)\preceq_{\infty}\varepsilon r$ for some
$\varepsilon<1$.
###### Remark 6.28.
While the setting of Proposition 6.2 is indeed rather limited, the situation
that both $\Gamma\subset\mathcal{N}_{u}(\Lambda)$ and
$\Lambda\subset\mathcal{N}_{u}(\Gamma)$ arises naturally from the motivating
example of SBE-rigidity in Theorem 3.7. Notice however that Theorem 3.7 is not
known for groups $G$ that admit $\mathbb{R}$-rank$\ 1$ factors, which is the
only setting for which I can prove Proposition 6.2.
#### 6.3.1 A Reduction
I start with the proof of Proposition 6.27. The first step is to establish the
fact that $\Lambda$ must preserve $\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
###### Lemma 6.29.
Let $G$ be a real finite-centre semisimple Lie group without compact factors,
$\Gamma\leq G$ an irreducible non-uniform lattice of $\mathbb{Q}$-rank $1$,
$\Lambda\leq G$ a discrete subgroup. Assume that
$\Gamma\subset\mathcal{N}_{u}(\Lambda)$ and that
$\Lambda\subset\mathcal{N}_{u^{\prime}}(\Gamma)$ for sublinear functions
$u,u^{\prime}$. Then
$\Lambda\cdot\mathcal{W}_{\mathbb{Q}}(\Gamma)\subset\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
Moreover, if $G$ is of $\mathbb{R}$-rank$\ 1$, the conclusion holds under the
relaxed assumption that $u^{\prime}(r)\preceq_{\infty}\varepsilon r$ for some
$\varepsilon<1$.
###### Proof.
The proof is similar to the argument of Lemma 6.13, and uses the linear
penetration rate of a geodesic into a horoball. Let
$\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$, and let $\mathcal{H}^{\Gamma}$ be a
horosphere bounding a $\Gamma$-free horoball $\mathcal{HB}^{\Gamma}$ with
$\mathcal{H}^{\Gamma}\cap\Gamma\cdot x_{0}$ a metric lattice in
$\mathcal{H}^{\Gamma}$. Assume first that $u^{\prime}$ is sublinear. Since
$\mathcal{HB}^{\Gamma}$ is $\Gamma$-free and
$\Lambda\subset\mathcal{N}_{u^{\prime}}(\Gamma)$, I can conclude that
$\Lambda\cdot
x_{0}\cap\mathcal{HB}^{\Gamma}\subset\mathcal{N}_{u^{\prime}}(\mathcal{H}^{\Gamma})$.
Recall (Lemma 2.32) that every geodesic ray $\eta$ with limit point
$\xi^{\prime}\in\mathcal{N}_{\frac{\pi}{2}}(\xi)$ penetrates
$\mathcal{HB}^{\Gamma}$ at linear rate. Therefore for every such geodesic ray
$\eta$ and every sublinear function $v$ there is $R=R(\eta,v)>0$ for which
$\mathcal{N}_{v}(\eta_{\restriction_{r>R}})$ is $\Lambda$-free.
On the other hand, let $\lambda\in\Lambda$, and assume towards contradiction
that $\lambda\xi\notin\mathcal{W}_{\mathbb{Q}}(\Gamma)$. Then by Proposition
2.31 there is a $\Gamma$-conical limit point
$\xi^{\prime}\in\mathcal{N}_{\frac{\pi}{2}}(\lambda\xi)$. The hypothesis that
$\Gamma\subset\mathcal{N}_{u}(\Lambda)$ then implies that for every $R>0$,
$\mathcal{N}_{u}(\eta_{\restriction{r>R}})\cap\Lambda\cdot
x_{0}\neq\emptyset$. Translating by $\lambda^{-1}$ yields a contradiction to
the previous paragraph. I conclude that
$\lambda\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
I now modify the argument to include $u^{\prime}(r)\preceq_{\infty}\varepsilon
r$ when $G$ is of $\mathbb{R}$-rank$\ 1$. In this case, the only point
$\xi^{\prime}\in\mathcal{N}_{\frac{\pi}{2}}(\lambda\xi)$ is $\lambda\xi$
itself. Therefore by the same argument as above, the assumption that
$\lambda\xi\notin\mathcal{W}_{\mathbb{Q}}(\Gamma)$ implies that the
$u$-sublinear neighbourhood of every geodesic ray with limit point $\xi$
intersects $\Lambda\cdot x_{0}$. I.e., for every $\eta$ with limit point $\xi$
and every $R>0$ it holds that
$\mathcal{N}_{u}(\eta_{\restriction_{r>R}})\cap\Lambda\cdot
x_{0}\neq\emptyset$. On the other hand, every such geodesic penetrates
$\mathcal{HB}^{\Gamma}$ at $1$-linear rate. This amounts to the following
fact: if $v^{\prime}(r)=\varepsilon r$ for some $\varepsilon\in(0,1)$, then
for some $R>0$, the set
$\mathcal{N}_{u}(\eta_{\restriction_{r>R}})\cap\mathcal{N}_{v^{\prime}}(\mathcal{H}^{\Gamma})=\emptyset$.
This is a contradiction to $\Lambda\subset\mathcal{N}_{u^{\prime}}(\Gamma)$. ∎
###### Proof of Proposition 6.27.
Assume towards contradiction that there is a sequence $\lambda_{n}$ such that
$d(\lambda_{n}x_{0},\Gamma\cdot x_{0})>n$. Recall that $\Gamma\cdot x_{0}$ is
a cocompact metric lattice in the compact core of $\Gamma$. This implies that
there is a number $D^{\prime}>0$ such that any $\lambda\in\Lambda$ for which
$\lambda x_{0}\notin\mathcal{N}_{D^{\prime}}(\Gamma\cdot x_{0})$ must lie at
least $\frac{1}{2}D^{\prime}$-deep inside a horoball of $\Gamma$. I can assume
that for all $n\in\mathbb{N}$ there are corresponding horoballs of $\Gamma$,
which I denote $\mathcal{HB}^{\Gamma}_{n}$, for which $\lambda_{n}\cdot
x_{0}\in\mathcal{HB}^{\Gamma}_{n}$. The fact that
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ then implies that
$\mathcal{N}_{D}(\Lambda\cdot x_{0})$ covers a cocompact metric lattice in
$\mathcal{H}^{\Gamma}_{n}$, namely the metric lattice $\Gamma\cdot
x_{0}\cap\mathcal{H}^{\Gamma}_{n}$. In the terminology of Section 6.2,
$\mathcal{H}^{\Gamma}_{n}$ is almost $\Lambda$-cocompact, or $D$-almost
$\Lambda$-cocompact.
I first prove that every horoball of $\Gamma$ contains a $\Lambda$-free
horoball (this is of course immediate if
$\Lambda\subset\mathcal{N}_{C}(\Gamma)$ for some $C>0$). Assume towards
contradiction that there is a horoball $\mathcal{HB}^{\Gamma}$ of $\Gamma$
that does not contain a $\Lambda$-free horoball. Denote
$\mathcal{H}^{\Gamma}:=\partial\mathcal{HB}^{\Gamma}$. In the notations of the
previous paragraph, I can assume without loss of generality that
$\mathcal{HB}^{\Gamma}=\mathcal{HB}^{\Gamma}_{n}$ for all $n\in\mathbb{N}$.
Denote by $\xi$ the base point of $\mathcal{HB}^{\Gamma}$, fix some arbitrary
$x\in\mathcal{H}^{\Gamma}$ and consider the geodesic ray $\eta:=[x,\xi)$. The
constraint that $\Lambda\subset\mathcal{N}_{u}(\Gamma)$ implies that for every
$R>0$ there is some $L>0$ for which the ball $B\big{(}\eta(L+t),R\big{)}$ is
$\Lambda$-free, for all $t\geq 0$. In particular, for all large enough
$n\in\mathbb{N}$ (depending on $R$), the horosphere
$\mathcal{H}(\xi,\lambda_{n}x_{0})$ that is parallel to $\mathcal{H}^{\Gamma}$
and that passes through $\lambda_{n}x_{0}$ contains a point that is the centre
of $\Lambda$-free ball of radius $R$. This property is $\Lambda$-invariant, as
well as the fact that $\mathcal{HB}^{\Gamma}$ is based at
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$. In particular, these two properties hold
for the horoballs
$\mathcal{HB}_{n}:=\lambda_{n}^{-1}\cdot\mathcal{HB}^{\Gamma}$, whose
respective base points I denote
$\xi_{n}:=\lambda_{n}^{-1}\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
Fix $R=D+2D_{\Gamma}$ (recall that $D_{\Gamma}$ is such that every horosphere
$\mathcal{H}$ of $\Gamma$ admits
$\mathcal{H}\subset\mathcal{N}_{D_{\Gamma}}(\Gamma\cdot
x_{0}\cap\mathcal{H})$). Let $L=L(D+2D_{\Gamma})$ be the corresponding bound
from the previous paragraph. For every $n>L$ the horoball $\mathcal{HB}_{n}$
has bounding horosphere $\mathcal{H}_{n}$ that admits a point
$z_{n}\in\mathcal{H}_{n}$ for which $B(z_{n},D+2D_{\Gamma})$ is
$\Lambda$-free. Moreover, the same is true for every horosphere that is
parallel to $\mathcal{H}_{n}$ which lies inside $\mathcal{HB}_{n}$. Since
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$, this means that every horosphere that
lies inside $\mathcal{HB}_{n}$ admits a point that is the centre of a
$\Gamma$-free ball of radius $2D_{\Gamma}$. I conclude that none of those
horospheres could be the horosphere of $\Gamma$ corresponding to the parabolic
limit point $\xi_{n}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$. Since
$x_{0}\in\mathcal{H}_{n}$ it must therefore be that $\mathcal{H}_{n}$ is a
horosphere of $\Gamma$. But this contradicts the fact that
$z_{n}\in\mathcal{H}_{n}$ and $B(z_{n},2D_{\Gamma})$ is $\Gamma$-free. This
shows that no horoball of $\Gamma$ contains a sequence of $\Lambda$-orbit
points that lie deeper and deeper in that horoball. Put differently, it shows
that every horoball of $\Gamma$ contains a $\Lambda$-free horoball.
I remark that the above argument shows something a bit stronger, which I will
not use but which I find illuminating. It proves that as soon as $d(\lambda
x_{0},\Gamma\cdot x_{0})$ is uniformly large enough, say more than $M$, then
$\lambda x_{0}$ must lie on a $(D+2D_{\Gamma}$)-almost $\Lambda$-cocompact
horosphere parallel to $\mathcal{H}^{\Gamma}$, where $\mathcal{H}^{\Gamma}$ is
the bounding horosphere of any horoball of $\Gamma$ in which $\lambda x_{0}$
lies (recall that it must lie in at least one such horoball). On the other
hand if $d(\lambda x_{0},\Gamma\cdot x_{0})<M$, then since every point in the
$\Gamma$-orbit lies on a horosphere of $\Gamma$ one concludes that $\lambda
x_{0}$ lies on a horosphere $\mathcal{H}$ based at
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$ that is $(M+D)$-almost $\Lambda$-cocompact.
I can now assume that every $\mathcal{HB}^{\Gamma}_{n}$ contains a
$\Lambda$-free horoball. In particular it contains a $1$-almost maximal
$\Lambda$-free horoball $\mathcal{HB}^{\Lambda}_{n}$ (see Definition 6.23). By
definition there is a point $\lambda_{n}^{\prime}x_{0}$ that is at distance at
most $1$ from $\mathcal{H}^{\Lambda}_{n}=\partial\mathcal{HB}^{\Lambda}_{n}$.
Up to enlarging $d(\lambda_{n}x_{0},\Gamma\cdot x_{0})$ or decreasing it by at
most $1$, I can assume $\lambda_{n}=\lambda_{n}^{\prime}$ to begin with.
Consider $\mathcal{HB}_{n}:=\lambda_{n}^{-1}\cdot\mathcal{HB}^{\Lambda}_{n}$
with $\mathcal{H}_{n}=\partial\mathcal{HB}_{n}$. This is a sequence of
horoballs, each of which contains a $\Lambda$-free horoball at depth at most
$1$, based at corresponding parabolic limit points
$\xi_{n}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$, and tangent to points that are
at distance at most $1$ from $x_{0}$, i.e., $\mathcal{H}_{n}\cap
B(x_{0},1)\neq\emptyset$.
Since $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ I conclude that each of the
$\mathcal{HB}_{n}$ contain a horoball of depth at most $D+1$ that is
$\Gamma$-free. Therefore the horoball of $\Gamma$ that is based at $\xi_{n}$
must have its bounding horosphere intersecting $B(x_{0},D+2)$. By Corollary
2.23 there are only finitely such horoballs. I conclude that there are
finitely many points
$\xi^{\prime}_{1},\dots,\xi^{\prime}_{K}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$
such that for every $n\in\mathbb{N}$ there is $i(n)\in\\{1,\dots,K\\}$ with
$\xi_{n}=\xi^{\prime}_{i(n)}$. From the Pigeonhole Principle there is some
$\xi^{\prime}\in\\{\xi^{\prime}_{1}\dots,\xi^{\prime}_{K}\\}$ for which
$\xi_{n}=\xi^{\prime}$ for infinitely many $n\in\mathbb{N}$. Passing to a
subsequence I assume that this is the case for all $n\in\mathbb{N}$.
To begin with the $\mathcal{HB}^{\Gamma}_{n}$ are horoballs of $\Gamma$, and
therefore as in the first case the bounding horospheres
$\mathcal{H}^{\Gamma}_{n}$ are $D+2D_{\Gamma}$-almost $\Lambda$-cocompact.
This is a $\Lambda$-invariant property and therefore the same holds for the
$\lambda_{n}^{-1}$ translate of it. These are the horospheres which are based
at $\xi^{\prime}$ and lie outside $\mathcal{HB}_{n}$ at distance
$d(\lambda_{n}x_{0},\Gamma\cdot x_{0})>n-1$ from $\mathcal{H}_{n}$. They form
a sequence of outer and outer horospheres based at the same point at
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$, all of which are $D+2D_{\Gamma}$-almost
$\Lambda$-cocompact. This is a contradiction, since the union of such
horospheres intersect every horoball of $X$, contradicting the existence of
$\Lambda$-free horoballs. Formally, take some
$\zeta\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$ different from $\xi^{\prime}$.
Since both $\xi^{\prime}$ and $\zeta$ lie in
$\mathcal{W}_{\mathbb{Q}}(\Gamma)$, they admit $d_{T}(\zeta,\xi^{\prime})=\pi$
and there is a geodesic $\eta$ with $\eta(-\infty)=\xi^{\prime}$ and
$\eta(\infty)=\zeta$. Let $\mathcal{HB}^{\Gamma}_{\zeta}$ be the horoball of
$\Gamma$ that is based at $\zeta$. By the first step of this proof, every such
horoball must contain a $\Lambda$-free horoball
$\mathcal{HB}^{\Lambda}_{\zeta}$. Therefore there is some $T>0$ such that for
all $t>T$ the point $\eta(t)$ lies $2(D+2D_{\Gamma})$ deep in
$\mathcal{HB}^{\Lambda}_{\zeta}$. I conclude that for all $t>T$,
$B\big{(}\eta(t),2D\big{)}$ is $\Lambda$-free. On the other hand, for
arbitrarily large $t$ it holds that the horosphere based at $\xi^{\prime}$ and
tangent to $\eta(t)$ is $D+2D_{\Gamma}$-almost $\Lambda$-cocompact, and in
particular $d\big{(}\eta(t),\Lambda\cdot x_{0}\big{)}<D+2D_{\Gamma}$, a
contradiction.
I conclude that $\Lambda\subset\mathcal{N}_{D^{\prime}}(\Gamma)$ for some
$D^{\prime}>0$, as claimed. ∎
###### Proof of Theorem 1.8.
The theorem follows immediately from Proposition 6.27 together with Theorems
6.8 and 6.9. ∎
#### 6.3.2 The Arguments of Schwartz and Eskin
##### The $\mathbb{R}$-rank$\ 1$ case.
The statement of Theorem 6.9 is a slight modification of Schwartz’s original
formulation. His framework leads to a discrete subgroup $\Delta\leq G$ such
that:
1. 1.
Every element of $\Delta$ _quasi-preserves_ the compact core of the lattice
$\Gamma$. Namely, each element of $\Delta$ is an isometry of $X$ that
preserves $\mathcal{W}_{\mathbb{Q}}(\Gamma)$ and that maps every horosphere of
$\Gamma$ to within the $D=D(\Delta)$ neighbourhood of some other horosphere of
$\Gamma$.
2. 2.
It holds that $\Gamma\subset\mathcal{N}_{D}(\Delta)$.
From these two properties Schwartz is able to deduce that $\Delta$ has finite
covolume, i.e. that $\Delta$ is a lattice in $G$. Here is a sketch of his
argument, which works whenever $\Gamma$ is a $\mathbb{Q}$-rank $1$ lattice.
###### Theorem 6.30.
In the setting described above, $\Delta$ is a lattice in $G$.
###### Proof sketch.
Consider $X_{0}^{\prime}:=\bigcup_{g\in\Delta}g\cdot X_{0}$, where $X_{0}$ is
the compact core of $\Gamma$. This space serves as a ‘compact core’ for
$\Delta$: the fact that $\Delta$ quasi-preserves $X_{0}$ implies that
$X_{0}^{\prime}\subset\mathcal{N}_{D}(X_{0})$. It is a $\Delta$-invariant
space, and therefore one gets an isometric action of $\Delta$ on
$X_{0}^{\prime}$. This action is cocompact: the reason is that $\Gamma$ acts
cocompactly on $X_{0}$, and $\Gamma\subset\mathcal{N}_{D}(\Delta)$. Formally,
every point in $X_{0}^{\prime}$ is $D$-close to a point in $X_{0}$. Every
point in $X_{0}$ is $D_{\Gamma}$-close to a point in $\Gamma\cdot x_{0}$.
Every point in $\Gamma\cdot x_{0}$ is $D$-close to a point in $\Delta\cdot
x_{0}$. Therefore the ball of radius $2D+D_{\Gamma}$ contains a fundamental
domain for the action of $\Delta$ on $X_{0}^{\prime}$.
It remains to see that the action of $\Delta$ on $X\setminus X_{0}^{\prime}$
is of finite covolume. As a result of the cocompact action of $\Delta$ on
$X_{0}^{\prime}$, there is $B:=B(x_{0},R)$ so that
$X_{0}^{\prime}\subset\Delta\cdot B$. $X_{0}^{\prime}$ is the complement of a
union of horoballs, which one may call _horoballs of $\Delta$_, with bounding
_horospheres of $\Lambda$_. The fact that $\Gamma$ is of $\mathbb{Q}$-rank $1$
means that the horoballs of $\Gamma$ are disjoint, and therefore those of
$\Lambda$ are almost disjoint: there is some $C>0$ such that for every
horosphere $\mathcal{H}$ of $\Lambda$ and every point $x\in\mathcal{H}$,
$d(x,X_{0}^{\prime})<C$. Up to enlarging the radius of $B$ by $C$, I can
assume that $\mathcal{H}\subset\Delta\cdot B$ for every horosphere
$\mathcal{H}$ of $\Lambda$.
Each horoball of $\Lambda$ is based at $\mathcal{W}_{\mathbb{Q}}(\Gamma)$, and
each lies uniformly boundedly close to the corresponding horoballs of
$\Gamma$. From Corollary 2.23 one therefore sees that there are finitely many
horoballs of $\Delta$ that intersect $B$. Denote them by
$\mathcal{HB}_{1},\dots,\mathcal{HB}_{N}$, their bounding horospheres by
$\mathcal{H}_{i}=\partial\mathcal{HB}_{i}$, and their intersection with $B$ by
$B_{i}:=B\cap\mathcal{HB}_{i}$. Let also
$\xi_{i}\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$ denote the base point of each
$\mathcal{HB}_{i}$. Each $B_{i}$ is pre-compact and therefore the projection
of each $B_{i}$ on $\mathcal{H}_{i}$ is pre-compact as well (this is a
consequence e.g. of the results of Heintze-Im hof recalled in Remark 2.6). Let
$D_{i}\subset\mathcal{H}_{i}$ be a compact set that contains this projection,
i.e. $P_{\mathcal{H}_{i}}(B_{i})\subset D_{i}\subset\mathcal{H}_{i}$. In
particular $B\cap\mathcal{H}_{i}\subset D_{i}$.
Observe now that for every horoball $\mathcal{HB}$ of $\Delta$, with bounding
horosphere $\mathcal{H}=\partial\mathcal{HB}$, the $\Delta$-orbit of every
point $x\in\mathcal{H}$ intersects some $D_{i}$. First notice that for
$x\in\mathcal{H}$ the choice of $B$ implies that the $\Delta$-orbit of $x$
must intersect $B$, say $gx\in B$. In particular $g\mathcal{H}\cap
B\neq\emptyset$, and since $g\mathcal{HB}$ is a horoball of $\Delta$ then by
definition $g\mathcal{H}=\mathcal{H}_{i}$ for some $i\in\\{1,\dots,N\\}$. One
concludes that indeed $gx\subset D_{i}$.
Moreover, let $y\in X$ is any point that lies inside a horoball $\mathcal{HB}$
of $\Delta$, and $x=P_{\mathcal{H}}(y)$ its projection on the bounding
horosphere $\mathcal{H}=\partial\mathcal{HB}$. By the previous paragraph there
is some $g\in\Delta$ and $i\in\\{1,\dots,N\\}$ for which $gx\in D_{i}$, and
therefore it is clear that $gy$ lies on a geodesic emanating from $D_{i}$ to
$\xi_{i}$.
Finally, define $\mathrm{Cone}(D_{i})$ to be the set of all geodesic rays that
emanate from $D_{i}$ and with limit point $\xi_{i}$. The previous paragraph
proves that $\bigcup_{i=1}^{N}\mathrm{Cone}(D_{i})$ contains a fundamental
domain for the action of $\Delta$ on $X\setminus X_{0}^{\prime}$. Moreover,
the fact that $D_{i}\subset\mathcal{H}_{i}$ is compact readily implies that
each $\mathrm{Cone}(D_{i})$ has finite volume, and so this fundamental domain
is of finite volume.
To conclude, $B\cup\big{(}\bigcup_{i=1}^{N}\mathrm{Cone}(D_{i})\big{)}$ is a
set of finite volume and it contains a fundamental domain for the
$\Delta$-action on $X$, as claimed. The proof of commensurability of $\Delta$
and $\Gamma$ is given in full in [53].
∎
There is one essential difference between Theorem 6.9 and Theorem 6.30, namely
the assumption that $\Lambda\subset\mathcal{N}_{D}(\Gamma)$ rather than that
$\Lambda$ quasi-preserves the compact core of $\Gamma$. In Schwartz’s work,
the fact that $\Delta\subset\mathcal{N}_{D}(\Gamma)$ is not relevant (even
though it easily follows from the construction of his embedding of $\Delta$ in
$G$). He only uses the two properties described above, namely the quasi-
preservation of $X_{0}$ and $\Gamma\subset\mathcal{N}_{D}(\Delta)$.
The assumption that $\Lambda$ quasi-preserves the compact core of $\Gamma$
does not feel appropriate in the context of sublinear rigidity, while the
metric condition $\Lambda\subset\mathcal{N}_{D}(\Gamma)$ seems much more
natural. It is a stronger condition as I now show. By Lemma 6.29,
$\Lambda\cdot\mathcal{W}_{\mathbb{Q}}(\Gamma)\subset\mathcal{W}_{\mathbb{Q}}(\Gamma)$.
Let $\mathcal{H}^{\Gamma}_{1}$ be a horosphere of $\Gamma$, based at
$\xi\in\mathcal{W}_{\mathbb{Q}}(\Gamma)$, and let $\gamma
x_{0}\in\mathcal{H}^{\Gamma}_{1}$ be some point on the metric lattice of
$\Gamma\cdot x_{0}$ on $\mathcal{H}^{\Gamma}_{1}$. There is an element
$\lambda\in\Lambda$ such that $d(\lambda x_{0},\gamma x_{0})<D$. Moreover,
since $\Lambda\subset\mathcal{N}_{D}(\Gamma)$ one knows that the parallel
horoball that lies $D$-deep inside $\mathcal{HB}^{\Gamma}_{1}$ is
$\Lambda$-free.
Let $\lambda^{\prime}\in\Lambda$ be an arbitrary element of $\Lambda$, and
consider $\lambda^{\prime}\cdot\mathcal{H}^{\Gamma}_{1}$. The last statement
in the previous paragraph is $\Lambda$-invariant, and so the horoball that
lies $D$-deep inside $\lambda^{\prime}\cdot\mathcal{HB}^{\Gamma}_{1}$ is
$\Lambda$-free. The fact that $\Gamma\subset\mathcal{N}_{D}(\Lambda)$ then
implies that the parallel horoball that lies $2D$ deep inside
$\lambda^{\prime}\mathcal{HB}^{\Gamma}_{1}$ is $\Gamma$-free. Let
$\mathcal{H}^{\Gamma}_{2}$ be the horosphere of $\Gamma$ that is based at
$\lambda^{\prime}\xi$. The last statement amounts to saying that
$\mathcal{H}^{\Gamma}_{2}$ lies at most $2D$-deep inside
$\lambda^{\prime}\mathcal{HB}^{\Gamma}_{1}$. On the other hand, one has
$d(\lambda^{\prime}\lambda
x_{0},\lambda^{\prime}\mathcal{H}^{\Gamma}_{1})=d(\lambda
x_{0},\mathcal{H}^{\Gamma}_{1})\leq D$, so there is a $\Lambda$-orbit point
that lies within $D$ of $\lambda^{\prime}\mathcal{H}^{\Gamma}_{1}$. The
parallel horoball that lies $D$-deep inside $\mathcal{HB}^{\Gamma}_{2}$ must
also be $\Lambda$-free, so I conclude that $\mathcal{H}^{\Gamma}_{2}$ must be
contained in the parallel horoball to
$\lambda^{\prime}\mathcal{HB}^{\Gamma}_{1}$ which contains it and that is at
distance $D$ from it. I conclude that
$d(\lambda^{\prime}\mathcal{H}^{\Gamma}_{1},\mathcal{H}^{\Gamma}_{2})\leq 2D$,
and so that $\Lambda$ quasi-preserves $X_{0}$.
###### Remark 6.31.
It is interesting to note that Schwartz’s arguments are similar in spirit to
my arguments in Section 6.2. In fact, one could also prove Theorem 6.9 using
the same type of arguments that appear repeatedly in section 6.2, namely by
moving $\Lambda$-free horoballs around the space, specifically the proof of
Proposition 6.27. I do not present this alternative proof here.
##### Higher rank.
Eskin’s proof is ergodic, and based on results of Mozes [41] and Shah [55]. I
produce it here without the necessary preliminaries, which are standard.
###### Proof of Theorem 6.8.
To prove that $\Lambda$ is a lattice amounts to finding a finite non-zero
$G$-invariant measure on $\Lambda\backslash G$. By Theorem $2$ in [41], if
$P\leq G$ is a parabolic subgroup then every $P$-invariant measure on
$\Lambda\backslash G$ is automatically $G$-invariant. Fix a minimal parabolic
subgroup $P\leq G$ and let $\mu_{0}$ be some fixed probability measure on
$\Lambda\backslash G$. Since $P$ is amenable it admits a tempered Følner
sequence $F_{n}\subset P$, and one can average $\mu_{0}$ along each $F_{n}$ to
get a sequence of probability measures $\mu_{n}$. The weak* compactness of the
unit ball in the space of measures on $\Lambda\backslash G$ implies that there
exists a weak* limit $\mu$ of the $\mu_{n}$. The measure $\mu$ is
automatically a finite $P$-invariant measure. It remains to show that $\mu$ is
not the zero measure. To see this it is enough to show that for some compact
set $C_{\Lambda}\subset\Lambda\backslash G$ and some $\Lambda
g=x\in\Lambda\backslash G$, one has
$0<\liminf_{n}\frac{1}{|F_{n}|}\int_{F_{n}}{\mathbbm{1}_{C_{\Lambda}}(xp^{-1})dp}$
(7)
Fix some compact neighbourhood $C_{\Gamma}\subset\Gamma\backslash G$ of the
trivial coset $\Gamma e$. The hypothesis
$\Gamma\subset\mathcal{N}_{D}(\Lambda)$ implies that there is a corresponding
compact neighbourhood $C_{\Lambda}\subset\Lambda\backslash G$ of the trivial
coset $\Lambda e$ such that for any $p\in P$, it holds that $\Gamma gp^{-1}\in
C_{\Gamma}\Rightarrow\Lambda gp^{-1}\in C_{\Lambda}$ (simply take
$C_{\Lambda}$ to be the $D+1$-blowup of $C_{\Gamma}$). The action of $P$ on
$\Gamma\backslash G$ is uniquely ergodic, therefore
$0<\mu_{\Gamma}(C_{\Gamma})=\lim_{n}\frac{1}{|F_{n}|}\int_{F_{n}}{\mathbbm{1}_{C_{\Gamma}}(\Gamma
p^{-1})dp}$
where $\mu_{\Gamma}$ denotes the natural $G$-invariant measure on
$\Gamma\backslash G$. The defining property of $C_{\Lambda}$ ensures that
Inequality (7) is satisfied, implying that $\mu$ is a non-zero $P$-invariant
probability measure on $\Lambda\backslash G$. I conclude that $\mu$ is also
$G$-invariant, and that $\Lambda$ is a lattice. If moreover
$\Lambda\subset\mathcal{N}_{D}(\Gamma)$, one may use Shah’s Corollary [55] to
conclude that $\Lambda$ is commensurable to $\Gamma$. ∎
### 6.4 Translating Geometry into Algebra
The goal of this section is to prove that the results of Section 6.2 imply
that $\Lambda$ satisfies the hypotheses of the Benoist-Miquel criterion
Theorem 6.4. Namely, that $\Lambda$ is Zariski dense, and that it intersects a
horospherical subgroup in a cocompact indecomposable lattice. These are
algebraic properties, and the proof that $\Lambda$ satisfies them is in
essence just a translation of the geometric results of Section 6.2 to an
algebraic language. The geometric data given by Section 6.2 is that for some
horosphere $\mathcal{H}$ bounding a $\Lambda$-free horoball,
$\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H})\cdot x_{0}$ intersects
$\mathcal{H}$ in a cocompact metric lattice (Proposition 6.11), and that the
set of $\Lambda$-conical limit points contains the set of $\Gamma$-conical
limit points (Corollary 6.26). Note that since $K$ is compact the former
implies that $\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H})$ is a uniform lattice
in $\mathrm{Stab}_{G}(\mathcal{H})$.
#### 6.4.1 A Horospherical Lattice
I assume that $\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H})$ is a lattice in
$\mathrm{Stab}_{G}\mathcal{H}$, and I want to show that $\Lambda$ intersects a
horospherical subgroup $U$ of $G$ in a lattice. This step requires quite a bit
of algebraic background, which I give below in full. In short, the first goal
is to show that $\mathrm{Stab}_{G}(\mathcal{H})$ admits a subgroup
$U\leq\mathrm{Stab}_{G}(\mathcal{H})$ that is a horospherical subgroup of $G$.
A lemma of Mostow (Lemma 6.32 below) allows to conclude that $\Lambda$
intersects $U$ in a lattice.
###### Lemma 6.32 (Lemma $3.9$ in [39]).
Let H be a Lie group having no compact connected normal semisimple non-trivial
Lie subgroups, and let $N$ be the maximal connected nilpotent normal Lie
subgroup of $H$. Let $\Gamma\leq H$ be a lattice. Then $N/N\cap\Gamma$ is
compact.
###### Remark 6.33.
In the original statement Mostow uses the term ‘analytic group’, which I
replaced here with ‘connected Lie subgroup’. This appears to be Mostow’s
definition of an analytic group. See e.g. Section $10$, Chapter $1$ in [32].
In Chevalley’s _Theory of Lie Groups_ , he defines a _Lie group_ as a locally
connected topological group whose identity component is an analytic group
(Definition $1$, Section $8$, Chapter $4$ in [12]), and proves (Theorem $1$,
Section $4$, Chapter $4$ therein) a 1-1 correspondence between analytic
subgroups of an analytic group and Lie subalgebras of the corresponding Lie
algebra.
Lemma 6.32 lays the rationale for the rest of this section. Explicitly, I
prove that $\mathrm{Stab}_{G}(\mathcal{H})$ admits a subgroup that is a
horospherical subgroup $U$ of $G$ (Corollary 6.36), and that $U$ is maximal
connected nilpotent normal Lie subgroup of $\mathrm{Stab}_{G}(\mathcal{H})$
(Corollary 6.42).
In order to use Lemma 6.32, I show that the horospherical subgroup $N_{\xi}$
is a maximal normal nilpotent connected Lie subgroup of
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$, and that
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ admits no compact normal factors.
This requires to establish the structure of
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$.
###### Definition 6.34.
In the notation $h_{\xi}^{t}=\exp(tX)$ and
$A_{\xi}=\exp\big{(}Z(X)\cap\mathfrak{p}\big{)}$ of Proposition 2.8, define
$A_{\xi}^{\perp}$ to be the codimension-$1$ submanifold of $A_{\xi}$ that is
orthogonal to $\\{h_{\xi}(t)\\}_{t\in\mathbb{R}}$ (with respect to the Killing
form in the Lie algebra).
###### Claim 6.35.
Every element $a\in A_{\xi}^{\perp}$ stabilizes
$\mathcal{H}=\mathcal{H}(x_{0},\xi)$.
###### Proof.
An element in $A_{\xi}$ is an element that maps $x_{0}$ to a point on a flat
$F\subset X$ that contains the geodesic ray $[x_{0},\xi)$. If $a\in
A_{\xi}^{\perp}$, then the geodesic $[x_{0},ax_{0}]$ is orthogonal to
$[x_{0},\xi)$, and lies in $F$. From Euclidean geometry and structure of
horospheres in Euclidean spaces, it is clear that
$ax_{0}\in\mathcal{H}(x_{0},\xi)$. Since $a\in G_{\xi}$, this means
$a\mathcal{H}=\mathcal{H}(ax_{0},\xi)=\mathcal{H}(x_{0},\xi)=\mathcal{H}$. ∎
###### Corollary 6.36.
Let $\mathcal{H}$ be a horosphere based at $\xi$. Then
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}=(K_{\xi}A_{\xi}^{\perp})^{\circ}N_{\xi}$,
and in particular it contains a horospherical subgroup of $G$. Moreover,
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ is normal in
$\mathrm{Stab}_{G}(\xi)^{\circ}$ and acts transitively on $\mathcal{H}$.
###### Proof.
Clearly $(K_{\xi}A_{\xi}^{\perp})^{\circ}N_{\xi}$ is a codimension-$1$
subgroup of $\mathrm{Stab}_{G}(\xi)^{\circ}$. Since
$\mathrm{Stab}_{G}(\mathcal{H})\neq\mathrm{Stab}_{G}(\xi)$ (e.g.
$h_{\xi}^{t}\notin\mathrm{Stab}_{G}(\mathcal{H})$ for $t\neq 0$), it is enough
to show that
$(K_{\xi}A_{\xi}^{\perp})^{\circ}N_{\xi}\leq\mathrm{Stab}_{G}(\mathcal{H})$.
Let $kan\in(K_{\xi}A_{\xi}^{\perp})^{\circ}N_{\xi}$. It fixes $\xi$, so it is
enough to show that $kanx_{0}\in\mathcal{H}$. Since $k\in K_{\xi}$ and
$kx_{0}=x_{0}$, it stabilizes $\mathcal{H}$. From Claim 6.35
$a\in\mathrm{Stab}_{G}(\mathcal{H})$. So it remains to check that $N_{\xi}$
stabilizes $\mathcal{H}$, but this is more or less the definition: fixing a
base point $x_{0}$, the horospheres based at $\xi$ are parameterized by
$\mathbb{R}$. Denote them by $\\{\mathcal{H}_{t}\\}_{t\in\mathbb{R}}$, where
$\mathcal{H}=\mathcal{H}_{0}$. In this parameterization, any element $g\in
G_{\xi}$ acts on $\\{\mathcal{H}_{t}\\}_{t\in\mathbb{R}}$ by translation. I
can thus define for $g\in\mathrm{Stab}_{G}(\xi)$ the real number $l(g)$ to be
that number for which $g\mathcal{H}_{t}=\mathcal{H}_{t+l(g)}$. Clearly
$l\big{(}h_{\xi}(t)\big{)}=t$. The element $n$ fixes $\xi$, so one has
$h_{\xi}^{-t}nh_{\xi}^{t}\mathcal{H}_{0}=h_{\xi}^{-t}\mathcal{H}_{t+l(n)}=\mathcal{H}_{t+l(n)-t}=\mathcal{H}_{l(n)}$
The fact that $n\in Ker(T_{\xi})$, i.e. that
$\lim_{t\rightarrow\infty}h_{\xi}^{-t}nh_{\xi}^{t}=e_{G}$ readily implies that
necessarily $l(n)=0$. I conclude that
$(K_{\xi}A_{\xi}^{\perp})^{\circ}N_{\xi}=\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$,
as wanted.
Next recall that $\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ acts transitively on
$X$. Let $x,y\in\mathcal{H}$, and consider
$g\in\mathrm{Stab}_{G}(\xi)^{\circ}$ with $gx=y$. Writing an element $g\in
G_{\xi}$ as $ka_{t}a_{\perp}n\in K_{\xi}h_{\xi}^{t}A_{\xi}^{\perp}N_{\xi}$,
the argument above shows that
$kh_{\xi}^{t}a_{\perp}n\mathcal{H}_{0}=\mathcal{H}_{0}$ if and only if $t=0$,
i.e., if and only if $g\in\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$.
Finally, let $g\in\mathrm{Stab}_{G}(\xi)$ and
$h\in\mathrm{Stab}_{G}(\mathcal{H})$. By the discussion above
$h\cdot\mathcal{H}_{t}=\mathcal{H}_{t}$ for all $t\in\mathbb{R}$. Clearly
$-l(g)=l(g^{-1})$, and therefore
$ghg^{-1}\mathcal{H}_{0}=gh\mathcal{H}_{-l(g)}=g\cdot\mathcal{H}_{-l(g)}=\mathcal{H}_{0}$
Therefore $\mathrm{Stab}_{G}(\mathcal{H})$ is normal in
$\mathrm{Stab}_{G}{\xi}$, and the same is true for the respective identity
components. ∎
###### Corollary 6.37.
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ is a connected Lie group with no
connected compact normal semisimple non-trivial Lie subgroups.
###### Proof.
Every compact subgroup of $G$ fixes a point. Let $H\leq G$ be some closed
subgroup. It is standard to note that a normal $N\leq H$ that fixes a point
$x\in X$ must fix every point in the orbit $H\cdot x$: $hnh^{-1}hx=hx$. Since
$H=\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ acts transitively on $\mathcal{H}$,
it shows that a normal compact subgroup of
$\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$ fixes every point in $\mathcal{H}$.
An isometry fixing a horosphere pointwise while fixing its base point is
clearly the identity, proving the claim. ∎
The following fact is well known but I could not find it in the literature.
###### Corollary 6.38.
A horosphere in $X$ is not convex.
###### Proof.
Let $\mathcal{H}^{\prime}$ be some horosphere in $X$, with base point
$\zeta\in X(\infty)$, and assume towards contradiction that it is convex. Fix
$x\in\mathcal{H}^{\prime}$ and $a_{t}^{\prime}$ the one parameter subgroup
with $\eta^{\prime}(\infty)=a_{t}^{\prime}x$, and denote
$\mathcal{H}^{\prime}_{t}=\mathcal{H}(a_{t}^{\prime}x,\zeta)$. Let $e_{G}\neq
n\in N_{\zeta}$ ($N_{\zeta}$ defined with respect to $a_{t}^{\prime}$ in a
corresponding Langlands decomposition), and consider the curve
$\eta_{n}^{\prime}(t):=a_{t}^{\prime}nx$. I claim that this is a geodesic. On
the one hand, the fact that $\mathcal{H}^{\prime}$ is convex implies that the
geodesic segment $[x,nx]$ is contained in $\mathcal{H}^{\prime}$. Therefore
$a_{t}^{\prime}[x,nx]=[a_{t}^{\prime}x,a_{t}^{\prime}nx]\subset\mathcal{H}^{\prime}_{t}$.
More generally it is clear that because
$a_{t}^{\prime}\mathcal{H}^{\prime}_{s}=\mathcal{H}^{\prime}_{s+t}$ it holds
that $\mathcal{H}^{\prime}_{t}$ is convex for every $t$ as soon as it is
convex for some $t$.
On the other hand, for every point $y\in[x,nx]$,
$d(y,\mathcal{H}^{\prime}_{t})=t$, and more generally for any
$y\in[a_{s}^{\prime}nx,a_{s}^{\prime}x]$ it holds that
$d(y,\mathcal{H}_{t}^{\prime})=|s-t|$. In particular this is true for
$\eta_{n}^{\prime}(t)=y_{t}:=a_{t}^{\prime}nx$. I get that
$d\big{(}\eta_{n}^{\prime}(t),\eta_{n}^{\prime}(s)\big{)}=|s-t|$. Therefore
$\eta_{n}^{\prime}$ is a geodesic (to be pedantic one has to show that
$\eta_{n}^{\prime}$ is a continuous curve, which is a result of the fact that
$a_{t}^{\prime}$ is a one parameter subgroup of isometries). Clearly
$d\big{(}\eta_{n}^{\prime}(t),\eta^{\prime}(t)\big{)}=d(a_{t}^{\prime}nx,a_{t}^{\prime}x)=d(nx,x)$
and therefore $\eta_{n}^{\prime}$ is at uniformly bounded distance to
$\eta^{\prime}$. This bounds $d(\eta_{n},\eta_{n}^{\prime})$ as bi-infinite
geodesics, i.e. for all $t\in\mathbb{R}$, not just as infinite rays. The Flat
Strip Theorem (Theorem $2.13$, Chapter $2.2$ in [11]), then implies that the
geodesics $\eta_{n},\eta_{n}^{\prime}$ bound a flat strip: an isometric copy
of $\mathbb{R}\times[0,l]$ (where $l=d(x,nx)$).
Up to now I did not use the fact that $n\in N_{\xi}$, only that the point $nx$
lies on a geodesic that is contained in
$\mathcal{H}^{\prime}=\mathcal{H}^{\prime}_{0}$. Therefore the entire bi-
infinite geodesic that is determined by $[x,nx]$ lies on a $2$-dimensional
flat $F$ that contains $\eta^{\prime}$. The two elements $n,a_{t}^{\prime}$
therefore admit $nx,a_{t}^{\prime}x\in F$. It is a fact that two such elements
must commute. I can conclude therefore that $[n,a_{t}^{\prime}]=e_{G}$, which
contradicts the fact that that $n\in N_{\zeta}=Ker(T_{\zeta})$. ∎
###### Lemma 6.39 (Theorem $11.13$ in [52]).
Let $N$ be a connected real Lie group. Then $\mathrm{Lie}(N)$ is a nilpotent
Lie algebra if and only if $N$ is a nilpotent Lie group.
###### Proposition 6.40 (Proposition $13$, Section $4$, Chapter $1$ in [7]).
In the notation of Proposition 2.8, $\mathfrak{n}_{\xi}=\mathrm{Lie}(N_{\xi})$
is a maximal nilpotent ideal in $\mathfrak{g}_{\xi}=\mathrm{Lie}(G_{\xi})$.
###### Remark 6.41.
1. 1.
The presentation of $\mathfrak{n}_{\xi}$ in [8] is given by means of the root
space decomposition of $\mathrm{Stab}_{G}(\xi)$, that appears in Proposition
$2.17.13$ in [19].
2. 2.
There are two main objects in the literature that are referred to as the
_nilpotent radical_ or the _nilradical_ of a Lie algebra. These are: (a) the
maximal nilpotent ideal of the Lie algebra, and (b) the intersection of the
kernels of all irreducible finite-dimensional representations. Proposition
$13$ in Section $4$ of Chapter $9$ in [7] shows that in the case of Lie
algebras of parabolic Lie groups, these notions coincide.
###### Corollary 6.42.
$N_{\xi}$ is a maximal connected nilpotent normal Lie subgroup of the identity
connected component $\mathrm{Stab}_{G}(\mathcal{H})^{\circ}$.
###### Proof.
Lemma 6.39 implies $N_{\xi}$ is nilpotent. Since
$\mathrm{Stab}_{G}{\mathcal{H}}\vartriangleleft\mathrm{Stab}_{G}(\xi)$, every
normal subgroup of $\mathrm{Stab}_{G}(\mathcal{H})$ containing $N_{\xi}$ is in
fact a normal subgroup of $\mathrm{Stab}_{G}(\xi)$, still containing
$N_{\xi}$. It remains to prove maximality of $N_{\xi}$ among all connected
nilpotent normal Lie subgroups of $\mathrm{Stab}_{G}(\xi)$. Any such subgroup
$N^{\prime}\vartriangleleft\mathrm{Stab}_{G}(\xi)$ gives rise to an ideal
$\mathfrak{n}^{\prime}$ of
$\mathfrak{g}_{\xi}=\mathrm{Lie}\big{(}\mathrm{Stab}_{G}(\xi)\big{)}$, and by
Lemma 6.39 it is a nilpotent ideal. Therefore by Proposition 6.40 it is
contained in $\mathfrak{n}_{\xi}=\mathrm{Lie}(N_{\xi})$, implying that
$N^{\prime}\leq N_{\xi}$.
∎
###### Corollary 6.43.
A lattice in $\mathrm{Stab}_{G}(\mathcal{H})$ intersects the horospherical
subgroup $N_{\xi}$ in a lattice.
###### Proof.
Corollaries 6.37 and 6.42 imply that the pair
$N_{\xi}\vartriangleleft\mathrm{Stab}_{G}(\mathcal{H})$ satisfy the hypotheses
of Mostow’s Lemma 6.32. ∎
#### 6.4.2 Indecomposable Horospherical Lattices
It is shown in [5] that if a horospherical lattice is contained in a Zariski
dense discrete subgroup, then the indecomposability condition is equivalent to
the irreducibility of the ambient group. The latter is imposed on $\Lambda$ as
a hypothesis in Theorem 6.1. The precise definitions and statements are as
follows.
###### Definition 6.44 (Definition $2.14$ in [5]).
For a semisimple real algebraic Lie group $G$ and $U$ a horospherical subgroup
of $G$, let $\Delta_{U}$ be a lattice in $U$.
1. 1.
$\Delta_{U}$ is _irreducible_ if for any proper normal subgroup $N$ of
$G^{\circ}$, one has $\Delta_{U}\cap N=\\{e\\}$.
2. 2.
$\Delta_{U}$ is _indecomposable_ if one cannot write $G^{\circ}$ as a product
$G^{\circ}=N^{\prime}N^{\prime\prime}$ of two proper normal subgroups
$N^{\prime},N^{\prime\prime}\vartriangleleft G$ with finite intersection such
that the group
$\Delta_{U}^{\prime}:=(\Delta_{U}\cap N^{\prime})(\Delta_{U}\cap
N^{\prime\prime})$
has finite index in $\Delta_{U}$.
###### Definition 6.45 (See Section $2.4.1$ in [5]).
Let $G$ be a semisimple real algebraic Lie group. A discrete subgroup
$\Lambda\leq G$ is said to be _irreducible_ if, for all proper normal
subgroups $N\vartriangleleft G$, the intersection $\Lambda\cap N$ is finite.
###### Lemma 6.46 (Lemma $4.3$ in [5]).
Let $G$ be a semisimple real algebraic Lie group, $U\subset G$ a non-trivial
horospherical subgroup, and $\Delta_{U}\leq U$ a lattice of $U$ which is
contained in a discrete Zariski dense subgroup $\Delta$ of $G$. Then the
following are equivalent:
1. 1.
$\Delta$ is irreducible.
2. 2.
$\Delta_{U}$ is irreducible.
3. 3.
$\Delta_{U}$ is indecomposable.
#### 6.4.3 Zariski Density
The last requirement is for $\Lambda$ to be Zariski dense. I use a geometric
criterion which is well known to experts.
###### Lemma 6.47 (Proposition $2$ in [30]).
Let $X$ be a symmetric space of noncompact type, $G=\mathrm{Isom}(X)^{\circ}$.
A subgroup $\Delta\leq G$ is Zariski dense if and only if:
1. 1.
The group $\Delta$ does not globally fix a point in $X(\infty)$, i.e.
$\Delta\not\leq\mathrm{Stab}_{G}(\zeta)$ for any $\zeta\in X(\infty)$.
2. 2.
The identity component of the Zariski closure of $\Delta$ does not leave
invariant any proper totally geodesic submanifold in $X$.
In the proof I use several facts - mostly algebraic, and two geometric. I
warmly thank Elyasheev Leibtag for his help and erudition in algebraic groups.
The first property I need is very basic.
###### Lemma 6.48.
Let $\Delta\leq G$ be a discrete subgroup, and let $H\leq G$ be the Zariski
closure of $\Delta$. Then $\Delta\cap H^{\circ}$ is of finite index in
$\Delta$.
###### Proof.
The subgroup $H^{\circ}$ is normal and of finite index in $H$. ∎
The following fact is probably known to experts. It appears in a recent work
by Bader and Leibtag [2].
###### Lemma 6.49 (Lemma $3.9$ in [2]).
Let $k$ be a field, $\mathbf{G}$ a connected $k$ algebraic group, $P\leq
G=\mathbf{G}(\mathbb{R})$ a parabolic subgroup. Then the centre of $G$
contains the centre of $P$.
Still on the algebraic side, I need a Theorem of Dani, generalizing the Borel
Density Theorem.
###### Theorem 6.50 (See [14]).
Let $\mathbf{S}$ be a real solvable algebraic group. If
$S=\mathbf{S}(\mathbb{R})$ is $\mathbb{R}$-split, then every lattice
$\Gamma_{S}\leq S$ is Zariski dense.
###### Remark 6.51.
It is a fact (see Theorem $15.4$ and Section $18$ in [6]) that:
1. 1.
Every unipotent group over $\mathbb{R}$ is $\mathbb{R}$-split.
2. 2.
For a field $k$ of characteristic $0$, a solvable linear algebraic $k$-group
is $k$-split if and only if its maximal torus is $k$-split.
Finally I need two geometric facts. The first is a characterization
determining when does a unipotent element belongs to $N_{\zeta}$ for some
$\zeta\in X(\infty)$.
###### Proposition 6.52 (Proposition $4.1.8$ in [19]).
Let $X$ be a symmetric space of noncompact type and of higher rank, $n\in
G=\mathrm{Isom}(X)^{\circ}$ a unipotent element, and $\zeta\in X(\infty)$. The
following are equivalent:
1. 1.
For $N_{\zeta}$ as in Proposition 2.8, $n\in N_{\zeta}$.
2. 2.
For some geodesic ray $\eta$ with $\eta(\infty)=\zeta$ it holds that
$\lim_{t\rightarrow\infty}d\big{(}n\eta(t),\eta(t)\big{)}=0$.
3. 3.
For every geodesic ray $\eta$ with $\eta(\infty)=\zeta$ it holds that
$\lim_{t\rightarrow\infty}d\big{(}n\eta(t),\eta(t)\big{)}=0$.
The last property I need is a characterization of the displacement function
for unipotent elements.
###### Proposition 6.53 (See proof of Proposition $3.4$ in [4]).
Let $X$ be a symmetric space of noncompact type, $\zeta\in X(\infty)$ some
point and $n\in N_{\zeta}$ an element of the unipotent radical of
$\mathrm{Stab}_{G}(\zeta)$. The displacement function $x\mapsto d(nx,x)$ is
constant on horospheres based at $\zeta$, and for every $\varepsilon>0$ there
is a horoball $\mathcal{HB}_{\varepsilon}$ based at $\zeta$ such that
$d(nx,x)<\varepsilon$ for every $x\in\mathcal{HB}_{\varepsilon}$.
###### Corollary 6.54.
Assume that:
1. 1.
$\big{(}\Lambda\cap\mathrm{Stab}_{G}(\mathcal{H})\big{)}\cdot x_{0}$ is a
cocompact metric lattice in a horosphere $\mathcal{H}\subset X$ bounding a
$\Lambda$-free horoball.
2. 2.
Every $\Gamma$-conical limit point is a $\Lambda$-conical limit point.
Then $\Lambda$ is Zariski dense.
###### Proof.
I show the criteria of Lemma 6.47 are met, starting with
$\Lambda\not\leq\mathrm{Stab}_{G}(\zeta)$ for any $\zeta\in X(\infty)$. To
this end, I first prove that $\Lambda\cdot x_{0}$ is not contained in any
bounded neighbourhood of any horosphere $\mathcal{H}^{\prime}$. Let
$\xi^{\prime}$ be the base point of $\mathcal{H}^{\prime}$. By Hattori’s Lemma
2.32 (and Remark 2.33), it is enough to find a $\Lambda$-conical limit point
$\zeta^{\prime}$ with $d_{T}(\xi^{\prime},\zeta^{\prime})\neq\frac{\pi}{2}$.
Take some $\zeta^{\prime\prime}\in X(\infty)$ at Tits distance $\pi$ of
$\xi^{\prime}$, i.e. take a flat $F$ on which $\xi^{\prime}$ lies and let
$\zeta^{\prime\prime}$ be the antipodal point to $\xi^{\prime}$ in $F$. Fix
$\varepsilon=\frac{\pi}{4}$. By Proposition 2.4, there are neighbourhoods of
the cone topology $U,V\subset X(\infty)$ of
$\xi^{\prime},\zeta^{\prime\prime}$ (respectively) so that every point
$\zeta^{\prime}\in V$ admits $d_{T}(\xi^{\prime},\zeta^{\prime})\geq
d_{T}(\xi^{\prime},\zeta^{\prime\prime})-\frac{\pi}{4}=\frac{3}{4}\pi$. Recall
that the set of $\Gamma$-conical limit points is dense (in the cone topology),
so the second hypothesis implies there is indeed a $\Lambda$-conical limit
point in $V$ and therefore at Tits distance different (in this case larger)
than $\frac{\pi}{2}$ from $\xi^{\prime}$. I conclude that $\Lambda\cdot x_{0}$
is not contained in any bounded metric neighbourhood of any horosphere of $X$.
Assume towards contradiction that $\Lambda\leq\mathrm{Stab}_{G}(\zeta)$. I
show that this forces $\Lambda\cap N_{\zeta}\neq\emptyset$. By Proposition
6.52 it is enough to find a unipotent element $\lambda\in\Lambda$ and a
geodesic $\eta$ with $\eta(\infty)=\zeta$ such that
$\lim_{t\rightarrow\infty}d\big{(}\lambda\eta(t),\eta(t)\big{)}=0$. Let $F$ be
a maximal flat with $\xi,\zeta\in F(\infty)$, $x\in F$ some point and
$X,Y\in\mathfrak{a}\leq\mathfrak{p}$ two vectors such that $\exp(tY)=\eta(t)$
for the unit speed geodesic $\eta=[x,\zeta)$, and $\exp(tX)=\eta^{\prime}(t)$
for the unit speed geodesic $\eta^{\prime}=[x,\xi)$ (where
$\mathfrak{a}\leq\mathfrak{p}$ a maximal abelian subalgebra in a suitable
Cartan decomposition $\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{k}$). Let
$\mathrm{Stab}_{G}(\xi)=K_{\xi}A_{\xi}N_{\xi}$ be the decomposition described
in Proposition 2.8 with respect to $X$ (notice that $N_{\xi}$ does not depend
on choice of $X$, see item $3$ of Proposition $2.17.7$ in [19]).
The assumption that $\Lambda\leq\mathrm{Stab}_{G}(\zeta)$ implies that for any
$\lambda\in\Lambda$ the distance $d\big{(}\lambda\eta(t),\eta(t)\big{)}$
either tends to $0$ as $t\rightarrow\infty$ or is uniformly bounded for
$t\in\mathbb{R}$. In the latter case there is some constant $c>0$ for which
$d\big{(}\lambda\eta(t),\eta(t)\big{)}=c$ for all $t\in\mathbb{R}$. As in the
proof of Corollary 6.38, the Flat Strip Theorem (Theorem $2.13$, Chapter $2.2$
in [11]) implies that $\lambda$ and $a_{t}:=\exp(tY)$ commute.
From the first hypothesis of the statement and Mostow’s result (Corollary
6.43) I know that $\Lambda\cap N_{\xi}$ is a cocompact lattice in $N_{\xi}$
(attention to subscripts). Therefore $\Lambda\cap N_{\xi}$ is Zariski dense in
$N_{\xi}$ (Theorem 6.50). Moreover, since commuting with an element is an
algebraic property, an element $g\in G$ that commutes with $\Lambda\cap
N_{\xi}$ must also commute with its Zariski closure, namely with $N_{\xi}$.
This means that if $a_{t}$ commutes with all $\Lambda\cap N_{\xi}$ then it
commutes with $N_{\xi}$, i.e. $a_{t}n=na_{t}$ for all $t\in\mathbb{R}$ and all
$n\in N_{\xi}$. I know that $a_{t}\in A_{\xi}$ commutes with both $K_{\xi}$
and $A_{\xi}$ (Proposition 2.8) therefore if $a_{t}$ also commutes with
$N_{\xi}$ then $a_{t}$ lies in the centre of $\mathrm{Stab}_{G}(\xi)$. This
means that $a_{t}$ is central in $G$ (Lemma 6.49). For a group $G$ with
compact centre this cannot happen, so there is indeed some unipotent element
$\lambda\in\Lambda\cap N_{\xi}$ for which
$\lim_{t\rightarrow\infty}d\big{(}\lambda\eta(t),\eta(t)\big{)}=0$. I conclude
from Proposition 6.52 that $\Lambda\cap N_{\zeta}\neq\emptyset$.
The first paragraph of the proof implies in particular that $\Lambda\cdot
x_{0}$ does not lie in any bounded neighbourhood of a horosphere
$\mathcal{H}^{\prime}$ based at $\zeta$. The assumption that
$\Lambda\subset\mathrm{Stab}_{G}(\zeta)$ implies that every
$\lambda\in\Lambda$ acts by translation on the filtration
$\\{\mathcal{H}^{\prime}_{t}\\}_{t\in\mathbb{R}}$ by horospheres based at
$\zeta$. Therefore as soon as $\Lambda\cdot x_{0}\not\subset\mathcal{H}_{t}$
for some $t\in\mathbb{R}$ one concludes that $\zeta$ is a horospherical limit
point of $\Lambda$, i.e. that every horoball based at $\zeta$ intersects the
orbit $\Lambda\cdot x_{0}$.
By Proposition 6.53 it holds that for a unipotent element $g\in N_{\zeta}$ the
displacement function $x\mapsto d(gx,x)$ depends only on the horosphere
$\mathcal{H}^{\prime}_{t}$ in which $x$ lies and that, for
$x_{t}\in\mathcal{H}^{\prime}_{t}$ it holds that
$\lim_{t\rightarrow\infty}d(gx_{t},x_{t})=0$ (up to reorienting the filtration
$t\in\mathbb{R}$ so that $\eta(t)\in\mathcal{H}^{\prime}_{t})$. For a non-
trivial element $\lambda_{\zeta}\in\Lambda\cap N_{\zeta}$ the previous
paragraph therefore yields a sequence of elements $\lambda_{n}\in\Lambda$ such
that
$\lim_{n\rightarrow\infty}d(\lambda_{\zeta}\lambda_{n}x_{0},\lambda_{n}x_{0})=0$,
contradicting the discreteness of $\Lambda$. I conclude that
$\Lambda\not\leq\mathrm{Stab}_{G}(\zeta)$ for every $\zeta\in X(\infty)$.
Assume that $H:=\big{(}\overline{\Lambda}^{Z}\big{)}^{\circ}$, the identity
connected component of the Zariski closure of $\Lambda$, stabilizes a totally
geodesic submanifold $Y\subset X$. By Lemma 6.48, $\Lambda_{0}:=\Lambda\cap H$
is of finite index in $\Lambda$, therefore
$\Lambda_{0}\cap\mathrm{Stab}_{G}(\mathcal{H})$ is also a cocompact lattice in
$\mathrm{Stab}_{G}(\mathcal{H})$. The fact that
$\big{(}\Lambda_{0}\cap\mathrm{Stab}_{G}(\mathcal{H})\big{)}\cdot x_{0}$ is a
cocompact metric lattice in $\mathcal{H}$ readily implies that
$\big{(}\Lambda_{0}\cap\mathrm{Stab}_{G}(\mathcal{H})\big{)}\cdot y$ is a
cocompact metric lattice in $\mathcal{H}_{y}=\mathcal{H}(y,\xi)$. This goes to
show that there is no loss of generality in assuming $x_{0}\in\mathcal{H}\cap
Y$. It follows that $\Lambda_{0}\cap\mathrm{Stab}_{G}(\mathcal{H})\cdot
x_{0}\subset Y\cap\mathcal{H}$, and therefore
$\mathcal{H}\subset\mathcal{N}_{D}(Y)$ for some $D>0$. A horosphere is a
codimension-$1$ submanifold, implying that $Y$ is either all of $X$ or of
codimension-$1$. The latter forces $Y=\mathcal{H}$, which is impossible since
$\mathcal{H}$ is not totally geodesic ($\mathcal{H}$ is not convex, see
Corollary 6.38). I conclude that $H$ does not stabilize any totally geodesic
proper submanifold, and hence that $\Lambda$ is Zariski dense. ∎
### 6.5 Proof of Theorem 6.1
I now complete the proof of the main sublinear rigidity theorem for
$\mathbb{Q}$-rank $1$ lattices.
###### Proof of Theorem 6.1.
If $\\{d_{\gamma}\\}_{\gamma\in\Gamma}$ is bounded, then $\Lambda$ is a
lattice by Corollary 6.6 or Theorem 6.8, depending on the $\mathbb{R}$-rank of
$G$.
If $\\{d_{\gamma}\\}_{\gamma\in\Gamma}$ is unbounded, then Proposition 6.11
and Corollary 6.26 both hold. In $\mathbb{R}$-rank$\ 1$ the proof again
follows immediately from Corollary 6.6 using Lemma 2.17 and Corollary 6.26.
In higher rank, the results of Section 6.4 allows one to conclude that
$\Lambda$ is an irreducible, discrete, Zariski dense subgroup that contains a
horospherical lattice. By Theorem 6.4, this renders $\Lambda$ a lattice. It is
a $\mathbb{Q}$-rank $1$ lattice as a result of Theorem 2.21. ∎
###### Remark 6.55.
The sublinear nature of the hypothesis in Theorem 1.6 induces coarse metric
constraints. A horospherical lattice on the other hand is a very precise
object. It is not clear how to produce unipotent elements in $\Lambda$, or
even general elements that preserve some horosphere. The proof above produces
a whole lattice of unipotent elements in $\Lambda$ (this is Corollary 6.43);
it is also the only proof that I know which produces even a single unipotent
element in $\Lambda$.
## References
* [1] Paul Albuquerque. Patterson-Sullivan theory in higher rank symmetric spaces. GAFA Geom. Funct. Anal., 9(1):1–28, March 1999.
* [2] U. Bader and E. Leibtag. Homomorphic images of algebraic groups. arXiv preprint arXiv:2212.03055, 2022.
* [3] W. Ballmann, M. Gromov, and V. Schroeder. Manifolds of Nonpositive Curvature. Birkhäuser Boston, 1985.
* [4] Werner Ballmann. Lectures on spaces of nonpositive curvature, volume 25. Springer Science & Business Media, 1995.
* [5] Yves Benoist and Sébastien Miquel. Arithmeticity of discrete subgroups containing horospherical lattices. Duke Mathematical Journal, 169(8):1485 – 1539, 2020.
* [6] Armand Borel. Linear Algebraic Groups. Springer New York, 1991.
* [7] N. Bourbaki. Lie Groups and Lie Algebras: Chapters 1-3. Bourbaki, Nicolas: Elements of mathematics. Springer-Verlag, 1989.
* [8] N. Bourbaki. Lie Groups and Lie Algebras: Chapters 7-9. Number 7-9 in Elements of mathematics. Springer Berlin Heidelberg, 2004\.
* [9] Brian H. Bowditch. Geometrical finiteness for hyperbolic groups. Journal of Functional Analysis, 113(2):245–317, 1993.
* [10] Brian H Bowditch. Geometrical finiteness with variable negative curvature. Duke Mathematical Journal, 77(1):229–274, 1995.
* [11] Martin R Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319. Springer Science & Business Media, 2013.
* [12] Claude Chevalley. Theory of Lie Groups. Princeton University Press, 1946.
* [13] Yves Cornulier. On sublinear bilipschitz equivalence of groups. Ann. ENS, 52:1201–1242, 2019.
* [14] SG Dani. On ergodic quasi-invariant measures of group automorphism. Israel Journal of Mathematics, 43(1):62–74, 1982.
* [15] Cornelia Druţu. Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank. Geometric and Functional Analysis - GAFA, 10, 06 2000.
* [16] Cornelia Druţu and Michael Kapovich. Geometric group theory. American Mathematical Society, 2017.
* [17] Cornelia Druţu and Mark Sapir. Tree-graded spaces and asymptotic cones of groups. Topology, 44(5):959–1058, 2005.
* [18] Patrick Eberlein. Lattices in spaces of nonpositive curvature. Annals of Mathematics, 111(3):435–476, 1980.
* [19] Patrick Eberlein. Geometry of Nonpositively Curved Manifolds. Chicago Lectures in Mathematics. University of Chicago Press, 1996.
* [20] A. Eskin and B. Farb. Quasi-flats and rigidity in higher rank symmetric spaces. Journal of the American Mathematical Society, 10:653–692, 1997\.
* [21] Alex Eskin. Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces. Journal of the American Mathematical Society, 11(2):321–361, 1998\.
* [22] Benson Farb. The quasi-isometry classification of lattices in semisimple lie groups. Mathematical Research Letters, 4:705–717, 1997.
* [23] Mikolaj Fraczyk and Tsachik Gelander. Infinite volume and infinite injectivity radius. Annals of Mathematics, 197(1):389–421, 2023.
* [24] Toshiaki Hattori. Geometric Limit Sets of Higher Rank Lattices. Proceedings of the London Mathematical Society, 90(3):689–710, 05 2005.
* [25] Ernst Heintze and Hans-Christoph Im Hof. Geometry of horospheres. Journal of Differential Geometry, 12(4):481 – 491, 1977.
* [26] Sigurdur Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. ISSN. Elsevier Science, 1979.
* [27] Sigurdur Helgason. Differential geometry and symmetric spaces, volume 341. American Mathematical Soc., 2001.
* [28] Lizhen Ji. From symmetric spaces to buildings, curve complexes and outer spaces. Innovations in Incidence Geometry, 10(none):33 – 80, 2009.
* [29] M. Kapovich and B. Liu. Geometric finiteness in negatively pinched hadamard manifolds. Annales Academiae Scientiarum Fennicae Mathematica, 2019.
* [30] Inkang Kim. Rigidity on symmetric spaces. Topology, 43(2):393–405, 2004.
* [31] B. Kleiner and B. Leeb. Rigidity of quasi-isometries for symmetric spaces and euclidean buildings. Publ. Math. IHES, 86:115–197, 1997.
* [32] Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics. Birkhäuser Boston, 2002.
* [33] Enrico Leuzinger. An exhaustion of locally symmetric spaces by compact submanifolds with corners. Inventiones Mathematicae, 121(1):389–410, December 1995.
|
# AGN feedback duty cycle in Planck SZ selected clusters using Chandra
observations
V. Olivares1,Y. Su1, P. Nulsen2,3, R. Kraft2, T. Somboonpanyakul4, F. Andrade-
Santos2, C. Jones2, W. Forman2
1 Department of Physics and Astronomy, University of Kentucky, 505 Rose
Street, Lexington, KY 40506, USA
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA
02138, USA
3 ICRAR, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009,
Australia
3 Kavli Institute for Particle Astrophysics & Cosmology, P.O. Box 2450,
Stanford University, Stanford, CA 94305, USA
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present a systematic study of X-ray cavities using archival Chandra
observations of nearby galaxy clusters selected by their Sunyaev-Zel’dovich
(SZ) signature in the Planck survey, which provides a nearly unbiased mass-
selected sample to explore the entire AGN feedback duty cycle. Based on X-ray
image analysis, we report that 30 of the 164 clusters show X-ray cavities,
which corresponds to a detection fraction of 18%. After correcting for spatial
resolution to match the high-$z$ SPT–SZ sample, the detection fraction
decreases to 9%, consistent with the high-z sample, hinting that the AGN
feedback has not evolved across almost 8 Gyrs. Our finding agrees with the
lack of evolution of cool-core clusters fraction. We calculate the cavity
power, $P_{\rm cav}$, and find that most systems of our sample have enough AGN
heating to offset the radiative losses of the intracluster medium.
###### keywords:
galaxies: clusters: general – intergalactic medium – X-rays: galaxies
††pubyear: 2022
## 1 Introduction
Feedback from active galactic nuclei (AGN) jets has been proposed to solve the
cooling flow problem (see Fabian 1994; McNamara et al. 2005, for a review).
Although the details of how AGN feedback counteracts the radiative losses of
the intracluster medium (ICM) in clusters are still not fully understood.
Early observations with the X-ray satellite ROSAT revealed surface brightness
deficits that appear to be spatially aligned with regions of radio emission in
the ICM of a few galaxy clusters (Boehringer et al., 1993; Carilli et al.,
1994). Nowadays, with the superb resolution of the X-ray Chandra observatory,
it has become clear that central AGN located in the Brightest Cluster Galaxy
(hereafter BCG) continuously interacts with the surrounding ICM, producing,
not solely, the X-ray surface brightness depressions known as X-ray cavities
(or bubbles), but also shocks and ripples (e.g., Fabian et al., 2006). In
addition, high-resolution radio observations by the Jansky Very Large Array
(JVLA) have shown that extended radio lobes inflated by the central AGN may
excavate these X-ray cavities by pushing aside the surrounding hot gas.
Accordingly, they are expected to be filled with radio emission (e.g., Bîrzan
et al., 2020). There have also been detections of the so-called “ghost
cavities” at low radio frequency, which are believed to trace a past AGN
outburst, for which the radio emission has faded away. More importantly, the
X-ray cavities and bubbles may provide a direct measurement of the work done
by the radio-mode feedback on the ICM (e.g., Gitti et al., 2010). The X-ray
cavities are not only supposed to carry enough energy to balance the cooling
losses of the X-ray emitting plasma (Bîrzan et al., 2008), but also play a key
role in the formation of extended multiphase filaments observed in cooling
flow clusters (e.g., Olivares et al., 2019, 2022; Russell et al., 2019).
Therefore, investigating the physical properties of X-ray cavities can improve
our understanding of AGN feedback and its impact on galaxy formation and
evolution.
Currently, most X-ray cavity studies of clusters, groups and elliptical
galaxies are based on Chandra X-ray observations for both individual systems
and dedicated surveys (see Bîrzan et al. 2004, 2008; Rafferty et al. 2006;
Nulsen et al. 2009; Dong et al. 2010; O’Sullivan et al. 2011; Hlavacek-
Larrondo et al. 2013; Shin et al. 2016; Panagoulia et al. 2014; Bîrzan et al.
2017). One of the main limitations of existing studies based on X-ray
selection methods is that they are often biased towards bright cool-core
systems and, consequently, against X-ray faint clusters. A complete unbiased
sample of galaxy clusters is desirable to obtain heating and cooling balance
constraints (Gitti et al., 2010), and to understand the duty cycle of AGN
feedback, which is estimated as the fraction of systems displaying bubbles
inflated by the central AGN.
Millimeter-wave surveys utilizing the SZ effect have the advantage of
providing nearly mass-limited samples, as the impact of SZ effect on the CMB
(cosmic microwave background) brightness temperature is independent of
redshift. This allows us to explore the entire AGN feedback duty cycle.
Examples of instruments used to undertake SZ surveys include the Planck
satellite (Planck Collaboration et al., 2011), the South Pole Telescope (SPT)
(Bleem et al., 2015, 2020), and the Atacama Cosmology Telescope (Hincks et
al., 2010; Hilton et al., 2018). For the high-$z$ Universe, Hlavacek-Larrondo
et al. (2015, hereafter HL15) performed a Chandra study of 83 clusters
selected from the SPT-SZ survey, and found X-ray surface brightness depression
in 6 clusters consistent with radio jets inflating X-ray cavities in their
ICM. Here, we present a study of X-ray cavities in the Planck SZ survey, which
provides a unique and unbiased view of AGN feedback in the nearby ($z<0.35$)
Universe and anchors the evolution of AGN feedback over the past 8 Gyrs.
This paper examines Chandra observations of 164 Planck SZ clusters with the
aim of identifying X-ray bubbles. In section 2 we describe the Planck SZ
sample. Section 3 presents the X-ray Chandra observations and describes the
methods used to identify X-ray surface brightness depressions. Section 4 is
devoted to the results and their implications. Section 5 presents the
limitations of the present study. Finally, section 6 summarizes our findings.
Throughout this paper, we adopted a standard cosmology with H0=70 km s-1 Mpc-1
and $\Omega_{\rm m}$=0.3.
## 2 Sample
The Chandra-Planck Legacy Program for Massive Clusters of Galaxies (Jones,
2012) is a deep X-ray survey of massive Planck clusters with redshift $\leq
0.35$ detected over almost the full sky (and $|$b$|>15\deg$) through the
Sunyaev-Zel’dovich effect by the first Planck mission released in early 2011
(Planck Collaboration et al., 2011). The observations are constructed by
combining the Chandra XVP (PI: Jones) and HRC Guaranteed Time Observations
(PI: Murray). At least 10,000 source counts have been collected for each
cluster to derive its gas properties out to $R_{\rm 500}$ (Andrade-Santos et
al., 2017, hereafter AS17). The Chandra Planck sample is nearly an unbiased,
mass-selected sample, covering the mass range $7\times 10^{13}{\rm
M}_{\odot}\leq M_{500}\leq 2\times 10^{15}{\rm M}_{\odot}$. The sample
consists of 164 clusters, of which a small fraction contain pronounced
substructures (35, subclusters), visually identified in the X-ray images
(AS17). Central density is the best known proxy for the central cooling time
(e.g., Su et al., 2020), and has been widely used to classify CC and NCC
clusters (e.g., Ruppin et al., 2021; Andrade-Santos et al., 2017). Based on
the central density classification of $n_{\rm core}=1.5\times$10-2 cm-3, as
presented in AS17, 63 clusters are classified as CC and 101 as NCC clusters.
Deprojected temperature and density profiles of clusters in this sample are
taken from AS17, to which we refer the reader for a detailed description.
## 3 Observation and Analysis
For each cluster, we used all available Chandra observations, including both
CCDs ACIS-I and ACIS-S. The data reduction and calibration of Chandra
observations were carried out using Chandra Interactive Analysis of
Observations software (CIAO) 4.12, and Chandra Calibration Database (CALDB)
4.9.2.1. The observations were reprocessed using the chandra_repro tool of
CIAO. Standard blank sky background files and readout artifacts were
subtracted. Point sources were detected in the 0.5-8.0 keV energy band, then
masked before performing the spectral and imagining analysis of the clusters.
Exposure corrected images were produced in the 0.5–2.0 keV band energy, and
used for the X-ray cavity analysis.
Unsharp masked images were produced to help identify X-ray cavities using CIAO
tool aconvolve. The original image was smoothed twice using a small- and
large-scale Gaussian kernel. The highly smoothed image was then subtracted
from the less smoothed image, enhancing the inhomogeneities in the residual
image. We tried different smoothing lengths for the more heavily smoothed
images based on the large scale of the cluster emission, starting on 10 up to
60 kpc. For the less smoothed images, we tried smoothing lengths comparable to
the physical size of a cavity, from 1 up to 20 kpc (e.g. Rafferty et al.,
2006). We also examined the residual image after subtracting an elliptical
double beta model, which was obtained by fitting a slightly smoothed 0.5–2.0
keV image. The second beta model is to account for excess emission from the
cool core.
We classified each cavity identified as Certain (C) or Potential (P). The
first two co-author independently looked for X-ray cavities and then
classified them based on the significance of each cavity.
Cavities were classified as certain if they appear as a clear visible
depression in the original image, but also the unsharp-masked image or double
$\beta$-subtracted image. In figure 1 we present an example of the methods
employed to identify cavities for a clusters with certain (C), potential (P),
and without cavities. A cavity was classified as potential if there was only a
hint of an X-ray depression in the original X-ray image, but visible in the
unsharp-masked or double $\beta$-model subtracted image. The number of counts
of the central region ($<$20 kpc) of the clusters with potential cavities is
too low for the cavities to be certain (see also Section 5). Clusters without
depressions were classified as lacking cavities. We also consider clusters
with dark annuli or rings created by bright excesses and asymmetries of the
cluster distribution as lacking cavities, as such surface brightness
depressions are not consistent with bubbles inflated by radio jets.
Figure 1: Example of the methods employed to look for X-ray cavities for a
cluster with Certain (top), Potential (middle) detected cavities, and without
(bottom) cavities. From left to right: 0.5-2.0 keV original image, double
$\beta$-subtracted image, and unsharp image. Detected cavities are displayed
with green ellipses.
Figure 2: Left panel: Detection fraction of cavities as a function of
redshift. The detection fraction for the entire sample is shown with a green
square. The detection fraction corrected by resolution to match the SPT-SZ
sample is shown with a green circle. The SPT-SZ detection fraction is
displayed with an orange circle (HL15). The grey arrows correspond to the
detection fraction when only “certain” cavities are taken into account. Right
panel: Cavity size versus redshift color-coded by the number of counts within
the central 20 kpc. We have also included cavity measurements from the
high-$z$ SPT–SZ sample (HL15), shown with black symbols. Certain (C) cavities
are displayed with circles, whereas potential (P) cavities with square
symbols. The dashed black line corresponds to two times the size of the
Chandra PSF as a function of redshift. In both panels, the upper axis gives
the lookback times in Gyr.
## 4 Results and Discussion
### 4.1 Detection fraction of cavities and evolution
Overall, we detected 67 X-ray cavities in 30 clusters, of which 32 are
classified as certain (C) and 35 as potential (P) cavities. From the CC
cluster sub-sample, we find 29 clusters with cavities, of which 12 clusters
reveal mostly certain cavities and 17 potential cavities. The remaining 34 CC
clusters lack X-ray depressions. We also find in one NCC cluster,
G269.51+26.42, two potential cavities located in opposite sides of the cluster
center. The rest of the NCC clusters show no hint of X-ray depressions. We
find that most of the detected cavities come in pairs, and they are usually
located on opposite sides of the cluster core, as expected, considering that
X-ray cavities are believed to be inflated by radio jets. It is worth
mentioning that 29/30 of clusters with cavities also have radio emission
associated with the central source (Olivares et al. in prep). Some clusters
show multiple X-ray cavities, likely due to either multiple AGN outbursts or
the disintegration of large cavities, while five clusters have single cavities
(e.g., Morsony et al., 2010; Canning et al., 2013).
In total, 18% of all clusters in our sample, including both CC and NCC
clusters contain X-ray cavities (see Fig 2, left panel), a few times smaller
than the fractions found by previous studies of nearby clusters and about
twice as high as that of the high-$z$ SPT–SZ sample (7%–9%; HL15). We have
included uncertainties associated with the fraction of clusters with cavities
using the Wilson interval method (Brown et al., 2001). We note that previous
studies of nearby clusters tend to be biased towards X-ray bright clusters.
Furthermore, our findings suggest a slightly lower duty cycle of $\sim$46%, as
28 of the 63 CC clusters ($n_{\rm core}$>1.5$\times$10-2 cm-3) have detected
cavities (see Fig. 2, left panel), compared to previous studies which predict
AGN feedback duty cycle to be high (60–90%, Bîrzan et al. 2012; Fabian 2012;
Panagoulia et al. 2014). We stress, however, that different definitions have
been used to classify cool core clusters. At high-$z$, HL15 found a lower
limit of $\sim$11% on the duty cycle for the SPT-SZ sample, as only 6 of the
52 clusters with signs of cooling reveal cavities.
To explore the evolution of the detection fraction of cavities, we compare our
results with those found in the high-$z$ SPT-SZ sample (HL15). Bear in mind
that the SPT–SZ sample is limited by resolution (see Fig 2, right panel), with
the smaller cavities detected in this sample having sizes of $\lesssim$10 kpc.
The latter is due to a combination of larger Chandra PSF at high $z$ and lower
number of counts compared to low-$z$ clusters. To account for that limitation,
we compute the detection fraction taking only clusters with cavities sizes
larger than $\gtrsim$10 kpc to match the observing bias of the SPT–SZ sample.
That yields a detection fraction of 9%, which is in good agreement with the
SPT–SZ sample (HL15). In the same vein, if we consider only clusters with
“certain” (C) cavities and sizes $\gtrsim$10 kpc, the detection fraction of
the Planck sample drops to 3%, close to the 2% obtained in the high-$z$ SPT–SZ
sample when only clearly detected cavities are taken into account. These
findings suggest that the AGN feedback duty cycle has remained constant over
almost 8 Gyrs. This trend strongly agrees with the lack of evolution in the
fraction of cool-cores clusters (of $\sim$40–60% across the same redshift
range Ruppin et al. 2021), which is linked to the ICM cooling. An absence of
evolution on the detection fraction of cavities could imply that the
mechanical feedback in CC clusters has been in place and maintained across
almost 8 Gyrs.
All the above is quite intriguing given that the AGN-hosting BCG fraction in
the SPT-SZ cluster sample, selected from infrared WISE observations, appears
to be strongly evolving with redshift (see Somboonpanyakul et al. 2022; Bîrzan
et al. 2017; Hlavacek-Larrondo et al. 2013; also Silverman et al. 2009;
Haggard et al. 2010 for related studies). The authors argue that nearby
clusters may grow by dry mergers without increasing the AGN activity. Whereas,
high-$z$ clusters may accrete cold gas from gas-rich mergers and satellite
interactions, which could drive a massive inflow of cold gas towards the
central region, increasing the AGN activity. With more fuel available at
high-$z$, the accretion rate is more likely to reach the Eddington limit,
leading to a transition from a mechanical feedback state to a radiative
feedback mode (e.g., Churazov et al., 2005; Dubois et al., 2012; Russell et
al., 2013). Therefore, the lack of evolution on the cavity fraction at
high-$z$ may be due to the dominance of BCGs with radiatively efficient AGNs
(see also Hlavacek-Larrondo et al. 2013).
### 4.2 Cooling luminosity versus Cavity power
Figure 3: Comparison between the mechanical power being injected by the AGN in
the BCG ($P_{\rm cav}$) and the cooling luminosity ($L_{\rm cool}$) of the
cluster at 7.7 Gyrs. The dotted lines are from bottom to top are pV, 4pV,
16pV, per cavity, respectively. Certain (C) cavities are shown with filled
green circles, whereas Potential (P) cavities with open green circles for
Planck selected clusters. We included X-ray cavities from the high-$z$ SPT–SZ
sample (HL15) with filled and open orange circles for the certain and
potential cavities, respectively. We have also added nearby clusters with
X-ray cavities from Rafferty et al. (2006) sample shown with purple circles.
One goal of this work is to test whether the AGN is able to compensate for the
cooling losses of the ICM by heating caused from radio jets. We used the
cavity power ($P_{\rm cav}$) as a proxy for the mechanical power released by
the AGN. The $P_{\rm cav}$ was estimated by dividing the total enthalpy of
each cavity ($E_{\rm cav}=4pV$) by its age. Here $p$ is the thermal pressure
of the ICM at the projected location of the cavity, defined as the center of
each ellipse, and $V$ is the cavity volume. We assumed that the cavities have
a prolate shape. The age of the cavity can be given by the buoyant rise time,
the refill time, or the sound crossing time (McNamara et al., 2000; Bîrzan et
al., 2004; McNamara et al., 2005). For the purpose of this work, we used the
buoyant rise time ($t_{\rm buoy}$) as the age of the cavity, as done in
previous studies. The $t_{\rm buoy}$ corresponds to time for the cavity to
rise buoyantly at its terminal velocity, and is defined as $t_{\rm
buoy}=R/v_{\rm t}=R\sqrt{SC/2gV}$, where $S$ is the cross-section of the
bubble ($S=\pi r_{\rm b}^{2}$), $C$ (= 0.75) is the drag coefficient (Churazov
et al., 2001). Lastly, the local gravitational potential, $g$, was derived
assuming hydrostatic equilibrium.
We drew on top of each identified X-ray cavity an ellipse model, as done in
previous works (e.g., Dong et al., 2010; Shin et al., 2016). Accordingly, the
volume of the cavities is $V=4\pi r_{\rm b}^{2}r_{\rm a}/3$, where $r_{\rm a}$
is the semi-major axis, $r_{\rm b}$ is the semi-minor axis of each X-ray
cavity (see Table 1). We also include in Table 1 the significance of the
detection for each cavity. The significance was calculate as the surface
brightness ratio of the surrounding “blackground”, measured within the same
aperture size as the cavity , and the cavity. Certain cavities have average
significance of 2.2, while potential cavities have average significance of
1.5.
Table 1: Cavity properties
Cluster name | Class | $r_{\rm a}$ | $r_{\rm b}$ | R | PA | $t_{\rm bouy}$ | $P_{\rm cav}$ | $L_{\rm cool}$ | cav.
---|---|---|---|---|---|---|---|---|---
| | (kpc) | (kpc) | (kpc) | (deg.) | ( 107 yr) | ( 1044 erg s-1) | (1044 erg s-1) | significance
G021.09+33.25 | C | 4.1 | 2.6 | 6 | 140 | 0.9${}^{+0.6}_{-3.3}$ | 0.9${}^{+1.6}_{-0.2}$ | 10.1 | 2.2
G021.09+33.25 | C | 5.4 | 3.6 | 7 | 70 | 1.6${}^{+2.2}_{-1.3}$ | 1.2${}^{+1.1}_{-1.0}$ | 10.1 | 2.7
(This table is available in its entirety in machine-readable form.)
Motivated from the previous studies (e.g., Rafferty et al., 2006), we
calculate the X-ray cooling luminosity, $L_{cool}$, within a volume where the
deprojected (isobaric) cooling time, $t_{\rm cool}$, is 7.7 Gyrs (see Table
1). It is representative of the epoch of the last major. Since then, clusters
have been relaxed and a cooling flow could develop (Rafferty et al., 2006).
For the cooling luminosity, we used $L_{\rm cool}=\int n_{\rm e}n_{\rm
H}\Lambda(T,Z)dV$, where $\Lambda(Z,T)$ is the cooling function which depends
on the temperature, $T$, and metallicity, $Z$, of the hot gas. We use the
cooling functions from Gnat & Sternberg (2007), assuming metallicity of
$Z=1~{}Z\odot$, since typical CC clusters have solar or nearly solar
metallicity within their cores (e.g., Molendi & Pizzolato, 2001; Mernier et
al., 2016).
In Figure 3 we compare the mechanical power released by the AGN located in the
central BCG and the cooling luminosity ($L_{\rm cool}$) of each cluster. As a
matter of comparison, we included galaxy groups and elliptical galaxies from
(Rafferty et al., 2006), as well as high-$z$ clusters from the SPT–SZ (HL15).
Notably, our Planck sample has systematically lower mechanical powers than the
high-$z$ SPT–SZ sample, indicating our ability to detect smaller cavities. The
smallest X-ray cavities detected in the SPT–SZ sample have sizes on the order
of $\sim$10 kpc, whereas the resolution for the Planck clusters, as they are
at lower $z$ clusters, is on the order of $\sim$2.5 kpc. The scatter on the
$L_{\rm cool}$ versus $P_{\rm cav}$ relation is slightly smaller in our sample
than in the high-$z$ SPT–SZ sample by $\sim$20%. To make a fair comparison
with the high-$z$ SPT-SZ sample taking only cavities with sizes $\gtrsim$10
kpcs, we find that the scatter is 65% smaller compared to the SPT-SZ sample.
The higher scatter on this relation for high-$z$ clusters is consistent with
being fueled mainly through wet mergers (Somboonpanyakul et al., 2022).
As shown in Fig 3 the $L_{\rm cool}$ and $P_{\rm cav}$ realized from the jets
are positively correlated, indicating that the energy realized by the AGN is
sufficient to balance the radiative losses of the ICM within the cooling
radius for most of the sources in the sample. Some of those objects may
require additional heating from another mechanism, such as thermal conduction
and shocks (e.g., Pope et al., 2005). However, we expect that some of these
clusters may be in a cooling phase discarding the need for an additional
heating source. The nature of the AGN feedback is cyclic and does not always
require a balance between the cooling luminosity and the AGN heating (McNamara
& Nulsen, 2007). This is likely displayed on the scatter of the $L_{\rm cool}$
versus $P_{\rm cav}$ relation. In that sense, the AGN power is variable, and
the objects change their $P_{\rm cav}$ depending on what phase of the AGN
feedback cycle they are observed in. Another source of scatter comes from
dynamically disturbed clusters due to either sloshing motions or mergers.
These mechanisms move the hot gas out of the central BCG to large distances
producing lower $L_{\rm cool}$ for a given $P_{\rm cav}$ value, as found for
clusters with higher centroid shifts indicative of dynamically disturbed
atmospheres (Olivares et al. in prep).
## 5 Limitations
One of this work’s limitations is the fact that we are probably missing
cavities due to shallow X-ray observations, in particular in high-$z$ clusters
(e.g., Diehl et al., 2008; Bîrzan et al., 2012). As pointed out by several
studies, X-ray cavities are more easily detected in clusters with stronger
cool cores, as the contrast between the depression and surroundings is
sharper, and it is more difficult to find bubbles in high-redshift clusters
due to the lack of counts. The detectability of cavities also decreases with
their radius (Enßlin & Heinz, 2002).
More importantly, cavities that have sizes below the resolution (e.g., cavity
size $\leq 2$ kpc for clusters at $z$=0.1) will be undetectable in such an
analysis. As shown in Figure 2, this effect will increase at high-$z$. For
example, cluster at $z$>0.5 only cavities with sizes $\geq 6$ kpc can be
detected. We also note that sources with more than 2000 counts, in the central
region, tend to have “certain” detected X-ray cavities (circle symbols).
Therefore, we stress that deeper Chandra follow-up observations are required
to confirm the presence of any potential X-ray cavity, especially in high-$z$
clusters. Aside from the data quality, other effects may be also interfering
with the cavity detectability, such as orientation, location, and angular size
(see Enßlin & Heinz 2002; Diehl et al. 2008; Brüggen et al. 2009 for more
details). As pointed out by Enßlin & Heinz (2002), the detectability of
cavities decreases with distance to the center, and for a cavity moving in the
plane of the cavities, the contrast decreases slowly. Cavities that lie off
the plane of the sky have reduced contrast and therefore are harder to detect.
To quantify this projection effect, we assume a random distribution of the
angle of the cavity relative to the plane of the sky, a typical cavity size of
10 kpc, and a typical beta profile for the ICM distribution (rc=20, beta=3/4).
At an average projected distance of 30 kpc, 20%–30% of the cavities would have
a contrast below our detection limit and would have been missed in our study.
It should be noted that the $P_{\rm cav}-L_{\rm cool}$ relation is also
affected by projection effects, likely introducing scatter. As pointed out by
Diehl et al. (2008), all the physical quantities involving the cavity power,
$P_{\rm cav}$, such as density, temperature, and pressure, are measured at the
projected distance from the cavity to the center rather than the true
distance. The former corresponds to a lower limit. The pressure increases
towards the center, leading to an overestimation of the ambient pressure at
the cavity position. On the other hand, the cavity ages will be underestimated
as they are proportional to the cavity distance. Both mentioned effects will
bias the cavity power upwards.
## 6 Conclusions
We have investigated the mechanical AGN feedback mechanism in central cluster
galaxies using archival X-ray Chandra observations of 164 Planck selected
clusters to search for X-ray cavities.
(i) Using several techniques to look for X-ray cavities, including inspection
of the original image, a model subtracted image and an unsharp masked image,
we find 65 X-ray cavities in 29 systems out of 63 CC clusters. Among them, 12
systems have clearly detected cavities, whereas 17 have only potential
depressions. Two potential cavities were also found in one NCC cluster.
(ii) We measured a total detection fraction of X-ray cavities of $\sim$18%,
twice the detection rate of the high-$z$ SPT–SZ sample, indicating that
clusters have radio-mode feedback only 18% of the time. Nevertheless, our
detection fraction of 9% is close to the high-$z$ SPT–SZ sample when taking
only cavities with sizes $\gtrsim$10 kpc to match the resolution of the SPT-SZ
sample. We interpreted this as an lack of evolution of the AGN feedback cycle
across cosmic time.
(iii) We find that the AGN heating traced by the power of the X-ray cavities
alone is able to balance the radiative losses of the ICM in our sample. Our
sources have slightly lower cavity power per cavity than high-$z$ massive
clusters from the SPT-SZ sample due to smaller cavities being detected in our
sample.
Future high-resolution X-ray observations from Chandra satellite and the
upcoming Advanced X-ray Imaging Satellite (AXIS) telescope will be needed to
find more cavities in the faintest clusters and confirm the discussed findings
in high-$z$ clusters.
## Acknowledgments
This research has made use of software provided by the Chandra X-ray Center
(CXC) in the application packages CIAO. V.O. and Y.S. were supported by NSF
grant 2107711, Chandra X-ray Observatory grant GO1-22126X, and NASA grant
80NSSC21K0714.
## Data Availability
The Chandra raw data used in this paper are available to download at the
HEASARC Data Archive website111https://heasarc.gsfc.nasa.gov/docs/archive.htm.
## References
* Andrade-Santos et al. (2017) Andrade-Santos F., et al., 2017, ApJ, 843, 76
* Bîrzan et al. (2004) Bîrzan L., Rafferty D. A., McNamara B. R., Wise M. W., Nulsen P. E. J., 2004, ApJ, 607, 800
* Bîrzan et al. (2008) Bîrzan L., McNamara B. R., Nulsen P. E. J., Carilli C. L., Wise M. W., 2008, ApJ, 686, 859
* Bîrzan et al. (2012) Bîrzan L., Rafferty D. A., Nulsen P. E. J., McNamara B. R., Röttgering H. J. A., Wise M. W., Mittal R., 2012, MNRAS, 427, 3468
* Bîrzan et al. (2017) Bîrzan L., Rafferty D. A., Brüggen M., Intema H. T., 2017, MNRAS, 471, 1766
* Bîrzan et al. (2020) Bîrzan L., et al., 2020, MNRAS, 496, 2613
* Bleem et al. (2015) Bleem L. E., et al., 2015, ApJS, 216, 27
* Bleem et al. (2020) Bleem L. E., et al., 2020, ApJS, 247, 25
* Boehringer et al. (1993) Boehringer H., Voges W., Fabian A. C., Edge A. C., Neumann D. M., 1993, MNRAS, 264, L25
* Brown et al. (2001) Brown L. D., Cai T., DasGupta A., 2001, Statistical Stadistical science, 16, 101
* Brüggen et al. (2009) Brüggen M., Scannapieco E., Heinz S., 2009, MNRAS, 395, 2210
* Canning et al. (2013) Canning R. E. A., et al., 2013, MNRAS, 435, 1108
* Carilli et al. (1994) Carilli C. L., Perley R. A., Harris D. E., 1994, MNRAS, 270, 173
* Churazov et al. (2001) Churazov E., Brüggen M., Kaiser C. R., Böhringer H., Forman W., 2001, ApJ, 554, 261
* Churazov et al. (2005) Churazov E., Sazonov S., Sunyaev R., Forman W., Jones C., Bohringer H., 2005, MNRAS: Letters, 363, L91
* Diehl et al. (2008) Diehl S., Li H., Fryer C. L., Rafferty D., 2008, ApJ, 687, 173
* Dong et al. (2010) Dong R., Rasmussen J., Mulchaey J. S., 2010, The Astrophysical Journal, 712, 883
* Dubois et al. (2012) Dubois Y., Devriendt J., Slyz A., Teyssier R., 2012, MNRAS, 420, 2662
* Enßlin & Heinz (2002) Enßlin T. A., Heinz S., 2002, A&A, 384, L27
* Fabian (1994) Fabian A. C., 1994, ARA&A, 32, 277
* Fabian (2012) Fabian A., 2012, ARA&A, 50, 455
* Fabian et al. (2006) Fabian A. C., Sanders J. S., Taylor G. B., Allen S. W., Crawford C. S., Johnstone R. M., Iwasawa K., 2006, MNRAS, 366, 417
* Gitti et al. (2010) Gitti M., O’Sullivan E., Giacintucci S., David L. P., Vrtilek J., Raychaudhury S., Nulsen P. E. J., 2010, ApJ, 714, 758
* Gnat & Sternberg (2007) Gnat O., Sternberg A., 2007, ApJS, 168, 213
* Haggard et al. (2010) Haggard D., Green P. J., Anderson S. F., Constantin A., Aldcroft T. L., Kim D.-W., Barkhouse W. A., 2010, ApJ, 723, 1447
* Hilton et al. (2018) Hilton M., et al., 2018, ApJS, 235, 20
* Hincks et al. (2010) Hincks A. D., et al., 2010, ApJS, 191, 423
* Hlavacek-Larrondo et al. (2013) Hlavacek-Larrondo J., Fabian A. C., Edge A. C., Ebeling H., Allen S. W., Sanders J. S., Taylor G. B., 2013, MNRAS, 431, 1638
* Hlavacek-Larrondo et al. (2015) Hlavacek-Larrondo J., et al., 2015, ApJ, 805, 35
* Jones (2012) Jones C., 2012, A Chandra-Planck Legacy Program for Massive Clusters of Galaxies, Chandra Proposal
* McNamara & Nulsen (2007) McNamara B., Nulsen P., 2007, ARA&A, 45, 117
* McNamara et al. (2000) McNamara B. R., et al., 2000, ApJ, 534, L135
* McNamara et al. (2005) McNamara B. R., Nulsen P. E. J., Wise M. W., Rafferty D. A., Carilli C., Sarazin C. L., Blanton E. L., 2005, Nature, 433, 45
* Mernier et al. (2016) Mernier F., de Plaa J., Pinto C., Kaastra J. S., Kosec P., Zhang Y. Y., Mao J., Werner N., 2016, A&A, 592, A157
* Molendi & Pizzolato (2001) Molendi S., Pizzolato F., 2001, ApJ, 560, 194
* Morsony et al. (2010) Morsony B. J., Heinz S., Brüggen M., Ruszkowski M., 2010, MNRAS, 407, 1277
* Nulsen et al. (2009) Nulsen P., Jones C., Forman W., Churazov E., McNamara B., David L., Murray S., 2009, in Heinz S., Wilcots E., eds, American Institute of Physics Conference Series Vol. 1201, The Monster’s Fiery Breath: Feedback in Galaxies, Groups, and Clusters. pp 198–201 (arXiv:0909.1809), doi:10.1063/1.3293033
* O’Sullivan et al. (2011) O’Sullivan E., Giacintucci S., David L. P., Gitti M., Vrtilek J. M., Raychaudhury S., Ponman T. J., 2011, ApJ, 735, 11
* Olivares et al. (2019) Olivares V., et al., 2019, A&A, 631, A22
* Olivares et al. (2022) Olivares V., et al., 2022, arXiv e-prints, p. arXiv:2201.07838
* Panagoulia et al. (2014) Panagoulia E. K., Fabian A. C., Sanders J. S., Hlavacek-Larrondo J., 2014, MNRAS, 444, 1236
* Planck Collaboration et al. (2011) Planck Collaboration et al., 2011, A&A, 536, A1
* Pope et al. (2005) Pope E. C. D., Pavlovski G., Kaiser C. R., Fangohr H., 2005, MNRAS, 364, 13
* Rafferty et al. (2006) Rafferty D. A., McNamara B. R., Nulsen P. E. J., Wise M. W., 2006, ApJ, 652, 216
* Ruppin et al. (2021) Ruppin F., McDonald M., Bleem L. E., Allen S. W., Benson B. A., Calzadilla M., Khullar G., Floyd B., 2021, ApJ, 918, 43
* Russell et al. (2013) Russell H. R., McNamara B. R., Edge A. C., Hogan M. T., Main R. A., Vantyghem A. N., 2013, MNRAS, 432, 530
* Russell et al. (2019) Russell H. R., et al., 2019, MNRAS, 490, 3025
* Shin et al. (2016) Shin J., Woo J.-H., Mulchaey J. S., 2016, ApJS, 227, 31
* Silverman et al. (2009) Silverman J. D., et al., 2009, ApJ, 695, 171
* Somboonpanyakul et al. (2022) Somboonpanyakul T., et al., 2022, arXiv e-prints, p. arXiv:2201.08398
* Su et al. (2020) Su Y., et al., 2020, MNRAS, 498, 5620
|
# Interactive Region-of-Interest Discovery using Exploratory Feedback
Behrooz Omidvar-Tehrani
Grenoble AI Institute
###### Abstract
In this paper, we propose a geospatial data management framework called IRIDEF
which captures and analyzes user’s exploratory feedback for an enriched
guidance mechanism in the context of interactive analysis. We discuss that
exploratory feedback can be a proxy for decision-making feedback when the
latter is scarce or unavailable. IRIDEF identifies regions of interest (ROIs)
via exploratory feedback, and highlights a few interesting and out-of-sight
POIs in each ROI. These highlights enable the user to shape up his/her future
interactions with the system. We detail the components of our proposed
framework in the form of a data analysis pipeline, and present the aspects of
efficiency and effectiveness for each component. We also discuss evaluation
plans and future directions for IRIDEF.
## 1 Introduction
Background. Nowadays, geospatial data are ubiquitous in various fields of
science, such as transportation, smart city management [1, 2], travel planning
[3], bike sharing [4], localized advertising [5], and regional health-care
[6]. A recent solution for an improved geospatial data management is to
provide means for interactive analysis, where users in the loop are guided
towards interesting subsets of data in an exploratory iterative manner [7, 8].
Typically, the guidance is performed through learning user’s preferences using
a decision-making feedback received from the user in each iteration, e.g.,
picking (clicking on) a favorite point of interest (POI). However, it is often
the case in geospatial scenarios that users forget or don’t feel necessary to
explicitly express their feedback in what they find interesting. As a result,
the interactive dialog will be broken and no guidance can be delivered. In
this paper, we focus on the following question: Is it possible to perform
interactive analysis on geospatial data without having access to decision-
making interactions?
Proposal. In the absence of decision-making interactions, we propose to focus
on exploratory feedback, i.e., patterns in signals captured from the user in
the background which provide hints on user’s interests. For instance, users
often hover their mouse (or make frequent touch actions on a touch screen,
such as scroll, pinch and zoom) over a region of interest to collect
information on the map (e.g., touristic places and hidden gems presented in
the form of map layers and tooltips) before landing on a decision about
picking a POI in that region, such as a home-stay. Hence it is possible to
infer the interest towards that region even without decision-making
interactions. This inferred knowledge should be leveraged in the guidance
mechanism. An instance of such guidance is to highlight a few interesting POIs
in the region of interest. We advocate a geospatial data management framework
(called IRIDEF) which captures and analyzes user’s exploratory feedback for an
enriched guidance mechanism in the context of interactive analysis.
Scenario. Lindsey is a visiting researcher from the US. She wants to rent a
home-stay in Paris via the Airbnb website. She likes to discover the city,
hence she is open to any type of lodging in any region with an interest to
stay in the center of Paris. Her exploration starts with a query which
expresses the preliminary set of her interests. The website returns 1500
different home-stays for her query. While scanning the very first items, she
shows (an exploratory) interest towards the region of Trocadero by hovering
her mouse around the Eiffel tower and checking the amenities within that
region. However, she forgets or doesn’t feel the necessity to click a POI
(i.e., a home-stay) in that region. While typical recommendation and
exploration systems do not necessarily focus on this implicit interest in the
future iterations, our framework ensures that Lindsey receives home-stay
recommendations related to the Trocadero region even if she didn’t provide any
decision-making feedback.
Challenges. Analyzing exploratory feedback is challenging. First, it is not
clear how this feedback should be interpreted in terms of the user
preferences. Exploratory feedback on geospatial data can be enabled via
different signals, such as mouse hovering [9], touch actions [10], voice [11],
and gaze [12]. Translating such enablers into geospatial semantics is
challenging. Second, all exploratory signals are not necessarily useful and
some may introduce false positives. For instance, a small mouse move on a
typical screen would yield more than 14,000 points (assuming 1600 DPI) which
may turn out to be just a random futile move. Beyond the first two challenges,
guiding users towards interesting POIs is also challenging, as it requires an
exhaustive scan over the geospatial data against the evolving user
preferences.
Contributions. We propose a guidance approach for interactive exploration of
geospatial data. Our approach identifies regions of interest (ROIs) without
the need for any decision-making feedback. Our proposed guidance mechanism is
to highlight a few interesting and out-of-sight POIs in each ROI, and let the
user investigate those POIs in his/her future interactions with the system.
The following list summarizes the contributions and claims discussed in this
paper:
* •
We define the notion of “exploratory user feedback” which enables a seamless
navigation in the geospatial data.
* •
We define the notion of “information highlighting”, a mechanism to highlight
important spatial information that is out-of-sight.
* •
We employ an efficient polygon-based approach to discover ROIs.
* •
We propose an approach to compute highlights on-the-fly in an efficient
manner.
To the best of our knowledge, our contributions have not been investigated
before in the literature. Popular map-based applications such as Google Maps
and Bing Maps do not offer interactive functionalities for feedback capturing.
In the literature, information highlighting [13, 14, 15] and spatial
recommendation approaches [16, 17] often assume that the user’s preferences
are static and will never change in time. This limits their functionality for
serving the scenarios of an interactive analysis. The process of feedback
capturing is mostly formulated for decision-making interactions [18, 19, 20,
21, 22, 23]. While a few fuse decision-making and exploratory feedbacks [24,
25, 26], our approach is not dependent on decision-making feedback and is able
to operate purely on exploratory feedback. It is to state the obvious that a
straightforward extension of our system is to incorporate decision-making
feedback (if available) to improve the effectiveness of the system.
Paper outline. The rest of this paper is organized as follows. In Section 2,
we elaborate on different instances of decision-making and exploratory
feedbacks in the literature. We discuss the data model and introduce in the
problem in Section 3. We present our proposed approach in Section 4, and
discuss evaluation plans in Section 5. Last, we conclude and present future
directions in Section 6.
Figure 1: Examples of decision-making and exploratory feedbacks in realistic
geospatial scenarios [6, 27, 3]
## 2 Decision-making Feedback versus and Exploratory Feedback
We briefly discuss a few examples in the literature to clarify the distinction
between decision-making and exploratory feedback types in realistic geospatial
applications. These examples are illustrated in Figure 1. In summary, we argue
that different types of decision-making feedback have been already employed,
but the exploratory feedback is often missing.
Medical domain. COVIZ [6] is an interactive web-based application which
enables medical experts to form and compare medical cohorts. In Figure 1-A,
the expert clicks on the Auvergne-Rhône-Alpes region (as a decision-making
feedback) to compare the patient cohort in this particular region with the
whole France. In Figure 1-B, the expert adds the air pollution layer to the
analysis to examine any potential correlation between the patients’ health
status and the abundance of the air pollution. While the expert explores the
cohort comparisons and pollution correlations, the tool does not collect any
exploratory feedback, such as mouse hover and gaze.
Aviation domain. DV8 [27] is an interactive aviation data analysis tool. When
several flight trajectories are visualized (Figure 1-C), the expert can click
on one trajectory to retrieve its information (departure, destination, etc.),
and double-click to solely focus on that single trajectory and analyze it
further (Figure 1-D). The interaction is always through the decision-making
feedback (single-click and double-click) and the exploratory feedback is not
supported. DV8 also supports touch gestures, such as pinch and zoom (Figure
1-E) and brush (Figure 1-F). However the touch actions are all considered as
decision-making feedback with an immediate resulting action. Hence there is no
support for exploratory feedback. The virtual reality (VR) version of DV8
(Figures 1-G and 1-H) enhances the exploration experience of the aviation
expert, particularly for analyzing flights in different altitudes. While the
gaze signal is an exploratory feedback which can be captured through VR, DV8
employs the signal only for navigating the geospatial data, and not for
guidance.
Travel domain. Simurgh [3] is an interactive travel package generation tool.
The user can ask for a new day plan using a drag-and-drop action over a region
of interest (the drag-and-drop in Figure 1-I and the resulting day plan in
Figure 1-J). She can also replace a point of interest by clicking on the point
(the selection in Figure 1-K and the replacement in Figure 1-L). All the
interactions are defined as the decision-making feedback. In other words,
Simurgh does not detect the regions of interest by following the exploratory
feedback.
## 3 Data Model and Problem Definition
To enable feedback capturing, we consider two different layers on a
geographical map: spatial layer and interaction layer. The spatial layer
contains POIs from a spatial database $\mathcal{P}$. The interaction layer
contains exploratory feedback points $\mathcal{M}$. These layers are explained
below.
Spatial layer. Each POI
$p=\langle\mathit{lat},\mathit{lon}\rangle\in\mathcal{P}$ is described using
its geographical coordinates. POIs are also associated to a set of domain-
specific attributes $\mathcal{A}$. For instance, in the dataset of a real
estate agency, POIs are properties (houses and apartments) and $\mathcal{A}$
contains attributes such as surface, number of rooms and price. The set of all
possible values for an attribute $a\in\mathcal{A}$ is denoted as $dom(a)$. We
also define user’s feedback $F$ as a vector over all attribute values (i.e.,
facets), i.e., $F\in\prod_{a\in\mathcal{A}}dom(a)$. The vector $F$ is
initialized by zeros and will be updated to express the user’s preferences.
The facet-based schema of $F$ ensures that learned feedback is always
transparent and interpretable by the user using the facets, and hence reduces
algorithmic anxiety [28].
Interaction layer. We assume that an exploratory signal addresses one specific
point $m$ on the screen, e.g., hovering at, gazing at, or providing a voice
command about $m$. When an exploratory signal is received, the point $m$ is
appended to the set $\mathcal{M}$. Each point is a tuple $m=\langle
x,y,t\rangle$, where $x$ and $y$ specify the affected pixel location and $t$
is a timestamp. To conform with geographical standards, we assume $m=\langle
0,0,t\rangle$ sits at the middle of the interaction layer, both horizontally
and vertically, for any $t$.
Transitioning between the layers. The user is in contact with the interaction
layer. To update the feedback vector $F$, we need to translate pixel locations
in the interaction layer to latitudes and longitudes in the spatial layer. We
employ equirectangular projection to obtain the best possible approximation of
a point $m=\langle x,y,t\rangle\in\mathcal{M}$ in the spatial layer, denoted
as $p(m)$.
$p(m=\langle
x,y,t\rangle)=\langle\mathit{lat}=y+\gamma,\mathit{lon}=\frac{x}{\mathit{cos}\gamma}+\theta\rangle$
(1)
The inverse operation, i.e., transforming a point
$p=\langle\mathit{lat},\mathit{lon}\rangle$ from the spatial layer to the
interaction is done using Equation 2.
$m(p=\langle\mathit{lat},\mathit{lon}\rangle)=\langle
x=(\mathit{lon}-\theta)\times\mathit{cos}\gamma,y=\mathit{lat}-\gamma\rangle$
(2)
The reference point for the transformation is the center of both layers. In
Equations 1 and 2, we assume that $\gamma$ is the latitude and $\theta$ is the
longitude of a point in the spatial layer corresponding to the center of the
interaction layer, i.e., $m=\langle 0,0\rangle$.
Problem definition. Given the user’s feedback $F$, we are interested in
solving two consecutive problems: $(i)$ discover regions of interest in the
form of geospatial clusters whose centroids correlate with $F$ (with respect
to the POI attributes in which the user is interested in), and $(ii)$ for each
discovered region, find at most $k$ POIs ($k$ is an input parameter) which are
relevant to $F$ and have high exploration quality. We define relevance and
exploration quality in Section 4.
Figure 2: IRIDEF framework.
## 4 Proposed Approach
We propose IRIDEF (Interactive Region-of-Interest Discovery using Exploratory
Feedback), a framework for exploiting exploratory feedback to highlight
interesting POIs as future analysis directions. As depicted in Figure 2, our
approach consists of a pipeline with three main components: CAPTURE, DISCOVER,
and HIGHLIGHT. After the user has explored the map for a while, IRIDEF
captures exploratory feedback from the exploration (i.e., the CAPTURE
component detailed in Section 4.2). Then a set of regions of interest (ROIs)
will be discovered using the captured feedback (i.e., the DISCOVER component
detailed in Section 4.3). Finally some out-of-sight interesting POIs will be
highlighted for each discovered ROI (i.e., HIGHLIGHTcomponent detailed in
Section 4.4). In the following, we first discuss the desiderata behind our
approach, and then detail each component of the pipeline.
### 4.1 Principles
In order to maximize the usability of IRIDEF, we believe that the framework
should be generic and fluid, as discussed below.
Genericness. IRIDEF’s pipeline is applicable to different datasets and
different types of exploratory feedback. This enables IRIDEF to cover
different exploration scenarios. The minimal requirement is that the input
dataset and the feedback signal match with our data model (Section 3).
Fluidity. A fluid interactive system does not break the user’s train of
thought. The fluidity is ensured by rendering results in an efficient and
effective manner. In the CAPTURE component, effectiveness is satisfied by
disregarding irrelevant signals. In the DISCOVER and HIGHLIGHT components,
effectiveness is interpreted as delivering meaningful and useful regions
(ROIs) and highlights (POIs), respectively. In all of the components,
efficiency is to return results instantaneously, often considered to be $\leq
500ms$ [29].
### 4.2 CAPTURE Component
Exploratory feedback can be captured using different latent signals, e.g.,
time dedicated to item details, touch actions, gaze, mouse moves, scrolling
speed, etc. Without loss of generality, we focus on mouse moves as an instance
of exploratory feedback signal. A particular challenge in capturing mouse
moves as the exploratory feedback is that the user may mindlessly move the
mouse everywhere on the map. Obviously, this should not signify that all the
locations are equally important to the user. An effective approach should only
capture a subset of this feedback which is then useful for discovering ROIs.
Also an efficient approach should capture this feedback without any
interruption in the fluidity of the user experience. For an effective and
efficient feedback capturing, IRIDEF performs the two following actions:
1. 1.
First, it records the exploratory signals (by adding the coordinates of the
screen points they were applied on to $\mathcal{M}$) only every $\varepsilon$
milliseconds to prevent adding redundant points.
2. 2.
After a given period of feedback capturing time, it groups the recorded
signals into $g$ different segments, $\mathcal{M}_{1}$ to $\mathcal{M}_{g}$.
The first segment starts at time zero (where the system started to operate),
and the last segment ends at the current time.
The choice of $\varepsilon$ depends on various parameters such as the
application (e.g., tourism, delivery, transportation) and the user’s
expertise. For instance, a larger $\varepsilon$ seems more appropriate for
novice users, as they might perform many random moves to get acquainted with
the data. In conformance with progressive data analytics [29], we set
$\varepsilon=100ms$ as the default value to ensure continuity preserving
latency.
Input: Mouse move points $\mathcal{M}$, time gap $\varepsilon$, segmentation
strategy $\psi$
Output: Segments $\mathcal{M}_{i}$, $i\in[1,g]$
1 $\mathit{segment\\_count}\leftarrow 0$
2 for _$m\in\mathcal{M}$ captured every $\varepsilon$ milliseconds_ do
3
$\mathcal{M}[\mathit{segment\\_count}]\leftarrow\mathcal{M}[\mathit{segment\\_count}]\cup\\{m\\}$
4
$\mathit{segment\\_change}\leftarrow\mathit{check\\_strategy}(\psi,m,\mathcal{M})$
5 if _$\mathit{segment\\_change}=\mathit{true}$_ then
6 $\mathit{segment\\_count}\leftarrow\mathit{segment\\_count}+1$
7 $\mathcal{M}[\mathit{segment\\_count}]\leftarrow\emptyset$
8
9 end if
10
11 end for
12return _$\mathcal{M}_{i}$ , $i\in[1,g]$ where $g=\mathit{segment\\_count}$_
Algorithm 1 CAPTURE algorithm
Moreover, the end of a segment is determined by one of the following
approaches:
* •
$\psi_{1}$: End the current segment after a fixed amount of time (i.e., fixed-
length segments). In this case, the value of $g$ is selected based on the
spatial density of the dataset under investigation.
* •
$\psi_{2}$: End the current segment if the mouse location is unchanged for a
certain amount of time.
* •
$\psi_{3}$: End the current segment after a drastic change in the signal,
where the drift is captured using signal segmentation approaches. We employ
the Wedding Cake technique for the dynamic segmentation of our signals [30,
31].
Algorithm 1 summarizes the CAPTURE process.
### 4.3 DISCOVER Component
The objective of this step in the IRIDEF pipeline is to obtain one or several
ROIs in which the user has expressed his/her exploratory feedback. We
conjecture that a region is more interesting for the user if it is denser,
i.e., the user moves the mouse in that region frequently, to collect
information from the background map. Hence ROIs can be simply discovered as
dense clusters of mouse move points. We denote the set of all ROIs as
$\mathcal{R}$ and we refer to the $i$-th ROI as $R_{i}\in\mathcal{R}$.
Algorithm 2 summarizes the DISCOVER process.
We employ ST-DBSCAN [32], a space-aware variant of DB-SCAN, to cluster points
in each segment (line 2 in Algorithm 2). For each subset of mouse move points
$\mathcal{M}_{i}$, $i\in[1,g]$, ST-DBSCAN begins with a random point
$m_{0}\in\mathcal{M}_{i}$ and collects all density-reachable points from
$m_{0}$ using a distance metric. As mouse move points are in the 2-dimensional
pixel space (i.e., the screen), we choose euclidean distance as the distance
metric. A density-reachable point $m_{i}$ is either directly reachable from
$m_{0}$, i.e., the distance between $m_{i}$ and $m_{0}$ is lower than a
distance threshold (an input parameter for the ST-DBSCAN algorithm), or
reachable via a path $m_{0}\dots m_{j-1},m_{j}\dots m_{i}$ where each point
$m_{j}$ in the path is directly reachable from its immediately prior point in
the path $m_{j-1}$. If $m_{0}$ turns out to be a core point, a cluster will be
generated. A point is core if there exist a certain amount of points in its
vicinity, i.e., with a distance lower than the distance threshold. The minimum
number of points for a core point is yet another input parameter for ST-
DBSCAN. If $m_{0}$ is not a core point, the algorithm picks another random
point in $\mathcal{M}_{i}$. The process is repeated until all points have been
processed. We denote the set of all resulting clusters for $\mathcal{M}_{i}$
as $\mathcal{C}_{i}=\\{C_{1},C_{2},\dots\\}$.
Input: Segments $\mathcal{M}_{1}$ to $\mathcal{M}_{g}$, user feedback vector
$F$, number of interactions performed so far $T$
Output: Set of discovered ROIs $\mathcal{R}$
$O\leftarrow\emptyset$
// the set of all polygons initialized as empty
$\mathcal{R}\leftarrow\emptyset$
// the set of all ROIs initialized as empty
1 for _each segment $\mathcal{M}_{i}$_ do
$\mathcal{C}_{i}\leftarrow\mathit{ST\\_DBSCAN}(\mathcal{M}_{i})$
// all clusters inside $\mathcal{M}_{i}$
2 $\mathcal{C}_{i}\leftarrow\mathit{AklToussaint}(\mathcal{C}_{i})$
$O_{i}\leftarrow\mathit{Graham\\_scan}(\mathcal{C}_{i})$
// all polygons inside $\mathcal{M}_{i}$
$O_{i}.\mathit{expand}(\mathit{confidence}(F,T))$
// Equation 3
3 $O\leftarrow O\cup O_{i}$
4 end for
5for _each pair of polygons $O_{x}\in O$ and $O_{y}\in O$_ do
6 $S\leftarrow\mathit{intersect}(O_{x},O_{y})$
7 if _$S.\mathit{size} >0$_ then
$\mathcal{R}\leftarrow\mathcal{R}\cup\\{S\\}$
8
9 end for
10 return _$\mathcal{R}$_
Algorithm 2 DISCOVER algorithm
Once the clusters are obtained for all the subsets of $\mathcal{M}$, we find
their intersections to locate recurring regions. Note that we don’t aim to
directly consider the clusters $\mathcal{C}$ as the ROIs, as they may contain
noisy signals. Their intersection counts as a confirmation of user
preferences. To obtain intersections, we need to clearly define the spatial
boundaries of each cluster. For this aim, we discover the polygons which cover
the points inside each cluster. We employ Graham scan algorithm (line 2 in
Algorithm 2) which is an efficient method to compute the convex hull for a
given set of points in a 2D plane [33]. We reduce the typical complexity of
Graham scan (i.e., $\mathcal{O}(|C_{i}|\times\mathit{log}|C_{i}|)$, $|C_{i}|$
being the number of points in the $i$-th cluster) to $\mathcal{O}(|C_{i}|)$ by
ordering the cluster members by their spatial coordinates. For more
efficiency, we perform Akl-Toussaint heuristics [34] before the polygon
computation to prune the points which are unnecessary for shaping the polygons
(line 2 in Algorithm 2). The intersections between the polygons constitute the
ROIs (lines 2 to 2 in Algorithm 2).
Personalizing discovered ROIs. By default, our ROI discovery approach creates
strictly tight ROIs, i.e., the area of the polygons is exactly inferred by the
points it covers. However in exploratory scenarios, the feedback points do not
necessarily reflect the exact interests of the user. The user exposes his/her
interests in a gradual manner using exploratory feedback captured in several
iterations. We believe that the user’s confidence (interpreted as the richness
of the user feedback vector $F$) should impact the way ROIs are computed,
hence personalized ROIs. In case the user is less confident (e.g., the user is
in early stages of his/her exploration), ROIs should be expanded in their area
(up to twice their original size) to let more opportunities arise (line 2 in
Algorithm 2). The user confidence is computed as follows.
$\mathit{confidence}(F,T)=\mathit{min}(1.0,\frac{||F||_{0}}{\xi\times T})$ (3)
In Equation 3, $\xi$ is a feedback frequency, and $T$ is the number of
interactions performed so far. For instance, given $|F|=50$, $T=10$, and
assuming that a typical user provides $7$ exploratory signals per iteration,
the confidence will be equal to $0.71$. The confidence is a coefficient for
stretching the ROI area. Let $A_{1}$ denote the area of the ROI $R_{1}$, the
confidence-aware area $A^{\prime}_{1}$ is computed as follows:
$A^{\prime}_{1}=(A_{1}+A_{1}\times\mathit{confidence})$. This process is shown
in line 2 of Algorithm 2.
Example. Figure 3 shows the steps that Lindsey follows to explore home-stays
in Paris. For the sake of simplicity, we assume Lindsey’s confidence is $1.0$.
Figure 3.A shows the mouse moves of Lindsey in different time stages. In this
example, we consider $g=3$ and capture Lindsey’s feedback in three different
time segments with fixed-length, i.e., $\psi_{1}$ (progressing from Figures
3.B to 3.D). It shows that Lindsey started her search around Eiffel Tower and
Arc de Triomphe (Figure 3.B) and gradually showed interest in areas located
south (Figure 3.C) and north (Figure 3.D) as well. All intersections between
those clusters are discovered (hatched regions in Figure 3.E) which will
contribute to the set of interesting regions (Figure 3.F), i.e., ROI1 to ROI4.
Figure 3: An example of discovering ROIs [9].
### 4.4 HIGHLIGHT Component
We define highlights as a subset of POIs in the form of suggestions for
directions of future analysis of the user. The highlights are generated by
performing the three following steps: matching points, updating feedback, and
highlighting POIs. First, we find POIs which fit into the polygons obtained in
the DISCOVER component. Then we update the user feedback $F$ according to
those POIs. Finally we highlight a set of POIs based on the updated content of
$F$.
Matching points. Being a function of mouse move points, ROIs are discovered in
the interaction layer. We then need to find out which POIs in $\mathcal{P}$
fall into ROIs. We employ Equation 2 to transform those POIs from the spatial
layer to the interaction layer. Then a simple spatial containment function can
verify whether a given POI fits into a given ROI.111Typically, we use the
implementation of $\mathit{ST\\_Within}()$ module in PostGIS for the
containment verification. To improve efficiency, we employ Quadtrees [35] in a
two-step approach: $(i)$ In an offline process, we build a Quadtree index for
all POIs in $\mathcal{P}$. We record the membership relations between POIs and
Quadtree grid cells in the index. $(ii)$ Once ROIs are discovered, we record
which cells in the Quadtree index intersect with the ROIs. For matching POIs,
we only check a subset which is inside the cells associated to ROIs and ignore
the ones outside, hence a drastic pruning of POIs in $\mathcal{P}$. Given an
ROI $R_{i}$, we denote the set of its matching points as $\mathcal{P}_{i}$. We
also define the binary vector $\overrightarrow{\mathcal{P}_{i}}$ whose cell of
$\langle a_{j},v_{w}\rangle$ is $1$ if at least one point in $\mathcal{P}_{i}$
gets the value $v_{w}\in\mathit{dom}(a_{j})$ for the attribute
$a_{j}\in\mathcal{A}$, otherwise $0$.
Updating feedback. The matching points depict the exploratory preferences of
the user. To memorize these preferences, we update the feedback vector $F$
using the attributes of the matching points. We consider an increment value
$\delta$ to update $F$. If $p$ is a matching point and gets
$v_{w}\in\mathit{dom}(a_{j})$ for attribute $a_{j}\in\mathcal{A}$, we augment
the value in the $F$’s cell of $\langle a_{j},v_{w}\rangle$ by the factor
$\delta$. Note that we only consider incremental feedback, i.e., we never
decrease a value in $F$. The vector $F$ will become normalized after each
update using a softmax function. The updated feedback vector is fully
transparent and the user can easily apprehend what has been learned from
his/her previous actions. Our current update model considers the feedback
vector to be recency-agnostic. We leave the integration of recency as future
work.
Highlighting POIs. The updated feedback vector $F$ is the input to the
highlighting phase. The objective is to select $k$ POIs out of all POIs inside
ROIs whose relevance and exploration quality are maximal. We denote the set of
highlights as $\mathcal{H}$. We propose two approaches to achieve our
objective, depending on how we define relevance and quality:
Input: Discovered ROIs $\mathcal{R}$, user feedback vector $F$, $k$,
$\mathit{time\\_limit}$, $\mathit{similarity\\_threshold}$
Output: Highlights $\mathcal{H}$
$\mathcal{H}\leftarrow\emptyset$
// highlights
1 for _each discovered ROI $R_{i}\in\mathcal{R}$_ do
2 $\mathcal{P}_{i}\leftarrow\mathit{match\\_points}(R_{i})$
3 $F.\mathit{update}(\mathcal{P}_{i})$
4 $\mathcal{L}_{i}\leftarrow$ sort the POIs in $\mathcal{P}_{i}$ in decreasing
order of their similarity with $F$
5 $p^{*}\leftarrow\mathit{most\\_similar\\_point}(\mathcal{P}_{i},F)$
6 $k^{\prime}\leftarrow k\times\mathit{peculiarity}(R_{i})$
7 $\mathcal{H}[R_{i}]\leftarrow\mathit{top}(\mathcal{L}_{i},k^{\prime})$
8 $p_{\mathit{next}}\leftarrow\mathit{get\\_next}(\mathcal{L}_{i})$
9 while _$\mathit{time\\_limit}$ not exceeded and
$\mathit{similarity}(p_{\mathit{next}},p^{*})\leq\mathit{similarity\\_threshold}$_
do
10 for _$p_{\mathit{current}}\in\mathcal{H}[R_{i}]$_ do
11 if
_$\mathit{diversity\\_improved}(\mathcal{H}[R_{i}],p_{\mathit{next}},p_{\mathit{current}})$_
then
$\mathcal{H}[R_{i}]\leftarrow\mathcal{H}[R_{i}]\cup\\{p_{\mathit{next}}\\}\setminus\\{p_{\mathit{current}}\\}$
12
13 end for
14 $p_{\mathit{next}}\leftarrow\mathit{get\\_next}(\mathcal{L}_{i})$
15
16 end while
17
18 end for
return _$\mathcal{H}$_
Algorithm 3 Greedy HIGHLIGHT algorithm
Greedy approach. Inspired from [9, 36, 37], we define the relevance as the
Cosine similarity between $F$ and the POIs (note that the feedback vector $F$
and the POIs are defined over the same schema), and the quality as the
diversity between the POIs. The diversity is computed using Cosine distance
between the POI attribute values. We then follow a greedy approach for each
ROI to maximize diversity while respecting a lower bound on similarity.
Algorithm 3 summarizes this approach. The similarity values are preprocessed
and organized in $\mathcal{L}_{i}$ for all POIs in $\mathcal{P}_{i}$ (line 3
in the algorithm). The algorithm starts the greedy process by initializing a
list $\mathcal{H}[R_{i}]$ with $k^{\prime}$ POIs at the top of
$\mathcal{L}_{i}$, i.e., the most similar POIs in $\mathcal{P}_{i}$ to $F$
(line 3 in the algorithm). While a time limit is not exceeded (time limit is
an input parameter which is often set to values $\leq 500ms$ [29]), the
algorithm scans $\mathcal{L}_{i}$ sequentially to find appropriate POI
replacements in $\mathcal{H}[R_{i}]$ to improve diversity (line 3 of the
algorithm). Once the greedy loop is done, the set $\mathcal{H}$ will be
returned by the algorithm, containing the highlights for all the discovered
ROIs.
Fuzzy approach. Inspired from [38, 39, 40, 3], we employ fuzzy clustering to
process all ROIs simultaneously. Algorithm 4 summarizes this approach. The
relevance is defined in the same way as the greedy approach, and the
exploration quality is defined using two factors: cohesiveness between POIs of
the same ROI (opposite of diversity, hence measured using Cosine similarity),
and representativeness, i.e., the sum of euclidean distances between ROI
centroids. We use a weighted sum over relevance and quality where the weights
are user-defined parameters ($w_{1}$ to $w_{3}$ in line 4 of Algorithm 4).
Through several trial-and-error tests and user studies in previous works [40,
39], we found that the most ideal set of weights are $w_{1}=0.5$, $w_{2}=0.25$
and $w_{3}=0.25$. The algorithm refines the centroids of ROIs iteratively
until convergence (lines 4 to 4 in Algorithm 4). Then $k^{\prime}$ most
probable points (in fuzzy clustering semantics) will be returned as highlights
for each centroid (line 4 in Algorithm 4).
Which approach to choose? We conjecture that the greedy approach is more
appropriate for the bird’s-eye view exploration, which mainly refers to early
stages of the exploration where the user is trying to get acquainted with the
geospatial data by random explorations. In this case, ROIs do not necessarily
need to be related and may represent independent future directions. However,
in the case of more focused exploration scenarios, the fuzzy approach would be
able to deliver highlights with more coverage over the whole regions of
interest. We plan to validate these hypotheses via extensive qualitative
evaluations.
Peculiar highlighting. Recall the main objective of the highlighting component
is to return out-of-sight POIs as future analysis directions. This simply
means that the neighborhoods that have been already investigated by the user
are less peculiar, and the POIs within those regions may not be as interesting
as the ones in unexplored regions. Given an ROI $R_{i}$, we define its
peculiarity score as follows.
$\mathit{peculiarity}(R_{i})=\mathit{Cosine\\_similarity}(F,\overrightarrow{\mathcal{P}_{i}})$
(4)
Input: Discovered ROIs $\mathcal{R}$, user feedback vector $F$, $k$
Output: Highlights $\mathcal{H}$
1 $\mathcal{P}_{\mathit{all}}\leftarrow\emptyset$
2 for _each discovered ROI $R_{i}\in\mathcal{R}$_ do
3
$\mathcal{P}_{\mathit{all}}\leftarrow\mathcal{P}_{\mathit{all}}\cup\mathit{match\\_points}(R_{i})$
4 $\mathit{centroid}_{\mathit{old}}\leftarrow\emptyset$
5
$\mathit{centroid}_{\mathit{current}}\leftarrow\mathit{get\\_centroid}(R_{i})$
6
7 end for
8$k^{\prime}\leftarrow k\times\mathit{peculiarity}(R_{i})$
9 while
_$\delta(\mathit{centroid}_{\mathit{old}},\mathit{centroid}_{\mathit{current}})$
is significant_ do
10
$\mathit{centroid}_{\mathit{old}}\leftarrow\mathit{centroid}_{\mathit{current}}$
11
$\mathit{centroid}_{\mathit{current}}\leftarrow\mathit{argmax}_{k^{\prime}}(w_{1}\times\mathit{relevance}(\mathcal{P}_{\mathit{all}}),$
$w_{2}\times\mathit{cohesiveness}(\mathcal{P}_{\mathit{all}}),$
$w_{3}\times\mathit{representativeness}(\mathcal{P}_{\mathit{all}}))$
12
13 end while
14
$\mathcal{H}\leftarrow\mathit{fuzzy\\_clusters}(\mathit{centroid}_{\mathit{current}})$
return _$\mathcal{H}$_
Algorithm 4 Fuzzy HIGHLIGHT algorithm
We then enrich the traditional $k$ parameter with the peculiarity semantics as
follows: $k^{\prime}=\lfloor k\times\mathit{peculiarity}(R_{i})\rfloor$ (line
3 in Algorithm 3 and line 4 in Algorithm 4). Note that $k^{\prime}$ is the
peculiarity-aware version of the $k$. This simply means that $k^{\prime}$ is
lower for less peculiar ROIs, and hence less POIs will be highlighted in them.
For instance, in case $F$ has already captured feedback about two-bedroom
home-stays and an ROI has only amenities with two bedrooms, that ROI will
receive a low peculiarity score, and hence very few POIs will be highlighted
in it.
## 5 Discussion on Evaluation
We plan to perform the following evaluation strategies to validate the
usefulness of IRIDEF:
Single-shot quantitative analysis. Although our approach is multi-shot, we can
consider only one iteration of our approach (CAPTURE $\rightarrow$ DISCOVER
$\rightarrow$ HIGHLIGHT) and see how the components behave in this single
iteration. The behavior can be captured through execution time and memory
consumption, as well as precision. We average over several single-shot runs.
The feedback will be captured through crowdsourcing campaigns.
Simulation study. We simulate interactive scenarios using virtual agents and
measure accumulated quality such as precision, hit ratio, and diversity.
User study. We also perform an in-depth lab study and an in-breadth
crowdsourcing study to survey real users about their perception on the
resulting regions (ROIs) and the highlights (POIs).
## 6 Conclusion and Future Work
In this paper, we present IRIDEF, an approach to interactively discover
regions of interest (ROIs) using exploratory feedback. The exploratory
feedback is captured from mouse moves over the geographical map while
analyzing spatial data. We propose a novel polygon-based mining algorithm
which returns a few highlights (POIs) in conformance with user’s exploratory
preferences. The highlights enable users to have a better understanding of
what to focus on in the followup steps in their analysis scenarios. We plan to
extend IRIDEF in several directions, such as the incorporation of multi-modal
exploratory feedback and the generation of sequential highlights as a
mobility-aware guidance.
## Acknowledgment
The author thanks Thibaut Thonet, Sruthi Viswanathan, Fabien Guillot, Jean-
Michel Renders, and Placido Neto for their constructive comments in the
process of writing this paper.
## References
* [1] John F. Roddick, Max J. Egenhofer, Erik G. Hoel, Dimitris Papadias, and Betty Salzberg. Spatial, temporal and spatio-temporal databases - hot issues and directions for phd research. SIGMOD Record, 33(2):126–131, 2004.
* [2] Zheng Xu, Yunhuai Liu, Neil Yen, Lin Mei, Xiangfeng Luo, Xiao Wei, and Chuanping Hu. Crowdsourcing based description of urban emergency events using social media big data. TCC, 2016.
* [3] Sihem Amer-Yahia, Ria M Borromeo, Shady Elbassuoni, Behrooz Omidvar-Tehrani, and Sruthi Viswanathan. Interactive generation and customization of travel packages for individuals and groups. In IUI, 2020.
* [4] Hangil Chung, Daniel Freund, and David B Shmoys. Bike angels: An analysis of citi bike’s incentive program. In SIGCAS. ACM, 2018.
* [5] Kaiyu Feng, Gao Cong, Sourav S Bhowmick, Wen-Chih Peng, and Chunyan Miao. Towards best region search for data exploration. In SIGMOD, 2016.
* [6] Cicero A. L. Pahin, Behrooz Omidvar-Tehrani, Sihem Amer-Yahia, Valerie Siroux, Jean-Louis Pepin, Jean-Christian Botel, and Comba Joao. COVIZ: A system for visual formation and exploration of patient cohorts. In VLDB, 2019.
* [7] Ori Bar El, Tova Milo, and Amit Somech. Towards autonomous, hands-free data exploration. In CIDR, 2020.
* [8] Arnab Nandi and H. V. Jagadish. Guided interaction: Rethinking the query-result paradigm. Proc. VLDB Endow., 4(12):1466–1469, 2011.
* [9] Behrooz Omidvar-Tehrani, Plácido A. Souza Neto, Francisco B. Silva Júnior, and Felipe M. Freire Pontes. Exploration of interesting dense regions on spatial data. In Proceedings of the Workshops of the EDBT/ICDT 2020 Joint Conference. CEUR-WS.org, 2020.
* [10] Lilong Jiang, Michael Mandel, and Arnab Nandi. Gesturequery: A multitouch database query interface. Proc. VLDB Endow., 6(12):1342–1345, 2013.
* [11] Sruthi Viswanathan, Fabien Guillot, and Maria Antonietta Grasso. What is natural?: Challenges and opportunities for conversational recommender systems. In María Inés Torres, Stephan Schlögl, Leigh Clark, and Martin Porcheron, editors, Proceedings of the 2nd Conference on Conversational User Interfaces, CUI 2020, Bilbao, Spain, July 22-24, 2020, pages 40:1–40:4. ACM, 2020.
* [12] Georg Buscher, Andreas Dengel, Ralf Biedert, and Ludger V Elst. Attentive documents: Eye tracking as implicit feedback for information retrieval and beyond. ACM Transactions on Interactive Intelligent Systems (TiiS), 1(2):1–30, 2012.
* [13] J. Liang and M. L. Huang. Highlighting in information visualization: A survey. In 2010 14th International Conference Information Visualisation, July 2010.
* [14] Anthony C. Robinson. Highlighting in geovisualization. Cartography and Geographic Information Science, 38(4):373–383, 2011\.
* [15] Kanit Wongsuphasawat, Dominik Moritz, Anushka Anand, Jock Mackinlay, Bill Howe, and Jeffrey Heer. Voyager: Exploratory analysis via faceted browsing of visualization recommendations. TVCG, 22(1), 2016.
* [16] Jie Bao, Yu Zheng, David Wilkie, and Mohamed Mokbel. Recommendations in location-based social networks: a survey. GeoInformatica, 19(3):525–565, 2015.
* [17] Justin J. Levandoski, Mohamed Sarwat, Ahmed Eldawy, and Mohamed F. Mokbel. Lars: A location-aware recommender system. In ICDE, pages 450–461, 2012.
* [18] Mansurul Bhuiyan, Snehasis Mukhopadhyay, and Mohammad Al Hasan. Interactive pattern mining on hidden data: a sampling-based solution. In Proceedings of the 21st ACM international conference on Information and knowledge management, pages 95–104. ACM, 2012.
* [19] Dong Xin, Xuehua Shen, Qiaozhu Mei, and Jiawei Han. Discovering interesting patterns through user’s interactive feedback. In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 773–778. ACM, 2006.
* [20] Kyriaki Dimitriadou, Olga Papaemmanouil, and Yanlei Diao. Aide: an active learning-based approach for interactive data exploration. IEEE Transactions on Knowledge and Data Engineering, 28(11):2842–2856, 2016.
* [21] Niranjan Kamat, Prasanth Jayachandran, Karthik Tunga, and Arnab Nandi. Distributed and interactive cube exploration. In ICDE, 2014.
* [22] Behrooz Omidvar-Tehrani, Sihem Amer-Yahia, and Alexandre Termier. Interactive user group analysis. In CIKM, pages 403–412. ACM, 2015.
* [23] Mario Boley, Michael Mampaey, Bo Kang, Pavel Tokmakov, and Stefan Wrobel. One click mining: Interactive local pattern discovery through implicit preference and performance learning. In Proceedings of the ACM SIGKDD Workshop on Interactive Data Exploration and Analytics, pages 27–35. ACM, 2013.
* [24] Eoin Mac Aoidh, Michela Bertolotto, and David C. Wilson. Analysis of implicit interest indicators for spatial data. In 15th ACM International Symposium on Geographic Information Systems, ACM-GIS 2007, November 7-9, 2007, Seattle, Washington, USA, Proceedings, page 47, 2007.
* [25] Andrea Ballatore and Michela Bertolotto. Semantically enriching vgi in support of implicit feedback analysis. In Katsumi Tanaka, Peter Fröhlich, and Kyoung-Sook Kim, editors, Web and Wireless Geographical Information Systems, pages 78–93, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg.
* [26] Nathan N. Liu, Evan W. Xiang, Min Zhao, and Qiang Yang. Unifying explicit and implicit feedback for collaborative filtering. In Proceedings of the 19th ACM International Conference on Information and Knowledge Management, CIKM ’10, pages 1445–1448, New York, NY, USA, 2010. ACM.
* [27] Behrooz Omidvar-Tehrani, Arnab Nandi, Nicholas Meyer, Dalton Flanagan, and Seth Young. DV8: interactive analysis of aviation data. In 33rd IEEE International Conference on Data Engineering, ICDE 2017, San Diego, CA, USA, April 19-22, 2017, pages 1411–1412. IEEE Computer Society, 2017.
* [28] Shagun Jhaver, Yoni Karpfen, and Judd Antin. Algorithmic anxiety and coping strategies of airbnb hosts. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems, page 421. ACM, 2018.
* [29] Jean-Daniel Fekete and Romain Primet. Progressive analytics: A computation paradigm for exploratory data analysis. arXiv preprint arXiv:1607.05162, 2016.
* [30] John Krumm and Eric Horvitz. Predestination: Inferring destinations from partial trajectories. In UbiComp, 2006.
* [31] Sobhan Moosavi, Behrooz Omidvar-Tehrani, R Bruce Craig, Arnab Nandi, and Rajiv Ramnath. Characterizing driving context from driver behavior. In Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 1–4, 2017.
* [32] Derya Birant and Alp Kut. ST-DBSCAN: An algorithm for clustering spatial-temporal data. Data Knowl. Eng., 60(1):208–221, January 2007.
* [33] Ronald L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Info. Pro. Lett., 1:132–133, 1972.
* [34] Luc Devroye and Godfried T Toussaint. A note on linear expected time algorithms for finding convex hulls. Computing, 26(4):361–366, 1981.
* [35] Raphael A. Finkel and Jon Louis Bentley. Quad trees a data structure for retrieval on composite keys. Acta informatica, 4(1):1–9, 1974.
* [36] Behrooz Omidvar-Tehrani, Plácido A. Souza Neto, Felipe M. Freire Pontes, and Francisco Bento da Silva Júnior. Geoguide: An interactive guidance approach for spatial data. In IEEE Smart Data, pages 1112–1117, 2017.
* [37] Behrooz Omidvar-Tehrani, Sruthi Viswanathan, and Jean-Michel Renders. Interactive and explainable point-of-interestrecommendation using look-alike groups. In SIGSPATIAL, 2020.
* [38] Vincent Leroy, Sihem Amer-Yahia, Eric Gaussier, and Hamid Mirisaee. Building representative composite items. In Proceedings of the 24th ACM International on Conference on Information and Knowledge Management, pages 1421–1430, 2015.
* [39] Manish Singh, Ria Mae Borromeo, Anas Hosami, Sihem Amer-Yahia, and Shady Elbassuoni. Customizing travel packages with interactive composite items. In DSAA, pages 137–145. IEEE, 2017.
* [40] Sihem Amer-Yahia, Shady Elbassuoni, Behrooz Omidvar-Tehrani, Ria Borromeo, and Mehrdad Farokhnejad. Grouptravel: Customizing travel packages for groups. In EDBT, 2019.
|
11institutetext: Department of Computing, Imperial College London, United
Kingdom
# On Polymorphic Sessions and Functions
A Tale of Two (Fully Abstract) Encodings
Bernardo Toninho Nobuko Yoshida
###### Abstract
This work exploits the logical foundation of session types to determine what
kind of type discipline for the $\pi$-calculus can exactly capture, and is
captured by, $\lambda$-calculus behaviours. Leveraging the proof theoretic
content of the soundness and completeness of sequent calculus and natural
deduction presentations of linear logic, we develop the first _mutually
inverse_ and _fully abstract_ processes-as-functions and functions-as-
processes encodings between a polymorphic session $\pi$-calculus and a linear
formulation of System F. We are then able to derive results of the session
calculus from the theory of the $\lambda$-calculus: (1) we obtain a
characterisation of inductive and coinductive session types via their
algebraic representations in System F; and (2) we extend our results to
account for _value_ and _process_ passing, entailing strong normalisation.
## 1 Introduction
Dating back to Milner's seminal work [29], encodings of $\lambda$-calculus
into $\pi$-calculus are seen as essential benchmarks to examine expressiveness
of various extensions of the $\pi$-calculus. Milner's original motivation was
to demonstrate the power of link mobility by decomposing higher-order
computations into pure name passing. Another goal was to analyse functional
behaviours in a broad computational universe of concurrency and non-
determinism. While _operationally_ correct encodings of many higher-order
constructs exist, it is challenging to obtain encodings that are precise wrt
behavioural equivalence: the semantic distance between the $\lambda$-calculus
and the $\pi$-calculus typically requires either restricting process
behaviours [45] (e.g. via typed equivalences [5]) or enriching the
$\lambda$-calculus with constants that allow for a suitable characterisation
of the term equivalence induced by the behavioural equivalence on processes
[43].
Encodings in $\pi$-calculi also gave rise to new typing disciplines: Session
types [20, 22], a typing system that is able to ensure deadlock-freedom for
communication protocols between two or more parties [23], were originally
motivated ``from process encodings of various data structures in an
asynchronous version of the $\pi$-calculus'' [21]. Recently, a propositions-
as-types correspondence between linear logic and session types [8, 9, 54] has
produced several new developments and logically-motivated techniques [49, 54,
7, 26] to augment both the theory and practice of session-based message-
passing concurrency. Notably, parametric session polymorphism [7] (in the
sense of Reynolds [41]) has been proposed and a corresponding abstraction
theorem has been shown.
Our work expands upon the proof theoretic consequences of this propositions-
as-types correspondence to address the problem of how to exactly match the
behaviours induced by session $\pi$-calculus encodings of the
$\lambda$-calculus with those of the $\lambda$-calculus. We develop mutually
inverse and fully abstract encodings (up to typed observational congruences)
between a polymorphic session-typed $\pi$-calculus and the polymorphic
$\lambda$-calculus. The encodings arise from the proof theoretic content of
the equivalence between sequent calculus (i.e. the session calculus) and
natural deduction (i.e. the $\lambda$-calculus) for _second-order_
intuitionistic linear logic, greatly generalising [49]. While fully abstract
encodings between $\lambda$-calculi and $\pi$-calculi have been proposed (e.g.
[5, 43]), our work is the first to consider a two-way, _both_ mutually inverse
_and_ fully abstract embedding between the two calculi by crucially exploiting
the linear logic-based session discipline. This also sheds some definitive
light on the nature of concurrency in the (logical) session calculi, which
exhibit ``don't care'' forms of non-determinism (e.g. processes may race on
stateless replicated servers) rather than ``don't know'' non-determinism
(which requires less harmonious logical features [2]).
In the spirit of Gentzen [14], we use our encodings as a tool to study non-
trivial properties of the session calculus, deriving them from results in the
$\lambda$-calculus: We show the existence of inductive and coinductive
sessions in the polymorphic session calculus by considering the representation
of initial $F$-algebras and final $F$-coalgebras [28] in the polymorphic
$\lambda$-calculus [1, 19] (in a linear setting [6]). By appealing to full
abstraction, we are able to derive processes that satisfy the necessary
algebraic properties and thus form adequate _uniform_ representations of
inductive and coinductive session types. The derived algebraic properties
enable us to reason about standard data structure examples, providing a
logical justification to typed variations of the representations in [30].
We systematically extend our results to a session calculus with $\lambda$-term
and process passing (the latter being the core calculus of [50], inspired by
Benton's LNL [4]). By showing that our encodings naturally adapt to this
setting, we prove that it is possible to encode higher-order process passing
in the first-order session calculus fully abstractly, providing a typed and
proof-theoretically justified re-envisioning of Sangiorgi's encodings of
higher-order $\pi$-calculus [46]. In addition, the encoding instantly provides
a strong normalisation property of the higher-order session calculus.
Contributions and the outline of our paper are as follows:
§ 3.1
develops a functions-as-processes encoding of a linear formulation of System
F, Linear-F, using a logically motivated polymorphic session $\pi$-calculus,
Poly$\pi$, and shows that the encoding is operationally sound and complete.
§ 3.2
develops a processes-as-functions encoding of Poly$\pi$ into Linear-F, arising
from the completeness of the sequent calculus wrt natural deduction, also
operationally sound and complete.
§ 3.3
studies the relationship between the two encodings, establishing they are
_mutually inverse_ and _fully abstract_ wrt typed congruence, the first two-
way embedding satisfying _both_ properties.
§ 4
develops a _faithful_ representation of inductive and coinductive session
types in Poly$\pi$ via the encoding of initial and final (co)algebras in the
polymorphic $\lambda$-calculus. We demonstrate a use of these algebraic
properties via examples.
§ 4.2,4.3
study term-passing and process-passing session calculi, extending our
encodings to provide embeddings into the first-order session calculus. We show
full abstraction and mutual inversion results, and derive strong normalisation
of the higher-order session calculus from the encoding.
In order to introduce our encodings, we first overview Poly$\pi$, its typing
system and behavioural equivalence (§ 2). We discuss related work and conclude
with future work (§ 5). Detailed proofs can be found in [52].
## 2 Polymorphic Session $\pi$-Calculus
This section summarises the polymorphic session $\pi$-calculus [7], dubbed
Poly$\pi$, arising as a process assignment to second-order linear logic [15],
its typing system and behavioural equivalences.
### 2.1 Processes and Typing
Syntax. Given an infinite set $\Lambda$ of names $x,y,z,u,v$, the grammar of
processes $P,Q,R$ and session types $A,B,C$ is defined by:
$\begin{array}[]{l}\begin{array}[]{lclllllllllllllllll}P,Q,R&::=&x\langle
y\rangle.P&\mid&x(y).P&\mid&P\mid
Q&\mid&(\mathbf{\nu}y)P&\mid&[x\leftrightarrow y]&\mid&{\bf 0}\\\\[2.84526pt]
&\mid&x\langle
A\rangle.P&\mid&x(Y).P&\mid&x.\mathsf{inl};P&\mid&x.\mathsf{inr};P&\mid&x.\mathsf{case}(P,Q)&\mid&!x(y).P\\\\[2.84526pt]
\end{array}\\\\[2.84526pt] \begin{array}[]{lcl}A,B&::=&\mathbf{1}\mid
A\multimap B\mid A\otimes B\mid A\with B\mid A\oplus B\mid\,\,!A\mid\forall
X.A\mid\exists X.A\mid X\end{array}\end{array}$
$x\langle y\rangle.P$ denotes the output of channel $y$ on $x$ with
continuation process $P$; $x(y).P$ denotes an input along $x$, bound to $y$ in
$P$; $P\mid Q$ denotes parallel composition; $(\mathbf{\nu}y)P$ denotes the
restriction of name $y$ to the scope of $P$; ${\bf 0}$ denotes the inactive
process; $[x\leftrightarrow y]$ denotes the linking of the two channels $x$
and $y$ (implemented as renaming); $x\langle A\rangle.P$ and $x(Y).P$ denote
the sending and receiving of a _type_ $A$ along $x$ bound to $Y$ in $P$ of the
receiver process; $x.\mathsf{inl};P$ and $x.\mathsf{inr};P$ denote the
emission of a selection between the $\mathsf{l}$eft or $\mathsf{r}$ight branch
of a receiver $x.\mathsf{case}(P,Q)$ process; $!x(y).P$ denotes an input-
guarded replication, that spawns replicas upon receiving an input along $x$.
We often abbreviate $(\mathbf{\nu}y)x\langle y\rangle.P$ to
$\overline{x}\langle y\rangle.P$ and omit trailing ${\bf 0}$ processes. By
convention, we range over linear channels with $x,y,z$ and shared channels
with $u,v,w$.
The syntax of session types is that of (intuitionistic) linear logic
propositions which are assigned to channels according to their usages in
processes: $\mathbf{1}$ denotes the type of a channel along which no further
behaviour occurs; $A\multimap B$ denotes a session that waits to receive a
channel of type $A$ and will then proceed as a session of type $B$; dually,
$A\otimes B$ denotes a session that sends a channel of type $A$ and continues
as $B$; $A\with B$ denotes a session that offers a choice between proceeding
as behaviours $A$ or $B$; $A\oplus B$ denotes a session that internally
chooses to continue as either $A$ or $B$, signalling appropriately to the
communicating partner; $!A$ denotes a session offering an unbounded (but
finite) number of behaviours of type $A$; $\forall X.A$ denotes a polymorphic
session that receives a type $B$ and behaves uniformly as $A\\{B/X\\}$;
dually, $\exists X.A$ denotes an existentially typed session, which emits a
type $B$ and behaves as $A\\{B/X\\}$.
Operational Semantics. The operational semantics of our calculus is presented
as a standard labelled transition system (Fig. 1) in the style of the _early_
system for the $\pi$-calculus [46].
In the remainder of this work we write $\equiv$ for a standard $\pi$-calculus
structural congruence extended with the clause $[x\leftrightarrow
y]\equiv[y\leftrightarrow x]$. In order to streamline the presentation of
observational equivalence [36, 7], we write $\equiv_{!}$ for structural
congruence extended with the so-called sharpened replication axioms [46],
which capture basic equivalences of replicated processes (and are present in
the proof dynamics of the exponential of linear logic). A transition
$P\xrightarrow{~{}\alpha~{}}Q$ denotes that $P$ may evolve to $Q$ by
performing the action represented by label $\alpha$. An action $\alpha$
($\overline{\alpha}$) requires a matching $\overline{\alpha}$ ($\alpha$) in
the environment to enable progress. Labels include: the silent internal action
$\tau$, output and bound output actions ($\overline{x\langle y\rangle}$ and
$\overline{(\nu z)x\langle z\rangle}$); input action $x(y)$; the binary choice
actions ($x.\mathsf{inl}$, $\overline{x.\mathsf{inl}}$, $x.\mathsf{inr}$, and
$\overline{x.\mathsf{inr}}$); and output and input actions of types
($\overline{x\langle A\rangle}$ and $x(A)$).
The labelled transition relation is defined by the rules in Fig. 1, subject to
the side conditions: in rule $(\mathsf{res})$, we require $y\not\in\mbox{\it
fn}(\alpha)$; in rule $(\mathsf{par})$, we require $\mbox{\it
bn}(\alpha)\cap\mbox{\it fn}(R)=\emptyset$; in rule $(\mathsf{close})$, we
require $y\not\in\mbox{\it fn}(Q)$. We omit the symmetric versions of
$(\mathsf{par})$, $(\mathsf{com})$, $(\mathsf{lout})$, $(\mathsf{lin})$,
$(\mathsf{close})$ and closure under $\alpha$-conversion. We write
$\rho_{1}\rho_{2}$ for the composition of relations $\rho_{1},\rho_{2}$. We
write $\xrightarrow{}$ to stand for $\xrightarrow{\tau}\equiv$. Weak
transitions are defined as usual: we write
$\stackrel{{\scriptstyle}}{{\Longrightarrow}}$ for the reflexive, transitive
closure of $\xrightarrow{\tau}$ and $\xrightarrow{}^{+}$ for the transitive
closure of $\xrightarrow{\tau}$. Given $\alpha\neq\tau$, notation
$\stackrel{{\scriptstyle\alpha}}{{\Longrightarrow}}$ stands for
$\stackrel{{\scriptstyle~{}}}{{\Longrightarrow}}\xrightarrow{\alpha}\stackrel{{\scriptstyle~{}}}{{\Longrightarrow}}$
and $\stackrel{{\scriptstyle\tau}}{{\Longrightarrow}}$ stands for
$\stackrel{{\scriptstyle}}{{\Longrightarrow}}$.
$\begin{array}[]{ccccc}{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{out}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
22.60664pt\vbox{\vbox{}\hbox{\hskip-22.60663pt\hbox{\hbox{$\displaystyle\displaystyle{x\langle
y\rangle.P\xrightarrow{\overline{x\langle y\rangle}}P}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{in}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
40.2386pt\vbox{\vbox{}\hbox{\hskip-40.2386pt\hbox{\hbox{$\displaystyle\displaystyle{x(y).P\xrightarrow{x(z)}P\\{\raisebox{1.93748pt}{\small$\displaystyle
z$}\\!/\mbox{\small$\displaystyle y$}\\}}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{outT}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
23.61392pt\vbox{\vbox{}\hbox{\hskip-23.61392pt\hbox{\hbox{$\displaystyle\displaystyle{x\langle
A\rangle.P\xrightarrow{\overline{x\langle A\rangle}}P}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{inT}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
45.42505pt\vbox{\vbox{}\hbox{\hskip-45.42505pt\hbox{\hbox{$\displaystyle\displaystyle{x(Y).P\xrightarrow{x(B)}P\\{\raisebox{1.93748pt}{\small$\displaystyle
B$}\\!/\mbox{\small$\displaystyle Y$}\\}}$}}}}}}$}}}\\\\[2.84526pt]
\begin{array}[]{ll}\begin{array}[]{c}{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{lout}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
22.60002pt\vbox{\vbox{}\hbox{\hskip-22.60002pt\hbox{\hbox{$\displaystyle\displaystyle{x.\mathsf{inl};P\xrightarrow{\overline{x.\mathsf{inl}}}P}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{id}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
52.91936pt\vbox{\vbox{}\hbox{\hskip-52.91934pt\hbox{\hbox{$\displaystyle\displaystyle{(\nu
x)([x\leftrightarrow y]\mid
P)\xrightarrow{\tau}P\\{\raisebox{1.93748pt}{\small$\displaystyle
y$}\\!/\mbox{\small$\displaystyle x$}\\}}$}}}}}}$}}}\\\\[2.84526pt]
{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{lin}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
38.5044pt\vbox{\vbox{}\hbox{\hskip-38.5044pt\hbox{\hbox{$\displaystyle\displaystyle{x.\mathsf{case}(P,Q)\xrightarrow{x.\mathsf{inl}}P}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{rep}$)}}}}}}\hbox{$\displaystyle\vbox{\hbox{\hskip
58.69226pt\vbox{\vbox{}\hbox{\hskip-58.69226pt\hbox{\hbox{$\displaystyle\displaystyle{!x(y).P\xrightarrow{x(z)}P\\{\raisebox{1.93748pt}{\small$\displaystyle
z$}\\!/\mbox{\small$\displaystyle y$}\\}\mid!x(y).P}$}}}}}}$}}}\\\\[2.84526pt]
\end{array}\quad\begin{array}[]{l}{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{open}$)}}}}}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
11.82158pt\vbox{\hbox{\hskip-11.82156pt\hbox{\hbox{$\displaystyle\displaystyle{P\xrightarrow{\overline{x\langle
y\rangle}}Q}$}}}\vbox{}}}\over\hbox{\hskip
19.91222pt\vbox{\vbox{}\hbox{\hskip-19.91222pt\hbox{\hbox{$\displaystyle\displaystyle{(\mathbf{\nu}y)P\xrightarrow{\overline{(\mathbf{\nu}y)x\langle
y\rangle}}Q}$}}}}}}$}}}\\\ \end{array}\\\ \end{array}\\\
{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{close}$)}}}}}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
32.46872pt\vbox{\hbox{\hskip-32.46872pt\hbox{\hbox{$\displaystyle\displaystyle{P\xrightarrow{\overline{(\mathbf{\nu}y)x\langle
y\rangle}}P^{\prime}\,\,Q\xrightarrow{x(y)}Q^{\prime}}$}}}\vbox{}}}\over\hbox{\hskip
36.33694pt\vbox{\vbox{}\hbox{\hskip-36.33694pt\hbox{\hbox{$\displaystyle\displaystyle{P\mid
Q\xrightarrow{\tau}(\mathbf{\nu}y)(P^{\prime}\mid
Q^{\prime})}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{par}$)}}}}}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
12.20023pt\vbox{\hbox{\hskip-12.20021pt\hbox{\hbox{$\displaystyle\displaystyle{P\xrightarrow{\alpha}Q}$}}}\vbox{}}}\over\hbox{\hskip
25.10335pt\vbox{\vbox{}\hbox{\hskip-25.10333pt\hbox{\hbox{$\displaystyle\displaystyle{P\mid
R\xrightarrow{\alpha}Q\mid R}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{com}$)}}}}}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
26.90778pt\vbox{\hbox{\hskip-26.90778pt\hbox{\hbox{$\displaystyle\displaystyle{P\xrightarrow{\overline{\alpha}}P^{\prime}\,\,Q\xrightarrow{\alpha}Q^{\prime}}$}}}\vbox{}}}\over\hbox{\hskip
24.7463pt\vbox{\vbox{}\hbox{\hskip-24.74629pt\hbox{\hbox{$\displaystyle\displaystyle{P\mid
Q\xrightarrow{\tau}P^{\prime}\mid Q^{\prime}}$}}}}}}$}}}\hskip
8.5359pt{\vbox{\hbox{\hbox{\small\sc\mbox{{\scriptsize{{($\displaystyle\mathsf{res}$)}}}}}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
12.20023pt\vbox{\hbox{\hskip-12.20021pt\hbox{\hbox{$\displaystyle\displaystyle{P\xrightarrow{\alpha}Q}$}}}\vbox{}}}\over\hbox{\hskip
28.38152pt\vbox{\vbox{}\hbox{\hskip-28.3815pt\hbox{\hbox{$\displaystyle\displaystyle{(\mathbf{\nu}y)P\xrightarrow{\alpha}(\mathbf{\nu}y)Q}$}}}}}}$}}\par}\end{array}\vspace{-4ex}$
Figure 1: Labelled Transition System.
Typing System. The typing rules of Poly$\pi$ are given in Fig. 2, following
[7]. The rules define the judgment $\Omega;\Gamma;\Delta\vdash P::z{:}A$,
denoting that process $P$ offers a session of type $A$ along channel $z$,
using the _linear_ sessions in $\Delta$, (potentially) using the unrestricted
or _shared_ sessions in $\Gamma$, with polymorphic type variables maintained
in $\Omega$. We use a well-formedness judgment $\Omega\vdash A\,\mathsf{type}$
which states that $A$ is well-formed wrt the type variable environment
$\Omega$ (i.e. $\mathit{fv}(A)\subseteq\Omega$). We often write $T$ for the
right-hand side typing $z{:}A$, $\cdot$ for the empty context and
$\Delta,\Delta^{\prime}$ for the union of contexts $\Delta$ and
$\Delta^{\prime}$, only defined when $\Delta$ and $\Delta^{\prime}$ are
disjoint. We write $\cdot\vdash P::T$ for $\cdot;\cdot;\cdot\vdash P::T$.
$\begin{array}[]{c}\raise
10.5pt\hbox{$\hbox{$\hbox{\small\sc$({{\multimap}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
48.95317pt\vbox{\hbox{\hskip-48.95316pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\vdash
P::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
58.4938pt\vbox{\vbox{}\hbox{\hskip-58.4938pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z(x).P::z{:}A\multimap B}$}}}}}}$}}\hbox{}$}\quad\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\otimes}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
76.38904pt\vbox{\hbox{\hskip-76.38904pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1}\vdash
P::y{:}A\quad\Omega;\Gamma;\Delta_{2}\vdash
Q::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
82.05086pt\vbox{\vbox{}\hbox{\hskip-82.05084pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1},\Delta_{2}\vdash(\mathbf{\nu}x)z\langle
y\rangle.(P\mid Q)::z{:}A\otimes B}$}}}}}}$}}\hbox{}$}\\\\[9.00002pt] \raise
10.5pt\hbox{$\hbox{$\hbox{\small\sc$({{\forall}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
43.07347pt\vbox{\hbox{\hskip-43.07346pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega,X;\Gamma;\Delta\vdash
P::z{:}A}$}}}\vbox{}}}\over\hbox{\hskip
57.94534pt\vbox{\vbox{}\hbox{\hskip-57.94534pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z(X).P::z{:}\forall X.A}$}}}}}}$}}\hbox{}$}\quad\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\forall}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
90.71165pt\vbox{\hbox{\hskip-90.71165pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega\vdash
B\,\mathsf{type}\quad\Omega;\Gamma;\Delta,x{:}A\\{B/X\\}\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
70.03331pt\vbox{\vbox{}\hbox{\hskip-70.03331pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}\forall
X.A\vdash x\langle B\rangle.P::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[9.00002pt] \raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\exists}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
79.97665pt\vbox{\hbox{\hskip-79.97665pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega\vdash
B\,\mathsf{type}\quad\Omega;\Gamma;\Delta\vdash
P::z{:}A\\{B/X\\}}$}}}\vbox{}}}\over\hbox{\hskip
57.89207pt\vbox{\vbox{}\hbox{\hskip-57.89206pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z\langle B\rangle.P::z{:}\exists X.A}$}}}}}}$}}\hbox{}$}\quad\raise
10.5pt\hbox{$\hbox{$\hbox{\small\sc$({{\exists}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
54.93346pt\vbox{\hbox{\hskip-54.93346pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega,X;\Gamma;\Delta,x{:}A\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
70.0866pt\vbox{\vbox{}\hbox{\hskip-70.08658pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}\exists
X.A\vdash x(X).P::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[9.00002pt] \raise
10.5pt\hbox{$\hbox{$\hbox{\small\sc$(\mathsf{id})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
0.75pt\vbox{\hbox{\hskip-0.75pt\hbox{\hbox{$\displaystyle\displaystyle{\,}$}}}\vbox{}}}\over\hbox{\hskip
53.78755pt\vbox{\vbox{}\hbox{\hskip-53.78754pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;x{:}A\vdash[x\leftrightarrow
z]::z{:}A}$}}}}}}$}}\hbox{}$}\quad\raise
10.5pt\hbox{$\hbox{$\hbox{\small\sc$(\mathsf{cut})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
91.93913pt\vbox{\hbox{\hskip-91.93912pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1}\vdash
P::x{:}A\quad\Omega;\Gamma;\Delta_{2},x{:}A\vdash
Q::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
62.52763pt\vbox{\vbox{}\hbox{\hskip-62.52763pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1},\Delta_{2}\vdash(\mathbf{\nu}x)(P\mid
Q)::z{:}C}$}}}}}}$}}\hbox{}$}\end{array}$ Figure 2: Typing Rules (Abridged –
See [52] for all rules).
As in [8, 9, 36, 54], the typing discipline enforces that channel outputs
always have as object a _fresh_ name, in the style of the internal mobility
$\pi$-calculus [44]. We clarify a few of the key rules: Rule
${{\forall}\mathsf{R}}$ defines the meaning of (impredicative) universal
quantification over session types, stating that a session of type $\forall
X.A$ inputs a type and then behaves uniformly as $A$; dually, to use such a
session (rule ${{\forall}\mathsf{L}}$), a process must output a type $B$ which
then warrants the use of the session as type $A\\{B/X\\}$. Rule
${{\multimap}\mathsf{R}}$ captures session input, where a session of type
$A\multimap B$ expects to receive a session of type $A$ which will then be
used to produce a session of type $B$. Dually, session output (rule
${{\otimes}\mathsf{R}}$) is achieved by producing a fresh session of type $A$
(that uses a disjoint set of sessions to those of the continuation) and
outputting the fresh session along $z$, which is then a session of type $B$.
Linear composition is captured by rule $\mathsf{cut}$ which enables a process
that offers a session $x{:}A$ (using linear sessions in $\Delta_{1}$) to be
composed with a process that _uses_ that session (amongst others in
$\Delta_{2}$) to offer $z{:}C$. As shown in [7], typing entails Subject
Reduction, Global Progress, and Termination.
Observational Equivalences. We briefly summarise the typed congruence and
logical equivalence with polymorphism, giving rise to a suitable notion of
relational parametricity in the sense of Reynolds [41], defined as a
contextual logical relation on typed processes [7]. The logical relation is
reminiscent of a typed bisimulation. However, extra care is needed to ensure
well-foundedness due to impredicative type instantiation. As a consequence,
the logical relation allows us to reason about process equivalences where type
variables are not instantiated with _the same_ , but rather _related_ types.
Typed Barbed Congruence ($\cong$). We use the typed contextual congruence
from [7], which preserves _observable_ actions, called barbs. Formally,
_barbed congruence_ , noted $\cong$, is the largest equivalence on well-typed
processes that is $\tau$-closed, barb preserving, and contextually closed
under typed contexts; see [7] and [52] for the full definition.
Logical Equivalence ($\approx_{\mathtt{L}}$). The definition of logical
equivalence is no more than a typed contextual bisimulation with the following
intuitive reading: given two open processes $P$ and $Q$ (i.e. processes with
non-empty left-hand side typings), we define their equivalence by inductively
closing out the context, composing with equivalent processes offering
appropriately typed sessions. When processes are closed, we have a single
distinguished session channel along which we can perform observations, and
proceed inductively on the structure of the offered session type. We can then
show that such an equivalence satisfies the necessary fundamental properties
(Theorem 2.1).
The logical relation is defined using the candidates technique of Girard [16].
In this setting, an _equivalence candidate_ is a relation on typed processes
satisfying basic closure conditions: an equivalence candidate must be
compatible with barbed congruence and closed under forward and converse
reduction.
###### Definition 2.1 (Equivalence Candidate)
An _equivalence candidate_ $\mathcal{R}$ at $z{:}A$ and $z{:}B$, noted
$\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$, is a binary relation on processes
such that, for every $(P,Q)\in\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$ both
$\cdot\vdash P::z{:}A$ and $\cdot\vdash Q::z{:}B$ hold, together with the
following (we often write $(P,Q)\in\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$
as $P\,\mathcal{R}\,Q::z{:}A\\!\Leftrightarrow\\!B$):
1. 1.
If $(P,Q)\in\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$, $\cdot\vdash P\cong
P^{\prime}::z{:}A$, and $\cdot\vdash Q\cong Q^{\prime}::z{:}B$ then
$(P^{\prime},Q^{\prime})\in\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$.
2. 2.
If $(P,Q)\in\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$ then, for all $P_{0}$
such that $\cdot\vdash P_{0}::z{:}A$ and
$P_{0}\stackrel{{\scriptstyle}}{{\Longrightarrow}}P$, we have
$(P_{0},Q)\in\,\mathcal{R}::z{:}A\\!\Leftrightarrow\\!B$. Symmetrically for
$Q$.
To define the logical relation we rely on some auxiliary notation, pertaining
to the treatment of type variables arising due to impredicative polymorphism.
We write $\omega:\Omega$ to denote a mapping $\omega$ that assigns a closed
type to the type variables in $\Omega$. We write $\omega(X)$ for the type
mapped by $\omega$ to variable $X$. Given two mappings $\omega:\Omega$ and
$\omega^{\prime}:\Omega$, we define an equivalence candidate assignment $\eta$
between $\omega$ and $\omega^{\prime}$ as a mapping of equivalence candidate
$\eta(X)::{-}{:}\omega(X)\\!\Leftrightarrow\\!\omega^{\prime}(X)$ to the type
variables in $\Omega$, where the particular choice of a distinguished right-
hand side channel is _delayed_ (i.e. to be instantiated later on). We write
$\eta(X)(z)$ for the instantiation of the (delayed) candidate with the name
$z$. We write $\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}$ to denote that
$\eta$ is a candidate assignment between $\omega$ and $\omega^{\prime}$; and
$\hat{\omega}(P)$ to denote the application of mapping $\omega$ to $P$.
We define a sequent-indexed family of process relations, that is, a set of
pairs of processes $(P,Q)$, written $\Gamma;\Delta\vdash
P\approx_{\mathtt{L}}Q::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$,
satisfying some conditions, typed under $\Omega;\Gamma;\Delta\vdash T$, with
$\omega:\Omega$, $\omega^{\prime}:\Omega$ and
$\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}$. Logical equivalence is
defined inductively on the size of the typing contexts and then on the
structure of the right-hand side type. We show only select cases (see [52] for
the full definition).
###### Definition 2.2 (Logical Equivalence)
(Base Case) Given a type $A$ and mappings $\omega,\omega^{\prime},\eta$, we
define _logical equivalence_ , noted
$P\approx_{\mathtt{L}}Q::z{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$,
as the smallest symmetric binary relation containing all pairs of processes
$(P,Q)$ such that (i) $\cdot\vdash\hat{\omega}(P)::z{:}\hat{\omega}(A)$; (ii)
$\cdot\vdash\hat{\omega}^{\prime}(Q)::z{:}\hat{\omega}^{\prime}(A)$; and (iii)
satisfies the conditions given below:
* •
$P\approx_{\mathtt{L}}Q::z{:}X[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]\text{
iff }(P,Q)\in\eta(X)(z)$
* •
$P\approx_{\mathtt{L}}Q::z{:}A\multimap
B[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$ iff $\forall
P^{\prime},y.~{}(P\xrightarrow{z(y)}P^{\prime})\Rightarrow\exists
Q^{\prime}.Q\stackrel{{\scriptstyle z(y)}}{{\Longrightarrow}}Q^{\prime}$ s.t.
$\forall
R_{1},R_{2}.R_{1}\approx_{\mathtt{L}}R_{2}::y{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}](\nu
y)(P^{\prime}\,|\,R_{1})\approx_{\mathtt{L}}(\nu
y)(Q^{\prime}\,|\,R_{2})::z{:}B[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$
* •
$P\approx_{\mathtt{L}}Q::z{:}A\otimes
B[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$ iff $\forall
P^{\prime},y.~{}~{}(P\xrightarrow{\overline{(\nu y)z\langle
y\rangle}}P^{\prime})\Rightarrow\exists
Q^{\prime}.Q\stackrel{{\scriptstyle\overline{(\nu y)z\langle
y\rangle}}}{{\Longrightarrow}}Q^{\prime}$ s.t. $\exists
P_{1},P_{2},Q_{1},Q_{2}.\ P^{\prime}\equiv_{!}P_{1}\mid P_{2}\wedge
Q^{\prime}\equiv_{!}Q_{1}\mid Q_{2}\wedge
P_{1}\approx_{\mathtt{L}}Q_{1}::y{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]\wedge
P_{2}\approx_{\mathtt{L}}Q_{2}::z{:}B[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$
* •
$P\approx_{\mathtt{L}}Q::z{:}\forall
X.A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$ iff $\forall
B_{1},B_{2},P^{\prime},\mathcal{R}::{-}{:}B_{1}\\!\Leftrightarrow\\!B_{2}.~{}~{}(P\xrightarrow{z(B_{1})}P^{\prime})$
implies $\exists Q^{\prime}.Q\stackrel{{\scriptstyle
z(B_{2})}}{{\Longrightarrow}}Q^{\prime},~{}P^{\prime}\approx_{\mathtt{L}}Q^{\prime}::z{:}A[\eta[X\mapsto\mathcal{R}]:\omega[X\mapsto
B_{1}]\\!\Leftrightarrow\\!\omega^{\prime}[X\mapsto B_{2}]]$
(Inductive Case) Let $\Gamma,\Delta$ be non empty. Given
$\Omega;\Gamma;\Delta\vdash P::T$ and $\Omega;\Gamma;\Delta\vdash Q::T$, the
binary relation on processes $\Gamma;\Delta\vdash
P\approx_{\mathtt{L}}Q::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$
(with $\omega,\omega^{\prime}:\Omega$ and
$\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}$) is inductively defined as:
$\begin{array}[]{lcl}\Gamma;\Delta,y:A\vdash
P\approx_{\mathtt{L}}Q::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]&\mbox{
iff }&\forall R_{1},R_{2}.\mbox{ s.t.
}R_{1}\approx_{\mathtt{L}}R_{2}::y{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}],\\\
&&\Gamma;\Delta\vdash(\nu
y)(\hat{\omega}(P)\mid\hat{\omega}(R_{1}))\approx_{\mathtt{L}}(\nu
y)(\hat{\omega}^{\prime}(Q)\mid\hat{\omega}^{\prime}(R_{2}))::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]\\\\[2.84526pt]
\Gamma,u:A;\Delta\vdash
P\approx_{\mathtt{L}}Q::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]&\mbox{
iff }&\forall R_{1},R_{2}.\mbox{ s.t.
}R_{1}\approx_{\mathtt{L}}R_{2}::y{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}],\\\
&&\Gamma;\Delta\vdash(\mathbf{\nu}u)(\hat{\omega}(P)\mid!u(y).\hat{\omega}(R_{1}))\approx_{\mathtt{L}}(\mathbf{\nu}u)(\hat{\omega}^{\prime}(Q)\mid!u(y).\hat{\omega}^{\prime}(R_{2}))::T[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]\end{array}$
For the sake of readability we often omit the
$\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}$ portion of
$\approx_{\mathtt{L}}$, which is henceforth implicitly universally quantified.
Thus, we write $\Omega;\Gamma;\Delta\vdash P\approx_{\mathtt{L}}Q::z{:}A$ (or
$P\approx_{\mathtt{L}}Q$) iff the two given processes are logically equivalent
for all consistent instantiations of its type variables.
It is instructive to inspect the clause for type input ($\forall X.A$): the
two processes must be able to match inputs of any pair of _related_ types
(i.e. types related by a candidate), such that the continuations are related
at the open type $A$ with the appropriate type variable instantiations,
following Girard [16]. The power of this style of logical relation arises from
a combination of the extensional flavour of the equivalence and the fact that
polymorphic equivalences do not require the same type to be instantiated in
both processes, but rather that the types are _related_ (via a suitable
equivalence candidate relation).
###### Theorem 2.1 (Properties of Logical Equivalence [7])
Parametricity:
If $\Omega;\Gamma;\Delta\vdash P::z{:}A$ then, for all
$\omega,\omega^{\prime}:\Omega$ and
$\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}$, we have
$\Gamma;\Delta\vdash\hat{\omega}(P)\approx_{\mathtt{L}}\hat{\omega^{\prime}}(P)::z{:}A[\eta:\omega\\!\Leftrightarrow\\!\omega^{\prime}]$.
Soundness:
If $\Omega;\Gamma;\Delta\vdash P\approx_{\mathtt{L}}Q::z{:}A$ then
$\mathcal{C}[P]\cong\mathcal{C}[Q]::z{:}A$, for any closing $\mathcal{C}[-]$.
Completeness:
If $\Omega;\Gamma;\Delta\vdash P\cong Q::z{:}A$ then
$\Omega;\Gamma;\Delta\vdash P\approx_{\mathtt{L}}Q::z{:}A$.
## 3 To Linear-F and Back
We now develop our mutually inverse and fully abstract encodings between
Poly$\pi$ and a linear polymorphic $\lambda$-calculus [55] that we dub
Linear-F. We first introduce the syntax and typing of the linear
$\lambda$-calculus and then proceed to detail our encodings and their
properties (we omit typing ascriptions from the existential polymorphism
constructs for readability).
###### Definition 3.1 (Linear-F)
The syntax of terms $M,N$ and types $A,B$ of Linear-F is given below.
$\small\begin{array}[]{lcl}M,N&::=&\lambda x{:}A.M\mid M\,N\mid\langle
M\otimes N\rangle\mid\mathsf{let}\,x\otimes
y=M\,\mathsf{in}\,N\mid\,!M\mid\mathsf{let}\,!u=M\,\mathsf{in}\,N\mid\Lambda
X.M\\\\[2.84526pt]
&\mid&M[A]\mid\mathsf{pack}\,A\,\mathsf{with}\,M\mid\mathsf{let}\,(X,y)=M\,\mathsf{in}\,N\mid\mathsf{let}\,\mathbf{1}=M\,\mathsf{in}\,N\mid\langle\rangle\mid\mathsf{T}\mid\mathsf{F}\\\\[4.5pt]
A,B&::=&A\multimap B\mid A\otimes B\mid\,!A\mid\forall X.A\mid\exists X.A\mid
X\mid\mathbf{1}\mid\mathbf{2}\end{array}$
The syntax of types is that of the multiplicative and exponential fragments of
second-order intuitionistic linear logic: $\lambda x{:}A.M$ denotes linear
$\lambda$-abstractions; $M\,N$ denotes the application; $\langle M\otimes
N\rangle$ denotes the multiplicative pairing of $M$ and $N$, as reflected in
its elimination form $\mathsf{let}\,x\otimes y=M\,\mathsf{in}\,N$ which
simultaneously deconstructs the pair $M$, binding its first and second
projection to $x$ and $y$ in $N$, respectively; $!M$ denotes a term $M$ that
does not use any linear variables and so may be used an arbitrary number of
times; $\mathsf{let}\,!u=M\,\mathsf{in}\,N$ binds the underlying exponential
term of $M$ as $u$ in $N$; $\Lambda X.M$ is the type abstraction former;
$M[A]$ stands for type application; $\mathsf{pack}\,A\,\mathsf{with}\,M$ is
the existential type introduction form, where $M$ is a term where the
existentially typed variable is instantiated with $A$;
$\mathsf{let}\,(X,y)=M\,\mathsf{in}\,N$ unpacks an existential package $M$,
binding the representation type to $X$ and the underlying term to $y$ in $N$;
the multiplicative unit $\mathbf{1}$ has as introduction form the nullary pair
$\langle\rangle$ and is eliminated by the construct
$\mathsf{let}\,\mathbf{1}=M\,\mathsf{in}\,N$, where $M$ is a term of type
$\mathbf{1}$. Booleans (type $\mathbf{2}$ with values $\mathsf{T}$ and
$\mathsf{F}$) are the basic observable.
The typing judgment in Linear-F is given as $\Omega;\Gamma;\Delta\vdash M:A$,
following the DILL formulation of linear logic [3], stating that term $M$ has
type $A$ in a linear context $\Delta$ (i.e. bindings for linear variables
$x{:}B$), intuitionistic context $\Gamma$ (i.e. binding for intuitionistic
variables $u{:}B$) and type variable context $\Omega$. The typing rules are
standard [7]. The operational semantics of the calculus are the expected call-
by-name semantics with commuting conversions [27]. We write $\Downarrow$ for
the evaluation relation. We write $\cong$ for the largest typed congruence
that is consistent with the observables of type $\mathbf{2}$ (i.e. a so-called
Morris-style equivalence as in [5]).
### 3.1 Encoding Linear-F into Session $\pi$-Calculus
We define a translation from Linear-F to Poly$\pi$ generalising the one from
[49], accounting for polymorphism and multiplicative pairs. We translate
typing derivations of $\lambda$-terms to those of $\pi$-calculus terms (we
omit the full typing derivation for the sake of readability).
Proof theoretically, the $\lambda$-calculus corresponds to a proof term
assignment for natural deduction presentations of logic, whereas the session
$\pi$-calculus from § 2 corresponds to a proof term assignment for sequent
calculus. Thus, we obtain a translation from $\lambda$-calculus to the session
$\pi$-calculus by considering the proof theoretic content of the constructive
proof of soundness of the sequent calculus wrt natural deduction. Following
Gentzen [14], the translation from natural deduction to sequent calculus maps
introduction rules to the corresponding right rules and elimination rules to a
combination of the corresponding left rule, cut and/or identity.
Since typing in the session calculus identifies a distinguished channel along
which a process offers a session, the translation of $\lambda$-terms is
parameterised by a ``result'' channel along which the behaviour of the
$\lambda$-term is implemented. Given a $\lambda$-term $M$, the process
$\llbracket M\rrbracket_{z}$ encodes the behaviour of $M$ along the session
channel $z$. We enforce that the type $\mathbf{2}$ of booleans and its two
constructors are consistently translated to their polymorphic Church encodings
before applying the translation to Poly$\pi$. Thus, type $\mathbf{2}$ is first
translated to $\forall X.!X\\!\multimap\,!X\\!\multimap X$, the value
$\mathsf{T}$ to $\Lambda X.\lambda u{:}!X.\lambda
v{:}!X.\mathsf{let}\,!x=u\,\mathsf{in}\,\mathsf{let}\,!y=v\,\mathsf{in}\,x$
and the value $\mathsf{F}$ to $\Lambda X.\lambda u{:}!X.\lambda
v{:}!X.\mathsf{let}\,!x=u\,\mathsf{in}\,\mathsf{let}\,!y=v\,\mathsf{in}\,y$.
Such representations of the booleans are adequate up to parametricity [6] and
suitable for our purposes of relating the session calculus (which has no
primitive notion of value or result type) with the $\lambda$-calculus
precisely due to the tight correspondence between the two calculi.
###### Definition 3.2 (From Linear-F to Poly$\pi$)
$\llbracket\Omega\rrbracket;\llbracket\Gamma\rrbracket;\llbracket\Delta\rrbracket\vdash\llbracket
M\rrbracket_{z}::z{:}A$ denotes the translation of contexts, types and terms
from Linear-F to the polymorphic session calculus. The translations on
contexts and types are the identity function. Booleans and their values are
first translated to their Church encodings as specified above. The translation
on $\lambda$-terms is given below:
$\begin{array}[]{lcllcl}\llbracket
x\rrbracket_{z}&\triangleq&[x\leftrightarrow z]&\llbracket
M\,N\rrbracket_{z}\triangleq&&(\mathbf{\nu}x)(\llbracket
M\rrbracket_{x}\mid(\mathbf{\nu}y)x\langle y\rangle.(\llbracket
N\rrbracket_{y}\mid[x\leftrightarrow z]))\\\ \llbracket
u\rrbracket_{z}&\triangleq&(\mathbf{\nu}x)u\langle x\rangle.[x\leftrightarrow
z]&\llbracket\mathsf{let}\,!u=M\,\mathsf{in}\,N\rrbracket_{z}&\triangleq&(\mathbf{\nu}x)(\llbracket
M\rrbracket_{x}\mid\llbracket N\rrbracket_{z}\\{x/u\\})\\\ \llbracket\lambda
x{:}A.M\rrbracket_{z}&\triangleq&z(x).\llbracket
M\rrbracket_{z}&\llbracket\langle M\otimes
N\rangle\rrbracket_{z}&\triangleq&(\mathbf{\nu}y)z\langle y\rangle.(\llbracket
M\rrbracket_{y}\mid\llbracket N\rrbracket_{z})\\\
\llbracket!M\rrbracket_{z}&\triangleq&!z(x).\llbracket
M\rrbracket_{x}&\llbracket\mathsf{let}\,x\otimes
y=M\,\mathsf{in}\,N\rrbracket_{z}&\triangleq&(\mathbf{\nu}w)(\llbracket
M\rrbracket_{y}\mid y(x).\llbracket N\rrbracket_{z})\\\ \llbracket\Lambda
X.M\rrbracket_{z}&\triangleq&z(X).\llbracket M\rrbracket_{z}&\llbracket
M[A]\rrbracket_{z}&\triangleq&(\mathbf{\nu}x)(\llbracket M\rrbracket_{x}\mid
x\langle A\rangle.[x\leftrightarrow z])\\\
\llbracket\mathsf{pack}\,A\,\mathsf{with}\,M\rrbracket_{z}&\triangleq&z\langle
A\rangle.\llbracket
M\rrbracket_{z}&\llbracket\mathsf{let}\,(X,y)=M\,\mathsf{in}\,N\rrbracket_{z}&\triangleq&(\mathbf{\nu}x)(\llbracket
M\rrbracket_{y}\mid y(X).\llbracket N\rrbracket_{z})\\\
\llbracket\langle\rangle\rrbracket_{z}&\triangleq&{\bf
0}&\llbracket\mathsf{let}\,\mathbf{1}=M\,\mathsf{in}\,N\rrbracket_{z}&\triangleq&(\mathbf{\nu}x)(\llbracket
M\rrbracket_{x}\mid\llbracket N\rrbracket_{z})\par\par\end{array}$
To translate a (linear) $\lambda$-abstraction $\lambda x{:}A.M$, which
corresponds to the proof term for the introduction rule for $\multimap$, we
map it to the corresponding ${{\multimap}\mathsf{R}}$ rule, thus obtaining a
process $z(x).\llbracket M\rrbracket_{z}$ that inputs along the result channel
$z$ a channel $x$ which will be used in $\llbracket M\rrbracket_{z}$ to access
the function argument. To encode the application $M\,N$, we compose (i.e.
$\mathsf{cut}$) $\llbracket M\rrbracket_{x}$, where $x$ is a fresh name, with
a process that provides the (encoded) function argument by outputting along
$x$ a channel $y$ which offers the behaviour of $\llbracket N\rrbracket_{y}$.
After the output is performed, the type of $x$ is now that of the function's
codomain and thus we conclude by forwarding (i.e. the $\mathsf{id}$ rule)
between $x$ and the result channel $z$.
The encoding for polymorphism follows a similar pattern: To encode the
abstraction $\Lambda X.M$, we receive along the result channel a type that is
bound to $X$ and proceed inductively. To encode type application $M[A]$ we
encode the abstraction $M$ in parallel with a process that sends $A$ to it,
and forwards accordingly. Finally, the encoding of the existential package
$\mathsf{pack}\,A\,\mathsf{with}\,M$ maps to an output of the type $A$
followed by the behaviour $\llbracket M\rrbracket_{z}$, with the encoding of
the elimination form $\mathsf{let}\,(X,y)=M\,\mathsf{in}\,N$ composing the
translation of the term of existential type $M$ with a process performing the
appropriate type input and proceeding as $\llbracket N\rrbracket_{z}$.
###### Example 3.1 (Encoding of Linear-F)
Consider the following $\lambda$-term corresponding to a polymorphic pairing
function (recall that we write $\overline{z}\langle w\rangle.P$ for
$(\mathbf{\nu}w)z\langle w\rangle.P$):
$\small\begin{array}[]{lcllcl}M\triangleq\Lambda X.\Lambda Y.\lambda
x{:}X.\lambda y{:}Y.\langle x\otimes
y\rangle&\mbox{and}&N\triangleq((M[A][B]\,M_{1})\,M_{2})\end{array}$
Then we have, with $\tilde{x}=x_{1}x_{2}x_{3}x_{4}$:
$\small\begin{array}[]{rcll}\llbracket
N\rrbracket_{z}&\equiv&(\mathbf{\nu}\tilde{x})(&\llbracket
M\rrbracket_{x_{1}}\mid x_{1}\langle A\rangle.[x_{1}\leftrightarrow x_{2}]\mid
x_{2}\langle B\rangle.[x_{2}\leftrightarrow x_{3}]\mid\\\
&&&\overline{x_{3}}\langle x\rangle.(\llbracket
M_{1}\rrbracket_{x}\mid[x_{3}\leftrightarrow
x_{4}])\mid\overline{x_{4}}\langle y\rangle.(\llbracket
M_{2}\rrbracket_{y}\mid[x_{4}\leftrightarrow z]))\\\
&\equiv&(\mathbf{\nu}\tilde{x})(&x_{1}(X).x_{1}(Y).x_{1}(x).x_{1}(y).\overline{x_{1}}\langle
w\rangle.([x\leftrightarrow w]\mid[y\leftrightarrow x_{1}])\mid x_{1}\langle
A\rangle.[x_{1}\leftrightarrow x_{2}]\mid\\\ &&&x_{2}\langle
B\rangle.[x_{2}\leftrightarrow x_{3}]\mid\overline{x_{3}}\langle
x\rangle.(\llbracket M_{1}\rrbracket_{x}\mid[x_{3}\leftrightarrow
x_{4}])\mid\overline{x_{4}}\langle y\rangle.(\llbracket
M_{2}\rrbracket_{y}\mid[x_{4}\leftrightarrow z]))\end{array}$
We can observe that $N\xrightarrow{}^{+}(((\lambda x{:}A.\lambda y{:}B.\langle
x\otimes y\rangle)\,M_{1})\,M_{2})\xrightarrow{}^{+}\langle M_{1}\otimes
M_{2}\rangle$. At the process level, each reduction corresponding to the redex
of type application is simulated by two reductions, obtaining:
$\small\begin{array}[]{lcll}\llbracket
N\rrbracket_{z}&\xrightarrow{}^{+}&(\mathbf{\nu}x_{3},x_{4})(&x_{3}(x).x_{3}(y).\overline{x_{3}}\langle
w\rangle.([x\leftrightarrow w]\mid[y\leftrightarrow x_{3}])\mid\\\
&&&\overline{x_{3}}\langle x\rangle.(\llbracket
M_{1}\rrbracket_{x}\mid[x_{3}\leftrightarrow
x_{4}])\mid\overline{x_{4}}\langle y\rangle.(\llbracket
M_{2}\rrbracket_{y}\mid[x_{4}\leftrightarrow z]))=P\end{array}$
The reductions corresponding to the $\beta$-redexes clarify the way in which
the encoding represents substitution of terms for variables via fine-grained
name passing. Consider $\llbracket\langle M_{1}\otimes
M_{2}\rangle\rrbracket_{z}\triangleq\overline{z}\langle w\rangle.(\llbracket
M_{1}\rrbracket_{w}\mid\llbracket M_{2}\rrbracket_{z})$ and
$\small\begin{array}[]{rcl}P&\xrightarrow{}^{+}&(\mathbf{\nu}x,y)(\llbracket
M_{1}\rrbracket_{x}\mid\llbracket M_{2}\rrbracket_{y}\mid\overline{z}\langle
w\rangle.([x\leftrightarrow w]\mid[y\leftrightarrow z]))\end{array}$
The encoding of the pairing of $M_{1}$ and $M_{2}$ outputs a fresh name $w$
which will denote the behaviour of (the encoding of) $M_{1}$, and then the
behaviour of the encoding of $M_{2}$ is offered on $z$. The reduct of $P$
outputs a fresh name $w$ which is then identified with $x$ and thus denotes
the behaviour of $\llbracket M_{1}\rrbracket_{w}$. The channel $z$ is
identified with $y$ and thus denotes the behaviour of $\llbracket
M_{2}\rrbracket_{z}$, making the two processes listed above equivalent. This
informal reasoning exposes the insights that justify the operational
correspondence of the encoding. Proof-theoretically, these equivalences simply
map to commuting conversions which push the processes $\llbracket
M_{1}\rrbracket_{x}$ and $\llbracket M_{2}\rrbracket_{z}$ under the output on
$z$.
###### Theorem 3.1 (Operational Correspondence)
* •
If $\Omega;\Gamma;\Delta\vdash M:A$ and $M\xrightarrow{}N$ then $\llbracket
M\rrbracket_{z}\stackrel{{\scriptstyle}}{{\Longrightarrow}}P$ such that
$\llbracket N\rrbracket_{z}\approx_{\mathtt{L}}P$
* •
If $\llbracket M\rrbracket_{z}\xrightarrow{}P$ then $M\xrightarrow{}^{+}N$ and
$\llbracket N\rrbracket_{z}\approx_{\mathtt{L}}P$
### 3.2 Encoding Session $\pi$-calculus to Linear-F
Just as the proof theoretic content of the soundness of sequent calculus wrt
natural deduction induces a translation from $\lambda$-terms to session-typed
processes, the _completeness_ of the sequent calculus wrt natural deduction
induces a translation from the session calculus to the $\lambda$-calculus.
This mapping identifies sequent calculus right rules with the introduction
rules of natural deduction and left rules with elimination rules combined with
(type-preserving) substitution. Crucially, the mapping is defined on _typing
derivations_ , enabling us to consistently identify when a process uses a
session (i.e. left rules) or, dually, when a process offers a session (i.e.
right rules).
${\footnotesize\begin{array}[]{l}\stretchleftright{\llparenthesis}{\raise
10.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\multimap}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
34.56946pt\vbox{\hbox{\hskip-34.56946pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta,x{:}A\vdash{\color[rgb]{0,0,1}P}::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
43.05003pt\vbox{\vbox{}\hbox{\hskip-43.05003pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta\vdash{\color[rgb]{0,0,1}z(x).P}::z{:}A\multimap
B}$}}}}}}$}}\hbox{}$}}{\rrparenthesis}\triangleq\raise
10.01332pt\hbox{$\hbox{$\hbox{\small\sc$(\multimap
I)$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
42.8808pt\vbox{\hbox{\hskip-42.8808pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta,x{:}A\vdash{\color[rgb]{0,0,1}\llparenthesis
P\rrparenthesis}_{\Delta,x{:}A\vdash z{:}B}:B}$}}}\vbox{}}}\over\hbox{\hskip
53.76971pt\vbox{\vbox{}\hbox{\hskip-53.76971pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta\vdash{\color[rgb]{0,0,1}\lambda
x{:}A.\llparenthesis P\rrparenthesis}_{\Delta,x{:}A\vdash z{:}B}:A\multimap
B}$}}}}}}$}}\hbox{}$}\\\\[16.00003pt]
\stretchleftright{\llparenthesis}{{\vbox{\hbox{\hbox{\small\sc$\displaystyle({{\multimap}\mathsf{L}})$}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
63.888pt\vbox{\hbox{\hskip-63.888pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta_{1}\vdash{\color[rgb]{0,0,1}P}::y{:}A\quad\Delta_{2},x{:}B\vdash{\color[rgb]{0,0,1}Q}::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
77.92137pt\vbox{\vbox{}\hbox{\hskip-77.92137pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta_{1},\Delta_{2},x{:}A\multimap
B\vdash{\color[rgb]{0,0,1}(\mathbf{\nu}y)x\langle y\rangle.(P\mid
Q)}::z{:}C}$}}}}}}$}}}}{\rrparenthesis}\par\triangleq\\\\[16.00003pt]
{\vbox{\hbox{\hbox{\small\sc(subst)}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
125.67091pt\vbox{\hbox{\hskip-125.6709pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta_{2},x{:}B\vdash{\color[rgb]{0,0,1}\llparenthesis
Q\rrparenthesis}_{\Delta_{2},x{:}B\vdash
z{:}C}:C\quad{\vbox{\hbox{\hbox{\small\sc$\displaystyle(\multimap
E)$}}\hbox{$\displaystyle\displaystyle{\hbox{\hskip
75.6542pt\vbox{\hbox{\hskip-75.65419pt\hbox{\hbox{$\displaystyle\displaystyle{x{:}A\multimap
B\vdash{\color[rgb]{0,0,1}x}{:}A\multimap
B\quad\Delta_{1}\vdash{\color[rgb]{0,0,1}\llparenthesis
P\rrparenthesis}_{\Delta_{1}\vdash y{:}A}:B}$}}}\vbox{}}}\over\hbox{\hskip
54.30835pt\vbox{\vbox{}\hbox{\hskip-54.30833pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta_{1},x{:}A\multimap
B\vdash{\color[rgb]{0,0,1}x\,\llparenthesis P\rrparenthesis}_{\Delta_{1}\vdash
y{:}A}:B}$}}}}}}$}}}}$}}}\vbox{}}}\over\hbox{\hskip
94.41118pt\vbox{\vbox{}\hbox{\hskip-94.41116pt\hbox{\hbox{$\displaystyle\displaystyle{\Delta_{1},\Delta_{2},x{:}A\multimap
B\vdash{\color[rgb]{0,0,1}\llparenthesis
Q\rrparenthesis}_{\Delta_{2},x{:}B\vdash
z{:}C}{\color[rgb]{0,0,1}\\{(x\,\llparenthesis
P\rrparenthesis}_{\Delta_{1}\vdash
y{:}A}{\color[rgb]{0,0,1})/x\\}}:C}$}}}}}}$}}}\end{array}}$ Figure 3:
Translation on Typing Derivations (Excerpt – See [52])
###### Definition 3.3 (From Poly$\pi$ to Linear-F)
We write
$\llparenthesis\Omega\rrparenthesis;\llparenthesis\Gamma\rrparenthesis;\llparenthesis\Delta\rrparenthesis\vdash\llparenthesis
P\rrparenthesis:A$ for the translation from typing derivations in Poly$\pi$ to
derivations in Linear-F. The translations on types and contexts are the
identity function. The translation on processes is given below, where the
leftmost column indicates the typing rule at the root of the derivation (see
Fig. 3 for an excerpt of the translation on typing derivations, where we write
$\llparenthesis P\rrparenthesis_{\Omega;\Gamma;\Delta\vdash z{:}A}$ to denote
the translation of $\Omega;\Gamma;\Delta\vdash P::z{:}A$. We omit $\Omega$ and
$\Gamma$ when unchanged).
$\begin{array}[]{llclllcl}({{\mathbf{1}}\mathsf{R}})&\llparenthesis{\bf
0}\rrparenthesis&\triangleq&\langle\rangle&({{\multimap}\mathsf{L}})&\llparenthesis(\mathbf{\nu}y)x\langle
y\rangle.(P\mid Q)\rrparenthesis&\triangleq&\llparenthesis
Q\rrparenthesis\\{(x\,\llparenthesis P\rrparenthesis)/x\\}\\\
(\mathsf{id})&\llparenthesis[x\leftrightarrow
y]\rrparenthesis&\triangleq&x&({{\multimap}\mathsf{R}})&\llparenthesis
z(x).P\rrparenthesis&\triangleq&\lambda x{:}A.\llparenthesis
P\rrparenthesis\\\ ({{\mathbf{1}}\mathsf{L}})&\llparenthesis
P\rrparenthesis&\triangleq&\mathsf{let}\,\mathbf{1}=x\,\mathsf{in}\,\llparenthesis
P\rrparenthesis&({{\otimes}\mathsf{R}})&\llparenthesis(\mathbf{\nu}x)z\langle
x\rangle.(P\mid Q)\rrparenthesis&\triangleq&\langle\llparenthesis
P\rrparenthesis\otimes\llparenthesis Q\rrparenthesis\rangle\\\
({{!}\mathsf{R}})&\llparenthesis!z(x).P\rrparenthesis&\triangleq&!\llparenthesis
P\rrparenthesis&({{\otimes}\mathsf{L}})&\llparenthesis
x(y).P\rrparenthesis&\triangleq&\mathsf{let}\,x\otimes
y=x\,\mathsf{in}\,\llparenthesis P\rrparenthesis\\\
({{!}\mathsf{L}})&\llparenthesis
P\\{u/x\\}\rrparenthesis&\triangleq&\mathsf{let}\,!u=x\,\mathsf{in}\,\llparenthesis
P\rrparenthesis&(\mathsf{copy})&\llparenthesis(\mathbf{\nu}x)u\langle
x\rangle.P\rrparenthesis&\triangleq&\llparenthesis P\rrparenthesis\\{u/x\\}\\\
({{\forall}\mathsf{R}})&\llparenthesis z(X).P\rrparenthesis&\triangleq&\Lambda
X.\llparenthesis P\rrparenthesis&({{\forall}\mathsf{L}})&\llparenthesis
x\langle B\rangle.P\rrparenthesis&\triangleq&\llparenthesis
P\rrparenthesis\\{(x[B])/x\\}\\\ ({{\exists}\mathsf{R}})&\llparenthesis
z\langle
B\rangle.P\rrparenthesis&\triangleq&\mathsf{pack}\,B\,\mathsf{with}\,\llparenthesis
P\rrparenthesis&({{\exists}\mathsf{L}})&\llparenthesis
x(Y).P\rrparenthesis&\triangleq&\mathsf{let}\,(Y,x)=x\,\mathsf{in}\,\llparenthesis
P\rrparenthesis\\\ (\mathsf{cut})&\llparenthesis(\mathbf{\nu}x)(P\mid
Q)\rrparenthesis&\triangleq&\llparenthesis Q\rrparenthesis\\{\llparenthesis
P\rrparenthesis/x\\}&(\mathsf{cut}^{!})&\llparenthesis(\mathbf{\nu}u)(!u(x).P\mid
Q)\rrparenthesis&\triangleq&\llparenthesis Q\rrparenthesis\\{\llparenthesis
P\rrparenthesis/u\\}\\\ \end{array}$
For instance, the encoding of a process $z(x).P::z{:}A\multimap B$, typed by
rule ${{\multimap}\mathsf{R}}$, results in the corresponding $\multimap\\!I$
introduction rule in the $\lambda$-calculus and thus is $\lambda
x{:}A.\llparenthesis P\rrparenthesis$. To encode the process
$(\mathbf{\nu}y)x\langle y\rangle.(P\mid Q)$, typed by rule
${{\multimap}\mathsf{L}}$, we make use of substitution: Given that the sub-
process $Q$ is typed as $\Omega;\Gamma;\Delta^{\prime},x{:}B\vdash Q::z{:}C$,
the encoding of the full process is given by $\llparenthesis
Q\rrparenthesis\\{(x\,\llparenthesis P\rrparenthesis)/x\\}$. The term
$x\,\llparenthesis P\rrparenthesis$ consists of the application of $x$ (of
function type) to the argument $\llparenthesis P\rrparenthesis$, thus ensuring
that the term resulting from the substitution is of the appropriate type. We
note that, for instance, the encoding of rule ${{\otimes}\mathsf{L}}$ does not
need to appeal to substitution – the $\lambda$-calculus $\mathsf{let}$ style
rules can be mapped directly. Similarly, rule ${{\forall}\mathsf{R}}$ is
mapped to type abstraction, whereas rule ${{\forall}\mathsf{L}}$ which types a
process of the form $x\langle B\rangle.P$ maps to a substitution of the type
application $x[B]$ for $x$ in $\llparenthesis P\rrparenthesis$. The encoding
of existential polymorphism is simpler due to the $\mathsf{let}$-style
elimination. We also highlight the encoding of the $\mathsf{cut}$ rule which
embodies parallel composition of two processes sharing a linear name, which
clarifies the use/offer duality of the intuitionistic calculus – the process
that offers $P$ is encoded and substituted into the encoded user $Q$.
###### Theorem 3.2 ()
If $\Omega;\Gamma;\Delta\vdash P::z{:}A$ then
$\llparenthesis\Omega\rrparenthesis;\llparenthesis\Gamma\rrparenthesis;\llparenthesis\Delta\rrparenthesis\vdash\llparenthesis
P\rrparenthesis:A$.
###### Example 3.2 (Encoding of Poly$\pi$)
Consider the following processes
$\small\begin{array}[]{l}P\triangleq z(X).z(Y).z(x).z(y).\overline{z}\langle
w\rangle.([x\leftrightarrow w]\mid[y\leftrightarrow z])\quad Q\triangleq
z\langle\mathbf{1}\rangle.z\langle\mathbf{1}\rangle.\overline{z}\langle
x\rangle.\overline{z}\langle y\rangle.z(w).[w\leftrightarrow r]\end{array}$
with $\vdash P::z{:}\forall X.\forall Y.X\multimap Y\multimap X\otimes Y$ and
$z{:}\forall X.\forall Y.X\multimap Y\multimap X\otimes Y\vdash
Q::r{:}\mathbf{1}$. $\small\begin{array}[]{ll}\mbox{Then:}&\llparenthesis
P\rrparenthesis=\Lambda X.\Lambda Y.\lambda x{:}X.\lambda y{:}Y.\langle
x\otimes y\rangle\quad\quad\llparenthesis
Q\rrparenthesis=\mathsf{let}\,x\otimes
y=z[\mathbf{1}][\mathbf{1}]\,\langle\rangle\,\langle\rangle\,\mathsf{in}\,\mathsf{let}\,\mathbf{1}=y\,\mathsf{in}\,x\\\
&\llparenthesis(\mathbf{\nu}z)(P\mid Q)\rrparenthesis=\mathsf{let}\,x\otimes
y=(\Lambda X.\Lambda Y.\lambda x{:}X.\lambda y{:}Y.\langle x\otimes
y\rangle)[\mathbf{1}][\mathbf{1}]\,\langle\rangle\,\langle\rangle\,\mathsf{in}\,\mathsf{let}\,\mathbf{1}=y\,\mathsf{in}\,x\end{array}$
By the behaviour of $(\mathbf{\nu}z)(P\mid Q)$, which consists of a sequence
of cuts, and its encoding, we have that $\llparenthesis(\mathbf{\nu}z)(P\mid
Q)\rrparenthesis\xrightarrow{}^{+}\langle\rangle$ and $(\mathbf{\nu}z)(P\mid
Q)\xrightarrow{}^{+}{\bf 0}=\llparenthesis\langle\rangle\rrparenthesis$.
In general, the translation of Def. 3.3 can introduce some distance between
the immediate operational behaviour of a process and its corresponding
$\lambda$-term, insofar as the translations of cuts (and left rules to non
$\mathsf{let}$-form elimination rules) make use of substitutions that can take
place deep within the resulting term. Consider the process at the root of the
following typing judgment
$\Delta_{1},\Delta_{2},\Delta_{3}\vdash(\mathbf{\nu}x)(x(y).P_{1}\mid(\mathbf{\nu}y)x\langle
y\rangle.(P_{2}\mid w(z).{\bf 0}))::w{:}\mathbf{1}\multimap\mathbf{1}$,
derivable through a $\mathsf{cut}$ on session $x$ between instances of
${{\multimap}\mathsf{R}}$ and ${{\multimap}\mathsf{L}}$, where the
continuation process $w(z).{\bf 0}$ offers a session
$w{:}\mathbf{1}\multimap\mathbf{1}$ (and so must use rule
${{\mathbf{1}}\mathsf{L}}$ on $x$). We have that:
$(\mathbf{\nu}x)(x(y).P_{1}\mid(\mathbf{\nu}y)x\langle y\rangle.(P_{2}\mid
w(z).{\bf 0}))\xrightarrow{}(\mathbf{\nu}x,y)(P_{1}\mid P_{2}\mid w(z).{\bf
0})$. However, the translation of the process above results in the term
$\lambda z{:}\mathbf{1}.\mathsf{let}\,\mathbf{1}=((\lambda
y{:}A.\llparenthesis P_{1}\rrparenthesis)\,\llparenthesis
P_{2}\rrparenthesis)\,\mathsf{in}\,\mathsf{let}\,\mathbf{1}=z\,\mathsf{in}\,\langle\rangle$,
where the redex that corresponds to the process reduction is present but
hidden under the binder for $z$ (corresponding to the input along $w$). Thus,
to establish operational completeness we consider full $\beta$-reduction,
denoted by $\xrightarrow{}_{\beta}$, i.e. enabling $\beta$-reductions under
binders.
###### Theorem 3.3 (Operational Completeness)
Let $\Omega;\Gamma;\Delta\vdash P::z{:}A$. If $P\xrightarrow{}Q$ then
$\llparenthesis P\rrparenthesis\xrightarrow{}_{\beta}^{*}\llparenthesis
Q\rrparenthesis$.
In order to study the soundness direction it is instructive to consider typed
process $x{:}\mathbf{1}\multimap\mathbf{1}\vdash\overline{x}\langle
y\rangle.(\mathbf{\nu}z)(z(w).{\bf 0}\mid\overline{z}\langle w\rangle.{\bf
0})::v{:}\mathbf{1}$ and its translation:
$\small\begin{array}[]{c}\llparenthesis\overline{x}\langle
y\rangle.(\mathbf{\nu}z)(z(w).{\bf 0}\mid\overline{z}\langle w\rangle.{\bf
0})\rrparenthesis=\llparenthesis(\mathbf{\nu}z)(z(w).{\bf
0}\mid\overline{z}\langle w\rangle.{\bf
0})\rrparenthesis\\{(x\,\langle\rangle)/x\\}\\\
=\mathsf{let}\,\mathbf{1}=(\lambda
w{:}\mathbf{1}.\mathsf{let}\,\mathbf{1}=w\,\mathsf{in}\,\langle\rangle)\,\langle\rangle\,\mathsf{in}\,\mathsf{let}\,\mathbf{1}=x\,\langle\rangle\,\mathsf{in}\,\langle\rangle\end{array}$
The process above cannot reduce due to the output prefix on $x$, which cannot
synchronise with a corresponding input action since there is no provider for
$x$ (i.e. the channel is in the left-hand side context). However, its encoding
can exhibit the $\beta$-redex corresponding to the synchronisation along $z$,
hidden by the prefix on $x$. The corresponding reductions hidden under
prefixes in the encoding can be _soundly_ exposed in the session calculus by
appealing to the commuting conversions of linear logic (e.g. in the process
above, the instance of rule ${{\multimap}\mathsf{L}}$ corresponding to the
output on $x$ can be commuted with the $\mathsf{cut}$ on $z$).
As shown in [36], commuting conversions are sound wrt observational
equivalence, and thus we formulate operational soundness through a notion of
_extended_ process reduction, which extends process reduction with the
reductions that are induced by commuting conversions. Such a relation was also
used for similar purposes in [5] and in [26], in a classical linear logic
setting. For conciseness, we define extended reduction as a relation on
_typed_ processes modulo $\equiv$.
###### Definition 3.4 (Extended Reduction [5])
We define $\mapsto$ as the type preserving relations on typed processes modulo
$\equiv$ generated by:
1. 1.
$\mathcal{C}[(\mathbf{\nu}y)x\langle y\rangle.P]\mid
x(y).Q\mapsto\mathcal{C}[(\mathbf{\nu}y)(P\mid Q)]$;
2. 2.
$\mathcal{C}[(\mathbf{\nu}y)x\langle
y\rangle.P]\mid\,!x(y).Q\mapsto\mathcal{C}[(\mathbf{\nu}y)(P\mid
Q)]\mid\,!x(y).Q$; and (3) $(\mathbf{\nu}x)(!x(y).Q)\mapsto\mathbf{0}$
where $\mathcal{C}$ is a (typed) process context which does not capture the
bound name $y$.
###### Theorem 3.4 (Operational Soundness)
Let $\Omega;\Gamma;\Delta\vdash P::z{:}A$ and $\llparenthesis
P\rrparenthesis\xrightarrow{}M$, there exists $Q$ such that $P\mapsto^{*}Q$
and $\llparenthesis Q\rrparenthesis=_{\alpha}M$.
### 3.3 Inversion and Full Abstraction
Having established the operational preciseness of the encodings to-and-from
Poly$\pi$ and Linear-F, we establish our main results for the encodings.
Specifically, we show that the encodings are mutually inverse up-to
behavioural equivalence (with fullness as its corollary), which then enables
us to establish full abstraction for _both_ encodings.
###### Theorem 3.5 (Inverse)
If $\Omega;\Gamma;\Delta\vdash M:A$ then
$\Omega;\Gamma;\Delta\vdash\llparenthesis\llbracket
M\rrbracket_{z}\rrparenthesis\cong M:A$. Also, if $\Omega;\Gamma;\Delta\vdash
P::z{:}A$ then $\Omega;\Gamma;\Delta\vdash\llbracket\llparenthesis
P\rrparenthesis\rrbracket_{z}\approx_{\mathtt{L}}P::z{:}A$
###### Corollary 3.1 (Fullness)
Let $\Omega;\Gamma;\Delta\vdash P::z{:}A$. $\exists M$ s.t.
$\Omega;\Gamma;\Delta\vdash M:A$ and $\Omega;\Gamma;\Delta\vdash\llbracket
M\rrbracket_{z}\approx_{\mathtt{L}}P::z{:}A$ Also, let
$\Omega;\Gamma;\Delta\vdash M:A$. $\exists P$ s.t. $\Omega;\Gamma;\Delta\vdash
P::z{:}A$ and $\Omega;\Gamma;\Delta\vdash\llparenthesis P\rrparenthesis\cong
M:A$
We now state our full abstraction results. Given two Linear-F terms of the
same type, equivalence in the image of the $\llbracket{-}\rrbracket_{z}$
translation can be used as a proof technique for contextual equivalence in
Linear-F. This is called the _soundness_ direction of full abstraction in the
literature [18] and proved by showing the relation generated by $\llbracket
M\rrbracket_{z}\approx_{\mathtt{L}}\llbracket N\rrbracket_{z}$ forms $\cong$;
we then establish the _completeness_ direction by contradiction, using
fullness.
###### Theorem 3.6 (Full Abstraction)
$\Omega;\Gamma;\Delta\vdash M\cong N:A$ iff
$\Omega;\Gamma;\Delta\vdash\llbracket
M\rrbracket_{z}\approx_{\mathtt{L}}\llbracket N\rrbracket_{z}::z{:}A$.
We can straightforwardly combine the above full abstraction with Theorem 3.5
to obtain full abstraction of the $\llparenthesis{-}\rrparenthesis$
translation.
###### Theorem 3.7 (Full Abstraction)
$\Omega;\Gamma;\Delta\vdash P\approx_{\mathtt{L}}Q::z{:}A$ iff
$\Omega;\Gamma;\Delta\vdash\llparenthesis P\rrparenthesis\cong\llparenthesis
Q\rrparenthesis:A$.
## 4 Applications of the Encodings
In this section we develop applications of the encodings of the previous
sections. Taking advantage of full abstraction and mutual inversion, we apply
non-trivial properties from the theory of the $\lambda$-calculus to our
session-typed process setting.
In § 4.1 we study inductive and coinductive sessions, arising through
encodings of initial $F$-algebras and final $F$-coalgebras in the polymorphic
$\lambda$-calculus.
In § 4.2 we study encodings for an extension of the core session calculus with
term passing, where terms are derived from a simply-typed $\lambda$-calculus.
Using the development of § 4.2 as a stepping stone, we generalise the
encodings to a _higher-order_ session calculus (§ 4.3), where processes can
send, receive and execute other processes.We show full abstraction and mutual
inversion theorems for the encodings from higher-order to first-order. As a
consequence, we can straightforwardly derive a strong normalisation property
for the higher-order process-passing calculus.
### 4.1 Inductive and Coinductive Session Types
The study of polymorphism in the $\lambda$-calculus [1, 19, 40, 6] has shown
that parametric polymorphism is expressive enough to encode both inductive and
coinductive types in a precise way, through a faithful representation of
initial and final (co)algebras [28], without extending the language of terms
nor the semantics of the calculus, giving a logical justification to the
Church encodings of inductive datatypes such as lists and natural numbers. The
polymorphic session calculus can express fairly intricate communication
behaviours, including generic protocols through both existential and universal
polymorphism (i.e. protocols that are parametric in their sub-protocols).
Using our fully abstract encodings between the two calculi, we show that
session polymorphism is expressive enough to encode inductive and coinductive
sessions, ``importing'' the results for the $\lambda$-calculus, which may then
be instantiated to provide a session-typed formulation of the encodings of
data structures in the $\pi$-calculus of [30].
Inductive and Coinductive Types in System F. Exploring an algebraic
interpretation of polymorphism where types are interpreted as functors, it can
be shown that given a type $F$ with a free variable $X$ that occurs only
positively (i.e. occurrences of $X$ are on the left-hand side of an even
number of function arrows), the polymorphic type $\forall X.((F(X)\rightarrow
X)\rightarrow X)$ forms an initial $F$-algebra [42, 1] (we write $F(X)$ to
denote that $X$ occurs in $F$). This enables the representation of
_inductively_ defined structures using an algebraic or categorical
justification. For instance, the natural numbers can be seen as the initial
$F$-algebra of $F(X)=\mathbf{1}+X$ (where $\mathbf{1}$ is the unit type and
$+$ is the coproduct), and are thus _already present_ in System F, in a
precise sense, as the type $\forall X.((\mathbf{1}+X)\rightarrow X)\rightarrow
X$ (noting that both $\mathbf{1}$ and $+$ can also be encoded in System F). A
similar story can be told for _coinductively_ defined structures, which
correspond to final $F$-coalgebras and are representable with the polymorphic
type $\exists X.(X\rightarrow F(X))\times X$, where $\times$ is a product
type. In the remainder of this section we assume the positivity requirement on
$F$ mentioned above.
While the complete formal development of the representation of inductive and
coinductive types in System F would lead us to far astray, we summarise here
the key concepts as they apply to the $\lambda$-calculus (the interested
reader can refer to [19] for the full categorical details).
|
---
(a)
|
---
(b)
Figure 4: Diagrams for Initial $F$-algebras and Final $F$-coalgebras
To show that the polymorphic type $T_{i}\triangleq\forall X.((F(X)\rightarrow
X)\rightarrow X)$ is an initial $F$-algebra, one exhibits a pair of
$\lambda$-terms, often dubbed $\mathsf{fold}$ and $\mathsf{in}$, such that the
diagram in Fig. 4(a) commutes (for any $A$, where $F(f)$, where $f$ is a
$\lambda$-term, denotes the functorial action of $F$ applied to $f$), and,
crucially, that $\mathsf{fold}$ is _unique_. When these conditions hold, we
are justified in saying that $T_{i}$ is a least fixed point of $F$. Through a
fairly simple calculation, it is easy to see that:
$\small\begin{array}[]{rcl}\mathsf{fold}&\triangleq&\Lambda X.\lambda
x{:}F(X)\rightarrow X.\lambda t{:}T_{i}.t[X](x)\\\
\mathsf{in}&\triangleq&\lambda x{:}F(T_{i}).\Lambda X.\lambda
y{:}F(X)\rightarrow X.y\,(F(\mathsf{fold}[X](x))(x))\end{array}$
satisfy the necessary equalities. To show uniqueness one appeals to
_parametricity_ , which allows us to prove that any function of the
appropriate type is equivalent to $\mathsf{fold}$. This property is often
dubbed initiality or universality.
The construction of final $F$-coalgebras and their justification as _greatest_
fixed points is dual. Assuming products in the calculus and taking
$T_{f}\triangleq\exists X.(X\rightarrow F(X))\times X$, we produce the
$\lambda$-terms
$\small\begin{array}[]{rcl}\mathsf{unfold}&\triangleq&\Lambda X.\lambda
f{:}X\rightarrow F(X).\lambda
x{:}T_{f}.\mathsf{pack}\,X\,\mathsf{with}\,(f,x)\\\
\mathsf{out}&\triangleq&\lambda
t:T_{f}.\mathsf{let}\,(X,(f,x))=t\,\mathsf{in}\,F(\mathsf{unfold}[X](f))\,(f(x))\end{array}$
such that the diagram in Fig. 4(b) commutes and $\mathsf{unfold}$ is unique
(again, up to parametricity). While the argument above applies to System F, a
similar development can be made in Linear-F [6] by considering
$T_{i}\triangleq\forall X.!(F(X)\multimap X)\multimap X$ and
$T_{f}\triangleq\exists X.!(X\multimap F(X))\otimes X$. Reusing the same names
for the sake of conciseness, the associated _linear_ $\lambda$-terms are:
$\small\begin{array}[]{rcl}\mathsf{fold}&\triangleq&\Lambda X.\lambda
u{:}!(F(X)\multimap X).\lambda y{:}T_{i}.(y[X]\,u):\forall X.!(F(X)\multimap
X)\multimap T_{i}\multimap X\\\ \mathsf{in}&\triangleq&\lambda
x{:}F(T_{i}).\Lambda X.\lambda y{:}!(F(X)\multimap
X).\mathsf{let}\,!u=y\,\mathsf{in}\,k\,(F\,(\mathsf{fold}[X](!u))(x)):F(T_{i})\multimap
T_{i}\\\ \mathsf{unfold}&\triangleq&\Lambda X.\lambda u{:}!(X\multimap
F(X)).\lambda x{:}X.\mathsf{pack}\,X\,\mathsf{with}\,\langle u\otimes
x\rangle:\forall X.!(X\multimap F(X))\multimap X\multimap T_{f}\\\
\mathsf{out}&\triangleq&\lambda
t:T_{f}.\mathsf{let}\,(X,(u,x))=t\,\mathsf{in}\,\mathsf{let}\,!f=u\,\mathsf{in}\,F(\mathsf{unfold}[X](!f))\,(f(x)):T_{f}\multimap
F(T_{f})\end{array}$
Inductive and Coinductive Sessions for Free. As a consequence of full
abstraction we may appeal to the $\llbracket{-}\rrbracket_{z}$ encoding to
derive representations of $\mathsf{fold}$ and $\mathsf{unfold}$ that satisfy
the necessary algebraic properties. The derived processes are (recall that we
write $\overline{x}\langle y\rangle.P$ for $(\mathbf{\nu}y)x\langle
y\rangle.P$):
$\small\begin{array}[]{rcl}\llbracket\mathsf{fold}\rrbracket_{z}&\triangleq&z(X).z(u).z(y).(\mathbf{\nu}w)((\mathbf{\nu}x)([y\leftrightarrow
x]\mid x\langle X\rangle.[x\leftrightarrow w])\mid\overline{w}\langle
v\rangle.([u\leftrightarrow v]\mid[w\leftrightarrow z]))\\\
\llbracket\mathsf{unfold}\rrbracket_{z}&\triangleq&z(X).z(u).z(x).z\langle
X\rangle.\overline{z}\langle y\rangle.([u\leftrightarrow
y]\mid[x\leftrightarrow z])\\\ \end{array}$
We can then show universality of the two constructions. We write $P_{x,y}$ to
single out that $x$ and $y$ are free in $P$ and $P_{z,w}$ to denote the result
of employing capture-avoiding substitution on $P$, substituting $x$ and $y$ by
$z$ and $w$. Let:
$\small\begin{array}[]{rcl}\mathsf{foldP}(A)_{y_{1},y_{2}}&\triangleq&(\mathbf{\nu}x)(\llbracket\mathsf{fold}\rrbracket_{x}\mid
x\langle A\rangle.\overline{x}\langle v\rangle.(\overline{u}\langle
y\rangle.[y\leftrightarrow v]\mid\overline{x}\langle
z\rangle.([z\leftrightarrow y_{1}]\mid[x\leftrightarrow y_{2}])))\\\
\mathsf{unfoldP}(A)_{y_{1},y_{2}}&\triangleq&(\mathbf{\nu}x)(\llbracket\mathsf{unfold}\rrbracket_{x}\mid
x\langle A\rangle.\overline{x}\langle v\rangle.(\overline{u}\langle
y\rangle.[y\leftrightarrow v]\mid\overline{x}\langle
z\rangle.([z\leftrightarrow y_{1}]\mid[x\leftrightarrow y_{2}])))\end{array}$
where $\mathsf{foldP}(A)_{y_{1},y_{2}}$ corresponds to the application of
$\mathsf{fold}$ to an $F$-algebra $A$ with the associated morphism
$F(A)\multimap A$ available on the shared channel $u$, consuming an ambient
session $y_{1}{:}T_{i}$ and offering $y_{2}{:}A$. Similarly,
$\mathsf{unfoldP}(A)_{y_{1},y_{2}}$ corresponds to the application of
$\mathsf{unfold}$ to an $F$-coalgebra $A$ with the associated morphism
$A\multimap F(A)$ available on the shared channel $u$, consuming an ambient
session $y_{1}{:}A$ and offering $y_{2}{:}T_{f}$.
###### Theorem 4.1 (Universality of $\mathsf{foldP}$)
$\forall Q$ such that $X;u{:}F(X)\multimap X;y_{1}{:}T_{i}\vdash Q::y_{2}{:}X$
we have $X;u{:}F(X)\multimap X;y_{1}{:}T_{i}\vdash
Q\approx_{\mathtt{L}}\mathsf{foldP}(X)_{y_{1},y_{2}}::y_{2}{:}X$
###### Theorem 4.2 (Universality of $\mathsf{unfoldP}$)
$\forall Q$ and $F$-coalgebra $A$ s.t $\cdot;\cdot;y_{1}{:}A\vdash
Q::y_{2}{:}T_{f}$ we have that $\cdot;u{:}F(A)\multimap A;y_{1}{:}A\vdash
Q\approx_{\mathtt{L}}\mathsf{unfoldP}(A)_{y_{1},y_{2}}::y_{2}::T_{f}$.
###### Example 4.1 (Natural Numbers)
We show how to represent the natural numbers as an inductive session type
using $F(X)=\mathbf{1}\oplus X$, making use of $\mathsf{in}$:
$\small\begin{array}[]{c}\mathsf{zero}_{x}\triangleq(\mathbf{\nu}z)(z.\mathsf{inl};{\bf
0}\mid\llbracket\mathsf{in}(z)\rrbracket_{x})\quad\mathsf{succ}_{y,x}\triangleq(\mathbf{\nu}s)(s.\mathsf{inr};[y\leftrightarrow
s]\mid\llbracket\mathsf{in}(s)\rrbracket_{x})\end{array}$
with $\mathsf{Nat}\triangleq\forall X.!((\mathbf{1}\oplus X)\multimap
X)\multimap X$ where $\vdash\mathsf{zero}_{x}::x{:}\mathsf{Nat}$ and
$y{:}\mathsf{Nat}\vdash\mathsf{succ}_{y,x}::x{:}\mathsf{Nat}$ encode the
representation of $0$ and successor, respectively. The natural $1$ would thus
be represented by
$\mathsf{one}_{x}\triangleq(\mathbf{\nu}y)(\mathsf{zero}_{y}\mid\mathsf{succ}_{y,x})$.
The behaviour of type $\mathsf{Nat}$ can be seen as a that of a sequence of
internal choices of arbitrary (but finite) length. We can then observe that
the $\mathsf{foldP}$ process acts as a recursor. For instance consider:
$\small\begin{array}[]{l}\mathsf{stepDec}_{d}\triangleq
d(n).n.\mathsf{case}(\mathsf{zero}_{d},[n\leftrightarrow
d])\quad\mathsf{dec}_{x,z}\triangleq(\mathbf{\nu}u)(!u(d).\mathsf{stepDec}_{d}\mid\mathsf{foldP}(\mathsf{Nat})_{x,z})\end{array}$
with
$\mathsf{stepDec}_{d}::d{:}(\mathbf{1}\oplus\mathsf{Nat})\multimap\mathsf{Nat}$
and $x{:}\mathsf{Nat}\vdash\mathsf{dec}_{x,z}::z{:}\mathsf{Nat}$, where
$\mathsf{dec}$ decrements a given natural number session on channel $x$. We
have that:
$\small\begin{array}[]{l}(\mathbf{\nu}x)(\mathsf{one}_{x}\mid\mathsf{dec}_{x,z})\equiv(\mathbf{\nu}x,y.u)(\mathsf{zero}_{y}\mid\mathsf{succ}_{y,x}!u(d).\mathsf{stepDec}_{d}\mid\mathsf{foldP}(\mathsf{Nat})_{x,z})\approx_{\mathtt{L}}\mathsf{zero}_{z}\end{array}$
We note that the resulting encoding is reminiscent of the encoding of lists of
[30] (where $\mathsf{zero}$ is the empty list and $\mathsf{succ}$ the cons
cell). The main differences in the encodings arise due to our primitive
notions of labels and forwarding, as well as due to the generic nature of
$\mathsf{in}$ and $\mathsf{fold}$.
###### Example 4.2 (Streams)
We build on Example 4.1 by representing _streams_ of natural numbers as a
coinductive session type. We encode infinite streams of naturals with
$F(X)=\mathsf{Nat}\otimes X$. Thus: $\mathsf{NatStream}\triangleq\exists
X.!(X\multimap(\mathsf{Nat}\otimes X))\otimes X$. The behaviour of a session
of type $\mathsf{NatStream}$ amounts to an infinite sequence of outputs of
channels of type $\mathsf{Nat}$. Such an encoding enables us to construct the
stream of all naturals $\mathsf{nats}$ (and the stream of all non-zero
naturals $\mathsf{oneNats}$):
$\small\begin{array}[]{lcl}\mathsf{genHdNext}_{z}&\triangleq&z(n).\overline{z}\langle
y\rangle.(\overline{n}\langle n^{\prime}\rangle.[n^{\prime}\leftrightarrow
y]\mid\,!z(w).\overline{n}\langle
n^{\prime}\rangle.\mathsf{succ}_{n^{\prime},w})\\\
\mathsf{nats}_{y}&\triangleq&(\mathbf{\nu}x,u)(\mathsf{zero}_{x}\mid\,!u(z).\mathsf{genHdNext}_{z}\mid\mathsf{unfoldP}(!\mathsf{Nat})_{x,y})\\\
\mathsf{oneNats}_{y}&\triangleq&(\mathbf{\nu}x,u)(\mathsf{one}_{x}\mid\,!u(z).\mathsf{genHdNext}_{z}\mid\mathsf{unfoldP}(!\mathsf{Nat})_{x,y})\end{array}$
with
$\mathsf{genHdNext}_{z}::z{:}!\mathsf{Nat}\multimap\mathsf{Nat}\otimes!\mathsf{Nat}$
and both $\mathsf{nats}_{y}$ and $\mathsf{oneNats}::y{:}\mathsf{NatStream}$.
$\mathsf{genHdNext}_{z}$ consists of a helper that generates the current head
of a stream and the next element. As expected, the following process
implements a session that ``unrolls'' the stream once, providing the head of
the stream and then behaving as the rest of the stream (recall that
$\mathsf{out}:T_{f}\multimap F(T_{f})$).
$(\mathbf{\nu}x)(\mathsf{nats}_{x}\mid\llbracket\mathsf{out}(x)\rrbracket_{y})::y{:}\mathsf{Nat}\otimes\mathsf{NatStream}$
We note a peculiarity of the interaction of linearity with the stream
encoding: a process that begins to deconstruct a stream has no way of
``bottoming out'' and stopping. One cannot, for instance, extract the first
element of a stream of naturals and stop unrolling the stream in a well-typed
way. We can, however, easily encode a ``terminating'' stream of all natural
numbers via $F(X)=(\mathsf{Nat}\otimes!X)$ by replacing the
$\mathsf{genHdNext}_{z}$ with the generator given as:
$\small\begin{array}[]{rcl}\mathsf{genHdNextTer}_{z}&\triangleq&z(n).\overline{z}\langle
y\rangle.(\overline{n}\langle n^{\prime}\rangle.[n^{\prime}\leftrightarrow
y]\mid\,!z(w).!w(w^{\prime}).\overline{n}\langle
n^{\prime}\rangle.\mathsf{succ}_{n^{\prime},w^{\prime}})\end{array}$
It is then easy to see that a usage of
$\llbracket\mathsf{out}(x)\rrbracket_{y}$ results in a session of type
$\mathsf{Nat}\otimes!\mathsf{NatStream}$, enabling us to discard the stream as
needed. One can replay this argument with the operator
$F(X)=(!\mathsf{Nat}\otimes X)$ to enable discarding of stream elements.
Assuming such modifications, we can then show:
$(\mathbf{\nu}y)((\mathbf{\nu}x)(\mathsf{nats}_{x}\mid\llbracket\mathsf{out}(x)\rrbracket_{y})\mid
y(n).[y\leftrightarrow
z])\approx_{\mathtt{L}}\mathsf{oneNats}_{z}::z{:}\mathsf{NatStream}$
### 4.2 Communicating Values – Sess$\pi\lambda$
We now consider a session calculus extended with a data layer obtained from a
$\lambda$-calculus (whose terms are ranged over by $M,N$ and types by
$\tau,\sigma$). We dub this calculus Sess$\pi\lambda$.
$\small\begin{array}[]{ll}\begin{array}[]{lcl}P,Q&::=&\dots\mid x\langle
M\rangle.P\mid x(y).P\\\ M,N&::=&\lambda x{:}\tau.M\mid M\,N\mid
x\end{array}&\quad\quad\begin{array}[]{lcl}A,B&::=&\dots\mid\tau\wedge
A\mid\tau\supset A\\\
\tau,\sigma&::=&\dots\mid\tau\rightarrow\sigma\end{array}\end{array}$
Without loss of generality, we consider the data layer to be simply-typed,
with a call-by-name semantics, satisfying the usual type safety properties.
The typing judgment for this calculus is $\Psi\vdash M:\tau$. We omit session
polymorphism for the sake of conciseness, restricting processes to
communication of data and (session) channels. The typing judgment for
processes is thus modified to $\Psi;\Gamma;\Delta\vdash P::z{:}A$, where
$\Psi$ is an intuitionistic context that accounts for variables in the data
layer. The rules for the relevant process constructs are (all other rules
simply propagate the $\Psi$ context from conclusion to premises):
$\small\begin{array}[]{c}\Psi;\Gamma;\Delta\vdash z\langle
M\rangle.P::z{:}\tau\wedge A\lx@proof@logical@and\Psi\vdash
M:\tau\Psi;\Gamma;\Delta\vdash P::z{:}A\quad\Psi;\Gamma;\Delta,x{:}\tau\wedge
A\vdash x(y).Q::z{:}C\Psi,y{:}\tau;\Gamma;\Delta,x{:}A\vdash
Q::z{:}C\\\\[4.5pt] \Psi;\Gamma;\Delta\vdash z(x).P::z{:}\tau\supset
A\Psi,x{:}\tau;\Gamma;\Delta\vdash
P::z{:}A\quad\Psi;\Gamma;\Delta,x{:}\tau\supset A\vdash x\langle
M\rangle.Q::z{:}C\lx@proof@logical@and\Psi\vdash
M:\tau\Psi;\Gamma;\Delta,x{:}A\vdash Q::z{:}C\end{array}$
With the reduction rule given by:111For simplicity, in this section, we define
the process semantics through a reduction relation. $x\langle M\rangle.P\mid
x(y).Q\xrightarrow{}P\mid Q\\{M/y\\}$. With a simple extension to our
encodings we may eliminate the data layer by encoding the data objects as
processes, showing that from an expressiveness point of view, data
communication is orthogonal to the framework. We note that the data language
we are considering is _not_ linear, and the usage discipline of data in
processes is itself also not linear.
#### To First-Order Processes
We now introduce our encoding for Sess$\pi\lambda$, defined inductively on
session types, processes, types and $\lambda$-terms (we omit the purely
inductive cases on session types and processes for conciseness). As before,
the encoding on processes is defined on _typing derivations_ , where we
indicate the typing rule at the root of the typing derivation.
$\small\begin{array}[]{l}\llbracket\tau\wedge
A\rrbracket\triangleq!\llbracket\tau\rrbracket\otimes\llbracket
A\rrbracket\qquad\llbracket\tau\supset
A\rrbracket\triangleq!\llbracket\tau\rrbracket\multimap\llbracket
A\rrbracket\qquad\llbracket\tau\rightarrow\sigma\rrbracket\triangleq!\llbracket\tau\rrbracket\multimap\llbracket\sigma\rrbracket\end{array}$
$\small\begin{array}[]{lrcllrcl}({{\wedge}\mathsf{R}})&\llbracket z\langle
M\rangle.P\rrbracket&\triangleq&\overline{z}\langle x\rangle.(!x(y).\llbracket
M\rrbracket_{y}\mid\llbracket P\rrbracket)&({{\wedge}\mathsf{L}})&\llbracket
x(y).P\rrbracket&\triangleq&x(y).\llbracket P\rrbracket\\\
({{\supset}\mathsf{R}})&\llbracket z(x).P\rrbracket&\triangleq&z(x).\llbracket
P\rrbracket&({{\supset}\mathsf{L}})&\llbracket x\langle
M\rangle.P\rrbracket&\triangleq&\overline{x}\langle y\rangle.(!y(w).\llbracket
M\rrbracket_{w}\mid\llbracket P\rrbracket)\end{array}$
$\small\begin{array}[]{lll}\llbracket
x\rrbracket_{z}\triangleq\overline{x}\langle y\rangle.[y\leftrightarrow
z]\quad\quad\quad\llbracket\lambda x{:}\tau.M\rrbracket_{z}\triangleq
z(x).\llbracket M\rrbracket_{z}\\\ \llbracket
M\,N\rrbracket_{z}\triangleq(\mathbf{\nu}y)(\llbracket
M\rrbracket_{y}\mid\overline{y}\langle x\rangle.(!x(w).\llbracket
N\rrbracket_{w}\mid[y\leftrightarrow z]))\end{array}$
The encoding addresses the non-linear usage of data elements in processes by
encoding the types $\tau\wedge A$ and $\tau\supset A$ as
$!\llbracket\tau\rrbracket\otimes\llbracket A\rrbracket$ and
$!\llbracket\tau\rrbracket\multimap\llbracket A\rrbracket$, respectively.
Thus, sending and receiving of data is codified as the sending and receiving
of channels of type $!$, which therefore can be used non-linearly. Moreover,
since data terms are themselves non-linear, the $\tau\rightarrow\sigma$ type
is encoded as $!\llbracket\tau\rrbracket\multimap\llbracket\sigma\rrbracket$,
following Girard's embedding of intuitionistic logic in linear logic [15].
At the level of processes, offering a session of type $\tau\wedge A$ (i.e. a
process of the form $z\langle M\rangle.P$) is encoded according to the
translation of the type: we first send a _fresh_ name $x$ which will be used
to access the encoding of the term $M$. Since $M$ can be used an arbitrary
number of times by the receiver, we guard the encoding of $M$ with a
replicated input, proceeding with the encoding of $P$ accordingly. Using a
session of type $\tau\supset A$ follows the same principle. The input cases
(and the rest of the process constructs) are completely homomorphic.
The encoding of $\lambda$-terms follows Girard's decomposition of the
intuitionistic function space [49]. The $\lambda$-abstraction is translated as
input. Since variables in a $\lambda$-abstraction may be used non-linearly,
the case for variables and application is slightly more intricate: to encode
the application $M\,N$ we compose $M$ in parallel with a process that will
send the ``reference'' to the function argument $N$ which will be encoded
using replication, in order to handle the potential for $0$ or more usages of
variables in a function body. Respectively, a variable is encoded by
performing an output to trigger the replication and forwarding accordingly.
Without loss of generality, we assume variable names and their corresponding
replicated counterparts match, which can be achieved through
$\alpha$-conversion before applying the translation. We exemplify our encoding
as follows:
$\small\begin{array}[]{c}\llbracket z(x).z\langle x\rangle.z\langle(\lambda
y{:}\sigma.x)\rangle.{\bf 0}\rrbracket=z(x).\overline{z}\langle
w\rangle.(!w(u).\llbracket x\rrbracket_{u}\mid\overline{z}\langle
v\rangle.(!v(i).\llbracket\lambda y{:}\sigma.x\rrbracket_{i}\mid{\bf 0}))\\\
\qquad\qquad\qquad\qquad=z(x).\overline{z}\langle
w\rangle.(!w(u).\overline{x}\langle y\rangle.[y\leftrightarrow
u]\mid\overline{z}\langle v\rangle.(!v(i).i(y).\overline{x}\langle
t\rangle.[t\leftrightarrow i]\mid{\bf 0}))\end{array}$
Properties of the Encoding. We discuss the correctness of our encoding. We
can straightforwardly establish that the encoding preserves typing.
To show that our encoding is operationally sound and complete, we capture the
interaction between substitution on $\lambda$-terms and the encoding into
processes through logical equivalence. Consider the following reduction of a
process:
$\displaystyle(\mathbf{\nu}z)(z(x).z\langle x\rangle.z\langle(\lambda
y{:}\sigma.x)\rangle.{\bf 0}\mid z\langle\lambda w{:}\tau_{0}.w\rangle.P)$
$\displaystyle\hskip 85.35826pt\xrightarrow{}(\mathbf{\nu}z)(z\langle\lambda
w{:}\tau_{0}.w\rangle.z\langle(\lambda y{:}\sigma.\lambda
w{:}\tau_{0}.w)\rangle.{\bf 0}\mid P)$ (1)
Given that substitution in the target session $\pi$-calculus amounts to
renaming, whereas in the $\lambda$-calculus we replace a variable for a term,
the relationship between the encoding of a substitution $M\\{N/x\\}$ and the
encodings of $M$ and $N$ corresponds to the composition of the encoding of $M$
with that of $N$, but where the encoding of $N$ is guarded by a replication,
codifying a form of explicit non-linear substitution.
###### Lemma 4.1 (Compositionality)
Let $\Psi,x{:}\tau\vdash M:\sigma$ and $\Psi\vdash N:\tau$. We have that
$\llbracket
M\\{N/x\\}\rrbracket_{z}\approx_{\mathtt{L}}(\mathbf{\nu}x)(\llbracket
M\rrbracket_{z}\mid!x(y).\llbracket N\rrbracket_{y})$
Revisiting the process to the left of the arrow in Equation 1 we have:
$\small\begin{array}[]{l}\llbracket(\mathbf{\nu}z)(z(x).z\langle
x\rangle.z\langle(\lambda y{:}\sigma.x)\rangle.{\bf 0}\mid z\langle\lambda
w{:}\tau_{0}.w\rangle.P)\rrbracket\\\ =(\mathbf{\nu}z)(\llbracket
z(x).z\langle x\rangle.z\langle(\lambda y{:}\sigma.x)\rangle.{\bf
0}\rrbracket_{z}\mid\overline{z}\langle x\rangle.(!x(b).\llbracket\lambda
w{:}\tau_{0}.w\rrbracket_{b}\mid\llbracket P\rrbracket))\\\
\xrightarrow{}(\mathbf{\nu}z,x)(\overline{z}\langle
w\rangle.(!w(u).\overline{x}\langle y\rangle.[y\leftrightarrow
u]\mid\overline{z}\langle v\rangle.(!v(i).\llbracket\lambda
y{:}\sigma.x\rrbracket_{i}\mid{\bf 0})\mid\,!x(b).\llbracket\lambda
w{:}\tau_{0}.w\rrbracket_{b}\mid\llbracket P\rrbracket))\end{array}$
whereas the process to the right of the arrow is encoded as:
$\small\begin{array}[]{l}\llbracket(\mathbf{\nu}z)(z\langle\lambda
w{:}\tau_{0}.w\rangle.z\langle(\lambda y{:}\sigma.\lambda
w{:}\tau_{0}.w)\rangle.{\bf 0}\mid P)\rrbracket\\\
=(\mathbf{\nu}z)(\overline{z}\langle w\rangle.(!w(u).\llbracket\lambda
w{:}\tau_{0}.w\rrbracket_{u}\mid\overline{z}\langle
v\rangle.(!v(i).\llbracket\lambda y{:}\sigma.\lambda
w{:}\tau_{0}.w\rrbracket_{i}\mid\llbracket P\rrbracket)))\end{array}$
While the reduction of the encoded process and the encoding of the reduct
differ syntactically, they are observationally equivalent – the latter inlines
the replicated process behaviour that is accessible in the former on $x$.
Having characterised substitution, we establish operational correspondence for
the encoding.
###### Theorem 4.3 (Operational Correspondence)
1. 1.
If $\Psi\vdash M:\tau$ and $\llbracket M\rrbracket_{z}\xrightarrow{}Q$ then
$M\xrightarrow{}^{+}N$ such that $\llbracket
N\rrbracket_{z}\approx_{\mathtt{L}}Q$
2. 2.
If $\Psi;\Gamma;\Delta\vdash P::z{:}A$ and $\llbracket
P\rrbracket\xrightarrow{}Q$ then $P\xrightarrow{}^{+}P^{\prime}$ such that
$\llbracket P^{\prime}\rrbracket\approx_{\mathtt{L}}Q$
3. 3.
If $\Psi\vdash M:\tau$ and $M\xrightarrow{}N$ then $\llbracket
M\rrbracket_{z}\stackrel{{\scriptstyle}}{{\Longrightarrow}}P$ such that
$P\approx_{\mathtt{L}}\llbracket N\rrbracket_{z}$
4. 4.
If $\Psi;\Gamma;\Delta\vdash P::z{:}A$ and $P\xrightarrow{}Q$ then $\llbracket
P\rrbracket\xrightarrow{}^{+}R$ with $R\approx_{\mathtt{L}}\llbracket
Q\rrbracket$
The process equivalence in Theorem 4.3 above need not be extended to account
for data (although it would be relatively simple to do so), since the
processes in the image of the encoding are fully erased of any data elements.
#### Back to $\lambda$-Terms.
We extend our encoding of processes to $\lambda$-terms to Sess$\pi\lambda$.
Our extended translation maps processes to linear $\lambda$-terms, with the
session type $\tau\wedge A$ interpreted as a pair type where the first
component is replicated. Dually, $\tau\supset A$ is interpreted as a function
type where the domain type is replicated. The remaining session constructs are
translated as in § 3.2.
$\small\begin{array}[]{c}\llparenthesis\tau\wedge A\rrparenthesis\triangleq\
!\llparenthesis\tau\rrparenthesis\otimes\llparenthesis
A\rrparenthesis\quad\quad\llparenthesis\tau\supset A\rrparenthesis\triangleq\
!\llparenthesis\tau\rrparenthesis\multimap\llparenthesis
A\rrparenthesis\quad\quad\llparenthesis\tau\rightarrow\sigma\rrparenthesis\triangleq\
!\llparenthesis\tau\rrparenthesis\multimap\llparenthesis\sigma\rrparenthesis\end{array}$
$\small\begin{array}[]{llclllcl}({{\wedge}\mathsf{L}})&\llparenthesis
x(y).P\rrparenthesis&\triangleq&\mathsf{let}\,y\otimes
x=x\,\mathsf{in}\,\mathsf{let}\,!y=y\,\mathsf{in}\,\llparenthesis
P\rrparenthesis&({{\wedge}\mathsf{R}})&\llparenthesis z\langle
M\rangle.P\rrparenthesis&\triangleq&\langle!\llparenthesis
M\rrparenthesis\otimes\llparenthesis P\rrparenthesis\rangle\\\
({{\supset}\mathsf{R}})&\llparenthesis x(y).P\rrparenthesis&\triangleq&\lambda
x{:}!\llparenthesis\tau\rrparenthesis.\mathsf{let}\,!x=x\,\mathsf{in}\,\llparenthesis
P\rrparenthesis&({{\supset}\mathsf{L}})&\llparenthesis x\langle
M\rangle.P\rrparenthesis&\triangleq&\llparenthesis
P\rrparenthesis\\{(x\,!\llparenthesis M\rrparenthesis)/x\\}\end{array}$
$\begin{array}[]{c}\llparenthesis\lambda
x{:}\tau.M\rrparenthesis\triangleq\lambda
x{:}!\llparenthesis\tau\rrparenthesis.\mathsf{let}\,!x=x\,\mathsf{in}\,\llparenthesis
M\rrparenthesis\par\quad\llparenthesis
M\,N\rrparenthesis\triangleq\llparenthesis M\rrparenthesis\,!\llparenthesis
N\rrparenthesis\quad\llparenthesis x\rrparenthesis\triangleq x\end{array}$
The treatment of non-linear components of processes is identical to our
previous encoding: non-linear functions $\tau\rightarrow\sigma$ are translated
to linear functions of type $!\tau\multimap\sigma$; a process offering a
session of type $\tau\wedge A$ (i.e. a process of the form $z\langle
M\rangle.P$, typed by rule ${{\wedge}\mathsf{R}}$) is translated to a pair
where the first component is the encoding of $M$ prefixed with $!$ so that it
may be used non-linearly, and the second is the encoding of $P$. Non-linear
variables are handled at the respective binding sites: a process using a
session of type $\tau\wedge A$ is encoded using the elimination form for the
pair and the elimination form for the exponential; similarly, a process
offering a session of type $\tau\supset A$ is encoded as a
$\lambda$-abstraction where the bound variable is of type
$!\llparenthesis\tau\rrparenthesis$. Thus, we use the elimination form for the
exponential, ensuring that the typing is correct. We illustrate our encoding:
$\small\begin{array}[]{rcl}\llparenthesis z(x).z\langle
x\rangle.z\langle(\lambda y{:}\sigma.x)\rangle.{\bf 0}\rrparenthesis&=&\lambda
x{:}!\llparenthesis\tau\rrparenthesis.\mathsf{let}\,!x=x\,\mathsf{in}\,\langle!x\,\otimes\langle!\llparenthesis\lambda
y{:}\sigma.x\rrparenthesis\,\otimes\langle\rangle\rangle\rangle\\\ &=&\lambda
x{:}!\llparenthesis\tau\rrparenthesis.\mathsf{let}\,!x=x\,\mathsf{in}\,\langle!x\,\otimes\langle!(\lambda
y{:}!\llparenthesis\sigma\rrparenthesis.\mathsf{let}\,!y=y\,\mathsf{in}\,x)\,\otimes\langle\rangle\rangle\rangle\end{array}$
Properties of the Encoding. Unsurprisingly due to the logical correspondence
between natural deduction and sequent calculus presentations of logic, our
encoding satisfies both type soundness and operational correspondence (c.f.
Theorems 3.2, 3.3, and 3.4). The full development can be found in [52].
#### Relating the Two Encodings.
We prove the two encodings are mutually inverse and preserve the full
abstraction properties (we write $=_{\beta}$ and $=_{\beta\eta}$ for $\beta$\-
and $\beta\eta$-equivalence, respectively).
###### Theorem 4.4 (Inverse)
If $\Psi;\Gamma;\Delta\vdash P::z{:}A$ then $\llbracket\llparenthesis
P\rrparenthesis\rrbracket_{z}\approx_{\mathtt{L}}\llbracket P\rrbracket$.
Also, if $\Psi\vdash M:\tau$ then $\llparenthesis\llbracket
M\rrbracket_{z}\rrparenthesis=_{\beta}\llparenthesis M\rrparenthesis$.
The equivalences above are formulated between the composition of the encodings
applied to $P$ (resp. $M$) and the process (resp. $\lambda$-term) _after_
applying the translation embedding the non-linear components into their linear
counterparts. This formulation matches more closely that of § 3.3, which
applies to linear calculi for which the _target_ languages of this section are
a strict subset (and avoids the formalisation of process equivalence with
terms). We also note that in this setting, observational equivalence and
$\beta\eta$-equivalence coincide [3, 31]. Moreover, the extensional flavour of
$\approx_{\mathtt{L}}$ includes $\eta$-like principles at the process level.
###### Theorem 4.5 ()
Let $\cdot\vdash M:\tau$ and $\cdot\vdash N:\tau$. $\llparenthesis
M\rrparenthesis=_{\beta\eta}\llparenthesis N\rrparenthesis$ iff $\llbracket
M\rrbracket_{z}\approx_{\mathtt{L}}\llbracket N\rrbracket_{z}$. Also, let
$\cdot\vdash P::z{:}A$ and $\cdot\vdash Q::z{:}A$. We have that $\llbracket
P\rrbracket\approx_{\mathtt{L}}\llbracket Q\rrbracket$ iff $\llparenthesis
P\rrparenthesis=_{\beta\eta}\llparenthesis Q\rrparenthesis$.
We establish full abstraction for the encoding of $\lambda$-terms into
processes (Theorem 4.5) in two steps: The completeness direction (i.e. from
left-to-right) follows from operational completeness and strong normalisation
of the $\lambda$-calculus. The soundness direction uses operational soundness.
The proof of Theorem 4.5 uses the same strategy of Theorem 3.7, appealing to
the inverse theorems.
### 4.3 Higher-Order Session Processes – Sess$\pi\lambda^{+}$
We extend the value-passing framework of the previous section, accounting for
process-passing (i.e. the higher-order) in a session-typed setting. As shown
in [50], we achieve this by adding to the data layer a _contextual monad_ that
encapsulates (open) session-typed processes as data values, with a
corresponding elimination form in the process layer. We dub this calculus
Sess$\pi\lambda^{+}$.
$\begin{array}[]{rclrcl}P,Q&::=&\dots\mid x\leftarrow
M\leftarrow\overline{y_{i}};Q&\quad\quad M.N&::=&\dots\mid\\{x\leftarrow
P\leftarrow\overline{y_{i}{:}A_{i}}\\}\\\
\tau,\sigma&::=&\dots\mid\\{\overline{x_{j}{:}A_{j}}\vdash
z{:}A\\}\end{array}$
The type $\\{\overline{x_{j}{:}A_{j}}\vdash z{:}A\\}$ is the type of a term
which encapsulates an open process that uses the linear channels
$\overline{x_{j}{:}A_{j}}$ and offers $A$ along channel $z$. This formulation
has the added benefit of formalising the integration of session-typed
processes in a functional language and forms the basis for the concurrent
programming language SILL [37, 50]. The typing rules for the new constructs
are (for simplicity we assume no shared channels in process monads):
$\begin{array}[]{c}\Psi\vdash\\{z\leftarrow
P\leftarrow\overline{x_{i}{:}A_{i}}\\}:\\{\overline{x_{i}{:}A_{i}}\vdash
z{:}A\\}\Psi;\cdot;\overline{x_{i}{:}A_{i}}\vdash P::z{:}A\\\\[4.5pt]
\Psi;\Gamma;\Delta_{1},\Delta_{2}\vdash x\leftarrow
M\leftarrow\overline{y_{i}};Q::z{:}C\lx@proof@logical@and\Psi\vdash
M:\\{\overline{x_{i}{:}A_{i}}\vdash
x{:}A\\}\Delta_{1}=\overline{y_{i}{:}A_{i}}\Psi;\Gamma;\Delta_{2},x{:}A\vdash
Q::z{:}C\end{array}$
Rule $\\{\\}I$ embeds processes in the term language by essentially quoting an
open process that is well-typed according to the type specification in the
monadic type. Dually, rule $\\{\\}E$ allows for processes to use monadic
values through composition that _consumes_ some of the ambient channels in
order to provide the monadic term with the necessary context (according to its
type). These constructs are discussed in substantial detail in [50]. The
reduction semantics of the process construct is given by (we tacitly assume
that the names $\overline{y}$ and $c$ do not occur in $P$ and omit the
congruence case):
$\begin{array}[]{c}(c\leftarrow\\{z\leftarrow
P\leftarrow\overline{x_{i}{:}A_{i}}\\}\leftarrow\overline{y_{i}};Q)\xrightarrow{}(\mathbf{\nu}c)(P\\{\overline{y}/\overline{x_{i}}\\{c/z\\}\\}\mid
Q)\end{array}$
The semantics allows for the underlying monadic term $M$ to evaluate to a
(quoted) process $P$. The process $P$ is then executed in parallel with the
continuation $Q$, sharing the linear channel $c$ for subsequent interactions.
We illustrate the higher-order extension with following typed process (we
write $\\{x\leftarrow P\\}$ when $P$ does not depend on any linear channels
and assume $\vdash Q::d{:}\mathsf{Nat}\wedge\mathbf{1}$):
$P\triangleq(\mathbf{\nu}c)(c\langle\\{d\leftarrow Q\\}\rangle.c(x).{\bf
0}\mid c(y).d\leftarrow y;d(n).c\langle n\rangle.{\bf 0})$ (2)
Process $P$ above gives an abstract view of a communication idiom where a
process (the left-hand side of the parallel composition) sends another process
$Q$ which potentially encapsulates some complex computation. The receiver then
_spawns_ the execution of the received process and inputs from it a result
value that is sent back to the original sender. An execution of $P$ is given
by:
$\small\begin{array}[]{rcl}P\xrightarrow{}(\mathbf{\nu}c)(c(x).{\bf 0}\mid
d\leftarrow\\{d\leftarrow Q\\};d(n).c\langle n\rangle.{\bf
0})&\xrightarrow{}&(\mathbf{\nu}c)(c(x).{\bf 0}\mid(\mathbf{\nu}d)(Q\mid
d(n).c\langle n\rangle.{\bf 0}))\\\
&\xrightarrow{}^{+}&(\mathbf{\nu}c)(c(x).{\bf 0}\mid c\langle 42\rangle.{\bf
0})\xrightarrow{}{\bf 0}\end{array}$
Given the seminal work of Sangiorgi [46], such a representation naturally begs
the question of whether or not we can develop a _typed_ encoding of higher-
order processes into the first-order setting. Indeed, we can achieve such an
encoding with a fairly simple extension of the encoding of § 4.2 to
Sess$\pi\lambda^{+}$ by observing that monadic values are processes that need
to be potentially provided with extra sessions in order to be executed
correctly. For instance, a term of type $\\{x{:}A\vdash y{:}B\\}$ denotes a
process that given a session $x$ of type $A$ will then offer $y{:}B$.
Exploiting this observation we encode this type as the session $A\multimap B$,
ensuring subsequent usages of such a term are consistent with this
interpretation.
$\small\begin{array}[]{lcl}\llbracket\\{\overline{x_{j}{:}A_{j}}\vdash
z{:}A\\}\rrbracket&\triangleq&\overline{\llbracket
A_{j}\rrbracket}\multimap\llbracket A\rrbracket\\\\[4.5pt]
\llbracket\\{x\leftarrow
P\rightarrow\overline{y_{i}}\\}\rrbracket_{z}&\triangleq&z(y_{0}).\dots.z(y_{n}).\llbracket
P\\{z/x\\}\rrbracket\quad(z\not\in\mbox{\it fn}(P))\\\\[4.5pt] \llbracket
x\leftarrow
M\leftarrow\overline{y_{i}};Q\rrbracket&\triangleq&(\mathbf{\nu}x)(\llbracket
M\rrbracket_{x}\mid\overline{x}\langle a_{0}\rangle.([a_{0}\leftrightarrow
y_{0}]\mid\dots\mid x\langle a_{n}\rangle.([a_{n}\leftrightarrow
y_{n}]\mid\llbracket Q\rrbracket)\dots))\end{array}$
To encode the monadic type $\\{\overline{x_{j}{:}A_{j}}\vdash z{:}A\\}$,
denoting the type of process $P$ that is typed by
$\overline{x_{j}{:}A_{j}}\vdash P::z{:}A$, we require that the session in the
image of the translation specifies a sequence of channel inputs with
behaviours $\overline{A_{j}}$ that make up the linear context. After the
contextual aspects of the type are encoded, the session will then offer the
(encoded) behaviour of $A$. Thus, the encoding of the monadic type is
$\llbracket A_{0}\rrbracket\multimap\dots\multimap\llbracket
A_{n}\rrbracket\multimap\llbracket A\rrbracket$, which we write as
$\overline{\llbracket A_{j}\rrbracket}\multimap\llbracket A\rrbracket$. The
encoding of monadic expressions adheres to this behaviour, first performing
the necessary sequence of inputs and then proceeding inductively. Finally, the
encoding of the elimination form for monadic expressions behaves dually,
composing the encoding of the monadic expression with a sequence of outputs
that instantiate the consumed names accordingly (via forwarding). The encoding
of process $P$ from Equation 2 is thus:
$\small\begin{array}[]{l}\llbracket P\rrbracket=(\mathbf{\nu}c)(\llbracket
c\langle\\{d\leftarrow Q\\}\rangle.c(x).{\bf 0}\rrbracket\mid\llbracket
c(y).d\leftarrow y;d(n).c\langle n\rangle.{\bf 0}\rrbracket)\\\
=(\mathbf{\nu}c)(\overline{c}\langle w\rangle.(!w(d).\llbracket
Q\rrbracket\mid c(x).{\bf 0})c(y).(\mathbf{\nu}d)(\overline{y}\langle
b\rangle.[b\leftrightarrow d]\mid d(n).\overline{c}\langle
m\rangle.(\overline{n}\langle e\rangle.[e\leftrightarrow m]\mid{\bf
0})))\end{array}$
#### Properties of the Encoding.
As in our previous development, we can show that our encoding for
Sess$\pi\lambda^{+}$ is type sound and satisfies operational correspondence.
The full development is omitted but can be found in [52].
We encode Sess$\pi\lambda^{+}$ into $\lambda$-terms, extending § 4.2 with:
$\small\begin{array}[]{l}\llparenthesis\\{\overline{x_{i}{:}A_{i}}\vdash
z{:}A\\}\rrparenthesis\triangleq\overline{\llparenthesis
A_{i}\rrparenthesis}\multimap\llparenthesis A\rrparenthesis\\\\[1.42262pt]
\llparenthesis x\leftarrow
M\leftarrow\overline{y_{i}};Q\rrparenthesis\triangleq\llparenthesis
Q\rrparenthesis\\{(\llparenthesis M\rrparenthesis\,\overline{y_{i}})/x\\}\
\quad\llparenthesis\\{x\leftarrow
P\leftarrow\overline{w_{i}}\\}\rrparenthesis\triangleq\lambda
w_{0}.\dots.\lambda w_{n}.\llparenthesis P\rrparenthesis\end{array}$
The encoding translates the monadic type $\\{\overline{x_{i}{:}A_{i}}\vdash
z{:}A\\}$ as a linear function $\overline{\llparenthesis
A_{i}\rrparenthesis}\multimap\llparenthesis A\rrparenthesis$, which captures
the fact that the underlying value must be provided with terms satisfying the
requirements of the linear context. At the level of terms, the encoding for
the monadic term constructor follows its type specification, generating a
nesting of $\lambda$-abstractions that closes the term and proceeding
inductively. For the process encoding, we translate the monadic application
construct analogously to the translation of a linear $\mathsf{cut}$, but
applying the appropriate variables to the translated monadic term (which is of
function type). We remark the similarity between our encoding and that of the
previous section, where monadic terms are translated to a sequence of inputs
(here a nesting of $\lambda$-abstractions). Our encoding satisfies type
soundness and operational correspondence, as usual. Further showcasing the
applications of our development, we obtain a novel strong normalisation result
for this higher-order session-calculus ``for free'', through encoding to the
$\lambda$-calculus.
###### Theorem 4.6 (Strong Normalisation)
Let $\Psi;\Gamma;\Delta\vdash P::z{:}A$. There is no infinite reduction
sequence starting from $P$.
###### Theorem 4.7 (Inverse Encodings)
If $\Psi;\Gamma;\Delta\vdash P::z{:}A$ then $\llbracket\llparenthesis
P\rrparenthesis\rrbracket_{z}\approx_{\mathtt{L}}\llbracket P\rrbracket$.
Also, if $\Psi\vdash M:\tau$ then $\llparenthesis\llbracket
M\rrbracket_{z}\rrparenthesis=_{\beta}\llparenthesis M\rrparenthesis$.
###### Theorem 4.8
Let $\vdash M:\tau$, $\vdash N:\tau$, $\vdash P::z{:}A$ and $\vdash Q::z{:}A$.
$\llparenthesis M\rrparenthesis=_{\beta\eta}\llparenthesis N\rrparenthesis$
iff $\llbracket M\rrbracket_{z}\approx_{\mathtt{L}}\llbracket N\rrbracket_{z}$
and $\llbracket P\rrbracket\approx_{\mathtt{L}}\llbracket Q\rrbracket$ iff
$\llparenthesis P\rrparenthesis=_{\beta\eta}\llparenthesis Q\rrparenthesis$.
## 5 Related Work and Concluding Remarks
Process Encodings of Functions. Toninho et al. [49] study encodings of the
simply-typed $\lambda$-calculus in a logically motivated session
$\pi$-calculus, via encodings to the linear $\lambda$-calculus. Our work
differs since they do not study polymorphism nor reverse encodings; and we
provide deeper insights through applications of the encodings. Full
abstraction or inverse properties are not studied.
Sangiorgi [43] uses a fully abstract compilation from the higher-order
$\pi$-calculus (HO$\pi$) to the $\pi$-calculus to study full abstraction for
Milner's encodings of the $\lambda$-calculus. The work shows that Milner's
encoding of the lazy $\lambda$-calculus can be recovered by restricting the
semantic domain of processes (the so-called _restrictive_ approach) or by
enriching the $\lambda$-calculus with suitable constants. This work was later
refined in [45], which does not use HO$\pi$ and considers an operational
equivalence on $\lambda$-terms called _open applicative bisimulation_ which
coincides with Lévy-Longo tree equality. The work [47] studies general
conditions under which encodings of the $\lambda$-calculus in the
$\pi$-calculus are fully abstract wrt Lévy-Longo and Böhm Trees, which are
then applied to several encodings of (call-by-name) $\lambda$-calculus. The
works above deal with untyped calculi, and so reverse encodings are
unfeasible. In a broader sense, our approach takes the restrictive approach
using linear logic-based session typing and the induced observational
equivalence. We use a $\lambda$-calculus with booleans as observables and
reason with a Morris-style equivalence instead of tree equalities. It would be
an interesting future work to apply the conditions in [47] in our typed
setting.
Wadler [54] shows a correspondence between a linear functional language with
session types GV and a session-typed process calculus with polymorphism based
on classical linear logic CP. Along the lines of this work, Lindley and Morris
[26], in an exploration of inductive and coinductive session types through the
addition of least and greatest fixed points to CP and GV, develop an encoding
from a linear $\lambda$-calculus with session primitives (Concurrent $\mu$GV)
to a pure linear $\lambda$-calculus (Functional $\mu$GV) via a CPS
transformation. They also develop translations between $\mu$CP and Concurrent
$\mu$GV, extending [25]. Mapping to the terminology used in our work [17],
their encodings are shown to be operationally complete, but no results are
shown for the operational soundness directions and neither full abstraction
nor inverse properties are studied. In addition, their operational
characterisations do not compose across encodings. For instance, while strong
normalisation of Functional $\mu$GV implies the same property for Concurrent
$\mu$GV through their operationally complete encoding, the encoding from
$\mu$CP to $\mu$GV does not necessarily preserve this property.
Types for $\pi$-calculi delineate sequential behaviours by restricting
composition and name usages, limiting the contexts in which processes can
interact. Therefore typed equivalences offer a coarser semantics than untyped
semantics. Berger et al. [5] study an encoding of System F in a polymorphic
linear $\pi$-calculus, showing it to be fully abstract based on game semantics
techniques. Their typing system and proofs are more complex due to the fine-
grained constraints from game semantics. Moreover, they do not study a reverse
encoding.
Orchard and Yoshida [33] develop embeddings to-and-from PCF with parallel
effects and a session-typed $\pi$-calculus, but only develop operational
correspondence and semantic soundness results, leaving the full abstraction
problem open.
Polymorphism and Typed Behavioural Semantics. The work of [7] studies
parametric session polymorphism for the intuitionistic setting, developing a
behavioural equivalence that captures parametricity, which is used (denoted as
$\approx_{\mathtt{L}}$) in our paper. The work [39] introduces a typed
bisimilarity for polymorphism in the $\pi$-calculus. Their bisimilarity is of
an intensional flavour, whereas the one used in our work follows the
extensional style of Reynolds [41]. Their typing discipline (originally from
[53], which also develops type-preserving encodings of polymorphic
$\lambda$-calculus into polymorphic $\pi$-calculus) differs significantly from
the linear logic-based session typing of our work (e.g. theirs does not ensure
deadlock-freedom). A key observation in their work is the coarser nature of
typed equivalences with polymorphism (in analogy to those for IO-subtyping
[38]) and their interaction with channel aliasing, suggesting a use of typed
semantics and encodings of the $\pi$-calculus for fine-grained analyses of
program behaviour.
F-Algebras and Linear-F. The use of initial and final (co)algebras to give a
semantics to inductive and coinductive types dates back to Mendler [28], with
their strong definability in System F appearing in [1] and [19]. The
definability of inductive and coinductive types using parametricity also
appears in [40] in the context of a logic for parametric polymorphism and
later in [6] in a linear variant of such a logic. The work of [55] studies
parametricity for the polymorphic linear $\lambda$-calculus of this work,
developing encodings of a few inductive types but not the initial (or final)
algebraic encodings in their full generality. Inductive and coinductive
session types in a logical process setting appear in [51] and [26]. Both works
consider a calculus with built-in recursion – the former in an intuitionistic
setting where a process that offers a (co)inductive protocol is composed with
another that consumes the (co)inductive protocol and the latter in a classical
framework where composed recursive session types are dual each other.
Conclusion and Future Work. This work answers the question of what kind of
type discipline of the $\pi$-calculus can exactly capture and is captured by
$\lambda$-calculus behaviours. Our answer is given by showing the first
mutually inverse and fully abstract encodings between two calculi with
polymorphism, one being the Poly$\pi$ session calculus based on intuitionistic
linear logic, and the other (a linear) System F. This further demonstrates
that the linear logic-based articulation of name-passing interactions
originally proposed by [8] (and studied extensively thereafter e.g. [50, 51,
36, 9, 54, 7, 25]) provides a clear and applicable tool for message-passing
concurrency. By exploiting the proof theoretic equivalences between natural
deduction and sequent calculus we develop mutually inverse and fully abstract
encodings, which naturally extend to more intricate settings such as process
passing (in the sense of HO$\pi$). Our encodings also enable us to derive
properties of the $\pi$-calculi ``for free''. Specifically, we show how to
obtain adequate representations of least and greatest fixed points in
Poly$\pi$ through the encoding of initial and final (co)algebras in the
$\lambda$-calculus. We also straightforwardly derive a strong normalisation
result for the higher-order session calculus, which otherwise involves non-
trivial proof techniques [13, 12, 36, 7, 5]. Future work includes extensions
to the classical linear logic-based framework, including multiparty session
types [10, 11]. Encodings of session $\pi$-calculi to the $\lambda$-calculus
have been used to implement session primitives in functional languages such as
Haskell (see a recent survey [32]), OCaml [34, 35, 24] and Scala [48].
Following this line of work, we plan to develop encoding-based implementations
of this work as embedded DSLs. This would potentially enable an exploration of
algebraic constructs beyond initial and final co-algebras in a session
programming setting. In particular, we wish to further study the meaning of
functors, natural transformations and related constructions in a session-typed
setting, both from a more fundamental viewpoint but also in terms of
programming patterns.
Acknowledgements. The authors would like to thank the reviewers for their
comments, suggestions and pointers to related works. This work is partially
supported by EPSRC EP/K034413/1, EP/K011715/1, EP/L00058X/1, EP/N027833/1 and
EP/N028201/1.
## References
* [1] Bainbridge, E.S., Freyd, P.J., Scedrov, A., Scott, P.J.: Functorial polymorphism. Theor. Comput. Sci. 70(1), 35–64 (1990)
* [2] Balzer, S., Pfenning, F.: Manifest sharing with session types. In: ICFP (2017)
* [3] Barber, A.: Dual intuitionistic linear logic. Tech. Rep. ECS-LFCS-96-347, School of Informatics, University of Edinburgh (1996)
* [4] Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models (extended abstract). In: CSL. pp. 121–135 (1994)
* [5] Berger, M., Honda, K., Yoshida, N.: Genericity and the pi-calculus. Acta Inf. 42(2-3), 83–141 (2005)
* [6] Birkedal, L., Møgelberg, R.E., Petersen, R.L.: Linear Abadi and Plotkin Logic. Logical Methods in Computer Science 2(5) (2006)
* [7] Caires, L., Pérez, J.A., Pfenning, F., Toninho, B.: Behavioral polymorphism and parametricity in session-based communication. In: ESOP 2013\. pp. 330–349 (2013)
* [8] Caires, L., Pfenning, F.: Session types as intuitionistic linear propositions. In: CONCUR 2010. pp. 222–236 (2010)
* [9] Caires, L., Pfenning, F., Toninho, B.: Linear logic propositions as session types. Mathematical Structures in Computer Science 26(3), 367–423 (2016)
* [10] Carbone, M., Lindley, S., Montesi, F., Schuermann, C., Wadler, P.: Coherence generalises duality: a logical explanation of multiparty session types. In: CONCUR'16. vol. 59, pp. 33:1–33:15. Sch. Dag. (2016)
* [11] Carbone, M., Montesi, F., Schurmann, C., Yoshida, N.: Multiparty session types as coherence proofs. In: CONCUR 2015. vol. 42, pp. 412–426. Sch. Dag. (2015)
* [12] Demangeon, R., Hirschkoff, D., Sangiorgi, D.: Mobile processes and termination. In: Semantics and Algebraic Specification. pp. 250–273 (2009)
* [13] Demangeon, R., Hirschkoff, D., Sangiorgi, D.: Termination in higher-order concurrent calculi. J. Log. Algebr. Program. 79(7), 550–577 (2010)
* [14] Gentzen, G.: Untersuchungen über das logische schließen. Mathematische Zeitschrift 39, 176–210 (1935)
* [15] Girard, J.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)
* [16] Girard, J., Lafont, Y., Taylor, P.: Proofs and Types. C. U. P. (1989)
* [17] Gorla, D.: Towards a unified approach to encodability and separation results for process calculi. Inf. Comput. 208(9), 1031–1053 (2010)
* [18] Gorla, D., Nestmann, U.: Full abstraction for expressiveness: history, myths and facts. Mathematical Structures in Computer Science 26(4), 639–654 (2016)
* [19] Hasegawa, R.: Categorical data types in parametric polymorphism. Mathematical Structures in Computer Science 4(1), 71–109 (1994)
* [20] Honda, K.: Types for dyadic interaction. In: CONCUR'93. pp. 509–523 (1993)
* [21] Honda, K.: Session types and distributed computing. In: ICALP (2012)
* [22] Honda, K., Vasconcelos, V.T., Kubo, M.: Language primitives and type disciplines for structured communication-based programming. In: ESOP'98. pp. 22–138 (1998)
* [23] Honda, K., Yoshida, N., Carbone, M.: Multiparty asynchronous session types. In: POPL'08. pp. 273–284 (2008)
* [24] Imai, K., Yoshida, N., Yuen, S.: Session-ocaml: a session-based library with polarities and lenses . In: COORDINATION. LNCS, vol. 10319, pp. 99–118 (2017)
* [25] Lindley, S., Morris, J.G.: A semantics for propositions as sessions. In: ESOP'15. pp. 560–584 (2015)
* [26] Lindley, S., Morris, J.G.: Talking bananas: structural recursion for session types. In: ICFP 2016. pp. 434–447 (2016)
* [27] Maraist, J., Odersky, M., Turner, D.N., Wadler, P.: Call-by-name, call-by-value, call-by-need and the linear lambda calculus. T. C. S. 228(1-2), 175–210 (1999)
* [28] Mendler, N.P.: Recursive types and type constraints in second-order lambda calculus. In: LICS'87. pp. 30–36 (1987)
* [29] Milner, R.: Functions as processes. Math. Struct. in C.S. 2(2), 119–141 (1992)
* [30] Milner, R., Parrow, J., Walker, D.: A calculus of mobile processes, I and II. Inf. Comput. 100(1), 1–77 (1992)
* [31] Ohta, Y., Hasegawa, M.: A terminating and confluent linear lambda calculus. In: RTA'06. pp. 166–180 (2006)
* [32] Orchard, D., Yoshida, N.: Session types with linearity in Haskell. In: Behavioural Types: from Theory to Tools. River Publishers (2017)
* [33] Orchard, D.A., Yoshida, N.: Effects as sessions, sessions as effects. In: POPL 2016. pp. 568–581 (2016)
* [34] Padovani, L.: A Simple Library Implementation of Binary Sessions. JFP 27 (2016)
* [35] Padovani, L.: Context-Free Session Type Inference. In: ESOP 2017 (2017)
* [36] Pérez, J.A., Caires, L., Pfenning, F., Toninho, B.: Linear logical relations for session-based concurrency. In: ESOP. pp. 539–558 (2012)
* [37] Pfenning, F., Griffith, D.: Polarized substructural session types. In: FoSSaCS. pp. 3–22 (2015)
* [38] Pierce, B.C., Sangiorgi, D.: Typing and subtyping for mobile processes. Mathematical Structures in Computer Science 6(5), 409–453 (1996)
* [39] Pierce, B.C., Sangiorgi, D.: Behavioral equivalence in the polymorphic pi-calculus. J. ACM 47(3), 531–584 (2000)
* [40] Plotkin, G.D., Abadi, M.: A logic for parametric polymorphism. In: TLCA '93. pp. 361–375 (1993)
* [41] Reynolds, J.C.: Types, abstraction and parametric polymorphism. In: IFIP Congress. pp. 513–523 (1983)
* [42] Reynolds, J.C., Plotkin, G.D.: On functors expressible in the polymorphic typed lambda calculus. Inf. Comput. 105(1), 1–29 (1993)
* [43] Sangiorgi, D.: An investigation into functions as processes. In: MFPS (1993)
* [44] Sangiorgi, D.: Pi-calculus, internal mobility, and agent-passing calculi. Theor. Comput. Sci. 167(1&2), 235–274 (1996)
* [45] Sangiorgi, D.: Lazy functions and mobile processes. In: Proof, Language, and Interaction, Essays in Honour of Robin Milner. pp. 691–720 (2000)
* [46] Sangiorgi, D., Walker, D.: The pi-calculus: A theory of mobile processes (2001)
* [47] Sangiorgi, D., Xu, X.: Trees from functions as processes. In: CONCUR (2014)
* [48] Scalas, A., Dardha, O., Hu, R., Yoshida, N.: A Linear Decomposition of Multiparty Sessions for Safe Distributed Programming. In: ECOOP'17 (2017)
* [49] Toninho, B., Caires, L., Pfenning, F.: Functions as session-typed processes. In: FOSSACS 2012. pp. 346–360 (2012)
* [50] Toninho, B., Caires, L., Pfenning, F.: Higher-order processes, functions, and sessions: A monadic integration. In: ESOP. pp. 350–369 (2013)
* [51] Toninho, B., Caires, L., Pfenning, F.: Corecursion and non-divergence in session-typed processes. In: TGC 2014. pp. 159–175 (2014)
* [52] Toninho, B., Yoshida, N.: On polymorphic sessions and functions: A tale of two (fully abstract) encodings (long version). CoRR abs/1711.00878 (2017)
* [53] Turner, D.: The polymorphic pi-calculus: Theory and implementation. Tech. Rep. ECS-LFCS-96-345, School of Informatics, University of Edinburgh (1996)
* [54] Wadler, P.: Propositions as sessions. J. Funct. Program. 24(2-3), 384–418 (2014)
* [55] Zhao, J., Zhang, Q., Zdancewic, S.: Relational parametricity for a polymorphic linear lambda calculus. In: APLAS. pp. 344–359 (2010)
Appendix
On Polymorphic Sessions and Functions
A Tale of Two (Fully Abstract) Encodings
Additional definitions and proofs of the main materials.
## Appendix 0.A Appendix
### 0.A.1 Additional Definitions for § 2 – Structural Congruence
###### Definition 0.A.1 (Structural congruence)
is the least congruence relation generated by the following laws: $P\mid{\bf
0}\equiv P$; $P\equiv_{\alpha}Q\Rightarrow P\equiv Q$; $P\mid Q\equiv Q\mid
P$; $P\mid(Q\mid R)\equiv(P\mid Q)\mid R$;
$(\mathbf{\nu}x)(\mathbf{\nu}y)P\equiv(\mathbf{\nu}y)(\mathbf{\nu}x)P$;
$x\not\in\mbox{\it fn}(P)\Rightarrow
P\mid(\mathbf{\nu}x)Q\equiv(\mathbf{\nu}x)(P\mid Q)$; $(\mathbf{\nu}x){\bf
0}\equiv{\bf 0}$; and $[x\leftrightarrow y]\equiv[y\leftrightarrow x]$.
###### Definition 0.A.2 (Extended Structural Congruence)
We write $\equiv_{!}$ for the least congruence relation on processes which
results from extending structural congruence $\equiv$ (Def. 0.A.1) with the
following axioms, dubbed the Sharpened Replication Axioms [46]:
1. 1.
$(\mathbf{\nu}u)(!u(z).P\mid(\nu y)(Q\mid R))\equiv_{!}(\nu y)((\nu
u)(!u(z).P\mid Q)\mid(\nu u)(!u(z).P\mid R))$
2. 2.
$(\nu u)(!u(y).P\mid(\nu v)(!v(z).Q\mid R))\equiv_{!}(\nu v)((!v(z).(\nu
u)(!u(y).P\mid Q))\mid(\nu u)(!u(y).P\mid R))$
3. 3.
$(\nu u)(!u(y).Q\mid P)\equiv_{!}P$ if $u\not\in\mbox{\it fn}(P)$
Axioms (1) and (2) represent principles for the distribution of shared servers
among processes, while (3) formalises the garbage collection of shared servers
which cannot be invoked by any process. The axioms embody distributivity,
contraction and weakening of shared resources and are sound wrt (typed)
observational equivalence [36].
### 0.A.2 Additional Definitions for § 2 – Typing Rules
Below we list the typing rules for the calculus of section § 2. We note that
the judgment $\Omega\vdash B\,\mathsf{type}$ simply requires that free
variables in $B$ be in $\Omega$. Moreover, typing treats processes quotiented
by structural congruence – given a well-typed process
$\Omega;\Gamma;\Delta\vdash P::T$, subject reduction ensures that for all
possible reductions $P\xrightarrow{\tau}P^{\prime}$, there exists a process
$Q$ where $P^{\prime}\equiv Q$ such that $\Omega;\Gamma;\Delta\vdash Q::T$.
Related properties hold wrt general transitions
$P\xrightarrow{\alpha}P^{\prime}$. We refer the reader to [9, 8] for
additional details on this matter.
$\begin{array}[]{c}\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$(\mathsf{id})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
0.83331pt\vbox{\hbox{\hskip-0.83331pt\hbox{\hbox{$\displaystyle\displaystyle{\,}$}}}\vbox{}}}\over\hbox{\hskip
59.7636pt\vbox{\vbox{}\hbox{\hskip-59.7636pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;x{:}A\vdash[x\leftrightarrow
z]::z{:}A}$}}}}}}$}}\hbox{}$}\quad\raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\mathbf{1}}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
0.83331pt\vbox{\hbox{\hskip-0.83331pt\hbox{\hbox{$\displaystyle\displaystyle{\,}$}}}\vbox{}}}\over\hbox{\hskip
32.05894pt\vbox{\vbox{}\hbox{\hskip-32.05893pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\cdot\vdash{\bf
0}::z{:}\mathbf{1}}$}}}}}}$}}\hbox{}$}\quad\raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\mathbf{1}}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
41.2835pt\vbox{\hbox{\hskip-41.2835pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
50.25224pt\vbox{\vbox{}\hbox{\hskip-50.25223pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}\mathbf{1}\vdash
P::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[10.00002pt] \raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\multimap}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
54.3921pt\vbox{\hbox{\hskip-54.3921pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\vdash
P::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
64.99284pt\vbox{\vbox{}\hbox{\hskip-64.99283pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z(x).P::z{:}A\multimap B}$}}}}}}$}}\hbox{}$}\quad\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\multimap}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
100.8318pt\vbox{\hbox{\hskip-100.8318pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1}\vdash
P::y{:}A\quad\Omega;\Gamma;\Delta_{2},x{:}B\vdash
Q::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
107.60963pt\vbox{\vbox{}\hbox{\hskip-107.60962pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1},\Delta_{2},x{:}A\multimap
B\vdash(\mathbf{\nu}y)x\langle y\rangle.(P\mid
Q)::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[10.00002pt] \raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\otimes}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
87.65411pt\vbox{\hbox{\hskip-87.65411pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1}\vdash
P::y{:}A\quad\Omega;\Gamma;\Delta_{2}\vdash
Q::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
90.73523pt\vbox{\vbox{}\hbox{\hskip-90.73521pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta_{1},\Delta_{2}\vdash(\mathbf{\nu}x)z\langle
y\rangle.(P\mid Q)::z{:}A\otimes B}$}}}}}}$}}\hbox{}$}\quad\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\otimes}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
67.34296pt\vbox{\hbox{\hskip-67.34296pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,y{:}A,x{:}B\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
76.03397pt\vbox{\vbox{}\hbox{\hskip-76.03395pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\otimes
B\vdash x(y).P::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[10.00002pt] \raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\with}\mathsf{R}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
87.54617pt\vbox{\hbox{\hskip-87.54617pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
P::z{:}A\quad\Omega;\Gamma;\Delta\vdash
Q::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
72.03941pt\vbox{\vbox{}\hbox{\hskip-72.03941pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z.\mathsf{case}(P,Q)::z{:}A\with B}$}}}}}}$}}\hbox{}$}\quad\raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\with}\mathsf{L}}_{1})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
54.27995pt\vbox{\hbox{\hskip-54.27994pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
74.88237pt\vbox{\vbox{}\hbox{\hskip-74.88237pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\with
B\vdash x.\mathsf{inl};P::z{:}C}$}}}}}}$}}\hbox{}$}\\\\[10.00002pt] \raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\with}\mathsf{L}}_{2})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
54.27995pt\vbox{\hbox{\hskip-54.27994pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\vdash
P::z{:}C}$}}}\vbox{}}}\over\hbox{\hskip
75.45181pt\vbox{\vbox{}\hbox{\hskip-75.45181pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta,x{:}A\with
B\vdash x.\mathsf{inr};P::z{:}C}$}}}}}}$}}\hbox{}$}\quad\raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\oplus}\mathsf{R}}_{1})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
39.71341pt\vbox{\hbox{\hskip-39.7134pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
P::z{:}A}$}}}\vbox{}}}\over\hbox{\hskip
62.96855pt\vbox{\vbox{}\hbox{\hskip-62.96855pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z.\mathsf{inl};P::z{:}A\oplus B}$}}}}}}$}}\hbox{}$}\\\\[10.00002pt] \raise
10.44444pt\hbox{$\hbox{$\hbox{\small\sc$({{\oplus}\mathsf{R}}_{2})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
41.39566pt\vbox{\hbox{\hskip-41.39565pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
P::z{:}B}$}}}\vbox{}}}\over\hbox{\hskip
63.538pt\vbox{\vbox{}\hbox{\hskip-63.538pt\hbox{\hbox{$\displaystyle\displaystyle{\Omega;\Gamma;\Delta\vdash
z.\mathsf{inr};P::z{:}A\oplus B}$}}}}}}$}}\hbox{}$}\quad\raise
11.0pt\hbox{$\hbox{$\hbox{\small\sc$({{\oplus}\mathsf{L}})$}\,$}{\hbox{$\displaystyle\displaystyle{\hbox{\hskip
|
# Compression of volume-surface integral equation matrices via Tucker
decomposition for magnetic resonance applications
Ilias I. Giannakopoulos, , Georgy D. Guryev, José E. C. Serrallés, , Ioannis
P. Georgakis, Luca Daniel, , Jacob K. White, , Riccardo Lattanzi This work was
supported by NIH R01 EB024536 and by NSF 1453675. It was performed under the
rubric of the Center for Advanced Imaging Innovation and Research (CAI2R,
www.cai2r.net), a NIBIB Biomedical Technology Resource Center (NIH P41
EB017183).Ilias I. Giannakopoulos, Ioannis P. Georgakis and Riccardo Lattanzi
are with Center for Advanced Imaging Innovation and Research (CAI2R),
Department of Radiology, New York University Grossman School of Medicine, NY,
USA.Georgy D. Guryev, José E.C. Serrallés, Luca Daniel, and Jacob K. White are
with the Research Laboratory of Electronics, Department of Electrical
Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, MA, USA.Riccardo Lattanzi is also with the Bernard and Irene
Schwartz Center for Biomedical Imaging, Department of Radiology, New York
University Grossman School of Medicine, NY, USA and the Vilcek Institute of
Graduate Biomedical Sciences, New York University Grossman School of Medicine,
NY, USA.
###### Abstract
In this work, we propose a method for the compression of the coupling matrix
in volume-surface integral equation (VSIE) formulations. VSIE methods are used
for electromagnetic analysis in magnetic resonance imaging (MRI) applications,
for which the coupling matrix models the interactions between the coil and the
body. We showed that these effects can be represented as independent
interactions between remote elements in 3D tensor formats, and subsequently
decomposed with the Tucker model. Our method can work in tandem with the
adaptive cross approximation technique to provide fast solutions of VSIE
problems. We demonstrated that our compression approaches can enable the use
of VSIE matrices of prohibitive memory requirements, by allowing the effective
use of modern graphical processing units (GPUs) to accelerate the arising
matrix-vector products. This is critical to enable numerical MRI simulations
at clinical voxel resolutions in a feasible computation time. In this paper,
we demonstrate that the VSIE matrix-vector products needed to calculate the
electromagnetic field produced by an MRI coil inside a numerical body model
with $1$ mm3 voxel resolution, could be performed in $\sim 33$ seconds in a
GPU, after compressing the associated coupling matrix from $\sim 80$ TB to
$\sim 43$ MB.
###### Index Terms:
Cross approximation, Global Maxwell Tomography, graphical processing unit,
magnetic resonance imaging, matrix-vector product, Tucker decomposition,
volume-surface integral equation.
## I Introduction
Magnetic resonance (MR) imaging (MRI) provides high-resolution images of the
interior anatomical and physiological structure of the human body, with
exquisite soft-tissue contrast. The quality of MR images, as well as the
achievable spatial and temporal resolution, depend on the available signal-to-
noise ratio (SNR). SNR increases with the main magnetic field strength. This
fact motivated the recent development of $7$ Tesla (T) clinical MR scanners
and research-only scanners with field strengths as high as $11.7$ T [1]. At
ultra-high-field (UHF) MRI ($\geq 7$ T), the radio frequency (RF) wavelength
is short. This results in strong interactions between biological tissues and
the electromagnetic (EM) field generated by the RF coils [2, 3, 4, 5]. Such
interactions could compromise image quality and patient safety. To address
these issues, EM modeling is often used to predict and manipulate the EM field
distribution during RF coil design.
Integral equation (IE) methods are suitable options for EM analysis in MRI.
First, they do not suffer from grid dispersion errors [6, 7], in contrast with
the finite-difference-time-domain (FDTD) and finite-element-methods (FEM),
because the Green’s functions in the underlying IE formulation act as an exact
EM field propagator from a source to an observation point. Second, for the
case of single-frequency problems, IE algorithms can be extensively customized
with the use of numerical linear algebra techniques for fast and accurate
simulations, tailored to specific applications [8, 9, 10, 11].
For example, the MAgnetic-Resonance Integral Equation (MARIE) suite [11, 12]
was developed to numerically compute the EM field distribution generated by RF
coils in the human body during MRI. MARIE combines surface and volume integral
equations (SIE,VIE), employing a triangular tessellation for the RF coils’
conductors and a uniform voxelized grid discretization for the body models.
RWG basis functions [13] and polynomial basis functions [12, 14] are used to
compute the unknowns of the surface and volume IE, respectively. Matrix-vector
products are accelerated using the fast Fourier transform (FFT).
The VIE computational engine of MARIE has been recently employed for the
forward problem in Global Maxwell Tomography (GMT) [15], a technique that
iteratively solves an ill-conditioned inverse problem to extract electrical
properties from volumetric MR measurements. In the first experimental
demonstration of GMT with a uniform phantom, constant incident fields were
used for all iterations [15]. More recently, it was shown in simulation that
GMT could accurately reconstruct brain electrical properties at $7$ T using a
tailored RF coil array [16]. However, in order to confirm this with in-vivo
experiments, the currents on the coil conductors cannot be simply considered
constant as in the initial experiment with a uniform phantom. Instead, the
incident fields must be updated at each iteration of GMT to account for
changes in the sample electrical properties distribution. Therefore, GMT must
be implemented with a volume-surface IE (VSIE) framework, in which a coupling
matrix represents the coil-body interactions in the IE system of equations
[11]. Such approach requires a large amount of memory, which could prevent
using clinically-relevant voxel resolutions and fine coil meshes.
The aim of this work is to use Tucker decomposition [17] to perform a column-
wise compression of the VSIE coupling matrix, in order to limit the associated
memory demand and enable the computation of the relevant matrix-vector
products in GPUs. Our approach was motivated by previous work [10] on the
reduction of the memory footprint of FFT-based VIE Green’s function tensors
and the acceleration of matrix-vector products in VIE using GPU. Tucker
decomposition belongs to a larger family of tensor decompositions and have
been used successfully in the past for matrix compression inside IE-based
simulations for EM applications. Examples include EM simulations of realistic
body models simulations for UHF MRI [18, 10] and capacitance extraction [19,
20, 21]. Other tensor decompositions could be used [22, 23, 24], but for the
intrinsic 3D nature of the problem at hand, Tucker optimizes operations and
memory complexity. In cases where the coil is placed far from the scatterer,
the coupling matrix can be first compressed with a 2D cross approximation
method [25, 26, 27] and then further compressed by applying our proposed
technique to the resulting matrices. Towards this direction, we developed an
algorithm based on the adaptive cross approximation (ACA) [28, 29] to
efficiently perform our compression approach within the iterative loop of ACA.
The remainder of this paper is organized as follows. In Section II, we
summarize the relevant technical background and equations related to the
standard VSIE method. We also show, as an example application, the use of the
VSIE coupling matrix in the GMT inverse problem formulation. In addition, we
outline the Tucker decomposition and the ACA method. In Section III, we
introduce the Tucker-based assembly technique of the coupling matrix along
with the novel algorithms for the matrix-vector products implementation.
Moreover, we present a new algorithm for a memory friendly assembly of the
coupling matrix, based on the combination of ACA with the Tucker
decomposition. Section IV describes a series of numerical experiments aimed at
investigating the multilinear rank and the memory requirements of the
compressed matrix in different scenarios, along with time footprint of the
matrix-vector product for various mesh discretizations. Section V discusses
the results. Section VI summarizes the work and provides a few take home
points. The following TABLE I lists the notations used in this work.
TABLE I: Notation Notation | Description
---|---
$a$ | Scalar
$\bm{a}$ | Vector in $\mathbb{C}^{3}$
$\mathbf{a}$ | Vector in $\mathbb{C}^{n}$
$A$ | Matrix in $\mathbb{C}^{n_{1}\times n_{2}}$
$A^{T}$ | Transpose of matrix
$A^{*}$ | Conjugate transpose of matrix
${\mathcal{A}}$ | Tensor in $\mathbb{C}^{n_{1}\times n_{2}\times n_{3}}$
$\mathscr{A}$ | Struct containing Tucker compressed tensors
${\cal{A}}$ | Operator acting on vectors in $\mathbb{C}^{3}$
$\times_{i}$ | n-mode products
${\mathrm{i}}$ | Imaginary unit ${\mathrm{i}}^{2}=-1$
## II Technical Background
### II-A Volume-Surface Integral Equation
#### II-A1 Coupled Linear System
Let us consider an IE-based solver for the EM wave analysis in MRI
applications. The body domain $\Omega$ can be modeled with the current-based
VIE (JVIE), while the conducting surfaces (receive and transmit RF coil
arrays, coil shields, and gradient coils) with the SIE. The IEs are solved
with the Method of Moments (MoM) [30]. The resulting system of equations can
be written in the following block matrix form.
$\begin{bmatrix}Z_{\rm cc}&T^{T}_{\rm bc}\\\ T_{\rm bc}&Z_{\rm
bb}\end{bmatrix}\begin{bmatrix}J_{\rm s}\\\ J_{\rm
p}\end{bmatrix}=\begin{bmatrix}V\\\ 0\end{bmatrix}.$ (1)
Here, $Z_{\rm cc}\in\mathbb{C}^{m\times m}$ is the Galerkin matrix that models
interactions between the coil’s discretization elements (triangles’ common
edges), with the aid of the free-space Green’s function that appears in the
SIE formulation. It associates the equivalent surface currents on the coil
($J_{\rm s}\in\mathbb{C}^{m\times p}$), with the voltage excitation matrix
$V\in\mathbb{C}^{m\times p}$. The coil conductors are modeled with triangular
meshes, and the unknown surface equivalent currents are approximated with RWG
basis functions [13]. $m$ is the number of RWG, or non-boundary, triangle
edges that appear on the mesh, whereas $p$ is the number of excitation ports
(i.e., number of coil’s channels) of the conducting surfaces.
The matrix $Z_{\rm bb}\in\mathbb{C}^{qn_{\rm v}\times qn_{\rm v}}$ is another
Galerkin matrix which models the EM interactions of an external volumetric EM
field, produced by the coil, with the body. Specifically, the matrix relates
the polarization currents in $\Omega$ to an external incident EM field
produced by the conducting surfaces. $n_{\rm v}$ is the number of voxels and
$q$ the number of basis functions per voxel. Differently than $Z_{\rm cc}$,
$Z_{\rm bb}$ requires a large amount of memory, even for coarse resolutions.
To handle that, $\Omega$ can be discretized over a voxelized uniform grid,
giving $Z_{\rm bb}$ a three-level Block-Toeplitz with Toeplitz Blocks (BTTB)
structure. As a result, only the defining columns of the BTTB matrix need to
be stored and the matrix-vector product can be accelerated using the FFT, as
in [31, 32, 33, 34, 35, 36, 37, 12]. The unknown polarization currents
($J_{\rm p}\in\mathbb{C}^{qn_{\rm v}\times p}$) can be discretized with
polynomial basis functions, either piecewise constant [12] (PWC, $3$ unknowns
per voxel) or piecewise linear [14] (PWL, $12$ unknowns per voxel), and a
single-voxel support.
The presence of conductive tissue near the coil conductors perturbs $J_{\rm
s}$ from their values in free-space. In fact, the voltage excitations at the
coil’s ports create incident EM fields that scatter from the dielectric body
back to the coil conductors, changing their current distribution. The coupling
matrix $T_{\rm bc}\in\mathbb{C}^{qn_{\rm v}\times m}$ is used to account for
this effect, by modeling the coupling interactions between the dyadic Green’s
function [38] of the SIE and the VIE formulations. Specifically, in equation
(1), $T_{\rm bc}$ maps electric surface equivalent currents to electric fields
through the ${\cal{N}}$ Green’s function operator of VIE:
${\cal{N}}\left(\bm{s}\right)\triangleq\nabla\times\nabla\times\int\limits_{\Omega}g\left(\bm{r}-\bm{r}^{\prime}\right)\bm{s}\left(\bm{r}^{\prime}\right)d^{3}\bm{r}^{\prime}.$
(2)
$g$ is the free-space Green’s function, or fundamental Helmholtz solution, and
it is equal to
$g\left(\bm{r}-\bm{r}^{\prime}\right)=\frac{e^{-{\mathrm{i}}k_{0}\lvert\bm{r}-\bm{r}^{\prime}\rvert}}{4\pi\lvert\bm{r}-\bm{r}^{\prime}\rvert},$
(3)
where $k_{0}$ is the free-space wavenumber, $\bm{r}$ the source point, and
$\bm{r}^{\prime}$ the observation point. Each element of the VSIE coupling
matrix is a 5D integral formed from the inner product between the discretized
${\cal{N}}$ operator applied on a VIE basis function, and an RWG basis
function.
#### II-A2 VSIE implementation of GMT
GMT estimates tissue electrical properties from MR measurements by solving an
inverse problem [15]. In GMT, the cost function compares actual measurements
against simulated measurements of the relative $b_{1}^{+}$ fields generated by
multiple sources (e.g., multiport transmit coils) inside a sample and
iteratively updates the estimate of the sample’s electrical properties. GMT
was initially demonstrated using the JVIE formulation for the solutions of the
forward problem, therefore ignoring the effect of the dielectric sample on the
incident fields. However, these interactions must be taken into account for
accurate in-vivo experiments with close-fitting RF coils. In other words, the
GMT framework must be ported from a VIE to a VSIE formulation, in which the
incident fields are not constant but calculated at each GMT iteration as
$\displaystyle E^{\rm inc}\left(\mathbf{\epsilon_{r}}\right)$
$\displaystyle=T_{\rm bc}J_{\rm s}\left(\mathbf{\epsilon_{r}}\right),$ (4)
$\displaystyle H^{\rm inc}\left(\mathbf{\epsilon_{r}}\right)$
$\displaystyle=Z_{\rm bc}J_{\rm s}\left(\mathbf{\epsilon_{r}}\right).$
where $E^{\rm inc}$ and $H^{\rm inc}$ are the discretized incident electric
and magnetic fields respectively. $\epsilon_{r}$ is the complex permittivity.
$Z_{\rm bc}$ maps the equivalent surface electric currents to the magnetic
fields with the aid of the ${\cal{K}}$ operator:
${\cal{K}}\left(\bm{s}\right)\triangleq\nabla\times\int\limits_{\Omega}g\left(\bm{r}-\bm{r}^{\prime}\right)\bm{s}\left(\bm{r}^{\prime}\right)d^{3}\bm{r}^{\prime}.$
(5)
In addition, in the new implementation, the gradient of the GMT’s cost
function will require to solve a Hermitian adjoint system of equations that
includes multiplications with the conjugate transpose of $Z_{\rm bc}$.
Matrix-vector products involving the coupling matrix are typically performed
without storing the full matrix, due to its intractably large size. In the
case of iterative inverse problem solutions, such as in GMT, this approach
could considerably increase the computation time, because it requires the re-
assembly of the full matrix at each iteration. In the next sections, we
propose a compression algorithm that reduces the computational burden, by
enabling one to assembly the full coupling matrix only once and then perform
just the matrix-vector multiplications in each of GMT’s iterations.
### II-B Numerical Linear Algebra Methods
#### II-B1 Tucker Decomposition
A 3D tensor ${\mathcal{A}}\in\mathbb{C}^{n_{1}\times n_{2}\times n_{3}}$ can
be decomposed with the Tucker model [17] in the following form:
$\displaystyle{\mathcal{A}}$
$\displaystyle={\mathcal{G}}\times_{1}U^{1}\times_{2}U^{2}\times_{3}U^{3},\>\text{or}$
(6) $\displaystyle{\mathcal{A}}_{ijk}$
$\displaystyle=\sum_{\chi=1}^{r_{1}}\sum_{\psi=1}^{r_{2}}\sum_{\zeta=1}^{r_{3}}{\mathcal{G}}_{\chi\psi\zeta}U^{1}_{i\chi}U^{2}_{j\psi}U^{3}_{k\zeta}.$
Here $U^{\gamma}\in\mathbb{C}^{n_{\gamma}\times r_{\gamma}},\gamma=1,2,3$, are
unitary matrices, dubbed as Tucker factors, while
${\mathcal{G}}\in\mathbb{C}^{r_{1}\times r_{2}\times r_{3}}$ is the Tucker
core. The dimensions of the Tucker core indicate the multilinear (or Tucker)
ranks of ${\mathcal{A}}$. The symbols $\times_{\gamma}$ are called $n$-mode
products and perform a convolution over the $\times_{\gamma}$ axis, for
example, the $\times_{1}$ product performs the following operation:
${\mathcal{P}}={\mathcal{G}}\times_{1}U^{1}\Leftrightarrow{\mathcal{P}}_{i\psi\zeta}=\sum\limits_{\chi=1}^{r_{1}}{\mathcal{G}}_{\chi\psi\zeta}U^{1}_{i\chi}.$
(7)
Here, ${\mathcal{P}}\in\mathbb{C}^{n_{1}\times r_{2}\times r_{3}}$. The
expansion of ${\mathcal{A}}$ in equation (6) can be truncated to a desired
tolerance, and return an approximation of ${\mathcal{A}}$. A visual
representation of Tucker decomposition can be seen in Fig. 1.
Figure 1: Visual representation of Tucker decomposition.
To compute the above-mentioned Tucker components, one has to choose a suitable
compression algorithm. The higher order singular value decomposition (HOSVD)
is an orthogonal Tucker decomposition, widely used because it has a proven
upper error bound [39]. Moreover, the algorithm is based entirely on singular
value decomposition (SVD), which provides a robust and stable approximation of
the initial tensor. Note that SVD requires the assembly of the initial array,
which could be challenging for large tensors. In such cases, one could
implement the Tucker decomposition using a 3D cross approximation algorithm
[40].
#### II-B2 Cross Approximation
A matrix $A\in\mathbb{C}^{n_{1}\times n_{2}}$ can be approximated with the so-
called 2D cross approximation method [25, 26] as follows:
$A\approx UV^{*}.$ (8)
Here, $U\in\mathbb{C}^{n_{1}\times r_{c}}$ and $V\in\mathbb{C}^{n_{2}\times
r_{c}}$. $r_{c}$ represents the column rank of matrix $A$. Cross approximation
algorithms construct the decomposition of $A$ by using only some rows and
columns of it, differently than SVD, which depends on the availability of the
full matrix. Several algorithms have been developed over the previous decades
for the implementation of cross approximation, with two being the most used
ones: the ACA [28, 29], and the maximum volume-based cross algorithm [27, 41].
The latter requires the implementation of LU, QR and SVD for its efficient
implementation. Therefore, the memory demand of the algorithm increases
drastically for large tall matrices, such as the coupling matrices $T_{\rm
bc}$ and $Z_{\rm bc}$ in the case fine voxel resolutions. On the other hand,
the memory demand of ACA is only dictated by the size of matrices $U$ and $V$.
## III Tucker-Based Compression Algorithm
In VSIE, the columns of the coupling matrix describe interactions between coil
and body basis functions through the Green’s functions of equations (2) and
(5); therefore, they represent well-separated geometrical blocks. Due to the
3D nature of the problem, the key idea of our proposed compression algorithm
is to reshape these columns as tensors and approximate them with the low
multilinear Tucker model. This compression strategy enabled us to develop a
new method to efficiently perform the matrix-vector product and an extension
to ACA, which are described later in this section.
### III-A Matrix Assembly
Each of the $m$ columns of the coupling matrices $Z_{\rm bc}$ and $T_{\rm bc}$
can be seen as the concatenation of $q$ vectors, where each vector represents
the component-wise interaction between one RWG element on the coil and the
basis functions of all the voxels in the body domain. For PWC, $q=3$, whereas
for PWL, $q=12$. Since these vectors model interactions between remote
discretization elements, they have low-rank properties [42]. To exploit the
low-rank, each column of the coupling matrix can be reshaped as $q$ 3D tensors
${\mathcal{Z}}^{kj}\in\mathbb{C}^{n_{1}\times n_{2}\times n_{3}}$, $k=1:q$,
$n_{\rm v}=n_{1}\times n_{2}\times n_{3}$, which are compressible with the
Tucker decomposition [43]. A graphical description of the algorithm is shown
in Fig. 2 for $Z_{\rm bc}$ and PWC basis functions,.
Figure 2: Visual representation of the Tucker-based algorithm for the
compression of the $Z_{\rm bc}$ matrix, in the case of PWC basis functions.
Each vector can be reshaped into a 3D tensor that is then compressed via
Tucker decomposition.
If the coupling matrix is first approximated with the ACA as $UV^{*}$, then
our approach can still be used to compress the $r_{c}$ columns of $U$. In
fact, cross approximation is a well-conditioned operation, therefore the
Tucker ranks of the reshaped columns of $U$ will be similar to the ones of the
reshaped columns of the coupling matrix. The $V$ matrix here is usually much
smaller than $U$ and does not require any additional compression.
TABLE II shows the total memory footprint associated with the assembly of the
coupling matrix: Full assembly, assembly with ACA, and assembly with our
proposed method by compressing either the columns of the coupling matrix
(Tucker) or the columns of $U$ (ACA+Tucker). The memory required after
compressing the coupling matrix with Tucker is $n_{\rm
v}/\left(r^{3}+3nr\right)$ times smaller than the memory required by the full
matrix, where $n$ and $r$ refer to the tensor’s linear dimension and Tucker
rank, respectively. If our Tucker-based compression method is instead applied
after the ACA assembly, then the total compression improves by a factor of
$\sim m/r_{c}$, given that $r_{c}$ is small. TABLE II also shows the
computational complexity of the assembly operations. The multiplicative
constant factor $c_{1}$, which is present in all cases, represents the cost to
compute the elements of the coupling matrix and is usually large. In fact,
each element requires a 5D integration, whose computational cost depends on
the number of both surface and volume quadrature integration points. As a
result, the assembly of the matrix is extremely inefficient and should be
implemented in parallel for multiple voxel-RWG basis function interactions.
Note that in certain cases, for example when the coil is close to the body,
ACA may not achieve a meaningful compression and would not be advantageous to
combine it with Tucker decomposition. In such cases, the preferable approach
would be to divide the coupling matrix in $q$ blocks of size $n_{\rm v}\times
m$, assembled them in parallel, and then compress their tensor components with
a Tucker-based method like HOSVD. Alternatively, if the coupling matrix is
sufficiently small, one could assemble it in its full form and then apply
Tucker directly to compress it.
TABLE II: Complexity for Constructing the Coupling Matrix Assembly Method | Operations | Memory
---|---|---
Full | ${\mathcal{O}}\left(c_{1}qn_{\rm v}m\right)$ | $qn_{\rm v}m$
ACA | ${\mathcal{O}}\left(c_{1}r_{c}^{2}\left(qn_{\rm v}+m\right)\right)$ | $qn_{\rm v}r_{c}+mr_{c}$
Tucker | Full + ${\mathcal{O}}\left(rqn_{\rm v}m\right)$ | $q\left(r^{3}+3nr\right)m$
ACA+Tucker | ACA + ${\mathcal{O}}\left(rqn_{\rm v}r_{c}\right)$ | $q\left(r^{3}+3nr\right)r_{c}+mr_{c}$
### III-B Matrix-Vector Product
Decompressing the coupling matrix to compute the matrix-vector product
$\mathbf{y}=Z_{\rm bc}\mathbf{x}$, like in equations (4), may not be possible
due to computer or GPU memory limitations. To address this, we propose a novel
approach to efficiently compute the matrix-vector product without fully
decompressing the coupling matrix. We initiate $\mathbf{y}$ as a vector of
zeros. Inside a loop that cycles over the RWG basis functions, we decompress
the $q$ tensors of a column $j\in[1,m]$, reshape them as vectors, and
concatenate them to form the $j$-th column of the original $Z_{\rm bc}$
matrix. The vector-scalar product between the $j$-th column and
$\mathbf{x}_{j}$ is then computed, and the result increments the elements of
$\mathbf{y}$. The same algorithm can be followed for the matrix-matrix product
$Y=Z_{\rm bc}X$.
The conjugate transpose matrix-vector product $\mathbf{y}=Z^{*}_{\rm
bc}\mathbf{x}$ is required for the computation of the gradient of the cost
function in the VSIE-based GMT implementation. This case is slightly different
than the standard matrix-vector product: inside the loop cycling through the
RWG functions, a row-column vector product must be computed between the
conjugate transpose of the decompressed $j$-th column of $Z_{\rm bc}$ and
$\mathbf{x}$, which yields the scalar $\mathbf{y}_{j}$. The algorithm remains
instead the same for the conjugate transpose matrix-matrix products. Both
algorithms (for $p$-columned matrices $X$ and $Y$) are summarized in the
pseudocode below:
Algorithm 1: $\>Y=Z_{\rm bc}X$
1:for k=1:$q$ do
2: $Y^{k}=0$
3:for j=1:$m$ do
4: for k=1:$q$ do
5: $\text{Decompress}\>{\mathcal{Z}}^{kj}$
6: $Y^{k}+={\mathcal{Z}}^{kj}(:)\mathbf{X_{j:}}$
7:$Y=\begin{bmatrix}Y^{1}\\\ \cdots\\\ Y^{q}\end{bmatrix}$
Algorithm 2: $\>Y=Z^{*}_{\rm bc}X$
1:$Y=0$
2:for j=1:$m$ do
3: for k=1:$q$ do
4: $\text{Decompress}\>{\mathcal{Z}}^{kj}$
5: $Y_{j:}=\begin{bmatrix}{\mathcal{Z}}^{1j}(:)\\\ \cdots\\\
{\mathcal{Z}}^{qj}(:)\end{bmatrix}^{*}X$
In Algorithm 1, $X\in\mathbb{C}^{m\times p}$, $Y\in\mathbb{C}^{qn_{\rm
v}\times p}$ (vice-versa for Algorithm 2), and ${\mathcal{Z}}^{kj}(:)$ is the
reshaped column vector of the tensor component ${\mathcal{Z}}^{kj}$. The
algorithms remain the same if $Z_{\rm bc}$ is compressed with ACA first
($Z_{\rm bc}=UV^{*}$). One has to replace $Z_{\rm bc}$ with $U$, $m$ with
$r_{c}$, and assign $X=V^{*}X$ for Algorithm 1, and $Y=VY$ for Algorithm 2.
Both the standard and the conjugate transpose matrix-vector products have the
same complexity, shown in TABLE III, for the full, ACA, Tucker, and ACA+Tucker
compressed cases. The full matrix-vector product is faster than the Tucker-
compressed approach by a factor of $(r+p)/p$, which depends on the number of
columns of $X$ and $Y$ and the Tucker rank. ACA can be faster than the full
case for small values of $r_{c}$. Although the approach based on Tucker
decomposition is slower because it requires additional flops compared to the
other methods, it is more likely to fit in GPUs, thanks to its small memory
footprint.
TABLE III: Matrix-Vector Product Complexity Matrix Form | Operations Complexity
---|---
Full | ${\mathcal{O}}\left(qn_{\rm v}mp\right)$
ACA | ${\mathcal{O}}\left(qn_{\rm v}r_{c}p\right)$ \+ ${\mathcal{O}}\left(r_{c}mp\right)$
Tucker | ${\mathcal{O}}\left(rqn_{\rm v}m\right)$ \+ ${\mathcal{O}}\left(qn_{\rm v}mp\right)$
ACA+Tucker | ${\mathcal{O}}\left(rqn_{\rm v}r_{c}\right)$ \+ ${\mathcal{O}}\left(qn_{\rm v}r_{c}p\right)$ \+ ${\mathcal{O}}\left(r_{c}mp\right)$
### III-C Tucker-based ACA
If the coupling matrix is first compressed with ACA, the previous methods for
matrix assembly and matrix-vector product could still be applied to the matrix
$U$ of the cross approximation. However, for the case of realistic body models
discretized with fine voxel resolutions, the traditional implementation of ACA
(a detailed description can be found in [44]) might fail due to random access
memory (RAM) overflow because of the size of $U$ (see section IV.B.2). To
address this, we propose an extension of ACA in which the matrix $U$ is
assembled directly in a compressed form, based on our proposed Tucker
decomposition technique. The algorithm is summarized in pseudocode bellow:
Algorithm 3: ACA of $Z\in\mathbb{C}^{m_{1}\times m_{2}}$. Assembly with
compressed $U$
1:$\epsilon=\text{tolerance}$, $i=1$, $\mathbf{s}_{1}=0$
2:$\mathscr{U}=[]$, $V=[]$
3:for $k=1:\text{min}(m_{1},m_{2})$ do
4: $\mathbf{r}\leftarrow Z_{i:}$
5: if $k>1$ then
6: $[f_{1},f_{2},f_{3},p]\leftarrow i$
7: for $l=1:size(\mathscr{U},2)$ do
8: $[{\mathcal{G}},U^{1},U^{2},U^{3}]\leftarrow U_{pl}$
9:
$\mathbf{t}_{l}={\mathcal{G}}\times_{1}U_{f_{1},:}^{1}\times_{2}U_{f_{2},:}^{2}\times_{3}U_{f_{3},:}^{3}$
10: $\mathbf{r}\mathrel{+}=-\mathbf{t}V^{*}$
11: $j\leftarrow\text{index of max element of}\>\mathbf{r}$
12: $\mathbf{y}\leftarrow\left(\mathbf{r}/\mathbf{r}_{j}\right)^{*}$
13: $\mathbf{x}\leftarrow Z_{:j}$
14: if $k>1$ then
15: $\mathbf{x}\mathrel{+}=-\textbf{Alg1}(\mathscr{U},V_{j:}^{*})$
16:
$\mathbf{s}_{k+1}\leftarrow\mathbf{s}_{k}+(\left\lVert\mathbf{x}\right\rVert\left\lVert\mathbf{y}\right\rVert)^{2}$
17: if $k>1$ then
18:
$\mathbf{s}_{k+1}\mathrel{+}=2\sum\left[\mathrm{Re}\,\\{\left(\textbf{Alg2}(\mathscr{U},\mathbf{x})\right)\odot\left(V^{*}\mathbf{y}\right)\\}\right]$
19: Reshape $\mathbf{x}$ to $q$ ${\mathcal{X}}^{q}$ tensors.
20:
$\mathscr{U}=[\mathscr{U}\>\text{HOSVD}({\mathcal{X}}^{1},3\epsilon)\>\cdots\>\text{HOSVD}({\mathcal{X}}^{q},3\epsilon)]$
21: $V=[V\>y]$
22: if
$\left\lVert\mathbf{x}\right\rVert\left\lVert\mathbf{y}\right\rVert\leq\epsilon\sqrt{\mathbf{s}_{k+1}}$
then break
23: $\mathbf{x}=\lvert\mathbf{x}\rvert$, $\mathbf{x}_{i}=0$
24: $i\leftarrow\text{index of max element of}\>\mathbf{x}$
Here $\mathscr{U}$ is a struct of size $q\times r_{c}$ ($q=3$ for PWC, $q=12$
for PWL, $r_{c}$ is the rank of $Z$) than contains tensors. Each time a new
column of $U$ is computed, it is reshaped to $q$ tensors, which are then
compressed with a truncated HOSVD of tolerance $3\epsilon$ (line 19,20). The
HOSVD tolerance has to be higher than the ACA tolerance since the irrelevant
numerical digits ($<1e-3$) appearing in $U$ are incompressible. We found that
a $3$ times higher tolerance is a good choice for our numerical examples. To
perform matrix- and conjugate transpose matrix-vector products with the
compressed $U$ we followed Algorithms 1 (line 15) and 2 (line 18). Finally,
when a row of $U$ is requested in Algorithm 3, we first calculate the voxel
and basis function component corresponding to that row (line 6) and then
decompress, using equation (6), only the required elements from the Tucker
compressed components of $U$ (line 8,9). The proposed algorithm avoids RAM
overflowing, but it is slower than the traditional ACA due to the multiple
tensor decompressions. Nevertheless, it could always be accelerated via GPU,
since its memory demand is as low as for one column of the coupling matrix.
## IV Numerical Experiments
### IV-A Tucker Rank Behavior
In this section, we study the low-Tucker rank properties of the $Z_{\rm bc}$
coupling matrix. We considered multiple geometrical scenarios and altered the
distance between the conductive surface (coil) and the VIE domain, the
operating frequency and the conductive surface’s discretization. The tensor
components of the columns of the coupling matrix were compressed with the
HOSVD algorithm and a tolerance of $1e-8$, which yielded a relative error
similar to the tolerance for all cases, due to the robustness of the SVD
itself. Such error can be considered negligible, because the tolerance of the
iterative solver used in FFT-based VIE systems is usually orders of magnitude
higher.
#### IV-A1 Tucker Rank vs. Distance
It is well established that the Green’s function integro-differential
operators between well-separated geometries present low-rank properties [42].
Here we studied the relation between the low multilinear rank of the
compressed coupling matrix $Z_{\rm bc}$ and the distance between the body’s
domain and the coil. We set the frequency to $298.06$ MHz, the operating
frequency of $7$ Tesla MRI. We modeled a single perfectly electric conducting
(PEC) loop coil of radius $\rho=0.50$ m and discretized it with $125$
triangular elements. The coil was centered in $\left(0,d,0\right)$, where $d$
was varied as $0.55$, $0.6$, $\dots$, $1$ m. The domain was a cuboid with edge
length of $1$ m, centered at $(0,0,0)$ and discretized with voxels of $1$ cm
isotropic resolution and PWC basis functions (Fig. 3). As a result, the
tensor’s size was $101\times 101\times 101$ and the memory required by the
fully assembled $Z_{\rm bc}$ was $5848$ MBs.
Figure 3: Loop-cubic domain geometry. The loop coil was shifted on the
$\hat{y}$ direction, for $10$ discrete distances between $0.55$ to $1$ m from
the center of the cube.
Fig. 4 illustrates the reduction of the maximum rank (maximum of all Tucker
ranks for all components) of the coupling matrix (right axis), and the total
memory of the compressed matrix using the algorithm described in Section IV
(left axis). It is evident that the greater the distance between the domain
and the coil, the lower the multilinear ranks and the required memory. The
compression factor varied between $\sim 50$ and $190$, depending on the
distance.
Figure 4: Memory footprint (left) and maximum rank (rank) of the compressed
$Z_{\rm bc}$ matrix, for different distances between the loop and the cubic
domain.
#### IV-A2 Tucker Rank vs. Frequency
The work presented in [42] showed that the rank of the integral operators for
3D problems increases linearly with respect to the operating frequency. This
was confirmed in [10], for the BTTB defining tensors of the FFT-based VIE
systems (discretized integral operators). These tensors were columns of the
corresponding Galerkin MoM matrices and modeled the interactions between one
voxel’s basis function and the whole domain via the ${\cal{N}}$ or ${\cal{K}}$
operators. In the present study, the tensors are columns of the coupling
matrix and model the interactions between one RWG basis function and the whole
body domain via the same operators. Since in both cases the interactions
between separated geometry blocks are modeled, one can expect a similar
behavior for the Tucker ranks.
To confirm this, we performed a frequency sweep ($300$, $600$, $\dots$,
$2700$MHz) for the setup in Fig. 3. The coil was discretized with $125$
elements, whereas the voxel’s isotropic resolution was set to $\lambda/20$,
with $\lambda$ being the wavelength. We repeated the calculations for three
positions of the coil ($d=0.55$, $0.65$, and $0.8$ m). The memory footprint
(left) of the compressed matrix, along with the maximum rank (right), are
shown in the dual axis chart of Fig. 5. The memory footprint increased
linearly with frequency, whereas the maximum rank grew at different rates for
the three investigated cases. This is expected because the maximum rank
represents the worst-case scenario among all ranks, whereas the memory
footprint summarizes the overall effect of all ranks.
Figure 5: Memory footprint (left) and maximum rank (rank) of the compressed
$Z_{\rm bc}$ matrix, for different operating frequencies. Results are shown
for three different distances between the loop and the domain.
#### IV-A3 Tucker Rank vs. Surface Mesh
Let us consider a fixed mesh for the domain and refine only the surface mesh
of the coil. As the coil mesh is refined, the edges of the triangular elements
become smaller and the number of columns of the coupling matrix increases,
with each column representing more remote element interactions between an edge
of the triangular mesh and the voxelized domain. As a result, we should expect
a drop in the Tucker ranks. To verify this, we used the same domain of the
previous sections, and a PEC equilateral triangle with centroid at
$(0,0.55\text{m},0)$ and one vertex at $(0,0.55\text{m},0.5\text{m})$. The
triangle was discretized with $10$ different meshes, starting from triangular
element’s edge of $0.5$m and reducing it by a factor of $\sqrt{2}$, which
resulted in $4$, $6$, $11$, $30$, $48$, $102$, $184$, $358$, $727$, and $1480$
elements. Fig. 6 reports the maximum rank as a function of the length of the
triangular element’s edge, confirming that the rank is smaller when the PEC
triangle’s mesh is finer.
Figure 6: Maximum rank of the compressed $Z_{\rm bc}$ matrix, for various PEC
triangle’s meshes. The rank drops as we refine the mesh.
### IV-B Application to VSIE-based MRI Simulations
Here we aim to validate the performance of the proposed algorithms for the
assembly of the coupling matrix $Z_{\rm bc}$, and the matrix-vector
implementation for two VSIE-based MRI applications. Both numerical experiments
were implemented in Matlab, except for the matrix assembly part which was
written in C++. For the GPU computations, we used an NVIDIA Quadro Volta GV100
32GB HBM2 PCIe. For the CPU computations, in the first experiment we used a
server with CentOS 6.9 operating system and an Intel(R) Xeon(R) CPU E5-2699 v3
at 2.30GHz, while for the second experiment we used a server with Ubuntu
18.04.5 LTS operating system and an Intel(R) Xeon(R) Gold 6248 CPU at 2.50GHz.
We parallelized on $12$ workers where needed.
#### IV-B1 Head Coil Experiments
We first demonstrated the proposed compression method for an 8-ports close-
fitting head coil, previously designed for GMT [16], which we loaded with the
“Billie” realistic head model from the virtual family population [45] (Fig.
7). The operating frequency was set to $298$ MHz.
Figure 7: Coil-head geometry. The RF coil (discretized with $2380$ triangular
element edges) was loaded with the voxelized realistic human head model
“Billie” (discretized with voxels of $1$ mm isotropic resolution).
The VSIE-based implementation of GMT requires performing operations on the
coupling matrix $Z_{\rm bc}$ and its conjugate transpose. We analyzed the
memory footprint reduction for the compressed coupling matrix and measured the
computation time for both the matrix- and conjugate transpose matrix-vector
products using the algorithms presented in section III. The coil was
discretized with both a coarse ($516$ RWG) and a fine ($2380$ RWG) mesh
resolution. For the VIE domain enclosing the head, we tested three different
voxel resolutions, namely $5$, $2$, and $1$ mm3, which resulted in $34\times
38\times 45$, $84\times 96\times 116$, and $168\times 188\times 222$ voxels,
respectively. Both PWC ($3$ unknowns per voxel) and PWL ($12$ unknown per
voxel) VIE basis functions were considered.
We used a tolerance of $1e-8$ for HOSVD, which would ensure accurate
estimation of electrical properties in an actual GMT experiment. Since the
coil closely fits the head, ACA (or SVD-based methods in general) are expected
to provide negligible compression with a tolerance of $1e-8$. We confirmed
this for the case of PWC basis functions, $5$ mm3 voxel resolution, and fine
coil discretization, for which, in fact, we found that $2238$ of the $2380$
singular values would be needed to accurately represent $Z_{\rm bc}$,
compressing the matrix from $6.18$ GB to $6.07$ GB. Consequently, for the head
coil experiments we did not use the Tucker-based ACA algorithm, but instead we
compressed the columns of the coupling matrix only with the HOSVD-based
method.
##### Memory Compression
The memory footprint for the assembly of the coupling matrix $Z_{\rm bc}$ is
shown in TABLE IV. The memory required to assemble the full matrix was
considerably larger than for the HOSVD-compressed matrix. For example, for PWC
basis functions, voxel resolution of $1$ mm3, and fine coil mesh, the required
memory in the full matrix case was $>740$ GBs, whereas the compressed matrix
required only $2.6$ GBs. Note that in the challenging case of PWL basis
functions, $1$ mm3 voxel resolution, and fine coil mesh, it was not feasible
to apply our compression method. In fact, the memory requirements even for
just one of the $q$ blocks of the matrix (see Section IV. A.) were
prohibitively large for our server. While we could have still implemented the
HOSVD compression by further dividing the matrix in smaller blocks, that would
have required $\sim 1$ month of computations. An alternative method for such
costly cases is mentioned in the discussion and will be pursued in future
work.
TABLE IV: Memory Requirements (GBs) of $Z_{\rm bc}$ Voxel Res. | Assembly | PWC-coarse | PWC-fine | PWL-coarse | PWL-fine
---|---|---|---|---|---
$5$ mm3 | Full | 1.34 | 6.18 | 5.36 | 24.74
HOSVD | 0.20 | 0.88 | 0.86 | 3.77
$2$ mm3 | Full | 21.57 | 99.52 | 86.30 | 398.09
HOSVD | 0.40 | 1.74 | 1.79 | 7.49
$1$ mm3 | Full | 161.73 | 745.99 | 646.95 | 2983.99
HOSVD | 0.63 | 2.62 | 2.85 | N/A
Fig. 8 shows that the compression factor, defined as the memory of the full
matrix over the memory of the compressed one, decreased as the voxel
resolutions ($h$) of the VIE domain’s grid became coarser. The behavior of the
compression factor was similar for PWC or PWL basis functions, either with
fine or coarse mesh discretization. This confirms the excellent stability of
our compression method.
Figure 8: Compression factor of the compressed matrix $Z_{\rm bc}$. Results
are shown for all investigated head and coil discretizations.
Fig. 9 shows the maximum Tucker rank, obtained with HOSVD, for all tensor
components of the coupling matrix. The rank decreased slowlier than the
compression factor (Fig. 8) for coarser discretizations of the VIE domain. For
example, when PWC basis functions and fine coil resolution were used (PWC-
fine), the maximum rank decreased by only $1.5$ times (from $42$ to $28$) when
the isotropic voxel size increased from $1$ to $5$ mm3, which corresponds to a
$5$ times smaller grid in all directions. For all cases, the maximum rank was
smaller for finer coil meshes, which is in agreement with the results shown in
section V.A.3.
Figure 9: Maximum rank as a function of voxel resolution of the VIE domain.
The maximum rank was calculated among all Tucker ranks of the decomposed
tensors, which were obtained with HOSVD. Results are shown for all
investigated head and coil discretizations. For each voxel resolution, the
size of the corresponding VIE discretization grid is indicated.
##### Computation Time
TABLE V reports the computation time for the assembly of the full and the
HOSVD-compressed coupling matrix (rounded to the nearest higher second). For
PWC, we used one quadrature integration point per triangle and voxel, while
for PWL, two for each triangle and eight for each voxel. For the low
resolution matrices, the assembly time for the compressed matrix was larger
than the one for the full matrix by a factor $\leq q$, since in the HOSVD case
the compressed matrix was assembled as $q$ sequential compressed matrix
blocks. For the larger matrices, our server could not perform the assembly of
the full matrix, due to the prohibitively large memory requirements, but it
was able to assemble the compressed matrix using our proposed method.
TABLE V: Time Footprint (hh:mm:ss) of $Z_{\rm bc}$ Assembly Voxel Res. | Assembly | PWC-coarse | PWC-fine | PWL-coarse | PWL-fine
---|---|---|---|---|---
$5$ mm3 | Full | 00:00:12 | 00:00:45 | 00:03:31 | 00:14:17
HOSVD | 00:00:34 | 00:02:33 | 00:32:27 | 02:21:21
$2$ mm3 | Full | 00:02:48 | N/A | 01:09:49 | N/A
HOSVD | 00:06:16 | 00:33:28 | 07:15:45 | 30:32:42
$1$ mm3 | Full | N/A | N/A | N/A | N/A
HOSVD | 00:27:59 | 02:31:52 | 52:17:11 | N/A
TABLE VI and VII summarize the computation times for the matrix- and conjugate
transpose matrix-vector products. Compared to the full form case (Full-CPU),
the compressed matrix-vector product requires additional operations for the
decompression of the tensors. While Algorithms 1 and 2 can reduce the memory
requirements of the matrix-vector products, the time footprint varies based on
how these algorithms are implemented. In particular, the nested loops over the
$m$ RWG functions can be either parallelized on a CPU, if the RAM can support
multiple tensor decompressions in parallel (HOSVD-CPU), or performed
sequentially using a GPU (HOSVD-GPU). In our tests, we multiplied $Z_{\rm bc}$
with $X\in\mathbb{C}^{m\times 8}$ (TABLE VI) and $Z^{*}_{\rm bc}$ with
$\Phi\in\mathbb{C}^{qn_{\rm v}\times 8}$ (TABLE VII), where both $X$ and
$\Phi$ were random matrices to keep the results general. The eight columns of
$X$ could correspond, for example, to the currents associated with the eight
channels of the coil in Fig. 7.
TABLE VI: Time Footprint (hh:mm:ss) of $Y=Z_{\rm bc}X$ Voxel Res. | Form | PWC-coarse | PWC-fine | PWL-coarse | PWL-fine
---|---|---|---|---|---
$5$ mm3 | Full-CPU | 00:00:01 | 00:00:02 | 00:00:02 | 00:00:06
HOSVD-CPU | 00:00:03 | 00:00:07 | 00:00:07 | 00:00:24
HOSVD-GPU | 00:00:03 | 00:00:11 | 00:00:10 | 00:00:44
$2$ mm3 | Full-CPU | 00:00:06 | N/A | 00:00:25 | N/A
HOSVD-CPU | 00:00:25 | 00:01:46 | 00:01:37 | 00:06:24
HOSVD-GPU | 00:00:04 | 00:00:14 | 00:00:13 | 00:00:56
$1$ mm3 | Full-CPU | N/A | N/A | N/A | N/A
HOSVD-CPU | 00:02:54 | 00:11:25 | 00:11:30 | N/A
HOSVD-GPU | 00:00:13 | 00:00:53 | 00:00:52 | N/A
TABLE VII: Time Footprint (hh:mm:ss) of $\Psi=Z^{*}_{\rm bc}\Phi$ Voxel Res. | Form | PWC-coarse | PWC-fine | PWL-coarse | PWL-fine
---|---|---|---|---|---
$5$ mm3 | Full-CPU | 00:00:01 | 00:00:02 | 00:00:02 | 00:00:06
HOSVD-CPU | 00:00:02 | 00:00:04 | 00:00:04 | 00:00:26
HOSVD-GPU | 00:00:02 | 00:00:07 | 00:00:05 | 00:00:21
$2$ mm3 | Full-CPU | 00:00:06 | N/A | 00:00:22 | N/A
HOSVD-CPU | 00:00:13 | 00:00:45 | 00:00:49 | 00:03:25
HOSVD-GPU | 00:00:03 | 00:00:11 | 00:00:10 | 00:00:42
$1$ mm3 | Full-CPU | N/A | N/A | N/A | N/A
HOSVD-CPU | 00:01:20 | 00:05:19 | 00:05:23 | N/A
HOSVD-GPU | 00:00:11 | 00:00:45 | 00:00:41 | N/A
For $5$ mm3 isotropic voxel resolution, the Full-CPU matrix-vector product was
the fastest for all cases, because the coupling matrix is small. For $2$ and
$1$ mm3 voxel resolution, the HOSVD-GPU implementation was the fastest. Note
that the Full-CPU case could not be performed for high voxel and coil mesh
resolutions, due to the excessive memory requirements. The HOSVD-CPU was
slower than HOSVD-GPU, except for the $5$ mm3 voxel resolution.
#### IV-B2 Body Coil Experiments
For the second MRI experiment, we simulated the volume bodycoil of a
commercial 3T MRI scanner [46, 47] and we loaded it with “Billie”, from the
virtual family population [45] (Fig. 10). The frequency was set to $123$ MHz,
corresponding to $3$ Tesla MRI. The coil has $32$ legs, a radius of $35.5$ cm,
length of $45$ cm, and is centered at $\left(0,0,0\right)$. We also modeled
the system conductive shield, which has a radius of $37.2$ cm, a length of
$1.5$ m and is centered at $\left(0,0,0\right)$. The distance between the coil
and the cuboid VIE domain enclosing “Billie” was $15.5$ cm and $24.5$ cm in
the $x$ and $y$, respectively. In contrast with the previous case where the
coil tightly fitted the head, here the coil is remote enough to allow a good
compression of the coupling matrix $Z_{\rm bc}$ with ACA. For this experiment,
we used PWC and PWL basis functions and three voxel resolutions ($5$, $2$, and
$1$ mm3), which corresponded to $81\times 44\times 108$, $205\times 109\times
270$, and $409\times 219\times 541$ voxels for the VIE domain. For the coil
and the shield we used $9450$ RWG basis functions. Two quadrature integration
points were used for each triangle and eight for each voxel, both for PWC and
PWL basis functions.
Figure 10: Coil-body geometry. The RF coil and shield (discretized with $9450$
triangular element edges) was loaded with part of the voxelized realistic
human body model “Billie” (discretized with voxels of $2$ mm isotropic
resolution).
##### Matrix Assembly
TABLE VIII, summarizes the memory requirements and the assembly time for the
coupling matrix $Z_{\rm bc}$. The ACA tolerance was set to $1e-3$ to achieve
good compression. The ACA rank of $Z_{\rm bc}$ was $250$ for the $5$ mm3 cases
and $287$ for the $2$ and $1$ mm3 cases. The maximum Tucker rank of $U$ was
between $15$ and $18$ for all cases. In the $5$ mm3 case, our results show
that ACA could offer an excellent compression of the coupling matrix and the
assembly could be rapidly performed in CPU. For $2$ mm3, ACA’s memory
requirements were large and ACA was outperformed in speed by our proposed
Algorithm 3 (Tucker-based ACA), for which the low memory footprint allowed
using a GPU. For $1$ mm3 resolution, the standard ACA algorithm could not be
performed even on a server equipped with hundreds of GB’s of RAM, due to
overwhelming memory requirements. On the other hand, our proposed ACA
extension in Algorithm 3 kept the memory demand small, enabling for fast
matrix assembly in GPU. Note that the full matrix assembly was only possible
for $5$ mm3 voxel resolution and PWC basis functions.
TABLE VIII: Memory Demand (GB) and Time Footprint (hh:mm:ss) of $Z_{\rm bc}$ Assembly Voxel Res. | Form | PWC | PWL
---|---|---|---
memory | time | memory | time
$5$ mm3 | Full-CPU | 162.60 | 00:15:49 | 650.42 | N/A
ACA-CPU | 4.33 | 00:01:53 | 17.24 | 00:06:39
Algorithm 3-GPU | 0.036 | 00:03:26 | 0.036 | 00:09:58
$2$ mm3 | Full-CPU | 2548 | N/A | 10195 | N/A
ACA-CPU | 77.44 | 00:37:16 | 309.65 | 04:30:36
Algorithm 3-GPU | 0.041 | 00:26:57 | 0.042 | 01:28:53
$1$ mm3 | Full-CPU | 20471 | N/A | 81884 | N/A
ACA-CPU | 621.75 | N/A | 2486 | N/A
Algorithm 3-GPU | 0.041 | 03:35:36 | 0.042 | 15:20:41
For the coarser case of $5$ mm3 voxel resolution and PWC basis functions, the
time footprint of Algorithm 3 for CPU (not shown in the table) was 00:17:50,
which is $\sim 5$ times slower than for the GPU execution.
##### Matrix-Vector Product Performance
The time footprints for the matrix-vector product between the compressed
coupling matrix $Z_{\rm bc}$ and a random vector
$\mathbf{x}\in\mathbb{C}^{m\times 1}$ are shown in TABLE IX. ACA-CPU
corresponds to performing the product $U(V^{*}\mathbf{x})$ in CPU. For
ACA+HOSVD-GPU, $Z_{\rm bc}$ was compressed with Algorithm 3, and the matrix-
vector product was performed with Algorithm 1 in GPU. For $5$ mm3 voxel
resolution, the efficiency is similar for both approaches. For $2$ mm3 voxel
resolution, ACA+HOSVD-GPU outperformed ACA by a factor of $3$, because, due to
its low memory demand, it could be executed on a GPU, whereas ACA could not.
TABLE IX: Time Footprint (hh:mm:ss) of $\mathbf{y}=Z_{\rm bc}\mathbf{x}$ Voxel Res. | Form | PWC | PWL
---|---|---|---
$5$ mm3 | Full-CPU | 00:00:11 | N/A
ACA-CPU | 00:00:01 | 00:00:01
ACA+HOSVD-GPU | 00:00:01 | 00:00:02
$2$ mm3 | Full-CPU | N/A | N/A
ACA-CPU | 00:00:05 | 00:00:18
ACA+HOSVD-GPU | 00:00:02 | 00:00:06
$1$ mm3 | Full-CPU | N/A | N/A
ACA-CPU | N/A | N/A
ACA+HOSVD-GPU | 00:00:09 | 00:00:33
The relative error of $\mathbf{y}$ obtained with ACA+HOSVD-GPU relative to
Full-CPU (ground truth) is shown on the right axis of Fig. 11 for the case of
$5$ mm3 voxel resolution and PWC basis functions. The plot shows how the error
changes as a function of the tolerance ($1e-3$, $\dots$, $1e-8$) used for ACA.
In particular, the relative error remained approximately an order of magnitude
higher than ACA’s tolerance. Fig. 11 also shows plots for the ACA rank and the
maximum Tucker rank of $Z_{\rm bc}$ (values on the left axis). Both ranks
increased as the tolerance of ACA was decreased. We expect similar results for
the other cases, but we were unable to assemble the full coupling matrix, due
to its vast memory footprint.
Figure 11: (left axis) ACA rank and maximum Tucker rank (obtained with HOSVD)
of $Z_{\rm bc}$. (right axis) Error in $\mathbf{y}=Z_{\rm bc}\mathbf{x}$
calculated with ACA+HOSVD-GPU relative to Full-CPU.
## V Discussion
We showed that our new Tucker-based algorithms could effectively compress the
coupling VSIE matrices in the case of fine voxel resolutions. Thanks to the
achieved compression factors, the matrix-vector products can be performed in
GPU’s, yielding faster execution. The proposed approach will be key for a
VSIE-based in vivo implementation of GMT, for which reducing memory
requirements and computation time are both critical factors.
For cases in which the coil is placed at some distance from the imaging
object, the coupling matrix is low-rank, thus can be compressed using ACA as
$UV^{*}$. For coarse voxel resolutions, such compression strategy alone is
effective and allows for a rapid CPU implementation of matrix-vector products.
However, as the voxel grid of the VIE domain is refined, the memory demand of
$U$ increases until the standard ACA becomes impractical for applications that
need high accuracy, such as GMT. In fact, in order to keep using ACA, one
would have to relax the tolerance, sacrificing accuracy. To avoid this, in
this work we introduced an extension of ACA (Algorithm 3), for which the
memory demand remains as a low as the memory required by one column of the
coupling matrix for any tolerance. Furthermore, Algorithm 3 can be executed in
GPU for rapid computations also in the case of fine voxel resolutions.
Other memory-friendly techniques are available for the fast implementation of
matrix-vector products in VSIE simulations: the magnetic resonance Green’s
function (MRGF), the fast multipole method (FMM), and the precorrected FFT
method (pFFT). The MRGF [11] is a model order reduction technique that can
considerably accelerate the solutions of the VSIE system. However, the
required computational time can be overwhelming when fine voxel resolutions
and PWL basis functions are used. The FMM has been extensively used for the
compression of the MoM matrix [48, 49, 50]. However, in our case, the FMM
could lead to unnecessary overheads, because we are only interested in the
compression of an off-diagonal block of the full MoM matrix (i.e., the
coupling matrix), since the remaining blocks can be handled efficiently with
other methods [13, 12, 10]. Finally, the pFFT method [8] could be used to
project the discretized coil’s elements onto an extended VIE domain, where the
Green’s function tensors are compressible with the Tucker decomposition
(pFFT+Tucker) [10]. However, this approach would be effective only when the
coil is close to the scatterer, like for the close-fitting coil in section
IV.B.1 [51]. In fact, in such situation the extended VIE domain would be
larger than the original one by only a few voxels in each direction, allowing
the matrix-vector products to fit in a GPU. As a result, and also because it
relies on the FFT, such pFFT+Tucker could be more efficient than our proposed
method, although more complex to implement.
An important aspect of the approach presented in this manuscript is that it
can effectively compress the coupling matrix both when the coil is close to or
far from the scatterer. Because of this, our method allows using GPUs to
accelerate the matrix-vector products for most applications. For example, if
the close-fitting coil geometry in Fig. 7 were integrated with a head gradient
insert with a surrounding conductive shield at a certain distance, with our
approach the coupling matrix would still be compressible and fit in the memory
of a GPU. In fact, the interactions between the shield and the head would have
lower Tucker ranks than the ones between the coil’s conductors and the head
(see Section IV.A.1). On the other hand, the pFFT+Tucker method would no
longer be efficient because it would require extending the VIE domain to fully
include the shield. Even though the Green’s function tensors of the extended
domain would still be compressible with Tucker decomposition, the unknowns
that multiply such tensors element-wise would not be. In fact, their
dimensions would have to significantly increase in order to perform the
relevant FFT-based matrix-vector products.
For the previous example, in order to fully exploit the highly parallel
architecture of the GPU with a minimal number of matrix-vector products
operations, one could use a hybrid method that combines pFFT and Algorithm 3.
To do that, the near and far interactions between the VIE domain and the
conducting surfaces would need to be separated. Then, the pFFT could be used
to model the near interactions between the VIE domain and the coil as in [51],
while the arising Green’s function operators could be compressed with the
Tucker decomposition as in [10]. Finally, the remaining far interactions
between the VIE domain and the coil could be modeled with a coupling matrix,
which would be vastly compressed with Algorithm 3. Such hybrid method could
enable us to rapidly execute the matrix-vector products in GPU for most cases,
including complex coil-shield geometries, fine voxel resolutions, and PWL
basis functions. The described hybrid method will be investigated in future
work.
The main limitation of our proposed method is that it requires a considerable
amount of time for the assembly of the coupling matrix, when the matrix is not
compressible with ACA. It is especially slow because each element of the
coupling matrix is a 5D integral. We showed that this could be addressed by
implementing matrix assembly and compression in parallel, but such approach is
not always possible due to memory limitations (IV.B.1). For such cases, one
could alternatively employ 3D cross-Tucker approximation algorithms [40, 18],
which are less efficient than HOSVD for the tensor dimensions in this work,
but do not suffer from memory constraints in case of large tensors. In fact,
3D cross-Tucker methods require only a small number of rows, columns, and
fibers of the tensor they approximate, and they can be implemented with linear
complexity (with respect to tensor’s linear size). In future work, we will
explore the execution of multiple 3D cross-Tucker steps in parallel to avoid
memory overflows when assembling a compressed coupling matrix in the case of
extremely fine resolutions.
## VI Conclusion
We presented a memory compression technique for the coupling matrix in VSIE
systems. Our method enables one to form and store the coupling matrix even
when its full form size is prohibitively large ($\sim 80$ TB). Specifically,
in this work we were able to achieve a compression between $\sim 0.5$ (PWC)
and $\sim 2$ (PWL) million times when simulating interactions between MRI
coils and realistic body models with some distance between them, in the case
of fine voxel resolutions of $1$ mm3. The error was around one order of
magnitude higher than the tolerance employed for the algorithm. The stored,
compressed matrices could be used multiple times without the need to repeat
the assembly. For example, this would allow one to rapidly perform EM
simulations for the same coil with different body geometries, as far as they
are contained in the original computational domain. For most cases, our
compression method enables fitting large coupling matrices in GPUs, resulting
in rapid execution of the VSIE matrix-vector product (from $1$ to $56$ seconds
for the studied scenarios). Finally, the proposed method could facilitate the
implementation of VSIE-based GMT for in vivo mapping of tissue electrical
properties at clinically-relevant voxel resolutions.
## References
* [1] S. W. Anderson _et al._ , “Effect of disease progression on liver apparent diffusion coefficient values in a murine model of NASH at 11.7 Tesla MRI,” _Journal of Magnetic Resonance Imaging_ , vol. 33, no. 4, pp. 882–888, 2011.
* [2] J. Jin and J. Chen, “On the SAR and field inhomogeneity of birdcage coils loaded with the human head,” _Magnetic resonance in medicine_ , vol. 38, no. 6, pp. 953–963, 1997.
* [3] R. Lattanzi _et al._ , “Electrodynamic constraints on homogeneity and radiofrequency power deposition in multiple coil excitations,” _Magnetic resonance in medicine_ , vol. 61, no. 2, pp. 315–334, 2009.
* [4] X. Zhang _et al._ , “From complex B1 mapping to local SAR estimation for human brain MR imaging using multi-channel transceiver coil at 7T,” _IEEE transactions on medical imaging_ , vol. 32, no. 6, pp. 1058–1067, 2013\.
* [5] M. Cosottini _et al._ , “Short-term side-effects of brain MR examination at 7 T: a single-centre experience,” _European radiology_ , vol. 24, no. 8, pp. 1923–1928, 2014.
* [6] A. Taflove and K. R. Umashankar, “Review of FD-TD numerical modeling of electromagnetic wave scattering and radar cross section,” _Proceedings of the IEEE_ , vol. 77, no. 5, pp. 682–699, 1989.
* [7] R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” _IEEE Transactions on Antennas and Propagation_ , vol. 40, no. 5, pp. 542–549, 1992.
* [8] J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , vol. 16, no. 10, pp. 1059–1072, 1997.
* [9] A. A. Tambova _et al._ , “On the generalization of directfn for singular integrals over quadrilateral patches,” _IEEE Transactions on Antennas and Propagation_ , vol. 66, no. 1, pp. 304–314, 2017.
* [10] I. I. Giannakopoulos, M. S. Litsarev, and A. G. Polimeridis, “Memory footprint reduction for the fft-based volume integral equation method via tensor decompositions,” _IEEE Transactions on Antennas and Propagation_ , vol. 67, no. 12, pp. 7476–7486, 2019.
* [11] J. F. Villena _et al._ , “Fast electromagnetic analysis of MRI transmit RF coils based on accelerated integral equation methods,” _IEEE Transactions on Biomedical Engineering_ , vol. 63, no. 11, pp. 2250–2261, 2016.
* [12] A. G. Polimeridis _et al._ , “Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects,” _Journal of Computational Physics_ , vol. 269, pp. 280–296, 2014.
* [13] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” _IEEE Transactions on antennas and propagation_ , vol. 30, no. 3, pp. 409–418, 1982.
* [14] I. P. Georgakis _et al._ , “A fast volume integral equation solver with linear basis functions for the accurate computation of electromagnetic fields in MRI,” _IEEE Transactions on Antennas and Propagation, Early Access_ , 2020.
* [15] J. E. Serrallés _et al._ , “Noninvasive Estimation of Electrical Properties from Magnetic Resonance Measurements via Global Maxwell Tomography and Match Regularization,” _IEEE Transactions on Biomedical Engineering_ , vol. 67, no. 1, pp. 3–15, 2019.
* [16] I. Giannakopoulos _et al._ , “Magnetic-resonance-based electrical property mapping using Global Maxwell Tomography with an 8-channel head coil at 7 Tesla: a simulation study,” _IEEE Transactions on Biomedical Engineering_ , vol. 68, no. 1, pp. 236–246, 2021.
* [17] L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” _Psychometrika_ , vol. 31, no. 3, pp. 279–311, 1966.
* [18] I. I. Giannakopoulos, M. S. Litsarev, and A. G. Polimeridis, “3D cross-Tucker approximation in FFT-based volume integral equation methods,” in _2018 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting_. IEEE, 2018, pp. 2507–2508.
* [19] J. Zhang, Y. Han, and J. Jiang, “Tucker decomposition-based tensor learning for human action recognition,” _Multimedia Systems_ , vol. 22, no. 3, pp. 343–353, 2016.
* [20] M. Wang _et al._ , “VoxCap: FFT-Accelerated and Tucker-Enhanced Capacitance Extraction Simulator for Voxelized Structures,” _arXiv preprint arXiv:2004.02609_ , 2020.
* [21] C. Qian and A. C. Yucel, “On the Compression of Translation Operator Tensors in FMM-FFT-Accelerated SIE Simulators via Tensor Decompositions,” _arXiv preprint arXiv:2010.00520_ , 2020.
* [22] I. V. Oseledets, “Tensor-train decomposition,” _SIAM Journal on Scientific Computing_ , vol. 33, no. 5, pp. 2295–2317, 2011.
* [23] B. N. Khoromskij and I. Oseledets, “Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs,” _Computational Methods in Applied Mathematics Comput. Methods Appl. Math._ , vol. 10, no. 4, pp. 376–394, 2010.
* [24] L. Grasedyck, “Hierarchical singular value decomposition of tensors,” _SIAM Journal on Matrix Analysis and Applications_ , vol. 31, no. 4, pp. 2029–2054, 2010.
* [25] E. Tyrtyshnikov, “Mosaic-skeleton approximations,” _Calcolo_ , vol. 33, no. 1-2, pp. 47–57, 1996.
* [26] E. Tyrtyshnikov, “Mosaic ranks and skeletons,” in _International Workshop on Numerical Analysis and Its Applications_. Springer, 1996, pp. 505–516.
* [27] S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, “A theory of pseudoskeleton approximations,” _Linear algebra and its applications_ , vol. 261, no. 1-3, pp. 1–21, 1997.
* [28] S. Kurz, O. Rain, and S. Rjasanow, “The adaptive cross-approximation technique for the 3D boundary-element method,” _IEEE transactions on Magnetics_ , vol. 38, no. 2, pp. 421–424, 2002.
* [29] M. Bebendorf and S. Rjasanow, “Adaptive low-rank approximation of collocation matrices,” _Computing_ , vol. 70, no. 1, pp. 1–24, 2003.
* [30] R. F. Harrington, _Field computation by moment methods_. Wiley-IEEE Press, 1993.
* [31] M. F. Catedra, E. Gago, and L. Nuno, “A numerical scheme to obtain the RCS of three-dimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform,” _IEEE transactions on antennas and propagation_ , vol. 37, no. 5, pp. 528–537, 1989.
* [32] P. Zwamborn and P. M. Van Den Berg, “The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 40, no. 9, pp. 1757–1766, 1992.
* [33] H. Gan and W. C. Chew, “A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems,” _Journal of Electromagnetic Waves and Applications_ , vol. 9, no. 10, pp. 1339–1357, 1995.
* [34] J. Jin _et al._ , “Computation of electromagnetic fields for high-frequency magnetic resonance imaging applications,” _Physics in Medicine & Biology_, vol. 41, no. 12, p. 2719, 1996.
* [35] M. Van Beurden and S. Van Eijndhoven, “Well-posedness of domain integral equations for a dielectric object in homogeneous background,” _Journal of Engineering Mathematics_ , vol. 62, no. 3, pp. 289–302, 2008.
* [36] J. Markkanen _et al._ , “Analysis of volume integral equation formulations for scattering by high-contrast penetrable objects,” _IEEE Transactions on Antennas and Propagation_ , vol. 60, no. 5, pp. 2367–2374, 2012.
* [37] P. Yla-Oijala _et al._ , “Surface and volume integral equation methods for time-harmonic solutions of Maxwell’s equations,” _Progress In Electromagnetics Research_ , vol. 149, pp. 15–44, 2014.
* [38] C.-T. Tai, _Dyadic Green functions in electromagnetic theory_. Institute of Electrical & Electronics Engineers (IEEE), 1994.
* [39] L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” _SIAM journal on Matrix Analysis and Applications_ , vol. 21, no. 4, pp. 1253–1278, 2000.
* [40] I. V. Oseledets, D. Savostianov, and E. E. Tyrtyshnikov, “Tucker dimensionality reduction of three-dimensional arrays in linear time,” _SIAM Journal on Matrix Analysis and Applications_ , vol. 30, no. 3, pp. 939–956, 2008.
* [41] S. A. Goreinov and E. E. Tyrtyshnikov, “The maximal-volume concept in approximation by low-rank matrices,” _Contemporary Mathematics_ , vol. 280, pp. 47–52, 2001.
* [42] W. Chai and D. Jiao, “Theoretical study on the rank of integral operators for broadband electromagnetic modeling from static to electrodynamic frequencies,” _IEEE Transactions on Components, Packaging and Manufacturing Technology_ , vol. 3, no. 12, pp. 2113–2126, 2013.
* [43] M. A. Francavilla _et al._ , “Maxwell parallel imaging,” _arXiv preprint arXiv:2008.09042_ , 2020.
* [44] K. Zhao, M. N. Vouvakis, and J.-F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems,” _IEEE transactions on electromagnetic compatibility_ , vol. 47, no. 4, pp. 763–773, 2005.
* [45] A. Christ _et al._ , “The Virtual Family-development of surface-based anatomical models of two adults and two children for dosimetric simulations,” _Physics in Medicine & Biology_, vol. 55, no. 2, p. N23, 2009\.
* [46] SIEMENS Healthineers, “Magnetom skyra.” [Online]. Available: https://www.siemens-healthineers.com/magnetic-resonance-imaging/3t-mri-scanner/magnetom-skyra
* [47] E. Milshteyn _et al._ , “Individualized sar calculations using computer vision-based mr segmentation and a fast electromagnetic solver,” _Magnetic Resonance in Medicine_ , vol. 85, no. 1, pp. 429–443, 2021.
* [48] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” _Journal of computational physics_ , vol. 73, no. 2, pp. 325–348, 1987.
* [49] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” _IEEE Antennas and Propagation Magazine_ , vol. 35, no. 3, pp. 7–12, 1993.
* [50] B. Shanker and H. Huang, “Accelerated Cartesian expansions–a fast method for computing of potentials of the form R- $\nu$ for all real $\nu$,” _Journal of Computational Physics_ , vol. 226, no. 1, pp. 732–753, 2007.
* [51] G. G. Guryev _et al._ , “Fast field analysis for complex coils and metal implants in MARIE 2.0.” in _Proc. ISMRM_ , 2019, p. 1035.
|
# Gröbner-Shirshov bases for some Lie algebras111Supported by the NNSF of
China (11171118), the Research Fund for the Doctoral Program of Higher
Education of China (20114407110007), the NSF of Guangdong Province
(S2011010003374) and the Program on International Cooperation and Innovation,
Department of Education, Guangdong Province (2012gjhz0007).
Yuqun Chen, Yu Li and Qingyan Tang
School of Mathematical Sciences, South China Normal University
Guangzhou 510631, P. R. China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract: We give Gröbner-Shirshov bases for Drinfeld-Kohno Lie algebra
$\textbf{L}_{n}$ in [27] and Kukin Lie algebra $A_{P}$ in [32], where $P$ is a
semigroup. As applications, we show that as $\mathbb{Z}$-module
$\textbf{L}_{n}$ is free and a $\mathbb{Z}$-basis of $\textbf{L}_{n}$ is
given. We give another proof of Kukin Theorem: if semigroup $P$ has the
undecidable word problem then the Lie algebra $A_{P}$ has the same property.
Key words: Gröbner-Shirshov basis, Lie algebra, Drinfeld-Kohno Lie algebra,
word problem, semigroup.
AMS 2000 Subject Classification: 17B01, 16S15, 3P10, 20M05, 03D15
## 1 Introduction
Gröbner bases and Gröbner-Shirshov bases were invented independently by A.I.
Shirshov for ideals of free (commutative, anti-commutative) non-associative
algebras [40, 41], free Lie algebras [39, 41] and implicitly free associative
algebras [39, 41] (see also [2, 4]), by H. Hironaka [29] for ideals of the
power series algebras (both formal and convergent), and by B. Buchberger [20]
for ideals of the polynomial algebras.
Gröbner bases and Gröbner-Shirshov bases theories have been proved to be very
useful in different branches of mathematics, including commutative algebra and
combinatorial algebra, see, for example, the books [1, 18, 21, 22, 24, 25],
the papers [2, 3, 4, 7, 9, 10, 11, 12, 13, 20, 23, 30, 35], and the surveys
[5, 6, 14, 15, 16, 17].
A.A. Markov [34], E. Post [37], A. Turing [42], P.S. Novikov [36] and W.W.
Boone [19] constructed finitely presented semigroups and groups with the
undecidable word problem. This result also follows from the Higman theorem
[28] that any recursive presented group is embeddable into finitely presented
group. A weak analogy of Higman theorem for Lie algebras was proved in [3]
that was enough for existence of a finitely presented Lie algebra with the
undecidable word problem.
In [32], Kukin constructed the Lie algebra $A_{P}$ for a semigroup $P$ such
that if $P$ has the undecidable word problem then $A_{P}$ has the same
property. In this paper we find a Gröbner-Shirshov basis for $A_{P}$ and then
give another proof of the above result.
The Drinfeld-Kohno Lie algebra $\textbf{L}_{n}$ appears in [26, 31] as the
holonomy Lie algebra of the complement of the union of the diagonals
$z_{i}=z_{j},\ i<j$. The universal Knizhnik-Zamolodchikov connection takes
values in this Lie algebra. In this paper, we give a Gröbner-Shirshov basis
for the Drinfeld-Kohno Lie algebra $\textbf{L}_{n}$ over $\mathbb{Z}$ and a
$\mathbb{Z}$-basis of $\textbf{L}_{n}$. As an application we get a simple
proof that $\textbf{L}_{n}$ is an iterated semidirect product of free Lie
algebras.
We are very grateful to Professor L.A. Bokut for his guidance and useful
discussions.
## 2 Composition-Diamond lemma for Lie algebras over a field
For the completeness of the paper, we formulate the Composition-Diamond lemma
for Lie algebras over a field in this section, see [6, 33, 38, 41] for
details.
Let $k$ be a filed, $I$ a well ordered index set, $X=\\{x_{i}|i\in I\\}$ a
set, $X^{*}$ the free monoid generated by $X$ and $Lie(X)$ the free Lie
algebra over $k$ generated by $X$.
We order $X=\\{x_{i}|i\in I\\}$ by $x_{i}>x_{t}$ if $i>t$ for any $i,t\in I$.
We use two linear orders on $X^{*}$: for any $u,v\in X^{*}$,
(i) (lex order) $1\succ t$ if $t\neq 1$ and, by induction, if $u=x_{i}u_{i}$
and $v=x_{j}v_{j}$ then $u\succ v$ if and only if $x_{i}>x_{j}$, or
$x_{i}=x_{j}$ and $u_{i}\succ v_{j}$;
(ii) (deg-lex order) $u>v$ if and only if $deg(u)>deg(v)$, or $deg(u)=deg(v)$
and $u\succ v$, where $deg(u)$ is the length of $u$.
We regard $Lie(X)$ as the Lie subalgebra of the free associative algebra
$k\langle X\rangle$, which is generated by $X$ under the Lie bracket
$[u,v]=uv-vu$. Given $f\in k\langle X\rangle$, denote $\bar{f}$ the leading
word of $f$ with respect to the deg-lex order. $f$ is monic if the coefficient
of $\bar{f}$ is 1.
###### Definition 2.1
An associative word $w\in X^{*}\setminus\\{1\\}$ is an associative Lyndon-
Shirshov word (ALSW for short) if
$(\forall u,v\in X^{*},u,v\neq 1)\ w=uv\Rightarrow w>vu.$
A non-associative word $(u)$ in $X$ is a non-associative Lyndon-Shirshov word
(NLSW for short), denoted by $[u]$, if
(i) $u$ is an ALSW;
(ii) if $[u]=[(u_{1})(u_{2})]$ then both $(u_{1})$ and $(u_{2})$ are NLSW’s;
(iii) if $[u]=[[[u_{11}][u_{12}]][u_{2}]]$ then $u_{12}\preceq u_{2}$.
We denote the set of all ALSW’s and NLSW’s in $X$ by $ALSW(X)$ and $NLSW(X)$
respectively.
For an ALSW $w$, there is a unique bracketing $[w]$ such that $[w]$ is NLSW:
$[w]=[[u][v]]$ if $deg(w)>1$, where $v$ is the longest proper associative
Lyndon-Shirshov end of $w$.
Shirshov Lemma Suppose that $w=aub$, where $w,u\in ALSW(X)$. Then
1. (i)
$[w]=[a[uc]d],$ where $b=cd$ and possibly $c=1$.
2. (ii)
Represent $c$ in the form $c=c_{1}c_{2}\ldots c_{n},$ where
$c_{1},\ldots,c_{n}\in ALSW(X)$ and $c_{1}\leq c_{2}\leq\ldots\leq c_{n}$.
Replacing $[uc]$ by $[\ldots[[u][c_{1}]]\ldots[c_{n}]]$ we obtain the word
$[w]_{u}=[a[\ldots[[[u][c_{1}]][c_{2}]]\ldots[c_{n}]]d]$
which is called the Shirshov special bracketing of $w$ relative to $u$.
3. (iii)
$\overline{[w]}_{u}=w.$
###### Definition 2.2
Let $S\subset Lie(X)$ with each $s\in S$ monic, $a,b\in{X^{*}}$ and $s\in S$.
If $a\bar{s}b$ is an ALSW, then we call
$[asb]_{\bar{s}}=[a\bar{s}b]_{\bar{s}}|_{[\bar{s}]\mapsto{s}}$ a special
normal $S$-word, where $[a\bar{s}b]_{\bar{s}}$ is defined as in Shirshov
Lemma. An $S$-word $(asb)$ is called a normal $S$-word if
$\overline{(asb)}=a\overline{s}b$.
Suppose that $f,\ g\in S$. Then, there are two kinds of compositions:
1. (i)
If $w=\bar{f}=a\bar{g}b$ for some $a,b\in X^{*}$, then the polynomial
$(f,g)_{w}=f-[agb]_{\bar{g}}$ is called the composition of inclusion of $f$
and $g$ with respect to $w$.
2. (ii)
If $w$ is a word such that $w=\bar{f}b=a\bar{g}$ for some $a,b\in X^{*}$ with
$deg(\bar{f})+deg(\bar{g})>deg(w)$, then the polynomial
$(f,g)_{w}=[fb]_{\bar{f}}-[ag]_{\bar{g}}$ is called the composition of
intersection of $f$ and $g$ with respect to $w$.
In (i) and (ii), $w$ is called an ambiguity.
Let $h$ be a Lie polynomial and $w\in X^{*}$. We shall say that $h$ is trivial
modulo $(S,w)$, denoted by $h\equiv_{Lie}0\ mod(S,w)$, if
$h=\sum_{i}\alpha_{i}(a_{i}s_{i}b_{i})$, where each $(a_{i}s_{i}b_{i})$ is a
normal $S$-word and $a_{i}\bar{s_{i}}b_{i}<w$.
The set $S$ is called a Gröbner-Shirshov basis in $Lie(X)$ if any composition
in $S$ is trivial modulo $S$ and corresponding $w$.
###### Theorem 2.3
(Composition-Diamond lemma for Lie algebras over a field) Let
$S\subset{Lie(X)}\subset{k\langle X\rangle}$ be nonempty set of monic Lie
polynomials. Let $Id(S)$ be the ideal of $Lie(X)$ generated by $S$. Then the
following statements are equivalent.
1. (i)
$S$ is a Gröbner-Shirshov basis in $Lie(X)$.
2. (ii)
$f\in{Id(S)}\Rightarrow{\bar{f}=a\bar{s}b}$ for some $s\in{S}$ and
$a,b\in{X^{*}}$.
3. (iii)
$Irr(S)=\\{[u]\in NLSW(X)\ |\ u\neq{a\bar{s}b},\ s\in{S},\ a,b\in{X^{*}}\\}$
is a linear basis for $Lie(X|S)=Lie(X)/Id(S)$.
Remark: The above Composition-Diamond lemma is also valid if we replace the
base field $k$ by an arbitrary commutative ring $K$ with identity. If this is
the case, as $K$-module, $Lie(X|S)$ is free with a $K$-basis $Irr(S)$.
## 3 Kukin’s construction of a Lie algebra with unsolvable word problem
Let $P=sgp\langle x,y|u_{i}=v_{i},\ i\in I\rangle$ be a semigroup. Consider
the Lie algebra
$A_{P}=Lie(x,\hat{x},y,\hat{y},z|S),$
where $S$ consists of the following relations:
1. (1)
$[\hat{x}x]=0,\ [\hat{x}y]=0,\ [\hat{y}x]=0,\ [\hat{y}y]=0$,
2. (2)
$[\hat{x}z]=-[zx],\ [\hat{y}z]=-[zy]$,
3. (3)
$\lfloor zu_{i}\rfloor=\lfloor zv_{i}\rfloor,\ i\in I$.
Here, $\lfloor zu\rfloor$ means the left normed bracketing.
In this section, we give a Gröbner-Shirshov basis for Lie algebra $A_{P}$ and
by using this result we give another proof for Kukin’s theorem, see Corollary
3.2.
Let the order $\hat{x}>\hat{y}>z>x>y$ and $>$ the deg-lex order on
$\\{\hat{x},\hat{y},x,y,z\\}^{*}$. Let $\rho$ be the congruence on
$\\{x,y\\}^{*}$ generated by $\\{(u_{i},v_{i}),\ i\in I\\}$. Let
1. $(3)^{\prime}$
$\lfloor zu\rfloor=\lfloor zv\rfloor,\ (u,v)\in\rho$ with $u>v$.
###### Theorem 3.1
With the above notation, the set $S_{1}=\\{(1),(2),(3)^{\prime}\\}$ is a
Gröbner-Shirshov basis in $Lie(\hat{x},\hat{y},x,y,z)$.
Proof: For any $u\in\\{x,y\\}^{*}$, by induction on $|u|$, $\overline{\lfloor
zu\rfloor}=zu$. All possible compositions in $S_{1}$ are intersection of (2)
and $(3)^{\prime}$, and inclusion of $(3)^{\prime}$ and $(3)^{\prime}$.
For $(2)\wedge(3)^{\prime},\ w=\hat{x}zu,\ (u,v)\in\rho,\ u>v,\
f=[\hat{x}z]+[zx],\ g=\lfloor zu\rfloor-\lfloor zv\rfloor$. We have
$\displaystyle([\hat{x}z]+[zx],\lfloor zu\rfloor-\lfloor
zv\rfloor)_{w}=[fu]_{\bar{f}}-[\hat{x}g]_{\bar{g}}$ $\displaystyle\equiv$
$\displaystyle\lfloor([\hat{x}z]+[zx])u\rfloor-[\hat{x}(\lfloor
zu\rfloor-\lfloor zv\rfloor)]$ $\displaystyle\equiv$ $\displaystyle\lfloor
zxu\rfloor+\lfloor\hat{x}zv\rfloor\equiv\lfloor zxu\rfloor-\lfloor
zxv\rfloor\equiv 0\ \ mod(S_{1},w).$
For $(3)^{\prime}\wedge(3)^{\prime},\ w=zu_{1}=zu_{2}e,\ e\in\\{x,y\\}^{*},\
(u_{i},v_{i})\in\rho,\ u_{i}>v_{i},\ i=1,2$. We have
$\displaystyle(\lfloor zu_{1}\rfloor-\lfloor zv_{1}\rfloor,\lfloor
zu_{2}\rfloor-\lfloor zv_{2}\rfloor)_{w}\equiv(\lfloor zu_{1}\rfloor-\lfloor
zv_{1}\rfloor)-\lfloor(\lfloor zu_{2}\rfloor-\lfloor zv_{2}\rfloor)e\rfloor$
$\displaystyle\equiv$ $\displaystyle\lfloor\lfloor zv_{2}\rfloor
e\rfloor-\lfloor zv_{1}\rfloor\equiv\lfloor zv_{2}e\rfloor-\lfloor
zv_{1}\rfloor\equiv 0\ \ mod(S_{1},w).$
Thus, the set $S_{1}=\\{(1),(2),(3)^{\prime}\\}$ is a Gröbner-Shirshov basis
in $Lie(\hat{x},\hat{y},x,y,z)$. $\blacksquare$
###### Corollary 3.2
(Kukin [32]) Let $u,v\in\\{x,y\\}^{*}$. Then
$u=v\ \mbox{ in the semigroup}\ P\Leftrightarrow\lfloor zu\rfloor=\lfloor
zv\rfloor\ \mbox{ in the Lie algebra }\ A_{P}.$
Proof: Suppose that $u=v\ \mbox{ in the semigroup}\ P$. Without loss of
generality, we may assume that $u=au_{1}b,\ v=av_{1}b$ for some
$a,b\in\\{x,y\\}^{*}$ and $(u_{1},v_{1})\in\rho$. For any $r\in\\{x,y\\}$, by
the relations (1), we have $[\hat{x}r]=0$ and so $\lfloor
zxc\rfloor=\lfloor[z\hat{x}]c\rfloor=[\lfloor zc\rfloor\hat{x}],\ \lfloor
zyc\rfloor=[\lfloor zc\rfloor\hat{y}]$ for any $c\in\\{x,y\\}^{*}$. From this
it follows that in $A_{P}$, $\lfloor zu\rfloor=\lfloor
zau_{1}b\rfloor=\lfloor\lfloor zau_{1}\rfloor b\rfloor=\lfloor\lfloor
zu_{1}\widehat{\overleftarrow{a}}\rfloor b\rfloor=\lfloor
zu_{1}\widehat{\overleftarrow{a}}b\rfloor=\lfloor
zv_{1}\widehat{\overleftarrow{a}}b\rfloor=\lfloor zav_{1}b\rfloor=\lfloor
zv\rfloor$, where $\overleftarrow{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}}=x_{i_{n}}x_{i_{n-1}}\cdots x_{i_{1}}$ and
$\widehat{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}}=\widehat{x_{i_{1}}}\widehat{x_{i_{2}}}\cdots\widehat{x_{i_{n}}}$,
$x_{i_{j}}\in\\{x,y\\}$. Moreover, $(3)^{\prime}$ holds in $A_{P}$.
Suppose that $\lfloor zu\rfloor=\lfloor zv\rfloor\ \mbox{ in the Lie algebra
}\ A_{P}$. Then both $\lfloor zu\rfloor$ and $\lfloor zv\rfloor$ have the same
normal form in $A_{P}$. Since $S_{1}$ is a Gröbner-Shirshov basis in $A_{P}$
by Theorem 3.1, both $\lfloor zu\rfloor$ and $\lfloor zv\rfloor$ can be
reduced to the same normal form of the form $\lfloor zc\rfloor$ for some
$c\in\\{x,y\\}^{*}$ only by the relations $(3)^{\prime}$. This implies that in
$P$, $u=c=v$. $\blacksquare$
By the above corollary, if the semigroup $P$ has the undecidable word problem
then so does the Lie algebra $A_{P}$.
## 4 Gröbner-Shirshov basis for the Drinfeld-Kohno Lie algebra
$\textbf{L}_{n}$
In this section we give a Gröbner-Shirshov basis for the Drinfeld-Kohno Lie
algebra $\textbf{L}_{n}$.
###### Definition 4.1
([27]) Let $n>2$ be an integer. The Drinfeld-Kohno Lie algebra
$\textbf{L}_{n}$ over $\mathbb{Z}$ is defined by generators $t_{ij}=t_{ji}$
for distinct indices $1\leq i,\ j\leq n-1$, and relations
$\displaystyle t_{ij}t_{kl}=0,$ $\displaystyle t_{ij}(t_{ik}+t_{jk})=0,$
where $i,\ j,\ k,\ l$ are distinct.
Clearly, $\textbf{L}_{n}$ has a presentation $Lie_{\mathbb{Z}}(T|S)$, where
$T=\\{t_{ij}|\ 1\leq i<j\leq n-1\\}$ and $S$ consists of the following
relations
$\displaystyle t_{ij}t_{kl}=\ 0,\ \ \ \ k<i<j,\ k<l,\ l\neq\ i,\ j$ (1)
$\displaystyle t_{jk}t_{ij}+t_{ik}t_{ij}=\ 0,\ \ \ \ i<j<k$ (2) $\displaystyle
t_{jk}t_{ik}-t_{ik}t_{ij}=\ 0,\ \ \ i<j<k$ (3)
Now we order $T$: $t_{ij}<t_{kl}$ if either $i<k$ or $i=k$ and $j<l$. Let $<$
be the deg-lex order on $T^{*}$.
###### Theorem 4.2
Let $S=\\{(\ref{a}),(\ref{b}),(\ref{c})\\}$ be as before, $<$ the deg-lex
order on $T^{*}$. Then $S$ is a Gröbner-Shirshov basis for $\textbf{L}_{n}$.
Proof. We list all the possible ambiguities. Denote $(i)\wedge(j)$ the
composition of the type $(i)$ and type $(j)$.
For $(1)\wedge(n)$, $1\leq n\leq 3$, the possible ambiguities $w$’s are:
$\displaystyle(1)\wedge(1)$ $\displaystyle t_{ij}t_{kl}t_{mr},\ (k<i<j,\ k<l,\
l\neq\ i,j,\ m<k<l,\ m<r,\ r\neq\ k,l),$ $\displaystyle(1)\wedge(2)$
$\displaystyle t_{ij}t_{kl}t_{mk},\ (\ k<i<j,\ k<l,\ l\neq\ i,j,\ m<k<l),$
$\displaystyle(1)\wedge(3)$ $\displaystyle t_{ij}t_{kl}t_{ml},\ (k<i<j,\ k<l,\
l\neq\ i,j,\ m<k<l).$
For $(2)\wedge(n)$, $1\leq n\leq 3$, the possible ambiguities $w$’s are:
$\displaystyle(2)\wedge(1)$ $\displaystyle t_{jk}t_{ij}t_{mr},\ (\ m<i<j<k,\
m<r,\ r\neq\ i,j),$ $\displaystyle(2)\wedge(2)$ $\displaystyle
t_{jk}t_{ij}t_{mi},\ (\ m<i<j<k),$ $\displaystyle(2)\wedge(3)$ $\displaystyle
t_{jk}t_{ij}t_{mj},\ (\ m<i<j<k).$
For $(3)\wedge(n)$, $1\leq n\leq 3$, the possible ambiguities $w$’s are:
$\displaystyle(3)\wedge(1)$ $\displaystyle t_{jk}t_{ik}t_{mr},\ (\ m<i<j<k,\
m<r,\ r\neq\ i,k),$ $\displaystyle(3)\wedge(2)$ $\displaystyle
t_{jk}t_{ik}t_{mi},\ (\ m<i<j<k),$ $\displaystyle(3)\wedge(3)$ $\displaystyle
t_{jk}t_{ik}t_{mk},\ (\ m<i<j<k).$
We claim that all compositions are trivial relative to $S$.
Here, we only prove cases $(1)\wedge(1)$, $(1)\wedge(2)$, $(2)\wedge(1)$,
$(2)\wedge(2)$, and the other cases can be proved similarly.
For $(1)\wedge(1)$, let $f=t_{ij}t_{kl},\ g=t_{kl}t_{mr},\ k<i<j,\ k<l,\
l\neq\ i,j,\ m<k<l,\ m<r,\ r\neq\ k,l.$ Then $w=t_{ij}t_{kl}t_{mr}$ and
$\displaystyle(f,g)_{w}$ $\displaystyle=$
$\displaystyle(t_{ij}t_{kl})t_{mr}-t_{ij}(t_{kl}t_{mr})$ $\displaystyle=$
$\displaystyle\ (t_{ij}t_{mr})t_{kl}\ mod(S,w).$
There are three subcases to consider: $r\neq\ i,j$, $r=i$, $r=j$.
Subcase 1. If $r\neq\ i,j$, then
$\displaystyle(t_{ij}t_{mr})t_{kl}\equiv\ 0\ mod(S,w).$
Subcase 2. If $r=i$, then
$\displaystyle(t_{ij}t_{mr})t_{kl}$ $\displaystyle=$
$\displaystyle(t_{ij}t_{mi})t_{kl}$ $\displaystyle\equiv$ $\displaystyle\
t_{kl}(t_{mj}t_{mi})$ $\displaystyle\equiv$ $\displaystyle\
(t_{kl}t_{mj})t_{mi}+t_{mj}(t_{kl}t_{mi})$ $\displaystyle\equiv$
$\displaystyle\ 0\ mod(S,w).$
Subcase 3. If $r=j$, then
$\displaystyle(t_{ij}t_{mr})t_{kl}$ $\displaystyle=$
$\displaystyle(t_{ij}t_{mj})t_{kl}$ $\displaystyle\equiv$ $\displaystyle\
-t_{kl}(t_{mj}t_{mi})$ $\displaystyle\equiv$ $\displaystyle\
-(t_{kl}t_{mj})t_{mi}-t_{mj}(t_{kl}t_{mi})$ $\displaystyle\equiv$
$\displaystyle\ 0\ mod(S,w).$
For $(1)\wedge(2)$, let $f=t_{ij}t_{kl},\ g=t_{kl}t_{mk}+t_{ml}t_{mk},\
k<i<j,\ k<l,\ l\neq\ i,j,\ m<k<l.\ $Then $w=t_{ij}t_{kl}t_{mk}$ and
$\displaystyle(f,g)_{w}$ $\displaystyle=$
$\displaystyle(t_{ij}t_{kl})t_{mk}-t_{ij}(t_{kl}t_{mk}+t_{ml}t_{mk})$
$\displaystyle=$ $\displaystyle\ (t_{ij}t_{mk})t_{kl}-t_{ij}(t_{ml}t_{mk})$
$\displaystyle\equiv$ $\displaystyle\ -t_{ij}(t_{ml}t_{mk})$
$\displaystyle\equiv$ $\displaystyle\
-(t_{ij}t_{ml})t_{mk}-t_{ml}(t_{ij}t_{mk})$ $\displaystyle\equiv$
$\displaystyle\ 0\ mod(S,w).$
For $(2)\wedge(1)$, let $f=t_{jk}t_{ij}+t_{ik}t_{ij},\ g=t_{ij}t_{mr},\
m<i<j<k,\ \ m<r,\ r\neq\ i,j.\ $Then $w=t_{jk}t_{ij}t_{mr}$ and
$\displaystyle(f,g)_{w}$ $\displaystyle=$
$\displaystyle(t_{jk}t_{ij}+t_{ik}t_{ij})t_{mr}-t_{jk}(t_{ij}t_{mr})$
$\displaystyle\equiv$ $\displaystyle\
(t_{jk}t_{mr})t_{ij}+(t_{ik}t_{mr})t_{ij}\ mod(S,w).$
There are two subcases to consider: $r\neq\ k$, $r=k$.
Subcase 1. If $r\neq\ k$, then
$\displaystyle(t_{jk}t_{mr})t_{ij}+(t_{ik}t_{mr})t_{ij}\equiv\ 0\ mod(S,w).$
Subcase 2. If $r=k$, then
$\displaystyle(t_{jk}t_{mr})t_{ij}+(t_{ik}t_{mr})t_{ij}$ $\displaystyle=$
$\displaystyle(t_{jk}t_{mk})t_{ij}+(t_{ik}t_{mk})t_{ij}$ $\displaystyle\equiv$
$\displaystyle\ -t_{ij}(t_{mk}t_{mj})-t_{ij}(t_{mk}t_{mi})$
$\displaystyle\equiv$ $\displaystyle\
(t_{ij}t_{mj})t_{mk}+(t_{ij}t_{mi})t_{mk}$ $\displaystyle\equiv$
$\displaystyle\ -t_{mk}(t_{mj}t_{mi})+t_{mk}(t_{mj}t_{mi})$
$\displaystyle\equiv$ $\displaystyle\ 0\ mod(S,w).$
For $(2)\wedge(2)$, let $f=t_{jk}t_{ij}+t_{ik}t_{ij},\
g=t_{ij}t_{mi}+t_{mj}t_{mi},\ m<i<j<k.\ $Then $w=t_{jk}t_{ij}t_{mi}$ and
$\displaystyle(f,g)_{w}$ $\displaystyle=$
$\displaystyle(t_{jk}t_{ij}+t_{ik}t_{ij})t_{mi}-t_{jk}(t_{ij}t_{mi}+t_{mj}t_{mi})$
$\displaystyle=$ $\displaystyle\
(t_{jk}t_{mi})t_{ij}+(t_{ik}t_{mi})t_{ij}+t_{ik}(t_{ij}t_{mi})-t_{jk}(t_{mj}t_{mi})$
$\displaystyle\equiv$ $\displaystyle\
t_{ij}(t_{mk}t_{mi})-t_{ik}(t_{mj}t_{mi})-t_{jk}(t_{mj}t_{mi})$
$\displaystyle\equiv$ $\displaystyle\
-(t_{ij}t_{mi})t_{mk}+(t_{ik}t_{mi})t_{mj}-(t_{jk}t_{mj})t_{mi}$
$\displaystyle\equiv$ $\displaystyle\
-t_{mk}(t_{mj}t_{mi})-(t_{mk}t_{mi})t_{mj}+(t_{mk}t_{mj})t_{mi}$
$\displaystyle\equiv$ $\displaystyle\ 0\ mod(S,w).$
So $S$ is a Gröbner-Shirshov basis for $\textbf{L}_{n}$. $\blacksquare$
Let $L$ be a Lie algebra over a commutative ring $K$, $L_{1}$ an ideal of $L$
and $L_{2}$ a subalgebra of $L$. We call $L$ a semidirect product of $L_{1}$
and $L_{2}$ if $L=L_{1}\oplus L_{2}$ as $K$-modules.
By Theorems 2.3 and 4.2, we have immediately the following corollaries.
###### Corollary 4.3
The Drinfeld-Kohno Lie algebra $\textbf{L}_{n}$ is a free $\mathbb{Z}$-module
with a $\mathbb{Z}$-basis
$Irr(S)=\\{[t_{ik_{1}}t_{ik_{2}}\cdots\ t_{ik_{m}}]\ |\
t_{ik_{1}}t_{ik_{2}}\cdots\ t_{ik_{m}}\ is\ an\mbox{ ALSW }in\ T^{*},\
m\in\mathbb{N}\\}.$
###### Corollary 4.4
([27]) $\textbf{L}_{n}$ is an iterated semidirect product of free Lie
algebras.
Proof. Let $A_{i}$ be the free Lie algebra generated by $\\{t_{ij}\ |\
i<j\leq\ n-1\\}$. Clearly,
$\textbf{L}_{n}=A_{1}\oplus\ A_{2}\oplus\ \cdots\ \oplus\ A_{n-2}$
as $\mathbb{Z}$-modules, and from the relations $(1),(2),(3)$, we have
$A_{i}\triangleleft\ A_{i}+A_{i+1}+\ \cdots\ +A_{n-2}.$
$\blacksquare$
Remark: In this section, if we replace the base ring $\mathbb{Z}$ by an
arbitrary commutative ring $K$ with identity, then all results hold.
## References
* [1] W.W. Adams, P. Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society (AMS), 1994.
* [2] G.M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978) 178-218.
* [3] L.A. Bokut, Insolvability of the word problem for Lie algebras and subalgebras of finitely presented Lie algebras, Izvestija AN USSR (mathem.) 36 (1972) 1173-1219.
* [4] L.A. Bokut, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976) 117-142.
* [5] L.A. Bokut, Y.Q. Chen, Gröbner-Shirshov bases: Some new results, Proceedings of the Second International Congress in Algebra and Combinatorics, World Scientific, (2008) 35-56.
* [6] L.A. Bokut, Y.Q. Chen, Gröbner-Shirshov bases and their calculation, arxiv.org/abs/1303.5366.
* [7] L.A. Bokut, Y.Q. Chen, Y.S. Chen, Composition-Diamond lemma for tensor product of free algebras, _Journal of Algebra_ 323 (2010) 2520-2537.
* [8] L.A. Bokut, Y.Q. Chen, Y.S. Chen, Groebner-Shirshov bases for Lie algebras over a commutative algebra, _Journal of Algebra_ 337 (2011) 82-102.
* [9] L.A. Bokut, Y.Q. Chen, X.M. Deng, Gröbner-Shirshov bases for Rota-Baxter algebras, _Siberian Math. J._ 51 (6) (2010) 978-988.
* [10] L.A. Bokut, Y.Q. Chen, Y. Li, Lyndon-Shirshov basis and anti-commutative algebras, _Journal of Algebra_ 378 (2013) 173-183.
* [11] L.A. Bokut, Y.Q. Chen, C.H. Liu, Gröbner-Shirshov bases for dialgebras,_International Journal of Algebra and Computation_ 20 (3) (2010) 391-415.
* [12] L.A. Bokut, Y.Q. Chen, Q.H. Mo, Gröbner-Shirshov bases and embeddings of algebras, _International Journal of Algebra and Computation_ 20 (2010) 875-900.
* [13] L.A. Bokut, Y.Q. Chen, Q.H. Mo, Gröbner-Shirshov bases for semirings, _Journal of Algebra_ 385 (2013) 47-63.
* [14] L.A. Bokut, Y.Q. Chen, K.P. Shum, Some new results on Groebner-Shirshov bases, in: Proceedings of International Conference on Algebra 2010, Advances in Algebraic Structures, (2012) 53-102.
* [15] L.A. Bokut, Y. Fong, W.-F. Ke, P.S. Kolesnikov, Gröbner and Gröbner-Shirshov bases in algebra and conformal algebras, Fundamental and Applied Mathematics 6 (3) (2000) 669-706.
* [16] L.A. Bokut, P.S. Kolesnikov, Gröbner-Shirshov bases: from their incipiency to the present, J. Math. Sci. 116 (1) (2003) 2894-2916.
* [17] L.A. Bokut, P.S. Kolesnikov, Gröbner-Shirshov bases, conformal algebras and pseudo-algebras, J. Math. Sci. 131 (5) (2005) 5962-6003.
* [18] L.A. Bokut, G. Kukin, Algorithmic and Combinatorial algebra, Kluwer Academic Publ., Dordrecht, 1994
* [19] W.W. Boone, The word problem, Ann. Math. 70 (1959) 207-265.
* [20] B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations, Aequationes Math. 4 (1970) 374-383.
* [21] B. Buchberger, G.E. Collins, R. Loos, R. Albrecht, Computer algebra, symbolic and algebraic computation, Computing Supplementum, Vol.4, New York: Springer-Verlag, 1982.
* [22] B. Buchberger, Franz Winkler, Gröbner bases and applications, London Mathematical Society Lecture Note Series, Vol.251, Cambridge: Cambridge University Press, 1998.
* [23] Y.S. Chen, Y.Q. Chen, Groebner-Shirshov bases for matabelian Lie algebras, _Journal of Algebra_ 358 (2012) 143-161.
* [24] D.A. Cox, J. Little, D. O’Shea, Ideals, varieties and algorithms: An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, New York: Springer-Verlag, 1992.
* [25] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math., Vol.150, Berlin and New York: Springer-Verlag, 1995.
* [26] V.G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz 1 (1989) 114-148.
* [27] P. Etingof, A. Henriques, J. Kamnitzer, E.M. Rains, The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points, Ann. Math. 171 (2010) 731-777.
* [28] G. Higman, Subgroups of finitely presented groups. Proc. Royal Soc. London (Series A) 262 (1961) 455-475.
* [29] H. Hironaka, Resolution of singularities of an algebraic variety over a field if characteristic zero, I, II, Ann. Math. 79 (1964) 109-203, 205-326.
* [30] S.-J. Kang, K.-H. Lee, Gröbner-Shirshov bases for irreducible $sl_{n+1}$-modules, Journal of Algebra 232 (2000) 1-20.
* [31] T. Kohno, Serie de Poincare Koszul associee aux groupes de tresses pures, Invent. Math. 82 (1985) 57-75.
* [32] G.P. Kukin, On the word problem for Lie algebras, Sibirsk. Mat. Zh. 18 (1977) 1194-1197.
* [33] Lyndon, R.C.: On Burnside’s problem I, Trans. Amer. Math. Soc. 77 (1954) 202-215.
* [34] A.A. Markov, Impossibility of some algorithms in the theory of some associative system, Dokl. Akad. Nauk SSSR 55 (1947) 587-590.
* [35] A.A. Mikhalev, A.A. Zolotykh, Standard Gröbner-Shirshov bases of free algebras over rings, I. Free associative algebras, _International Journal of Algebra and Computation_ 8 (6) (1998) 689-726.
* [36] P.S. Novikov, On algorithmic undecidability of the word problem in the theory of groups, Trudy MKat. Inst. Steklov. 44 (1955) 1-144.
* [37] E. Post, A variant of a recursively unsolvable problem, Bull. Amer. Math. Soc. 52 (1946) 264-268.
* [38] A.I. Shirshov, On free Lie rings, Mat. Sb. 45 (2) (1958) 113-122.
* [39] A.I. Shirshov, Some algorithmic problem for Lie algebras, Sibirsk. Mat. Zh. 3 (2) (1962) 292-296; English translation in SIGSAM Bull. 33 (1999) 3-6.
* [40] A.I. Shirshov, Some algorithmic problem for $\varepsilon$-algebras, Sibirsk. Mat. Z. 3 (1962) 132-137.
* [41] Selected works of A.I. Shirshov, Eds L.A. Bokut, V. Latyshev, I. Shestakov, E. Zelmanov, Trs M. Bremner, M. Kochetov, Birkhäuser, Basel, Boston, Berlin, 2009.
* [42] A.M. Turing, The word problem in semi-groups with cancellation, Ann. Math. 52 (1950) 191-505.
|
# On the Multiway Principal Component Analysis∗
Jialin Ouyang and Ming Yuan
Department of Statistics
Columbia University
(())
###### Abstract
Multiway data are becoming more and more common. While there are many
approaches to extending principal component analysis (PCA) from usual data
matrices to multiway arrays, their conceptual differences from the usual PCA,
and the methodological implications of such differences remain largely
unknown. This work aims to specifically address these questions. In
particular, we clarify the subtle difference between PCA and singular value
decomposition (SVD) for multiway data, and show that multiway principal
components (PCs) can be estimated reliably in absence of the eigengaps
required by the usual PCA, and in general much more efficiently than the usual
PCs. Furthermore, the sample multiway PCs are asymptotically independent and
hence allow for separate and more accurate inferences about the population
PCs. The practical merits of multiway PCA are further demonstrated through
numerical, both simulated and real data, examples.
11footnotetext: This research was supported in part by NSF Grants DMS-2015285
and DMS-2052955.
## 1 Introduction
More and more often in practice, we need to deal with data of rich and complex
structures that are more appropriately organized as multiway arrays rather
than the usual data matrices. Examples of such multiway data are ubiquitous in
many fields such as chemometrics, economics, psychometrics, and signal
processing among others (see, e.g., Kroonenberg, 2008; Anandkumar et al.,
2014; Zhang and Xia, 2018; Chen et al., 2020a, b; Han et al., 2020; Xia et
al., 2020; Bi et al., 2021; Chen et al., 2021; Han et al., 2022). In this
paper, we investigate the methodological implications and statistical
properties of principal component analysis (PCA) for this type of data and
pinpoint the benefits and challenges of doing so.
PCA is among the most popular statistical methods for multivariate data
analysis when data are organized as matrices. See, e.g., Anderson (1984);
Jolliffe (2002). With each column vector of a data matrix as an observation,
PCA seeks orthogonal linear transformations of these vectors into a new
coordinate system so that the variance of each coordinate is maximized
successively. It allows us to represent most of the variation in the data by a
small number of coordinates and therefore can guide us in reducing the
dimensionality. As such, PCA often serves as a critical first step to capture
the essential features in a dataset for many downstream analyses and is widely
used in many scientific and engineering fields. Moving beyond matrices, for
multiway data, each observation itself forms a matrix or more generally a
multiway array. For example, when repeated measurements are made across
different combinations of location and time, each observation can be more
naturally organized as a matrix with each row corresponding to a certain
location and each column a time point. To apply PCA to this type of data, it
is tempting to neglect the multiway nature of the observations and treat each
observation as a vector nonetheless, a practice often referred to as
_stringing_. However, as observed in numerous practical applications,
appropriately accounting for the additional structure when applying PCA can
greatly enhance interpretability and improve efficiency. See, e.g.,
Kroonenberg (2008).
There is a long and illustrious history of developing suitable methods for
such a purpose and it can be traced back at least to the pioneering work of
Tucker, Harshman, and Carroll in the 1960s. Since then, numerous approaches
have also been developed. Examples include Kroonenberg and De Leeuw (1980); De
Lathauwer et al. (2000); Vasilescu and Terzopoulos (2002); Yang et al. (2004);
Kong et al. (2005); Zhang and Zhou (2005); Lu et al. (2006, 2008); Li et al.
(2010); Liu et al. (2017); Taguchi (2018) among many others. See, e.g., Lu et
al. (2011); Cichocki et al. (2015) for recent surveys of existing techniques.
Most of these developments are outside the mainstream statistics literature
and often with a strong algorithmic flavor and exploratory data analysis
focus. These approaches are intuitive and often yield more interpretable
insights than naively applying PCA after stringing. However, their statistical
underpinnings are largely unknown. The main goal of this article is to fill in
this void. Indeed, as we shall demonstrate, a careful and rigorous statistical
treatment allows for a better understanding of the operating characteristics
of multiway PCA, leads to improved methodology, and reveals new opportunities
and challenges in analyzing multiway data.
More specifically, we focus on a simple and natural approach to multiway PCA:
when seeking linear transformations that maximize the variance, we impose the
additional constraint that they conform to the multiway structure of the data.
Doing so not only allows for enhanced interpretability but also inherits many
nice properties of the usual PCA. Just as the usual principal components (PCs)
are the eigenvectors of the covariance matrix, the multiway PCs can be
identified with certain eigenvectors of the covariance operator. To better
understand the impact of multiway structure on our ability to recover and make
inferences about the multiway PCs, we also investigate the properties of
multiway PCA under a spiked covariance model.
Statistical properties of the usual PCA are well understood in the classical
setting where the sample size is large whereas the number of variables is
small (see, e.g., Anderson, 1984). More and more often in today’s
applications, however, the dimensionality can also be large. There are
abundant theoretical results concerning the usual PCA in such a high-
dimensional setting as well, especially in the context of the spiked
covariance model. For example, Johnstone and Lu (2009) first demonstrated the
critical role of dimensionality in PCA by showing that, with fixed signal
strength, the sample PCA is consistent if and only if the number of variables
is of a smaller order than the sample size. In another influential paper, Paul
(2007) established the asymptotic distribution of sample PCs. Other related
treatments include Baik and Silverstein (2006); Nadler (2008); Johnstone and
Lu (2009); Jung and Marron (2009); Bai and Silverstein (2010); Lee et al.
(2010); Benaych-Georges and Nadakuditi (2011); Bai and Yao (2012); Shen et al.
(2013); Koltchinskii and Lounici (2014); Koltchinskii et al. (2017); Wang and
Fan (2017); Koltchinskii et al. (2020) among numerous others. In a sense, our
results naturally extend these earlier works to multiway PCA. However, the
need to work with higher-order covariance operators rather than covariance
matrices creates new and fundamental challenges and requires us to develop a
different proof strategy and several new technical tools. More importantly,
our analysis also reveals fundamental differences in behavior between the
usual PCA and multiway PCA and inspires new methodological development for the
latter.
Firstly, we establish the rates of convergence for the sample multiway PCs
under mild regularity conditions. These rates explain why it is essential that
we account for the inherent data structure when applying PCA to multiway data,
and why naively applying PCA after stringing could be problematic.
Intuitively, multiway PCA uses fewer parameters than the usual PCA and
therefore is easier for estimation. This is described precisely by our result
in that the estimation error of multiway PCs is determined by the dimension of
each mode of the data array rather than the total number of entries, and
therefore multiway PCs can be estimated accurately even if the latter far
exceeds the sample size. But a more important observation is that how well a
multiway PC can be estimated is determined by the corresponding eigenvalue of
the covariance operator, and not the gap between its eigenvalues like the
usual PCA. This somewhat surprising finding has far-reaching implications. In
particular, it means that for multiway data the PCs can be estimated well even
if their corresponding eigenvalues are not simple.
Moreover, to facilitate making statistical inferences about multiway PCs, we
derive asymptotic distributions of the sample multiway PCs. Our results again
reveal unexpected but important distinctions between multiway PCA and usual
PCA. For example, the estimated multiway PCs are asymptotically independent of
each other, and their asymptotic distribution is determined by their
corresponding eigenvalues instead of eigengaps. Furthermore, we show that bias
correction is important for the sample multiway PCs. Similar to the usual PCA,
sample multiway PC can exhibit significant bias when the dimension (of each
mode) is high. But there is also another source of bias that may arise due to
the inherent ambiguity in ordering the PCs in absence of eigengaps.
Nonetheless, we show that both types of bias can be eliminated, enabling us to
make inferences about and construct confidence intervals for the multiway PCs.
The rest of the paper is organized as follows. In Section 2, we introduce the
notion of multiway PCA both at a population level and how it works on a finite
sample. Section 3 investigates the rates of convergence for the sample
multiway PCs. Turning our attention to the asymptotic distribution of multiway
PCA in section 4, we show how to make valid inferences about the multiway PCs.
The merits of the multiway PCA and our proposed approaches are further
demonstrated through numerical experiments, both simulated and real, in
Section 5. We conclude with a summary in Section 6. Due to the space limit,
all proofs are relegated to supplementary material.
## 2 Multiway PCA
Multiway PCA can be viewed through the lens of usual PCA with the additional
multiway structure imposed on the PCs. Let
$\mathscr{X}\in\mathbb{R}^{d_{1}\times\cdots\times d_{p}}$ be an order-$p$
random array. To simplify, we shall assume in what follows that $\mathscr{X}$
is centered, i.e., $\mathbb{E}\mathscr{X}=0$, unless otherwise indicated. The
idea behind PCA is to look for a linear transformation of $\mathscr{X}$ that
maximizes the variance:
$\max_{\mathscr{W}\in\mathbb{R}^{d_{1}\times\cdots\times
d_{p}}:\|\mathscr{W}\|_{\rm F}=1}{\rm
var}(\langle\mathscr{X},\mathscr{W}\rangle).$ (1)
Here $\|\mathscr{W}\|_{\rm F}=\langle\mathscr{W},\mathscr{W}\rangle^{1/2}$ and
$\langle\mathscr{X},\mathscr{W}\rangle=\sum_{j_{1}=1}^{d_{1}}\cdots\sum_{j_{p}=1}^{d_{p}}x_{j_{1},\ldots,j_{p}}w_{j_{1},\ldots,j_{p}}.$
Denote by $\mathscr{U}_{1}$ the solution to (1). The basic premise of multiway
PCA is that $\mathscr{U}_{1}$ conforms to the multiway structure underlying
$\mathscr{X}$ in that it is a rank-one tensor and can be expressed as
$\mathscr{U}_{1}=\mathbf{u}_{1}^{(1)}\otimes\mathbf{u}_{1}^{(2)}\otimes\cdots\otimes\mathbf{u}_{1}^{(p)},$
(2)
where $\mathbf{u}_{1}^{(q)}$ is a unit length vector $\mathbb{R}^{d_{q}}$ and
$\otimes$ stands for the outer product, i.e., the $(i_{1},\ldots,i_{p})$ entry
of $\mathscr{U}_{1}$ is given by
$\left[\mathscr{U}_{1}\right]_{i_{1},\ldots,i_{p}}=u_{1i_{1}}^{(1)}u_{1i_{2}}^{(2)}\cdots
u_{1i_{p}}^{(p)}.$
In other words, $\mathscr{U}_{1}$ is also the solution to
$\max_{\mathscr{W}\in\Theta}{\rm var}(\langle\mathscr{X},\mathscr{W}\rangle),$
(3)
where $\Theta$ is the collection of all unit length rank-one tensors of
conformable dimensions, i.e.,
$\Theta=\\{\mathscr{W}=\mathbf{w}^{(1)}\otimes\mathbf{w}^{(2)}\otimes\cdots\otimes\mathbf{w}^{(p)}:\mathbf{w}^{(q)}\in\mathbb{R}^{d_{q}},\|\mathbf{w}^{(q)}\|=1,\forall
q=1,\ldots,p\\}.$
Even if the solution to (1) is not strictly rank-one as described by (2),
imposing such a constraint when seeking variance-maximizing transformation can
nonetheless be desirable because of the enhanced interpretability: the
additional rank-one constraint allows us to separate the effect along each
mode, and help address questions such as “who does what to whom and when”
which are often central to multiway data analysis. See, e.g., Kroonenberg
(2008) for further discussion and numerous motivating examples.
Subsequent PCs can be defined successively:
$\max_{\begin{subarray}{c}\mathscr{W}\in\mathbb{R}^{d_{1}\times\cdots\times
d_{p}}:\|\mathscr{W}\|_{\rm F}=1\\\
\mathscr{W}\perp\mathscr{U}_{l},l=1,\ldots,k-1\end{subarray}}{\rm
var}(\langle\mathscr{X},\mathscr{W}\rangle).$ (4)
As before, we shall consider the case when the solution has rank one. A key
requirement in defining PCs is that the $k$th PC is orthogonal to all other
PCs, i.e., $\mathscr{W}\perp\mathscr{U}_{l}$. In vector case, i.e., $p=1$,
this simply means that $\langle\mathscr{W},\mathscr{U}_{l}\rangle=0$. In
multiway case, however, there are many different notions of orthogonality.
See, e.g., Kolda (2001) for a detailed discussion on this subject. Each notion
has its own subtleties and caveats that may have different statistical
implications. In this work we shall focus on the notion of _complete
orthogonality_ : two rank-one tensors
$\mathscr{W}_{1}=\mathbf{w}_{1}^{(1)}\otimes\mathbf{w}_{1}^{(2)}\otimes\cdots\otimes\mathbf{w}_{1}^{(p)}$
and
$\mathscr{W}_{2}=\mathbf{w}_{2}^{(1)}\otimes\mathbf{w}_{2}^{(2)}\otimes\cdots\otimes\mathbf{w}_{2}^{(p)}$
are complete orthogonal if and only if
$\langle\mathbf{w}_{1}^{(q)},\mathbf{w}_{2}^{(q)}\rangle=(\mathbf{w}_{1}^{(q)})^{\top}\mathbf{w}_{2}^{(q)}=0$
for all $q=1,\ldots,p$. More specifically, the $k$th multiway PC, denoted by
$\mathscr{U}_{k}$, solves
$\max_{\mathscr{W}\in\Theta:\mathscr{W}\perp_{c}\mathscr{U}_{l},\forall
l<k}{\rm var}(\langle\mathscr{X},\mathscr{W}\rangle),$ (5)
where $\perp_{c}$ stands for complete orthogonality.
As in the case of the usual PCA, multiway PCs can also be equivalently defined
using the covariance matrix of ${\rm vec}(\mathscr{X})$. In fact, it is more
convenient to think of a covariance operator when it comes to multiway data.
More specifically, we shall view
$\Sigma:={\rm cov}(\mathscr{X})=\mathbb{E}(\mathscr{X}\otimes\mathscr{X})$
as a $d_{1}\times d_{2}\times\cdots\times d_{p}\times d_{1}\times\cdots\times
d_{p}$ array. Then for any $\mathscr{W}\in\Theta$,
${\rm
var}(\langle\mathscr{X},\mathscr{W}\rangle)=\langle\Sigma,\mathscr{W}\otimes\mathscr{W}\rangle=\langle\Sigma,\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(p)}\otimes\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(p)}\rangle.$
Write
$\lambda_{k}={\rm var}(\langle\mathscr{X},\mathscr{U}_{k}\rangle).$
Because of the symmetry of $\Sigma$,
$\mathscr{U}_{1}\otimes\mathscr{U}_{1}=\operatorname{argmax}_{\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(2p)}:\|\mathbf{w}^{(q)}\|=1}\langle\Sigma,\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(2p)}\rangle$
so that $\lambda_{1}\mathscr{U}_{1}\otimes\mathscr{U}_{1}$ is also the best
rank-one approximation to $\Sigma$ (see, e.g., Friedland, 2013). Similarly,
$\mathscr{U}_{k}\otimes\mathscr{U}_{k}=\operatorname{argmax}_{\begin{subarray}{c}\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(2p)}:\|\mathbf{w}^{(q)}\|=1\\\
\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(2p)}\perp_{c}\mathscr{U}_{l}\otimes\mathscr{U}_{l},\quad
l<k\end{subarray}}\langle\Sigma,\mathbf{w}^{(1)}\otimes\cdots\otimes\mathbf{w}^{(2p)}\rangle$
In vector case, e.g. $p=1$, $\\{(\lambda_{k},\mathscr{U}_{k}):k\geq 1\\}$ are
the eigenpairs of the covariance matrix $\Sigma$ and
$\Sigma_{r}:=\sum_{k=1}^{r}\lambda_{k}\mathscr{U}_{k}\otimes\mathscr{U}_{k}$
is the best rank-$r$ approximation to $\Sigma$, i.e.,
$\Sigma_{r}=\operatorname{argmin}_{A\in\mathbb{R}^{d_{1}\times d_{1}}:{\rm
rank}(A)\leq r}\|A-\Sigma\|.$
When $p>1$, this characterization becomes tenuous because the notion of best
low-rank approximation becomes precarious. For matrices, best low-rank
approximations can be identified with singular value decomposition thanks to
the Eckart-Young theorem. Low-rank approximation to tensors is much more
subtle and the best low-rank approximation may not exist in general. See,
e.g., Hackbusch (2012). Nonetheless, by construction, $\Sigma_{r}$ is the so-
called best rank-$r$ _greedy orthogonal approximation_ to $\Sigma$. See, e.g.,
Kolda (2001). In particular, when the multiway structure does manifest itself
in a way such that the usual PCs are rank-one tensors, for example, the
solution to (1) and (4) has rank one, then $\Sigma_{r}$ is the best low-rank
approximation to $\Sigma$.
Sample multiway PCs can also be defined in a similar fashion. Specifically,
given a sample $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ of independent copies
of $\mathscr{X}$, $\mathscr{U}_{k}$s can be estimated by maximizing the sample
variances:
$\widehat{\mathscr{U}}_{k}:=\operatorname{argmax}_{\mathscr{W}\in\Theta:\mathscr{W}\perp_{c}\mathscr{U}_{l},\forall
l<k}\frac{1}{n}\sum_{i=1}^{n}\langle\mathscr{X}_{i},\mathscr{W}\rangle^{2}.$
(6)
Let
$\widehat{\Sigma}=\frac{1}{n}\sum_{i=1}^{n}\mathscr{X}_{i}\otimes\mathscr{X}_{i}$
be the sample covariance operator. Then $\widehat{\mathscr{U}}_{1}$ can be
defined via the best rank-one approximation to $\widehat{\Sigma}$
$\widehat{\mathscr{U}}_{1}=\operatorname{argmax}_{\mathscr{W}\in\Theta}\langle\widehat{\Sigma},\mathscr{W}\otimes\mathscr{W}\rangle.$
And other PCs can also be equivalently defined as
$\widehat{\mathscr{U}}_{k}=\operatorname{argmax}_{\mathscr{W}\in\Theta:\mathscr{W}\perp_{c}\widehat{\mathscr{U}}_{l},\forall
l<k}\langle\widehat{\Sigma},\mathscr{W}\otimes\mathscr{W}\rangle.$
Note that $\widehat{\mathscr{U}}_{k}$ can also be identified with the best
rank-one approximation to a deflated covariance operator:
$\widehat{\mathscr{U}}_{k}=\operatorname{argmax}_{\mathscr{W}\in\Theta}\langle\check{\Sigma}_{k},\mathscr{W}\otimes\mathscr{W}\rangle,$
where
$\check{\Sigma}_{k}=\widehat{\Sigma}\times_{1}\widehat{{\cal
P}}_{k}^{(1)}\times_{2}\cdots\times_{p}\widehat{{\cal
P}}_{k}^{(p)}\times_{p+1}\widehat{{\cal
P}}_{k}^{(1)}\times_{p+2}\cdots\times_{2p}\widehat{{\cal P}}_{k}^{(p)}$
and $\widehat{{\cal P}}_{k}^{(1)}$ is the projection matrix of the linear
subspace spanned by $\\{\widehat{\mathbf{u}}_{l}^{(q)}:1\leq l<k\\}$.
Hereafter $\times_{q}$ represents the mode $q$ product between a tensor
$\mathscr{T}\in\mathbb{R}^{d_{1}\times d_{2}\times\dots\times d_{k}}$ and a
matrix $A\in\mathbb{R}^{m\times d_{q}}$ so that
$\mathscr{T}\times_{q}A\in\mathbb{R}^{d_{1}\times\dots d_{q-1}\times m\times
d_{q+1}\dots\times d_{k}}$ with elements
$[\mathscr{T}\times_{q}A]_{i_{1}\dots i_{q-1}ji_{q+1}\dots
i_{k}}=\sum_{i_{q}=1}^{d_{q}}\mathscr{T}_{i_{1}\dots i_{q}\dots
i_{k}}A_{ji_{q}}.$
Computing the best rank-one approximation to a tensor is a classical problem
in numerical linear algebra, and casting the sample multiway PCA as such
allows us to take advantage of the many existing algorithms for doing so. In
this work, we focus on the statistical properties of multiway PCA. Readers
interested in further discussions about the computational aspect are referred
to, e.g., Zhang and Golub (2001); Hackbusch (2012); Janzamin et al. (2019) and
references therein.
Similar to the usual PCs, multiway PCs can be used to construct low-rank
approximations of the original data. However, there are also fundamental,
albeit sometimes subtle, differences between the two types of PCA. The usual
sample PCs coincide with the leading singular vectors of the data matrix after
appropriate centering and therefore can be computed via singular value
decomposition (SVD). In contrast, multiway PCA is, while closely related to,
not equivalent to the best low-rank approximations of the original data array
in general. More specifically, consider stacking the observations into a
higher-order tensor $\mathbf{X}\in\mathbb{R}^{n\times d_{1}\times\cdots\times
d_{p}}$ whose $i$th frontal slice is $\mathscr{X}_{i}$. In the case when
$\mathscr{X}$ is a vector, i.e., $p=1$, $\mathbf{X}$ is a matrix and the
sample PC, $\widehat{\mathscr{U}}_{k}$ as defined above, is its $k$th right
singular vector. It is therefore tempting to do the same and estimate
$\mathscr{U}_{k}$s by seeking the best orthogonal low-rank approximation to
$\mathbf{X}$ directly:
$\min_{\begin{subarray}{c}\mathscr{W}_{1},\ldots,\mathscr{W}_{r}\in\Theta,\mathbf{a}_{1},\ldots,\mathbf{a}_{r}\in\mathbb{R}^{n}\\\
\mathscr{W}_{l}\perp_{c}\mathscr{W}_{k},\mathbf{a}_{l}\perp\mathbf{a}_{k},\forall
l\neq
k\end{subarray}}\left\|\mathbf{X}-\left(\mathbf{a}_{1}\otimes\mathscr{W}_{1}+\cdots+\mathbf{a}_{r}\otimes\mathscr{W}_{r}\right)\right\|_{\rm
F}$ (7)
See, e.g., Harshman and Lundy (1984). This problem, often known as the tensor
SVD problem, has attracted a lot of attention in recent years. See, e.g.,
Richard and Montanari (2014); Hopkins et al. (2015); Liu et al. (2017); Zhang
and Xia (2018); Auddy and Yuan (2020). However, the sample multiway PCs are
generally not the solution to (7). First of all, the difference between the
best orthogonal rank-$r$ and rank-$(r-1)$ approximations to $\mathbf{X}$ is
generally not a rank-one tensor and therefore cannot be associated with a
multiway PC. See, e.g., Hackbusch (2012). To overcome this challenge, one may
consider solving (7) in a greedy fashion, i.e, optimizing (7) over
$\mathscr{W}_{k}$ and $\mathbf{a}_{k}$ only while fixing the other ones. In
general, however, this still results in a different set of PCs because of the
extra orthogonality constraint on $\mathbf{a}_{k}$s imposed by (7). As we
shall see, this subtle distinction between multiway PCA and low-rank
approximations to a data tensor not only means that a treatment different from
that for the tensor SVD is needed for multiway PCA but also leads to different
statistical behavior between the two.
## 3 Rates of Convergence
A natural question one first asks is how well $\mathscr{U}_{k}$ and its
components $\mathbf{u}_{k}^{(q)}$s can be estimated by their sample
counterparts. We shall now turn our attention to this question and study the
rate of convergence for the sample multiway PCs. On the one hand, we provide
further justification for the superiority of multiway PCA to the usual PCA
with stringing, in addition to enhanced interpretability. On the other hand,
our investigation also leads to new insights into the operating
characteristics of sample multiway PCA and its intriguing distinction from the
usual PCA. To fix ideas, we shall consider the so-called spiked covariance
model as a working model for our theoretical development.
Suppose that a random array $\mathscr{X}\in\mathbb{R}^{d_{1}\times\cdots\times
d_{p}}$ follows a linear factor model:
$\mathscr{X}=\sum_{k=1}^{r}\sigma_{k}\theta_{k}\mathscr{U}_{k}+\sigma_{0}\mathscr{E},$
(8)
where $(\theta_{1},\ldots,\theta_{r})^{\top}\sim N(0,I_{r})$ are the random
factor loadings, $\mathscr{U}_{k}$s ($\in\Theta$) are unit length rank-one
principal components such that $\mathscr{U}_{k}\perp_{c}\mathscr{U}_{l}$ for
any $k\neq l$, and $\mathscr{E}$ is a noise tensor with independent $N(0,1)$
entries. It is worth pointing out that our results and arguments can be
extended beyond normality and applied to general subgaussian distributions. We
opt for the normality assumption for ease of presentation. Without loss of
generality, we shall also assume that eigenvalues of the signal are nontrivial
and sorted in non-increasing order, i.e.,
$\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}>0$. Note that we do not
require $\sigma_{k}$s to be distinct. It is not hard to see that the
covariance operator of the aforementioned $\mathscr{X}$ is given by
$\Sigma=\sum_{k=1}^{r}\sigma_{k}^{2}\mathscr{U}_{k}\otimes\mathscr{U}_{k}+\sigma_{0}^{2}\mathscr{I},$
where $\mathscr{I}$ is the identity tensor, i.e., $\mathscr{I}_{j_{1}\ldots
j_{p}j_{1}^{\prime}\ldots j_{p}^{\prime}}=1$ if $j_{q}=j_{q}^{\prime}$ for all
$q=1,\ldots,p$ and $0$ otherwise. The spiked covariance model such as (8) is
widely used as a working model to study PCA in the case of vector
observations, i.e., $p=1$. See, e.g., Johnstone (2001) and Paul (2007).
In this section, we shall establish the rates of convergence of the sample
multiway PCs. To this end, denote by $\angle(\mathbf{w}_{1},\mathbf{w}_{2})$
the angle between two vectors $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ taking
value in $[0,\pi/2]$, and similarly for two arrays $\mathscr{W}_{1}$ and
$\mathscr{W}_{2}$, $\angle(\mathscr{W}_{1},\mathscr{W}_{2})$ denotes the angle
between their vectorizations ${\rm vec}(\mathscr{W}_{1})$ and ${\rm
vec}(\mathscr{W}_{2})$.
It is instructive to begin with the classical setting where the dimensionality
$d_{1},\ldots,d_{p}$ as well as all other parameters, e.g. $\sigma_{0}$,
$\sigma_{k}$s and $r$, are held fixed as the sample size $n$ diverges. Our
first result shows that the sample PC $\widehat{\mathscr{U}}_{k}$ and its
components $\widehat{\mathbf{u}}_{k}^{(q)}$s are root-$n$ consistent in this
regime.
###### Theorem 3.1.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Assume that all parameters are fixed as the sample size
$n$ increases. Let
$\widehat{\mathscr{U}}_{k}=\widehat{\mathbf{u}}_{k}^{(1)}\otimes\cdots\otimes\widehat{\mathbf{u}}_{k}^{(p)}$
be the sample multiway PC as defined by (6). Then there exists a permutation
$\pi$ over $[r]:=\\{1,\ldots,r\\}$ such that
$\max_{1\leq q\leq
p}\sin\angle(\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{u}_{\pi(k)}^{(q)})=O_{p}(n^{-1/2}),$
(9)
for all $k\in[r]$, and hence
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})=O_{p}(n^{-1/2}),\qquad
k=1,\ldots,r$
as $n\to\infty$.
The most notable difference between the above result and those for the usual
PCA (e.g., Anderson, 1984) is that fact that the root-$n$ consistency of the
sample multiway PCs does not require that the eigenvalues
($(\sigma_{k}^{2}+\sigma_{0}^{2})$s or equivalently $\sigma_{k}$s) of the
covariance matrix be simple, i.e., $\sigma_{k}\neq\sigma_{k+1}$. Note that,
without the multiway structural constraint, the usual PCs are only uniquely
defined and hence can possibly be estimated if their corresponding eigenvalues
are simple. As Theorem 3.1 indicates, such a restriction is not necessary for
multiway PCA. For multiway PCA, each sample PC is root-$n$ consistent
regardless of the other eigenvalues. It is also worth noting that, since we do
not require the $\sigma_{k}$s to be distinct, there is no guarantee that
$\widehat{\mathscr{U}}_{k}$ estimates $\mathscr{U}_{k}$. This is not a
deficiency of multiway PCA, but rather a necessity due to the possible
indeterminacy of the $k$th largest eigenvalue. In fact, if
$\sigma_{k+1}<\sigma_{k}<\sigma_{k-1}$, then we can choose $\pi(k)=k$ in
Theorem 3.1. In general, Theorem 3.1 shows that each of the sample PCs is
necessarily a root-$n$ consistent estimate of one of the multiway PCs.
To further understand the operating characteristics and merits of multiway
PCA, we now consider the more general case and further highlight the role of
dimensionality and signal-to-noise ratio. For brevity, in what follows, we
shall assume that $\mathscr{X}$ is “nearly cubic” in that there exist
constants $0<c_{1},c_{2}<\infty$ such that $c_{1}d\leq d_{1},\ldots,d_{p}\leq
c_{2}d$ for some natural number $d$ which may diverge with $n$. General cases
can be treated similarly but incur considerably more cumbersome notation and
tedious derivation.
###### Theorem 3.2.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$.
Let
$\widehat{\mathscr{U}}_{k}=\widehat{\mathbf{u}}_{k}^{(1)}\otimes\cdots\otimes\widehat{\mathbf{u}}_{k}^{(p)}$
be the sample multiway PC as defined by (6). Suppose that
$r\log r\leq c_{0}\min\\{n,d\\}\quad{\rm
and}\quad\left(\frac{\sigma_{0}}{\sigma_{r}}+\frac{\sigma_{0}^{2}}{\sigma_{r}^{2}}\right)\cdot\max\left\\{\sqrt{\frac{d}{n}},{\frac{d}{n}}\right\\}\leq\frac{c_{0}}{\sqrt{r}},$
(10)
for a sufficiently small constant $c_{0}>0$. Then there exist a constant $C>0$
and a permutation $\pi$ over $[r]$ such that
$\max_{1\leq q\leq
p}\sin\angle(\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{u}_{\pi(k)}^{(q)})\leq
C\left(\frac{\sigma_{0}}{\sigma_{\pi(k)}}+\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}\right)\cdot\max\left\\{\sqrt{\frac{d}{n}},{\frac{d}{n}}\right\\},$
(11)
for all $k\in[r]$, and hence
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})\leq
C\left(\frac{\sigma_{0}}{\sigma_{\pi(k)}}+\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}\right)\cdot\max\left\\{\sqrt{\frac{d}{n}},{\frac{d}{n}}\right\\},\qquad
k=1,\ldots,r,$
with probability tending to one as $n$ diverges.
Theorem 3.2 can be viewed as a generalization of Theorem 3.1. Its proof is
rather involved and we shall brief discuss some of the challenges and the main
ideas for resolving them. The proof proceeds by induction over $k$. Special
attention is needed to deal with the case when an eigenvalue is not simple or
the eigengap is small. This creates difficulty in identifying which multiway
PC a sample multiway PC estimates, or equivalently the permutation $\pi$. To
this end, we shall define
$\pi(1)=\operatorname{argmax}_{1\leq l\leq
r}\left\\{\sigma_{l}^{2}\left|\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{1}^{(q)}\rangle\right|\right\\},$
and for $k>1$,
$\pi(k):=\operatorname{argmax}_{l\notin\pi([k-1])}\left\\{\sigma_{l}^{2}\left|\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right|\right\\}.$
To remove the influence of eigengaps altogether, we need to carefully quantify
the impact of estimation error of
$\widehat{\mathscr{U}}_{1},\ldots,\widehat{\mathscr{U}}_{k-1}$ on the $k$th
sample multiway PC. To this end, we shall derive bounds for both
$\max_{1\leq q\leq
p}\sin\angle(\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}),$
and
$\max_{1\leq q\leq
p}\max_{l\notin\pi([k])}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle,$
and leverage the fact that the latter can be much smaller than the former.
When $d=O(n)$, the convergence rate given in Theorem 3.2 is
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})\leq
C\left(\frac{\sigma_{0}}{\sigma_{\pi(k)}}+\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}\right)\cdot\sqrt{\frac{d}{n}};$
and when $d\gg n$, we have
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})\leq
C\left(\frac{\sigma_{0}}{\sigma_{\pi(k)}}+\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}\right)\cdot\frac{d}{n}.$
In particular, $\widehat{\mathscr{U}}_{k}$ is consistent, e.g.,
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})=o_{p}(1),$
whenever $\sigma_{\pi(k)}/\sigma_{0}\gg\max\\{d/n,(d/n)^{1/4}\\}$.
Of particular interest here is the role of dimensionality. The rates of
convergence given by Theorem 3.2 depend on the dimensionality through $d$
rather than the ambient dimension $D:=d_{1}d_{2}\cdots d_{p}$. This is because
multiway PCA restricts PCs to be rank-one tensors and therefore has fewer
parameters. Such dimensionality reduction is especially important for multiway
data. Consider, for example, the case when $\sigma_{0}$ and $\sigma_{k}$s are
fixed, then by virtue of the results from Johnstone and Lu (2009), direct
application of the usual PCA after stringing necessarily leads to an
inconsistent estimate of $\mathscr{U}_{k}$ whenever $D\gg n$. Yet, our result
indicates that multiway PCA is consistent whenever $d\ll n$.
To draw further comparisons with the usual PCA, we now focus on the case when
$d\ll n$ and $r$, $\sigma_{0},\sigma_{1},\ldots,\sigma_{r}$ are fixed. As
shown by Birnbaum et al. (2013), in this regime, the usual PCA (i.e., $p=1$)
satisfies
$\sin\angle(\widehat{\mathscr{U}}_{k},\mathscr{U}_{\pi(k)})\asymp\left(\frac{\sigma_{0}}{\sigma_{\pi(k)}}+\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}\right)\cdot\sqrt{\frac{d}{n}}+{1\over\sqrt{n}}\left(\sum_{k^{\prime}\neq
k}{(\sigma_{0}+\sigma_{\pi(k)})(\sigma_{0}+\sigma_{\pi(k^{\prime})})\over\sigma_{\pi(k)}^{2}-\sigma_{\pi(k^{\prime})}^{2}}\right)$
with probability tending to one. Comparing the above rate with that from
Theorem 3.2, it is clear that the difference between the two lie at the second
term on the right hand side. Its presence for the usual PCA dictates that
there should be no ties among $\sigma_{k}$s. Even if the $\sigma_{k}$s are all
distinct, how well we can estimate a PC crucially depends on the gap between
its corresponding eigenvalue and the other eigenvalues when $p=1$. In
contrast, the bounds given by Theorem 3.2 are determined by $\sigma_{\pi(k)}$
alone and not the eigengap
$\min\\{\sigma_{\pi(k)-1}^{2}-\sigma_{\pi(k)}^{2},\sigma_{\pi(k)}^{2}-\sigma_{\pi(k)+1}^{2}\\}$
as in the usual PCA case.
It is also instructive to compare the convergence rate for the multiway PCA
from Theorem 3.2 with those for tensor SVD. Recall that
$\mathbf{X}=\sum_{k=1}^{r}\sigma_{k}\Theta_{k}\otimes\mathscr{U}_{k}+\sigma_{0}\mathbf{E},$
where $\Theta_{k}=(\theta_{1k},\ldots,\theta_{nk})^{\top}$ is a vector
containing the $n$ realizations of $\theta_{k}$ and $\mathbf{E}$ is a $n\times
d_{1}\times\cdots\times d_{p}$ tensor whose $i$th frontal slice is
$\mathscr{E}_{i}$. In contrast, $\Theta_{k}$s are deterministic in a tensor
SVD model. If $\Theta_{k}$s are orthogonal to each other, then
$\mathscr{U}_{k}$ can be estimated at the rate of
$\sigma_{0}\|\mathbf{E}\|/(\sigma_{k}\|\Theta_{k}\|)$ which is of the order
$(\sigma_{0}/\sigma_{k})\max\\{\sqrt{d/n},1\\}$. This is a direct consequence
of the perturbations bounds from Auddy and Yuan (2020) and a similar bound was
also derived by Richard and Montanari (2014) in the rank-one case, i.e.,
$r=1$. In our case, however, $\Theta_{k}$ and $\Theta_{l}$ are random and in
general not orthogonal to each other. As a result, the rates we obtained are
different in their dependence on the signal-to-noise ratio
$\sigma_{k}/\sigma_{0}$. Similar phenomenon has also been observed for the
usual PCA (see, e.g., Birnbaum et al., 2013).
## 4 Asymptotic Normality and Bias Correction
We now turn to the distributional properties of multiway PCA. This requires us
to further delineate the role of bias in the sample PCs. It is known that the
usual PCA is biased when the dimension ($D$) is large when compared with the
sample size. See, e.g., Koltchinskii and Lounici (2014); Koltchinskii et al.
(2020) and the references therein. The same phenomenon is observed for the
sample multiway PCs and a non-negligible bias arise when the dimension of each
mode ($d$) is large when compared with the sample size. In addition, there is
a more subtle source of bias for the sample multiway PCs due to the ambiguity
in ordering the multiway PCs in the absence of eigengaps. As noted before, the
lack of an eigengap means that the $k$th PC may not necessarily be estimated
by the $k$th sample multiway PC. As a more concrete example, consider the case
when $r=2$ and $\lambda_{1}=\lambda_{2}$. Then $\mathscr{U}_{1}$ can be
estimated by either $\widehat{\mathscr{U}}_{1}$ or
$\widehat{\mathscr{U}}_{2}$, and as Theorem 3.2 shows, the rate of convergence
remains the same in both cases. But the asymptotic distribution may differ
between the two scenarios: $\widehat{\mathscr{U}}_{2}$ is required to be
orthogonal to $\widehat{\mathscr{U}}_{1}$ and estimating $\mathscr{U}_{1}$ by
$\widehat{\mathscr{U}}_{2}$ may incur extra bias.
In this section, we shall introduce ways to correct for both types of bias and
establish the asymptotic normality of the bias-corrected sample PCs. As is
customary in the literature, we shall assume that $r$ and
$\sigma_{1},\ldots,\sigma_{r}$ are fixed for brevity. In light of the results
from the previous section, the sample PCs are consistent if $d\ll n$ in this
setting. We shall therefore focus on this regime in the current section.
### 4.1 When $d=o(\sqrt{n})$
When $d$ is not too large, the bias is solely due to the possibility of
repeated eigenvalues and thus ambiguity of the ordering of PCs. Indeed if
$\sigma_{k}$s are distinct, then there is no need for bias correction when
$d=o(\sqrt{n})$ and all of our results in this subsection will hold for the
sample multiway PCs. But in practice, we may not know or want to assume that
the eigenvalues are simple. Fortunately, we can remove any possible bias
fairly easily by a simple one-step update of the sample PCs. More
specifically, we shall consider estimating $\mathbf{u}_{\pi(k)}^{(q)}$ by
$\tilde{\mathbf{u}}_{k}^{(q)}$, the leading eigenvector of
$\widehat{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)}).$
The additional step frees up the orthogonality constraints imposed on the
$k$th sample multiway PC and therefore allows us to suppress any adverse
influence of $\widehat{\mathscr{U}}_{1},\ldots,\widehat{\mathscr{U}}_{k-1}$.
We now consider the asymptotic distribution of the bias-corrected sample PCs.
We again start with the classical regime when all parameters are fixed as $n$
increases.
###### Theorem 4.1.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Assume that all parameters are fixed as the sample size
$n$ increases. Let
$\tilde{\mathbf{u}}_{1}^{(q)},\ldots,\tilde{\mathbf{u}}_{r}^{(q)}$ be defined
as above. Then there exists a permutation $\pi:[r]\to[r]$ such that
$\displaystyle\sqrt{n}\left[{\rm vec}(\tilde{\mathbf{U}}^{(q)})-{\rm
vec}(\mathbf{U}^{(q)}_{\pi})\right]$
$\displaystyle\overset{d}{\to}N\left(0,{\rm
diag}\left(\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(1)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(1)}^{4}}\right){\cal
P}_{\mathbf{u}_{\pi(1)}^{(q)}}^{\perp},\ldots,\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(r)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(r)}^{4}}\right){\cal
P}_{\mathbf{u}_{\pi(r)}^{(q)}}^{\perp}\right)\right),$
as $n\to\infty$, where
$\tilde{\mathbf{U}}^{(q)}=[\tilde{\mathbf{u}}_{1}^{(q)},\ldots,\tilde{\mathbf{u}}_{r}^{(q)}]$,
$\mathbf{U}_{\pi}^{(q)}=[\mathbf{u}_{\pi(1)}^{(q)},\ldots,\mathbf{u}_{\pi(r)}^{(q)}]$
and ${\cal
P}_{\mathbf{u}_{k}^{(q)}}^{\perp}=I_{d_{q}}-\mathbf{u}_{k}^{(q)}\otimes\mathbf{u}_{k}^{(q)}$.
Theorem 4.1 indicates that
$n\cdot{\rm
var}\left(\tilde{\mathbf{u}}_{k}^{(q)}\right)\to{\sigma_{0}^{2}(\sigma_{\pi(k)}^{2}+\sigma_{0}^{2})\over\sigma_{\pi(k)}^{4}}\left(I-\mathbf{u}_{\pi(k)}^{(q)}\otimes\mathbf{u}_{\pi(k)}^{(q)}\right),$
and
$n\cdot{\rm
cov}\left(\tilde{\mathbf{u}}_{k}^{(q)},\tilde{\mathbf{u}}_{l}^{(q)}\right)\to
0.$
Namely, all estimates of the multiway PCs are asymptotically normal and
independent of each other. Note also that the asymptotic distribution of
$\tilde{\mathbf{u}}_{k}^{(q)}$ does not depend on other eigenvalues or PCs. In
other words, it can be estimated to the same precision as if all other
components $\mathscr{U}_{l}$, $l\neq k$ are known! This is to be contrasted
with the usual PCA where the asymptotic distribution of $\mathbf{u}_{k}^{(q)}$
depends on all other eigenvectors and eigenvalues.
More specifically, it is well known that in vector case, i.e., when $p=1$,
under the additional assumption that $\sigma_{1}^{2},\ldots,\sigma_{r}^{2}$
are distinct, the sample PCs satisfy
$\displaystyle n\cdot{\rm var}\left(\widehat{\mathbf{u}}_{k}^{(1)}\right)$
$\displaystyle\to\sum_{1\leq l\leq r,l\neq
k}{(\sigma_{k}^{2}+\sigma_{0}^{2})(\sigma_{l}^{2}+\sigma_{0}^{2})\over(\sigma_{k}^{2}-\sigma_{l}^{2})^{2}}\mathbf{u}_{l}^{(1)}\otimes\mathbf{u}_{l}^{(1)}+{(\sigma_{k}^{2}+\sigma_{0}^{2})\sigma_{0}^{2}\over\sigma_{k}^{4}}\left(I-\sum_{1\leq
l\leq r}\mathbf{u}_{l}^{(1)}\otimes\mathbf{u}_{l}^{(1)}\right)$
and for any $1\leq l\leq r$ and $l\neq k$,
$n\cdot{\rm
cov}\left(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{l}^{(1)}\right)\to-{(\sigma_{k}^{2}+\sigma_{0}^{2})(\sigma_{l}^{2}+\sigma_{0}^{2})\over(\sigma_{k}^{2}-\sigma_{l}^{2})^{2}}\cdot\mathbf{u}_{k}^{(1)}\otimes\mathbf{u}_{l}^{(1)}.$
See, e.g., Anderson (1984). It is clear that the sample PCs are always
correlated with each other. Moreover, note that
${\sigma_{0}^{2}(\sigma_{k}^{2}+\sigma_{0}^{2})\over\sigma_{k}^{4}}\leq{(\sigma_{k}^{2}+\sigma_{0}^{2})(\sigma_{l}^{2}+\sigma_{0}^{2})\over(\sigma_{k}^{2}-\sigma_{l}^{2})^{2}}.$
and the strict inequality holds for any $k\neq l\leq r$. This suggests that
the estimated multiway PCs have smaller variations than the usual PCs with the
same set of eigenvalues.
We now turn our attention to the more general case when the dimensionality and
other parameters are allowed to diverge with $n$. Because the PCs now may have
different dimensions for different sample sizes, it is more natural to
consider their linear forms, e.g.
$\langle\mathbf{u}^{(q)}_{k},\mathbf{v}\rangle$, for some fixed vector
$\mathbf{v}\in\mathbb{R}^{d_{q}}$. If the dimensions are fixed, Theorem 4.1
immediately suggests that
$\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle$ estimates
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$, and
$\sqrt{n}\left(\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\to_{d}N\left(0,\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)\|{\cal
P}_{\mathbf{u}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\|^{2}\right)$
The following result shows that this continues to hold as long as
$d=o(\sqrt{n})$.
###### Theorem 4.2.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Assume that (10) holds, $d=o(\sqrt{n})$. Then there exists
a permutation $\pi:[r]\to[r]$ such that
$\sqrt{n}\left(\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\to_{d}N\left(0,\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)\|{\cal
P}_{\mathbf{u}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\|^{2}\right)$
as $n\to\infty$.
Theorem 4.2 shows that the same asymptotic behavior of
$\tilde{\mathbf{u}}_{j}^{(q)}$ as in the fixed dimension case can be expected
whenever $d=o(\sqrt{n})$.
### 4.2 When $d=o(n)$
For higher dimension, the simple bias-correction described above is no longer
sufficient and a close inspection reveals that $\tilde{\mathbf{u}}_{j}^{(q)}$
still incurs a non-negligible bias when $d\gg\sqrt{n}$. Thankfully, both types
of bias can be corrected with a sample-splitting approach similar in spirit to
the scheme developed by Koltchinskii and Lounici (2014) for the usual PCA.
Without loss of generality, assume that $n$ is an even number and we randomly
split the $n$ observations into two halves:
$(\mathscr{X}_{1},\ldots,\mathscr{X}_{n/2})$ and
$(\mathscr{X}_{n/2+1},\ldots,\mathscr{X}_{n})$. Denote by
$\widehat{\Sigma}^{[1]}$ and $\widehat{\Sigma}^{[2]}$ the sample covariance
operator based on the two halves of data respectively. Similarly, we shall
write
$\widehat{\mathscr{U}}_{k}^{[h]}=\widehat{\mathbf{u}}_{k}^{(1),[h]}\otimes\cdots\otimes\widehat{\mathbf{u}}_{k}^{(p),[h]}$
the $k$th sample PC based on the $h$ ($=1$ or $2$) halves of the data.
However, as noted before, $\widehat{\mathscr{U}}_{k}^{[1]}$ and
$\widehat{\mathscr{U}}_{k}^{[2]}$ may not estimate the same PC. To this end,
we shall reorder $\widehat{\mathscr{U}}_{k}^{[1]}$s and
$\widehat{\mathscr{U}}_{k}^{[2]}$s with $\widehat{\mathscr{U}}_{k}$s (i.e.,
the estimators derived from the entire dataset) as reference points.
Specifically, without loss of generality, we assume that
$\widehat{\mathscr{U}}_{k}^{[1]}=\operatorname{argmin}_{k\leq l\leq
r}\sin\angle\left(\widehat{\mathscr{U}}_{l}^{[1]},\widehat{\mathscr{U}}_{k}\right).$
The same procedure is applied to relabel $\widehat{\mathscr{U}}_{k}^{[2]}$s.
Note also that the sign of a PC is irrelevant in that $\mathscr{U}_{k}$ and
$-\mathscr{U}_{k}$ represent the same transformation. We shall therefore also
assume hereafter, without loss of generality, that
$\langle\widehat{\mathbf{u}}_{k}^{(q),[1]},\widehat{\mathbf{u}}_{k}^{(q),[2]}\rangle\geq
0$.
Recall that
$\displaystyle\sigma_{k}^{2}\mathbf{u}_{k}^{(q)}{\mathbf{u}_{k}^{(q)}}^{\top}+\sigma_{0}^{2}I_{d_{q}}=\Sigma(\mathbf{u}_{k}^{(1)},\ldots,\mathbf{u}_{k}^{(q-1)},\cdot,\mathbf{u}_{k}^{(q+1)},\ldots,\mathbf{u}_{k}^{(p)},$
$\displaystyle\mathbf{u}_{k}^{(1)},\ldots,\mathbf{u}_{k}^{(q-1)},\cdot,\mathbf{u}_{k}^{(q+1)},\ldots,\mathbf{u}_{k}^{(p)}).$
We shall then update the sample PC using the above identity with $\Sigma$ and
$\mathbf{u}_{k}^{(q)}$s estimated from separate halves. Denote by
$\check{\mathbf{u}}_{k}^{(q),[1]}$ the leading eigenvector of
$\displaystyle\widehat{\Sigma}^{[1]}$
$\displaystyle(\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]},$
$\displaystyle\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]}),$
and similarly $\check{\mathbf{u}}_{k}^{(q),[2]}$ the leading eigenvector of
$\displaystyle\widehat{\Sigma}^{[2]}$
$\displaystyle(\widehat{\mathbf{u}}_{k}^{(1),[1]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[1]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[1]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[1]},$
$\displaystyle\widehat{\mathbf{u}}_{k}^{(1),[1]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[1]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[1]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[1]}).$
To avoid losing efficiency due to sample splitting, we consider a new estimate
$\check{\mathscr{U}}_{k}=\check{\mathbf{u}}_{k}^{(1)}\otimes\cdots\otimes\check{\mathbf{u}}_{k}^{(p)}$
where
$\check{\mathbf{u}}_{k}^{(q)}={\check{\mathbf{u}}_{k}^{(q),[1]}+\check{\mathbf{u}}_{k}^{(q),[2]}\over\left\|\check{\mathbf{u}}_{k}^{(q),[1]}+\check{\mathbf{u}}_{k}^{(q),[2]}\right\|}.$
The following theorem shows that we can construct an unbiased estimate of
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$ by appropriately
rescaling $\langle\check{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle$, as long as
$d=o(n^{2/3})$.
###### Theorem 4.3.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Let
$\check{\mathscr{U}}_{k}=\check{\mathbf{u}}_{k}^{(1)}\otimes\cdots\otimes\check{\mathbf{u}}_{k}^{(p)}$
be the estimated PC as defined above. Assume $r$ and
$\sigma_{1},\dots,\sigma_{r}$ are fixed, and $d=o(n^{2/3})$. Then there exists
a permutation $\pi:[r]\to[r]$ such that
$\sqrt{n}\left((1+b_{k}^{(q)})\langle\check{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\to_{d}N\left(0,\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)\|{\cal
P}_{\mathbf{u}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\|^{2}\right)$
as $n\to\infty$ where
$b_{k}^{(q)}=\sqrt{1+\frac{d_{q}}{n}\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)}-1.$
(12)
It is worth pointing out that when $d=o(n^{1/2})$, the bias correction factor
described by (12) obeys $b_{k}^{(q)}=o(n^{-1/2})$ and therefore can be
neglected. This agrees with our earlier observation and of course also
suggests that sample-splitting is unnecessary if $d\ll n^{1/2}$. When $d\gg
n^{1/2}$, bias correction becomes essential. In particular, Theorem 4.3
suggests that, as long as $d=o(n^{2/3})$, an explicit bias correction factor
can be applied. For higher dimensions, it is unclear if a similar explicit
expression exists for the debiasing factor. Nonetheless, we can derive a
suitable bias correction factor for all $d\ll n$ via additional sample
splitting.
More specifically, we first randomly split the observations into two halves.
The first half of the data is then further split into two equal-sized groups
to compute the sample covariance operators $\widehat{\Sigma}^{[1][1]}$ and
$\widehat{\Sigma}^{[1][2]}$, then we compute
$\widehat{\mathbf{u}}_{k}^{(q),[1][1]}$ and
$\widehat{\mathbf{u}}_{k}^{(q),[1][2]}$ as the leading eigenvectors of
$\displaystyle\widehat{\Sigma}^{[1][1]}$
$\displaystyle(\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]},$
$\displaystyle\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]}),$
$\displaystyle\widehat{\Sigma}^{[1][2]}$
$\displaystyle(\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]},$
$\displaystyle\widehat{\mathbf{u}}_{k}^{(1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(q-1),[2]},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1),[2]},\dots,\widehat{\mathbf{u}}_{k}^{(p),[2]}),$
Similarly, we used the second half of the data to compute
$\widehat{\mathbf{u}}_{k}^{(q),[2][1]}$s, and
$\widehat{\mathbf{u}}_{k}^{(q),[2][2]}$s. As before, we shall sort these
estimates in compatible order and sign. Let
$\widehat{b}_{k}^{(q)}=\frac{\left\|\check{\mathbf{u}}_{k}^{(q),[1]}+\check{\mathbf{u}}_{k}^{(q),[2]}\right\|}{\sqrt{\left\langle\widehat{\mathbf{u}}_{k}^{(q),[1][1]},\widehat{\mathbf{u}}_{k}^{(q),[1][2]}\right\rangle}+\sqrt{\left\langle\widehat{\mathbf{u}}_{k}^{(q),[2][1]},\widehat{\mathbf{u}}_{k}^{(q),[2][2]}\right\rangle}}-1.$
(13)
###### Theorem 4.4.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Let
$\check{\mathscr{U}}_{k}=\check{\mathbf{u}}_{k}^{(1)}\otimes\cdots\otimes\check{\mathbf{u}}_{k}^{(p)}$
be the estimated PC as defined above. Assume $r$ and
$\sigma_{1},\dots,\sigma_{r}$ are fixed, and $d=o(n)$. Then there exists a
permutation $\pi:[r]\to[r]$ such that
$\sqrt{n}\left((1+\widehat{b}_{k}^{(q)})\langle\check{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\to_{d}N\left(0,\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)\|{\cal
P}_{\mathbf{u}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\|^{2}\right),$
as $n\to\infty$ where $\widehat{b}_{k}^{(q)}$ is given by (13). Moreover,
$\displaystyle\widehat{b}_{k}^{(q)}=\sqrt{1+\frac{d_{q}}{n}\left(\frac{\sigma_{0}^{2}}{\sigma_{\pi(k)}^{2}}+\frac{\sigma_{0}^{4}}{\sigma_{\pi(k)}^{4}}\right)}-1+O_{p}\left(\frac{d^{3/2}}{n^{3/2}}\right)+o_{p}\left(\frac{1}{\sqrt{n}}\right).$
In light of Theorem 4.4, the double sample splitting approach can be employed
to derive confidence intervals for linear forms of the multiway PCs as long as
$d=o(n)$. This robustness, however, comes at the expense of increased
computational cost and could incur a loss of efficiency in finite samples. In
practice, one may still prefer the explicit bias correction as described by
Theorem 4.3 if $d$ is not very large, or the one-step update if $d$ is small.
### 4.3 Inference about multiway PCs
The asymptotic normality we showed earlier in the section forms the basis for
making inferences about linear forms
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$. In particular, one of
the most interesting and also simplest examples of linear forms of PCs is
their coordinates, i.e., $\mathbf{v}$ is a column vector of the identity
matrix. To derive confidence intervals of or testing hypotheses about
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$, however, we need to also
estimate its variance. Specifically, its asymptotic distribution depends only
on $\sigma_{0}$, $\sigma_{\pi(k)}$, and $\mathbf{u}_{k}^{(q)}$, all of which
can be consistently estimated by their sample counterpart. Let
$\widehat{\sigma}_{0}^{2}={1\over\prod_{q=1}^{p}(d_{q}-r)}\sum_{1\leq
i_{q}\leq d_{q},1\leq q\leq p}[\check{\Sigma}_{r+1}]_{i_{1}\cdots
i_{p}i_{1}\cdots i_{p}}$
and
$\widehat{\sigma}_{\pi(k)}^{2}=\langle\widehat{\Sigma},\widehat{\mathscr{U}}_{k}\otimes\widehat{\mathscr{U}}_{k}\rangle-\widehat{\sigma}_{0}^{2}.$
The following theorem suggest that the asymptotic normality remains valid if
we replace the variance of linear forms
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$ with these estimates:
###### Theorem 4.5.
Let $\mathscr{X}_{1},\ldots,\mathscr{X}_{n}$ be independent observations
following the spiked covariance model (8) with $p>1$ such that
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\cdots\otimes\mathbf{u}_{k}^{(p)}$
and $\sigma_{k}>0$. Assume $r$ and $\sigma_{1},\dots,\sigma_{r}$ are fixed.
There exists a permutation $\pi:[r]\to[r]$ such that
* (a)
If $d=o(\sqrt{n})$, then
${\sqrt{n}\left(\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\over\sqrt{\frac{\widehat{\sigma}_{0}^{2}}{\widehat{\sigma}_{\pi(k)}^{2}}+\frac{\widehat{\sigma}_{0}^{4}}{\widehat{\sigma}_{\pi(k)}^{4}}}\
\big{\|}{\cal
P}_{\tilde{\mathbf{u}}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\big{\|}}\to_{d}N\left(0,1\right),$
* (b)
If $d=o(n^{2/3})$, then
${\sqrt{n}\left((1+b_{k}^{(q)})\langle\check{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\over\sqrt{\frac{\widehat{\sigma}_{0}^{2}}{\widehat{\sigma}_{\pi(k)}^{2}}+\frac{\widehat{\sigma}_{0}^{4}}{\widehat{\sigma}_{\pi(k)}^{4}}}\
\big{\|}{\cal
P}_{\check{\mathbf{u}}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\big{\|}}\to_{d}N\left(0,1\right),$
where $b_{k}^{(q)}$ is given by (12).
* (c)
If $d=o(n)$, then
${\sqrt{n}\left((1+\widehat{b}_{k}^{(q)})\langle\check{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\over\sqrt{\frac{\widehat{\sigma}_{0}^{2}}{\widehat{\sigma}_{\pi(k)}^{2}}+\frac{\widehat{\sigma}_{0}^{4}}{\widehat{\sigma}_{\pi(k)}^{4}}}\
\big{\|}{\cal
P}_{\check{\mathbf{u}}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\big{\|}}\to_{d}N\left(0,1\right),$
where $\widehat{b}_{k}^{(q)}$ is given by (13).
Theorem 4.5 is an immediate consequence of Slutsky’s Theorem and Theorems
4.2-4.4. It allows us to make inference or construct confidence intervals for
$\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$. Consider, for example,
testing hypothesis that
$H_{0}:\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle=0\qquad vs\qquad
H_{a}:\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\neq 0,$
when $d=o(\sqrt{n})$. We can proceed to reject $H_{0}$ if and only if
$\left|\sqrt{n}\left(\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle-\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle\right)\right|\geq
z_{\alpha/2}\sqrt{\frac{\widehat{\sigma}_{0}^{2}}{\widehat{\sigma}_{\pi(k)}^{2}}+\frac{\widehat{\sigma}_{0}^{4}}{\widehat{\sigma}_{\pi(k)}^{4}}}\big{\|}{\cal
P}_{\tilde{\mathbf{u}}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\big{\|},$
where $z_{\alpha/2}$ is the upper $\alpha/2$ quantile of the standard normal
distribution. Theorem 4.5 guarantees this is a level-$\alpha$ test
asymptotically. Similarly, we can also construct $(1-\alpha)$ confidence
interval for $\langle\mathbf{u}^{(q)}_{\pi(k)},\mathbf{v}\rangle$:
$\left(\langle\tilde{\mathbf{u}}^{(q)}_{k},\mathbf{v}\rangle\pm\frac{z_{\alpha/2}}{\sqrt{n}}\sqrt{\frac{\widehat{\sigma}_{0}^{2}}{\widehat{\sigma}_{\pi(k)}^{2}}+\frac{\widehat{\sigma}_{0}^{4}}{\widehat{\sigma}_{\pi(k)}^{4}}}\big{\|}{\cal
P}_{\tilde{\mathbf{u}}_{\pi(k)}^{(q)}}^{\perp}\mathbf{v}\big{\|}\right).$
In particular, by taking
$\mathbf{v}\in\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{d_{q}}\\}$, we can use the
above formula to derive confidence intervals for the coordinates of
$\mathbf{u}_{\pi(k)}^{(q)}$. Situations with larger $d$ can also be treated
accordingly.
## 5 Numerical Experiments
To complement our theoretical analyses and further demonstrate the merits of
multiway PCA, we conducted several sets of numerical experiments.
### 5.1 Simulation Studies
We first present a set of simulation studies to illustrate the finite-sample
behavior of the sample PCs. These experiments are specifically designed to
assess the role of bias correction, and robustness to deviation from the
normal distribution. Throughout this subsection, unless otherwise noted,
samples were generated according to the spike covariance model (8) with $p=2$,
e.g., each $\mathscr{X}_{i}$ is a matrix. Since the two modes are
exchangeable, we only focus on the first mode $q=1$ for brevity. We also fixed
the number of spikes at $r=2$. In each case, we shall set the singular values
$\sigma_{1}=\sigma_{2}$. In other words, for each of our examples, the usual
PCA (with stringing) will not be able to identify the PCs because of the
multiplicity. As mentioned before, without loss of generality and for the sake
of brevity, we reordered $\check{\mathbf{u}}_{1}^{(1)}$ and
$\check{\mathbf{u}}_{2}^{(1)}$ such that
$\sin\angle(\check{\mathbf{u}}_{1}^{(1)},\mathbf{u}_{1})\leq\sin\angle(\check{\mathbf{u}}_{2}^{(1)},\mathbf{u}_{1})$.
In addition, we replaced $\check{\mathbf{u}}_{k}^{(1)}$ with
$-\check{\mathbf{u}}_{k}^{(1)}$ whenever
$\langle\check{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{k}^{(1)}\rangle<0$. For low-
dimensional setup, $\tilde{\mathbf{u}}_{1}^{(1)}$ and
$\tilde{\mathbf{u}}_{2}^{(1)}$ are treated similarly.
In the first set of experiments, we considered a low-dimensional setup with
$d_{1}=d_{2}=10$, $n=200$, $\sigma_{1}=\sigma_{2}=2$, and the true PCs were
given by
$\displaystyle\mathbf{u}_{1}^{(1)}=(\sqrt{3}/2,1/2,0,\dots,0)^{\top},\quad\mathbf{u}_{1}^{(2)}=(1,0,\dots,0)^{\top},$
$\displaystyle\mathbf{u}_{2}^{(1)}=(-1/2,\sqrt{3}/2,0,\dots,0)^{\top},\quad\mathbf{u}_{2}^{(2)}=(0,1,0,\dots,0)^{\top}.$
(14)
Figure 1(a) reports the histograms of the first two (nonzero) entries of
$\tilde{\mathbf{u}}_{1}^{(1)}$ based on 300 simulation runs. The histograms
are overlaid with the asymptotic distributions derived in Theorem 4.1. The
agreement between the two confirms the accuracy of the asymptotic distribution
when the dimensionality is low.
(a) $d_{1}=d_{2}=10$
(b) $d_{1}=d_{2}=50$
Figure 1: Multiway PCA for data generated from normal distribution.
To demonstrate the need and effectiveness of bias correction, we increased the
dimension to $d_{1}=d_{2}=50$. Correspondingly we set $n=400$ and
$\sigma_{1}=\sigma_{2}=3$. We repeated the experiment another 300 times and as
before, Figure 1(b) reports the histograms of the first two entries of
$\check{\mathbf{u}}_{1}^{(1)}$ along with the asymptotic distribution derived
in Theorem 4.3, plotted in red lines. The dashed black line overlaid with the
histogram of the first entries corresponds to the asymptotic distribution
without bias correction as given by Theorem 4.1. It is clear that in this
setting, debiasing is necessary and the bias correction of Theorem 4.3 indeed
leads to a more precise approximation of the finite sample distribution.
Our next set of simulations aims to explore the robustness of our approach to
deviation from normality. To this end, $\\{\theta_{k},k\in[r]\\}$ and the
entries of $\mathscr{E}$ were simulated independently from ${\rm
Poisson}(1)-1$ (so that they still have mean $0$ and variance $1$). Again we
set $n=400$ and $\sigma_{1}=\sigma_{2}=3$. Figures 2(a) and 2(b) summarize
results based on 300 runs, for dimensions $d_{1}=d_{2}=10$ and
$d_{1}=d_{2}=50$, respectively. We overlay them with the theoretical
asymptotic distributions given by Theorems 4.1 and 4.3. The results are
qualitatively similar to those from the previous setting.
(a) $d_{1}=d_{2}=10$
(b) $d_{1}=d_{2}=50$
Figure 2: Multiway PCA for data generated from Poisson distribution.
### 5.2 World Bank Data
We now consider a real-world data example – the open source global development
data from the World Bank***https://data.worldbank.org/. The world Bank offers
access to annual country-level data of a number of development indicators. In
particular, we shall focus on the following nine most common and important
economic and demographic indicators:
* `GDP`: gross domestic product (GDP) based on purchasing power parity;
* `Import`: import volume index (year 2000=100);
* `Export`: export volume index (year 2000=100);
* `CO2`: total CO2 emissions, in kilo-ton;
* `CPI`: Consumer price index (year 2010 = 100);
* `Life Span`: Life expectancy at birth;
* `Urban Population`: Urban population, percentage of total population;
* `Tourism`: number of international inbound tourists;
* `Birth Rate`: Birth rate, crude (per 1,000 people).
Yearly data for these indicators have been recorded and we focus on data from
Year 2000 through 2018, as considerable data are missing outside this range.
We also discarded countries that have more than 5% of missing data in our
analysis, resulting in a total of 160 countries under consideration.
These indicators are all positive but of vastly different magnitudes. To this
end, a log transformation was first applied. Each log-transformed indicator
was then standardized so that the log-transformed indicator has a mean $0$ and
a mean absolute deviation $1$ for all countries. The use of mean absolute
deviation, instead of variance, for standardization allows more robust
analysis in the presence of outlying observations. Denote by
$\mathbf{X}_{k,t,i}$ the resulting indicator $i$ for country $k$ at time $t$.
There remain a handful of missing values and for convenience, they are
replaced with $0$ in our analysis. The data tensor $\mathbf{X}$ of dimensions
$160\times 19\times 9$. Each frontal slice
$\mathscr{X}_{k}=\mathbf{X}_{k,\cdot,\cdot}$
corresponds to a country and is a $19\times 9$ matrix. Note that its ambient
dimension is $19\times 9=171$ and greater than the number of countries so it
is problematic to apply the usual PCA with stringing. Accounting for the
multiway structure, we can consider the multiway PCs of the form
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\mathbf{u}_{k}^{(2)}\in\mathbb{R}^{19\times
9}.$
These PC carry a clear meaning: each $\mathscr{U}_{k}$ represents a shared
_development pattern_ , where $\mathbf{u}_{k}^{(1)}$ is the corresponding
shared _temporal trend_ , and $\mathbf{u}_{k}^{(2)}$ is the corresponding
_comovement pattern_.
Figure 3 plots the estimated leading PC along both modes, namely
$\check{\mathbf{u}}_{1}^{(1)}$ and $\check{\mathbf{u}}_{k}^{(2)}$, together
with the $95\%$ confidence intervals for each of their coordinates. It is by
far the most significant component, explaining 57.6% of the total variation.
It is also evident from the temporal component that the first PC describes a
roughly constant growth trend. The only year with a decrease is 2008 when the
Global Financial Crisis took place. Correspondingly, except for the entry
corresponding to birth rate, all other entries of
$\check{\mathbf{u}}_{k}^{(2)}$ are positive. This suggests a general economic
development during this period, with the birth rate in decline.
Figure 3: The first PC, general economic development: $\mathbf{u}_{1}^{(1)}$
and $\mathbf{u}_{1}^{(2)}$ plotted with 95% confidence intervals.
Similarly, Figure 4 shows the second multiway PC in the two modes along with
their $95\%$ confidence bands. This PC captures a change of developmental
direction at the year of 2008. In particular, CPI, life span, urban
population, and tourism steadily decreased prior to 2008 but reversed course
after the financial crisis. In contrast, GDP, import, export, CO2 emission and
birth rate followed an opposite pattern. There are many plausible explanations
for this pattern. It is possible that the quantitative easing policies applied
by most major economies since 2008 led to growth in the domestic market, thus
enhancing the life-quality indicators. It is also possible that the growing
inequality after 2008, also caused by quantitative easing among other factors
(see, e.g., Montecino and Epstein, 2015), has in turn caused the increase in
life quality among the upper and the upper middle class. The tourism indicator
is the number of international inbound tourists, which most likely is driven
by the upper middle class and beyond. The continuous increase in life
expectancy in the USA is also reported to be driven primarily by the well-off
(see, e.g., Chetty et al., 2016).
Figure 4: The second PC, life quality: $\mathbf{u}_{2}^{(1)}$ and
$\mathbf{u}_{2}^{(2)}$ plotted with 95% confidence intervals.
Finally, Figure 5 shows the third multiway PC. We begin to see much wider
confidence intervals as the signal becomes weaker. In fact, only the period
around 2008 are significantly different from zero, and likewise, the entries
corresponding to life span, urbanization, and tourism are statistically
insignificant. This indicates that these patterns likely focus on the impact
of the 2008 financial crisis: it caused an immediate economic downturn but
recovered not long after.
Figure 5: The third PC, international trade: $\mathbf{u}_{3}^{(1)}$ and
$\mathbf{u}_{3}^{(2)}$ plotted with 95% confidence intervals.
### 5.3 NYC Bike Rental Data
Another data example we considered is the Citibike trip
data†††https://ride.citibikenyc.com/system-data. In particular, all the
Citibike trips from January 1, 2018 to December 31, 2019, on weekdays (522
days in total) that started in Manhattan and lasted for at least 60 seconds
were used in our analysis. During this period, there are 35 zip codes in
Manhattan with at least one Citibike station. There are a total of 29,515,527
trips and we form a data tensor $\mathbf{Y}$ of dimension $522\times 24\times
35$ where $Y_{kij}$ denotes the number of trips starting during the $i$th hour
of the $k$th day from the $j$th zip code.
The number of counts at different zip codes are of drastically different
magnitudes, and the total counts during the two years also display a clear
seasonal trend. To facilitate our analysis, we standardized the counts from
each zip code $j$ at each day $k$ so that they have mean $0$ and mean absolute
deviation $1$. As in the previous example, a direct application of the usual
PCA can be misleading as the ambient dimension of the daily observation is
$24\times 35=840$ and greater than the number of days. Nonetheless, it is
helpful to consider multiway PCs of the form
$\mathscr{U}_{k}=\mathbf{u}_{k}^{(1)}\otimes\mathbf{u}_{k}^{(2)}\in\mathbb{R}^{24\times
35},$
where $\mathbf{u}_{k}^{(1)}$ captures the time-of-the-day effect of bike
rental, and $\mathbf{u}_{k}^{(2)}$ the location pattern.
Figure 6 plots the first multiway PC. The spatial pattern clearly indicates
that this represents an overall pattern across Manhattan with all 35 entries
of $\mathbf{u}_{k}^{(2)}$ being estimated as positive. The temporal pattern
indicates that bike rental strongly coincides with the rush hours with two
peaks during the morning and afternoon rush hours. The blank area downtown is
zip code 10006, the big blank rectangular is Central Park, the small blank
underneath is zip code 10020, and the blank area to the north of Central Park
has zip codes 10030 and 10031. At the time of the recorded period, no Citibike
station existed in these areas.
Figure 6: The first PC: overall pattern.
The second PC, as shown in Figure 7, reveals differences in rental patterns
across neighborhoods. While the first PC suggests increased rental activities
both in the morning and afternoon rush hours, the second PC captures the
difference between morning and evening rental patterns as indicated by the
positive peak during the evening rush hours and the negative peak during the
morning rush hours. As such, a neighborhood with positive loadings may see
more evening rentals than morning rentals. These are the downtown Financial
District, Lower Manhattan, and Midtown, largely corresponding to the business
area of Manhattan. On the other hand, zip codes corresponding to negative
loadings represent mostly residential areas of Manhattan, including the East
Village, Upper West Side, and Upper East Side.
Figure 7: The second PC: rush hour differences.
Figure 8 depicts the third PC. The temporal pattern has a narrow and tall peak
during the afternoon rush hours suggesting that this PC captures the subtle
spatial difference during this time of the day. In particular, the zip codes
with large positive values (purple color) are the area around Wall Street (the
small purple block in Lower Manhattan), the area around Grand Central
Terminal, and an area in Upper East Side. The negative zip codes in this
pattern include the areas around SoHo, Greenwich Village, and Harlem.
Figure 8: The third PC: afternoon rush hour details.
## 6 Summary
In this paper, we study PCA under the settings that each observation is a
matrix or more generally a multiway array. We investigate how to extract
multiway PCs and study their statistical properties. In addition to the
obvious advantages of increased efficiency and enhanced interpretability, our
analysis provides a number of new insights into the operating characteristics
of multiway PCA and their methodological implications.
First, we show that multiway PCs can be estimated without the eigengap
requirement. Specifically, under a spike covariance model, we establish rates
of convergence for the sample multiway PCs. In particular, they are consistent
whenever the signal-to-noise ratio
$\sigma_{k}/\sigma_{0}\gg\max\\{d/n,(d/n)^{1/4}\\}$ where $d$ is the dimension
of one mode. Perhaps more interestingly, we prove that the sample multiway PCs
are asymptotically independent of each other, at least when the dimension
$d=o(\sqrt{n})$. In higher dimensions, the sample PCs can be biased and the
bias can be corrected via sample-splitting to lead to asymptotically normal
estimates of the multiway PCs, which enables us to construct confidence
intervals or conduct hypothesis testing for linear forms of the PCs.
Our theoretical developments are complemented by numerical experiments, both
simulated and real. In particular, meaningful findings can be inferred when
applying our methods to two real-world datasets, further demonstrating the
merits of our methodology.
## References
* Anandkumar et al. (2014) Animashree Anandkumar, Rong Ge, Daniel Hsu, Sham M Kakade, and Matus Telgarsky. Tensor decompositions for learning latent variable models. _Journal of machine learning research_ , 15:2773–2832, 2014\.
* Anderson (1984) T. W. Anderson. _An Introduction to Multivariate Statistical Analysis_. Wiley, New York, NY, second edition, 1984.
* Auddy and Yuan (2020) Arnab Auddy and Ming Yuan. Perturbation bounds for (nearly) orthogonally decomposable tensors. _arXiv preprint arXiv:2007.09024_ , 2020.
* Bai and Silverstein (2010) Zhidong Bai and Jack W Silverstein. _Spectral analysis of large dimensional random matrices_ , volume 20. Springer, 2010.
* Bai and Yao (2012) Zhidong Bai and Jianfeng Yao. On sample eigenvalues in a generalized spiked population model. _Journal of Multivariate Analysis_ , 106:167–177, 2012\.
* Baik and Silverstein (2006) Jinho Baik and Jack W Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. _Journal of multivariate analysis_ , 97(6):1382–1408, 2006.
* Benaych-Georges and Nadakuditi (2011) Florent Benaych-Georges and Raj Rao Nadakuditi. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. _Advances in Mathematics_ , 227(1):494–521, 2011\.
* Bi et al. (2021) Xuan Bi, Xiwei Tang, Yubai Yuan, Yanqing Zhang, and Annie Qu. Tensors in statistics. _Annual review of statistics and its application_ , 8:345–368, 2021.
* Birnbaum et al. (2013) Aharon Birnbaum, Iain M Johnstone, Boaz Nadler, and Debashis Paul. Minimax bounds for sparse pca with noisy high-dimensional data. _Annals of statistics_ , 41(3):1055, 2013.
* Chen et al. (2020a) Elynn Y Chen, Jianqing Fan, and Ellen Li. Statistical inference for high-dimensional matrix-variate factor model. _arXiv preprint arXiv:2001.01890_ , 2020a.
* Chen et al. (2020b) Elynn Y Chen, Dong Xia, Chencheng Cai, and Jianqing Fan. Semiparametric tensor factor analysis by iteratively projected svd. _arXiv preprint arXiv:2007.02404_ , 2020b.
* Chen et al. (2021) Rong Chen, Dan Yang, and Cun-Hui Zhang. Factor models for high-dimensional tensor time series. _Journal of the American Statistical Association_ , pages 1–23, 2021\.
* Chetty et al. (2016) Raj Chetty, Michael Stepner, Sarah Abraham, Shelby Lin, Benjamin Scuderi, Nicholas Turner, Augustin Bergeron, and David Cutler. The association between income and life expectancy in the united states, 2001-2014. _Jama_ , 315(16):1750–1766, 2016.
* Cichocki et al. (2015) Andrzej Cichocki, Danilo Mandic, Lieven De Lathauwer, Guoxu Zhou, Qibin Zhao, Cesar Caiafa, and Huy Anh Phan. Tensor decompositions for signal processing applications: From two-way to multiway component analysis. _IEEE signal processing magazine_ , 32(2):145–163, 2015.
* De Lathauwer et al. (2000) Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. A multilinear singular value decomposition. _SIAM journal on Matrix Analysis and Applications_ , 21(4):1253–1278, 2000.
* Friedland (2013) Shmuel Friedland. Best rank one approximation of real symmetric tensors can be chosen symmetric. _Frontiers of Mathematics in China_ , 8(1):19–40, 2013.
* Hackbusch (2012) Wolfgang Hackbusch. _Tensor spaces and numerical tensor calculus_ , volume 42. Springer, 2012.
* Han et al. (2022) Rungang Han, Rebecca Willett, and Anru R Zhang. An optimal statistical and computational framework for generalized tensor estimation. _The Annals of Statistics_ , 50(1):1–29, 2022\.
* Han et al. (2020) Yuefeng Han, Rong Chen, Dan Yang, and Cun-Hui Zhang. Tensor factor model estimation by iterative projection. _arXiv preprint arXiv:2006.02611_ , 2020.
* Harshman and Lundy (1984) Richard A Harshman and Margaret E Lundy. The parafac model for three-way factor analysis and multidimensional scaling. _Research methods for multimode data analysis_ , 46:122–215, 1984.
* Hopkins et al. (2015) Samuel B Hopkins, Jonathan Shi, and David Steurer. Tensor principal component analysis via sum-of-square proofs. In _Conference on Learning Theory_ , pages 956–1006, 2015.
* Janzamin et al. (2019) Majid Janzamin, Rong Ge, Jean Kossaifi, Anima Anandkumar, et al. Spectral learning on matrices and tensors. _Foundations and Trends® in Machine Learning_ , 12(5-6):393–536, 2019.
* Johnstone (2001) Iain M Johnstone. On the distribution of the largest eigenvalue in principal components analysis. _Annals of statistics_ , pages 295–327, 2001.
* Johnstone and Lu (2009) Iain M Johnstone and Arthur Yu Lu. On consistency and sparsity for principal components analysis in high dimensions. _Journal of the American Statistical Association_ , 104(486):682–693, 2009.
* Jolliffe (2002) I. Jolliffe. _Principal Component Analysis_. Springer, 2002.
* Jung and Marron (2009) Sungkyu Jung and J Stephen Marron. Pca consistency in high dimension, low sample size context. _The Annals of Statistics_ , 37(6B):4104–4130, 2009.
* Kolda (2001) Tamara G Kolda. Orthogonal tensor decompositions. _SIAM Journal on Matrix Analysis and Applications_ , 23(1):243–255, 2001.
* Koltchinskii and Lounici (2014) Vladimir Koltchinskii and Karim Lounici. Asymptotics and concentration bounds for spectral projectors of sample covariance. _arXiv preprint arXiv:1408.4643_ , 2014.
* Koltchinskii et al. (2017) Vladimir Koltchinskii, Karim Lounici, et al. Concentration inequalities and moment bounds for sample covariance operators. _Bernoulli_ , 23(1):110–133, 2017.
* Koltchinskii et al. (2020) Vladimir Koltchinskii, Matthias Löffler, and Richard Nickl. Efficient estimation of linear functionals of principal components. _The Annals of Statistics_ , 48(1):464–490, 2020\.
* Kong et al. (2005) Hui Kong, Lei Wang, Eam Khwang Teoh, Xuchun Li, Jian-Gang Wang, and Ronda Venkateswarlu. Generalized 2d principal component analysis for face image representation and recognition. _Neural Networks_ , 18(5-6):585–594, 2005.
* Kroonenberg (2008) Pieter M Kroonenberg. _Applied multiway data analysis_. John Wiley & Sons, 2008.
* Kroonenberg and De Leeuw (1980) Pieter M Kroonenberg and Jan De Leeuw. Principal component analysis of three-mode data by means of alternating least squares algorithms. _Psychometrika_ , 45(1):69–97, 1980.
* Lee et al. (2010) Seunggeun Lee, Fei Zou, and Fred A Wright. Convergence and prediction of principal component scores in high-dimensional settings. _Annals of statistics_ , 38(6):3605, 2010.
* Li et al. (2010) Xuelong Li, Yanwei Pang, and Yuan Yuan. L1-norm-based 2dpca. _IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics)_ , 40(4):1170–1175, 2010.
* Liu et al. (2017) Tianqi Liu, Ming Yuan, and Hongyu Zhao. Characterizing spatiotemporal transcriptome of human brain via low rank tensor decomposition. _arXiv preprint arXiv:1702.07449_ , 2017.
* Lu et al. (2006) Haiping Lu, Konstantinos N Plataniotis, and Anastasios N Venetsanopoulos. Multilinear principal component analysis of tensor objects for recognition. In _18th International Conference on Pattern Recognition (ICPR’06)_ , volume 2, pages 776–779. IEEE, 2006.
* Lu et al. (2008) Haiping Lu, Konstantinos N Plataniotis, and Anastasios N Venetsanopoulos. Mpca: Multilinear principal component analysis of tensor objects. _IEEE transactions on Neural Networks_ , 19(1):18–39, 2008.
* Lu et al. (2011) Haiping Lu, Konstantinos N Plataniotis, and Anastasios N Venetsanopoulos. A survey of multilinear subspace learning for tensor data. _Pattern Recognition_ , 44(7):1540–1551, 2011\.
* Montecino and Epstein (2015) Juan Montecino and Gerald Epstein. Did quantitative easing increase income inequality? _Institute for New Economic Thinking working paper series_ , (28), 2015.
* Nadler (2008) Boaz Nadler. Finite sample approximation results for principal component analysis: A matrix perturbation approach. _The Annals of Statistics_ , 36(6):2791–2817, 2008.
* Paul (2007) Debashis Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. _Statistica Sinica_ , pages 1617–1642, 2007.
* Richard and Montanari (2014) Emile Richard and Andrea Montanari. A statistical model for tensor pca. In _Advances in Neural Information Processing Systems_ , pages 2897–2905, 2014.
* Shen et al. (2013) Dan Shen, Haipeng Shen, Hongtu Zhu, and JS Marron. Surprising asymptotic conical structure in critical sample eigen-directions. _arXiv preprint arXiv:1303.6171_ , 2013.
* Taguchi (2018) Y-H Taguchi. Tensor decomposition-based and principal-component-analysis-based unsupervised feature extraction applied to the gene expression and methylation profiles in the brains of social insects with multiple castes. _BMC bioinformatics_ , 19(4):99, 2018.
* Vasilescu and Terzopoulos (2002) M Alex O Vasilescu and Demetri Terzopoulos. Multilinear analysis of image ensembles: Tensorfaces. In _European conference on computer vision_ , pages 447–460. Springer, 2002.
* Vershynin (2010) Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. _arXiv preprint arXiv:1011.3027_ , 2010.
* Wang and Fan (2017) Weichen Wang and Jianqing Fan. Asymptotics of empirical eigenstructure for high dimensional spiked covariance. _Annals of statistics_ , 45(3):1342, 2017.
* Xia et al. (2020) Dong Xia, Anru R Zhang, and Yuchen Zhou. Inference for low-rank tensors–no need to debias. _arXiv preprint arXiv:2012.14844_ , 2020.
* Yang et al. (2004) Jian Yang, David Zhang, Alejandro F Frangi, and Jing-yu Yang. Two-dimensional pca: a new approach to appearance-based face representation and recognition. _IEEE transactions on pattern analysis and machine intelligence_ , 26(1):131–137, 2004.
* Zhang and Xia (2018) Anru Zhang and Dong Xia. Tensor svd: Statistical and computational limits. _IEEE Transactions on Information Theory_ , 64(11):7311–7338, 2018.
* Zhang and Zhou (2005) Daoqiang Zhang and Zhi-Hua Zhou. (2d) 2pca: Two-directional two-dimensional pca for efficient face representation and recognition. _Neurocomputing_ , 69(1-3):224–231, 2005.
* Zhang and Golub (2001) Tong Zhang and Gene H Golub. Rank-one approximation to high order tensors. _SIAM Journal on Matrix Analysis and Applications_ , 23(2):534–550, 2001.
## Appendix A Notations and Preliminary Bounds
Write $a\vee b:=\max\\{a,b\\}$ and $a\wedge b:=\min\\{a,b\\}$. For a positive
integer $n$, let $[n]:=\\{1,2,\dots,n\\}$. For a vector
$\mathbf{x}\in\mathbb{R}^{d}$, denote $\|\mathbf{x}\|$ to be its
$\ell_{2}$-norm, $\|\mathbf{x}\|_{1}$ to be its $\ell_{1}$-norm, and
$\|\mathbf{x}\|_{\infty}=\max_{i}|x_{i}|$ to be its $\ell_{\infty}$-norm. For
two sequences of real numbers $\\{a_{n}\\}$ and $\\{b_{n}\\}$, write
$a_{n}=O(b_{n})$ if $\exists C,\exists M$, such that $\forall n>M$,
$|a_{n}|\leq C|b_{n}|$. Write $a_{n}=o(b_{n})$ if
$\lim_{n\to\infty}a_{n}/b_{n}=0$. For two sequences of real-valued random
variables $X_{n}$ and $Y_{n}$, write $X_{n}=O_{p}(Y_{n})$ if
$X_{n}=R_{n}Y_{n}$ and $R_{n}$ is uniformly tight. Write $X_{n}=o_{p}(Y_{n})$
if $X_{n}=R_{n}Y_{n}$ and $R_{n}\overset{p}{\to}0$. For linear subspace $U$ of
$\mathbb{R}^{d}$, denote $P_{U}$ and $P_{U}^{\perp}$ to be the orthogonal
projection onto $U$ and its orthogonal complement $U^{\perp}$, respectively.
For a non-zero vector $u\in\mathbb{R}^{d}$, denote $P_{u}:=P_{{\rm
span}\\{u\\}}$ and $P_{u}^{\perp}:=P_{{\rm span}\\{u\\}}^{\perp}$.
For an order-$k$ tensor $\mathscr{T}\in\mathbb{R}^{d_{1}\times
d_{2}\times\dots\times d_{k}}$, define its _tensor operator norm_ as:
$\displaystyle\|\mathscr{T}\|:=\sup_{\mathbf{u}_{j}\in\mathbb{R}^{d_{j}},\|\mathbf{u}_{j}\|=1}\mathscr{T}(\mathbf{u}_{1},\mathbf{u}_{2},\dots,\mathbf{u}_{k}).$
(15)
Specifically, when $k=2$ so that $\mathscr{T}\in\mathbb{R}^{d_{1}\times
d_{2}}$ is a matrix, $\|\mathscr{T}\|$ is the matrix spectral norm of
$\mathscr{T}$. For tensor $\mathscr{T}\in\mathbb{R}^{d_{1}\times
d_{2}\times\dots\times d_{k}}$, write
$\|\mathscr{T}\|_{\max}=\max_{i_{1},\dots,i_{k}}\left|\mathscr{T}_{i_{1},\dots,i_{k}}\right|$
to be its $\ell_{\infty}$-norm.
With a slight abuse of notation, the mode $q$ product of
$\mathscr{T}\in\mathbb{R}^{d_{1}\times d_{2}\times\dots\times d_{k}}$ with a
vector $\mathbf{a}\in\mathbb{R}^{d_{q}}$, denoted by
$\mathscr{T}\times_{q}\mathbf{a}$, is defined as an order-$(k-1)$ tensor of
size $d_{1}\times\dots d_{q-1}\times d_{q+1}\dots\times d_{k}$, with elements
$[\mathscr{T}\times_{q}\mathbf{a}]_{i_{1}\dots i_{q-1}i_{q+1}\dots
i_{k}}=\sum_{i_{q}=1}^{d_{q}}\mathscr{T}_{i_{1}\dots i_{q}\dots
i_{k}}\mathbf{a}_{i_{q}}.$
Write
$\widehat{\Sigma}_{\theta}={1\over
n}\sum_{i=1}^{n}\theta_{i}\otimes\theta_{i},\qquad\widehat{\Sigma}_{\mathscr{E}}={1\over
n}\sum_{i=1}^{n}\mathscr{E}_{i}\otimes\mathscr{E}_{i},$
and
$\widehat{\Sigma}_{\theta,\mathscr{E}}={1\over
n}\sum_{i=1}^{n}\theta_{i}\otimes\mathscr{E}_{i},$
the sample covariance matrices of $\theta$, $\mathscr{E}$ and between them
respectively. Correspondingly denote by $\Sigma_{\theta}$,
$\Sigma_{\mathscr{E}}$ and $\Sigma_{\theta,\mathscr{E}}$ their population
counterpart. It is clear $\Sigma_{\theta,\mathscr{E}}=0$. Recall also that
$\widehat{\Sigma}={1\over
n}\sum_{i=1}^{n}\mathscr{X}_{i}\otimes\mathscr{X}_{i}.$
and
$\Sigma=\sum_{l=1}^{r}\sigma_{l}\mathbf{u}_{l}^{(1)}\otimes\cdots\otimes\mathbf{u}_{l}^{(p)}\otimes\mathbf{u}_{l}^{(1)}\otimes\cdots\otimes\mathbf{u}_{l}^{(p)}+\sigma_{0}^{2}\mathscr{I}$
are the sample and population covariance matrices of $\mathscr{X}$.
The proof relies on the following technical lemmas.
###### Lemma 1.
There exists a numerical constant $C>0$ such that for any $t\geq 1$,
$\|\widehat{\Sigma}-\Sigma\|\leq
C(\sigma_{1}^{2}+\sigma_{0}^{2})\max\left\\{\sqrt{d\over n},{d\over
n},\sqrt{t\over n},{t\over n}\right\\},$
$\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\leq
C\sigma_{0}\max\left\\{\sqrt{d\over n},{d\over n},\sqrt{t\over n},{t\over
n}\right\\},$ $\|\widehat{\Sigma}_{\mathscr{E}}-\Sigma_{\mathscr{E}}\|\leq
C\sigma_{0}^{2}\max\left\\{\sqrt{d\over n},{d\over n},\sqrt{t\over n},{t\over
n}\right\\},$
and
$\|\widehat{\Sigma}_{\theta}-\Sigma_{\theta}\|\leq C\max\left\\{\sqrt{r\over
n},{r\over n},\sqrt{t\over n},{t\over n}\right\\},$
with probability at least $1-e^{-t}$.
Note that we shall use $C$ to denote a constant that may take different values
at each appearance. We shall also make use the following bounds:
###### Lemma 2.
There exists a numerical constant $C>0$ such that for any $t>0$,
$\|\widehat{\Sigma}_{\theta}-\Sigma_{\theta}\|_{\max}\leq
C\max\left\\{\sqrt{\log r\over n},{\log r\over n},\sqrt{t\over n},{t\over
n}\right\\},$ $\max_{1\leq l_{1},l_{2}\leq
r}\left|\widehat{\Sigma}_{\mathscr{E},\theta}(\mathbf{u}_{l_{1}}^{(1)},\mathbf{u}_{l_{2}}^{(2)},\ldots,\mathbf{u}_{l_{2}}^{(p)},{\bf
e}_{l_{2}})\right|\leq C\sigma_{0}\max\left\\{\sqrt{\log r\over n},{\log
r\over n},\sqrt{t\over n},{t\over n}\right\\}$
where ${\bf e}_{l_{2}}$ is the $l_{2}$th canonical basis of
${\mathbb{R}}^{r}$, and
$\max_{1\leq l\leq
r}\left|(\widehat{\Sigma}_{\mathscr{E}}-\Sigma_{\mathscr{E}})(\mathbf{u}_{l}^{(1)},\ldots,\mathbf{u}_{l}^{(p)},\mathbf{u}_{l}^{(1)},\ldots,\mathbf{u}_{l}^{(p)})\right|\leq
C\sigma_{0}^{2}\max\left\\{\sqrt{\log r\over n},{\log r\over n},\sqrt{t\over
n},{t\over n}\right\\},$
with probability at least $1-e^{-t}$.
Both Lemmas are well known and follow immediately from an application of union
bounds and $\chi^{2}$ tail bounds. See, e.g., Vershynin (2010).
## Appendix B Proof of Theorems 3.1 and 3.2
Theorem 3.1 follows immediately from Theorem 3.2 and it suffices to prove the
latter. For brevity, we shall focus on the case when $d\leq n$ and $r$
diverges with $n$. Denote by ${\cal E}$ the event that
$\|\widehat{\Sigma}_{\theta}-\Sigma_{\theta}\|\leq C\sqrt{r\over n},\quad{\rm
and}\quad\sigma_{0}^{-2}\|\widehat{\Sigma}_{\mathscr{E}}-\Sigma_{\mathscr{E}}\|,\sigma_{0}^{-1}\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\leq
C\sqrt{d\over n}$
and
$\|\widehat{\Sigma}_{\theta}-\Sigma_{\theta}\|_{\max},\
\sigma_{0}^{-1}\max_{1\leq l_{1},l_{2}\leq
r}\left|\widehat{\Sigma}_{\mathscr{E},\theta}(\mathbf{u}_{l_{1}}^{(1)},\mathbf{u}_{l_{2}}^{(2)},\ldots,\mathbf{u}_{l_{2}}^{(p)},{\bf
e}_{l_{2}})\right|\leq C\sqrt{\log r\over n}$
By Lemmas 1 and 2, ${\cal E}$ holds with probability tending to one. It
suffices to proceed conditional on the event ${\cal E}$.
As noted, the $k$th sample PCs may not correspond to the $k$th population PCs
because we do not assume the existence of eigengap and $\sigma_{k}$s may not
even be distinct. Nonetheless, we can match the sample PCs with population PCs
as follows. Define
$\pi(1)=\operatorname{argmax}_{1\leq l\leq
r}\left\\{\sigma_{l}^{2}\left|\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{1}^{(q)}\rangle\right|\right\\},$
and for $k>1$,
$\pi(k):=\operatorname{argmax}_{l\notin\pi([k-1])}\left\\{\sigma_{l}^{2}\left|\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right|\right\\}.$
The goal is to show that with high probability,
$\eta_{k}:=\max_{1\leq q\leq
p}\sin\angle(\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)})\leq
C\left({\sigma_{0}\over\sigma_{\pi(k)}}+{\sigma_{0}^{2}\over\sigma_{\pi(k)}^{2}}\right)\max\left\\{\sqrt{d\over
n},{d\over n}\right\\}=:\delta_{k},$ (16)
for $k=1,\ldots,r$. Our proof proceeds by induction over $k$. To facilitate
the induction, we shall also prove that
$\displaystyle\tilde{\eta}_{k}$ $\displaystyle:=$ $\displaystyle\max_{1\leq
q\leq
p}\max_{l\notin\pi([k])}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle$
(17) $\displaystyle\leq$ $\displaystyle
C\left({\sigma_{0}\over\sigma_{\pi(k)}}+{\sigma_{0}^{2}\over\sigma_{\pi(k)}^{2}}\right)^{2}\max\left\\{{d\over
n},{d^{2}\over
n^{2}}\right\\}+C\left({\sigma_{0}\over\sigma_{\pi(k)}}+{\sigma_{0}^{2}\over\sigma_{\pi(k)}^{2}}\right)\sqrt{\log
r\over n}$ $\displaystyle=:$ $\displaystyle\tilde{\delta}_{k}.$
In addition to (16) and (17), we shall also prove that
$\sigma_{\pi(k)}^{2}\geq\max_{l\notin\pi([k])}\sigma_{l}^{2}(1-C\delta_{k}^{2}).$
(18)
This immediately implies that
$\sum_{l=1}^{k}\tilde{\delta}_{k}^{2}\leq C\delta_{k}^{2}\quad{\rm
and}\quad\max_{1\leq l\leq k}\\{\sigma_{\pi(l)}\delta_{l}\\}\leq
C\sigma_{\pi(k)}\delta_{k},$
by the taking the constant $c_{0}$ in (10) small enough. We shall make use of
these bounds repeatedly.
As noted, we shall proceed by induction over $k$. In particular, we shall
denote by $\delta_{0}=\tilde{\delta}_{0}=0$ so that the the base case holds
trivially when $k=0$. Now assume the induction hypotheses (16) and (17) holds
for $1,\ldots,k-1$. We want to show that they continue to hold for $k$. The
general architect of the argument is similar to that for the base case, but
additional challenges arise with the need to control the impact of estimation
error of $\widehat{\mathbf{u}}_{l}^{(q)}$s for $1\leq l<k$.
Denote by $\widehat{{\cal P}}^{(q)}$ the projection matrix onto the linear
space spanned by
$\\{\widehat{\mathbf{u}}_{1}^{(q)},\dots,\widehat{\mathbf{u}}_{k-1}^{(q)}\\}$,
for $q\in[p]$. Note that in the case when $k=1$, $\widehat{{\cal
P}}^{(q)}={\bf 0}_{d\times d}$. Then
$\displaystyle(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})$
$\displaystyle=$
$\displaystyle\operatorname{argmax}_{\begin{subarray}{c}\|\mathbf{w}^{(q)}\|\leq
1,\langle\mathbf{w}^{(q)},\widehat{\mathbf{u}}_{l}^{(q)}\rangle=0,\\\ \forall
l\leq
k-1,q\in[p]\end{subarray}}\widehat{\Sigma}(\mathbf{w}^{(1)},\dots,\mathbf{w}^{(p)},\mathbf{w}^{(1)},\dots,\mathbf{w}^{(p)})$
$\displaystyle=$
$\displaystyle\operatorname{argmax}_{\begin{subarray}{c}\|\mathbf{w}^{(q)}\|=1,\forall
q\in[p]\end{subarray}}\widehat{\Sigma}(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)})$
where $\widehat{{\cal P}}^{(q)}_{\perp}=I-\widehat{{\cal P}}^{(q)}$. Observe
that
$\widehat{{\cal
P}}_{\perp}^{(q)}\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\cdot,\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\propto\widehat{\mathbf{u}}_{k}^{(q)},$
where $\tilde{\Sigma}=\widehat{\Sigma}-\sigma_{0}^{2}\mathscr{I}$. This
implies that
$\langle\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{w}\rangle={\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\over\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})}.$
In particular, for $l\neq k$,
$\sin\angle(\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{u}_{l}^{(q)})={\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\over\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})},$
and
$\sin\angle(\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{u}_{k}^{(q)})={\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(q-1)},\widehat{{\cal
P}}_{\perp}^{(q)}(\widehat{\mathbf{u}}_{k}^{(q)}-\langle\widehat{\mathbf{u}}_{k}^{(q)},\mathbf{u}_{\pi(k)}^{(q)}\rangle\mathbf{u}_{\pi(k)}^{(q)}),\widehat{\mathbf{u}}_{k}^{(q+1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\over\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})},$
We shall derive lower bounds for the nominators and an upper bound for the
denominator. It suffices to consider the case $q=1$. Other indices can be
treated in an identical fashion.
### B.1 Lower Bound for
$\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})$
Denote by $\tilde{\Sigma}=\Sigma-\sigma_{0}^{2}\mathscr{I}$. Observe that
$\displaystyle\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})$
(19) $\displaystyle\geq$
$\displaystyle\max_{l\notin\pi([k-1])}\tilde{\Sigma}(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)})$ $\displaystyle\geq$
$\displaystyle\max_{l\notin\pi([k-1])}(\Sigma-\sigma_{0}^{2}\mathscr{I})\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)})$
$\displaystyle-\sup_{\|\mathbf{w}^{(q)}\|\leq 1,1\leq q\leq
p}\left|(\widehat{\Sigma}-\Sigma)(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})\right|.$
Next we bound the two terms on the rightmost hand side.
Starting with the first term, note that for any $l\notin\pi([k-1])$,
$\displaystyle(\Sigma-\sigma_{0}^{2}\mathscr{I})(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)})$ $\displaystyle=$
$\displaystyle\sum_{1\leq l^{\prime}\leq
r}\left[\sigma_{l^{\prime}}^{2}\prod_{q=1}^{p}\langle\widehat{{\cal
P}}^{(q)}_{\perp}\mathbf{u}_{l}^{(q)},\mathbf{u}_{l^{\prime}}^{(q)}\rangle^{2}\right]$
$\displaystyle\geq$
$\displaystyle\sigma_{l}^{2}\prod_{q=1}^{p}\langle\widehat{{\cal
P}}^{(q)}_{\perp}\mathbf{u}_{l}^{(q)},\mathbf{u}_{l}^{(q)}\rangle^{2}$
$\displaystyle=$ $\displaystyle\sigma_{l}^{2}\prod_{q=1}^{p}\|\widehat{{\cal
P}}^{(q)}_{\perp}\mathbf{u}_{l}^{(q)}\|^{4}.$
By the induction hypothesis (17),
$\|\widehat{{\cal P}}^{(q)}\mathbf{u}_{l}^{(q)}\|^{2}=\sum_{1\leq
l^{\prime}<k}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{l^{\prime}}^{(q)}\rangle^{2}\leq\sum_{1\leq
l^{\prime}<k}\tilde{\delta}^{2}_{l^{\prime}}\leq C\delta_{k}^{2}.$
By taking the constant $c_{0}$ of (10) small enough, we can ensure that
$\max_{l\notin\pi([k-1])}(\Sigma-\sigma_{0}^{2}\mathscr{I})(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{u}_{l}^{(1)},\ldots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{u}_{l}^{(p)})\geq(1-C\delta_{k}^{4})\tau^{2}.$ (20)
where
$\tau^{2}=\max_{l\notin\pi([k-1])}\sigma_{l}^{2}.$
Next we derive a bound for
$\sup_{\|\mathbf{w}^{(q)}\|\leq 1,1\leq q\leq
p}(\Sigma-\widehat{\Sigma})(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)}).$
Note that
$\displaystyle(\Sigma-\widehat{\Sigma})(\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)},\widehat{{\cal
P}}^{(1)}_{\perp}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}^{(p)}_{\perp}\mathbf{w}^{(p)})$ (21) $\displaystyle=$
$\displaystyle\sum_{l_{1}=1}^{r}\sum_{l_{2}=1}^{r}\sigma_{l_{1}}\sigma_{l_{2}}\left(\widehat{\Sigma}_{\theta,l_{1}l_{2}}-\Sigma_{\theta,l_{1}l_{2}}\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)$
$\displaystyle+\frac{2}{n}\sum_{i=1}^{n}\sum_{l=1}^{r}\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})$
$\displaystyle+\left(\frac{1}{n}\sum_{i=1}^{n}[\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})]^{2}-\sigma_{0}^{2}\prod_{q=1}^{p}\|\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\|^{2}\right).$
We bound the three terms on the right hand side separately.
The first term can be bounded by
$\displaystyle\biggl{|}\sum_{l_{1}=1}^{r}\sum_{l_{2}=1}^{r}\sigma_{l_{1}}\sigma_{l_{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il_{1}}\theta_{il_{2}}\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)$
$\displaystyle\qquad-\sum_{l=1}^{r}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}\biggr{|}$
$\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta}-I_{r}\|\left[\sum_{l=1}^{r}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}\right]$ $\displaystyle=$
$\displaystyle\|\widehat{\Sigma}_{\theta}-I_{r}\|\left[\sum_{l\in\pi([k-1])}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}+\sum_{l\notin\pi([k-1])}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}\right],$
Recall that for any $l\in\pi([k-1])$,
$|\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle|=|\langle\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{u}_{l}^{(q)},\mathbf{w}^{(q)}\rangle|\leq\|\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{u}_{l}^{(q)}\|\leq\delta_{l}.$
Therefore,
$\displaystyle\sum_{l\in\pi([k-1])}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}$ $\displaystyle\leq$
$\displaystyle\max_{l\in\pi([k-1])}\sigma_{l}^{2}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}$ $\displaystyle\leq$
$\displaystyle\max_{1\leq
l<k}\\{\sigma_{\pi(l)}^{2}\delta_{l}^{2p-2}\\}\leq\max_{1\leq
l<k}\\{\sigma_{\pi(l)}^{2}\delta_{l}^{2}\\}\leq\tau^{2},$
by taking $c_{0}$ of (10) small enough. On the other hand,
$\sum_{l\notin\pi([k-1])}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}\leq\max_{l\notin\pi([k-1])}\sigma_{l}^{2}=\tau^{2}.$
This implies that
$\left|\sum_{l_{1}=1}^{r}\sum_{l_{2}=1}^{r}\sigma_{l_{1}}\sigma_{l_{2}}\left(\widehat{\Sigma}_{\theta,l_{1}l_{2}}-\Sigma_{\theta,l_{1}l_{2}}\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\right|\leq
C\tau^{2}\sqrt{r\over n}.$ (22)
Similarly, the second term can be bounded by
$\displaystyle\left|\frac{1}{n}\sum_{i=1}^{n}\sum_{l=1}^{r}\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})\right|$ (23) $\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\left[\sum_{l=1}^{r}\sigma_{l}^{2}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\rangle\right)^{2}\right]^{1/2}$
$\displaystyle\leq$ $\displaystyle C\tau\sigma_{0}\sqrt{d\over n}.$
Finally, the third term can be bounded by
$\left|\frac{1}{n}\sum_{i=1}^{n}[\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})]^{2}-\sigma_{0}^{2}\prod_{q=1}^{p}\|\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{w}^{(q)}\|^{2}\right|\leq\|\widehat{\Sigma}_{\mathscr{E}}-\Sigma_{\mathscr{E}}\|\leq
C\sigma_{0}^{2}\sqrt{d\over n}.$ (24)
Combing (21)-(24), we get
$\displaystyle\sup_{\|\mathbf{w}^{(q)}\|\leq 1,1\leq q\leq
p}\left|(\widehat{\Sigma}-\Sigma)(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}^{(1)},\dots,\widehat{{\cal
P}}_{\perp}^{(p)}\mathbf{w}^{(p)})\right|$ $\displaystyle\leq$ $\displaystyle
C\tau^{2}\sqrt{r\over n}+C(\sigma_{0}\tau+\sigma_{0}^{2})\sqrt{d\over n}.$
Together with (19), this implies
$\tilde{\Sigma}(\widehat{\mathbf{u}}_{k}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\geq\tau^{2}\left(1-C\delta_{k}^{4}-C\sqrt{r\over
n}\right)-C(\sigma_{0}\tau+\sigma_{0}^{2})\sqrt{d\over
n}-C\sigma_{0}^{2}\delta_{k}^{2},$ (25)
by taking $c_{0}$ of (10) small enough.
### B.2 Upper Bounds for $\tilde{\Sigma}(\widehat{{\cal
P}}_{\perp}^{(1)}(\widehat{\mathbf{u}}_{k}^{(1)}-\langle\widehat{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{\pi(k)}^{(1)}\rangle\mathbf{u}_{\pi(k)}^{(1)}),\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})$
Observe that for any $\mathbf{w}$ orthogonal to $\mathbf{u}_{\pi(k)}^{(1)}$,
we have
$\displaystyle\tilde{\Sigma}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})$
(26) $\displaystyle=$
$\displaystyle\sigma_{\pi(k)}^{2}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}^{2}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle$
$\displaystyle+\sum_{l\neq\pi(k)}\sigma_{\pi(k)}\sigma_{l}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}\theta_{il}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle$
$\displaystyle+\sum_{l\neq\pi(k)}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle$
$\displaystyle+\sum_{l_{1},l_{2}\neq\pi(k)}\sigma_{l_{1}}\sigma_{l_{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il_{1}}\theta_{il_{2}}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l_{1}}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l=1}^{r}\sigma_{l}\theta_{il}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right]$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l=1}^{r}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})-\sigma_{0}^{2}\langle\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(p)}\rangle.$
Again each term on the right hand side needs to be bounded carefully.
The first term on the right hand side of (26) can be bounded by
$\displaystyle\left|\sigma_{\pi(k)}^{2}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}^{2}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right|$
$\displaystyle\leq\tau^{2}\|\widehat{\Sigma}_{\theta}\|_{\max}|\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}\mathbf{w}\rangle|.$
In particular, when
$\mathbf{w}=\widehat{\mathbf{u}}_{k}^{(1)}-\langle\widehat{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{\pi(k)}^{(1)}\rangle\mathbf{u}_{\pi(k)}^{(1)}$,
we have
$\displaystyle|\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle|$ $\displaystyle=$
$\displaystyle\left|\langle\widehat{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{\pi(k)}^{(1)}\rangle(1-\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{\pi(k)}^{(1)}\rangle)\right|$ $\displaystyle\leq$
$\displaystyle 1-\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{\pi(k)}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle\|\widehat{{\cal
P}}^{(1)}\mathbf{u}_{\pi(k)}^{(1)}\|^{2}\leq\sum_{1\leq
l<k}\tilde{\delta}_{l}^{2}\leq C\delta_{k}^{2},$
by taking $c_{0}$ small enough. Thus,
$\left|\sigma_{\pi(k)}^{2}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}^{2}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right|\leq C\tau^{2}\delta_{k}^{2}.$ (27)
The second term on the right hand side of (26) can be bounded as follows:
$\displaystyle\left|\sum_{l\neq\pi(k)}\sigma_{\pi(k)}\sigma_{l}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}\theta_{il}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right|$ $\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\left(\sum_{l\neq\pi(k)}\langle\mathbf{u}_{l}^{(1)},\widehat{\mathbf{u}}_{k}^{(1)}\rangle^{2}\right)^{1/2}\max_{l\neq\pi(k)}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}$
$\displaystyle=$
$\displaystyle\tau\delta_{k}\|\widehat{\Sigma}_{\theta}-I\|\max_{l\neq\pi(k)}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}$
$\displaystyle\leq$
$\displaystyle\tau\delta_{k}\|\widehat{\Sigma}_{\theta}-I\|\max\left\\{\max_{l\in\pi([k-1])}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\},\max_{l\notin\pi([k])}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}\right\\}.$
Note that
$\max_{l\in\pi([k-1])}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}\leq\max_{l\in\pi([k-1])}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\widehat{{\cal
P}}_{\perp}^{(q)}\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}\leq\max_{1\leq
l<k}\\{\sigma_{\pi(l)}\delta_{l}^{p-1}\\}\leq 2\tau\delta_{k}^{p-1},$
and
$\max_{l\notin\pi([k])}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle|\right\\}\leq\tau\delta_{k}^{p-1}.$
We get
$\left|\sum_{l\neq\pi(k)}\sigma_{\pi(k)}\sigma_{l}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}\theta_{il}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right|\leq
C\tau^{2}\delta_{k}^{p}\sqrt{r\over n}.$ (28)
And similarly, when
$\mathbf{w}=\widehat{\mathbf{u}}_{k}^{(1)}-\langle\widehat{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{\pi(k)}^{(1)}\rangle\mathbf{u}_{\pi(k)}^{(1)}$,
we bound the third term on the right hand side of (26) by
$\displaystyle\left|\sum_{l\neq\pi(k)}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right|$ (29) $\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\left(\sum_{l\neq\pi(k)}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle^{2}\right)^{1/2}\max_{l\neq\pi(k)}\left\\{\sigma_{l}\prod_{q=2}^{p}|\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right\\}$
$\displaystyle\leq$ $\displaystyle C\tau^{2}\delta_{k}^{p}\sqrt{r\over n},$
where in the last inequality we used the fact that
$\sum_{l\neq\pi(k)}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle^{2}\leq\|\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\|^{2}\leq\|\mathbf{w}\|^{2}\leq\delta_{k}^{2};$
the fourth term by
$\displaystyle\left|\sum_{l_{1},l_{2}\neq\pi(k)}\sigma_{l_{1}}\sigma_{l_{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il_{1}}\theta_{il_{2}}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l_{1}}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right|$ (30) $\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta}\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\tau^{2}\delta_{k}^{2(p-1)};$
the fifth term by
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l\neq\pi(k)}\sigma_{l}\theta_{il}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\right]$ (31) $\displaystyle=$
$\displaystyle\sum_{l\neq\pi(k)}\left\\{\sigma_{l}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle\left[{1\over
n}\sum_{i=1}^{n}\theta_{il}\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\right]\right\\}$
$\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\cdot\left[\sum_{l\neq\pi(k)}\left(\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)\right]^{1/2}$
$\displaystyle\leq$ $\displaystyle C\sigma_{0}\tau\delta_{k}^{p-1}\sqrt{d\over
n};$
and the sixth term by
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l=1}^{r}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
(32) $\displaystyle\leq$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{\pi(k)}\theta_{i\pi(k)}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l\neq\pi(k)}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
$\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|+\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\sigma_{0}\tau\sqrt{d\over n}.$
Finally the last term can be bounded by
$\left|\frac{1}{n}\sum_{i=1}^{n}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})-\sigma_{0}^{2}\langle\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{w},\widehat{\mathbf{u}}_{k}^{(1)}\rangle\right|\leq\|\widehat{\Sigma}_{\mathscr{E}}-\Sigma_{\mathscr{E}}\|\leq
C\sigma_{0}^{2}\sqrt{d\over n}.$
Together with (27)-(32), we get
$\displaystyle\tilde{\Sigma}(\widehat{{\cal
P}}_{\perp}^{(1)}(\widehat{\mathbf{u}}_{k}^{(1)}-\langle\widehat{\mathbf{u}}_{k}^{(1)},\mathbf{u}_{\pi(k)}^{(1)}\rangle\mathbf{u}_{\pi(k)}^{(1)}),\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})\leq
C\tau^{2}\delta_{k}^{2}+C(\sigma_{0}^{2}+\sigma_{0}\tau)\sqrt{d\over n}.$
### B.3 Upper Bounds for $\max_{l\notin\pi([k])}\tilde{\Sigma}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)})$
To derive the helper bound (17), we also need an upper bound for
$\max_{l\notin\pi([k])}\tilde{\Sigma}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)},\widehat{\mathbf{u}}_{k}^{(1)},\ldots,\widehat{\mathbf{u}}_{k}^{(p)}).$
We shall follow a similar step by bound each term on the right hand side of
(26), but now with $\mathbf{w}=\mathbf{u}_{m}^{(1)}$ ($m\notin\pi([k])$).
Specifically, the first term can be bounded by
$\displaystyle\left|\sigma_{\pi(k)}^{2}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}^{2}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$
$\displaystyle\leq\tau^{2}\|\widehat{\Sigma}_{\theta}\|_{\max}|\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle|.$
Note that
$\langle\mathbf{u}_{\pi(k)}^{(1)},\mathbf{u}_{m}^{(1)}\rangle=\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}^{(1)}\mathbf{u}_{m}^{(1)}\rangle+\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle=0.$
We get
$|\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle|=|\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}^{(1)}\mathbf{u}_{m}^{(1)}\rangle|\leq\|\widehat{{\cal
P}}\mathbf{u}_{\pi(k)}^{(1)}\|\|\widehat{{\cal
P}}\mathbf{u}_{m}^{(1)}\|\leq\sum_{1\leq l<k}\tilde{\delta}_{l}^{2}\leq
C\delta_{k}^{2},$
by Cauchy-Schwartz inequality. This implies that
$\left|\sigma_{\pi(k)}^{2}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}^{2}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|\leq
C\tau^{2}\delta_{k}^{2}.$
by taking $c_{0}$ small enough.
The second term can also be bounded by
$\displaystyle\left|\sum_{l\neq\pi(k)}\sigma_{\pi(k)}\sigma_{l}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{i\pi(k)}\theta_{il}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$ $\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\||\langle\mathbf{u}_{\pi(k)}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle\tau^{2}\delta_{k}^{p+1}\sqrt{r\over n}.$
We bound the third term by
$\displaystyle\left|\sum_{l\neq\pi(k)}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$ $\displaystyle\leq$
$\displaystyle\left|\sigma_{m}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{im}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{m}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{m}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$
$\displaystyle+\left|\sum_{l\notin\\{\pi(k),m\\}}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$
The first term on the right hand side can be further bounded by
$\tau^{2}\|\widehat{\Sigma}_{\theta}-\Sigma_{\theta}\|_{\max}\prod_{q=2}^{p}\left|\langle\mathbf{u}_{m}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right|\leq
C\tau^{2}\delta_{k}^{p-1}\sqrt{\log r\over n}.$
Now consider the second term:
$\displaystyle\left|\sum_{l\notin\\{\pi(k),m\\}}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$ $\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\left(\sum_{l\notin\\{\pi(k),m\\}}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$
$\displaystyle\tau\|\widehat{\Sigma}_{\theta}-I\|\Biggl{(}\sum_{l\in\pi([k-1])}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}$
$\displaystyle\hskip
100.0pt+\sum_{l\notin\pi([k])\cup\\{m\\}}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\Biggr{)}^{1/2}.$
The first term in the bracket on the rightmost hand side can be bounded by
$\displaystyle\sum_{l\in\pi([k-1])}\sigma_{l}^{2}\|\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{l}^{(1)}\|^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{l\in\pi([k-1])}\langle\mathbf{u}_{l}^{(2)},\widehat{\mathbf{u}}_{k}^{(2)}\rangle^{2}\right)\left(\max_{l\in\pi([k-1])}\sigma_{l}^{2}\|\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{l}^{(1)}\|^{2}\prod_{q=3}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)$
$\displaystyle\leq$
$\displaystyle\delta_{k}^{2}\left(\max_{l\in\pi([k-1])}\sigma_{l}^{2}\delta_{l}^{2}\delta_{k}^{2(p-2)}\right)\leq
C\tau^{2}\delta_{k}^{2p};$
the second term by
$\|\widehat{{\cal
P}}^{(1)}\mathbf{u}_{m}^{(1)}\|^{2}\left(\max_{l\notin\pi([k])\cup\\{m\\}}\sigma_{l}^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)\leq
C\tau^{2}\delta_{k}^{2p},$
so that
$\displaystyle\left|\sum_{l\neq\pi(k)}\sigma_{l}\sigma_{\pi(k)}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il}\theta_{i\pi(k)}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$ $\displaystyle\leq$
$\displaystyle C\left(\tau^{2}\delta_{k}^{p-1}\sqrt{\log r\over
n}+\tau^{2}\delta_{k}^{p}\sqrt{r\over n}\right).$
Similar to before, the fourth term on the right hand side of (26) can be
bounded by
$\displaystyle\left|\sum_{l_{1},l_{2}\neq\pi(k)}\sigma_{l_{1}}\sigma_{l_{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\theta_{il_{1}}\theta_{il_{2}}\right)\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l_{1}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l_{2}}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l_{1}}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right|$ $\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta}\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\tau^{2}\delta_{k}^{2(p-1)};$
and the fifth term by
$\displaystyle\left|\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l\neq\pi(k)}\sigma_{l}\theta_{il}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\right]\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{l\neq\pi(k)}\left\\{\sigma_{l}\left(\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle\left[{1\over
n}\sum_{i=1}^{n}\theta_{il}\mathscr{E}_{i}(\widehat{\mathbf{u}}_{k}^{(1)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\right]\right\\}\right|$
$\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\cdot\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\langle\mathbf{u}_{l}^{(1)},\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)}\rangle^{2}\prod_{q=2}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\sigma_{0}\tau\delta_{k}^{p-1}\sqrt{d\over
n}.$
We now turn to the sixth term on the right hand side of (26). Write
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l=1}^{r}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l\neq\pi(k)}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left[\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{\pi(k)}\theta_{i\pi(k)}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right].$
The first term again can be bounded by
$\displaystyle\left|\frac{1}{n}\sum_{i=1}^{n}\left[\sum_{l\neq\pi(k)}\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{l}\theta_{il}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]\right|$
$\displaystyle\leq$
$\displaystyle\|\widehat{\Sigma}_{\theta,\mathscr{E}}\|\left(\sum_{l\neq\pi(k)}\sigma_{l}^{2}\prod_{q=1}^{p}\langle\mathbf{u}_{l}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle^{2}\right)^{1/2}\leq
C\sigma_{0}\tau\delta_{k}^{p-1}\sqrt{d\over n}.$
For the second term, note that
$\displaystyle\left|\frac{1}{n}\sum_{i=1}^{n}\left[\mathscr{E}_{i}(\widehat{{\cal
P}}_{\perp}^{(1)}\mathbf{u}_{m}^{(1)},\widehat{\mathbf{u}}_{k}^{(2)},\dots,\widehat{\mathbf{u}}_{k}^{(p)})\sigma_{\pi(k)}\theta_{i\pi(k)}\left(\prod_{q=1}^{p}\langle\mathbf{u}_{\pi(k)}^{(q)},\widehat{\mathbf{u}}_{k}^{(q)}\rangle\right)\right]\right|$
|
# $k$-reduced groups and Milnor invariants
Benjamin Audoux Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille,
France<EMAIL_ADDRESS>, Jean-Baptiste Meilhan Univ. Grenoble
Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France jean-
<EMAIL_ADDRESS>and Akira Yasuhara Faculty of
Commerce, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050,
Japan<EMAIL_ADDRESS>
###### Abstract.
We characterize, in an algebraic and in a diagrammatic way, Milnor string link
invariants indexed by sequences where any index appears at most $k$ times, for
any fixed $k\geq 1$. The algebraic characterization is given in terms of an
Artin-like action on the so-called $k$–reduced free groups; the diagrammatic
characterization uses the langage of welded knot theory. The link case is also
addressed.
## Introduction
In his seminal works [20, 21], J. Milnor introduced a family of concordance
invariants for $n$-component links, which can be seen as a wide generalization
of the linking number. Indexed by sequences $I$ of possibly repeating indices
in $\\{1,\ldots,n\\}$, these _Milnor invariants_ $\mu(I)$ are integers
extracted from longitudes within the fundamental group of the link complement,
well-defined only modulo a subtle indeterminacy. The geometric meaning of this
indeterminacy was clarified by the work of N. Habegger and X.S. Lin [12], who
showed that Milnor invariants are actually well-defined integer-valued
invariants of _string links_ , which are pure tangles without closed
components.
Recall that two (string) links are _link-homotopic_ if they are related by a
sequence of isotopies and crossing changes involving two strands of a same
component. Milnor proved in [21] that, if a sequence $I$ is without
repetition, then Milnor invariant $\overline{\mu}(I)$ is in fact a link-
homotopy invariant. He further showed how these non-repeated invariants can be
extracted from the _reduced_ fundamental group of the link complement, which
is is the ‘maximal’ quotient where each meridian commutes with any of its
conjugates. Using an Artin-like action of string links on the reduced free
group, Habegger and Lin actually showed that two string links have same Milnor
invariants indexed by sequences without repetition, if and only if they are
link-homotopic [12].
The purpose of this paper is to give a higher-order version of this
classification result of Habegger and Lin. Namely, we characterize the
information contained by Milnor invariants $\mu(I)$ of string links with
$r(I)\leq k$, where $r(I)$ denotes the maximum number of time that any index
appears in $I$. (In particular, Milnor invariants $\mu(I)$ with $r(I)=1$ are
precisely Milnor link-homotopy invariants.)
Our main result is the following; explanations of terminologies and notation
shall follow.
###### Main Theorem.
Let $L$ and $L^{\prime}$ be two $n$-component string links. The following are
equivalent.
1. (1)
$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.
2. (2)
$L$ and $L^{\prime}$ induce the same $k$–reduced free action:
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$.
3. (3)
$L$ and $L^{\prime}$ are self $w_{k}$-concordant.
Let us first explain assertion (2). Let $G(L)$ be the fundamental group of the
complement of $L$. This group is normally generated by $n$ meridians, and for
each $i$, we denote by $N_{i}$ the normal subgroup of $G(L)$ generated by the
$i$th meridian. The _$k$ –reduced quotient_ of $G(L)$ is defined by
$\textnormal{R}_{k}G(L):=\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}},$
where $\Gamma_{q}N$ denotes the $q$th term of the lower central series of a
group $N$. This generalizes Milnor’s above-mentioned notion of reduced group
[20], which coincides with $R_{1}G(L)$. In particular, if $F_{n}$ is the free
group on $n$ generators, then $\textnormal{R}_{k}F_{n}$ is called the _$k$
–reduced free group_. Now, given an $n$-component string link $L$, we show in
Section 2.2 that, for each $k\geq 2$, there is an associated _$k$ –reduced
free action_
$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(R_{k}F_{n}),$
where $\operatorname{Aut}_{c}(R_{k}F_{n})$ denotes the set of automorphisms of
$R_{k}F_{n}$ that act by conjugation on each generator. This can be seen as a
generalization of Habegger-Lin’s representation for the group of link-homotopy
classes of string links [12, Thm. 1.7] in the case $k=1$. We stress, however,
that our proof is very different in nature from [12].
Let us now clarify assertion (3). We use the langage of _welded knot theory_ ,
which is a diagrammatic generalization of knot theory. We stress, firstly,
that the set of classical (string) links injects into the larger set
$w\textnormal{S}L(n)$ of welded (string) links and, secondly, that Milnor
invariants extend naturally to welded objects, so that they coincide with the
classical invariants on classical objects.
In [19] a diagrammatic calculus for welded knotted objects was developped,
called _arrow calculus_ , which can be seen as a generalization of the theory
of Gauss diagrams. A $w$-tree for a welded diagram $D$, is an oriented,
unitrivalent tree, whose univalent vertices lie disjointly on $D$. Given such
a $w$-tree $T$ for $D$, there is a ‘surgery’ procedure that yields a new
welded diagram $D_{T}$, which is roughly obtained by inserting an ‘iterated
commutator of crossings’ in a neighborhood of the head of $T$. Define the
degree of a $w$-tree to be half the number of its vertices. The _self
$w_{k}$-equivalence_ is the equivalence relation on welded string links
generated by surgeries on degree $\geq k$ $w$-trees whose univalent vertices
all lie on the _same_ string link component. For $k=1$, this notion turns out
to coincide with the usual link-homotopy relation for string links [1]. On the
other hand, there is a combinatorial equivalence relation of _welded
concordance_ for welded (string) links, which is generated by birth, death and
saddle moves [5, 10] and which naturally encompasses the topological
concordance for classical (string) links. The _self $w_{k}$-concordance_ is
the equivalence relation obtained by combining the above two relations, and
our main result states that this characterizes combinatorially the information
contained in Milnor invariants $\mu(I)$ with $r(I)\leq k$.
In fact, the $k$-reduced free action involved in assertion (2) is more
generally defined for $wSL(n)$, and our main theorem follows from a more
general characterization for welded string links, see Theorems 2.8 and 3.19.
We also show that the map sending a welded string link $L$ to its associated
$k$-reduced free action $\varphi^{(k)}$, descends to a group isomorphism
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}(R_{k}F_{n}),$
for any $k\geq 1$; see Proposition 3.21. The case $k=1$ was proved in [1], and
is a generalization of [12, Thm. 1.7]. This isomorphism suggests that welded
theory provides a sensible diagrammatic counterpart of the algebraic
constructions underlying Milnor invariants. Our main theorem also refines a
recent result of B. Colombari [7], who gave a diagrammatic characterization of
string links having same Milnor invariants of length $\leq q$. We note that
the geometric properties of Milnor link invariants $\overline{\mu}(I)$ with
$r(I)\leq k$ was previously investigated in [9, 27, 26], using clasper theory.
We address the (welded) link case in the final section of this paper. As
developped there, building on the proof of our main theorem for string links,
and using straightforward adaptations of the above mentioned work of Colombari
[7], we obtain in particular the following for classical links.
###### Theorem.
A link has vanishing Milnor invariants $\overline{\mu}(I)$ with $r(I)\leq k$,
if and only if it is self $w_{k}$-equivalent to the trivial link.
This follows from a more general result (Theorem 4.3) which characterizes the
so-called $k$–reduced peripheral system of _welded_ links, that is the
$k$-reduced link group endowed with peripheral elements, see Section 4.
###### Acknowledgments.
The authors thank Jacques Darné for bringing to their knowledge the result [8,
Lem. 2.37]. The first author is partially supported by the project SyTriQ
(ANR-20-CE40-0004) of the ANR, and thanks the IRL PIMS-Europe for its
hospitality during the period in which part of the work on this paper was
done. The second author is partially supported by the project AlMaRe
(ANR-19-CE40-0001-01) of the ANR. The third author is supported by the JSPS
KAKENHI grant 21K03237, and by the Waseda University grant for Special
Research Projects 2021C-120.
## 1\. Basic algebraic preliminaries
This section reviews algebraic tools that will be used in this paper, together
with basic and well-known results. Several notation used throughout the paper
will also be set in this section.
Let $n$ be a positive integer. We denote by $F_{n}$ the free group on $n$
generators $\alpha_{1},\cdots,\alpha_{n}$. For each $i\in\\{1,\cdots,n\\}$,
denote by $N_{i}$ the normal subgroup of $F_{n}$ generated by $\alpha_{i}$.
### 1.1. Commutators and the lower central series
We use the following convention for commutators and conjugates ($x,y\in
F_{n}$):
$[x,y]:=x\overline{y}\,\overline{x}y\quad\textrm{ and }\quad
x^{y}:=\overline{y}xy,$
where we write $\overline{a}$ for the inverse of an element $a$. This somewhat
nonstandard convention for commutators will be justified by the diagrammatic
counterpart of the theory, reviewed in Section 3.1. We note that with our
convention, for elements $a,b,c$ of $F_{n}$, we have the following basic
properties:
(C$0$) $\overline{[a,b]}=[\overline{b},\overline{a}];$
(C$1$) $[a,b]=\overline{b}^{\overline{a}}b=a\overline{a}^{b};$
(C$2$) $[a,b]^{c}=[a^{c},b^{c}];$
Moreover, recall the following commutator identies (compare with [18, Thm.
5.1]).
(C$3$) $[a,bc]=[a,c][a,b]^{c}\,\,\,\textrm{ and
}\,\,\,[ab,c]=[b,c]^{\overline{a}}[a,c];$ (C$4$)
$\big{[}[a,b],c\big{]}=\big{[}\overline{a},[\overline{c},\overline{b}]\big{]}^{b\overline{a}}\big{[}\overline{b},[\overline{a},\overline{c}]\big{]}^{c\overline{a}}.$
The lower central series of $F_{n}$ is the family of nested subgroups
$\\{\Gamma_{k}F_{n}\\}_{k}$ defined inductively by $\Gamma_{1}F_{n}=F_{n}$ and
$\Gamma_{k+1}F_{n}=[F_{n},\Gamma_{k}F_{n}]$. The following, less standard,
notion will also be useful.
###### Definition 1.1.
A length $k$ _linear commutator_ is a an element of $\Gamma_{k}F_{n}$ of the
form
$\left[x_{1},\left[x_{2},\left[x_{3},\cdots[x_{k-1},x_{k}]\cdots\rule{0.0pt}{8.5359pt}\right]\rule{0.0pt}{11.38092pt}\right]\rule{0.0pt}{14.22636pt}\right]$
for some elements $x_{i}\in F_{n}$.
We shall need the following basic results.
###### Lemma 1.2.
Let $C\in F_{n}$ be a length $l$ commutator, with $k$ entries in $N_{i}$ for
some $i$ ($k\leq l$). Then $C$ is a product of length $\geq l$ commutators
where each entry is an element of
$\\{\alpha_{j},\overline{\alpha}_{j}\\}_{j}$, and with at least $k$ entries
that are either $\alpha_{i}$ or its inverse.
###### Proof.
Combining (C$3$) with (C$0$) and (C$1$), gives the identities
(C$5$)
$[a,bc]=[a,c][a,b]\big{[}[\overline{b},\overline{a}],c\big{]}\,\,\,\textrm{
and }\,\,\,[ab,c]=\big{[}a,[\overline{c},\overline{b}]\big{]}[b,c][a,c].$
By assumption, $k$ entries of $C$ are products of conjugates of $\alpha_{i}$
or its inverse. Using (C$5$) on these $k$ entries, $C$ can be written as a
product of length $\geq l$ commutators, each having $\geq k$ entries that are
a single conjugate of $\alpha_{i}$ or its inverse. Since
$a^{b}=a[\overline{a},b]$ by (C$1$), we have by using (C$5$) that $C$ is
written as a product of length $\geq l$ commutators, each having at least $k$
entries that are either $\alpha_{i}$ or its inverse. Now consider one such
length $\geq l$ commutator $C^{\prime}$. One can apply (C$5$) iteratively on
all entries of $C^{\prime}$, until it is written as a product of iterated
commutators with entries in
$\big{\\{}\alpha_{j};\overline{\alpha}_{j}\big{\\}}_{j}$ and clearly, each of
these commutators necessarily contains at least $k$ entries that are either
$\alpha_{i}$ or its inverse, since $C^{\prime}$ does. ∎
###### Lemma 1.3.
Let $C\in F_{n}$ be a length $l$ commutator, with $k$ entries in $N_{i}$ for
some $i$ ($k\leq l$). Then $C\in\Gamma_{k}N_{i}$. More precisely, $C$ is a
product of length $k$ linear commutators whose entries are all conjugates of
$\alpha_{i}$ or its inverse.
###### Proof.
By assumption, $C$ is a length $l$ commutator with at least $k$ entries that
are products of conjugates of $\alpha_{i}$ or its inverse. Using (C$1$)
repeatedly, one can write $C$ as a length $k$ commutator, whose entries are
all products of conjugates of $\alpha_{i}$ or its inverse. This shows that
$C\in\Gamma_{k}N_{i}$. Now, using recursively (C$3$) and (C$2$), each such
commutator can be expressed as a product of length $k$ commutators, whose
entries are a single conjugate of $\alpha_{i}$ or its inverse. By (C$4$),
combined with (C$2$), a length $k$ commutator can be expressed as a product of
linear ones, and in our context all $k$ entries will remain a single conjugate
of $\alpha_{i}$ or its inverse. ∎
For the next two results, let $G$ be a group which is normally generated by
$n$ elements $\alpha_{1},\ldots,\alpha_{n}$. For each $i\in\\{1,\cdots,n\\}$,
denote by $N_{i}$ the normal subgroup of $G$ generated by the $i$th generator.
###### Lemma 1.4.
For any $k\in{\mathds{N}}$,
$\Gamma_{kn+1}G\subset\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}.$
###### Proof.
Using (C$5$), any element of $\Gamma_{kn+1}G$ can be expressed as a product of
length $\geq kn+1$ commutators with entries in
$\big{\\{}\alpha_{j}^{g},\overline{\alpha}_{j}^{g}\,;\,g\in G\big{\\}}_{j}$.
For any such commutator $C$, there exists some $i$ such that at least $k+1$
entries of $C$ are elements of $N_{i}$, and Lemma 1.3 implies that
$C\in\Gamma_{k+1}N_{i}$. ∎
Define a _conjugating_ endomorphism of $G$ as an endomorphism which sends each
generator to a conjugate of itself. The following is a simple adaptation of
[8, Lem. 2.37] to our setting.
###### Lemma 1.5.
For all $k\geq 2$, any conjugating endomorphism $\varphi$ of $G$ induces an
automorphism of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$. In particular, if $G$ is nilpotent,
then $\varphi$ is itself an automorphism of $G$.
###### Proof.
As a conjugating endomorphism, $\varphi$ induces the identity on
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{2}G$}}}$, and more generally on
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k-1}G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$ for all $k\geq 2$. An induction on
$k$, using the Five Lemma on the natural exact sequence
$0\longrightarrow\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k-1}G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}\longrightarrow\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}\longrightarrow\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k-1}G$}}}\longrightarrow 0,$
then shows that $\varphi$ induces an automorphism of $\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$, for any $k$.
In particular, if $G$ is nilpotent of order $N$, then $\varphi$ induces hence
an automorphism of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{N}G$}}}\cong G$. ∎
### 1.2. Basic commutators
We now recall the notion of basic commutators in a free group, which seems to
have first appeared in [14].
###### Definition 1.6.
A _set of basic commutators_ in the set $\\{\alpha_{1},\cdots,\alpha_{n}\\}$
is an infinite ordered set of commutators $\big{\\{}C_{i}\big{\\}}_{i}$,
defined inductively as follows. Basic commutators of length $1$ are the
$C_{i}=\alpha_{i}$ for $i=1,\cdots,n$. A basic commutator of length $m>1$ has
form $[C_{i},C_{j}]$, for some basic commutators $C_{i},C_{j}$ such that
* •
length$(C_{i})+$length$(C_{j})=m$;
* •
$C_{i}<C_{j}$, and $C_{j}=[C_{k},C_{l}]$ further implies that $C_{k}\leq
C_{i}$.
Basic commutators of length $m$ follow those of length $<m$, and are ordered
in an arbitrary fixed way with respect to each other.
What makes this notion significant is the following fundamental result from
[13, Thm. 11.2.4] (see also [18, Thm. 5.13.A]).
###### Theorem 1.7.
If $\big{\\{}C_{i}\big{\\}}_{i}$ is a set of basic commutators, and $k\geq 0$
is an integer, then any element $g$ in the free group $F_{n}$ can be written
in a unique way as a product
$g=C_{1}^{e_{1}}\cdots C_{N(k)}^{e_{N(k)}}h,$
with $e_{j}\in\mathbb{Z}$ and $h\in\Gamma_{k+1}F_{n}$, where $N(k)$ is the
number of basic commutators of length $\leq k$.
###### Remark 1.8.
In [13, 18], the convention $[a,b]=\overline{a}\overline{b}ab$ is used for
commutators. Although Definition 1.6 thus gives _different_ sets of basic
commutators as the ones used in [13, 18], it does share the same fundamental
properties, and in particular Theorem 1.7, as well as the coming Lemma 1.11.
### 1.3. The Magnus expansion
Denote by $\mathbb{Z}\langle\langle X_{1},\cdots,X_{n}\rangle\rangle$ the ring
of formal power series in the noncommuting variables $X_{1},\cdots,X_{n}$.
###### Definition 1.9.
The _Magnus expansion_ is the group homomorphism
$E:\leavevmode\nobreak\ F_{n}\longrightarrow\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$
defined by sending $\alpha_{i}$ to $1+X_{i}$.
It is well-known that $E$ is in fact injective [18, Thm. 5.6], and that it is
well-behaved with respect to the lower central series, in the sense of the
following fundamental result [17, 25].
###### Theorem 1.10.
For all $k\geq 1$, we have $f\in\Gamma_{k}F_{n}$ if and only if
$E(f)=1+\textrm{(terms of degree $\geq k$)}$.
Basic commutators are well-behaved with respect to the Magnus expansion, in
the sense of the following result, which is outlined by Levine in [16, pp.
365].
###### Lemma 1.11.
Let $C$ be a basic commutator of length $k$, such that $\alpha_{i}$ occurs
$r_{i}$ times in $C$ for each $i$. We have
$E(C)=1+P+\textrm{(terms of degree $\geq k+1$)},$
where $P\neq 0$ is a sum of degree $k$ terms, each involving $r_{i}$ times the
variable $X_{i}$ ($i=1,\cdots,n$).
In the above, the sum $P$ of lowest degree (non trivial) terms in $E(C)$ is
called the _principal part_ of $E(C)$.
###### Proof.
We proceed by induction on the length $k$, where the case $k=1$ is trivial.
Let $C$ be a basic commutator of length $k$ for some $k>1$. There exists basic
commutators $C_{1}$ and $C_{2}$, of respective length $k_{1}$ and $k_{2}$,
such that $C=[C_{1},C_{2}]$ and $k_{1}+k_{2}=k$. By induction hypothesis for
$i=1,2$ we have that $E(C_{i})=1+P_{i}+\textrm{(degree $>k_{i}$ terms)}$, with
$P_{i}$ a sum of degree $k_{i}$ terms with the appropriate occurences of each
variables. Thus $E(C)=1+P_{1}P_{2}-P_{2}P_{1}+\textrm{(degree $>k$ terms)}$,
and the conclusion would follow from the fact that $P_{1}P_{2}-P_{2}P_{1}\neq
0$. In order to see that $P_{1}P_{2}-P_{2}P_{1}$ is indeed nonzero, observe
that length $k$ basic commutators form a basis for $\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k}F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}$, as a consequence of Theorem
1.7. This in particular tells us that $C\notin\Gamma_{k+1}F_{n}$, hence by
Theorem 1.10 we have $P_{1}P_{2}-P_{2}P_{1}\neq 0$. ∎
## 2\. Milnor invariants and $k$–reduced free action
The main diagrammatic object of this paper will be the following.
###### Definition 2.1.
Consider the $2$-disk $[0,1]\times[0,1]$, equipped with fixed points
$p_{i}\times\\{\varepsilon\\}$ for $i\in\\{1,\cdots,n\\}$ and
$\varepsilon\in\\{0,1\\}$. An $n$-component _welded string link_ is the welded
equivalence class of an immersion of $n$ disjoint copies of the unit interval
into $[0,1]\times[0,1]$, such that the $i$th interval runs from
$p_{i}\times\\{0\\}$ to $p_{i}\times\\{1\\}$, and whose double points are
decorated either as a classical (as in usual knot diagrams) or a virtual
crossing (drawn as transverse double points); see Figure 1. Here, the _welded
equivalence_ is generated by the three usual Reidemeister moves involving
classical crossings, together with the OC move shown in Figure 1 and the
_Detour move_ , which replaces an arc with only virtual crossings (possibly
none) by another arc with same endpoints and only virtual crossings (possibly
none).
$\vbox{\hbox{\includegraphics[height=56.9055pt]{exsl.pdf}}}\hskip
71.13188pt\vbox{\hbox{\includegraphics[height=49.79231pt]{moves_3.pdf}}}\xleftrightarrow{\textrm{OC}}\vbox{\hbox{\includegraphics[height=49.79231pt]{moves_4.pdf}}}$
Figure 1. A $3$-component welded string link (left), and the OC move (right)
In particular, a welded string link without virtual crossing is merely a
diagram of a classical string link. A key point is that classical string links
inject in this way into welded string links: two diagrams without virtual
crossings, that are welded equivalent, represent isotopic objects, see [11,
Thm 1.B].
We denote by $\mathbf{1}$ the trivial string link diagram
$\cup_{i}p_{i}\times[0,1]$.
### 2.1. Brief review of welded Milnor invariants
Milnor invariants are classical link invariants defined by Milnor in the
fifties [20, 21], and extended to (classical) string links by Habbeger and Lin
[12]. We review here the welded extension of Milnor invariants. It was given
in [1, Sec. 6] using a topological approach, and was later reformulated in
[22] in a purely diagrammatic way.
Given a welded string link $L$, there is an associated group $G(L)$, which
coincides with the fundamental group of the complement when $L$ is classical,
see [15]. As in Wirtinger’s algorithm, a generator of $G(L)$ is associated
with each arc in a diagram of $L$, where now an arc is allowed to contain
virtual crossings, and each classical crossing gives a conjugacy relation:
$\vbox{\hbox{\includegraphics[height=42.67912pt]{Wir1.pdf}}}\ \leadsto\
\overline{a}\overline{b}a^{\prime}b\hskip
28.45274pt\vbox{\hbox{\includegraphics[height=42.67912pt]{Wir2.pdf}}}\
\leadsto\ \overline{a}ba^{\prime}\overline{b}.\\\ $
It is well-known that this is invariant under welded equivalence.
Denote by $L_{1},\ldots,L_{n}$ the components of $L$. For each $i$, label by
$a_{i,j}$ ($1\leq j\leq m_{i}+1$) the successive arcs met along $L_{i}$ when
following the orientation, where $m_{i}+1$ denotes the number of arcs of
$L_{i}$, et by $b_{i,j}$ the generator $a_{k,l}^{\pm 1}$ associated with the
overpassing arc that is met when running from $a_{i,j}$ to $a_{i,j+1}$, and
the local orientation:
$\vbox{\hbox{\includegraphics[height=49.79231pt]{Longitude.pdf}}}.$
Then the group of $L$ has a presentation of the form
(2.1) $G(L)=\big{\langle}a_{i,j},\ 1\leq i\leq n,\ 1\leq j\leq m_{i}\ \big{|}\
\overline{a}_{ij+1}\overline{b}_{i,j}a_{i,j}b_{i,j}\ 1\leq i\leq n,\ 1\leq
j\leq m_{i}-1\big{\rangle},$
###### Definition 2.2.
For $1\leq i\leq n$, the _preferred $i$th longitude_ $\lambda_{i}(L)$ is given
by $\overline{a}_{i,1}^{f_{i}}\ b_{i,1}\ b_{i,2}\ \ldots\ b_{i,m_{i}-1}$,
where $f_{i}$ is the sum of the signs of all classical crossings involving
only arcs of the $i$th component.
Let us fix some integer $q\geq 2$. It is well-known that the images
$\alpha_{1},\ldots,\alpha_{n}$ of the generators $a_{i,1}$ in the quotient
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}$, do generate this group [6]. In
particular, the image of the preferred $i$th longitude in
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}$, can be expressed as a word, which
we still denote by $\lambda_{i}(L)$, in the variables
$\alpha_{1},\ldots,\alpha_{n}$.
###### Definition 2.3.
For each sequence $I=j_{1}j_{2}\ldots j_{l}i$ of integers in
$\\{1,\cdots,n\\}$ ($l<q$), the coefficient $\mu_{L}(I)$ of $X_{j_{1}}\cdots
X_{j_{l}}$ in the Magnus expansion $E\big{(}\lambda_{i}(L)\big{)}$ is an
invariant of the welded string link $L$, called a _welded Milnor invariant_.
###### Remarks 2.4.
1. (1)
These welded Milnor invariants naturally coincides with the classical
construction in the case of classical string links.
2. (2)
The fact that $\lambda_{i}(L)$ represents the _preferred_ $i$th longitude
implies that $E\big{(}\lambda_{i}(L)\big{)}$ does not contain any term of the
form $X_{i}^{s}$, for $s\geq 1$, hence that $\mu_{L}(I)=0$ for any sequence of
the form $I=ii\cdots i$.
Recall from the introduction that, given a sequence $I$, we denote by $r(I)$
the maximum number of time that any index appears in $I$. For classical
(string) links, Milnor invariants $\mu(I)$ with $r(I)=1$, that is, non-
repeated Milnor invariants, are known to be link-homotopy invariants [21, 12].
Habegger and Lin actually showed that non-repeated Milnor invariants classify
string links up to link-homotopy [12], a result that was later extended to the
welded setting in [1], where non-repeated Milnor invariants are showed to
classify welded string links up to self-virtualization. Here, _self-
virtualization_ is the equivalence relation generated by the local replacement
of a classical self-crossing, by a virtual one – what turns out to coincide
with link-homotopy for classical string links (this was implicit in [1] and
formally stated in [2, Thm. 4.3]).
### 2.2. The $k$–reduced free action
As above, let $L$ be an $n$-component welded string link, with associated
group $G(L)$. By definition, $G(L)$ is normally generated by the initial arcs
$\alpha_{i}:=a_{i,1}$ of each component. As in the introduction, we can thus
consider the _$k$ –reduced quotient_ of $G(L)$
$\textnormal{R}_{k}G(L):=\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}},$
where $N_{i}$ denotes the normal subgroup generated by $\alpha_{i}$. As a
consequence of Lemma 1.4, we have that the group $\textnormal{R}_{k}G(L)$ is
nilpotent of order at most $kn+1$.
Let $F_{n}$ be the free group generated by $\alpha_{1},\cdots,\alpha_{n}$. We
have the following.
###### Lemma 2.5.
For each $k\geq 1$, we have an isomorphism
$\textnormal{R}_{k}G(L)\simeq\textnormal{R}_{k}F_{n}=\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}}.$
###### Proof.
As shown in [1, §5] (see also [19, §6.3]), for each $q\geq 1$ we have an
isomorphism
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}\simeq\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}F_{n}$}}}.$
Hence, by Lemma 1.4, we have
$\textnormal{R}_{k}G(L)=\textnormal{R}_{k}\\!\left(\\!\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{kn+1}G(L)$}}}\right)\simeq\textnormal{R}_{k}\\!\left(\\!\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{kn+1}F_{n}$}}}\right)=\textnormal{R}_{k}F_{n}.\qed$
For each $i$, consider the $i$th preferred longitude of $L$ from Definition
2.2. This defines an element $\lambda^{k}_{i}(L)$ in
$\textnormal{R}_{k}G(L)\simeq\textnormal{R}_{k}F_{n}$, called the _$k$
–reduced $i$th longitude_ of $L$.
Denote by $\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ the set of
conjugating automorphisms of $\textnormal{R}_{k}F_{n}$, which are
automorphisms sending each generator to a conjugate of itself. Since
$\textnormal{R}_{k}F_{n}$ is nilpotent, by Lemma 1.5, any conjugating
endomorphism of $R_{k}F_{n}$ is in
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$, for all $k\geq 1$.
###### Definition 2.6.
The _$k$ –reduced free action_ associated with $L$,
$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}),$
is defined by sending each generator $\alpha_{i}$ to its conjugate by
$\lambda^{k}_{i}(L)$.
This action can be seen as a generalization of Habegger-Lin’s representation
for the group of link-homotopy classes of string links [12, Thm. 1.7], in the
sense that the case $k=1$ recovers their construction.
### 2.3. Milnor invariants and $k$–reduced free action
The main purpose of this section is Theorem 2.8, which implies the equivalence
(1)$\Leftrightarrow$(2) in our main theorem.
Throughout the rest of this paper, we shall make use of the following.
###### Notation 2.7.
Recall that $N_{i}$ denotes the normal subgroup of $F_{n}$ generated by the
$i$th generator $\alpha_{i}$ ($i\in\\{1,\cdots,n\\}$). For $k\geq 1$, we set
$J^{k}:=\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}\,\,\,\textrm{ and
}\,\,\,J_{i}^{k}:=\Gamma_{k+1}N_{1}\cdots\Gamma_{k}N_{i}\cdots\Gamma_{k+1}N_{n}.$
Denote also by $R^{k}$ the two-sided ideal of $\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$ generated by all terms having at least $k+1$
occurences of some variable, and by $R^{k}_{i}$ the ideal generated by terms
having either at least $k+1$ occurences of some variable, or $k$ occurences of
the variable $X_{i}$.
###### Theorem 2.8.
Let $L$ and $L^{\prime}$ be two $n$-component welded string links. The
following are equivalent:
1. (i)
$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.
2. (ii)
the $k$–reduced $i$th longitudes $\lambda^{k}_{i}(L)$ and
$\lambda^{k}_{i}(L^{\prime})$ are congruent modulo $J_{i}^{k}$, for all $i$.
3. (iii)
$L$ and $L^{\prime}$ induce the same $k$–reduced free action
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$.
The rest of this section is devoted to the proof of Theorem 2.8, which is done
in three steps.
Firstly, we prove that (ii)$\Rightarrow$(iii). Suppose that the $k$–reduced
$i$th longitudes $\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are
congruent modulo $J_{i}^{k}$, for all $i$. We have
$\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}=Xg_{i}$, for some
$g_{i}\in\Gamma_{k}N_{i}$ and some $X\in\prod_{j\neq i}\Gamma_{k+1}N_{j}$. By
(C$3$) we thus have that
$\big{[}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})},\alpha_{i}\big{]}=[g_{i},\alpha_{i}]^{\overline{X}}[X,\alpha_{i}]\in
J^{k}.$
This readily implies that
$\big{[}\alpha_{i},\lambda^{k}_{i}(L)\big{]}\equiv\big{[}\alpha_{i},\lambda^{k}_{i}(L^{\prime})\big{]}$
mod $J^{k}$, which is equivalent to saying that $L$ and $L^{\prime}$ induce
the same $k$–reduced free action:
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$.
Secondly, we prove that (iii)$\Rightarrow$(i). If
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$,
then for each $i$ we have
$\alpha_{i}^{\lambda^{k}_{i}(L)}\equiv\alpha_{i}^{\lambda^{k}_{i}(L^{\prime})}$
mod $J^{k}$, which is equivalent to
$\alpha_{i}\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\overline{\alpha}_{i}\equiv\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\textrm{
mod $J^{k}$.}$
Taking the Magnus expansion then gives
(2.2)
$X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}-E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\big{(}X_{i}-X_{i}^{2}+\cdots\big{)}-X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\big{(}X_{i}-X_{i}^{2}+\cdots\big{)}\in
R^{k}.$
If
$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}$
lives in $R^{k}_{i}$, then $L$ and $L^{\prime}$ have same Milnor invariants
$\mu(I)$ with $r(I)\leq k$. Suppose by contradiction that
$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}\equiv
S_{q}+\left(\textrm{terms of degree $>q$}\right)\textrm{ mod $R^{k}_{i}$},$
for some $q$ and a sum $S_{q}$ of degree $q$ terms which are _not_ in
$R^{k}_{i}$. Multiplying the above by
$E\big{(}\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}$, which by definition
has constant term equal to $1$, we have
$E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\equiv
1+S_{q}+\left(\textrm{terms of degree $>q$}\right)\textrm{ mod $R^{k}_{i}$}.$
Equation (2.2) then gives
$X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}-E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}X_{i}\equiv
X_{i}S_{q}-S_{q}X_{i}+\left(\textrm{terms of degree $>q+1$}\right)\textrm{ mod
$R^{k}_{i}$}.$
Observe that $X_{i}S_{q}$ and $S_{q}X_{i}$ are not in $R^{k}$, since
$S_{q}\notin R_{i}^{k}$. Hence, if $X_{i}S_{q}\neq S_{q}X_{i}$, we obtain a
contradiction with (2.2). But if $X_{i}S_{q}=S_{q}X_{i}$, a simple
combinatorial argument shows that $S_{q}$ can be nothing else than the
monomial $X_{i}^{q}$. This tells us that
$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}$
contains the term $X_{i}^{q}$, which is in contradiction with the fact that
$\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are images of
_preferred_ $i$th longitudes, see Remark 2.4 (ii).
Thirdly and lastly, let us prove that (i)$\Rightarrow$(ii). For this purpose,
we prove the following, which should be thought of as a ‘$k$–reduced’ version
of the injective property of the Magnus expansion (Theorem 1.10).
###### Proposition 2.9.
Let $g$ be an element of $F_{n}$. Then $g\in J_{i}^{k}$ if and only if
$E(g)\in 1+R_{i}^{k}$.
Assuming this result momentarily, we immediately deduce the desired
implication (i)$\Rightarrow$(ii) of Theorem 2.8. Indeed, consider two welded
string links $L$ and $L^{\prime}$ with same Milnor invariants $\mu(I)$ with
$r(I)\leq k$. This precisely means that, for each $i$, their respective $i$th
preferred longitudes $\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$
satisfy
$E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}\equiv
E\big{(}\lambda^{k}_{i}(L)\big{)}\textrm{ mod $R_{i}^{k}$}.$
This rewrites as
$E\big{(}\overline{\lambda^{k}_{i}(L)}\lambda^{k}_{i}(L^{\prime})\big{)}=1+R_{i}^{k}$,
which by Proposition 2.9 gives that
$\overline{\lambda^{k}_{i}(L)}\lambda^{k}_{i}(L^{\prime})\in J_{i}^{k}$, as
desired.
###### Proof of Proposition 2.9.
The ‘only if’ part of the statement follows easily from well-known properties
of the Magnus expansion [18], so we prove here the other implication. By
Theorem 1.7, there is a set of basic commutators $\\{C_{i}\\}_{i}$ such that
$g=C_{1}^{e_{1}}\cdots C_{N(nk)}^{e_{N(nk)}}h,$
for some unique integers $e_{1},\cdots,e_{N(nk)}\in\mathbb{Z}$ and a unique
element $h\in\Gamma_{nk+1}F_{n}$. Set
$g_{j}:=\prod_{\textrm{length$(C_{j_{m}})=j$}}C_{j_{m}}^{e_{j_{m}}}$, so that
$g=g_{1}\cdots g_{kn}h$. Since $h\in J_{i}^{k}$, it remains to show that
$g_{j}\in J_{i}^{k}$ for all $j$. Suppose by contradiction that this is not
the case, and let $p$ be the smallest integer such that
$g_{p}=\prod_{m=1}^{N}C_{p_{m}}^{e_{p_{m}}}$ is not in $J_{i}^{k}$ ($N\geq
1$). This means that there is a nonempty subset $S$ of $\\{1,\cdots,N\\}$ such
that $e_{p_{l}}\neq 0$ and $C_{p_{l}}\notin J_{i}^{k}$, for all $l\in S$.
Lemma 1.11 tells us that, for each $m$, the principal part of $E(C_{p_{m}})$
is either in $R_{i}^{k}$, or is in the $\mathbb{Z}$-module generated by
monomials with $<(k-1)$ copies of $X_{i}$ and $<k$ copies of any other
variable. This implies in particular that the sum of the principal parts of
$E\big{(}\prod_{l\in S}C_{p_{l}}^{e_{p_{l}}}\big{)}$ is not in
$R_{i}^{k}\setminus\\{0\\}$. But as a property of basic commutators, we know
that these principal parts are linearly independent [18]; it follows that
$E(g_{p})-1$ is non trivial and not in $R_{i}^{k}$. By minimality of $p$, this
implies that $E(g)-1$ is not in $R_{i}^{k}$. This is a contradiction, which
concludes the proof. ∎
###### Remark 2.10.
The case $k=1$ of Proposition 2.9 was previously established in [28, Prop.
7.10]. One can actually further generalize this result as follows. Let
$\mathbf{r}=(r_{1},\cdots,r_{n})$ be an $n$-tuple of positive integers.
Consider the subgroup
$J^{\mathbf{r}}:=\Gamma_{r_{1}}N_{1}\cdots\Gamma_{r_{n}}N_{n}$ of $F_{n}$, and
the ideal $R^{\mathbf{r}}$ of $\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$ generated by all terms where the variable
$X_{i}$ appears at least $r_{i}$ times, for $i=1,\cdots,n$. Then the above
proof generalizes in a straightforward way to show that an element $g\in
F_{n}$ is in $J^{\mathbf{r}}$ if and only if $E(g)$ is in $1+R^{\mathbf{r}}$.
## 3\. Arrow calculus and self $w_{k}$-concordance
### 3.1. Arrow calculus for welded objects
Let us briefly review arrow calculus, which is diagrammatic calculus
developped in [19] for the study of welded objects. In Section 3.2, we explain
how this diagrammatic tool is intimately related to the commutator calculus
reviewed in Section 1.
Let $L$ be an $n$-component welded string link.
###### Definition 3.1.
A _$w$ -tree_ for $L$ is a planar immersion of an oriented, connected uni-
trivalent tree $T$, such that
* •
the set of all vertices of $T$ is embedded in the interior of
$[0,1]\times[0,1]$, such that trivalent vertices are disjoint from $L$ and
univalent vertices are in $L\setminus\\{\textrm{crossings of $L$}\\}$;
* •
at each trivalent vertex, the three incident edges are cyclically ordered, and
there are two ingoing and one outgoing edge;
* •
edges of $T$ may cross virtually $L$ or $T$ itself;
* •
edges are possibly decorated by some _twists_ $\bullet$, which are disjoint
from all crossings and vertices, and which satisfy the rule .
A univalent vertex of $T$ is a _head_ if $T$ is locally oriented towards $L$,
and it is a _tail_ otherwise. For a union of $w$-trees for $L$, we allow
virtual crossings among edges, and we require all vertices to be pairwise
disjoint. See Figure 2 for an example.
We note that a $w$-tree contains a single head, due to the orientation
convention at trivalent vertices.
Figure 2. An example of $w$-tree for the trivial $3$–component string link
###### Definition 3.2.
A $w$-tree is _linear_ , if it has the following shape as an abstract tree
with possibly a number of $\bullet$ on its edges.
###### Definition 3.3.
The _degree_ of $T$ is its number of tails or, equivalently, half the total
number of vertices. A degree $k$ $w$-tree is called a _$w_{k}$ -tree_, and a
$w_{1}$-tree is also called a $w$-arrow.
Given a union of $w$-arrows $A$ for $L$, there is a notion of _surgery along
$A$_, which produces a new welded string link $L_{A}$ as follows:
.
In general, if $A$ contains some virtual crossing, this likewise introduces
pairs of virtual crossings in $L_{A}$.
Now, given a union of $w_{k}$-trees $P$ for $L$, one can define the notion of
surgery along $P$ by first replacing $P$ by a union of $w$-arrows, called the
_expansion_ of $P$ and denoted by $E(P)$, defined recursively by the local
rule illustrated below, then performing surgery on $E(P)$. The result will be
denoted by $L_{P}$.
.
###### Remark 3.4.
The above figure suggests that the union $E(P)$ of $w$-arrows obtained from
$P$ by applying (E) recursively, has the shape of an ‘iterated commutator’ of
$w$-arrows. This observation is made rigourous and further discussed in
Section 3.2.
###### Definition 3.5.
A _$w$ -tree presentation_ $(\mathbf{1},P)$ for $L$ is a union $P$ of
$w$-trees for the trivial diagram $\mathbf{1}$, such that $L=\mathbf{1}_{P}$.
Two arrow presentations $(\mathbf{1},P)$ and $(\mathbf{1},P^{\prime})$
representing equivalent welded string links are called _equivalent_.
The main point of this notion is that any welded string link admits a $w$-tree
presentation [19, Prop. 4.2]. Moreover, in [19, Thm. 5.21], a set of moves on
$w$-trees is provided, which suffice to deform any $w$-tree presentation of
$L$ into any other. These moves imply further operations, hence a full
diagrammatic calculus called _arrow calculus_ , which can be used to study
welded objects and their invariants. We do not reproduce all these moves here,
but only provide below those that will be needed in this paper :
We refer the reader to [19] for more details on arrow calculus.
We will also use the following, which refines the notion of equivalence given
in Definition 3.5.
###### Definition 3.6.
Let $(\mathbf{1},P)$ be a $w$-tree presentation for $L$, and let $S$ be a
subset of $P$. Denote by $\sigma\subset\mathbf{1}$ a neighborhood of the
endpoints of $S$, which identifies with a trivial diagram, such that $\sigma$
is disjoint from $P\setminus S$. Let $S^{\prime}$ be a union of $w$-trees for
$\sigma$. We say that $S$ and $S^{\prime}$ are _locally equivalent_ whenever
$\sigma_{S^{\prime}}$ is welded equivalent to $\sigma_{S}$.
Note that, in this setting, $(\mathbf{1},P)$ and
$\big{(}\mathbf{1},(P\setminus S)\cup S^{\prime}\big{)}$ are equivalent
$w$-tree presentations.
### 3.2. Algebraic formalism for $w$-trees
Let $(\mathbf{1},P)$ be a $w$-tree presentation for a welded diagram.
Let $A$ be a union of $w$-arrows in $P$ that are _adjacent_ , in the sense
that _all_ heads in $A$ are met consecutively on a portion $h_{A}$ of
$\mathbf{1}$, called the _support_ of $A$, without meeting any crossing or
endpoint.111We shall also say in this situation that the heads of $A$ are
_adjacent_. The _complement_ of $A$ is then defined as the $w$-tree
presentation $(\mathbf{1},P\setminus A)$ obtained from $(\mathbf{1},P)$ by
removing all $w$-arrows in $A$. The support $h_{A}$ and the arrow heads of
$P\setminus A$ then cut the strands of $\mathbf{1}$ into intervals that we
label by letters $x_{1},\ldots,x_{p}$; we set $F_{(P;A)}:=\langle
x_{1},\ldots,x_{p}\rangle$, the free group generated by these letters.
$\vbox{\hbox{\includegraphics[height=85.35826pt]{Cut1.pdf}}}\ \leadsto\
\vbox{\hbox{\includegraphics[height=85.35826pt]{Cut2.pdf}}}\ \leadsto\ \
w(A)=\overline{x}_{3}x_{5}x_{4}\overline{x}_{6}$
Figure 3. An example of word associated to a set of adjacent $w$-arrows
Assuming that all arrow heads are met to the right side when traveling along
$h_{A}$ following its orientation (this is always possible thanks to the Head
reversal move), each head of an arrow in $A$ can be labeled222It should be
noted that replacing arrows by labels corresponds actually to the _cut-
diagram_ point of view on welded objects, introduced in [4]. by the letter
$x_{i}$ or $\overline{x}_{i}$, depending on whether the arrow contains an even
or an odd number of twists, where $x_{i}$ is the label of the interval on
which the tail lies. We can then define an element $w(A)\in F_{(P;A)}$ by
reading the labels in order when running along $h_{A}$ according to its
orientation; see Figure 3 for an example. Notice that this _word_ $w(A)$
remains unchanged under a Tails exchange move and that, conversely, it
determines the union of $w$-arrows $A$ up to Tails exchange moves. More
generally, we have the following.
###### Lemma 3.7.
Suppose that two $w$-tree presentations $(\mathbf{1},P)$ and
$(\mathbf{1},P^{\prime})$ only differ by replacing a union of adjacent arrows
$A$ by another union of adjacent arrows $A^{\prime}$ with same support. Then
$F_{(P;A)}=F_{(P^{\prime};A^{\prime})}$, and $w(A)=w(A^{\prime})$ if and only
if $A$ and $A^{\prime}$ differ by a sequence of Inverse moves and Tails
exchange moves.
###### Proof.
First note that $A$ and $A^{\prime}$ have same complement, hence we indeed
have that $F_{(P;A)}=F_{(P^{\prime};A^{\prime})}$. Since this is a free group,
we have $w(A)=w(A^{\prime})$ if and only if these two words differ by
inserting or deleting copies of $x_{j}\overline{x}_{j}$ or
$\overline{x}_{j}x_{j}$ for any $j$. But this is equivalent to saying that $A$
and $A^{\prime}$ differ by a sequence of Inverse moves (with pairs of
$w$-arrows) and Tails exchange moves. ∎
Lemma 3.7 provides a one-to-one correspondence between sets of adjacent arrows
with a fixed support and complement, up to equivalence, and elements in the
associated free group. Under this correspondence, our conventions for the
commutator notation $[x,y]$ and the conjugation notation $x^{y}$, given in
Section 1.1, have natural diagrammatic counterparts.
It is easily observed that a commutator corresponds to the expansion of a
$w$–tree:
$x\overline{y}\overline{x}y=[x,y]\quad\leftrightarrow\quad\vbox{\hbox{\includegraphics[height=42.67912pt]{com1.pdf}}}\,=\,\vbox{\hbox{\includegraphics[height=42.67912pt]{com0.pdf}}}.$
In general, to a $w$-tree $T$ which is part of a $w$-tree presentation
$(\mathbf{1},P)$, corresponds a word $w(T)\in F_{\left(P;E(T)\right)}$, which
is defined as the word corresponding to its expansion; in other words, we set
$w(T)=w\big{(}E(T)\big{)}$. From the definition, the word $w(T)$ can be
directly read from $T$ using the following procedure. Label each edge of $T$
containing a tail by the generator $x_{i}$ at the tail, then label the
remaining edges of $T$ by recursively applying the local rules of Figure 4;
the label at the edge containing the head is $w(T)$; see Example 3.9 (i).
Figure 4. Procedure to compute $w(T)$ from a $w$-tree
Note that, under this correspondence, the word associated to a linear $w$-tree
(Definition 3.2), is a linear commutator with entries in
$\\{x_{i},\overline{x}_{i}\\}_{i}$, in the sense of Definition 1.1.
The conjugation notation, on the other hand, corresponds to a situation where
the Slide move can be performed:
$\overline{y}xy=x^{y}\quad\leftrightarrow\quad\vbox{\hbox{\includegraphics[height=42.67912pt]{com2.pdf}}}\,=\,\vbox{\hbox{\includegraphics[height=42.67912pt]{com3.pdf}}},$
Since two $w$-tree presentations that differ by a Slide move are locally
equivalent, the above rightmost picture can be seen as the diagrammatic
counterpart of notation $x^{y}$ in our correspondence.
#### 3.2.1. Conjugated trees
Combining these two observations, we can thus extend our correspondence to a
wider range of $w$-tree presentations:
###### Definition 3.8.
A $w$-tree $T$ for $\mathbf{1}$ is called _conjugated_ if there exists a union
$U$ of _pairs of conjugating $w$-arrows_ for $T$. Here, a pair of conjugating
$w$-arrows for $T$ is a union of two parallel $w$-arrows that only differ by a
twist, and whose heads are on the same component of $\mathbf{1}$ and only
separated by a tail of $T$, and possibly other nested pairs of conjugating
$w$-arrows. See Example 3.9.
Let $T$ be a conjugated $w$-tree $T$ with union $U$ of conjugating $w$-arrows,
in a $w$-tree presentation $(\mathbf{1},P)$. One can define a corresponding
word $w^{U}(T)\in F_{(P\setminus U;E(T))}$, which is the word associated with
the union of adjacent $w$-arrows obtained by taking the expansion of $T$ and
applying Inverse moves and Slide moves333Since each tail of $T$ yields a
_union_ of $w$-arrows after expansion, one first need to use the Inverse move
several times between these tails to be able to perform the Slide moves. to
$w$-arrows in $U$. This word $w^{U}(T)$ can be directly read off $T\cup U$, by
substituting each letter $a$ in $w(T)\in F_{(P\setminus U;E(T))}$ by its
conjugate $a^{w(X)}$ whenever the corresponding tail of $T$ admits a union of
conjugating $w$-arrows $X\cup\overline{X}\subset U$, where $X$ denotes the
union of $w$-arrows met before the tail according to the orientation (and
$\overline{X}$ are the $w$-arrows met after the tail). Let us illustrate this
concretely on an example:
###### Example 3.9.
The $w_{4}$-tree $T$ shown below is a conjugated tree (here, the labels $a$,
$b$, $c$, $d$ and $e$ are not necessarily mutually distinct).
(i) Ignoring all conjugating $w$-arrow, we have that
$w(T)=\big{[}[a,b],[d,e]\big{]}$.
(ii) Denoting by $U$ the union of conjugating $w$-arrows for $T$, we have
$w^{U}(T)=\big{[}[a^{b},b^{\overline{c}}],[d^{ec},e]\big{]}.$
###### Remark 3.10.
We stress that, for a $w_{k}$–tree $T$, $w^{U}(T)$ is a length $k$ commutator
whose entries are conjugates of
$\big{\\{}x_{i},\overline{x}_{i}\big{\\}}_{i}$. Conversely, it follows from
the above that for any length $k$ commutator $w\in F_{n}$ whose entries are
conjugates of $\\{x_{i},\overline{x}_{i}\\}_{i}$, there exists a $w_{k}$-tree
$T$ and a union $U$ of conjugating arrows such that $w=w^{U}(T)$. This applies
more generaly to _products_ of such commutators, which then correspond to
_adjacent_ conjugated trees, _i.e._ conjugated trees with adjacent heads.
###### Remark 3.11.
Deleting a conjugated tree $T$, in the notation of Definition 3.8, yields the
union $U$ of conjugating $w$-arrows, which can in turn all be deleted by using
the Inverse move.
For this paper, we will need the following, which is a direct translation of
Lemma 1.3 via the above correspondence:
###### Lemma 3.12.
Let $T$ be a $w_{l}$-tree in some union of $w$-trees for $\mathbf{1}$, such
that the $i$th component of $\mathbf{1}$ contains $k$ tails of $T$ ($k\leq
l$). Then $T$ is locally equivalent to a union of adjacent conjugated linear
$w_{k}$-trees, with all tails on the $i$th component.
#### 3.2.2. Relation to the welded group
Finally, let us recall how, given a $w$-tree presentation $(\mathbf{1},P)$ for
$L$, one can associate a presentation for the group $G(L)$ defined in Section
2.1, using our algebraic formalism.
Again, by the Head reversal move, one can freely assume that all heads of $P$
are attached to the right side of $\mathbf{1}$ according to the orientation.
The heads of $P$ split $\mathbf{1}$ into a union of arcs, each of which yields
a generator $x_{i}$, and we denote by $F_{P}$ the free group generated by
$\big{\\{}x_{i}\big{\\}}_{i}$.
Now, let $T$ be a single $w_{k}$-tree in $P$, and denote by $a$ and $b$ the
two generators of $F_{P}$ associated with the arcs to the left and right of
its head, respectively. Following the above, a word $w(T)$ is defined by
expanding $T$ into a union $E(T)$ of adjacent $w$-arrows, and writing the
associated word. This $w(T)$ naturally lives in $F_{P}$, since the complement
of $h_{E(T)}$ is obtained by just deleting a neighborhood of the head of $T$.
Figure 5 then illustrates how $T$ yields a conjugation relation
$R_{T}$:$\leavevmode\nobreak\ b=\overline{w(T)}aw(T)$ among the two generators
$a,b$ separated by its head.
Figure 5. Conjugating relation $R_{T}$ associated with $T$
In fact, we obtain in this way the following presentation (see [19, § 6.1.1]):
$G(L)=\langle\\{x_{i}\\}_{i}\,|\,\\{R_{T}\\}_{T\in P}\rangle.$
In particular, the $i$th preferred longitude of $L$ from Definition 2.2, can
be written as $\lambda_{i}(L)=\alpha_{i}^{s_{i}}\prod_{T\in P_{i}}w(T)$, for
some $s_{i}\in\mathbb{Z}$, where $P_{i}$ is the subset of $w$-trees in $P$
whose heads are on the $i$th component of $\mathbf{1}$, ordered according to
their occurence on the $i$th component when following the orientation.
###### Remark 3.13.
Observe that, in the notation of Figure 5, if the head of $T$ is on the $i$th
component then $a,b\in N_{i}$, and if moreover $w(T)\in\Gamma_{k}N_{i}$ for
some $k$, then we have:
$b=\overline{w(T)}aw(T)=a\big{[}\overline{a},w(T)\big{]}\equiv a\textrm{ mod
$\Gamma_{k+1}N_{i}$}$.
#### 3.2.3. Self $w_{k}$-equivalence and $w^{(k+1)}$-equivalence
This section is concerned with the following families of equivalence relations
for welded objects.
###### Definition 3.14.
Let $k\geq 1$. The _$w_{k}$ -equivalence_, resp. _self $w_{k}$-equivalence_,
is the equivalence relation generated by welded equivalence and surgery along
$w$-trees, resp. self $w$-trees, of degree $\geq k$. Here, a _self $w$-tree_
is a $w$-tree whose endpoints all lie on a same component. The _$w^{(k+1)}$
-equivalence_ is the equivalence relation generated by welded equivalence and
surgery along $w$-trees having at least $k+1$ endpoints on a same component.
###### Remark 3.15.
Given a $w_{kn+1}$-tree $T$ for some $n$-component string link, there is
necessarily some index $i$ such that $T$ has at least $k+1$ ends on the $i$th
component. This elementary observation shows that the $w_{kn+1}$-equivalence
implies the $w^{(k+1)}$-equivalence.
The following will play a central role in proving our main theorem, but might
also be of independent interest in the study of arrow calculus.
###### Theorem 3.16.
Two welded string links are self $w_{k}$-equivalent, if and only if they are
$w^{(k+1)}$-equivalent.
###### Proof.
Since a self $w$-tree of degree $\geq k$ has at least $k+1$ endpoints, which
are all attached to a same component, we clearly have the ‘only if’ part of
the statement.
In order to prove the ‘if’ part, consider a $w$-tree $T^{\prime}$ for an
$n$-component welded string link $L$, with $k+1$ endpoints on the $i$th
component $L_{i}$ of $L$. We distinguish two cases. If the head of $T$ is on
$L_{i}$, then by Lemma 3.12, it is locally equivalent to a union of conjugated
self $w_{k}$-trees on the $i$th component. Hence in this case, $L_{T}$ is
clearly self $w_{k}$-equivalent to $L$ by Remark 3.11. Now, if the head of $T$
is on the $j$th component $L_{j}$ of $L$ for some $j\neq i$, then Lemma 3.12
more precisely tells us that $T$ is locally equivalent to a union of
conjugated linear $w_{k+1}$-trees, with all tails on the $i$th component and
with head on the $j$th component. Consider one such linear $w$-tree, as on the
left-hand side of the figure below:
This figure shows how applying the expansion move (E) to such a tree, followed
by the Slide move, yields a union of two $w$-arrows $A\cup A^{\prime}$ and two
self $w_{k}$-trees $B\cup B^{\prime}$ on the $i$th component. Deleting $B\cup
B^{\prime}$ up to self $w_{k}$-equivalence, then deleting $A\cup A^{\prime}$
by the Inverse move, yields the empty diagram. This observation, together with
Remark 3.11, shows that $L_{T}$ is self $w_{k}$-equivalent to $L$. ∎
###### Remark 3.17.
The argument of this proof can be used to show that the (self)
$w_{k}$-equivalence is generated by welded equivalence and surgery along
(self) $w_{k}$-trees, rather than (self) $w$-trees of degree $\geq k$.
### 3.3. Self $w_{k}$-concordance and $w^{(k+1)}$-concordance
The main purpose of this section is Theorem 3.19, which implies the
equivalence (1)$\Leftrightarrow$(3) in our main theorem. First, we introduce
the self $w_{k}$-concordance relation, along with the $w^{(k+1)}$-concordance
equivalence.
The notion of concordance for welded links was introduced and studied in [5,
10].
Two $n$-component welded string links $L$ and $L^{\prime}$ are _welded
concordant_ if one can be obtained from the other by a sequence of welded
equivalence and the birth/death and saddle moves of Figure 6, such that, for
each $i\in\\{1,\cdots,n\\}$, the number of birth/death moves used to deform
the $i$th component of $L$ into the $i$th component of $L^{\prime}$ is equal
to the number of saddle moves.
Figure 6. The birth/death and saddle moves
###### Definition 3.18.
Let $k\geq 1$. The _$w_{k}$ -concordance_, resp. _self $w_{k}$-concordance_,
is the equivalence relation generated by welded concordance and
$w_{k}$-equivalence, resp. self $w_{k}$-equivalence. The _$w^{(k+1)}$
-concordance_ is the equivalence relation generated by $w^{(k+1)}$-equivalence
and welded concordance.
The following is the main result of this section.
###### Theorem 3.19.
Let $L$ and $L^{\prime}$ be two $n$-component welded string links. The
following are equivalent:
1. (i)
$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.
2. (ii)
$L$ and $L^{\prime}$ are $w^{(k+1)}$-concordant.
3. (iii)
$L$ and $L^{\prime}$ are self $w_{k}$-concordant.
The rest of this section is devoted to the proof of Theorem 3.19.
We already have directly that (ii) and (iii) are equivalent, by Theorem 3.16.
In the rest of the proof, we use the fact from Theorem 2.8 that (i) is
equivalent to saying that the $k$–reduced $i$th longitudes
$\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are congruent modulo
$J_{i}^{k}=\Gamma_{k+1}N_{1}\cdots\Gamma_{k}N_{i}\cdots\Gamma_{k+1}N_{n}$ for
all $i$.
Let us prove that (iii) implies (i). It is shown in [4] that welded Milnor
invariants are invariant under welded concordance. So it suffices to show
that, if $L^{\prime}$ is obtained from $L$ by surgery along a self
$w_{k}$-tree $T$, we have
$\lambda^{k}_{i}(L^{\prime})\equiv\lambda^{k}_{i}(L)\mod J_{i}^{k}$ for all
$i$. Suppose that all endpoints of $T$ are on the $i_{0}$th component of $L$
for some $i_{0}$. Then $w(T)\in\Gamma_{k}N_{i_{0}}$, and by Remark 3.13 we
have directly that
$\lambda^{k}_{i}(L)\equiv\lambda^{k}_{i}(L)\mod\Gamma_{k+1}N_{i_{0}}(\subset
J_{i}^{k})$ for all $i\neq i_{0}$. Furthermore, there exists some words $a,b$
in $F_{n}$, such that $\lambda^{k}_{i_{0}}(L)=ab$ and, again by Remark 3.13,
$\lambda^{k}_{i_{0}}(L^{\prime})\equiv aw(T)b\mod\Gamma_{k+1}N_{i_{0}}$. This
implies that $\lambda^{k}_{i_{0}}(L^{\prime})\equiv\lambda^{k}_{i_{0}}(L)\mod
J_{i_{0}}^{k}$, as desired.
Finally, we prove that (i) implies (ii). By assumption, and using Lemmas 2.5
and 1.2, the $k$–reduced $i$th preferred longitudes
$\lambda^{k}_{i}(L^{\prime})$ and $\lambda^{k}_{i}(L)$ differ by a finite
sequence of the following operations:
* (a)
inserting or deleting copies of $\alpha_{j}\overline{\alpha}_{j}$ or
$\overline{\alpha}_{j}\alpha_{j}$ for any $j$;
* (b)
inserting or deleting length $\geq k+1$ commutators with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$, and with at least
$k+1$ entries with some index $j\neq i$.
* (c)
inserting or deleting length $\geq k$ commutators with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$, and with at least
$k$ entries with index $i$.
We aim at showing that, under this assumption, $L$ and $L^{\prime}$ are
$w^{(k+1)}$-equivalent.
As a first step, let us assume that both $L$ and $L^{\prime}$ are given by a
so-called sorted $w$-tree presentation:
###### Definition 3.20.
A $w$-tree presentation for an $n$-component welded string link is _sorted_
444This notion was first defined in [1] under the term _ascending_ , in the
context of Gauss diagrams. if, when running along each component following
the orientation, all tails are met before all heads.
Given a sorted presentation $(\mathbf{1},P)$, the tails of all $w_{k}$-trees
are contained in the initial arcs of $\mathbf{1}$, hence the words associated
to these $w_{k}$-trees are length $k$ commutator with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$. Denote by $P_{i}$
the subset of $w$-trees in $P$ whose heads are on the $i$th component.
According to Section 3.2.2, there exists some $s_{i}\in\mathbb{Z}$ such that
the $i$th preferred longitude is equal to $\alpha_{i}^{s_{i}}\prod_{T\in
P_{i}}w(T)$. Using the Isolated move, we can introduce a union $A_{i}$ of
$w$-arrows whose endpoints are all on the $i$th component, such that
$(\mathbf{1},P\cup A_{i})$ is sorted and equivalent to $(\mathbf{1},P)$, and
such that the $i$th preferred longitude is equal to $\prod_{T\in A_{i}\cup
P_{i}}w(T)$. By this observation, we can freely assume that the words
associated with our sorted presentations, are precisely the preferred
longitudes of our string links $L$ and $L^{\prime}$. Hence by Lemma 3.7, if
$\lambda_{i}(L)$ and $\lambda_{i}(L^{\prime})$ are equal as words in
$\alpha_{1},\cdots,\alpha_{n}$, then applying (E) recursively to these sorted
presentations, yields the _same_ union of $w$-arrows.
It thus remains to realize the above three operations (a), (b) and (c) by a
$w^{(k+1)}$-equivalence among sorted tree presentations. This is done as
follows:
* •
Operation (a) is simply achieved by Inverse moves, which insert a pair of
parallel $w$-arrows, one having a $\bullet$, running from the bottom of the
$j$th component and ending on the $i$th component.
* •
Operation (b), resp. (c), is achieved by inserting or deleting sorted
$w$-trees, with head on the $i$th component, and with at least $k+1$ tails on
the $j$th component, resp. with at least $k$ tails on the $i$th component.
This shows that $L$ and $L^{\prime}$ are $w^{(k+1)}$-equivalent, and we are
done in the sorted case.
In order to conclude the proof, it is now enough to show that any welded
string link is $w^{(k+1)}$-concordant to a string link with a sorted
presentation, since Milnor invariants $\mu(I)$ with $r(I)\leq k$ are
invariants of $w^{(k+1)}$-concordance. This can be seen as a direct corollary
of [7, Cor. 2.5], which tells us that any $w$-tree presentation can be
deformed into a sorted one by a sequence of welded concordance and surgeries
along $w_{nk+1}$-trees. The desired result then follows by Remark 3.15.
### 3.4. Artin-like isomorphisms
The stacking product endows the set $w\textnormal{S}L(n)$ of $n$-component
welded string links with a structure of monoid, with unit the trivial diagram
$\mathbf{1}$. As seen in subsection 2.2, a conjugating automorphism
$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is
associated to any welded string link $L$, which sends each generator
$\alpha_{i}$ of $R_{k}F_{n}$ to its conjugate by the $k$-reduced longitude
$\lambda^{k}_{i}(L)$. This yields, for each $k\geq 1$, a monoid homomorphism
$\varphi^{(k)}:w\textnormal{S}L(n)\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}).$
###### Proposition 3.21.
For each $k\geq 1$, the map $\varphi^{(k)}$ descends to a group isomorphism
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}).$
###### Proof.
Consider the quotient map $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$,
which will shall still denote by $\varphi^{(k)}$. The fact that this map is
injective is a consequence of Theorems 3.19 and 2.8. To prove surjectivity, it
is sufficient to observe that an element of
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is specified by an $n$-tuple
of conjugating elements in $\textnormal{R}_{k}F_{n}$, and that these
conjugating elements are easily realized as the preferred longitudes of a
sorted welded string link; see the paragraph following Def. 3.20. Since
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is a group, it follows that
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self $w_{k}$-concordance}$}}}$ is a group
too555This fact was already known, as any welded string link is invertible up
to concordance [10, Prop. 6]., and that the isomorphism is a group
isomorphism. ∎
###### Remark 3.22.
Using the result of Colombari [7] mentioned in the introduction, the proof of
Proposition 3.21 can be adapted in a straightfoward way to show that, for all
$k\geq 1$, we have an isomorphism
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}\\!\left(\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}\right),$
where $\operatorname{Aut}_{c}\\!\left(\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}\right)$ denotes the group of
conjugating automorphisms of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}$.
We conclude this section by a (long) remark addressing the 4-dimensional
counterpart of this study, that can be safely skipped by the reader who is not
interested in this matter.
###### Remark 3.23.
Welded theory is intimately connected to the topology of _ribbon knotted
surfaces_ in $4$–space, via the so-called _Tube map_ defined by Satoh in [23].
In our context, the Tube map is a surjective monoid homomorphism
$\textrm{Tube}:w\textnormal{S}L(n)\longrightarrow rT(n),$
where $rT(n)$ denotes the monoid of _ribbon tubes_ introduced in [1]. The
question of the injectivity of this Tube map is however still open, but it is
worth noting that our present work implies that any element in the kernel of
the Tube map is self $w_{k}$-concordant to $\mathbf{1}$ for all $k$.
Indeed, there is a $4$–dimensional analogue of arrow calculus, which was
developped (prior to [19]) by Watanabe in [24]. There, a notion of
$RC_{k}$-equivalence is defined for ribbon knotted surfaces, in terms of
surgery along degree $k$ oriented claspers, which are embedded surfaces that
realize topologically an oriented uni-trivalent tree with $2k$ vertices. A
refined notion of _self $RC_{k}$-equivalence_ is easily defined on $rT(n)$ by
further requesting that such degree $k$ oriented claspers only intersect a
single ribbon tube component. Combining furthermore this self
$RC_{k}$-equivalence with the topological concordance of ribbon tubes, one
defines a notion of _self $RC_{k}$-concordance_ for these objects. It is known
[5, Prop. 4.8] that the Tube map sends concordant welded links to concordant
ribbon tori, and it is easily verified that it sends self $w_{k}$-concordant
welded string links to self $RC_{k}$-concordant ribbon tubes. Hence, for each
$k\geq 1$, the Tube map induces a surjective homomorphism
$\textrm{Tube}_{k}:\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\longrightarrow\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$rT(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self $RC_{k}$-concordance}$}}}.$
These maps are actually isomorphisms, and the proof goes as follows. Following
[1, Sec. 2.2.4], which deals with the $k=1$ case, one can use Stallings
theorem to associate an action
$\varphi^{(k)}_{T}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ for any
ribbon tube $T$, which actually conjugates, in
$\textnormal{R}_{k}\pi_{1}(B^{4}\setminus T)\cong\textnormal{R}_{k}F_{n}$, the
$i$th meridians of $T$ by its $i$th preferred longitude. By another use of
Stallings theorem in dimension one more, this action is invariant under
concordance, and it can be directly seen that $RC_{k}$-equivalent ribbon tubes
have $k$-reduced $i$th longitudes which are congruent modulo $J_{i}^{k}$
(borrowing notation from Thm. 2.8). As the Tube map is known to preserve the
associated groups and longitudes, this leads to a map
$\varphi^{(k)}_{r}:\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$rT(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$RC_{k}$-concordance}$}}}\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}),$
which satisfies $\varphi^{(k)}=\varphi^{(k)}_{r}\circ\textrm{Tube}_{k}$. It
follows then from the injectivity of $\varphi^{(k)}$ that $\textrm{Tube}_{k}$
is injective, hence an isomorphism. This implies that an element in the kernel
of the Tube map is trivial up to self $w_{k}$-concordance.
## 4\. The link case
By Theorems 2.8 and 3.19, welded string links are classified up to self
$w_{k}$–equivalence by their $k$–reduced longitudes. Similar phenomena occur
in the classifications up to self-virtualization [1] and $w_{k}$–concordance
[7], and these results were extended to the case of welded links in [3] and
[7] respectively, in terms of (adaptations of) the peripheral system. In this
final section, we outline how a similar extension can be derived for links up
to self $w_{k}$–concordance.
_Welded links_ are defined in the very same way as welded string links, by
replacing the copies of the unit interval with copies of the circle $S^{1}$.
Let $L$ be a welded link. Using the same Wirtinger-like procedure as in
Section 2.1, a group $G(L)$ can be associated to any welded link $L$, which
agrees with the fundamental group of the complement if $L$ is a classical
link. A choice of one generic basepoint on each component determines a set of
meridians which normally generates $G(L)$. Up to Detour moves, these
basepoints also allow to cut open $L$ into a welded string links $S_{L}$, and
$G(L)$ is isomorphic to the quotient of $G(S_{L})$ which identifies, for each
strand, the meridians associated with its two endpoints. An _$i$ th longitude_
for $L$ can then be defined as the image of the $i$th longitude for $S_{L}$ in
this quotient. Of course, the result depends on the choice of the basepoints:
moving the $i$th basepoint actually results in conjugating simultaneously the
associated $i$th meridian and longitude.666Note that, since all the meridians
defined in this way are conjugated, all these normally generating sets of
meridians lead to the same notion of $k$–reduced quotient for $G(L)$.
###### Definition 4.1.
Let $k$ be a positive integer. The _$k$ –reduced peripheral system_ of $L$ is
defined as
$\big{(}\textnormal{R}_{k}G(L),\\{x_{i}\\}_{i},\\{\lambda^{k}_{i}\Gamma_{k}N_{i}\\}_{i}\big{)}$
where, for a given choice of basepoints, $(x_{i},\lambda^{k}_{i})$ are the
images in $\textnormal{R}_{k}G(L)$ of the associated meridians and preferred
$i$th longitudes, and $\lambda^{k}_{i}.\Gamma_{k}N_{i}$ is the image in
$\textnormal{R}_{k}G(L)$ of the coset of $\lambda^{k}_{i}$ modulo
$\Gamma_{k}N_{i}$, with $N_{i}$ the normal subgroup of
$\textnormal{R}_{k}G(L)$ generated by $x_{i}$. It is well-defined up to, for
each $i$, simultaneous conjugation of $x_{i}$ and $\lambda^{k}_{i}$ by an
element of $\textnormal{R}_{k}G(L)$.
###### Remark 4.2.
The $1$–reduced peripheral system coincides with the _reduced peripheral
system_ introduced by Milnor in his foundational paper [21] on links up to
link-homotopy, see also [3].
The arrow calculus reviewed in Section 3.1 applies in the exact same way to
welded links, and leads to the notions of _self $w_{k}$-equivalence_ and _self
$w_{k}$-concordance_ for these objects, as in Definitions 3.14 and 3.18.
Theorem 3.16 also holds for welded links, since the proof is purely local.
The main result of this section is the following.
###### Theorem 4.3.
Two (welded) links have equivalent $k$–reduced peripheral systems if and only
if they are self $w_{k}$–concordant.
The rest of this section is devoted to the proof of Theorem 4.3. We stress
that all ingredients of the proof are mostly corollaries of their string link
counterparts, and straightforward adaptations of techniques of [3, 7]. Hence
we will only sketch the proof of Theorem 4.3 below, only hinting to the main
arguments and outlining the specificities of the link case.
The fact that the $k$–reduced peripheral system is invariant under self
$w_{k}$–concordance is an easy consequence of the string link case addressed
in the previous sections. Indeed, a notion of $k$–reduced peripheral system
can be defined for welded string links as in Definition 4.1, except that, in
this case, there is a natural choice of meridians, and it follows from
Theorems 3.19 and 2.8 that this is invariant under self $w_{k}$–concordance.
By fixing a set of basepoints on a welded link $L$, and considering its
$k$–reduced peripheral system as a quotient of the $k$–reduced peripheral
system of the associated welded string link $S_{L}$, we obtain the desired
invariance property.
The proof of ‘only if’ part of Theorem 4.3 goes roughly along the same lines
as the proof of (i)$\Rightarrow$(ii) of Theorem 3.19, given in Section 3.3.
The analogue of Definition 3.20 in the link case is the following, see [3]: a
$w$–tree presentation for a welded link is _sorted_ if, on each component, all
the heads are adjacent. (A $w$–tree presentation for a welded link is a union
of $w$-trees for the trivial diagram of the unlink.)
Since the $w_{kn+1}$–equivalence implies the self $w_{k}$–equivalence by
Remark 3.15 and Theorem 3.16, we have the following as a direct corollary of
[7, Prop. 3.5].
###### Proposition 4.4.
Any welded link is self $w_{k}$–equivalent to a welded link admitting a sorted
presentation.
From this proposition, and the invariance of the $k$–reduced group under self
$w_{k}$–equivalence, the exact same argument as in [3, Lem. 1.18] proves the
following.
###### Proposition 4.5.
For every welded link $L$, its $k$–reduced group admits the following
presentation
$\textnormal{R}_{k}G(L)\cong\big{\langle}x_{1},\ldots,x_{n}\ |\
\Gamma_{k+1}N_{i}\textrm{ and }[x_{i},\lambda^{k}_{i}]\textrm{ for all
$i$}\big{\rangle},$
where, for each $i$, $x_{i}$ is a meridian on the $i$th component, $N_{i}$ is
the normal subgroup generated by $x_{i}$, and $\lambda^{k}_{i}$ is a
representative word for the corresponding longitude.
The rest of the proof then follows the exact same lines as [7, Prop. 3.7]. We
start with two welded links which have equivalent $k$–reduced peripheral
systems, and consider sorted presentations using Proposition 4.4. The main
difference with the string link case of Section 3.3 is that a fourth operation
is involved on the $i$th preferred longitudes: by Proposition 4.5, these
longitudes might differ by the insertion or deletion of a commutator
$[x_{j},\lambda^{k}_{j}]$ for some $j$. This extra operation can be achieved
by a $w_{k}$-concordance on sorted presentations, with the same trick as
illustrated in the first figure of [7, Proof of Prop. 3.7]. This concludes the
proof of Theorem 4.3.
Notice that, in the above argument, the welded concordance is only needed for
the fourth operation that inserts/deletes a commutator
$[x_{j},\lambda^{k}_{j}]$ for some $j$. This is because such relators appear
in the presentation of $\textnormal{R}_{k}G(L)$ given in Proposition 4.4. In
the special case of a link $L$ with vanishing Milnor invariants
$\overline{\mu}_{L}(I)$ with $r(I)\leq k$, we have that
$\textnormal{R}_{k}G(L)\cong RF_{n}$, meaning that this extra operation
involving welded concordance is not needed for the proof. This implies that a
(welded) link has vanishing Milnor invariants $\overline{\mu}(I)$ with
$r(I)\leq k$, if and only if it is self $w_{k}$-equivalent to the unlink, as
stated in the introduction. This is shown by applying verbatim the same
argument as in [7, §3.2].777More precisely, the exact same arguments showing
that [7, Thm. 3.8] implies [7, Cor. 3.11], shows that Theorem 4.3 implies the
theorem stated at the end of the introduction.
## References
* [1] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner. Homotopy classification of ribbon tubes and welded string links. Ann. Sc. Norm. Super. Pisa Cl. Sci., 17(1):713–761, 2017.
* [2] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner. On usual, virtual and welded knotted objects up to homotopy. J. Math. Soc. Japan, 69(3):1079–1097, 2017.
* [3] B. Audoux and J.-B. Meilhan. Characterization of the reduced peripheral system of links. arXiv:1904.04763, 2019.
* [4] B. Audoux, J.-B. Meilhan, and A. Yasuhara. Milnor concordance invariant for knotted surfaces and beyond. arXiv:2109.14578, 2021.
* [5] H. U. Boden and M. Chrisman. Virtual concordance and the generalized alexander polynomial. J. Knot Theory Ramifications, 30(5):2150030, 2021.
* [6] K.-T. Chen. Commutator calculus and link invariants. Proc. Amer. Math. Soc., 3(4):44–55, 1952.
* [7] B. Colombari. A diagrammatic characterization of milnor invariants. arXiv:2201.01499 , 2021.
* [8] J. Darné. On the stable Andreadakis problem. J. Pure Appl. Algebra, 223(12):5484–5525, 2019.
* [9] T. Fleming and A. Yasuhara. Milnor’s invariants and self $C_{k}$-equivalence. Proc. Amer. Math. Soc., 137(2):761–770, 2009.
* [10] R. Gaudreau. Classification of virtual string links up to cobordism. Ars Math. Contemp., 19(1):37–49, 2020.
* [11] M. Goussarov, M. Polyak, and O. Viro. Finite-type invariants of classical and virtual knots. Topology, 39(5):1045–1068, 2000.
* [12] N. Habegger and X.-S. Lin. The classification of links up to link-homotopy. J. Amer. Math. Soc., 3:389–419, 1990.
* [13] M. Hall. The Theory of Groups. AMS Chelsea Publishing, 1959.
* [14] P. Hall. A contribution to the theory of groups of prime-power order. Proc. Lond. Math. Soc. (2), 36:29–95, 1933.
* [15] L. H. Kauffman. Virtual knot theory. European J. Combin., 20(7):663–690, 1999.
* [16] J. P. Levine. An approach to homotopy classification of links. Trans. Am. Math. Soc., 306(1):361–387, 1988.
* [17] W. Magnus. Über Beziehungen zwischen höheren Kommutatoren. J. Reine Angew. Math., 177:105–115, 1937.
* [18] W. Magnus, A. Karrass, and D. Solitar. Combinatorial group theory : presentations of groups in terms of generators and relations. Dover books on advanced mathematics. Dover, New York, 1976.
* [19] J.-B. Meilhan and A. Yasuhara. Arrow calculus for welded and classical links. Alg. Geom. Topol., 19(1):397–456, 2019.
* [20] J. Milnor. Link groups. Ann. of Math. (2), 59:177–195, 1954.
* [21] J. Milnor. Isotopy of links. Algebraic geometry and topology. In A symposium in honor of S. Lefschetz, pages 280–306. Princeton University Press, Princeton, N. J., 1957.
* [22] H. A. Miyazawa, K. Wada, and A. Yasuhara. Combinatorial Approach to Milnor Invariants of Welded Links. Michigan Mathematical Journal, pages 1 – 30, 2021.
* [23] S. Satoh. Virtual knot presentation of ribbon torus-knots. J. Knot Theory Ramifications, 9(4):531–542, 2000.
* [24] T. Watanabe. Clasper-moves among ribbon 2-knots characterizing their finite type invariants. J. Knot Theory Ramifications, 15(9):1163–1199, 2006.
* [25] E. Witt. Treue Darstellung Liescher Ringe. J. Reine Angew. Math., 177:152–160, 1937.
* [26] A. Yasuhara. Classification of string links up to self delta-moves and concordance. Algebr. Geom. Topol., 9(1):265–275, 2009.
* [27] A. Yasuhara. Self delta-equivalence for links whose Milnor’s isotopy invariants vanish. Trans. Amer. Math. Soc., 361(9):4721–4749, 2009.
* [28] E. Yurasovskaya. Homotopy string links over surfaces. PhD Thesis, The University of British Columbia, 2008.
|
# HesScale: Scalable Computation of Hessian
Diagonals
Mohamed Elsayed & A. Rupam Mahmood
Department of Computing Science, Alberta Machine Intelligence Institute (Amii)
University of Alberta
Edmonton, Alberta, Canada
<EMAIL_ADDRESS>
CIFAR AI Chair
###### Abstract
Second-order optimization uses curvature information about the objective
function, which can help in faster convergence. However, such methods
typically require expensive computation of the Hessian matrix, preventing
their usage in a scalable way. The absence of efficient ways of computation
drove the most widely used methods to focus on first-order approximations that
do not capture the curvature information. In this paper, we develop HesScale,
a scalable approach to approximating the diagonal of the Hessian matrix, to
incorporate second-order information in a computationally efficient manner. We
show that HesScale has the same computational complexity as backpropagation.
Our results on supervised classification show that HesScale achieves high
approximation accuracy, allowing for scalable and efficient second-order
optimization.111Code is available at https://github.com/mohmdelsayed/HesScale.
## 1 Introduction
First-order optimization offers a cheap and efficient way of performing local
progress in optimization problems by using gradient information. However,
their performance suffers from instability or slow progress when used in ill-
conditioned landscapes. Such a problem is present because first-order methods
do not capture curvature information which causes two interrelated issues.
First, the updates in first-order have incorrect units (Duchi et al. 2011),
which creates a scaling issue. Second, first-order methods lack
parameterization invariance (Martens 2020) in contrast to second-order methods
such as natural gradient (Amari 1998) or Newton-Raphson methods. Therefore,
some first-order normalization methods were developed to address the
invariance problem (Ba et al. 2016, Ioffe & Szegedy 2015, Salimans & Kingma
2016). On the other hand, some recent adaptive step-size methods try to
alleviate the scaling issue by using gradient information for first-order
curvature approximation (Luo et al. 2019, Duchi et al. 2011, Zeiler 2012,
Reddi et al. 2018, Kingma & Ba 2015, Tran & Phong 2019, Tieleman et al. 2012).
Specifically, such methods use the empirical Fisher diagonals heuristic by
maintaining a moving average of the squared gradients to approximate the
diagonal of the Fisher information matrix. Despite the huge adoption of such
methods due to their scalability, they use inaccurate approximations. Kunstner
et al. (2019) showed that the empirical Fisher does not generally capture
curvature information and might have undesirable effects. They argued that the
empirical Fisher approximates the Fisher or the Hessian matrices only under
strong assumptions that are unlikely to be met in practice. Moreover, Wilson
et al. (2017) presented a counterexample where the adaptive step-size methods
are unable to reduce the error compared to non-adaptive counterparts such as
stochastic gradient descent.
Although second-order optimization can speed up the training process by using
the geometry of the landscape, its adoption is minimal compared to first-order
methods. The exact natural gradient or Newton-Raphson methods require the
computation, storage, and inversion of the Fisher information or the Hessian
matrices, making them computationally prohibitive in large-scale tasks.
Accordingly, many popular second-order methods attempt to approximate less
expensively. For example, a type of truncated-Newton method called Hessian-
free methods (Martens 2010) exploits the fact that the Hessian-vector product
is cheap (Bekas et al. 2007) and uses the iterative conjugate gradient method
to perform an update. However, such methods might require many iterations per
update or some tricks to achieve stability, adding computational overhead
(Martens & Sutskever 2011). Some variations try to approximate only the
diagonals of the Hessian matrix using stochastic estimation with matrix-free
computations (Chapelle & Erhan 2011, Martens et al. 2012, Yao et al. 2021).
Other methods impose probabilistic modeling assumptions and estimate a block
diagonal Fisher information matrix (Martens & Grosse 2015, Botev et al. 2017).
Such methods are invariant to reparametrization but are computationally
expensive since they need to perform matrix inversion for each block.
Deterministic diagonal approximations to the Hessian (LeCun et al. 1990,
Becker & Lecun 1989) provide some curvature information and are efficient to
compute. Specifically, they can be implemented to be as efficient as first-
order methods. We view this category of approximation as scalable second-order
methods. In neural networks, curvature backpropagation (Becker & Lecun 1989,
Mizutani & Dreyfus 2008) can be used to backpropagate the curvature vector.
Although these methods show a promising direction for scalable second-order
optimization, the approximation quality is sometimes poor with objectives such
as cross-entropy (Martens et al. 2012). A scalable second-order method with
high quality approximation is still needed.
In this paper, we present HesScale, a high-quality approximation method for
the Hessian diagonals. Our method is also scalable and has little memory
requirement with linear computational complexity while maintaining high
approximation accuracy.
## 2 Background
In this section, we describe the Hessian matrix for neural networks and some
existing methods for estimating it. Generally, Hessian matrices can be
computed for any scalar-valued function that are twice differentiable. If
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is such a function, then for its
argument ${\bm{\psi}}\in\mathbb{R}^{n}$, the Hessian matrix
${\bm{H}}\in\mathbb{R}^{n\times n}$ of $f$ with respect to ${\bm{\psi}}$ is
given by
$H_{i,j}=\nicefrac{{\partial^{2}f({\bm{\psi}})}}{{\partial\psi_{i}\partial\psi_{j}}}$.
Here, the $i$th element of a vector ${\bm{v}}$ is denoted by $v_{i}$, and the
element at the $i$th row and $j$th column of a matrix ${\bm{M}}$ is denoted by
$M_{i,j}$. When the need for computing the Hessian matrix arises for
optimization in deep learning, the function $f$ is typically the objective
function, and the vector ${\bm{\psi}}$ is commonly the weight vector of a
neural network. Computing and storing an $n\times n$ matrix, where $n$ is the
number of weights in a neural network, is expensive. Therefore, many methods
exist for approximating the Hessian matrix or parts of it with less memory
footprint, computational requirement, or both. A common technique is to
utilize the structure of the function to reduce the computations needed. For
example, assuming that connections from a certain layer do not affect other
layers in a neural network allows one to approximate a block diagonal Hessian.
The computation further simplifies when we have piece-wise linear activation
functions (e.g., ReLU), which result in a _Generalized Gauss-Newton_ (GGN)
(Schraudolph 2002) approximation that is equivalent to the block diagonal
Hessian matrix with linear activation functions. The GGN matrix is more
favored in second-order optimization since it is positive semi-definite.
However, computing a block diagonal matrix is still demanding.
Many approximation methods were developed to reduce the storage and
computation requirements of the GGN matrix. For example, under probabilistic
modeling assumptions, the _Kronecker-factored Approximate Curvature_ (KFAC)
method (Martens & Grosse 2015) writes the GGN matrix ${\bm{G}}$ as a Kronecker
product of two matrices of smaller sizes as:
${\bm{G}}={\bm{A}}\otimes{\bm{B}}$, where
${\bm{A}}=\mathbb{E}[{\bm{h}}{\bm{h}}^{\top}]$,
${\bm{B}}=\mathbb{E}[{\bm{g}}{\bm{g}}^{\top}]$, ${\bm{h}}$ is the activation
output vector, and ${\bm{g}}$ is the gradient of the loss with respect to the
activation input vector. The ${\bm{A}}$ and ${\bm{B}}$ matrices can be
estimated by Monte Carlo sampling and an exponential moving average. KFAC is
more efficient when used in optimization since it requires inverting only the
small matrices using the Kronecker-product property
$({\bm{A}}\otimes{\bm{B}})^{-1}={\bm{A}}^{-1}\otimes{\bm{B}}^{-1}$. However,
KFAC is still expensive due to the storage of the block diagonal matrices and
computation of Kronecker product, which prevent it from being used as a
scalable method.
Computing the Hessian diagonals can provide some curvature information with
relatively less computation. However, it has been shown that the exact
computation for diagonals of the Hessian typically has quadratic complexity
with the unlikely existence of algorithms that can compute the exact diagonals
with less than quadratic complexity (Martens et al. 2012). Some stochastic
methods provide a way to compute unbiased estimates of the exact Hessian
diagonals. For example, the AdaHessian (Yao et al. 2021) algorithm uses the
Hutchinson’s estimator
$\text{diag}({\bm{H}})=E[{\bm{z}}\circ({\bm{H}}{\bm{z}})]$, where ${\bm{z}}$
is a multivariate random variable with a Rademacher distribution and the
expectation can be estimated using Monte Carlo sampling with an exponential
moving average. Similarly, the GGN-MC method (Dangel et al. 2020) uses the
relationship between the Fisher information matrix and the Hessian matrix
under probabilistic modeling assumptions to have an MC approximation of the
diagonal of the GGN matrix. Although these stochastic approximation methods
are scalable due to linear or $O(n)$ computational and memory complexity, they
suffer from low approximation quality, improving which requires many sampling
and factors of additional computations.
## 3 The Proposed HesScale Method
In this section, we present our method for approximating the diagonal of the
Hessian at each layer in feed-forward networks, where a backpropagation rule
is used to utilize the Hessian of previous layers. We present the derivation
of the backpropagation rule for fully connected and convolutional neural
networks in supervised learning. Similar derivation for fully connected
networks with mean squared error is presented before (LeCun et al. 1990,
Becker & Lecun 1989). However, we use the exact diagonals of the Hessian
matrix at the last layer with some non-linear and non-element-wise output
activations such as softmax and show that it can still be computed in linear
computational complexity. We show the derivation for Hessian diagonals for
fully connected networks in the following and provide the derivation for the
convolutional neural networks in Appendix B.
We use the supervised classification setting where there is a collection of
data examples. These data examples are generated from some _target function_
${f}^{*}$ mapping the input ${\bm{x}}$ to the output $y$, where the $k$-th
input-output pair is $({\bm{x}}_{k},y_{k})$. In this task, the _learner_ is
required to predict the output class $y\in\\{1,2,...,m\\}$ given the input
vector ${\bm{x}}\in\mathbb{R}^{d}$ by estimating the target function
${f}^{*}$. The performance is measured with the cross-entropy loss,
$\mathcal{L}({\bm{p}},{\bm{q}})=-\sum_{i=1}^{m}p_{i}\log q_{i}$, where
${\bm{p}}\in\mathbb{R}^{m}$ is the vector of the target one-hot encoded class
and ${\bm{q}}\in\mathbb{R}^{m}$ is the predicted output. The learner is
required to reduce the cross-entropy by matching the target class.
Consider a neural network with $L$ layers that outputs the predicted output
${\bm{q}}$. The neural network is parametrized by the set of weights
$\\{{\bm{W}}_{1},...,{\bm{W}}_{L}\\}$, where ${\bm{W}}_{l}$ is the weight
matrix at the $l$-th layer, and its element at the $i$th row and the $j$th
column is denoted by $W_{l,i,j}$. During learning, the parameters of the
neural network are changed to reduce the loss. At each layer $l$, we get the
activation output ${\bm{h}}_{l}$ by applying the activation function
$\bm{\sigma}$ to the activation input ${\bm{a}}_{l}$:
${\bm{h}}_{l}=\bm{\sigma}({\bm{a}}_{l})$. We simplify notations by defining
${\bm{h}}_{0}\doteq{\bm{x}}$. The activation output ${\bm{h}}_{l}$ is then
multiplied by the weight matrix ${\bm{W}}_{l+1}$ of layer $l+1$ to produce the
next activation input:
${a}_{l+1,i}=\sum_{j=1}^{|{\bm{h}}_{l}|}{{W}_{l+1,i,j}}{h}_{l,j}$. We assume
here that the activation function is element-wise activation for all layers
except for the final layer $L$, where it becomes the softmax function. The
backpropagation equations for the described network are given as follows
Rumelhart et al. (1986):
$\displaystyle\frac{\partial\mathcal{L}}{\partial a_{l,i}}$
$\displaystyle=\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}\frac{\partial a_{l+1,k}}{\partial h_{l,i}}\frac{\partial
h_{l,i}}{\partial
a_{l,i}}=\sigma^{\prime}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}W_{l+1,k,i},$ (1) $\displaystyle\frac{\partial\mathcal{L}}{\partial
W_{l,i,j}}$ $\displaystyle=\frac{\partial\mathcal{L}}{\partial
a_{l,i}}\frac{\partial a_{l,i}}{\partial
W_{l,i,j}}=\frac{\partial\mathcal{L}}{\partial a_{l,i}}h_{l-1,j}.$ (2)
In the following, we write the equations for the exact Hessian diagonals with
respect to weights
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{W^{2}_{l,i,j}}}}$, which
requires the calculation of
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{a^{2}_{l,i}}}}$ first:
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{l,i}}}$
$\displaystyle=\frac{\partial}{\partial
a_{l,i}}\left(\sigma^{\prime}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}W_{l+1,k,i}\right)$
$\displaystyle={\sigma^{\prime}}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\sum_{p=1}^{|{\bm{a}}_{l+1}|}\frac{\partial^{2}\mathcal{L}}{\partial
a_{l+1,k}\partial a_{l+1,p}}\frac{\partial a_{l+1,p}}{\partial
a_{l,i}}W_{l+1,k,i}+\sigma^{\prime\prime}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}W_{l+1,k,i}$
$\displaystyle={\sigma^{\prime}}(a_{l,i})^{2}\sum_{k=1}^{|{\bm{a}}_{l+1}|}\sum_{p=1}^{|{\bm{a}}_{l+1}|}\frac{\partial^{2}\mathcal{L}}{\partial
a_{l+1,k}\partial
a_{l+1,p}}W_{l+1,p,i}W_{l+1,k,i}+\sigma^{\prime\prime}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}W_{l+1,k,i},$
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}$
$\displaystyle=\frac{\partial}{\partial
W_{l,i,j}}\left(\frac{\partial\mathcal{L}}{\partial
a_{l,i}}h_{l-1,j}\right)=\frac{\partial}{\partial
a_{l,i}}\left(\frac{\partial\mathcal{L}}{\partial
a_{l,i}}\right)\frac{\partial a_{l,i}}{\partial
W_{l,i,j}}h_{l-1,j}=\frac{\partial^{2}\mathcal{L}}{\partial
a^{2}_{l,i}}h^{2}_{l-1,j}.$ (3)
Since, the calculation of
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{a^{2}_{l,i}}}}$ depends on the
off-diagonal terms, the computation complexity becomes quadratic. Following
Becker and Lecun (1989), we approximate the Hessian diagonals by ignoring the
off-diagonal terms, which leads to a backpropagation rule with linear
computational complexity for our estimates
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}}$ and
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{l,i}}}}$:
$\displaystyle\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{l,i}}}}$
$\displaystyle\doteq{\sigma^{\prime}}(a_{l,i})^{2}\sum_{k=1}^{|{\bm{a}}_{l+1}|}\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{l+1,k}}}}W^{2}_{l+1,k,i}+\sigma^{\prime\prime}(a_{l,i})\sum_{k=1}^{|{\bm{a}}_{l+1}|}\frac{\partial\mathcal{L}}{\partial
a_{l+1,k}}W_{l+1,k,i},$ (4)
$\displaystyle\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}}$
$\displaystyle\doteq\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{l,i}}}}h^{2}_{l-1,j}.$
(5)
However, for the last layer, we use the exact Hessian diagonals
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{L,i}}}}\doteq\frac{\partial^{2}\mathcal{L}}{\partial{a^{2}_{L,i}}}$
since it can be computed in $O(n)$ for the softmax activation function and the
cross-entropy loss. More precisely, the exact Hessian diagonals for cross-
entropy loss with softmax is simply ${\bm{q}}-{\bm{q}}\circ{\bm{q}}$, where
${\bm{q}}$ is the predicted probability vector and $\circ$ denotes element-
wise multiplication. We found empirically that this small change makes a large
difference in the approximation quality, as shown in Fig. 1(a). Hence, unlike
Becker and Lecun (1989) who use a Hessian diagonal approximation of the last
layer by Eq. 4, we use the exact values directly to achieve more approximation
accuracy. We call this method for Hessian diagonal approximation _HesScale_
and provide its pseudocode for supervised classification in Algorithm 1.
Algorithm 1 HesScale: Computing Hessian diagonals of a neural network layer in
classification
Neural network $f$ and a layer number $l$
First and second order information
$\widehat{\frac{\partial\mathcal{L}}{\partial a_{l+1,i}}}$ and
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial a_{l+1,i,j}^{2}}}$, unless
$l=L$
Input-output pair $({\bm{x}},y)$
Set loss function $\mathcal{L}$ to cross-entropy loss
Compute preference vector ${\bm{a}}_{L}\leftarrow f({\bm{x}})$ and target one-
hot-encoded vector ${\bm{p}}\leftarrow\texttt{onehot}(y)$
Compute the predicted probability vector
${\bm{q}}\leftarrow{\bm{\sigma}}({\bm{a}}_{L})$ using softmax function
${\bm{\sigma}}$
Compute the error $\mathcal{L}({\bm{p}},{\bm{q}})$
if $l=L$ then $\triangleright$ Computing Hessian diagonals exactly for the
last layer
Compute
$\frac{\partial\mathcal{L}}{\partial{\bm{a}}_{L}}\leftarrow{\bm{q}}-{\bm{p}}$$\triangleright$
$\frac{\partial\mathcal{L}}{\partial{\bm{a}}_{L}}$ consists of elements
$\frac{\partial\mathcal{L}}{\partial a_{L,i}}$
Compute $\frac{\partial\mathcal{L}}{\partial{\bm{W}}_{L}}$ using Eq. 2
$\triangleright$ $\frac{\partial\mathcal{L}}{\partial{\bm{W}}_{L}}$ consists
of elements $\frac{\partial\mathcal{L}}{\partial W_{L,i,j}}$
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{a}}_{L}^{2}}}\leftarrow{\bm{q}}-{\bm{q}}\circ{\bm{q}}$
$\triangleright$
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{a}}_{L}^{2}}}$ consists
of elements $\widehat{\frac{\partial^{2}\mathcal{L}}{\partial a_{L,i}^{2}}}$
Compute $\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{W}}_{L}^{2}}}$
using Eq. 5$\triangleright$
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{W}}_{L}^{2}}}$ consists
of elements $\widehat{\frac{\partial^{2}\mathcal{L}}{\partial W_{L,i,j}^{2}}}$
else if $l\neq L$ then
Compute $\frac{\partial\mathcal{L}}{\partial{\bm{a}}_{l}}$ and
$\nicefrac{{\partial\mathcal{L}}}{{\partial{\bm{W}}_{l}}}$ using Eq. 1 and Eq.
2
Compute $\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{a}}_{l}^{2}}}$
and $\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{W}}_{l}^{2}}}$ using
Eq. 4 and Eq. 5
end if
return $\frac{\partial\mathcal{L}}{\partial{\bm{W}}_{l}}$,
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{W}}_{l}^{2}}}$,
$\frac{\partial\mathcal{L}}{\partial{\bm{a}}_{l}}$, and
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{a}}_{l}^{2}}}$
HesScale is not specific to cross-entropy loss as the exact Hessian diagonals
can be calculated in $O(n)$ for some other widely used loss functions as well.
We show this property for negative log-likelihood function with Gaussian and
softmax distributions in Appendix A. The computations can be reduced further
using a linear approximation for the activation functions (by dropping the
second term in Eq. 4), which corresponds to an approximation of the GGN
matrix. We call this variation of our method _HesScaleGN_.
Based on HesScale, we make a stable optimizer, which we call AdaHesScale,
given in Algorithm 2. We use the same style introduced in Adam (Kingma & Ba
2015), using the squared diagonal approximation instead of the squared
gradients to update the moving average. Moreover, we introduce another
optimizer based on HesScaleGN, which we call AdaHesScaleGN.
Algorithm 2 AdaHesScale for optimization
Neural network $f$ with weights $\\{{\bm{W}}_{1},...,{\bm{W}}_{L}\\}$ and a
dataset $\mathcal{D}$
Small number $\epsilon\leftarrow 10^{-8}$
Exponential decay rates $\beta_{1},\beta_{2}\in[0,1)$
step size $\alpha$
Initialize $\\{{\bm{W}}_{1},...,{\bm{W}}_{L}\\}$
Initialize time step $t\leftarrow 0$.
for $l$ in $\\{L,L-1,...,1\\}$ do $\triangleright$ Set exponential moving
averages at time step 0 to zero
${\bm{M}}_{l}\leftarrow 0;\quad{\bm{V}}_{l}\leftarrow 0$ $\triangleright$ Same
size as ${\bm{W}}_{l}$
end for
for $({\bm{x}},y)$ in $\mathcal{D}$ do
$t\leftarrow t+1$
${\bm{r}}_{L+1}\leftarrow{\bm{s}}_{L+1}\leftarrow\bm{\emptyset}$
$\triangleright$ ${\bm{r}}_{l}$ and ${\bm{s}}_{l}$ stand for
$\frac{\partial\mathcal{L}}{\partial{\bm{a}}_{l}}$ and
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{a}}_{l}^{2}}}$,
respectively
for $l$ in $\\{L,L-1,...,1\\}$ do
${\bm{F}}_{l},{\bm{S}}_{l},{\bm{r}}_{l},{\bm{s}}_{l}\;\leftarrow\;$
HesScale($f,{\bm{x}},y,l,{\bm{r}}_{l+1},{\bm{s}}_{l+1}$). $\triangleright$
Check Algorithm 1
${\bm{M}}_{l}\leftarrow\beta_{1}{\bm{M}}_{l}+(1-\beta_{1}){\bm{F}}_{l}$
$\triangleright$ ${\bm{F}}_{l}$ stands for
$\frac{\partial\mathcal{L}}{\partial{\bm{W}}_{l}}$
${\bm{V}}_{l}\leftarrow\beta_{2}{\bm{V}}_{l}+(1-\beta_{2}){\bm{S}}_{l}^{2}$
$\triangleright$ ${\bm{S}}_{l}$ stands for
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{\bm{W}}_{l}^{2}}}$
$\hat{{\bm{M}}}_{l}\leftarrow{\bm{M}}_{l}/(1-\beta_{1}^{t})$ $\triangleright$
Bias-corrected estimate for ${\bm{F}}_{l}$
$\hat{{\bm{V}}}_{l}\leftarrow{\bm{V}}_{l}/(1-\beta_{2}^{t})$ $\triangleright$
Bias-corrected estimate for ${\bm{S}}_{l}$
${\bm{W}}_{l}\leftarrow{\bm{W}}_{l}-\alpha\hat{{\bm{M}}}_{l}\oslash(\hat{{\bm{V}}}_{l}+\epsilon)^{\circ\frac{1}{2}}$
$\triangleright$ $\oslash$ is element-wise division
$\triangleright$ ${\bm{A}}^{\circ\frac{1}{2}}$ is element-wise square root of
${\bm{A}}$
end for
end for
## 4 Approximation quality & scalability of HesScale
In this section, we evaluate HesScale for its approximation quality and
computational cost and compare it with other methods. These measures
constitute the criteria we look for in scalable and efficient methods. For our
experiments, we implemented HesScale using the _BackPack_ framework (Dangel et
al. 2020), which allows easy implementation of backpropagation of statistics
other than the gradient.
We start by studying the approximation quality of Hessian diagonals compared
to the true values. To measure the approximation quality of the Hessian
diagonals for different methods, we use the $L^{1}$ distance between the exact
Hessian diagonals and their approximations. Our task here is supervised
classification, and data examples are randomly generated. We used a network of
three hidden layers with _tanh_ activations, each containing 16 units. The
network weights and biases are initialized randomly. The network has six
inputs and ten outputs. For each example pair, we compute the exact Hessian
diagonals for each layer and their approximations from each method. All
layers’ errors are summed and averaged over 1000 data examples for each
method. In this experiment, we used 40 different initializations for the
network weights, shown as colored dots in Fig. 1(a). Each point represents the
summed error over network layers, averaged over 1000 examples for each
different initialization. In this figure, we show the average error incurred
by each method normalized by the average error incurred by HesScale. Any
approximation that incurs an averaged error above 1 has a worse approximation
than HesScale, and any approximation with an error less than 1 has a better
approximation than HesScale. Moreover, we show the layer-wise error for each
method in Fig. 1(b).
(a) Normalized $L^{1}$ error with respect to HesScale
(b) Layer-wise $L^{1}$ error
Figure 1: The averaged error for each method is normalized by the averaged
error incurred by HesScale. We show 40 initialization points with the same
colors across all methods. The norm of the vector of Hessian diagonals
$|\text{diag}({\bm{H}})|$ is shown as a reference.
Different Hessian diagonal approximations are considered for comparison with
HesScale. We included several deterministic and stochastic approximations for
the Hessian diagonals. We also include the approximation of the Fisher
Information Matrix done by squaring the gradients and denoted by $g^{2}$,
which is highly adopted by many first-order methods (e.g., Kingma and Ba,
2015). We compare HesScale with three stochastic approximation methods:
AdaHessian (Yao et al. 2021), Kronecker-factored approximate curvature (KFAC)
(Martens & Grosse 2015), and the Monte-Carlo (MC) estimate of the GGN matrix
(GGN-MC) (Dangel et al. 2020). We also compare HesScale with two deterministic
approximation methods: the diagonals of the exact GGN matrix (Schraudolph
2002) ($\text{diag}({\bm{G}})$) and the diagonal approximation by Becker and
Lecun (1989) (BL89). In KFAC, we extract the diagonals from the block diagonal
matrix and show the approximation error averaged over 1 MC sample (KFAC-MC1)
and over 50 MC samples (KFAC-MC50), both per each data example. Since
AdaHessian and GGN-MC are already diagonal approximations, we use them
directly and show the error with 1 MC sample (GGN-MC1 & AdaHessian-MC1) and
with 50 MC samples (GGN-MC50 & AdaHessian-MC50).
HesScale provides a better approximation than the other deterministic and
stochastic methods. For stochastic methods, we use many MC samples to improve
their approximation. However, their approximation quality is still poor.
Methods approximating the GGN diagonals do not capture the complete Hessian
information since the GGN and Hessian matrices are different when the
activation functions are not piece-wise linear. Although these methods
approximate the GGN diagonals, their approximation is significantly better
than the AdaHessian approximation. And among the methods for approximating the
GGN diagonals, HesScaleGN performs the best and close to the exact GGN
diagonals. This experiment clearly shows that HesScale achieves the best
approximation quality compared to other stochastic and deterministic
approximation methods.
Next, we perform another experiment to evaluate the computational cost of our
optimizers. Our Hessian approximation methods and corresponding optimizers
have linear computational complexity, which can be seen from Eq. 4 and Eq. 5.
However, computing second-order information in optimizers still incurs extra
computations compared to first-order optimizers, which may impact how the
total computations scale with the number of parameters. Hence, we compare the
computational cost of our optimizers with others for various number of
parameters. More specifically, we measure the update time of each optimizer,
which is the time needed to backpropagate first-order and second-order
information and update the parameters.
We designed two experiments to study the computational cost of first-order and
second-order optimizers. In the first experiment, we used a neural network
with a single hidden layer. The network has 64 inputs and 512 hidden units
with tanh activations. We study the increase in computational time when
increasing the number of outputs exponentially, which roughly doubles the
number of parameters. The set of values we use for the number of outputs is
$\\{2^{4},2^{5},2^{6},2^{7},2^{8},2^{9}\\}$. The results of this experiment
are shown in Fig. 2(a). In the second experiment, we used a neural network
with multi-layers, each containing 512 hidden units with tanh activations. The
network has 64 inputs and 100 outputs. We study the increase in computational
time when increasing the number of layers exponentially, which also roughly
doubles the number of parameters. The set of values we use for the number of
layers is $\\{1,2,4,8,16,32,64,128\\}$. The results of this experiment are
shown in Fig. 2(b). The points in Fig. 2(a) and Fig. 2(b) are averaged over 30
updates. The standard errors of the means of these points are smaller than the
width of each line. On average, we notice that the cost of AdaHessian,
AdaHesScale, and AdaHesScaleGN are three, two, and 1.25 times the cost of
Adam, respectively.
(a) Increasing number of outputs in a neural network
(b) Increasing number of layers in a neural network
Figure 2: The average computation time for each step of an update is shown for
different optimizers. The computed update time is the time needed by each
optimizer to backpropagate gradients or second-order information and to update
the parameters of the network. GGN overlaps with H in (a).
It is clear that our methods are among the most computationally efficient
approximation method for Hessian diagonals.
## 5 Empirical performance of HesScale in Optimization
In this section, we compare the performance of our optimizers—AdaHesScale and
AdaHesScaleGN—with three second-order optimizers: BL89, GGNMC, and AdaHessian.
We also include comparisons to two first-order methods: Adam and SGD. We
exclude KFAC and the exact diagonals of the GGN matrix from our comparisons
due to their prohibitive computations.
Our optimizers are evaluated in the supervised classification problem with a
series of experiments using different architectures and three datasets: MNIST,
CIFAR-10, and CIFAR-100. Instead of attempting to achieve state-of-the-art
performance with specialized techniques and architectures, we follow the
DeepOBS benchmarking work (Schneider et al. 2019) and compare the optimizers
in their generic and pristine form using relatively simple networks. It allows
us to perform a more fair comparison without extensively utilizing specialized
knowledge for a particular task. In the first experiment, we use the MNIST-MLP
task from DeepOBS. The images are flattened and used as inputs to a network of
three fully connected layers (1000, 500, and 100 units) with tanh activations.
We train each method for 100 epochs with a batch size of 128. We show the
training plots in Fig. 6(a) with their corresponding sensitivity plots in
Appendix C, Fig. 8(a). In the second experiment, we use the CIFAR10-3C3D task
from the DeepOBS benchmarking tasks. The network consists of three
convolutional layers with _tanh_ activations, each followed by max pooling.
After that, two fully connected layers (512 and 256 units) with _tanh_
activations are used. We train each method for 100 epochs with a batch size of
128. We show the training plots in Fig. 6(b) with their corresponding
sensitivity plots in Fig. 8(b). In the third experiment, we use the
CIFAR100-3C-3D task from DeepOBS. The network is the same as the one used in
the second task except for the activations are _ELU_. We train each method for
200 epochs with a batch size of 128. We show the training plots in Fig. 7(b)
with their corresponding sensitivity plots in Fig. 9(b). In the fourth
experiment, we use the CIFAR100-ALL-CNN task from DeepOBS with the ALL-CNN-C
network, which consists of 9 convolutional layers (Springenberg et al. 2015).
We use ELU activations insead of ReLU to differentiate between the performance
of AdaHesScale and AdaHesScaleGN. We show the training plots in Fig. 7(a) with
their corresponding sensitivity plots in Fig. 9(a).
In the MNIST-MLP and CIFAR-10-3C3D experiments, we performed a hyperparameter
search for each method to determine the best set of $\beta_{1}$, $\beta_{2}$,
and $\alpha$. The range of $\beta_{2}$ is $\\{0.99,0.999,0.9999\\}$ and the
range of $\beta_{1}$ is $\\{0.0,0.9\\}$. The range of step size is selected
for each method to create a convex curve. Our criterion was to find the best
hyperparameter configuration for each method in the search space that
minimizes the area under the validation loss curve. The performance of each
method was averaged over 30 independent runs. Each independent run had the
same initial representation for the algorithms used in an experiment. Using
each method’s best hyperparameter configuration on the validation set, we show
the performance of each method against the time in seconds needed to complete
the required number of epochs, which better depicts the computational
efficiency of the methods. Fig. 3(a) and Fig. 3(b) show these results on
MNIST-MLP and CIFAR-10 tasks. Moreover, we show the sensitivity of each method
to the step size in Fig. 5(a) and Fig. 5(b).
(a) MNIST-MLP
(b) CIFAR-10 3C3D
Figure 3: MNIST-MLP and CIFAR-10 3C3D classification tasks. Each method is
trained for 100 epochs. We show the time taken by each algorithm in seconds
(left) and we show the learning curves in the number of epochs (right). The
performance of each method is averaged over 30 independent runs. The shaded
area represents the standard error.
(a) CIFAR-100 3C3D
(b) CIFAR-100 All-CNN-C
Figure 4: CIFAR-100 3C3D and CIFAR-100 ALL-CNN classification tasks. Each
method from the first task is trained for 200 epochs and each method from the
second task is trained for 350 epochs. We show the time taken by each
algorithm in seconds (left) and we show the learning curves in the number of
epochs (right). The performance of each method is averaged over 30 independent
runs. The shaded area represents the standard error.
(a) MNIST
(b) CIFAR-10
Figure 5: Sensitivity of the step size for each method on MNIST-MLP and
CIFAR-10 3C3D tasks. We select the best values of $\beta_{1}$ and $\beta_{2}$
for each step size $\alpha$.
In the CIFAR-100-ALL-CNN and CIFAR-100-3C3D experiments, we used the set of
$\beta_{1}$ and $\beta_{2}$ that achieved the best robustness in the previous
two tasks. The values used for $\beta_{1}$ and $\beta_{2}$ were $0.9$ and
$0.999$ respectively. We did a hyperparameter search for each method to
determine the best step size using the specified $\beta_{1}$ and $\beta_{2}$.
The range of step size is selected for each method to create a convex curve,
while ensuring that each method uses a similar search budget and the same
interval between search points. Our criterion was to find the best
hyperparameter configuration for each method in the search space that
minimizes the area under the validation loss curve. The performance of each
method was averaged over 30 independent runs. Each independent run had the
same initial representation for the algorithms used in an experiment. Using
each method’s best hyperparameter configuration on the validation set, we show
the performance of each method against the time in seconds needed to complete
the required number of epochs. Fig. 4(a) and Fig. 4(b) show these results on
CIFAR-100-ALL-CNN and CIFAR-100-3C3D tasks.
Our results show that all optimizers except for BL89 performed well on the
MNIST-MLP task. However, in CIFAR-10, CIFAR-100 3c3d, and CIFAR-100 ALL-CNN,
we notice that AdaHessian performed worse than all methods except BL89. This
result is aligned with AdaHessian’s inability to accurately approximate the
Hessian diagonals, as shown in Fig. 1. Moreover, AdaHessian required more
computational time compared to all methods, which is also reflected in Fig. 2.
While being time-efficient, AdaHesScaleGN consistently outperformed all
methods in CIFAR-10-3C3D and CIFAR-100-3C3D, and it outperformed all methods
except AdaHesScale in CIFAR-100 ALL-CNN. This result is aligned with our
methods’ accurate approximation of Hessian diagonals. Our experiments indicate
that incorporating HesScale and HesScaleGN approximations in optimization
methods can be of significant performance advantage in both computation and
accuracy. AdaHesScale and AdaHesScaleGN outperformed other optimizers likely
due to their accurate approximation of the diagonals of the Hessian and GGN,
respectively.
## 6 Conclusion
HesScale is a scalable and efficient second-order method for approximating the
diagonals of the Hessian at every network layer. Our work is based on the
previous work done by Becker and Lecun (1989). We performed a series of
experiments to evaluate HesScale against other scalable algorithms in terms of
computational cost and approximation accuracy. Moreover, we demonstrated how
HesScale can be used to build efficient second-order optimization methods. Our
results showed that our methods provide a more accurate approximation and
require small additional computations.
## 7 Broader Impact
Second-order information is used in domains other than optimization. For
example, some works alleviating catastrophic forgetting use a utility measure
for the network’s connections to protect them. Typically, an auxiliary loss is
used between such connections, and their old values are weighted by their
corresponding importance. Such methods (LeCun et al. 1990, Hassibi & Stork
1993, Dong et al. 2017, Kirkpatrick et al. 2017, Schwarz et al. 2018, Ritter
et al. 2018) use the diagonal of the Fisher information matrix or the Hessian
matrix as a utility measure for each weight. The quality of these algorithms
depends heavily on the approximation quality of the second-order
approximation. Second-order information can also be used in neural network
pruning. Molchanov et al. (2019) showed that second-order approximation with
the exact Hessian diagonals could closely represent the true measure of the
utility of each weight.
The accurate and efficient approximation for the diagonals of the Hessian at
each layer enables HesScale to be used in many important lines of research.
Using this second-order information provides a reliable measure of connection
utility. Therefore, using HesScale in these types of problems can potentially
improve the performance of neural network pruning methods and regularization-
based catastrophic forgetting prevention methods.
#### Acknowledgments
We gratefully acknowledge funding from the Canada CIFAR AI Chairs program, the
Reinforcement Learning and Artificial Intelligence (RLAI) laboratory, the
Alberta Machine Intelligence Institute (Amii), and the Natural Sciences and
Engineering Research Council (NSERC) of Canada. We would also like to thank
Compute Canada for providing the computational resources needed.
## References
Amari, S. (1998). Natural Gradient Works Efficiently in Learning. Neural
Computation, 10(2), 251–276.
Ba, J. L., Kiros, J. R., & Hinton, G. E. (2016). Layer normalization. arXiv
preprint arXiv:1607.06450.
Becker, S. & Lecun, Y. (1989). Improving the convergence of back-propagation
learning with second-order methods. Proceedings of the 1988 Connectionist
Models Summer School (pp. 29–37).
Bekas, C., Kokiopoulou, E., & Saad, Y. (2007). An estimator for the diagonal
of a matrix. Applied Numerical Mathematics, 57(11), 1214–1229.
Botev, A., Ritter, H., & Barber, D. (2017). Practical Gauss-Newton
optimisation for deep learning. Proceedings of the 34th International
Conference on Machine Learning, 70, 557–565.
Chan, A., Silva, H., Lim, S., Kozuno, T., Mahmood, A. R., & White, M. (2022).
Greedification operators for policy optimization: Investigating forward and
reverse KL divergences. Journal of Machine Learning Research, 23(253), 1-79.
Chapelle, O. & Erhan, D. (2011). Improved preconditioner for hessian free
optimization. NIPS Workshop on Deep Learning and Unsupervised Feature
Learning.
Dangel, F., Kunstner, F., & Hennig, P. (2020). BackPACK: Packing more into
backprop. International Conference on Learning Representations.
Dong, X., Chen, S., & Pan, S. J. (2017). Learning to prune deep neural
networks via layer-wise optimal brain surgeon. Proceedings of the 31st
International Conference on Neural Information Processing Systems (pp.
4860-4874).
Duchi, J., Hazan, E., & Singer, Y. (2011). Adaptive subgradient methods for
online learning and stochastic optimization. Journal of Machine Learning
Research, 12(61), 2121–2159.
Hassibi, B. & Stork, D. (1993). Second order derivatives for network pruning:
Optimal brain surgeon. Advances in Neural Information Processing Systems, 5.
Ioffe, S. and Szegedy, C. (2015). Batch normalization: Accelerating deep
network training by reducing internal covariate shift. International
conference on machine learning (pp. 448-456).
Kingma, D. P. & Ba, J. (2015). Adam: A method for stochastic optimization.
International Conference on Learning Representations.
Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G.,
Rusu, A. A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., et al.
(2017). Overcoming catastrophic forgetting in neural networks. Proceedings of
the national academy of sciences, 114(13), 3521–3526.
Kunstner, F., Hennig, P., & Balles, L. (2019). Limitations of the empirical
fisher approximation for natural gradient descent. Advances in Neural
Information Processing Systems, 32.
LeCun, Y., Denker, J., & Solla, S. (1990). Optimal brain damage. Advances in
Neural Information Processing Systems, 2.
Luo, L., Xiong, Y., & Liu, Y. (2019). Adaptive gradient methods with dynamic
bound of learning rate. International Conference on Learning Representations.
Martens, J. (2010). Deep learning via hessian-free optimization. International
Conference on Machine Learning (pp. 735–742).
Martens, J. (2020). New insights & perspectives on the natural gradient
method. Journal of Machine Learning Research, 21(146), 1-76.
Martens, J. & Grosse, R. (2015). Optimizing neural networks with kronecker-
factored approximate curvature. International Conference on Machine Learning
(pp. 2408–2417).
Martens, J. & Sutskever, I. (2011). Learning recurrent neural networks with
hessian-free optimization. International Conference on Machine Learning (pp.
1033-1040).
Martens, J., Sutskever, I., & Swersky, K. (2012). Estimating the hessian by
back-propagating curvature. International Conference on International
Conference on Machine Learning(pp. 963–970).
Mizutani, E. & Dreyfus, S. E. (2008). Second-order stagewise backpropagation
for hessian-matrix analyses & investigation of negative curvature. Neural
Networks, 21(2), 193–203.
Molchanov, P., Mallya, A., Tyree, S., Frosio, I., & Kautz, J. (2019).
Importance estimation for neural network pruning. IEEE/CVF Conference on
Computer Vision and Pattern Recognition (pp. 11264–11272).
Reddi, S. J., Kale, S., & Kumar, S. (2018). On the convergence of Adam and
beyond. International Conference on Learning Representations.
Ritter, H., Botev, A., and Barber, D. (2018). Online structured Laplace
approximations for overcoming catastrophic forgetting. International
Conference on Neural Information Processing Systems (pp. 3742–3752).
Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning
representations by back-propagating errors. Nature, 323(6088), 533–536.
Salimans, T. and Kingma, D. P. (2016). Weight normalization: A simple
reparameterization to accelerate training of deep neural networks. Advances in
neural information processing systems, 29, 901–909.
Schneider, F., Balles, L., and Hennig, P. (2019). DeepOBS: A deep learning
optimizer benchmark suite. International Conference on Learning
Representations.
Schraudolph, N. N. (2002). Fast curvature matrix-vector products for second-
order gradient descent. Neural Computation, 14(7), 1723–1738.
Schwarz, J., Czarnecki, W., Luketina, J., Grabska-Barwinska, A., Teh, Y. W.,
Pascanu, R., and Hadsell, R. (2018). Progress & compress: A scalable framework
for continual learning. International Conference on Machine Learning (pp.
4528–4537).
Springenberg, J., Dosovitskiy, A., Brox, T., and Riedmiller, M. (2015).
Striving for simplicity: The all convolutional net. International Conference
on Learning Representations [Workshop].
Tieleman, T., Hinton, G., et al. (2012). Lecture 6.5-RMSProp: Divide the
gradient by a running average of its recent magnitude. COURSERA: Neural
networks for machine learning, 4(2), 26–31.
Tran, P. T. and Phong, L. T. (2019). On the convergence proof of AMSGrad and a
new version. IEEE Access, 7, 61706–61716.
Wilson, A. C., Roelofs, R., Stern, M., Srebro, N., and Recht, B. (2017). The
marginal value of adaptive gradient methods in machine learning. International
Conference on Neural Information Processing Systems (pp. 4151–4161).
Yao, Z., Gholami, A., Shen, S., Keutzer, K., and Mahoney, M. W. (2021).
Adahessian: An adaptive second order optimizer for machine learning. AAAI
Conference on Artificial Intelligence, 35(12), 10665-10673.
Zeiler, M. D. (2012). Adadelta: An adaptive learning rate method. arXiv
preprint arXiv:1212.5701.
## Appendix A Hessian diagonals of the log-likelihood function for two common
distributions
Here, we provide the diagonals of the Hessian matrix of functions involving
the log-likelihood of two common distributions: a normal distribution and a
categorical distribution with probabilities represented by a softmax function,
which we refer to as a _softmax distribution_. We show that the exact
computations of the diagonal can be computed with linear complexity since
computing the diagonal elements does not depend on off-diagonals in these
cases. In the following, we consider the softmax and normal distributions, and
we write the exact Hessian diagonals in both cases.
### A.1 Softmax distribution
Consider a cross-entropy function for a discrete probability distribution as
$f\doteq-\sum_{i=1}^{|{\bm{q}}|}p_{i}\log q_{i}(\bm{\theta})$, where
${\bm{q}}$ is a probability vector that depends on a parameter vector
$\bm{\theta}$, and ${\bm{p}}$ is a one-hot vector for the target class. For
softmax distributions, ${\bm{q}}$ is parametrized by a softmax function
${\bm{q}}\doteq{e^{\bm{\theta}}/\sum_{i=1}^{|{\bm{q}}|}e^{\theta_{i}}}$. In
this case, we can write the gradient of the cross-entropy function with
respect to $\bm{\theta}$ as
$\nabla_{\bm{\theta}}f(\bm{\theta})={\bm{q}}-{\bm{p}}.$
Next, we write the exact diagonal elements of the Hessian matrix as follows:
$\text{diag}({\bm{H}}_{\bm{\theta}})=\text{diag}(\nabla_{\bm{\theta}}({\bm{q}}-{\bm{p}}))={\bm{q}}-{\bm{q}}^{2},$
where ${\bm{q}}^{2}$ denotes element-wise squaring of ${\bm{q}}$, and $\nabla$
operator applied to a vector denotes Jacobian. Computing the exact diagonals
of the Hessian matrix depends only on vector operations, which means that we
can compute it in $O(n)$. The cross-entropy loss is used with softmax
distribution in many important tasks, such as supervised classification and
discrete reinforcement learning control with parameterized policies (Chan et
al. 2022).
### A.2 Multivariate normal distribution with diagonal covariance
For a multivariate normal distribution with diagonal covariance, the parameter
vector $\bm{\theta}$ is determined by the mean-variance vector pair:
$\bm{\theta}\doteq(\bm{\mu},{\bm{\sigma}}^{2})$. The log-likelihood of a
random vector ${\bm{x}}$ drawn from this distribution can be written as
$\displaystyle\log q({\bm{x}};\bm{\mu},{\bm{\sigma}}^{2})$
$\displaystyle=-\frac{1}{2}({\bm{x}}-\bm{\mu})^{\top}\bm{D}({\bm{\sigma}}^{2})^{-1}({\bm{x}}-\bm{\mu})-\frac{1}{2}\log(|\bm{D}({\bm{\sigma}}^{2})|)+c$
$\displaystyle=-\frac{1}{2}({\bm{x}}-\bm{\mu})^{\top}\bm{D}({\bm{\sigma}}^{2})^{-1}({\bm{x}}-\bm{\mu})-\frac{1}{2}\log(\sum_{i=1}^{|{\bm{\sigma}}|}\sigma^{2}_{i})+c,$
where $\bm{D}({\bm{\sigma}}^{2})$ gives a diagonal matrix with
${\bm{\sigma}}^{2}$ in its diagonal, $|{\bm{M}}|$ is the determinant of a
matrix ${\bm{M}}$ and $c$ is some constant. We can write the gradients of the
log-likelihood function with respect to $\bm{\mu}$ and ${\bm{\sigma}}^{2}$ as
follows:
$\displaystyle\nabla_{\bm{\mu}}\log q({\bm{x}};\bm{\mu},{\bm{\sigma}}^{2})$
$\displaystyle=\bm{D}({\bm{\sigma}}^{2})^{-1}({\bm{x}}-\bm{\mu})=({\bm{x}}-\bm{\mu})\oslash{\bm{\sigma}}^{2},$
$\displaystyle\nabla_{{\bm{\sigma}}^{2}}\log
q({\bm{x}};\bm{\mu},{\bm{\sigma}}^{2})$
$\displaystyle=\frac{1}{2}\big{[}({\bm{x}}-\bm{\mu})^{2}\oslash{\bm{\sigma}}^{2}-\mathbf{1}\big{]}\oslash{\bm{\sigma}}^{2},$
where $\mathbf{1}$ is an all-ones vector, and $\oslash$ denotes element-wise
division. Finally, we write the exact diagonals of the Hessian matrix as
$\text{diag}({\bm{H}}_{\bm{\mu}})=\text{diag}(\nabla_{{\bm{\mu}}}({\bm{x}}-\bm{\mu})\oslash{\bm{\sigma}}^{2})=-\mathbf{1}\oslash{\bm{\sigma}}^{2},$
$\text{diag}({\bm{H}}_{{\bm{\sigma}}^{2}})=\text{diag}\Big{(}\nabla_{{\bm{\sigma}}^{2}}\big{[}\frac{1}{2}[({\bm{x}}-\bm{\mu})^{2}\oslash{\bm{\sigma}}^{2}-\mathbf{1}]\oslash{\bm{\sigma}}^{2}\big{]}\Big{)}=\big{[}0.5\mathbf{1}-({\bm{x}}-\bm{\mu})^{2}\oslash{\bm{\sigma}}^{2}\big{]}\oslash{\bm{\sigma}}^{4}.$
Clearly, the gradient and the exact Hessian diagonals can be computed in
$O(n)$. Log-likelihood functions for normal distributions are used in many
important problems, such as variational inference and continuous reinforcement
learning control.
## Appendix B HesScale with Convolutional Neural Networks
Here, we derive the Hessian propagation for convolutional neural networks
(CNNs). Consider a CNN with $L-1$ layers followed by a fully connected layer
that outputs the predicted output ${\bm{q}}$. The CNN filters are
parameterized by $\\{{\bm{W}}_{1},...,{\bm{W}}_{L}\\}$, where ${\bm{W}}_{l}$
is the filter matrix at the $l$-th layer with the dimensions $k_{l,1}\times
k_{l,2}$, and its element at the $i$th row and the $j$th column is denoted by
$W_{l,i,j}$. For the simplicity of this proof, we assume that the number of
filters at each layer is one; the proof can be extended easily to the general
case. The learning algorithm learns the target function $f^{*}$ by optimizing
the loss $\mathcal{L}$. During learning, the parameters of the neural network
are changed to reduce the loss. At the layer $l$, we get the activation output
matrix ${\bm{H}}_{l}$ by applying the activation function $\bm{\sigma}$ to the
activation input ${\bm{A}}_{l}$: ${\bm{H}}_{l}=\bm{\sigma}({\bm{A}}_{l})$. We
assume here that the activation function is element-wise activation for all
layers except for the final layer $L$, where it becomes the softmax function.
We simplify notations by defining ${\bm{H}}_{0}\doteq{\bm{X}}$, where
${\bm{X}}$ is the input sample. The activation output ${\bm{H}}_{l}$ is then
convoluted by the weight matrix ${\bm{W}}_{l+1}$ of layer $l+1$ to produce the
next activation input:
${A}_{l+1,i,j}=\sum_{m=0}^{k_{l,1}-1}\sum_{n=0}^{k_{l,2}-1}{{W}_{l+1,m,n}}{H}_{l,(i+m),(j+n)}$.
We denote the size of the activation output at the $l$-th layer by
$h_{l}\times w_{l}$. The backpropagation equations for the described network
are given following Rumelhart et al. (1986):
$\displaystyle\frac{\partial\mathcal{L}}{\partial A_{l,i,j}}$
$\displaystyle=\sum_{m=0}^{k_{l+1,1}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}\frac{\partial A_{l+1,(i-m),(j-n)}}{\partial A_{l,i,j}}$
$\displaystyle=\sum_{m=0}^{k_{l+1,1}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}\sum_{m^{\prime}=0}^{k_{l+1,1}-1}\sum_{n^{\prime}=0}^{k_{l+1,2}-1}W_{l+1,m^{\prime},n^{\prime}}\frac{\partial
H_{l,(i-m+m^{\prime}),(j-n+n^{\prime})}}{\partial A_{l,i,j}}$
$\displaystyle=\sum_{m=0}^{k_{l+1,1}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}W_{l+1,m,n}\sigma^{\prime}(A_{l,i,j})$
$\displaystyle=\sigma^{\prime}(A_{l,i,j})\sum_{m=0}^{k_{l+1,1}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}W_{l+1,m,n},$ (6)
$\displaystyle\frac{\partial\mathcal{L}}{\partial W_{l,i,j}}$
$\displaystyle=\sum_{m=0}^{h_{l}-k_{l,1}}\sum_{n=0}^{w_{l}-k_{l,2}}\frac{\partial\mathcal{L}}{\partial
A_{l,m,n}}\frac{\partial A_{l,m,n}}{\partial
W_{l,i,j}}=\sum_{m=0}^{h_{l}-k_{l,1}}\sum_{n=0}^{w_{l}-k_{l,2}}\frac{\partial\mathcal{L}}{\partial
A_{l,m,n}}H_{l-1,(i+m),(j+n)}.$ (7)
In the following, we write the equations for the exact Hessian diagonals with
respect to weights
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{W^{2}_{l,i,j}}}}$, which
requires the calculation of
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{A^{2}_{l,i,j}}}}$ first:
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial{A^{2}_{l,i,j}}}$
$\displaystyle=\frac{\partial}{\partial
A_{l,i,j}}\Bigg{[}\sigma^{\prime}(A_{l,i,j})\sum_{m=0}^{k_{l+1,1}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}W_{l+1,m,n}\Bigg{]}$
$\displaystyle=\sigma^{\prime}(A_{l,i,j})\sum_{m,p=0}^{k_{l+1,2}-1}\sum_{n,q=0}^{k_{l+1,2}-1}\frac{\partial^{2}\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}\partial A_{l+1,(i-p),(j-q)}}\frac{\partial
A_{l+1,(i-p),(j-q)}}{\partial A_{l,i,j}}W_{l+1,m,n}$
$\displaystyle+\sigma^{\prime\prime}(A_{l,i,j})\sum_{m=0}^{k_{l+1,2}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}W_{l+1,m,n}$
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}$
$\displaystyle=\frac{\partial}{\partial
W_{l,i,j}}\Bigg{[}\sum_{m=0}^{h_{l}-k_{l,1}}\sum_{n=0}^{w_{l}-k_{l,2}}\frac{\partial\mathcal{L}}{\partial
A_{l,m,n}}H_{l-1,(i+m),(j+n)}\Bigg{]}$
$\displaystyle=\sum_{m,p=0}^{h_{l}-k_{l,1}}\sum_{n,q=0}^{w_{l}-k_{l,2}}\frac{\partial^{2}\mathcal{L}}{\partial
A_{l,m,n}\partial A_{l,p,q}}\frac{\partial A_{l,p,q}}{\partial
W_{l,i,j}}H_{l-1,(i+m),(j+n)}$
Since the calculation of
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{A^{2}_{l,i,j}}}}$ and
$\nicefrac{{\partial^{2}\mathcal{L}}}{{\partial{W^{2}_{l,i,j}}}}$ depend on
the off-diagonal terms, the computation complexity becomes quadratic.
Following Becker and Lecun (1989), we approximate the Hessian diagonals by
ignoring the off-diagonal terms, which leads to a backpropagation rule with
linear computational complexity for our estimates
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}}$ and
$\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{A^{2}_{l,i,j}}}}$:
$\displaystyle\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{A^{2}_{l,i,j}}}}$
$\displaystyle\doteq\sigma^{\prime}(A_{l,i,j})^{2}\sum_{m=0}^{k_{l+1,2}-1}\sum_{n=0}^{k_{l+1,2}-1}\widehat{\frac{\partial^{2}\mathcal{L}}{\partial
A^{2}_{l+1,(i-m),(j-n)}}}W^{2}_{l+1,m,n}$
$\displaystyle+\sigma^{\prime\prime}(A_{l,i,j})\sum_{m=0}^{k_{l+1,2}-1}\sum_{n=0}^{k_{l+1,2}-1}\frac{\partial\mathcal{L}}{\partial
A_{l+1,(i-m),(j-n)}}W_{l+1,m,n},$ (8)
$\displaystyle\widehat{\frac{\partial^{2}\mathcal{L}}{\partial{W^{2}_{l,i,j}}}}$
$\displaystyle\doteq\sum_{m=0}^{h_{l}-k_{l,1}}\sum_{n=0}^{w_{l}-k_{l,2}}\widehat{\frac{\partial^{2}\mathcal{L}}{\partial
A^{2}_{l,m,n}}}H^{2}_{l-1,(i+m),(j+n)}.$ (9)
## Appendix C Optimization plots in the number of epochs
We give the training loss, training accuracy, validation loss, validation
accuracy, test loss, and test accuracy for each of the methods we include in
our comparison in Fig. 6 and Fig. 7. Moreover, we give the sensitivity plots
for $\beta_{1}$, $\beta_{2}$, and $\alpha$ for each method in Fig. 8 and Fig.
9.
(a) MNIST
(b) CIFAR-10
Figure 6: Learning curves of each algorithm on two tasks, MNIST-MLP and
CIFAR-10 3C3D, for 100 epochs. We show the best configuration for each
algorithm on the validation set. The best parameter configuration for each
algorithm is selected based on the area under the curve for the validation
loss.
(a) CIFAR-100 All CNN
(b) CIFAR-100 3C3D
Figure 7: Learning Curves of each algorithm on CIFAR-100 with All-CNN and 3C3D
architectures, for 100 epochs. We show the best configuration for each
algorithm on the validation set. The best parameter configuration for each
algorithm is selected based on the area under the curve for the validation
loss.
(a) MNIST
(b) CIFAR-10
Figure 8: Parameter Sensitivity study for each algorithm on two data sets,
MNIST and CIFAR-10. The range of $\beta_{2}$ is $\\{0.99,0.999,0.9999\\}$ and
the range of $\beta_{1}$ is $\\{0.0,0.9\\}$. Each point for each algorithm
represents the average test loss given a set of parameters.
(a) CIFAR-100 All-CNN
(b) CIFAR-100 3C3D
Figure 9: Parameter Sensitivity study for each algorithm on CIFAR-100 with
All-CNN and 3C3D architectures. The range of step size is
$\\{10^{-5},10^{-4},10^{-3},10^{-2},10^{-1},10^{0}\\}$. We choose $\beta_{1}$
to be equal to $0.9$ and $\beta_{2}$ to be equal to $0.999$. Each point for
each algorithm represents the average test loss given a set of parameters.
|
# Intriguing properties of neural networks
Christian Szegedy
Google Inc.
&Wojciech Zaremba
New York University
&Ilya Sutskever
Google Inc.
&Joan Bruna
New York University
&Dumitru Erhan
Google Inc.
&Ian Goodfellow
University of Montreal
&Rob Fergus
New York University
Facebook Inc.
###### Abstract
Deep neural networks are highly expressive models that have recently achieved
state of the art performance on speech and visual recognition tasks. While
their expressiveness is the reason they succeed, it also causes them to learn
uninterpretable solutions that could have counter-intuitive properties. In
this paper we report two such properties.
First, we find that there is no distinction between individual high level
units and random linear combinations of high level units, according to various
methods of unit analysis. It suggests that it is the space, rather than the
individual units, that contains the semantic information in the high layers of
neural networks.
Second, we find that deep neural networks learn input-output mappings that are
fairly discontinuous to a significant extent. We can cause the network to
misclassify an image by applying a certain hardly perceptible perturbation,
which is found by maximizing the network’s prediction error. In addition, the
specific nature of these perturbations is not a random artifact of learning:
the same perturbation can cause a different network, that was trained on a
different subset of the dataset, to misclassify the same input.
## 1 Introduction
Deep neural networks are powerful learning models that achieve excellent
performance on visual and speech recognition problems [9, 8]. Neural networks
achieve high performance because they can express arbitrary computation that
consists of a modest number of massively parallel nonlinear steps. But as the
resulting computation is automatically discovered by backpropagation via
supervised learning, it can be difficult to interpret and can have counter-
intuitive properties. In this paper, we discuss two counter-intuitive
properties of deep neural networks.
The first property is concerned with the semantic meaning of individual units.
Previous works [6, 13, 7] analyzed the semantic meaning of various units by
finding the set of inputs that maximally activate a given unit. The inspection
of individual units makes the implicit assumption that the units of the last
feature layer form a distinguished basis which is particularly useful for
extracting semantic information. Instead, we show in section 3 that random
projections of $\phi(x)$ are semantically indistinguishable from the
coordinates of $\phi(x)$. This puts into question the conjecture that neural
networks disentangle variation factors across coordinates. Generally, it seems
that it is the entire space of activations, rather than the individual units,
that contains the bulk of the semantic information. A similar, but even
stronger conclusion was reached recently by Mikolov et al. [12] for word
representations, where the various directions in the vector space representing
the words are shown to give rise to a surprisingly rich semantic encoding of
relations and analogies. At the same time, the vector representations are
stable up to a rotation of the space, so the individual units of the vector
representations are unlikely to contain semantic information.
The second property is concerned with the stability of neural networks with
respect to small perturbations to their inputs. Consider a state-of-the-art
deep neural network that generalizes well on an object recognition task. We
expect such network to be robust to small perturbations of its input, because
small perturbation cannot change the object category of an image. However, we
find that applying an _imperceptible_ non-random perturbation to a test image,
it is possible to arbitrarily change the network’s prediction (see figure 5).
These perturbations are found by optimizing the input to maximize the
prediction error. We term the so perturbed examples “adversarial examples”.
It is natural to expect that the precise configuration of the minimal
necessary perturbations is a random artifact of the normal variability that
arises in different runs of backpropagation learning. Yet, we found that
adversarial examples are relatively robust, and are shared by neural networks
with varied number of layers, activations or trained on different subsets of
the training data. That is, if we use one neural net to generate a set of
adversarial examples, we find that these examples are still statistically hard
for another neural network even when it was trained with different
hyperparameters or, most surprisingly, when it was trained on a different set
of examples.
These results suggest that the deep neural networks that are learned by
backpropagation have nonintuitive characteristics and intrinsic blind spots,
whose structure is connected to the data distribution in a non-obvious way.
## 2 Framework
Notation We denote by $x\in\mathbb{R}^{m}$ an input image, and $\phi(x)$
activation values of some layer. We first examine properties of the image of
$\phi(x)$, and then we search for its blind spots.
We perform a number of experiments on a few different networks and three
datasets :
* •
For the MNIST dataset, we used the following architectures [11]
* –
A simple fully connected network with one or more hidden layers and a Softmax
classifier. We refer to this network as “FC”.
* –
A classifier trained on top of an autoencoder. We refer to this network as
“AE”.
* •
The ImageNet dataset [3].
* –
Krizhevsky et. al architecture [9]. We refer to it as “AlexNet”.
* •
$\sim 10$M image samples from Youtube (see [10])
* –
Unsupervised trained network with $\sim$ 1 billion learnable parameters. We
refer to it as “QuocNet”.
For the MNIST experiments, we use regularization with a weight decay of
$\lambda$. Moreover, in some experiments we split the MNIST training dataset
into two disjoint datasets $P_{1}$, and $P_{2}$, each with 30000 training
cases.
## 3 Units of: $\phi(x)$
Traditional computer vision systems rely on feature extraction: often a single
feature is easily interpretable, e.g. a histogram of colors, or quantized
local derivatives. This allows one to inspect the individual coordinates of
the feature space, and link them back to meaningful variations in the input
domain. Similar reasoning was used in previous work that attempted to analyze
neural networks that were applied to computer vision problems. These works
interpret an activation of a hidden unit as a meaningful feature. They look
for input images which maximize the activation value of this single feature
[6, 13, 7, 4].
The aforementioned technique can be formally stated as visual inspection of
images $x^{\prime}$, which satisfy (or are close to maximum attainable value):
$\displaystyle
x^{\prime}=\operatorname*{arg\,max}_{x\in\mathcal{I}}\langle\phi(x),e_{i}\rangle$
where $\mathcal{I}$ is a held-out set of images from the data distribution
that the network was not trained on and $e_{i}$ is the natural basis vector
associated with the $i$-th hidden unit.
Our experiments show that any random direction $v\in\mathbb{R}^{n}$ gives rise
to similarly interpretable semantic properties. More formally, we find that
images $x^{\prime}$ are semantically related to each other, for many
$x^{\prime}$ such that
$\displaystyle
x^{\prime}=\operatorname*{arg\,max}_{x\in\mathcal{I}}\langle\phi(x),v\rangle$
This suggests that the natural basis is not better than a random basis for
inspecting the properties of $\phi(x)$. This puts into question the notion
that neural networks disentangle variation factors across coordinates.
First, we evaluated the above claim using a convolutional neural network
trained on MNIST. We used the MNIST test set for $\mathcal{I}$. Figure 1 shows
images that maximize the activations in the natural basis, and Figure 2 shows
images that maximize the activation in random directions. In both cases the
resulting images share many high-level similarities.
Next, we repeated our experiment on an AlexNet, where we used the validation
set as $\mathcal{I}$. Figures 3 and 4 compare the natural basis to the random
basis on the trained network. The rows appear to be semantically meaningful
for both the single unit and the combination of units.
(a) Unit sensitive to lower round stroke.
(b) Unit sensitive to upper round stroke, or lower straight stroke.
(c) Unit senstive to left, upper round stroke.
(d) Unit senstive to diagonal straight stroke.
Figure 1: An MNIST experiment. The figure shows images that maximize the
activation of various units (maximum stimulation in the natural basis
direction). Images within each row share semantic properties.
(a) Direction sensitive to upper straight stroke, or lower round stroke.
(b) Direction sensitive to lower left loop.
(c) Direction senstive to round top stroke.
(d) Direction sensitive to right, upper round stroke.
Figure 2: An MNIST experiment. The figure shows images that maximize the
activations in a random direction (maximum stimulation in a random basis).
Images within each row share semantic properties.
(a) Unit sensitive to white flowers.
(b) Unit sensitive to postures.
(c) Unit senstive to round, spiky flowers.
(d) Unit senstive to round green or yellow objects.
Figure 3: Experiment performed on ImageNet. Images stimulating single unit
most (maximum stimulation in natural basis direction). Images within each row
share many semantic properties.
(a) Direction sensitive to white, spread flowers.
(b) Direction sensitive to white dogs.
(c) Direction sensitive to spread shapes.
(d) Direction sensitive to dogs with brown heads.
Figure 4: Experiment performed on ImageNet. Images giving rise to maximum
activations in a random direction (maximum stimulation in a random basis).
Images within each row share many semantic properties.
Although such analysis gives insight on the capacity of $\phi$ to generate
invariance on a particular subset of the input distribution, it does not
explain the behavior on the rest of its domain. We shall see in the next
section that $\phi$ has counterintuitive properties in the neighbourhood of
almost every point form data distribution.
## 4 Blind Spots in Neural Networks
So far, unit-level inspection methods had relatively little utility beyond
confirming certain intuitions regarding the complexity of the representations
learned by a deep neural network [6, 13, 7, 4]. Global, network level
inspection methods _can_ be useful in the context of explaining classification
decisions made by a model [1] and can be used to, for instance, identify the
parts of the input which led to a correct classification of a given visual
input instance (in other words, one can use a trained model for weakly-
supervised localization). Such global analyses are useful in that they can
make us understand better the input-to-output mapping represented by the
trained network.
Generally speaking, the output layer unit of a neural network is a highly
nonlinear function of its input. When it is trained with the cross-entropy
loss (using the Softmax activation function), it represents a conditional
distribution of the label given the input (and the training set presented so
far). It has been argued [2] that the deep stack of non-linear layers in
between the input and the output unit of a neural network are a way for the
model to encode a _non-local generalization prior_ over the input space. In
other words, it is assumed that is possible for the output unit to assign non-
significant (and, presumably, non-epsilon) probabilities to regions of the
input space that contain no training examples in their vicinity. Such regions
can represent, for instance, the same objects from different viewpoints, which
are relatively far (in pixel space), but which share nonetheless both the
label and the statistical structure of the original inputs.
It is implicit in such arguments that $local$ generalization—in the very
proximity of the training examples—works as expected. And that in particular,
for a small enough radius $\varepsilon>0$ in the vicinity of a given training
input $x$, an $x+r$ satisfying $||r||<\varepsilon$ will get assigned a high
probability of the correct class by the model. This kind of smoothness prior
is typically valid for computer vision problems. In general, imperceptibly
tiny perturbations of a given image do not normally change the underlying
class.
Our main result is that for deep neural networks, the smoothness assumption
that underlies many kernel methods does not hold. Specifically, we show that
by using a simple optimization procedure, we are able to find adversarial
examples, which are obtained by imperceptibly small perturbations to a
correctly classified input image, so that it is no longer classified
correctly.
In some sense, what we describe is a way to traverse the manifold represented
by the network in an efficient way (by optimization) and finding _adversarial
examples_ in the input space. The adversarial examples represent low-
probability (high-dimensional) “pockets” in the manifold, which are hard to
efficiently find by simply randomly sampling the input around a given example.
Already, a variety of recent state of the art computer vision models employ
input deformations during training for increasing the robustness and
convergence speed of the models [9, 13]. These deformations are, however,
statistically inefficient, for a given example: they are highly correlated and
are drawn from the same distribution throughout the entire training of the
model. We propose a scheme to make this process adaptive in a way that
exploits the model and its deficiencies in modeling the local space around the
training data.
We make the connection with hard-negative mining explicitly, as it is close in
spirit: hard-negative mining, in computer vision, consists of identifying
training set examples (or portions thereof) which are given low probabilities
by the model, but which should be high probability instead, cf. [5]. The
training set distribution is then changed to emphasize such hard negatives and
a further round of model training is performed. As shall be described, the
optimization problem proposed in this work can also be used in a constructive
way, similar to the hard-negative mining principle.
### 4.1 Formal description
We denote by $f:\mathbb{R}^{m}\longrightarrow\\{1\dots k\\}$ a classifier
mapping image pixel value vectors to a discrete label set. We also assume that
$f$ has an associated continuous loss function denoted by
$\textrm{loss}_{f}:\mathbb{R}^{m}\times\\{1\dots
k\\}\longrightarrow\mathbb{R}^{+}$. For a given $x\in\mathbb{R}^{m}$ image and
target label $l\in\\{1\dots k\\}$, we aim to solve the following box-
constrained optimization problem:
* •
Minimize $\|r\|_{2}$ subject to:
1. 1.
$f(x+r)=l$
2. 2.
$x+r\in[0,1]^{m}$
The minimizer $r$ might not be unique, but we denote one such $x+r$ for an
arbitrarily chosen minimizer by $D(x,l)$. Informally, $x+r$ is the closest
image to $x$ classified as $l$ by $f$. Obviously, $D(x,f(x))=f(x)$, so this
task is non-trivial only if $f(x)\neq l$. In general, the exact computation of
$D(x,l)$ is a hard problem, so we approximate it by using a box-constrained
L-BFGS. Concretely, we find an approximation of $D(x,l)$ by performing line-
search to find the minimum $c>0$ for which the minimizer $r$ of the following
problem satisfies $f(x+r)=l$.
* •
Minimize $c|r|+\textrm{loss}_{f}(x+r,l)$ subject to $x+r\in[0,1]^{m}$
This penalty function method would yield the exact solution for $D(X,l)$ in
the case of convex losses, however neural networks are non-convex in general,
so we end up with an approximation in this case.
### 4.2 Experimental results
Figure 5: Adversarial examples generated for AlexNet [9].(Left) is a correctly
predicted sample, (center) difference between correct image, and image
predicted incorrectly magnified by 10x (values shifted by 128 and clamped),
(right) adversarial example. All images in the right column are predicted to
be an “ostrich, Struthio camelus”. Average distortion based on 64 examples is
0.006508. Plase refer to http://goo.gl/huaGPb for full resolution images. The
examples are strictly randomly chosen. There is not any postselection
involved.
Figure 6: Adversarial examples for QuocNet [10]. A binary car classifier was
trained on top of the last layer features without fine-tuning. The randomly
chosen examples on the left are recognized correctly as cars, while the images
in the middle are not recognized. The rightmost column is the magnified
absolute value of the difference between the two images.
Our “minimimum distortion” function $D$ has the following intriguing
properties which we will support by informal evidence and quantitative
experiments in this section:
1. 1.
For all the networks we studied (MNIST, QuocNet [10], AlexNet [9]), for each
sample, we have always managed to generate very close, visually hard to
distinguish, adversarial examples that are misclassified by the original
network (see figure 5 and http://goo.gl/huaGPb for examples).
2. 2.
Cross model generalization: a relatively large fraction of examples will be
misclassified by networks trained from scratch with different hyper-parameters
(number of layers, regularization or initial weights).
3. 3.
Cross training-set generalization a relatively large fraction of examples will
be misclassified by networks trained from scratch on a disjoint training set.
The above observations suggest that adversarial examples are somewhat
universal and not just the results of overfitting to a particular model or to
the specific selection of the training set. They also suggest that back-
feeding adversarial examples to training might improve generalization of the
resulting models. Our preliminary experiments have yielded positive evidence
on MNIST to support this hypothesis as well: We have successfully trained a
two layer 100-100-10 non-convolutional neural network with a test error below
$1.2\%$ by keeping a pool of adversarial examples a random subset of which is
continuously replaced by newly generated adversarial examples and which is
mixed into the original training set all the time. We used weight decay, but
no dropout for this network. For comparison, a network of this size gets to
$1.6\%$ errors when regularized by weight decay alone and can be improved to
around $1.3\%$ by using carefully applied dropout. A subtle, but essential
detail is that we only got improvements by generating adversarial examples for
each layer outputs which were used to train all the layers above. The network
was trained in an alternating fashion, maintaining and updating a pool of
adversarial examples for each layer separately in addition to the original
training set. According to our initial observations, adversarial examples for
the higher layers seemed to be significantly more useful than those on the
input or lower layers. In our future work, we plan to compare these effects in
a systematic manner.
For space considerations, we just present results for a representative subset
(see Table 1) of the MNIST experiments we performed. The results presented
here are consistent with those on a larger variety of non-convolutional
models. For MNIST, we do not have results for convolutional models yet, but
our first qualitative experiments with AlexNet gives us reason to believe that
convolutional networks may behave similarly as well. Each of our models were
trained with L-BFGS until convergence. The first three models are linear
classifiers that work on the pixel level with various weight decay parameters
$\lambda$. All our examples use quadratic weight decay on the connection
weights: $\textrm{loss}_{decay}=\lambda\sum w_{i}^{2}/k$ added to the total
loss, where $k$ is the number of units in the layer. Three of our models are
simple linear (softmax) classifier without hidden units (FC10($\lambda$)). One
of them, FC10($1$), is trained with extremely high $\lambda=1$ in order to
test whether it is still possible to generate adversarial examples in this
extreme setting as well.Two other models are a simple sigmoidal neural network
with two hidden layers and a classifier. The last model, AE400-10, consists of
a single layer sparse autoencoder with sigmoid activations and 400 nodes with
a Softmax classifier. This network has been trained until it got very high
quality first layer filters and this layer was not fine-tuned. The last column
measures the minimum average pixel level distortion necessary to reach $0\%$
accuracy on the training set. The distortion is measure by
$\sqrt{\frac{\sum(x_{i}^{\prime}-x_{i})^{2}}{n}}$ between the original $x$ and
distorted $x^{\prime}$ images, where $n=784$ is the number of image pixels.
The pixel intensities are scaled to be in the range $[0,1]$.
In our first experiment, we generated a set of adversarial instances for a
given network and fed these examples for each other network to measure the
proportion of misclassified instances. The last column shows the average
minimum distortion that was necessary to reach 0% accuracy on the whole
training set. The experimental results are presented in Table 2. The columns
of Table 2 show the error (proportion of misclassified instances) on the so
distorted training sets. The last two rows are given for reference showing the
error induced when distorting by the given amounts of Gaussian noise. Note
that even the noise with stddev 0.1 is greater than the stddev of our
adversarial noise for all but one of the models. Figure 7 shows a
visualization of the generated adversarial instances for two of the networks
used in this experiment The general conclusion is that adversarial examples
tend to stay hard even for models trained with different hyperparameters.
Although the autoencoder based version seems most resilient to adversarial
examples, it is not fully immune either.
(a) Even columns: adversarial examples for a linear (FC) classifier
(stddev=0.06)
(b) Even columns: adversarial examples for a 200-200-10 sigmoid network
(stddev=0.063)
(c) Randomly distorted samples by Gaussian noise with stddev=1. Accuracy: 51%.
Figure 7: Adversarial examples for a randomly chosen subset of MNIST compared with randomly distorted examples. Odd columns correspond to original images, and even columns correspond to distorted counterparts. The adversarial examples generated for the specific model have accuracy 0% for the respective model. Note that while the randomly distorted examples are hardly readable, still they are classified correctly in half of the cases, while the adversarial examples are never classified correctly. Model Name | Description | Training error | Test error | Av. min. distortion
---|---|---|---|---
FC10($10^{-4}$) | Softmax with $\lambda=10^{-4}$ | 6.7% | 7.4% | 0.062
FC10($10^{-2}$) | Softmax with $\lambda=10^{-2}$ | 10% | 9.4% | 0.1
FC10($1$) | Softmax with $\lambda=1$ | 21.2% | 20% | 0.14
FC100-100-10 | Sigmoid network $\lambda=10^{-5},10^{-5},10^{-6}$ | 0% | 1.64% | 0.058
FC200-200-10 | Sigmoid network $\lambda=10^{-5},10^{-5},10^{-6}$ | 0% | 1.54% | 0.065
AE400-10 | Autoencoder with Softmax $\lambda=10^{-6}$ | 0.57% | 1.9% | 0.086
Table 1: Tests of the generalization of adversarial instances on MNIST. | FC10($10^{-4}$) | FC10($10^{-2}$) | FC10($1$) | FC100-100-10 | FC200-200-10 | AE400-10 | Av. distortion
---|---|---|---|---|---|---|---
FC10($10^{-4}$) | 100% | 11.7% | 22.7% | 2% | 3.9% | 2.7% | 0.062
FC10($10^{-2}$) | 87.1% | 100% | 35.2% | 35.9% | 27.3% | 9.8% | 0.1
FC10($1$) | 71.9% | 76.2% | 100% | 48.1% | 47% | 34.4% | 0.14
FC100-100-10 | 28.9% | 13.7% | 21.1% | 100% | 6.6% | 2% | 0.058
FC200-200-10 | 38.2% | 14% | 23.8% | 20.3% | 100% | 2.7% | 0.065
AE400-10 | 23.4% | 16% | 24.8% | 9.4% | 6.6% | 100% | 0.086
Gaussian noise, stddev=0.1 | 5.0% | 10.1% | 18.3% | 0% | 0% | 0.8% | 0.1
Gaussian noise, stddev=0.3 | 15.6% | 11.3% | 22.7% | 5% | 4.3% | 3.1% | 0.3
Table 2: Cross-model generalization of adversarial examples. The columns of
the Tables show the error induced by distorted examples fed to the given
model. The last column shows average distortion wrt. original training set.
Still, this experiment leaves open the question of dependence over the
training set. Does the hardness of the generated examples rely solely on the
particular choice of our training set as a sample or does this effect
generalize even to models trained on completely different training sets?
Model | Error on $P_{1}$ | Error on $P_{2}$ | Error on Test | Min Av. Distortion
---|---|---|---|---
FC100-100-10: 100-100-10 trained on $P_{1}$ | 0% | 2.4% | 2% | 0.062
FC123-456-10: 123-456-10 trained on $P_{1}$ | 0% | 2.5% | 2.1% | 0.059
FC100-100-10’ trained on $P_{2}$ | 2.3% | 0% | 2.1% | 0.058
Table 3: Models trained to study cross-training-set generalization of the generated adversarial examples. Errors presented in Table correpond to original not-distorted data, to provide a baseline. | FC100-100-10 | FC123-456-10 | FC100-100-10’
---|---|---|---
Distorted for FC100-100-10 (av. stddev=0.062) | 100% | 26.2% | 5.9%
Distorted for FC123-456-10 (av. stddev=0.059) | 6.25% | 100% | 5.1%
Distorted for FC100-100-10’ (av. stddev=0.058) | 8.2% | 8.2% | 100%
Gaussian noise with stddev=$0.06$ | 2.2% | 2.6% | 2.4%
Distorted for FC100-100-10 amplified to stddev=$0.1$ | 100% | 98% | 43%
Distorted for FC123-456-10 amplified to stddev=$0.1$ | 96% | 100% | 22%
Distorted for FC100-100-10’ amplified to stddev=$0.1$ | 27% | 50% | 100%
Gaussian noise with stddev=$0.1$ | 2.6% | 2.8% | 2.7%
Table 4: Cross-training-set generalization error rate for the set of
adversarial examples generated for different models. The error induced by a
random distortion to the same examples is displayed in the last row.
To study cross-training-set generalization, we have partitioned the 60000
MNIST training images into two parts $P_{1}$ and $P_{2}$ of size 30000 each
and trained three non-convolutional networks with sigmoid activations on them:
Two, FC100-100-10 and FC123-456-10, on $P_{1}$ and FC100-100-10 on $P_{2}$.
The reason we trained two networks for $P_{1}$ is to study the cumulative
effect of changing the hypermarameters and the training sets at the same time.
Models FC100-100-10 and FC100-100-10 share the same hyperparameters: both of
them are 100-100-10 networks, while FC123-456-10 has different number of
hidden units. In this experiment, we were distorting the elements of the test
set rather than the training set. Table 3 summarizes the basic facts about
these models. After we generate adversarial examples with $100\%$ error rates
with minimum distortion for the test set, we feed these examples to the each
of the models. The error for each model is displayed in the corresponding
column of the upper part of Table 4. In the last experiment, we magnify the
effect of our distortion by using the examples
$x+0.1\frac{x^{\prime}-x}{\|x^{\prime}-x\|_{2}}$ rather than $x^{\prime}$.
This magnifies the distortion on average by 40%, from stddev $0.06$ to $0.1$.
The so distorted examples are fed back to each of the models and the error
rates are displayed in the lower part of Table 4. The intriguing conclusion is
that the adversarial examples remain hard for models trained even on a
disjoint training set, although their effectiveness decreases considerably.
### 4.3 Spectral Analysis of Unstability
The previous section showed examples of deep networks resulting from purely
supervised training which are unstable with respect to a peculiar form of
small perturbations. Independently of their generalisation properties across
networks and training sets, the adversarial examples show that there exist
small additive perturbations of the input (in Euclidean sense) that produce
large perturbations at the output of the last layer. This section describes a
simple procedure to measure and control the additive stability of the network
by measuring the spectrum of each rectified layer.
Mathematically, if $\phi(x)$ denotes the output of a network of $K$ layers
corresponding to input $x$ and trained parameters $W$, we write
$\phi(x)=\phi_{K}(\phi_{K-1}(\dots\phi_{1}(x;W_{1});W_{2})\dots;W_{K})~{},$
where $\phi_{k}$ denotes the operator mapping layer $k-1$ to layer $k$. The
unstability of $\phi(x)$ can be explained by inspecting the upper Lipschitz
constant of each layer $k=1\dots K$, defined as the constant $L_{k}>0$ such
that
$\forall\,x,\,r~{},~{}\|\phi_{k}(x;W_{k})-\phi_{k}(x+r;W_{k})\|\leq
L_{k}\|r\|~{}.$
The resulting network thus satsifies $\|\phi(x)-\phi(x+r)\|\leq L\|r\|$, with
$L=\prod_{k=1}^{K}L_{k}$.
A half-rectified layer (both convolutional or fully connected) is defined by
the mapping $\phi_{k}(x;W_{k},b_{k})=\max(0,W_{k}x+b_{k})$. Let $\|W\|$ denote
the operator norm of $W$ (i.e., its largest singular value). Since the non-
linearity $\rho(x)=\max(0,x)$ is contractive, i.e. satisfies
$\|\rho(x)-\rho(x+r)\|\leq\|r\|~{}$ for all $x,r$; it follows that
$\|\phi_{k}(x;W_{k})-\phi_{k}(x+r;W_{k})\|=\|\max(0,W_{k}x+b_{k})-\max(0,W_{k}(x+r)+b_{k})\|\leq\|W_{k}r\|\leq\|W_{k}\|\|r\|~{},$
and hence $L_{k}\leq\|W_{k}\|$. On the other hand, a max-pooling layer
$\phi_{k}$ is contractive:
$\forall\,x\,,\,r\,,~{}\|\phi_{k}(x)-\phi_{k}(x+r)\|\leq\|r\|~{},$
since its Jacobian is a projection onto a subset of the input coordinates and
hence does not expand the gradients. Finally, if $\phi_{k}$ is a contrast-
normalization layer
$\phi_{k}(x)=\frac{x}{\Big{(}\epsilon+\|x\|^{2}\Big{)}^{\gamma}}~{},$
one can verify that
$\forall\,x\,,\,r\,,~{}\|\phi_{k}(x)-\phi_{k}(x+r)\|\leq\epsilon^{-\gamma}\|r\|$
for $\gamma\in[0.5,1]$, which corresponds to most common operating regimes.
It results that a conservative measure of the unstability of the network can
be obtained by simply computing the operator norm of each fully connected and
convolutional layer. The fully connected case is trivial since the norm is
directly given by the largest singular value of the fully connected matrix.
Let us describe the convolutional case. If $W$ denotes a generic $4$-tensor,
implementing a convolutional layer with $C$ input features, $D$ output
features, support $N\times N$ and spatial stride $\Delta$,
$Wx=\left\\{\sum_{c=1}^{C}x_{c}\star
w_{c,d}(n_{1}\Delta,n_{2}\Delta)\,;d=1\,\dots,D\right\\}~{},$
where $x_{c}$ denotes the $c$-th input feature image, and $w_{c,d}$ is the
spatial kernel corresponding to input feature $c$ and output feature $d$, by
applying Parseval’s formula we obtain that its operator norm is given by
$\|W\|=\sup_{\xi\in[0,N\Delta^{-1})^{2}}\|A(\xi)\|~{},$ (1)
where $A(\xi)$ is a $D\times(C\cdot\Delta^{2})$ matrix whose rows are
$\forall~{}d=1\dots
D~{},~{}A(\xi)_{d}=\Big{(}\Delta^{-2}\widehat{w_{c,d}}(\xi+l\cdot
N\cdot\Delta^{-1})\,;\,c=1\dots C\,,\,l=(0\dots\Delta-1)^{2}\Big{)}~{},$
and $\widehat{w_{c,d}}$ is the 2-D Fourier transform of $w_{c,d}$:
$\widehat{w_{c,d}}(\xi)=\sum_{u\in[0,N)^{2}}w_{c,d}(u)e^{-2\pi
i(u\cdot\xi)/N^{2}}~{}.$
Layer | Size | Stride | Upper bound
---|---|---|---
Conv. $1$ | $3\times 11\times 11\times 96$ | $4$ | $2.75$
Conv. $2$ | $96\times 5\times 5\times 256$ | $1$ | $10$
Conv. $3$ | $256\times 3\times 3\times 384$ | $1$ | $7$
Conv. $4$ | $384\times 3\times 3\times 384$ | $1$ | $7.5$
Conv. $5$ | $384\times 3\times 3\times 256$ | $1$ | $11$
FC. 1 | $9216\times 4096$ | N/A | $3.12$
FC. 2 | $4096\times 4096$ | N/A | $4$
FC. 3 | $4096\times 1000$ | N/A | $4$
Table 5: Frame Bounds of each rectified layer of the network from [9].
Table 5 shows the upper Lipschitz bounds computed from the ImageNet deep
convolutional network of [9], using (1). It shows that instabilities can
appear as soon as in the first convolutional layer.
These results are consistent with the exsitence of blind spots constructed in
the previous section, but they don’t attempt to explain why these examples
generalize across different hyperparameters or training sets. We emphasize
that we compute upper bounds: large bounds do not automatically translate into
existence of adversarial examples; however, small bounds guarantee that no
such examples can appear. This suggests a simple regularization of the
parameters, consisting in penalizing each upper Lipschitz bound, which might
help improve the generalisation error of the networks.
## 5 Discussion
We demonstrated that deep neural networks have counter-intuitive properties
both with respect to the semantic meaning of individual units and with respect
to their discontinuities. The existence of the adversarial negatives appears
to be in contradiction with the network’s ability to achieve high
generalization performance. Indeed, if the network can generalize well, how
can it be confused by these adversarial negatives, which are indistinguishable
from the regular examples? Possible explanation is that the set of adversarial
negatives is of extremely low probability, and thus is never (or rarely)
observed in the test set, yet it is dense (much like the rational numbers),
and so it is found near every virtually every test case. However, we don’t
have a deep understanding of how often adversarial negatives appears, and thus
this issue should be addressed in a future research.
## References
* [1] David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and Klaus-Robert Müller. How to explain individual classification decisions. The Journal of Machine Learning Research, 99:1803–1831, 2010.
* [2] Yoshua Bengio. Learning deep architectures for ai. Foundations and trends® in Machine Learning, 2(1):1–127, 2009.
* [3] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 248–255. IEEE, 2009.
* [4] Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. Technical Report 1341, University of Montreal, June 2009. Also presented at the ICML 2009 Workshop on Learning Feature Hierarchies, Montréal, Canada.
* [5] Pedro Felzenszwalb, David McAllester, and Deva Ramanan. A discriminatively trained, multiscale, deformable part model. In Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, pages 1–8. IEEE, 2008.
* [6] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra Malik. Rich feature hierarchies for accurate object detection and semantic segmentation. arXiv preprint arXiv:1311.2524, 2013.
* [7] Ian Goodfellow, Quoc Le, Andrew Saxe, Honglak Lee, and Andrew Y Ng. Measuring invariances in deep networks. Advances in neural information processing systems, 22:646–654, 2009\.
* [8] Geoffrey E. Hinton, Li Deng, Dong Yu, George E. Dahl, Abdel rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N. Sainath, and Brian Kingsbury. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Process. Mag., 29(6):82–97, 2012.
* [9] Alex Krizhevsky, Ilya Sutskever, and Geoff Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems 25, pages 1106–1114, 2012.
* [10] Quoc V Le, Marc’Aurelio Ranzato, Rajat Monga, Matthieu Devin, Kai Chen, Greg S Corrado, Jeff Dean, and Andrew Y Ng. Building high-level features using large scale unsupervised learning. arXiv preprint arXiv:1112.6209, 2011.
* [11] Yann LeCun and Corinna Cortes. The mnist database of handwritten digits, 1998.
* [12] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013.
* [13] Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional neural networks. arXiv preprint arXiv:1311.2901, 2013.
|
Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2023.0322000
This work was funded (or partially funded) by the Center for Statistics and
Applications in Forensic Evidence (CSAFE) through Cooperative Agreements
70NANB15H176 and 70NANB20H019 between NIST and Iowa State University, which
includes activities carried out at Carnegie Mellon University, Duke
University, University of California Irvine, University of Virginia, West
Virginia University, University of Pennsylvania, Swarthmore College and
University of Nebraska, Lincoln.
Corresponding author: Abby Martin (e-mail: [email protected]).
# Forensic Camera Identification: Effects of Off-Nominal Exposures
ABBY MARTIN1 ROY MAXION2 and JENNIFER NEWMAN3 Department of Mathematics,
Iowa State University, Ames, IA 50011 USA (e-mail<EMAIL_ADDRESS>Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA
15213 USA (e-mail<EMAIL_ADDRESS> Department of Mathematics, Iowa State
University, Ames, IA 50011 USA (email<EMAIL_ADDRESS>
###### Abstract
Photo-response non-uniformity (PRNU) is a technology that can match a digital
photograph to the camera that took it. Due to its use in forensic
investigations and use by forensic experts in court, it is important that
error rates for this technology are reliable for a wide range of evidence
image types. In particular, images with off-nominal exposures are not
uncommon. This paper presents a preliminary investigation of the impact that
images with different exposure types — too dark or too light — have on error
rates for PRNU source camera identification. We construct a new dataset
comprised of 8400 carefully collected images ranging from under-exposed (too
dark) to nominally exposed to over-exposed (too bright). We first establish
baseline error rates using only nominally exposed images, resulting in a true-
positive rate of 100% and a true-negative rate of 99.92%. When off-nominal
images are tested, we find striking results: the true-negative rate for under-
exposed images is 99.46% (a false-positive rate of roughly one in two hundred,
typically unacceptable in a forensic context), and for over-exposed images the
true-positive rate falls to 82.90%. Our results highlight the importance of
continued study of error rates for the PRNU source camera identification to
assure adherence to the high standards set for admissibility of forensic
evidence in court.
###### Index Terms:
Camera Identification, Digital Forensics, Image Forensics, PRNU, Questioned
Images
=-21pt
## I Introduction
Consider a photo found at a crime scene. The photo can be traced to the camera
that took it using a phenomenon known as photo response non-uniformity (PRNU).
PRNU is a unique and persistent pattern (fingerprint) of variabilities of
voltage levels across the pixels in a digital camera’s photosensitive image
sensor, and this pattern appears in each photo the camera takes. The PRNU
fingerprint of the camera’s sensor is compared with the PRNU fingerprint of a
photo, producing a score measured against an accepted fixed threshold to
determine a match (or non-match) between the camera sensor and the photo.
The PRNU source-camera-identification algorithm is based on a large-scale
study of images from Flickr [1], and is considered accurate across a variety
of images. Flickr is a public website for users to share images. Due to self-
curation (users uploading their best images), Flickr has a negligible
proportion of off-nominal exposure images [2] (e.g., over- or under-exposed,
like the examples in Fig. 2 [3]). Thus, results from Flickr data may have
overlooked limitations, including generalizing error rates to data not
represented in the Flickr dataset (e.g., off-nominal exposure types).
The exclusion of off-nominal data when testing forensics tools has real-world
consequences. Studies performed by the forensic community play a critical role
supporting admissibility of expert witness testimony under federal law
(including estimating error rates), as introduced by Daubert v. Merrell Dow
Pharmaceuticals [4]. A large-scale study [1] established the court-approved
PRNU source camera identification algorithm using Flickr images to determine
error rates. However, many forensic tools have opportunities for improvement
due to a host of factors such as lack of standardize corpora [5], non-existent
tool validation procedures [6], etc. We were unable to find any studies that
determined error rates for off-nominal exposure images for the PRNU source
camera identification algorithm. Additionally, the data in [1] is no longer
available [7]. This motivated our investigation of the PRNU source-camera-
identification algorithm applied to known off-nominal image data.
Our contribution is a methodology to evaluate the response of the PRNU source-
camera-identification algorithm to off-nominal exposure images. Forensic
experts representing both the prosecution and defense can use publications
incorporating this methodology to better represent the error rates associated
with specific evidence images, especially the false-positive error rate
(incorrectly matching an image with a camera). Our methodology can be adapted
to estimate PRNU source-camera-identification error rates for other types of
off-nominal images as well. This process can be integrated into the estimation
of error rates for a tool or technique to satisfy the Daubert criteria.
The rest of the paper is organized as follows: Section II outlines the
research problem and approach; Section III clarifies language used throughout
the remainder of the paper; Section IV is a discussion of the forensic
development of PRNU source camera identification; Section V provides a
detailed overview of the PRNU source-camera-identification algorithm used in
this paper; Section VI describes the data and characterizes under- and over-
exposed images; Section VII outlines the methodology of the experiments
performed; and Section VIII presents the results. In Section IX, we share the
limitations of the experiments, followed by a discussion of implications of
our results in Section X. We conclude with a summary in Section XI.
## II Problem and Approach
Recognizing that prior work did not account for off-nominal images (e.g., too
light or too dark), the present work aims to determine whether off-nominal
images skew the true positive and true negative rates previously reported in
[1]. We implement the PRNU algorithm presented in that work to establish error
rates for off-nominal images. In Section VII, we present a rigorous evaluation
of the error rates for this source-camera-identification algorithm when the
image exposures are off-nominal, including a sensitivity analysis to determine
how proportions of off-nominal images in the dataset can impact error rate
estimates. In principle, this framework could also be applied to other types
of off-nominal image data (e.g., digital zoom or out-of-focus).
## III Terminology
The image of unknown origin, which could be considered evidence in a forensic
case, is referred to as the questioned image. The camera believed to have
taken the questioned image is referred to as the specific camera. To avoid
repetitious language, when we refer to an image, we mean the questioned image
(unless otherwise specified). Similarly, when we refer to a camera, we mean
the specific camera.
We focus on the source-camera-identification problem, which aims to identify
the specific camera that captured a questioned image. We do not address the
camera-model-identification problem, which attempts to identify only the model
of the camera used to collect the questioned image. As an example, source
camera identification could conclude an image came from a specific iPhone6s
camera, whereas camera-model identification would only be able to conclude the
image was from some iPhone6s camera (i.e., _this_ camera versus this _kind_ of
camera). For simplicity, when we refer to camera identification we are
referring to the source-camera-identification problem. We use PRNU cameraID
algorithm to refer specifically to the source-camera-identification algorithm
as established in the large-scale study [1], which is the algorithm used
throughout this paper.
## IV Background and Related Work
Photo response non-uniformity (PRNU) is a persistent artifact in digital
images due to imperfections in the camera-sensor manufacturing process [8].
When light impinges on the photosensitive portion of a pixel, called the
photodiode, the pixel responds by generating a current in proportion to the
number of photons striking it. However, the imperfections result in consistent
differences from the mean values of currents among the pixels, and it is this
pattern of responses – the PRNU – that is unique to each camera sensor. This
pattern remains constant and present in each image; therefore, PRNU can be
considered part of the sensor’s fixed pattern noise.
The first digital image sensor, a charged-coupled device (CCD), was invented
in 1970 [9], and within a year large variabilities in dark signal (thermal
noise when no light is falling on the sensor) were observed [10]. The first
use of fixed pattern noise to identify an individual digital camera sensor
appeared in [11] using the dark signal. This method assumes scenes that are
very dark, so it is not useful for images taken under typical lighting
conditions.
In 2005, the first computational method was introduced for extracting the PRNU
from an image for the purpose of source camera identification [12]. Since the
PRNU can be extracted from photos taken in typical lighting environments,
researchers shifted to the PRNU as a camera fingerprint. Several improvements
followed, including: introducing a maximum likelihood estimator to improve the
camera fingerprint calculation [13]; changing to the peak-to-correlation
energy (PCE) ratio as a similarity metric [14]; and discovering that (unlike
correlation scores) a threshold value based on the PCE does not need to change
for each camera fingerprint estimation [15].
PRNU-based camera identification became standardized for court use in a large-
scale study performed on 1,053,580 JPEG images from Flickr representing
approximately 6,896 cameras over 150 models [1]. This work used the
distribution of PCE scores between the questioned images and the camera
fingerprints to set a threshold based on the false positive rate (FPR) to
ensure acceptable true negative rates (TNR) and true positive rates (TPR),
resulting in a recommended PCE threshold of 60. The overall identification
rates given in [1] are a TPR of 97.62% and a TNR of 99.9976%. Our goal is to
use the standard algorithm set by this work to establish error rates for a
very different set of data. In Section V, we give details of the PRNU source-
camera-identification (PRNU cameraID) algorithm as established in [1].
Research since 2009 has continued to modify and question PRNU camera
identification. Use of the PRNU has expanded to fields such as forensic
countermeasures for forged PRNU information [16, 17], image anonymity
techniques [18], convolutional neural networks (CNN)-based forgery detection
[19], and user authentication [20]. Many papers have proposed changes to the
PRNU cameraID algorithm: enhancements to fingerprint estimation [21, 22, 23,
24]; different similarity measures [25]; and use of machine-learning methods
such as CNNs [26]. Other work has focused on the impact of various image
artifacts on PRNU camera identification, including: vignetting [27]; JPEG
compression [28]; proprietary image processing [29, 30, 31, 32]; color
saturation [33]; as well as gamma correction, contrast enhancement, histogram
equalization, and white balance [34]. One large-scale study of over 33,000
images from Flickr found several devices with low true negative rates (e.g.,
0.8% for the Fujifilm X-T30 camera) [2] for images from the same camera model
as the specific camera.
Concerns about PRNU-based camera identification have been raised in
preliminary investigations of exposure settings, specifically ISO. A brief
study of the effects of ISO values on PRNU camera identification found that
ISO impacts noise levels, gray levels, and correlation values [35]. The
Warwick database [36], which contains images with a variety of ISO values,
establishes that forgery detection and correlation-predictor values are
impacted by ISO [37]. However, the Warwick investigation of ISO relies on
correlation predictors to identify forged regions of images and does not
implement a decision threshold [37]. The standard PRNU cameraID algorithm in
[1] relies on PCE scores and a threshold of 60, not correlation predictors.
Additionally, the authors in [37] vary both the ISO and exposure time settings
to ensure similar exposure values between images of the same scene, meaning
all images have a consistent visual brightness. The research question in [37]
differs from ours: their paper asks whether correlation predictors and forgery
detection are impacted by ISO (the exposure type of the image remains constant
by changing the exposure time to compensate for the changes in ISO). We ask
whether the PCE score and camera identification are impacted by changing the
exposure type (i.e., auto-, under-, or over-exposed images). For our data, we
intentionally vary the ISO and exposure time to gather brighter and darker
versions of the same scene, thus varying the exposure value and brightness of
the image (see Fig. 2).
In order to establish error rates for off-nominal images, we must adhere to
the algorithm used by forensic experts. To the best of our knowledge, this is
the algorithm implemented in the large-scale study [1]. Whereas these recent
works [35, 37] [36] have addressed the impact of ISO on correlation, forgery
detection, and correlation predictor values, we do not know of any work that
has addressed the problem of over- or under-exposed images and their effect on
error rates for the PRNU cameraID algorithm. This paper tackles exactly that
problem.
## V Overview of PRNU Camera Identification
We refer to the protocol followed in [1] as the PRNU cameraID algorithm.
Generally, this algorithm consists of three parts: (1) estimating the PRNU
fingerprints of the camera and of the image; (2) calculating the peak-to-
correlation energy (PCE) ratio between these two fingerprints; (3) comparing
the PCE score with a threshold of 60 to determine whether (or not) the camera
captured the image. The details of this process are described in the remainder
of this section.
### V-A Estimate Fingerprints Using PRNU Noise Model
This section gives a summary of the fingerprint extraction algorithms as
presented in [38]. To compare a camera with an image, we must estimate the
PRNU noise from both the camera sensor and the image. Let $I$ be an image, and
let $I_{0}$ be the corresponding “noiseless image” which would result from a
sensor without any imperfections. Similarly, let $K$ be the true PRNU noise
component of the camera. Note that all multiplication in this section is
performed element-wise, thus the image $I$ is modeled:
$I=I_{0}+I_{0}K+\Theta,$ (1)
where all noise components besides the PRNU are denoted by $\Theta$ [38].
However, it is not feasible to obtain the noiseless image $I_{0}$ or the true
PRNU fingerprint $K$. In order to approximate the PRNU component of the image
noise, a high-pass filter is applied to suppress scene content and reduce non-
unique low-frequency patterns included in the noise fingerprint of an image,
such as intensity gradient and vignetting. The Daubechies wavelet denoising
filter [39] was originally chosen because it produced the best experimental
results, likely due to its superior scene suppression (particularly for edges
that appear within an image) [8]. Hence, $F$ is the Daubechies denoising
filter applied to the image.
The noise $W$ for image $I$ can be estimated [13]:
$W=I-F(I).$ (2)
This denoising step is followed by additional image processing of the
grayscale image to remove non-unique artifacts (NUAs) due to JPEG-compression
and camera-model fixed-pattern noise. NUAs can increase the similarity between
images from different cameras and thus contribute to a lower TNR [40]. The
image fingerprint estimate is therefore calculated as:
$Q=G(W),$ (3)
where $G$ represents these additional image processing operations.
The camera-fingerprint estimation follows a similar process, but is calculated
using a set of several images. This improves the PRNU noise estimate in (2) by
implementing a maximum likelihood estimate of the true camera fingerprint $K$
using several images. First, we apply $F$, the Daubechies denoising filter, to
all $N$ images. Let $I^{(i)}$ for $i\in\\{1,2,...,N\\}$ be the set of images
used to estimate $\hat{K}$, the camera fingerprint. First, estimate the image
fingerprint for each $I^{(i)}$ as before, $W^{(i)}=I^{(i)}-F(I^{(i)})$. Then
the camera fingerprint $\hat{K}$ is estimated [41]:
$\hat{K}=G\left(\frac{\sum_{i=1}^{N}W^{(i)}I^{(i)}}{\sum_{i=1}^{N}(I^{(i)})^{2}}\right),$
(4)
where we used $N=30$ images, and $G(\cdot)$ represents the additional
processing performed after calculating the maximum likelihood estimate of the
camera fingerprint to remove NUAs due to the camera model and JPEG
compression, as done for the fingerprint estimate for a single image.
From a careful reading of the available code [42] in MATLAB [43], we observe
that saturated pixels of an image are excluded from the camera fingerprint
estimate, where saturated pixels are characterized by the maximum intensity
value of the image (at least 250) with at least one neighboring pixel of equal
intensity. Although we intentionally collected very bright images, none of the
pixels in our over-exposed (or any exposure type) image set meet these
standards for a saturated pixel.
### V-B Peak-To-Correlation Energy (PCE) Calculation
When we calculate the similarity between camera and image fingerprints, we use
the signed PCE score given in Equation (8) of [28] as well as in the MATLAB
implementation [42]. Using the signed PCE score differs slightly from the PCE
calculation in [1], because if the peak correlation is negative, then the PCE
score will be negative. Negative PCE scores could change the estimated
probability density function (Equation (12) in [1]) used to set the threshold
of 60. An alternative distribution could impact the chosen PCE decision
threshold, which would change the estimated error rates. Although it is
unlikely that the signed PCE would impact the PRNU cameraID error rate
estimates, it is worth noting this change from the algorithm as initially
implemented [1].
### V-C PRNU Source Camera Identification Algorithm Overview
To replicate the algorithm in [1], we used the MATLAB [43] code provided by
the same authors [42]. An overview of the PRNU cameraID algorithm is shown in
Fig. 1. The inputs are two fingerprints, one from the camera (Box 1 in Fig. 1)
and one from an image (Box 2 in Fig. 1), and the questioned image (Box 3 in
Fig. 1). Recall that the image fingerprint is estimated from a single image,
and the camera fingerprint is estimated from 30 images.
Figure 1: PRNU cameraID Algorithm. Boxes 1 and 2 are the fingerprints
estimated for the camera and questioned image, respectively. Operation 4 is an
element-wise multiplication between two matrices representing the camera
fingerprint (Box 1) and questioned image (Box 3). If the PCE (Box 5) is above
60, conclude that the image was taken by the specific-camera (Box 6);
otherwise not (Box 7).
The PCE score is calculated between the image fingerprint and the product of
the camera fingerprint and the image pixel intensities (operation 4 on Boxes 1
and 3 in Fig. 1). If the PCE score is greater than 60, conclude that the image
came from the camera under test (Box 6 in Fig. 1). Otherwise, conclude that
the image originated elsewhere (Box 7 in Fig. 1).
## VI Data
The dataset from [1] is no longer available [7], so we collected 8400 images
from StegoAppDB [3] with specific ISO and exposure-time settings relative to
the auto-exposure settings. This section provides our characterization of off-
nominal exposure types, as well as the data sources, data protocol, and
theoretical and experimental support for our labeling decision of over- and
under-exposed images.
### VI-A Characterization of Over- and Under-Exposed Images
A selection of over-, under-, and auto-exposed images from our dataset is
shown in Fig. 2, along with their ISO and exposure time settings. These are
the three exposure types of data we use in our experiments. Perceived
brightness of objects varies, so we characterize an off-nominally exposed
image by comparing its level of brightness with its nominally exposed version.
Specifically, we say an image is over-exposed (third row in Fig. 2) if its
overall visual appearance is noticeably brighter than its auto-exposed
counterpart (second row in Fig. 2). Similarly, we say an image is under-
exposed (first row in Fig. 2) if its overall visual appearance is noticeably
darker than its auto-exposed counterpart. See Section VI-B2 for the formulaic
relation between auto- and off-nominal exposure settings.
Figure 2: Off-Nominal and Nominal Exposure Examples. The first row is a sample
of under-exposed images, the second row is a sample of auto-exposed images,
and the third row is a sample of over-exposed images from our dataset [3]. The
pair (ISO, EXP) denotes the auto-exposure ISO and exposure time settings,
respectively. The programmed (ISO, EXP) values relative to the auto-exposed
settings are given to the left of the images under the exposure type, and the
recorded values for the actual data are provided underneath the photo itself.
The model and device number that acquired the trio of images is given at the
top of each column of images (model-device number). Five different devices
provided the sample data.
The intentional selection of exposure settings relative to auto-exposure is
one supporting argument for an image being over- or under-exposed, but we
provide two additional arguments that strengthen our characterization of
exposure type: (1) the exposure value [44] quantifies the brightness of each
off-nominal image relative to its auto-exposed version; and (2) a visual
inspection by three human judges assesses the agreement between the captured
exposure settings and a human visual assessment of the brightness level. With
these additional arguments, we obtain an acceptable level of certainty that
the characterizations of exposure type for over-, under-, and auto-exposed
images in StegoAppDB are indeed consistent and are satisfactory for testing
the PRNU cameraID algorithm for nominal and off-nominal exposures.
### VI-B Apparatus & Instrumentation
#### VI-B1 Apparatus
Twenty-eight smartphone cameras were used to acquire the images used in this
research; they are cataloged in Table I.
#### VI-B2 Instrumentation
The controlled collection of images for hundreds of scenes using 28 separate
smartphone cameras in StegoAppDB [3] requires some amount of automation, not
least because the images were taken with specific ISOs and exposure times. For
this reason, an application (app) called Cameraw (pronounced camer-raw) was
written in Apple’s Swift language [45] for Apple devices, and in Java [46] for
Android devices.
TABLE I: Instruments, OS, protocols. Parenthetical numbers indicate multiple instances of the camera model. 100 over-, 100 auto-, and 100 under-exposed images (300 images total) per camera. | Camera (# of Devices) | OS | Protocol
---|---|---|---
1 | Pixel1 (4) | Android | A (Same Scene)
2 | Pixel2 (4) | Android | A (Same Scene)
3 | iPhone6s (4) | iOS | A (Same Scene)
4 | iPhone7 (4) | iOS | A (Same Scene)
5 | iPhone8 (2) | iOS | A (Same Scene)
6 | OnePlus5 (2) | Android | B (Unique Scene)
7 | SamsungS8 (2) | Android | B (Unique Scene)
8 | iPhone6sPlus (2) | iOS | B (Unique Scene)
9 | iPhone7Plus (2) | iOS | B (Unique Scene)
10 | iPhoneX (2) | iOS | B (Unique Scene)
Cameraw operates much like any other camera application, using one button to
capture a scene. When that button is pressed, Cameraw collects images with a
variety of exposure settings, stepping through a pre-selected sequence of
changing ISO and exposure times (EXP). From this sequence, we use the
following three settings: (1) auto-exposed/nominal (camera establishes ISO and
EXP automatically); (2) over-exposed/off-nominal (3 * ISO, 2 * EXP); and (3)
under-exposed/off-nominal (0.5 * ISO, 0.5 * EXP). Camera aperture for each
device remained constant (smartphone apertures cannot be changed).
### VI-C Images & Data-Collection Protocols
#### VI-C1 Images
The data comprise 8400 images from StegoAppDB [3], with 300 images captured
from each of 28 smartphones across ten brands/models (e.g., Apple/iPhone8);
see Table I. One third of all the images were auto-exposed; one third of all
images were intentionally over-exposed; the remaining third were intentionally
under-exposed. Full-sized images are used for the experiments described in
this paper.
#### VI-C2 Protocol
Table I shows two protocols, A and B, that were followed during data-
collection. One (A) is for same-scene content; the other (B) is for unique-
scene content.
Protocol A: Same-scene content. Each of the 18 protocol-A cameras was attached
to a tripod in a given scene location. Each camera took three images with
staggered exposure settings, as previously described. All cameras were
similarly oriented, so there were no upside-down images. This process was
repeated 100 times, with 100 unique scene positions for the tripod. The scene
content was the same for all 18 cameras, although the registration may have
been slightly imperfect.
Protocol B: Unique-scene content. Ten cameras follow a “unique” scene content
acquisition protocol. This is the same as Protocol A except the scene content
is not repeated from one camera to the next. In contrast to the Protocol-A
images, which comprised 100 different scenes, Protocol-B images comprised 1000
different scenes.
### VI-D Exposure Value Comparison
One clear justification for our characterization of auto-, under-, and over-
exposed images is the exposure value for the examples in Fig. 2. Consider the
three images of oranges in a bowl from the OnePlus5-1 camera (first column of
the figure). The top-row image is visually much darker than the middle-row
auto-exposed image, and the bottom-row image is visually much brighter. The
under-exposed image has an ISO of 500, half of 1000 (the auto-exposed image
ISO value), and an exposure time of 1/100 seconds (again, half of the auto-
exposed image exposure time of 1/50 seconds). Similarly, the over-exposed
image has an ISO of 3000, three times the auto-exposed ISO of 1000, and an
exposure time of 1/25 seconds (twice the auto-exposed image exposure time of
1/50 seconds).
The exposure value is a quantification of light on the sensor determined by
the $f$-stop (related to the aperture, which is constant in smartphone
cameras), exposure time, and ISO. Exposure value is lower when there is less
light and higher when there is more light on the sensor. We calculate exposure
value as described in [44] relative to the exposure value with ISO 100
($EV_{100}$). For the auto-exposed example image, the exposure value is
$EV_{100}+\log_{2}10$. The under-exposed example image has an exposure value
of $EV_{100}+\log_{2}5$ and the over-exposed example has an exposure value of
$EV_{100}+\log_{2}30$. Clearly, the under-exposed image has quantifiably less
and the over-exposed image has quantifiably more light on the sensor than the
auto-exposed image. This relationship holds for all images in our dataset,
which consists of only typically-lit indoor scenes.
### VI-E Visual Validation of Off-Nominal Settings
Another supporting argument for the three exposure types is a validation of
the images using human judgment. By assessing the agreement amongst three
human judges and one computer program, we can measure the consistency of the
responses. If the consistency is high enough, we are satisfied that this
validation supports using these images for testing PRNU camera identification
of nominal and off-nominal exposures.
We used the same 5600 off-nominal images described in Section VI-C1, half of
which are under-exposed and half are over-exposed. This is too many images for
a human rater to evaluate in a reasonable amount of time without fatigue, so
we randomly selected 5% (280 images) for a human-judgment study. Half of those
(140 images) were too dark and the other half were too light. Each half was
mixed with 140 randomly drawn, nominally exposed images to form two sets of
280 images each.
The two sets of 280 images were shown to human judges in a web-based tool
displaying a single image at a time. The judge clicked one of three text boxes
indicating their judgment as auto/over/under-exposed. When a text box was
clicked, the tool logged the choice, and advanced to the next image. The task
took about 17 minutes per set.
The resulting data were analyzed using Fleiss’s kappa [47] with four raters –
three human judges and the computer program that chose the images in the first
place. Using the “R” statistical software package irr and the “R” function
kappam.fleiss [48, 49], the kappa value was 0.9430212 with a z-statistic of
75.70786 (i.e., 75.7 standard deviations away from the mean) with a p-value of
0 – a nearly-perfect agreement amongst the four raters. This is clearly a
level of confidence that justifies using these images for testing a camera-
identification algorithm.
## VII Methods
We propose a methodical investigation of off-nominally exposed images. The
methodology consists of the following steps:
1. 1.
Define a careful data collection protocol which minimizes the differences
between nominal and off-nominal data, intentionally changing the
characteristic under investigation.
2. 2.
Isolate the points in the forensic algorithm where the characteristic under
investigation impact the error rate estimate.
3. 3.
Establish baseline error rates by executing the forensic algorithm using only
the collected nominal data.
4. 4.
For each point in the algorithm identified in step 2, incrementally and
independently exchange the nominal data with the off-nominal data.
We are investigating exposure settings, so our nominal dataset is our auto-
exposed images and the off-nominal dataset consists of our over- and under-
exposed images. Next, we identify estimation of the camera fingerprint (Box 1
in Fig. 1) and the questioned image (Boxes 2 and 3 in Fig. 1) as the two steps
of the PRNU cameraID algorithm directly impacted by image exposure settings.
Section VII-A details our baseline experiment, which generates camera
fingerprints using auto-exposed images and uses a set of questioned images
composed only of auto-exposed images. Finally, we iteratively change the
exposure type of the images used to generate the camera fingerprint and the
set of questioned images. This allows us to understand how off-nominal
exposure images alter the error rate estimates for the PRNU cameraID
algorithm, and provides baseline error rates for direct comparisons.
We investigate the impact of off-nominal exposure settings on the PRNU
cameraID algorithm by partitioning the data into three exposure types. Each
exposure type is used systematically to generate the camera fingerprints
and/or questioned images.
We performed 3 fundamental experiments:
1\. Auto-exposed images (baseline / nominal)
2\. Over-exposed images (too light / off-nominal)
3\. Under-exposed images (too dark / off-nominal)
We also implement two sensitivity analyses, one for the sensitivity of the TPR
to different proportions of off-nominal exposure images in the questioned
image set and one for the sensitivity of error rates relative to the PCE
decision threshold. A sensitivity analysis can show how even small, controlled
changes impact error rate estimates.
### VII-A Auto-exposed images - baseline / nominal
The baseline experiment is the PRNU cameraID algorithm (Section V) applied to
a set of nominally exposed images, which comprises 2800 auto-exposed images
(100 auto-exposed images from each of the 28 cameras). We repeat the baseline
experiment for five trials, where each trial is defined by a specific
partitioning of the images.
First, we randomly partition the 100 images per camera into two groups: 1) 30
images used to generate the camera fingerprint; 2) 70 questioned images. The
images are partitioned so that no camera fingerprint image shares scene
content with any questioned image. For each camera, estimate the camera
fingerprint (28 cameras $\times$ 1 camera fingerprint = 28 camera
fingerprints). For each questioned image, estimate the image fingerprint (28
cameras $\times$ 70 images = 1960 image fingerprints).
Second, we calculate the PCE scores between each camera fingerprint and its
own questioned images (28 camera fingerprints $\times$ 70 fingerprints for
images from the camera = 1960 PCE scores). Next, we compare the PCE scores for
images from the camera to the threshold of 60. If the PCE score is above 60,
the image is a true positive and contributes to the TPR. If the PCE score is
at most 60, the image is a false negative and contributes to the False
Negative Rate (FNR = $1-$ TPR).
Next, we calculate the PCE scores between each camera fingerprint and a set of
questioned images from a different camera (28 specific-camera fingerprints
$\times$ 70 fingerprints from images from a different camera $\times$ 27
different cameras for test = 52920 PCE scores). To calculate a balanced
accuracy (i.e., an accuracy that responds equally to the TNR and TPR), we
select a random subset of 1960 PCE scores from the 52920 PCEs for questioned
images from a different camera. In order to avoid lucky or unlucky subsets of
PCE scores, we perform the random selection of 1960 PCE scores 100 times. We
compare the PCE scores for images from other cameras to the threshold of 60.
If the PCE score is above 60, the image is a false positive and contributes to
the FPR (FPR = $1-$ TNR). If the PCE score is at most 60, the image is a true
negative and contributes to the TNR. We calculate the accuracy for each of the
100 PCE score subsets using the 1960 PCE scores used to calculate the TPR, and
the 1960 PCE scores used to calculate the TNR. Finally, we average the TPR,
TNR, and accuracy values for the 100 PCE score subsets to calculate the error
rates for each of the five trials.
The auto-exposed image baseline experiment is also subjected to two
sensitivity analyses regarding the TPR and TNR: 1) for different proportions
of off-nominal exposures in the questioned image set, and 2) for different PCE
thresholds. For the first sensitivity analysis, we incrementally replace 1% of
the images comprising the questioned image set with off-nominal data, thus
creating 101 questioned image sets (i.e., the first set has 100% auto-exposed
questioned images and 0% off-nominally exposed images, the second set has 99%
auto-exposed questioned images and 1% off-nominally exposed images, and so on
with the final set comprising 100% off-nominally exposed images). The
sensitivity analysis of the PCE decision threshold is performed by
incrementally shifting the threshold of 60 and recalculating the TPR and TNR
values.
### VII-B Over-exposed images - too light / off-nominal
The over-exposed image experiment differs from the baseline by estimating two
camera fingerprints (one using auto-exposed images and one using over-exposed
images) and the questioned image set comprises only over-exposed images. This
experiment uses 5600 images: 100 auto-exposed images from each of the 28
devices and 100 over-exposed images from each of the 28 devices.
We adapt the procedure used in the baseline experiment for the over-exposed
experiment by performing the same procedural steps but with different sets of
data: 1) 30 auto-exposed images per camera are used to estimate the camera
fingerprints and 70 over-exposed images from each camera comprise the set of
questioned images (auto-fingerprint vs. over-image); 2) 30 over-exposed images
per camera are used to estimate the camera fingerprints and 70 over-exposed
images from each camera comprise the set of questioned images (over-
fingerprint vs. over-image). We repeat the sensitivity analysis of the PCE
threshold on the auto-fingerprint vs. over-image and over-fingerprint vs.
over-image experiments.
We remark that the most likely scenario where a forensic practitioner might
encounter an over-exposed image is as a questioned image. The practitioner may
have access to the suspect camera and can therefore control the exposure
settings of images used to estimate the camera fingerprint, but the exposure
of the questioned image is already established. A forensic practitioner using
over-exposed images for the camera fingerprint when the questioned image is
auto-exposed is an unlikely scenario, so we omit the results of this
experiment. We investigate the over-fingerprint vs. over-image scenario as a
possible solution to the degraded error rates caused by comparing auto-exposed
fingerprints and over-exposed questioned images. This experiment examines
whether camera fingerprints estimated with the same exposure type as the
questioned image will have a more similar PRNU estimate than camera
fingerprints estimated solely from auto-exposed images, potentially reducing
the error rates.
The sensitivity analysis of the PCE decision threshold is performed for both
experiments by incrementally shifting the PCE threshold and recalculating the
TPR and TNR values.
### VII-C Under-exposed images - too dark / off-nominal
The under-exposed image experiment repeats the investigations in Section VII-B
except we replace the over-exposed images with under-exposed images, as
characterized in Section VI — Data. The purpose behind each of the two under-
exposed image scenarios corresponds to the motivations for the over-exposed
experiment scenarios. Each investigation was also subjected to a sensitivity
analysis of the PCE threshold.
## VIII Results
We present results from the experiments detailed in Section VII \- Methods for
images of each exposure type. The TPR decreases by at least 14.27% for the
off-nominal questioned images compared to the auto-exposed questioned-image
baseline, meaning many images from the specific camera are missed. Similarly,
the TNR decreases to 99.46% for under-exposed questioned images - an error of
approximately one in two hundred images incorrectly identified as a match with
the specific camera. Such mistakes can lead to increased false positives,
often connected to wrongful convictions. Our results are separated into
nominal and off-nominal questioned images, and end with an investigation of
possible remediations (Sections VIII-A, VIII-B, and VIII-C, respectively).
TABLE II: Aggregate results over 5 trials averaged over 100 repetitions Experiment | TPR (std.) | TNR (std.) | Accuracy (std.)
---|---|---|---
| Auto-fingerprint
---
vs. Auto-image Baseline
1 (0) | 0.9992 (0.0001) | 0.9996 (0.0001)
| Auto-fingerprint
---
vs. Over-image
0.8290 (0.0030) | 0.9998 (0.0001) | 0.9144 (0.0015)
| Auto-fingerprint
---
vs. Under-image
0.8573 (0.0003) | 0.9946 (0.0007) | 0.9260 (0.0004)
| Over-fingerprint
---
vs. Over-image
0.8888 (0.0070) | 0.9996 (0.0002) | 0.9442 (0.0035)
| Under-fingerprint
---
vs. Under-image
0.9999 (0.0002) | 0.8426 (0.0043) | 0.9213 (0.0020)
TABLE III: True Positive Rate (TPR), True Negative Rate (TNR), and standard deviation (STD) for auto-exposed camera fingerprint with auto-exposed questioned images (left column), over-exposed test data (middle column), and under-exposed test data (right column) by camera model. TPRs and TNRs lower than the baseline experiment are in BOLD for the off-nominal experiments. Camera Model | Auto-fingerprint vs. Auto-image Baseline | Auto-fingerprint vs. Over-image | Auto-fingerprint vs. Under-image
---|---|---|---
| TPR (STD) | TNR (STD) | TPR (STD) | TNR (STD) | TPR (STD) | TNR (STD)
Pixel1 | 1 (0) | 0.9998 (0.0002) | 0.9850 (0.0122) | 0.9995 (0.0003) | 1 (0) | 0.9857 (0.0021)
Pixel2 | 1 (0) | 0.9998 (0.0003) | 0.0007 (0.0016) | 0.9999 (0.0002) | 0.0014 (0.0020) | 0.9977 (0.0010)
iPhone6s | 1 (0) | 0.9999 (0.0001) | 0.9936 (0.0016) | 0.9997 (0.0001) | 1 (0) | 0.9984 (0.00003)
iPhone7 | 1 (0) | 0.9999 (0.0001) | 0.9750 (0.0067) | 0.9999 (0.0001) | 1 (0) | 0.9995 (0.0003)
iPhone8 | 1 (0) | 1 (0) | 0.9543 (0.0130) | 1.0000 (0.0001) | 1 (0) | 0.9998 (0.0003)
OnePlus5 | 1 (0) | 0.9984 (0.0009) | 0.9957 (0.0039) | 0.9993 (0.0006) | 1 (0) | 0.9673 (0.042)
SamsungS8 | 1 (0) | 0.9920 (0.0018) | 1 (0) | 0.9999 (0.0001) | 1 (0) | 0.9957 (0.0005)
iPhone6sPlus | 1 (0) | 1 (0) | 0.9914 (0.0032) | 0.9999 (0.0002) | 1 (0) | 0.9992 (0.0002)
iPhone7Plus | 1 (0) | 1 (0) | 0.8729 (0.0155) | 1.0000 (0.0001) | 1 (0) | 0.9994 (0.0004)
iPhoneX | 1 (0) | 1 (0) | 0.8829 (0.0139) | 1.0000 (0.0001) | 1 (0) | 0.9999 (0.0002)
### VIII-A Baseline (Nominal Images)
The case where the camera fingerprint and questioned images are auto-exposed
provides a baseline to compare with the off-nominal experiments. Results from
this baseline experiment are shown in the first row of Table II, and represent
the current scenario used by forensic practitioners. The TPR is 100% and the
TNR is 99.92%. Results for the same baseline data are listed by the 10
individual models, shown on the left-hand side of Table III. These results are
in line with the error rates from the study in [1].
### VIII-B Off-Nominal Questioned Image Sets
The most singular results using off-nominal data with the PRNU cameraID
algorithm are the TPR values shown in Table II. Rows two and three in Table II
are error rate estimates for the most common scenarios where a practitioner
might encounter an off-nominally exposed image: when the camera fingerprint is
estimated from auto-exposed images, but the questioned image is over- or
under-exposed. The TPR values for these two experiments are strikingly
different from the baseline results. When the camera fingerprint is composed
of auto-exposed images and all questioned images are over-exposed, the TPR is
82.90% and the TNR is 99.98% (row two, Table II). When the camera fingerprint
is composed of auto-exposed images and all questioned images are under-
exposed, the TPR is 85.73% and the TNR is 99.46% (row three, Table II). The
TPRs for these off-nominal images – a TPR decrease of 17.1% for over-exposed
and of 14.27% for under-exposed - are much lower than the 100% TPR for the
baseline experiment. The large reduction in TPR from our baseline experiment
warrants attention to the differing error rates between image exposure types.
Additionally, the TNR of 99.46% for the under-exposed questioned images (row
three, Table II) corresponds to an FPR of 0.54%. This is an error of roughly
one in two hundred associated with incorrectly matching an image with a
camera.
The sensitivity analysis of the TPR when the questioned image set consists of
various proportions of off-nominal images is given in Fig. 3. This sensitivity
analysis highlights the negative linear relationship between the proportion of
off-nominally exposed questioned images and the TPR. Both the over-exposed
questioned images (Fig. 3, solid blue line) and the under-exposed questioned
images (Fig. 3, dashed orange line) have a direct impact on the TPR estimate,
even when only a small percent of the questioned image set are off-nominally
exposed. The leftmost endpoints of the orange and blue lines in Fig. 3
correspond to TPRs of 100% (TPR for row one of Table II). Similarly, the
rightmost endpoints of the orange (dashed) and blue (solid) lines in Fig. 3
correspond to a TPR of 82.90% for over-exposed images (row two of Table II)
and 85.73% for under-exposed images (row three of Table II). Note that the TPR
for over-exposed images is always less than the TPR for under-exposed images,
regardless of the proportion of off-nominal test data. The consistency of this
linear relationship supports the notion that off-nominal exposure types do
indeed affect the error rates of the PRNU cameraID algorithm.
The results for the auto-exposed camera fingerprint experiments are listed for
the 10 models in Table III (auto-image in the left-hand column, over-image in
the middle column, and under-image in the right-hand column). Note that the
Pixel 2 performs dramatically poorly for both off-nominal exposures with a TPR
of 0.07% on over-exposed questioned images and TPR of 0.14% on under-exposed
questioned images. Although determining the cause of the Pixel 2 camera
model’s poor performance for off-nominally exposed images is outside the scope
of this paper, we theorize this could be caused by proprietary pipeline
processing, which is protected by manufacturers (in this case, Google). Poor
performance for specific camera models has been observed in prior research,
such as the 0.8% TNR of the Fujifilm X-T30 in [2]. Particularly poor
performance by individual models is another reason to encourage rigorous tool
validation procedures [6].
Figure 3: Sensitivity Analysis of Off-Nominally Exposed Questioned Images:
Both under-exposed (dashed orange line) and over-exposed (solid blue line)
questioned images have a roughly linear relationship with the degradation of
the True Positive Rate (TPR). This implies that even small percentages of off-
nominally exposed images in the questioned image set can have an impact on the
error rate estimates.
### VIII-C Possible Solutions for Off-Nominal Data
Figure 4: Sensitivity analysis of PCE threshold (off-nominal exposures). The
upper left graph (baseline) shows a near-perfect TPR and TNR for the PCE
threshold of 60 (dashed red line) when both the camera fingerprint and the
image fingerprint are estimated from nominal images. However, when
fingerprints are estimated from any off-nominal images, either the TPR or the
TNR degrades dramatically. Raising the PCE threshold does not improve the TPR
(solid blue line), although it does gradually improve the TNR (dotted orange
line). Lowering the PCE threshold slightly improves the TPR, but does degrade
the TNR.
One possible method to improve the error rates for off-nominal exposures is to
estimate the camera fingerprint from the same exposure type as the questioned
image. When the camera fingerprint uses only over-exposed images and the
questioned images are also over-exposed (row 4 in Table II), the TPR rises
slightly to 88.88%, but the TNR falls to 99.96%. Similarly, when the camera
fingerprint uses only under-exposed images and the questioned images are also
under-exposed (row 5 in Table II), the TPR increases considerably (99.99%),
yet the TNR decreases (84.26%). A drop in the TNR corresponds to an increase
in false positives, so this is a trade-off that forensic practitioners are
unlikely to accept.
Another possible solution to the reduced accuracy values in Table II for off-
nominal exposures is to use an alternative PCE threshold. We perform a
sensitivity analysis of the TPR and TNR to different PCE thresholds by
incrementally shifting the integer PCE threshold and recalculating the TPR and
TNR using the same set of PCE scores. Fig. 4 demonstrates that lowering the
threshold to increase the TPR for the over-exposed image experiments (plots
(b) and (d) in Fig. 4) or for the under-exposed image experiments (plots (c)
and (e) in Fig. 4) would also lower the TNR, meaning that the overall accuracy
would not improve.
Our final experiment also investigates an alternate PCE threshold. The PCE
threshold of 60 was initially set to produce a 0% FPR on the subset of images
from a different model than the specific camera [1]. We compute the lowest
integer threshold value to produce a 0% FPR (regardless of camera model) and
recalculate the accuracy (results for each experiment are given in Table IV).
Note that in each case the threshold must be raised, and in some cases more
than doubled (rows 1 and 3 through 5 in Table IV). For the over-exposed image
experiments, these accuracy values are all lower than the accuracy using the
PCE threshold of 60 (rows 2 and 4 in Table IV). The alternate threshold does
improve the accuracy slightly for the auto-fingerprint vs. under-image
experiment (row 3 in Table IV) compared with the results for the PCE threshold
of 60 (row 3 in Table II). Choosing a threshold based on a 0% FPR, however,
not only depends on the dataset and exposure type, but is problematic for
other reasons detailed in Section X.
TABLE IV: Lowest threshold resulting in a 100% TNR (0% FPR) for each experiment and the corresponding TPR and accuracy. Experiment | Threshold | TPR | TNR | Accuracy
---|---|---|---|---
| Auto-fingerprint
---
vs. Auto-image Baseline
210 | 1 | 1 | 1
| Auto-fingerprint
---
vs. Over-image
77 | 0.8213 | 1 | 0.9107
| Auto-fingerprint
---
vs. Under-image
143 | 0.8569 | 1 | 0.9285
| Over-fingerprint
---
vs. Over-image
442 | 0.2300 | 1 | 0.6150
| Under-fingerprint
---
vs. Under-image
539 | 0.7587 | 1 | 0.8794
## IX Limitations
Digital image forensics researchers typically encounter one of two problems
for data collection: resource exhaustion or quality of images. Manually taking
enough pictures from a large variety of cameras is often infeasible due to the
significant time and resources required. An alternative source of data
collection is scraping images from online collections (e.g., Flickr). However,
these images are of unknown provenance and their true origin and exposure
settings are frequently unknown.
The foundational PRNU cameraID research [1] prioritized the size of the
dataset by downloading over one million Flickr images from nearly seven
thousand cameras, enabling them to make universal claims. The absence of a
data-collection protocol makes it impossible to ascertain the proportion of
nominal/off-nominal images in that dataset. We chose instead to prioritize
control over the image collection and exposure settings by taking 8400
calibrated images from 28 cameras, as detailed in Section VI \- Data. Although
we are not claiming universality, we present our results as a demonstration
that off-nominal exposure settings can alter camera-identification error
rates, often quite dramatically. Our claims about our dataset are only
possible because of the exacting data-collection protocol followed. A similar,
but larger-scale study remains for future research.
A further limitation is that the cameras used in our research are not the most
recent models (e.g., the iPhone X was released in November 2017). We have
estimated that it would take at least 14 weeks to add just one more camera of
a new model. Adding a new camera would delay the preparation of a technical
report by roughly a quarter of a year, by which time yet another new model
would likely have been released. Therefore, it is simply not practical to
continually add up-to-date cameras for the purpose of one paper. Newer camera
models include additional complications that must be addressed in future work,
including high dynamic ranges, multiple lenses and sensors, new proprietary
processing, AI, etc. That said, we recognize the importance of maintaining
current datasets if camera-identification technology is to keep pace with
camera development.
## X Discussion
The off-nominal image experiments performed dramatically poorly when compared
with the auto-exposed image baseline. The 14-17% TPR decrease is quite large
(rows two and three, Table II) and cannot be attributed to chance, poor data
quality, or methodological errors. When used to investigate which camera
captured an evidence image, a TPR of 85.73% is roughly one out of seven,
meaning the correct camera might be missed as many as one out of seven times
if the questioned image is under-exposed (or more often if the questioned
image is over-exposed).
One goal of forensic research is to prepare methodologies for use in a court
of law. The PRNU camera-identification algorithm described in [1] forms the
foundation of an FBI application called FINDCamera [50]. The false-positive
error rate estimate of one in a million presented by an expert witness was
based on over one million images from Flickr [51]. Our results show that the
false-positive error rate can rise to one in two hundred for under-exposed
questioned images, and one in five thousand for over-exposed questioned
images. Differences in error rate estimates could impact how jurors weigh the
strength of evidence, which is particularly important in cases with severe
consequences. For example, the aforementioned 2011 trial resulted in a prison
sentence of 45 years [51].
We investigated two seemingly-obvious solution candidates that, unfortunately,
turned out not to fully address the aforementioned shift in error rates. The
first solution was to create camera fingerprints consistent with the exposure
type of the questioned image. However, this approach only modestly improves
the over-exposure true-positive error rates, while causing the false-positive
rate to skyrocket for under-exposed images (rows 4 and 5 of Table II). A
second solution is to use a PCE threshold other than 60, as demonstrated in
Fig. 4 and in Table IV. Note that the alternate thresholds only minimally
improve the error rates, and in some cases simply exacerbate the problem.
Additionally, recent work has raised concerns that examiners changing decision
thresholds based on subjective analysis of the evidence and forensic scenario
can negatively impact the legal system [52].
Although there is a good instinct to mitigate any error inherent to a
methodology, it can be difficult to change existing forensic tools. Further
refining error rates in context of the questioned image can help forensic
practitioners better understand and communicate the error associated with pre-
existing tools. Introducing new methodologies requires both acceptance by
forensic practitioners and rigorous research to meet the Daubert standard,
which requires time, use by others in the community, and passing another
Daubert challenge for the new tool. Methods which can be used in conjunction
with current tools (such as our proposed protocol to estimate more accurate
error rates with respect to the exposure type of the questioned image) can
introduce needed incremental changes between significant technological shifts.
Our experiments clearly demonstrate that off-nominal images (e.g., over- or
under-exposed) impact the error rates of the PRNU cameraID algorithm. We also
show that the two most obvious and straightforward modifications to the
existing algorithm do not adequately rectify the performance problems for off-
nominal exposure images. Proper tool validation and error rate estimation is a
crucial aspect of this forensic field that must continually be improved.
## XI Conclusions
We present a study of the PRNU source-camera-identification algorithm [1] for
off-nominal (over- and under-exposed) images using a meticulously-collected
dataset [3]. Our work implements a systematic investigation to show that error
rates are worse when off-nominal images are used for forensic source camera
identification. In particular, for over-exposed questioned images the true-
positive rate is 82.90%, as compared with 100% for nominal (auto-exposed)
images. Of note is the contrast between our nominal baseline’s false-positive
rate (0.08%) and the roughly one in two hundred false-positive rate for under-
exposed (too dark) images. This disparity is concerning, as it can have real-
life consequences in the criminal judicial process. Simple and obvious
mitigations, such as changing PCE thresholds, do not solve the error-rate
problem for off-nominal images. The insight gained from our methodology can
help forensic practitioners better understand and communicate the error rates
of forensic tools when applied to data representing off-nominal conditions.
## Appendix A Data and Code Availability
Data from this study are available upon request. Code is available from its
authors [42].
## Appendix B Attributions/Declarations
Author contributions:
AM: Conceptualization, analysis, initial/final drafts.
RM: Conceptualization, methodology, analysis, all drafts.
JN: Conceptualization, funding acquisition, initial draft.
Ethical approval:
The Iowa State University Institutional Review Board confirmed that the human-
subjects aspect of this study is exempt; the information collected contains no
personally identifiable information, and is not intended to contribute to
generalizable knowledge.
Conflicts:
The authors declare no conflicts of interest.
## Acknowledgment
We thank Huayun Huang for her design and implementation of the user-study
data-set validation software.
## References
* [1] M. Goljan, J. Fridrich, and T. Filler, “Large scale test of sensor fingerprint camera identification,” in IS&T/SPIE Electronic Imaging Science and Technology, pp. 1–12, 04 February 2009. San Jose, CA, USA, 18-22 January 2009.
* [2] M. Iuliani, M. Fontani, and A. Piva, “A leak in PRNU based source identification—questioning fingerprint uniqueness,” IEEE Access, vol. 9, pp. 52455–52463, 2021.
* [3] J. Newman, L. Lin, W. Chen, S. Reinders, Y. Wang, M. Wu, and Y. Guan, “StegoAppDB: A steganography apps forensics image database,” in Proceedings of the IS&T International Symposium on Electronic Imaging: Media Watermarking, Security, and Forensics (N. D. M. Adnan M. Alattar and G. Sharma, eds.), pp. 536–1 – 536–11, 13-17 January 2019. Burlingame, CA, USA, 13-17 January 2019.
* [4] Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579 (1993).
* [5] S. Garfinkel, P. Farrell, V. Roussev, and G. Dinolt, “Bringing science to digital forensics with standardized forensic corpora,” Digital Investigation, vol. 6, pp. S2–S11, 2009.
* [6] N. Hughes and U. Karabiyik, “Towards reliable digital forensics investigations through measurement science,” WIREs Forensic Science, vol. 2, no. 4, p. e1367, 2020.
* [7] M. Goljan, “Personal communication; email to R. Maxion, 04 March 2023,” 2023.
* [8] J. Lukáš, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Transactions on Information Forensics and Security, vol. 1, no. 2, pp. 205–214, 2006.
* [9] W. Boyle and G. Smith, “Charge coupled semiconductor devices,” The Bell System Technical Journal, vol. 49, no. 4, pp. 587–593, 1970.
* [10] M. Tompsett, G. Amelio, W. Bertram, R. Buckley, W. McNamara, J. Mikkelsen, and D. Sealer, “Charge-coupled imaging devices: Experimental results,” IEEE Transactions on Electron Devices, vol. 18, no. 11, pp. 992–996, 1971.
* [11] K. Kurosawa, K. Kuroki, and N. Saitoh, “CCD fingerprint method-identification of a video camera from videotaped images,” in International Conference on Image Processing, pp. 537–540, 1999. Kobe, Japan, 24-28 October 1999.
* [12] J. Lukas, J. Fridrich, and M. Goljan, “Determining digital image origin using sensor imperfections,” in Proceedings of SPIE IS&T Electronic Imaging Science and Technology: Image and Video Communications and Processing (A. Said and J. G. Apostolopoulos, eds.), vol. 5685, pp. 249–260, 16-20 January 2005. San Jose, CA, USA, 16-20 January 2005.
* [13] M. Chen, J. Fridrich, and M. Goljan, “Digital imaging sensor identification (further study),” in Sec., Steg., and Watermarking of Multimedia Contents IX (E. J. D. III and P. W. Wong, eds.), vol. 6505, pp. 258 – 270, 2007.
* [14] M. Goljan and J. Fridrich, “Camera identification from cropped and scaled images,” in Proceedings of the IS&T International Symposium on Electronic Imaging: Security, Forensics, Steganography, and Watermarking of Multimedia Contents X, vol. 6819, 68190E, pp. 68190E–1 – 68190E–13, 26-31 January 2008. San Jose, CA, USA, 26-31 January 2008.
* [15] M. Goljan, “Digital camera identification from images – estimating false acceptance probability,” in International Workshop on Digital Watermarking, p. 454—468, 2008. Busan, Korea (Republic of), 10-12 November 2008.
* [16] M. Goljan, J. Fridrich, and M. Chen, “Sensor noise camera identification: Countering counter-forensics,” in Proceedings of SPIE IS&T Electronic Imaging: Media Forensics and Security II (A. M. A. Nasir D. Memon, Jana Dittmann and E. J. D. III, eds.), vol. 7541, pp. 75410s–1 – 75410s–12, 18-20 January 2010. San Jose, CA, USA, 18-20 January 2010.
* [17] M. Goljan, J. Fridrich, and M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Transactions on Information Forensics and Security, vol. 6, no. 1, pp. 227–236, 2011.
* [18] M. Irshad, N.-F. Law, K. Loo, and S. Haider, “IMGCAT: An approach to dismantle the anonymity of a source camera using correlative features and an integrated 1D convolutional neural network,” Array, vol. 18, p. 100279ff, 2023.
* [19] D. Cozzolino and L. Verdoliva, “Noiseprint: A CNN-based camera model fingerprint,” IEEE Transactions on Information Forensics and Security, vol. 15, pp. 144–159, 2020.
* [20] D. Maier, H. Erb, P. Mullan, and V. Haupert, “Camera fingerprinting authentication revisited,” in International Symposium on Research in Attacks, Intrusions and Defenses (RAID 2020), pp. 31–46, October 2020. San Sebastian, Spain, 14-16 October 2020.
* [21] A. Cortiana, V. Conotter, G. Boato, and F. G. B. DeNatale, “Performance comparison of denoising filters for source camera identification,” in Proceedings of the SPIE IS&T International Symposium on Electronic Imaging:Media Watermarking, Security, and Forensics III (N. D. Memon, J. Dittmann, A. M. Alattar, and E. J. Delp III, eds.), vol. 7880, pp. 78807–1 – 78807–6, 23-27 January 2011. San Francisco, CA, USA, 23-27 January 2011.
* [22] J. Li, Y. Liu, B. Ma, C. Wang, C. Qin, X. Wu, and S. Li, “A novel PCA-based method for PRNU distillation to the benefit of source camera identification,” Applied Sciences, vol. 13, no. 11, 2023.
* [23] S. Fernández-Menduiña, F. Pérez-González, and M. Masciopinto, “Source camera attribution via PRNU emphasis: Towards a generalized multiplicative model,” Signal Processing: Image Communication, vol. 114, p. 116944, 2023.
* [24] A. Mehrish, A. V. Subramanyam, and S. Emmanuel, “Robust PRNU estimation from probabilistic raw measurements,” Signal Processing: Image Communication, vol. 66, pp. 30–41, 2018.
* [25] S. Reinders, L. Lin, W. Chen, Y. Guan, and J. Newman, “Score-based likelihood ratiosfor camera device identification,” in Proceedings of the IS&T International Symposium on Electronic Imaging: Media Watermarking, Security, and Forensics (N. D. M. Adnan M. Alattar and G. Sharma, eds.), pp. 1–7, 26-30 January 2020. Burlingame, CA, USA, 26-30 January 2020.
* [26] M. Fanfani, A. Piva, and C. Colombo, “PRNU registration under scale and rotation transform based on convolutional neural networks,” Pattern Recognition, vol. 124, no. 108413, 2022.
* [27] C.-T. Li and R. Satta, “Empirical investigation into the correlation between vignetting effect and the quality of sensor pattern noise,” IET Computer Vision, vol. 6, no. 6, pp. 560–566, 2012.
* [28] M. Goljan, M. Chen, P. Comesaña, and J. Fridrich, “Effect of compression on sensor-fingerprint based camera identification,” in IS&T International Symposium on Electronic Imaging Science and Technology: Media Watermarking, Security, and Forensics (A. M. Alattar and N. D. Memon, eds.), pp. 1–10, Society for Imaging Science and Technology (IS&T), 14-18 February 2016. San Francisco, CA, USA, 14-18 February 2016.
* [29] D. Baracchi, M. Iuliani, A. G. Nencini, and A. Piva, “Facing image source attribution on iPhone X,” in International Workshop on Digital Watermarking, pp. 196–207, 2020. Melbourne, Victoria, Australia, 25-27 November 2020.
* [30] A. Montibeller and F. Pérez-González, “Exploiting PRNU and linear patterns in forensic camera attribution under complex lens distortion correction,” in IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 1–5, 2023. Rhodes Island, Greece, 04-10 June 2023.
* [31] C. Albisani, M. Iuliani, and A. Piva, “Checking PRNU usability on modern devices,” in IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 2535–2539, 06-11 June 2021. Toronto, ON, Canada, 06-11 June 2021.
* [32] S. Joshi, P. Korus, N. Khanna, and N. Memon, “Empirical evaluation of PRNU fingerprint variation for mismatched imaging pipelines,” in IEEE International Workshop on Information Forensics and Security, pp. 1–6, 2020. New York, NY, USA, 06-11 December 2020.
* [33] K. Henry, Digital photography analysis: Analytical framework for measuring the effects of saturation on photo response non-uniformity. University of Colorado at Denver, 2016.
* [34] S. Samaras, V. Mygdalis, and I. Pitas, “Robustness in blind camera identification,” in International Conference on Pattern Recognition, pp. 3874–3879, 04 December 2016. Cancun, Mexico, 04-08 December 2016.
* [35] L. Lin, W. Chen, Y. Wang, S. Reinders, Y. Guan, J. Newman, and M. Wu, “The impact of exposure settings in digital image forensics,” in IEEE International Conference on Image Processing, pp. 540–544, 2018. Athens, Greece, 07-10 October 2018.
* [36] Y. Quan, C.-T. Li, Y. Zhou, and L. Li, “Warwick image forensics dataset for device fingerprinting in multimedia forensics,” in IEEE International Conference on Multimedia and Expo, pp. 1–6, 2020. London, UK, 06-10 July 2020.
* [37] Y. Quan and C.-T. Li, “On addressing the impact of ISO speed upon PRNU and forgery detection,” IEEE Transactions on Information Forensics and Security, vol. 16, pp. 190–202, 2021.
* [38] M. Goljan, M. Chen, and J. Fridrich, “Identifying common source digital camera from image pairs,” in 2007 IEEE International Conference on Image Processing, vol. 6, pp. VI–125, IEEE, 2007.
* [39] I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988.
* [40] J. Fridrich, “Sensor defects in digital image forensic [sic],” in Digital Image Forensics: There is More to a Picture than Meets the Eye (H. T. Sencar and N. Memon, eds.), pp. 179–218, New York, NY: Springer Science+Business Media, 2013.
* [41] M. Chen, J. Fridrich, M. Goljan, and J. Lukáš, “Determining image origin and integrity using sensor noise,” IEEE Transactions on Information and Security, vol. 3, no. 1, pp. 74–90, 2008.
* [42] M. Goljan, M. Chen, P. Comeaña, and J. Fridrich, “MATLAB/Python camera fingerprint implementation.” http://dde.binghamton.edu/download/camera_fingerprint/, 2016. Last date modified: February 2016.
* [43] The MathWorks Inc., “MATLAB version: 9.13.0 (R2022b),” 2022. The MathWorks Inc., Natick, MA, United States.
* [44] H.-C. Lee, Introduction to color imaging science. Cambridge University Press, 2005.
* [45] Apple Inc., “The Swift Programming Language,” 2014. Apple Inc., Cupertino, CA, United States.
* [46] K. Arnold, J. Gosling, and D. Holmes, “The Java programming language,” 2000. Addison-Wesley Longman Publishing Co., Inc, Boston, MA, United States.
* [47] J. L. Fleiss, “Measuring nominal scale agreement among many raters,” Psychological Bulletin, vol. 76, no. 5, pp. 378–382, 1971.
* [48] M. Gamer, J. Lemon, I. Fellows, and P. Singh, “irr: Various Coefficients of Interrater Reliability and Agreement,” 2012. R package, version 0.84, https://CRAN.R-project.org/package=irr.
* [49] R Core Team, “R: A Language and Environment for Statistical Computing,” 2017. R Foundation for Statistical Computing, Vienna, Austria.
* [50] A. MacKenzie and W. E. Bruehs, “Photo response nonuniformity (PRNU) meets Daubert standards,” Journal of Forensic Identification, vol. 68, no. 4, pp. 467–471, 2018.
* [51] US District Court, “United States of America v. Nathan Railey, United States District Court for the Southern District of Alabama, Case # 1:10-cr-00266-CG-N,” 2011. PACER Document 198.pdf.
* [52] W. C. Thompson, “Shifting decision thresholds can undermine the probative value and legal utility of forensic pattern-matching evidence,” Proceedings of the National Academy of Sciences, vol. 120, no. 41, p. e2301844120, 2023.
| Abby Martin received her B.A. degree in Computer Science/Software
Engineering and Mathematics from Augustana University, Sioux Falls, SD, in
2017 and is pursuing her Ph.D. degree in Mathematics and Computer Science from
Iowa State University, Ames, IA under the supervision of Professors Jennifer
Newman and Jin Tian. From 2020 to 2024, she was a Research Assistant with the
Center for Statistics and Applications in Forensic Evidence. Her research
interests include camera source identification, steganalysis, and digital
image forensics.
---|---
| Roy Maxion (IEEE Fellow) is a Research Professor emeritus in computer
science and machine learning at Carnegie Mellon University, where he is also
the director of the Dependable Systems Laboratory. He has long been a
passionate proponent of rigorous/foundational scientific methodology. He is an
IEEE Fellow, and recently served as a member of the US National Academy of
Sciences committee on Future Research Goals and Directions for Foundational
Science in Cybersecurity. He won the 2019 IEEE/IFIP Test-of-Time award for his
work using typing rhythms for user authentication. He recently won the Taiwan
Tamkang Panda Trophy for his lecture on the sensitivity of machine learning
systems to small irregularities in data. He is one of the founding members of
the US Center for Statistics and Applications in Forensic Evidence. He is a
member of the editorial boards of IEEE Security and Privacy and the
International Journal of Machine Learning.
---|---
| Jennifer Newman Jennifer L. Newman (Senior Member, IEEE) is a Professor of
Mathematics at Iowa State University, Ames, Iowa and holds a Scott Hanna
Faculty Fellow in Mathematics. She received her B.A. in Physics from Mount
Holyoke College, and an M.S. and Ph.D. in Mathematics from the University of
Florida, Gainesville. She has served as an Associate Editor for the Journal of
Electronic Imaging and for the Journal of Mathematical Imaging and has served
on numerous program committees for many professional conferences. Her current
research is in statistical forensic imaging, including steganography,
steganalysis and camera identification. Other research interests include image
algebra; object detection, modeling and machine learning for images;
morphological and other neural networks; and statistical modeling of textures
for forward and inverse problems. She has over 80 refereed publications and
has been a Principal Investigator or Co-Principal Investigator on over $24
million in grants. She has mentored over 32 graduate students as their major
professor, and many undergraduate students as well.
---|---
|
11institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, France 22institutetext: V. N. Karazin Kharkiv
National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine 33institutetext:
European Southern Observatory (ESO), Alonso de Cordova 3107, 1900, Casilla
Vitacura, Santiago, Chile 44institutetext: Department of Earth, Atmospheric
and Planetary Sciences, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139,
USA 55institutetext: Astronomical Institute, Academy of Sciences of the Czech
Republic, CZ-25165 Ondřejov, Czech Republic 66institutetext: INAF -
Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monte Porzio
Catone, Italy 77institutetext: Department of Earth, Atmospheric, and Planetary
Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA 88institutetext: INAF - Osservatorio Astronomico di
Roma, Monte Porzio Catone (RM), Italy 99institutetext: INAF - Department of
Physics and Astronomy, University of Padova, Vicolo dell’Osservatorio, 3,
I-35122 Padova, Italy 1010institutetext: Agenzia Spaziale Italiana (ASI), Via
del Politecnico 00133 Roma, Italy
# Compositional properties of planet-crossing asteroids from astronomical
surveys
A. V. Sergeyev Corresponding author<EMAIL_ADDRESS>B. Carry
11 M. Marsset 3344 P. Pravec 55 D. Perna 66 F. E. DeMeo 77 4 4 V. Petropoulou
88 M. Lazzarin 99 F. La Forgia 99 I. Di Petro 1010 the NEOROCKS team
The NEOROCKS team: E. Dotto, M. Banaszkiewicz, S. Banchi, M.A. Barucci, F.
Bernardi, M. Birlan, A. Cellino, J. De Leon, M. Lazzarin, E. Mazzotta Epifani,
A. Mediavilla, J. Nomen Torres, E. Perozzi, C. Snodgrass, C. Teodorescu, S.
Anghel, A. Bertolucci, F. Calderini, F. Colas, A. Del Vigna, A. Dell’Oro, A.
Di Cecco, L. Dimare, P. Fatka, S. Fornasier, E. Frattin, P. Frosini, M.
Fulchignoni, R. Gabryszewski, M. Giardino, A. Giunta, T. Hromakina, J.
Huntingford, S. Ieva, J.P. Kotlarz, M. Popescu, J. Licandro, H. Medeiros, F.
Merlin, F. Pinna, G. Polenta, A. Rozek, P. Scheirich, A. Sonka, G.B.
Valsecchi, P. Wajer, A. Zinzi.
###### Abstract
Context. The study of planet-crossing asteroids is of both practical and
fundamental importance. As they are closer than asteroids in the Main Belt, we
have access to a smaller size range, and this population frequently impacts
planetary surfaces and can pose a threat to life.
Aims. We aim to characterize the compositions of a large corpus of planet-
crossing asteroids and to study how these compositions are related to orbital
and physical parameters.
Methods. We gathered publicly available visible colors of near-Earth objects
(NEOs) from the Sloan Digital Sky Survey (SDSS) and SkyMapper surveys. We also
computed SDSS-compatible colors from reflectance spectra of the Gaia mission
and a compilation of ground-based observations. We determined the taxonomy of
each NEO from its colors and studied the distribution of the taxonomic classes
and spectral slope against the orbital parameters and diameter.
Results. We provide updated photometry for 470 NEOs from the SDSS, and
taxonomic classification of 7,401 NEOs. We classify 42 NEOs that are mission-
accessible, including six of the seven flyby candidates of the ESA Hera
mission. We confirm the perihelion dependance of spectral slope among S-type
NEOs, likely related to a rejuvenation mechanism linked with thermal fatigue.
We also confirm the clustering of A-type NEOs around 1.5–2 AU, and predict the
taxonomic distribution of small asteroids in the NEO source regions in the
Main Belt.
###### Key Words.:
Minor planets, asteroids: NEOs – Techniques: photometric – Surveys
## 1 Introduction
Asteroids are the remnants of the building blocks that accreted to form the
terrestrial planets and the core of the giant planets in the early Solar
System $4.6\text{\,}\mathrm{G}\mathrm{y}$ ago. Asteroids are also the origin
of the meteorites that fell on the planets, including the Earth. These
meteorites represent the only possibility to study in detail the composition
of asteroids in the laboratory (e.g., 2008-ChEG-68-Consolmagno;
2015-Icarus-252-Cloutis), with the exception of the tiny samples of rock,
provided by return-sample missions: JAXA Hayabusa (2011-Science-333-Yurimoto)
and Hayabusa-2 (2022Sci...375.1011T), as well as the soon due NASA OSIRIS-REx
(2017SSRv..212..925L).
In contrast to targeted sample collection, we cannot choose the origin of
meteorites striking the Earth. Identifying their source regions is therefore
crucial to determining the physical conditions and abundances in elements that
reigned in the protoplanetary nebula around the young Sun
(2006-MESS2-McSween). From the analysis of a bolide trajectory, it is possible
to reconstruct a meteorite’s heliocentric orbit (2006-MPS-41-Gounelle),
although such determinations have been limited to only a few meteorites
(2018-Icarus-311-Granvik).
Figure 1: Schematic view of the extraction, convertion, and merging of NEOs
from SDSS, SMSS, Gaia, and Classy catalogs.
Among the different dynamical classes of asteroids, the near-Earth and Mars-
crosser asteroids (NEAs and MCs), whose orbits cross that of the telluric
planets, form a transient population. Their typical lifetime is of only a few
million years before they are ejected from the Solar System, fall into the
Sun, or impact a planet (1997Sci...277..197G). We refer here to near-Earth
objects (NEOs) in a liberal sense, encompassing both asteroid-like and comet-
like objects whose orbits cross that of a terrestrial planet (hence including
NEAs, MCs, and some Hungarias).
These populations are of both scientific and pragmatic interest. As they are
closer to the Earth than the asteroid belt, we have access to smaller objects
from ground-based telescopes. Their orbital proximity implies a much smaller
impulsion to reach them with a spacecraft and make them favorable targets for
space exploration (2012-DPS-Abell). On the other hand, these objects could
potentially pose a threat, and studying their properties is a key aspect in
planning risk mitigation (2015hchp.book..763D), of which the National
Aeronautics and Space Administration (NASA) Demonstration for Autonomous
Rendezvous Technology (DART) and European Space Agency (ESA) Hera missions are
lively demonstrators (2021PSJ.....2..173R; 2022PSJ.....3..160M).
We focus here on the compositional properties of a large corpus of NEOs as
part of the NEOROCKS project (2021plde.confE.221D), whose goal is the
characterization of the NEO population. The article is organized as follows:.
In
|
# The SDSS-Gaia View of the Color–Magnitude Relation for Blue Horizontal-
branch Stars
Fabrícia O. Barbosa Universidade de São Paulo, Instituto de Astronomia,
Geofísica e Ciências Atmosféricas, Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil Rafael M. Santucci Universidade Federal de
Goiás, Instituto de Estudos Socioambientais, Planetário, Goiânia, GO
74055-140, Brazil Universidade Federal de Goiás, Campus Samambaia, Instituto
de Física, Goiânia, GO 74001-970, Brazil Silvia Rossi Universidade de São
Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas,
Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil Guilherme Limberg Universidade de São Paulo,
Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Departamento de
Astronomia,
SP 05508-090, São Paulo, Brazil Angeles Pérez-Villegas Instituto de
Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 106, C.
P. 22800, Ensenada, B. C., Mexico Hélio D. Perottoni Universidade de São
Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas,
Departamento de Astronomia,
SP 05508-090, São Paulo, Brazil Institut de Ciències del Cosmos (ICCUB),
Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona,
Spain
###### Abstract
We present an updated sample of blue horizontal-branch (BHB) stars selected
from the photometric and spectroscopic data from Sloan Digital Sky Survey and
its associated project Sloan Extension for Galactic Understanding and
Exploration (SEGUE). With this data, we selected candidates for A-type stars
in the color-color space and then a mixture modeling technique was implemented
in order to distinguish between BHB and main-sequence/blue-straggler stars
based on their surface gravity values ($\log\rm{g}$) estimated by the SEGUE
Stellar Parameter Pipeline. Our robust approach allows us to attribute
individual probabilities of each star truly being in the BHB stage. Hence, our
method is advantageous in comparison to previous SEGUE BHB selections that
adopted simple $\log\rm{g}$ cuts. We also revisit the color–magnitude relation
for these stars and propose two calibrations, based on updated distances for
Galactic globular clusters, to estimate absolute magnitudes with $(g-r)_{0}$
and $(u-r)_{0}$ colors.
Galaxy: stellar halo – stars: horizontal branch – stars: distances
## 1 Introduction
The Gaia mission (Gaia Collaboration et al., 2016) has provided a better
understanding of the Galaxy, in particular regarding the field of Galactic
Archaeology (Helmi, 2020; Brown, 2021). The astrometric information provided
for an unprecedented number of objects has dramatically changed the way we
study the Galactic halo (e.g., Belokurov et al., 2018; Myeong et al., 2018;
Koppelman et al., 2018; Malhan et al., 2018).
Despite the huge amount of direct measurements supplied by Gaia, distances
inferred from brightness are still of great value. At magnitude $G<15$, the
early third data release (EDR3) presents parallax uncertainties of
${\sim}0.02$ mas (Gaia Collaboration et al., 2021), and they increase
significantly for fainter stars. To overcome this limitation, we can use
various well-known distance tracers such as RR Lyrae (Shapley, 1916), Cepheids
(Leavitt & Pickering, 1912), and blue horizontal-branch (BHB) stars (Cacciari,
1999).
BHBs are metal-poor ([Fe/H]111[A/B]
$=\log\rm(N_{A}/N_{B})_{\star}-\log(N_{A}/N_{B})_{\odot}$, where $\rm N_{A}$
and $\rm N_{B}$ are the number density of atoms of the elements A and B,
respectively. $\star$ refers to the considered star, and $\odot$ refers to the
Sun. $\lesssim-0.5$; Santucci et al. 2015b) A or B-type stars that burn helium
in their cores. These evolved stars present a high and nearly constant
luminosity, making them perfect for investigating the outer regions of the
halo and the assembly history of our Galaxy (Xue et al., 2011, 2008; Deason et
al., 2011, 2017; Belokurov et al., 2014; Santucci et al., 2015b). In recent
works, BHBs were used to study dynamical substructures and stellar streams
(Yuan et al., 2019, 2020, 2022; Peñarrubia & Petersen, 2021; Li et al., 2022;
Wu et al., 2022), the connection between the apocenter pile-up of orbits and
the so-called “break-radius” of the stellar halo density profile (Deason et
al., 2018), the anisotropy of the halo velocity distribution (Lancaster et
al., 2019), the age gradient of the halo out to $\sim$35 kpc (Whitten et al.,
2019), to estimate the total dynamical mass of the Milky Way (Deason et al.,
2021; Bird et al., 2022), and even to demonstrate the influence of the Large
Magellanic Cloud in our Galaxy’s halo (Erkal et al., 2021; Petersen &
Peñarrubia, 2021).
The well-defined structure of the horizontal branch in the color-magnitude
diagram (CMD), a roughly constant luminosity, permits the development of a
distance calibration for these BHBs. The first approximation developed was a
linear fit, using the ($B-V$) color and absolute magnitude in the $V$ band,
for stars in globular clusters (Hayes & Philip, 1979). Likewise, Preston et
al. (1991) defined a smoother relation, a fourth degree polynomial, for the
same color-magnitude space. Two decades later, a widely used calibration was
presented by Deason et al. (2011, hereafter D11) based on magnitudes in the
$ugriz$ system for the Sloan Digital Sky Survey (SDSS; York et al., 2000)
eighth data release (DR8; Aihara et al., 2011), which had its color range
extended by Belokurov & Koposov (2016) afterwards. In the meantime, Fermani &
Schönrich (2013) argued that it is extremely important to take into account
the effect of the metallicity on the absolute magnitude estimation, proposing
a new calibration based on a statistical method. However, Santucci et al.
(2015a) and Utkin & Dambis (2020) showed that the differences between
considering or not the metallicity in the relations are negligible, with D11’s
estimates being $2.5\%$ higher, within (1-$\sigma$) errors of both
calibrations.
D11’s relation still remains the most used calibration for BHB stars (Santucci
et al., 2015a, b; Thomas et al., 2018; Whitten et al., 2019; Donlon et al.,
2020; Martin et al., 2022) even though photometric data have been updated
several times since then. Moreover, we can now compare photometric distances
of BHB stars with purely geometric estimates from Gaia’s parallaxes (e.g.,
Bailer-Jones et al. 2021) as well as new measurements for Galactic globular
clusters (Vasiliev & Baumgardt, 2021). These facts bring to light the
relevance of reviewing the D11’s calibration with recent data.
This paper is organized as follows. In Section 2, we describe the photometric
selection and revise a previous method to identify BHB stars. Section 3
presents the selection of stars in globular clusters and the method used to
define the absolute magnitude calibration. Finally, in Section 4 we discuss
our results.
## 2 Data
### 2.1 A-type stars
The initial selection of A-type stars was made using the photometry from the
sixteenth data release (DR16) of SDSS (Ahumada et al., 2020). For the
selection of BHB stars, we were specially interested in the spectroscopic data
obtained by the Sloan Extension for Galactic Understanding and Exploration
(SEGUE; Yanny et al., 2009) processed by the SEGUE Stellar Parameter Pipeline
(SSPP; Lee et al., 2008a, b)222Last run on DR9 (Ahn et al., 2012; Rockosi et
al., 2022)..
We implemented color cuts applying the following criteria:
$-0.3<(g-r)_{0}<0.1$ and $0.8<(u-g)_{0}<1.4$, similar to those used in
previous works (Sirko et al. 2004, D11). All the magnitudes were corrected
using the extinction coefficients ($A_{g}$, $A_{r}$, $A_{u}$) provided by the
SDSS catalog itself and we removed stars with relative errors in the $g$-band
magnitude greater than 1%.
The photometric selection is able to exclude several undesired objects, such
as white dwarfs, quasars and cooler spectral types (Yanny et al., 2009;
Vickers et al., 2012), but the major source of contamination, blue straggler
stars (BSSs), remains. The distinction between evolved and main-sequence
stars/BSSs is commonly made by investigating spectral features, specially
Balmer lines, whose depths are affected by effective temperature ($T_{\rm
eff}$) and widths by surface gravity ($\log\rm{g}$). With the output of SSPP,
we can directly inspect these stellar atmospheric parameters. Therefore, we
cross-matched the filtered sample with the SSPP catalog using
$5^{\prime\prime}$ radius. In addition to color filters, we restricted our
sample to stars with moderate signal-to-noise ratio ($S/N>10$) and $7500\,{\rm
K}<T_{\rm eff}<10000\,{\rm K}$ (Deason et al., 2012; Santucci et al., 2015a),
where $T_{\rm eff}$ is the estimate adopted by the pipeline. Duplicated stars
with the smallest $S/N$ were removed, which resulted in 16463 stars. The
restrictions above remove poor-quality data and cooler stars that could remain
after the color cut, which assures that contamination from other non-BSS stars
is minimal.
### 2.2 BHB stars
One of the techniques used to disentangle BSSs and BHB stars is the $f_{m}$
versus $D_{0.2}$ method (Pier, 1983), where $f_{m}$ is the minimum flux
relative to the continuum level and $D_{0.2}$ is a measurement of the line
width of Balmer lines, so it provides indirect information regarding both
$T_{\rm eff}$ and $\log\rm{g}$. Later, a different approach was proposed by
Clewley et al. (2002) based on the parameters of the Sérsic profile (Sersic,
1968), which describes the shape of the lines.
BSSs present a stronger $\log\rm{g}$ than those located in the horizontal
branch. Santucci et al. (2015a) showed that these stellar types are clearly
distinguishable for magnitudes $g_{0}<18$ with SEGUE/SDSS DR8 data, being
possible to classify them by fitting a combination of two Gaussian functions
to their $\log\rm{g}$ distributions. This method was proved to be in good
concordance with spectral analysis, with more than 90% of agreement. When
replicating this procedure with current SEGUE data, we noticed a change in the
peaks of both groups and a greater overlap in the $\log\rm{g}$ distribution as
presented in the left column of Fig. 1. This is observed even for relatively
bright A-type stars ($g_{0}<18$, top panel), which makes it more difficult to
separate these objects with that simple approach (dashed lines indicate the
Gaussian fits from Santucci et al. 2015a). The differences are probably due to
changes between the releases of SDSS on the $\log\rm{g}$ estimates considered
to obtain the final adopted parameter333See
https://www.sdss.org/dr16/spectro/sspp_changes/ for detailed information..
Figure 1: Histograms of $\log\rm{g}$. Left column: log g adopted by the
pipeline for stars with $g_{0}<18$ (top) and $g_{0}>18$ (bottom). Dashed lines
are the Gaussian distributions defined by Santucci et al. (2015a). Right
column: log g estimates provided by SSPP spectroscopically determined (top)
and from ANNRR method (bottom) for all stars.
### 2.3 Classification
Given the two-Gaussian-like morphology of the $\log\rm{g}$ distributions
observed for our sample of A-type stars (Fig. 1), we used a Gaussian Mixture
Model (GMM) unsupervised approach in order to distinguish BHBs and BSSs. For
this task, we utilize the scikit-learn (Pedregosa et al., 2011)
GaussianMixture444https://scikit-
learn.org/stable/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture.
package. In this GMM implementation, the expectation-maximization algorithm
(Dempster et al., 1977) is employed in the search for the best-fit model.
The GMM technique fits the data as a finite combination of $K$ Gaussian
distributions. As made previously by Santucci et al. (2015a), $K$ was defined
based on visual inspection of $\log\rm{g}$ estimates presented in Fig. 1 and
the assumption that the contamination is predominantly of main-sequence
stars/BSSs. Therefore, $K=2$ is an adequate value for the sample. Moreover,
GMM can be readily applied to data of arbitrary dimensionality. Therefore, we
take advantage of such flexibility and explore a suitable combination of
$\log\rm{g}$ estimates provided by SSPP (we refer the reader to Lee et al.
2008a for details about different approaches to determine $\log\rm{g}$ from
SEGUE spectra). We noticed that the distributions of both $\log\rm{g}_{\rm
ANNRR}$ and $\log\rm{g}_{\rm SPEC}$ exhibit clearly two peaks, as expected for
the BHBs/BSSs dichotomy, while it is not possible to observe this feature in
others. These two distributions are shown in the right column of Fig. 1.
Hence, we proceeded with the GMM separation within the two-dimensional space
defined by these $\log\rm{g}$ estimates. The final $\log\rm{g}$ adopted by the
pipeline was not considered an extra dimension as it consists of a weighted
mean of the valid estimates.
In order to guarantee the robustness of our method against uncertainties
reported by the SSPP, we constructed a set of $10^{4}$ realizations of each
star’s $\log\rm{g}$ estimates in a Monte Carlo framework. Then, we performed
the GMM classification for all iterations. Finally, the fraction of instances
that a star is attributed to a certain class (either BHB or BSS) is taken as
its membership probability for that given group. For this procedure, stars
without valid estimates of both $\log\rm{g}_{\rm ANNRR}$ and $\log\rm{g}_{\rm
SPEC}$ are removed. With this strategy, we achieved a sample of 5699/4590
stars classified as BHBs above 50%/99% probability555The full sample is
available at https://github.com/guilhermelimberg/bhb_dist..
The final classification obtained is shown in Fig. 2. The difference in the
uncertainties of the estimates greatly influences the classification, as the
$\log\rm{g}_{\rm ANNRR}$ presents more precise values ($\sim 0.06$) than
$\log\rm{g}_{\rm SPEC}$ ($\sim 0.21$). We cross-matched our sample with the
one from Santucci et al. (2015a) to evaluate the fraction of BSS
contamination. 10% of our BHB set was classified previously as BSSs, and,
among those with a probability greater than 99% of being BHB following the
method implemented here, 2% is possibly incorrectly assigned.
Figure 2: Distribution of classified stars with $g_{0}<18$ (left) and
$g_{0}>18$ (right) in the surface gravity space. Median errors are indicated
in the bottom right corner. Histograms show the distribution of log g from the
respective axis for stars classified as BHB and BSS. Colors indicate the
probability of being a BHB star.
## 3 Absolute magnitude calibration
### 3.1 Globular clusters stars
The procedure to construct an absolute magnitude relation follows previous
works (see Section 1), starting with the selection of BHB stars in globular
clusters. We used the photometric catalog from An et al. (2008), which
provides magnitudes for crowded fields observed by SDSS. The clusters
presenting a well defined horizontal branch were selected and their magnitudes
were corrected using the standard extinction ($E(B-V)$) from Schlegel et al.
(1998) along with the relative extinctions from Wang & Chen (2019) for $g-,u-$
and $r-$band. Vasiliev & Baumgardt (2021) attributed a membership probability
for stars in globular clusters based on proper motions and parallaxes from
Gaia EDR3. We selected stars from several globular clusters that were more
likely than 0.99 to belong to those clusters and we obtained their absolute
magnitude in the SDSS $g$-band ($M_{g}$) with the estimated distance for each
cluster given by these authors. The list of clusters, their heliocentric
distances and distance moduli are presented in Table 1.
Table 1: Heliocentric distances provided by Vasiliev & Baumgardt (2021) for each globular cluster, uncertainties, and their respective distance moduli. Cluster | D (kpc) | $\sigma_{D}$ (kpc) | $(m-M)_{0}$ (mag)
---|---|---|---
NGC2419 | 83.0 | 1.5 | 19.59
NGC4147 | 18.65 | 0.16 | 16.35
NGC5024, M53 | 18.59 | 0.15 | 16.35
NGC5053 | 17.30 | 0.14 | 16.19
NGC5272, M3 | 10.20 | 0.06 | 15.04
NGC5466 | 16.32 | 0.13 | 16.06
NGC5904, M5 | 7.49 | 0.05 | 14.37
NGC6205, M13 | 7.53 | 0.06 | 14.38
NGC6341, M92 | 8.60 | 0.05 | 14.67
NGC7078, M15 | 10.73 | 0.14 | 15.15
NGC7089, M2 | 11.62 | 0.13 | 15.33
To create the sample used to implement the calibration, we applied the limits
for colors as defined for the initial selection (see Section 2.1). Then, the
stars were selected in a single combined CMD, limiting the $M_{g}$ between
$-0.15$ and $1.15$. After this exercise, the remaining globular cluster
members were checked individually at the SIMBAD database (Wenger et al.,
2000), and those classified as variables, blue stragglers and other
undesirable types were removed. We also excluded stars with flags in the
magnitudes used, leaving us with 744 stars to derive the calibrations from.
### 3.2 Fitting the horizontal branch
Finding the best mathematical relationships to fit observable data is not an
easy task. In previous works, the absolute magnitudes for BHB stars have been
described as a high-degree polynomial (Preston et al., 1991; Deason et al.,
2011; Belokurov & Koposov, 2016). Instead of arbitrarily assuming that this
function is the best representation of the data, we explore the possible
combinations between colors and absolute magnitudes. For this task, we
employed the TuringBot software (Ashok et al., 2020), a code that performs
symbolic regression using a simulated annealing algorithm (Delahaye et al.,
2019; Chira & Plionis, 2019) in order to search for the best set of parameters
and mathematical operations to describe the data.
TuringBot is particularly interesting in this case, because it allows the
visualization of the estimated mathematical laws, allowing the user to choose
the most appropriate equations for their needs. Furthermore, the user is free
to choose the mathematical operations involved in the fitted functions, the
error metric for convergence, as well as the input variables. The best fits
are presented in a summarized box, combining the error and the complexity of
the equations. The complexity is defined by the sum of the “size” of the
mathematical operations, constants, and variables present in the solutions.
The program assumes that an input variable, constant, sum, subtraction, and
multiplication have size 1 each, division has size 2, and more complex
operations have higher sizes666More TuringBot details can be found in the
program documentation, available at:
https://turingbotsoftware.com/documentation.html.. We verified that the use of
very complex mathematical operations is unnecessary and does not improve the
average error of the equations presented by the software. Hence, we adopted
only the basic mathematical operations (sum, subtraction, multiplication, and
division) as input for the search for absolute magnitude calibrations. The
mean absolute error was used as a criterion for convergence and we tested the
dependence of all the most common available observable variables found in the
literature for estimates of this type, such as magnitudes, color indices, and
metallicity.
After evaluating all the combinations of input variables ($u_{0}$, $g_{0}$,
$r_{0}$, $(u-g)_{0}$, $(u-r)_{0}$, $(g-r)_{0}$, and [Fe/H]) presented in
Appendix A, we found that there is no significant dependence on metallicity in
the calibrations provided by the code, regardless of the mathematical
operations adopted and the algorithm convergence time. The colors $(u-g)_{0}$
and $(u-r)_{0}$ provided calibrations with smaller errors than the color
$(g-r)_{0}$, traditionally used in the absolute magnitude calibration of BHB
stars (D11; Belokurov & Koposov 2016), and also smaller than $(g-r)_{0}$ with
[Fe/H], which means we can achieve more accurate results that do not require
metallicity information. The observed improvement with $(u-g)_{0}$ and
$(u-r)_{0}$ color might be associated with the $u$ filter, whose transmission
curve is mostly between 3000Å and 4000Å, i.e., it is positioned in a region of
the spectrum where the Balmer discontinuity ($\sim$3645Å) is located, as well
as several Hydrogen lines from the Balmer series, which makes it a useful
indirect indicator of $T_{\rm eff}$ and $\log\rm{g}$ of the BHBs, atmospheric
parameters that are directly linked to the mass of the stars in the horizontal
branch (Valcarce & Catelan, 2008).
Fig. 3 shows the associated errors for each fit in the final BHB sample.
Clearly, $(u-r)_{0}$ presents a better performance than $(g-r)_{0}$ and
$(u-g)_{0}$. Using the same tool, we find that the relation proposed in D11 is
a function of complexity 33 and ${\rm error}=0.12$, whilst equations of lower
order present a much lower complexity with errors of ${\sim}0.10$.
We chose the first functions from which there is no significant decrease in
error, i.e., functions of complexity 6 in Fig. 3, as those that best describes
the data. Exists a singularity in the calibrations, however it is outside of
our color range. Hence, it does not imply an obstacle to their usage in the
context of this work.
Figure 3: Error comparison between fits for colors $(g-r)_{0}$, $(u-g)_{0}$
and $(u-r)_{0}$ from TuringBot.
$M_{g}=\frac{0.178}{0.537+(g-r)_{0}}$ (1)
$M_{g}=\frac{0.721}{(u-r)_{0}}-0.212$ (2)
### 3.3 Distances analysis
Left panels in Fig. 4 show the distribution of BHB stars in the CMD with color
$(g-r)_{0}$ (top) and $(u-r)_{0}$ (bottom). In the top left panel, we can
observe how the calibration proposed here (Eq. 1) provides magnitudes lower
than D11’s, which results in larger distances. The difference is minimal at
$(g-r)_{0}\sim-0.20$, where both equation come closer, and the smaller values
are a consequence of the inclusion of the cluster NGC7078, whose stars are
brighter and were not included in D11. On the other hand, the distribution
using $(u-r)_{0}$ has a lower dispersion (bottom left panel).
Figure 4: Left panels: color–magnitude diagrams showing BHBs used to define
the calibrations. Dash-dotted line represents the polynomial fit defined in
D11. Solid lines represent the calibration for $(g-r)_{0}$ and $(u-r)_{0}$
color presented in this work (Eq. 1 and 2, respectively, from top to bottom).
Median errors of the data are indicated in the bottom right corner of each
panel. Right panels: difference between distances calculated with D11’s
calibration and those presented in this work, respectively $(g-r)_{0}$ (top)
and $(u-r)_{0}$ (bottom).
In the right panels (Fig. 4), we also show the comparison between distances
estimated with the relation from D11 and each calibration defined in the
present work. For consistence with D11’s relation, only stars bluer than
$(g-r)_{0}=0$ were considered, and we rejected stars with BHB probabilities of
less than 0.99 to reduce the number of misclassified stars. The new
calibration using color $(g-r)_{0}$ provides distances about 5% larger than
D11’s for the reddest stars, while the other end of the color window attains a
relative difference of up to 9% (top right panel). For the color $(u-r)_{0}$,
the scatter is more uniform and much larger for the bluest stars (bottom right
panel).
When comparing with purely astrometric heliocentric distances, there is a
considerable scatter, even for stars closer than 5 kpc. For this comparison,
we selected stars with relative parallax uncertainty from Gaia EDR3777Gaia
Collaboration (2020). in the interval $0<\sigma_{\varpi}/\varpi<0.2$, re-
normalized unit weight errors within the recommended range
($\texttt{RUWE}<1.4$; Lindegren 2018), and also a BHB probability greater than
0.99 (${>}\ 300$ stars). In Fig. 5, we show the comparison between geometric
(left) and photogeometric (right) distances provided by Bailer-Jones et al.
(2021) and our calibration using $(u-r)_{0}$ color. For fainter stars, both
Bailer-Jones et al.’s (2021) distances are frequently underestimated. The gray
region indicates the interval within 20% of distance in the respective
horizontal axes, and we find 65% of the stars when using photogeometric
estimates and 54% with the purely geometric outside it. Gaia’s parallax
measurements potentially are not accurate enough for these BHBs and so the
final results are not representative of the sample (since the distances
inferred from Bayesian methods are strongly dependent on the measured
parallax). We also point out that this effect is unlikely the consequence of
an inappropriate classification since 94% of the stars possess
$\log\rm{g}_{\rm ADOP}<3.6$ and would receive the same label by Santucci et
al.’s (2015a) method. Finally, we cannot endorse the compatibility between
D11’s distances for BHB stars found in the Pristine survey (Starkenburg et
al., 2019) and Gaia DR2’s parallaxes. As the Pristine data are not publicly
available, it is not possible to evaluate whether the difference is due to the
BHB sample used.
Figure 5: Comparison between the distances estimated by Eq. 2
($D_{(u-r)_{0}}$), geometric ($D_{geo}$, left) and photogeometric ($D_{phot}$,
right) distances from Bailer-Jones et al. (2021). Colors indicate Gaia G
magnitude and gray area indicates the region of $1-\sigma$. Median errors are
indicated in the bottom right corner of the top panels.
Similar inconsistencies between photometric and astrometic-inferred distances
were also observed by previous works. Using OB stars, Shull & Danforth (2019)
noted an increase in the discrepancies at ${d>1.5}$ kpc, with B-type stars
showing smaller values of distances when considering parallaxes alone. Our
A-type stars sample seems to follow this same trend.
## 4 Discussion & summary
Since the last absolute magnitude calibration published for BHB stars we had
the advent of Gaia data, which allowed us to review the previous relationship
thanks to the better characterization of globular clusters (Vasiliev &
Baumgardt, 2021).
Using data from the SSPP catalog, we obtained a sample of ${\sim}5700$ BHB
stars implementing the GMM algorithm. This new approach is an alternative to
the previous individual Gaussian fits, as it is clear that they do not
distinguish the latest SSPP $\log{\rm g}$ distribution properly (Fig. 1).
To find which kind of function better describes the distribution of BHBs in
the CMD, we used a software for implementing symbolic regression. We suggest
two new color–magnitude calibrations based on photometry, including a relation
with $(u-r)_{0}$ color that has not been used before. This calibration
provides more accurate estimates than $(g-r)_{0}$ color in most cases. For the
bluest stars, the differences can exceed 10% of nominal values (Eq. 2).
However, this difference decreases for redder and more distant stars. Here, we
show that the calibrations can be simpler and achieve an acceptable result,
very similar to those from by D11’s relation.
We noted substantial differences between photometric and
geometric/photogeometric distances. A possibility would be inaccurate
estimates of $\log{\rm g}$ provided by SEGUE, as it was mainly designed for
cool stars ($T_{\rm eff}<7500\,{\rm K}$). However, Santucci et al. (2015a)
showed that this is unlikely to be the case. It could also be due to an
incorrect value of extinction for the SDSS photometry, yet it also does not
seem to explain the disparity. Most stars were observed in regions of low
extinction and we could not find any relation between the extinction values
and the inconsistency observed. The observed differences also lead us to
believe that measured parallaxes for these stars could be unreliable, as there
is no agreement between the distance estimates even for the closest stars. It
would be interesting to investigate whether the same discrepancy can be
observed with other halo tracers.
The new sample made available here can help to improve results already known
about the structures of the Milky Way stellar halo. For example, these BHBs
can be used to revisit the duality of the stellar halo or re-evaluate the
Galaxy’s mass estimate.
We thank the anonymous referee for the careful review and all the suggestions,
which greatly improved our work. This research was financed with public funds,
without which it would not have been possible. F.O.B. acknowledges CAPES
(PROEX; Proc. 88887.604787/2021-00). R.M.S. acknowledges CNPq (Proc.
306667/2020-7). S.R. acknowledges partial financial support from FAPESP (Proc.
2015/50374-0 and 2014/18100-4), CAPES, and CNPq. G.L. acknowledges FAPESP
(Proc. 2021/10429-0). A.P.-V. acknowledges the DGAPA-PAPIIT grant IA103122.
H.D.P. thanks FAPESP (Procs. 2018/21250-9 and 2022/04079-0). This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Funding for the Sloan
Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the
U.S. Department of Energy Office of Science, and the Participating
Institutions. SDSS acknowledges support and resources from the Center for
High-Performance Computing at the University of Utah. The SDSS web site is
www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, Center for Astrophysics — Harvard & Smithsonian (CfA), the Chilean
Participation Group, the French Participation Group, Instituto de Astrofísica
de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatório Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University. This research has also made use of RStudio (RStudio Team,
2022) and TOPCAT (http://www.starlink.ac.uk/topcat/, Taylor, 2005).
## References
* Ahn et al. (2012) Ahn, C. P., Alexandroff, R., Allende Prieto, C., et al. 2012, ApJS, 203, 21, doi: 10.1088/0067-0049/203/2/21
* Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3, doi: 10.3847/1538-4365/ab929e
* Aihara et al. (2011) Aihara, H., Allende Prieto, C., An, D., et al. 2011, ApJS, 193, 29, doi: 10.1088/0067-0049/193/2/29
* An et al. (2008) An, D., Johnson, J. A., Clem, J. L., et al. 2008, ApJS, 179, 326, doi: 10.1086/592090
* Ashok et al. (2020) Ashok, D., Scott, J., Wetzel, S., Panju, M., & Ganesh, V. 2020, arXiv e-prints, arXiv:2010.11328. https://arxiv.org/abs/2010.11328
* Bailer-Jones et al. (2021) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147, doi: 10.3847/1538-3881/abd806
* Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, MNRAS, 478, 611, doi: 10.1093/mnras/sty982
* Belokurov & Koposov (2016) Belokurov, V., & Koposov, S. E. 2016, MNRAS, 456, 602, doi: 10.1093/mnras/stv2688
* Belokurov et al. (2014) Belokurov, V., Koposov, S. E., Evans, N. W., et al. 2014, MNRAS, 437, 116, doi: 10.1093/mnras/stt1862
* Bird et al. (2022) Bird, S. A., Xue, X.-X., Liu, C., et al. 2022, arXiv e-prints, arXiv:2207.08839. https://arxiv.org/abs/2207.08839
* Brown (2021) Brown, A. G. A. 2021, ARA&A, 59, doi: 10.1146/annurev-astro-112320-035628
* Cacciari (1999) Cacciari, C. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 167, Harmonizing Cosmic Distance Scales in a Post-HIPPARCOS Era, ed. D. Egret & A. Heck, 140–160
* Chira & Plionis (2019) Chira, M., & Plionis, M. 2019, MNRAS, 490, 5904, doi: 10.1093/mnras/stz2885
* Clewley et al. (2002) Clewley, L., Warren, S. J., Hewett, P. C., et al. 2002, MNRAS, 337, 87, doi: 10.1046/j.1365-8711.2002.05864.x
* Deason et al. (2011) Deason, A. J., Belokurov, V., & Evans, N. W. 2011, MNRAS, 416, 2903, doi: 10.1111/j.1365-2966.2011.19237.x
* Deason et al. (2017) Deason, A. J., Belokurov, V., Koposov, S. E., et al. 2017, MNRAS, 470, 1259, doi: 10.1093/mnras/stx1301
* Deason et al. (2018) Deason, A. J., Belokurov, V., Koposov, S. E., & Lancaster, L. 2018, ApJ, 862, L1, doi: 10.3847/2041-8213/aad0ee
* Deason et al. (2012) Deason, A. J., Belokurov, V., Evans, N. W., et al. 2012, MNRAS, 425, 2840, doi: 10.1111/j.1365-2966.2012.21639.x
* Deason et al. (2021) Deason, A. J., Erkal, D., Belokurov, V., et al. 2021, MNRAS, 501, 5964, doi: 10.1093/mnras/staa3984
* Delahaye et al. (2019) Delahaye, D., Chaimatanan, S., & Mongeau, M. 2019, in International Series in Operations Research & Management Science (ISOR), Vol. 272, Handbook of Metaheuristics, ed. M. Gendreau & J.-Y. Potvin (Springer), 1–35.ISBN 978–3–319–91085–7, doi: 10.1007/978-3-319-91086-4_1
* Dempster et al. (1977) Dempster, A. P., Laird, N. M., & Rubin, D. B. 1977, J. R. Stat. Soc. B, 39, 1, doi: https://doi.org/10.1111/j.2517-6161.1977.tb01600.x
* Donlon et al. (2020) Donlon, Thomas, I., Newberg, H. J., Sanderson, R., & Widrow, L. M. 2020, ApJ, 902, 119, doi: 10.3847/1538-4357/abb5f6
* Erkal et al. (2021) Erkal, D., Deason, A. J., Belokurov, V., et al. 2021, MNRAS, 506, 2677, doi: 10.1093/mnras/stab1828
* Fermani & Schönrich (2013) Fermani, F., & Schönrich, R. 2013, MNRAS, 430, 1294, doi: 10.1093/mnras/sts703
* Gaia Collaboration (2020) Gaia Collaboration. 2020, Gaia Source Catalogue EDR3, IPAC, doi: 10.26131/IRSA541
* Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
* Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657
* Hayes & Philip (1979) Hayes, D. S., & Philip, A. G. D. 1979, PASP, 91, 71, doi: 10.1086/130444
* Helmi (2020) Helmi, A. 2020, ARA&A, 58, 205, doi: 10.1146/annurev-astro-032620-021917
* Koppelman et al. (2018) Koppelman, H., Helmi, A., & Veljanoski, J. 2018, ApJ, 860, L11, doi: 10.3847/2041-8213/aac882
* Lancaster et al. (2019) Lancaster, L., Koposov, S. E., Belokurov, V., Evans, N. W., & Deason, A. J. 2019, MNRAS, 486, 378, doi: 10.1093/mnras/stz853
* Leavitt & Pickering (1912) Leavitt, H. S., & Pickering, E. C. 1912, Harvard College Observatory Circular, 173, 1
* Lee et al. (2008a) Lee, Y. S., Beers, T. C., Sivarani, T., et al. 2008a, AJ, 136, 2022, doi: 10.1088/0004-6256/136/5/2022
* Lee et al. (2008b) —. 2008b, AJ, 136, 2050, doi: 10.1088/0004-6256/136/5/2050
* Li et al. (2022) Li, T. S., Ji, A. P., Pace, A. B., et al. 2022, ApJ, 928, 30, doi: 10.3847/1538-4357/ac46d3
* Lindegren (2018) Lindegren, L. 2018. http://www.rssd.esa.int/doc_fetch.php?id=3757412
* Malhan et al. (2018) Malhan, K., Ibata, R. A., & Martin, N. F. 2018, MNRAS, 481, 3442, doi: 10.1093/mnras/sty2474
* Martin et al. (2022) Martin, N. F., Venn, K. A., Aguado, D. S., et al. 2022, Nature, 601, 45, doi: 10.1038/s41586-021-04162-2
* Myeong et al. (2018) Myeong, G. C., Evans, N. W., Belokurov, V., Sanders, J. L., & Koposov, S. E. 2018, ApJ, 856, L26, doi: 10.3847/2041-8213/aab613
* Peñarrubia & Petersen (2021) Peñarrubia, J., & Petersen, M. S. 2021, MNRAS, 508, L26, doi: 10.1093/mnrasl/slab090
* Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
* Petersen & Peñarrubia (2021) Petersen, M. S., & Peñarrubia, J. 2021, Nature Astronomy, 5, 251, doi: 10.1038/s41550-020-01254-3
* Pier (1983) Pier, J. R. 1983, ApJS, 53, 791, doi: 10.1086/190910
* Preston et al. (1991) Preston, G. W., Shectman, S. A., & Beers, T. C. 1991, ApJ, 375, 121, doi: 10.1086/170175
* Rockosi et al. (2022) Rockosi, C. M., Lee, Y. S., Morrison, H. L., et al. 2022, The Astrophysical Journal Supplement Series, 259, 60, doi: 10.3847/1538-4365/ac5323
* RStudio Team (2022) RStudio Team. 2022, RStudio: Integrated Development Environment for R, RStudio, PBC, Boston, MA. http://www.rstudio.com/
* Santucci et al. (2015a) Santucci, R. M., Placco, V. M., Rossi, S., et al. 2015a, ApJ, 801, 116, doi: 10.1088/0004-637X/801/2/116
* Santucci et al. (2015b) Santucci, R. M., Beers, T. C., Placco, V. M., et al. 2015b, ApJ, 813, L16, doi: 10.1088/2041-8205/813/1/L16
* Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525, doi: 10.1086/305772
* Sersic (1968) Sersic, J. L. 1968, Atlas de Galaxias Australes (Universidad Nacional de Cordoba: Observatorio Astronomico)
* Shapley (1916) Shapley, H. 1916, ApJ, 43, 217, doi: 10.1086/142246
* Shull & Danforth (2019) Shull, J. M., & Danforth, C. W. 2019, ApJ, 882, 180, doi: 10.3847/1538-4357/ab357d
* Sirko et al. (2004) Sirko, E., Goodman, J., Knapp, G. R., et al. 2004, AJ, 127, 899, doi: 10.1086/381483
* Starkenburg et al. (2019) Starkenburg, E., Youakim, K., Martin, N., et al. 2019, MNRAS, 490, 5757, doi: 10.1093/mnras/stz2935
* Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
* Thomas et al. (2018) Thomas, G. F., McConnachie, A. W., Ibata, R. A., et al. 2018, MNRAS, 481, 5223, doi: 10.1093/mnras/sty2604
* Utkin & Dambis (2020) Utkin, N. D., & Dambis, A. K. 2020, MNRAS, 499, 1058, doi: 10.1093/mnras/staa2819
* Valcarce & Catelan (2008) Valcarce, A. A. R., & Catelan, M. 2008, A&A, 487, 185, doi: 10.1051/0004-6361:20078231
* Vasiliev & Baumgardt (2021) Vasiliev, E., & Baumgardt, H. 2021, MNRAS, 505, 5978, doi: 10.1093/mnras/stab1475
* Vickers et al. (2012) Vickers, J. J., Grebel, E. K., & Huxor, A. P. 2012, AJ, 143, 86, doi: 10.1088/0004-6256/143/4/86
* Wang & Chen (2019) Wang, S., & Chen, X. 2019, ApJ, 877, 116, doi: 10.3847/1538-4357/ab1c61
* Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9, doi: 10.1051/aas:2000332
* Whitten et al. (2019) Whitten, D. D., Beers, T. C., Placco, V. M., et al. 2019, ApJ, 884, 67, doi: 10.3847/1538-4357/ab4269
* Wu et al. (2022) Wu, W., Zhao, G., Xue, X.-X., Bird, S. A., & Yang, C. 2022, ApJ, 924, 23, doi: 10.3847/1538-4357/ac31ac
* Xue et al. (2008) Xue, X. X., Rix, H. W., Zhao, G., et al. 2008, ApJ, 684, 1143, doi: 10.1086/589500
* Xue et al. (2011) Xue, X.-X., Rix, H.-W., Yanny, B., et al. 2011, ApJ, 738, 79, doi: 10.1088/0004-637X/738/1/79
* Yanny et al. (2009) Yanny, B., Rockosi, C., Newberg, H. J., et al. 2009, AJ, 137, 4377, doi: 10.1088/0004-6256/137/5/4377
* York et al. (2000) York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513
* Yuan et al. (2020) Yuan, Z., Chang, J., Beers, T. C., & Huang, Y. 2020, ApJ, 898, L37, doi: 10.3847/2041-8213/aba49f
* Yuan et al. (2019) Yuan, Z., Smith, M. C., Xue, X.-X., et al. 2019, ApJ, 881, 164, doi: 10.3847/1538-4357/ab2e09
* Yuan et al. (2022) Yuan, Z., Malhan, K., Sestito, F., et al. 2022, ApJ, 930, 103, doi: 10.3847/1538-4357/ac616f
## Appendix A Fitting the horizontal branch with [Fe/H]
For the analysis of dependence of metallicity, we have a smaller sample of
stars than the one used for the calibrations, about 40 stars from the original
sample were available in the SSPP data. The same procedure was done with both
the pure photometric sample and this one cross-matched with SSPP.
Each solution evaluation (considering different input variables) was taken in
a period of approximately 10 min, which is enough for the convergence of
several solutions, as TuringBot needs less time to converge than other similar
softwares (Ashok et al., 2020). The hardware involved in this process is
highly important for the convergence time. In our case, the program was
executed in a computer with an AMD Ryzen 7 2700 processor, with 16 threads,
all used at once. Tests were also made by running the program longer and no
significant improvement was observed.
Figure 6: Error comparison between fits for some combinations of magnitudes,
colors, and [Fe/H] from TuringBot.
Figure 6 shows how the error decreases with the increase of the complexity of
the functions (see Section 3.2 for the definition of “complexity”). Some fits
coincide since the program can create colors from magnitudes, if it is better
than the magnitudes alone, and not use all the variables provided. This is the
reason why the line for all the parameters is not visible. We can see that,
for equations up to complexity 25, there is no advantage of adding the
metallicity information as we can achieve similar errors using only
magnitudes.
## Appendix B Globular cluster distances
Figure 7 presents a boxplot for the distances obtained with both calibrations
and the one from D11 compared to those provided by Vasiliev & Baumgardt
(2021). The size of each box represent the spread (25th and 75th quartiles)
and the center line indicates the median value. The number of stars in each
cluster is displayed above their NGC identifier.
The overall results indicate that Eq. 2 is in general more accurate than D11’s
relation. This conclusion is supported by better distance predictions where 10
of the 11 globular clusters were better constrained with Eq. 2. The
calibration with color $(u-r)_{0}$ revealed to be more accurate than using the
relation provided in this work for color $(g-r)_{0}$, as nine clusters present
lower dispersion and eight clusters show medians close to zero when using the
former color. NGC2416 was the only cluster where the color $(g-r)_{0}$
presented a better performance, however, this cluster was represented by only
9 members and therefore this result may be due to sub-sampling. The sample
considered has a bias to closer cluster, which is apparent by huge distance
gap between NGC2416, the farthest cluster, and NGC5466, the penultimate.
Therefore, we cannot ascertain whether or not the accuracy of distances
estimated using color $(u-r)_{0}$ varies with the distance.
Figure 7: Boxplot for relative distance difference using each calibration
presented in this work (Eq. 1 and Eq. 2) and that from D11. $D_{V21}$ is the
distance provided by Vasiliev & Baumgardt (2021), inferred from Gaia EDR3
data. The number of data points in each cluster is displayed at the bottom of
the panel.
|
# Hierarchical hub-filament structures and gas inflows on galaxy-cloud scales
J. W. Zhou Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121
Bonn, Germany<EMAIL_ADDRESS>Timothy A. Davis Cardiff Hub for
Astrophysics Research and Technology, School of Physics and Astronomy, Cardiff
University, Queens Buildings, Cardiff CF24 3AA, UK
###### Abstract
We investigated the kinematics and dynamics of gas structures on galaxy-cloud
scales in two spiral galaxies NGC5236 (M83) and NGC4321 (M100) using CO
(2$-$1) line. We utilized the FILFINDER algorithm on integrated intensity maps
for the identification of filaments in two galaxies. Clear fluctuations in
velocity and density were observed along these filaments, enabling the fitting
of velocity gradients around intensity peaks. The variations in velocity
gradient across different scales suggest a gradual and consistent increase in
velocity gradient from large to small scales, indicative of gravitational
collapse, something also revealed by the correlation between velocity
dispersion and column density of gas structures. Gas structures at different
scales in the galaxy may be organized into hierarchical systems through
gravitational coupling. All the features of gas kinematics on galaxy-cloud
scale are very similar to that on cloud-clump and clump-core scales studied in
previous works. Thus, the interstellar medium from galaxy to dense core scales
presents multi-scale/hierarchical hub-filament structures. Like dense core as
the hub in clump, clump as the hub in molecular cloud, now we verify that
cloud or cloud complex can be the hub in spiral galaxies. Although the scaling
relations and the measured velocity gradients support the gravitational
collapse of gas structures on galaxy-cloud scales, the collapse is much slower
than a pure free-fall gravitational collapse.
###### keywords:
Submillimeter: ISM; ISM: clouds; ISM: kinematics and dynamics; galaxies: ISM;
galaxies: structure; galaxies: star formation; techniques: image processing
## 1 Introduction
Measuring the dynamical coupling between density enhancements in giant
molecular clouds and gas motion of their local environment gives us a
perspective to understand the formation of hierarchical structures in high-
mass star formation regions McKee2007-45; Motte2018-56; Henshaw2020-4. High-
resolution observations of high-mass star-forming regions reveal the
structured arrangement of density enhancements within filamentary gas
networks, notably in hub-filament systems. Within these systems, converging
flows channel material into the central hub through the interconnected
filaments Peretto2013; Henshaw2014; Zhang2015; Liu2016; Yuan2018; Lu2018;
Issac2019; Dewangan2020; Liu2022-511; Zhou2022-514; Zhou2023-676. In
observations, gas inflow will naturally produce velocity gradients along
filaments Kirk2013; Liu2016; Yuan2018; Williams2018-613; Chen2019-875;
Chen2020-891; Pillai2020-4; Zhou2022-514; Zhou2023-676.
Henshaw2020-4 identified widespread velocity fluctuations spanning various
spatial scales and physical contexts within galaxies. They observed
oscillatory gas movements with wavelengths ranging from 0.3 to 400 parsecs,
intricately linked to regularly spaced density enhancements. These
enhancements are likely a result of gravitational instabilities
Henshaw2016-463; Elmegreen2018-863. Furthermore, the spatial correlation
between density enhancements and velocity gradient extrema may indicate
convergent motion due to gravitational collapse Hacar2011-533; Hacar2016-587;
Clarke2016-458; Misugi2019-881; Zhou2022-514; Zhou2023-676.
Zhou2022-514 investigated the physical properties and evolution of hub-
filament systems in $\sim$ 140 protoclusters, utilizing spectral lines
observed in the ATOMS (ALMA Three-millimeter Observations of Massive Star-
forming regions) survey Liu2020. Their findings indicate the presence of hub-
filament structures across a range of scales, spanning from 0.1 parsec to
several parsecs, in diverse Galactic environments. Additionally, slender
structures resembling filaments, such as spiral arms, have been identified at
scales below 0.1 parsecs in the vicinity of high-mass protostars Liu2015-804;
Maud2017-467; Izquierdo2018-478; Chen2020-4; Sanhueza2021-915. As proposed by
Zhou2022-514, self-similar hub-filament structures and filamentary accretion
seem to exist across scales, ranging from several thousand astronomical units
to several parsecs, within high-mass star-forming regions. This paradigm of
hierarchical/multi-scale hub-filament structures was generalized from clump-
core scale to cloud-clump scale in Zhou2023-676. Hierarchical collapse and
hub-filaments structures feeding the central regions are also described in
previous works, see Motte2018-56; Vazquez2019-490; Kumar2020-642 and
references therein.
Kinematically, the results in Zhou2023-676 also reveal the multi-scale hub-
filament structures in the G333 molecular cloud complex. The G333 complex
exhibits prevalent kinematic characteristics consistent with hub-filament
systems. Notably, the intensity peaks, acting as hubs, correlate with
converging velocities, indicating that the surrounding gas flows are
converging to dense structures. Specifically, there is a discernible increase
in velocity gradients at smaller scales. The filaments in the Large APEX sub-
Millimeter Array (LAsMA) and the Atacama Large Millimeter/submillimeter Array
(ALMA) observations show clear velocity gradients. The velocity gradients
fitted using the LAsMA and ALMA data exhibit consistent agreement over the
range of scales covered by ALMA observations in the ATOMS survey ($\textless$
5 pc). In Zhou2023-676, larger scale gas motions were investigated (the
longest filament $\sim$50 pc), yet similar results were obtained compared to
small-scale ALMA observations. Interestingly, the variations in velocity
gradients measured at distinct scales—small scales ($\textless$ 1 pc), medium
scales ($\sim$ 1-7.5 pc), and large scales ($\textgreater$ 7.5 pc)—align with
expectations from gravitational free-fall with central masses of $\sim$ 500
M⊙, $\sim$ 5000 M⊙ and $\sim$ 50000 M⊙ for the respective scales. This implies
that sustaining velocity gradients on larger scales necessitates higher
masses. The association of higher masses with larger scales suggests that the
inflow on a larger scale is driven by the larger-scale structure which may be
the gravitational clustering of smaller-scale structures. This observation
aligns with the hierarchical nature commonly found in molecular clouds and the
gas inflow from large to small scales. The large-scale gas inflow is likely
driven by gravity, indicating a state of global gravitational collapse within
the molecular clouds in the G333 complex. This supports the argument that
these molecular clouds serve as cloud-scale hub-filament structures. The
change in velocity gradients with the scale in the G333 complex indicates that
the morphology of the velocity field in position-position-velocity (PPV) space
resembles a ”funnel” structure. The funnel structure can be elucidated as the
acceleration of material converging towards the central hub and the
gravitational contraction of star-forming clouds or clumps. Large-scale
velocity gradients always associate with many intensity peaks, and the larger-
scale inflow is driven by the larger-scale structure, indicating that the
clustering of smaller-scale gravitational structures locally can serve as the
gravitational center on a larger scale. In a way, the funnel structure
provides insight into the gravitational potential well shaped by this
clustering.
In previous work, we have investigated gas dynamics and kinematics in
molecular clouds from cloud to core scales. The main task of this work is to
generalize the above physical pictures to galaxy-cloud scales, now the
smallest unit is the molecular cloud itself. In Zhou2022-514, dense core is
the hub in clump, in Zhou2023-676, clump is the hub in molecular cloud, here
we treat molecular cloud as the hub in galaxy. Self-similar or
hierarchical/multi-scale hub-filament structures and filamentary accretion
feeding the central regions exist from molecular cloud to dense core scales
Motte2018-56; Vazquez2019-490; Kumar2020-642; Zhou2022-514; Zhou2023-676, this
picture will be extended to galaxy-cloud scales in this work.
## 2 Data and target
We select two face-on spiral galaxies NGC5236 (M83) and NGC4321 (M100) from
the PHANGS-ALMA survey. We use the combined 12m+7m+TP PHANGS-ALMA CO (2$-$1)
data cubes to investigate gas kinematics and dynamics in the two galaxies,
which have a spectral resolution of 2.5 km s-1 and angular resolutions
$\sim$2.1 ′′ and $\sim$1.7 ′′, corresponding to linear resolutions $\sim$51 pc
and $\sim$123 pc for NGC5236 and NGC4321 at the distances 4.89 Mpc and 15.21
Mpc Leroy2021-257; Anand2021-501, respectively. The field-of-view (FOV) of
NGC5236 and NGC4321 in the ALMA observations are $\sim$ 10.5 Kpc $\times$ 10.8
Kpc and $\sim$ 17.1 Kpc $\times$ 15.5 Kpc, respectively. An overview of the
PHANGS-ALMA survey’s science goals, sample selection, observation strategy,
and data products is described in Leroy2021-257. A detailed description of
data calibration, imaging, and product creation procedures is presented in
Leroy2021-255. The PHANGS-ALMA CO (2$-$1) data cubes and other data products
(such as Moment maps) are available from the PHANGS team website
111https://sites.google.com/view/phangs/home. NGC5236 and NGC4321 were
selected for the following reasons:
1\. They are face-on galaxies. The inclination angles of NGC5236 and NGC4321
are 24o and 38.5o Lang2020-897, respectively. Thus, we can ignore the
projection effect in velocity gradient fitting in Sec.3.3;
2\. They have strong CO (2$-$1) emission, presenting large-scale continuous
structures (strong spiral arms), thus we can trace galaxy-scale gas motions;
3\. NGC5236 has similar morphology to NGC4321, but it is about 3 times closer
than NGC4321, thus we can resolve more detailed structures.
## 3 Results
### 3.1 Velocity component
Figure 1: An overview of gas kinematics in NGC4321 and NGC5236. (a) and (d)
The velocity distribution of the entire galaxies in PPV space decomposed by
GAUSSPY+; (b) and (e) The main filaments of NGC4321 and NGC5236 identified by
FILFINDER algorithm overlaid on the Moment 0 maps; (c) and (f) The integrated
intensity distributions extracted from the Moment 0 maps along the main
filaments in panels (b) and (e); (c2) and (f2) The velocity distributions
extracted from the Moment 1 maps along the main filaments in panels (b) and
(e); (c1) and (f1) The velocity residual distributions along the main
filaments in panels (b) and (e) after subtracting the large-scale velocity
gradients due to the galaxy rotation.
Before studying gas kinematics, we need to ascertain the distribution of gas
components in the galaxy. We utilized the fully automated Gaussian decomposer
GAUSSPY+ Lindner2015-149; Riener2019-628, designed to break down intricate
spectra of molecular lines into multiple Gaussian components. The parameter
settings for the decomposition align with those employed by Zhou2023-676. For
the CO (2$-$1) data cube of NGC4321, 50071 spectra were fitted by GAUSSPY+,
98% spectra are single component, 2% spectra have two velocity components.
323059 spectra were fitted for NGC5236, 87% spectra show single component,
11.5% spectra have two components. The reduced $\chi^{2}$ values output in the
fitting of NGC4321 and NGC5236 are around 1, which means that both of them get
a good fit. Generally, multiple velocity components ($>$1 velocity component)
mainly concentrate on the central region of the galaxy. In this work, we
mainly focus on the gas kinematics on the spiral arms. Therefore, the analysis
can be carried out based on the Moment maps.
In Fig.1, we can see ubiquitous velocity fluctuations, which may be attributed
to gravitational instability, as discussed below. In the central region of the
galaxy, there is almost no velocity fluctuations, clear velocity fluctuations
are mainly distributed on the spiral arms, revealing the interconnection
between velocity fluctuations and star formation activities.
### 3.2 Scaling relations
#### 3.2.1 Dendrogram
Figure 2: Dendrogram structures. (a) Leaves; (b) Branches. Red and orange eclipses represent type1 and type3 structures, respectively. Figure 3: Same as Fig.2, but for NGC5236. Table 1: Dendrogram structures in NGC 4321 and NGC 5236. | leaves | branches | type1 | type3 | total
---|---|---|---|---|---
NGC 4321 | 583 | 270 | 828 | 25 | 853
NGC 5236 | 1722 | 859 | 2339 | 242 | 2581
The issues of identifying structures in a PPV cube using the dendrogram
algorithm are discussed in Zhou2024-682-128. Consequently, we employed the
same approach as outlined in Zhou2024-682-128 and Zhou2024-682-173 to identify
dense gas structures. We conducted a direct identification of hierarchical
(sub-)structures based on the 2D integrated intensity (Moment 0) map of CO
(2$-$1) emission. Subsequently, we extracted the average spectrum of each
identified structure to delve into its velocity components and gas kinematics.
All the retained structures on the strictly masked Moment 0 map are reliable,
we therefore only require the smallest area of the identified structure larger
than 1 beam and do not set other parameters in the algorithm to reduce the
dependence of the identification on the parameter settings. Dendrogram
algorithm decomposes the intensity data (Moment 0 map) into hierarchical
structures. As illustrated in Fig.1 of Zhou2024-682-128, all identified
structures can be divided into three categories, i.e. i-leaves (isolated
leaves), c-leaves (clustered leaves) and branches. In this work, we merged
c-leaves and i-leaves. Finally, there are two kinds of structures, i.e. leaf
and branch structures. In Fig.2 and Fig.3, the structures identified by the
dendrogram algorithm exhibit a strong correspondence with the background
integrated intensity maps.
The algorithm characterizes the morphology of each structure by approximating
it as an ellipse. Within the dendrogram, the rms sizes (second moments) of the
intensity distribution along the two spatial dimensions define the long and
short axes of the ellipse, denoted as $a$ and $b$. As described in
Zhou2024-682-128, a smaller ellipse is obtained with $a$ and $b$,
necessitating a multiplication by a factor of two to appropriately enlarge the
ellipse. Then the effective physical radius of an ellipse is $R\rm_{eff}$
=$\sqrt{2a\times 2b}*D$, where $D$ is the distance of the galaxy. For a
structure with the area $A$, the total integrated intensity of the structure
is $I_{\rm CO}$, then the mass of the structure can be calculated by
$M=\alpha^{2-1}_{\rm CO}\times I_{\rm CO}\times A,$ (1)
here $\alpha^{2-1}_{\rm CO}\approx 6.7\rm M_{\odot}\rm pc^{-2}(\rm
K*kms^{-1})^{-1}$ Leroy2022-927.
The large-scale velocity gradients due to the galaxy rotation will contribute
to the non-thermal velocity dispersion. Before extracting the average spectra
of the identified structures, we subtracted the large-scale velocity gradients
in PPV data cube based on the constructed gas dynamical model in Sec.3.3.3.
According to their averaged spectra, the structures with absorption features
were eliminated firstly. Following Zhou2024-682-173, we only consider type1
(single velocity component) and type3 (blended velocity components) structures
in this work.
As detailed in Zhou2024-682-128, two screening criteria were applied for
structure refinement: (1) Removal of repetitive branch structures; and (2)
Exclusion of branch structures exhibiting complex morphology. Finally, there
are 853 and 2581 retained structures for NGC4321 and NGC5236, respectively, as
listed in Table 1 and marked in Fig.2 and Fig.3 by ellipses. NGC5236 is
$\sim$3 times nearer than NGC4321, the total number of the identified
structures in NGC5236 is also $\sim$3 times more than that in NGC4321.
#### 3.2.2 Scaling relations
Figure 4: Scaling relations of leaf and branch structures of NGC 4321 and NGC
5236. (a)-(c) and (a1)-(c1) $\sigma-N*R$; (d)-(f) and (d1)-(f1) $\sigma-N$;
(g)-(i) and (g1)-(i1) $\sigma-R$. $\sigma$, $R$ and $N$ are the velocity
dispersion, effective radius, and column density of each structure,
respectively. Here, “P” represents the Pearson coefficient. Dashed vertical
line marks the boundary of two fittings.
The scaling relations provide insights into the physical states of the
structures. Although the identified structures in NGC5236 are much more than
that in NGC4321, the scaling relations in Fig.4 are comparable. Due to the
further distance, some faint structures were filtered out in NGC4321, which
can be seen in NGC5236. In Fig.4, the faint or low-density structures produce
dispersive tails, similar to the Fig.14 of Zhou2024-682-128 for the G333
molecular cloud complex in the Milky Way.
In Fig.4(g)-(i), it appears that only branch structures demonstrate a
discernible correlation between velocity dispersion and scale. Clumps or cores
characterized by high column density exhibit greater velocity dispersion
relative to those with lower column density, a phenomenon attributed to gas
motions associated with gravitational collapse Ballesteros2011-411;
Traficante2018-473; Ballesteros2018-479; Li2023-949; Zhou2024-682-128. As
depicted in Fig.4(d)-(f), the positive correlations observed between velocity
dispersion and column density suggest a gravitational origin for the velocity
dispersion. In the context of pure free-fall, an anticipated $\sigma-N*R$
relation would exhibit a slope of 0.5. For a more convenient comparison with
$\sigma-R$ and $\sigma-N$ relations, we transform the Heyer relation,
$\sigma/R^{0.5}\propto N^{0.5}$, to the form $\sigma\propto(R*N)^{0.5}$
(Ballesteros2011-411, Eq. 3 in). Both relations should ideally have a slope of
0.5. However, in Fig.4(a)-(c), the slopes of the $\sigma-N*R$ relations are
notably shallower than 0.5, suggesting a deceleration from the expected
behavior of pure free-fall gravitational collapse.
### 3.3 Velocity gradient
The gravitational collapse of the structures has been revealed by the scaling
relations. This section gives a more detailed discussion of the gravitational
collapse from the perspective of the velocity gradient.
#### 3.3.1 Identification of filaments
The FILFINDER algorithm Koch2015 was used to identify and characterize
filaments in the galaxy. Following the method described in Zhou2022-514 and
Zhou2023-676, we used the Moment 0 maps of CO (2$-$1) emission from the data
products of the PHANGS-ALMA survey to identify filaments. Fig.1(b) and (e)
display the skeletons of the identified filaments superimposed on the Moment 0
maps. Notably, these skeletons exhibit a high degree of agreement with the gas
structures traced by CO (2$-$1) emission. In this work, we only consider the
filaments along the spiral arms (the main filaments), where have strong CO
(2$-$1) emission, which allows us to study large-scale continuous gas motions
in the galaxy.
Almost all algorithms have the issue of parameter settings, thus we try not to
discuss the identified filament itself in this work, such as its length and
width. Actually, we mainly regard the FILFINDER algorithm as a tool to draw
lines, these lines can make connections between discrete dense structures, or
put dense structures in gas networks. Then by extracting the velocity
information along these lines, we can study how the surrounding gas converges
to dense gas structures (gravitational centers). In Zhou2022-514 and
Zhou2023-676, the filaments traverse multiple local hub-filament structures,
characterized by fluctuations in velocity and density along the filaments.
These local dense structures, acting as gravitational centers, facilitate the
accretion of surrounding diffuse gas, culminating in the formation of local
hub-filament structures. In the PPV space, a hub-filament structure can be
represented as a funnel structure, as demonstrated in Fig.9 of Zhou2023-676.
The gradient of the funnel profile serves as an indicator of the gravitational
field’s strength. Consequently, examining the local velocity gradients along
the filaments provides insights into the strength of local gravitational
fields.
In Fig.1, although the velocity fluctuations at small scales can be seen
everywhere, they are hidden in global large-scale velocity gradients due to
the galaxy rotation. To derive the local velocity fluctuations, we must first
subtract the large-scale velocity gradients. We take two methods to solve this
issue.
#### 3.3.2 Linear fitting
Figure 5: A segment of NGC5236’s main filament, used to demonstrate the
velocity gradient fitting. (a) Linear fitting of the large-scale velocity
gradient due to the bulk motion; (b) Velocity field of the segment in panel
(a) after subtracting the fitted large-scale velocity gradient. Local velocity
gradients are fitted in the ranges defined by the red vertical dashed lines in
panel (c), and straight lines show the linear fitting results. The color-
coding of straight lines are meaningless, but it is convenient for the reader;
(c) Blue and red dotted lines show the normalized velocity and intensity,
respectively. Orange box marks a gravitational coupling structure.
Fig.1(c2)&(c) and (f2)&(f) show the intensity-weighted velocity (Moment 1) and
integrated intensity (Moment 0) of CO (2$-$1) line emission along the main
filaments with intense velocity and density fluctuations. A segment of
NGC5236’s main filament was used to demonstrate the velocity gradient fitting
in Fig.5. Here, we eliminated the large-scale velocity gradient by employing a
simple linear function for modeling. Subsequently, we subtracted this model
from the velocity field obtained from the Moment 1 map along the main
filament. We divided the main filaments into many segments to ensure that each
segment is as straight as possible, thus increasing the accuracy of the linear
fit, as shown in Fig.5(a). The fitted straight line can be the baseline, then
we subtracted the baseline for each segment to leave only the residual local
velocity fluctuations. The good correspondence between velocity gradients and
intensity peaks in Fig.5(b) indicates that the surrounding gases are
converging to the center of the gravitational potential well. For each segment
of the main filament, we firstly estimated the global velocity gradients
between velocity peaks and valleys at two sides of the intensity peaks
(representing gravitational centers), and ignored local velocity fluctuations.
Then we also derived additional velocity gradients over smaller distances
around the local intensity peaks, as illustrated in Fig.5(b). Generally,
large-scale velocity gradients are associated with multiple intensity peaks.
Given that NGC4321 and NGC5236 are good face-on galaxies, here we ignored the
projection effect in velocity gradient fitting.
#### 3.3.3 Gas dynamical model
Figure 6: (a) The velocity field (Moment 1) of NGC 4321 in observation; (b)
Gas dynamical model created by the Kinematic Molecular Simulation (KinMS)
package; (c) Velocity fluctuations after removing the created model.
As a comparison, we also removed the bulk motion or the galaxy rotation by
creating gas dynamical models using the Kinematic Molecular Simulation (KinMS)
package of Davis2013-429 based on the Moment-1 map, as shown in Fig.6(b). Then
we used the Moment 1 map to subtract the created model, and obtained the
velocity residual map. Finally, we extracted the velocity fluctuations from
the velocity residual map rather than the Moment 1 map. The subsequent
velocity gradient fitting is the same with Sec.3.3.2.
#### 3.3.4 Statistical analysis of the fitted velocity gradients
Figure 7: (a) The correlation between the fitted velocity gradient and the
scale; (b) The mass distribution of the identified structures. Figure 8:
Statistical analysis of all fitted velocity gradients. (a) Velocity gradient
vs. the length. The color lines show the freefall velocity gradients for
comparison. For the freefall model, red, magenta, green, cyan, blue and black
lines denote masses of 105 M⊙, 105.5 M⊙, 106 M⊙ , 106.5 M⊙, 107 M⊙ and 107.5
M⊙, respectively. Panels (b), (c), and (d): Zoomed maps with lengths
$\textless$ 500 pc (small scale), $\sim$ 500 – 1500 pc (medium scale), and
$\textgreater$ 1500 pc (large scale) in panel (a).
In Sec.3.3.2 and Sec.3.3.3, the galaxy centers were excluded in the fitting.
As shown in Fig.7, two methods give consistent fitting results. In Fig.8, we
can find the same gas kinematic modes presented in Zhou2023-676 and
Zhou2024-682-173. The variations in velocity gradients at both small and large
scales (with a boundary around 500 pc) align with expectations from
gravitational free-fall, with central masses of $\sim$ 105–106.5 M⊙ and $\sim$
106–107.5 M⊙. This implies that sustaining velocity gradients on larger scales
requires correspondingly larger masses, and larger masses imply larger scales,
suggesting that the larger-scale inflow is driven by the larger-scale
structure which may arise from the gravitational clustering of smaller-scale
structures, in harmony with the presence of hierarchical or multi-scale hub-
filament structures within the galaxy and the gas inflows from large to small
scales. In the orange box marked in Fig.5(b) and (c), multiple peaks are
coupled together to form a gravitational potential well on larger scale, and
each peak itself is also a local gravitational potential well.
In Fig.7(a), almost all measured velocity gradients can be fitted in the mass
range $\sim$ 105–107 M⊙, which is consistent with the mass distribution of the
identified structures shown in Fig.7(b), indicating that local dense
structures and their complex as gravitational centers will accrete the
surrounding diffuse gas and then produce the observed velocity gradients at
different scales.
#### 3.3.5 Deviation of the free-fall model
We can do a deeper analysis for the fitted velocity gradients with a simple
model. Assuming free-fall,
$\nabla v=-\frac{d}{dR}\sqrt{\frac{2GM}{R}}=\sqrt{\frac{GM}{2R^{3}}},$ (2)
in this equation, velocity gradient is more sensitive to scale. Thus we only
fit the correlation between velocity gradient and scale in Fig.7(a). In
equation.2, $\nabla v\propto R^{-1.5}$, however, the linear fitting in
Fig.7(a) gives $\nabla v\propto R^{-0.8}$, thus $\sim$2 times smaller than the
slope of the free-fall model, also indicating the slowing down of a pure free-
fall gravitational collapse presented in Sec.3.2.2.
## 4 Discussion
Gas kinematics on galaxy-cloud scales clearly deviate from the free-fall
model. The deviation may come from measurement or calculation biases. We only
measured the projection of the realistic velocity vector and hence
acceleration, thus tends to underestimate the observed acceleration. Moreover,
we only considered the molecular line CO (2$-$1), which only traces part of
the baryonic matters in the galaxy, thus underestimated the mass of the
gravitational center.
The shallower slope in Fig.7(a) may have physical correlation with the flat
rotation curve of the galaxy. Gas motions on galaxy-cloud scales may still
couple with the galactic potential, a comprehensive model should account for
the interplay between motions within the galactic potential and the self-
gravitational potential of the cloud. Dobbs2013-432; Meidt2018-854;
Meidt2020-892; Utreras2020-892. In the model proposed by Meidt2020-892, the
transition to free-fall collapse can only happen once the gas has completely
decoupled from the galactic potential. However, it is not clear down to which
spatial scales gas motions remain dynamically relevant to galactic potential
or start decoupling with galactic potential.
Tidal forces have been advocated in previous works to restrict or trigger star
formation. Previous studies by Ballesteros2009-393; Ballesteros2009-395 have
shown the impact of tidal forces arising from an effective Galactic potential
on molecular clouds. These forces have the potential to either compress or
disrupt molecular clouds, influencing the overall star formation efficiency.
Thilliez2014-31 examined the stability of molecular clouds within the Large
Magellanic Cloud (LMC) against shear and the galactic tide. However, their
findings indicate that star formation in the LMC is not impeded by either
tidal or shear instability. Moreover, in Ramirez2022-515, tidal stresses from
neighbouring molecular cloud complexes may increase interstellar turbulence,
rather than the galactic potential.
Gravity is a long-range force. A local dense structure evolves under its self-
gravity, but as a gravitational center, its influence can also affect
neighboring structures. At the same time, it also experiences the
gravitational pull from other nearby sources. The tidal and gravitational
fields are mutually interdependent. As described above, the
hierarchical/multi-scale gravitational coupling of gas structures means the
extensive tidal interactions in the galaxy. Whether the galactic potential has
effect on molecular clouds and their complexes or not, molecular clouds should
be affected by the cumulative tidal interactions of many nearby materials,
which may prevent gravitational collapse and growth of instabilities or star
formation in the cloud. Due to the diffuse and complex morphology of matter
distribution in the galaxy, a complete tidal calculation would be very
complex. One should derive the gravitational potential distribution from the
observed density distribution and then calculate the tidal field according to
the gravitational potential distribution, as presented in Li2024-528. Diverse
manifestations of gravitational effects on gas within molecular clouds were
unveiled in Li2024-528: Dense regions experience gravitational collapse, while
the surrounding gas is subject to significant tidal forces, suppressing
fragmentation. This gas, influenced by extensive tides, is directed towards
the dense regions, serving as fuel for star formation. The spatial
distribution of regions experiencing varying tidal influences elucidates the
observed hierarchical and localized pattern of star formation. Similar
mechanisms may also exist on galaxy-cloud scales, we will discuss this topic
in detail in future work.
Figure 9: The average number density distribution of the identified
structures.
In addition to the factors discussed above, magnetic fields may have a
significant impact on the gas kinematics of molecular clouds as suggested by
simulations Kim2021-911; Seifried2020-497; Ibanez2022-925; Ganguly2023-525 and
observations Crutcher2010-725; Li2011-479; Crutcher2012-50; Stephens2022-926;
Ngoc2023-953; Rawat2024-528. Especially, in simulations, the diffuse gas with
a number density ($n$) less than 100 cm-3 in the envelopes of molecular clouds
may be upheld against gravitational collapse due to magnetic support
Ibanez2022-925; Ganguly2023-525. Assuming the spherical geometry of the
identified clouds, the average number density can be calculated by
$\overline{n}=M/(\frac{4}{3}\pi R_{\rm eff}^{3})/(\mu m_{\rm H}),$ (3)
where $\mu$ = 2.37 is the mean molecular weight per ‘free particle’ (H2 and
He, the number of metal particles is negligible), mH is atomic hydrogen mass.
As shown in Fig.9, almost all of structures have the average number density ¡
100 cm-3. However, the real cloud shape should be more sheetlike instead of
spherical Shetty2006-647; Inutsuka2015-580; Arzoumanian2018-70; Kohno2021-73;
Arzoumanian2022-660; Rezaei2022-930; Zhou2023-519; Zhou2023-676;
Clarke2023-519; Ganguly2023-525. Therefore, the average number density of the
clouds should be significantly underestimated. Even if the average number
density estimates differ by an order of magnitude, what revealed by Fig.9
remains promising. The supportive role of the magnetic field may be a
significant factor contributing to the deviation from free fall motions, and
further detailed investigation is warranted in future research.
## 5 Summary
We investigated the kinematics and dynamics of gas structures on galaxy-cloud
scales in two spiral galaxies NGC5236 (M83) and NGC4321 (M100) using the CO
(2$-$1) line. The main conclusions are as follows:
1\. We directly identified hierarchical (sub-)structures according to the 2D
integrated intensity (Moment 0) map of CO (2$-$1) emission. Subsequently, we
extracted the average spectrum for each structure, delving into its velocity
components and gas kinematics. Considering that the large-scale velocity
gradients due to the galaxy rotation will contribute to the non-thermal
velocity dispersion, before extracting the average spectra of the identified
structures, we subtracted the large-scale velocity gradients in PPV data cube
based on the constructed gas dynamical model.
2\. In examining the scaling relation among velocity dispersion ($\sigma$),
effective radius ($R$), and column density ($N$) across all structures, it
becomes evident that $\sigma-N*R$ consistently exhibits a stronger correlation
compared to $\sigma-N$ and $\sigma-R$. The observed correlations between
velocity dispersion and column density suggest a potential link to
gravitational collapse, corroborated by the measured velocity gradients.
However, it is noteworthy that the slopes of the $\sigma-N*R$ relations appear
to be significantly shallower than the anticipated value of 0.5, implying a
deceleration from the characteristic behavior of a pure free-fall
gravitational collapse.
3\. We employed the FILFINDER algorithm to identify and characterize filaments
within the galaxy using integrated intensity maps. Observable velocity and
density fluctuations along these filaments enabled us to fit local velocity
gradients around intensity peaks, a process performed after removing the
global large-scale velocity gradients attributed to the galaxy’s rotation.
4\. Statistical analysis of the fitted velocity gradients on galaxy-cloud
scales shows the same gas kinematic modes presented on cloud-clump and clump-
core scales. The variations in velocity gradients at both small and large
scales (with a boundary around 500 pc) align with expectations from
gravitational free-fall, with central masses of $\sim$ 105–106.5 M⊙ and $\sim$
106–107.5 M⊙. This implies that sustaining velocity gradients on larger scales
requires correspondingly larger masses, and larger masses imply larger scales,
suggesting that the larger-scale inflow is driven by the larger-scale
structure which may arise from the gravitational clustering of smaller-scale
structures, in harmony with the presence of hierarchical or multi-scale hub-
filament structures within the galaxy and the gas inflows from large to small
scales.
5\. In free-fall model, the velocity gradient and scale satisfy $\nabla
v\propto R^{-1.5}$. However, in the observation, $\nabla v\propto R^{-0.8}$,
also indicating the slowing down of a pure free-fall gravitational collapse.
J. W. Zhou thanks V. Kalinova for the comments. It is a pleasure to thank the
PHANGS team, the data cubes and other data products shared by the team make
this work can be carried out easily. This paper makes use of the following
ALMA data:
ADS/JAO.ALMA#2013.1.01161.S,
ADS/JAO.ALMA#2015.1.00121.S,
ADS/JAO.ALMA#2015.1.00956.S,
ADS/JAO.ALMA#2016.1.00386.S,
ADS/JAO.ALMA#2017.1.00886.L.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), NSTC and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ.
## Data Availability
The PHANGS-ALMA CO (2$-$1) data cubes and other data products (such as moment
maps) are available from the PHANGS team website
222https://sites.google.com/view/phangs/home.
|
11institutetext: 1Aarhus University, 2University of Copenhagen
# Text-Driven Stylization of Video Objects
Sebastian Loeschcke1 Serge Belongie2 Sagie Benaim2
# Text-Driven Stylization of Video Objects
Sebastian Loeschcke1 Serge Belongie2 Sagie Benaim2
###### Abstract
We tackle the task of stylizing video objects in an intuitive and semantic
manner following a user-specified text prompt. This is a challenging task as
the resulting video must satisfy multiple properties: (1) it has to be
temporally consistent and avoid jittering or similar artifacts, (2) the
resulting stylization must preserve both the global semantics of the object
and its fine-grained details, and (3) it must adhere to the user-specified
text prompt. To this end, our method stylizes an object in a video according
to two target texts. The first target text prompt describes the global
semantics and the second target text prompt describes the local semantics. To
modify the style of an object, we harness the representational power of CLIP
to get a similarity score between (1) the local target text and a set of local
stylized views, and (2) a global target text and a set of stylized global
views. We use a pretrained atlas decomposition network to propagate the edits
in a temporally consistent manner. We demonstrate that our method can generate
consistent style changes over time for a variety of objects and videos, that
adhere to the specification of the target texts. We also show how varying the
specificity of the target texts and augmenting the texts with a set of
prefixes results in stylizations with different levels of detail. Full results
are given on our project webpage: https://sloeschcke.github.io/Text-Driven-
Stylization-of-Video-Objects/.
###### Keywords:
Video Editing, Text-Guided Stylization, CLIP
## 1 Introduction
Manipulating semantic object entities in videos using human instructions
requires skilled workers with domain knowledge. We seek to eliminate these
requirements by specifying a desired edit or stylization through an easy,
intuitive, and semantic user instruction in the form of a text-prompt.
However, manipulating video content semantically is a challenging task. One
challenge is in generating consistent content or style changes in time, that
adhere to the target text specification. Another challenge is to manipulate
the content of an object such that it preserves the content of the original
video and the global semantics while also adhering to fine-grained details in
the target text.
| |
---|---|---
| |
Input video | “Swan made out of cactus” | “Swan with crocodile skin”
Figure 1: Two representative video frames and the edited video frames together
with the global target text.
In recent years, advances in computational methods emerged that enable
manipulation of appearances and style in images and allow novice users to
perform realistic image editing. These methods include manipulation tools that
use natural language (text prompts) to express the desired stylization of
images or 3D objects [18, 6]. The text-driven manipulation is facilitated by
recent developments in models for joint embeddings of text and images, e.g the
Contrastive Language Image Pretraining (CLIP [23]) model. Instead of
manipulating images or 3D objects, we use CLIP in the context of video
manipulation. This is not a straightforward task since simply maximizing the
semantic (CLIP-based) similarity between a valid target text and each 2D frame
in the video often leads to degenerate solutions. Also, applying methods for
image manipulation to each frame in a video results in edits that lack
temporal consistency.
The recently introduced Neural Layered Atlases (NLA) work [14], demonstrates
the ability to separate a moving object in a video from its background by
decomposing the video into a set of 2D atlases. Each atlas provides a unified
image representation of an object or background over the video. Edits applied
to the image representation are automatically mapped back to the video in a
temporally consistent manner. However, editing an image still requires manual
effort and editing skills from the user. Another problem with this approach is
that the 2D atlas representation can be hard to edit due to local
deformations.
We propose a method for performing intuitive and consistent video editing with
multiple capabilities by using the representational power of CLIP to express a
desired edit through a text-prompt. An example could be to change the style of
a swan swimming in a lake according to a target text: “A swan with cactus
skin.” Text is easily modifiable and allows users to express complex and
abstract stylizations intuitively. Using text to express edits reduces the
need for manual editing skills and also avoids the problems related to
deformation in the 2D atlas representations. An illustration is shown in Fig.
1.
To apply temporally consistent edits to an object in a video, our method uses
the atlas decomposition method presented in NLA [14]. We train a generator on
a single input video by sampling local and global views of each frame in the
video and applying various augmentations to each view. Our method uses a
global loss that compares each global view with a global target text and a
local loss that compares each local view with a local target text. The global
loss then focuses on the global semantics and the local views focus on the
fine-grained details. To regularize our learning, we use a sparsity loss that
encourages sparse representation and a temporal triplet loss that encourages
frames that are close in time to be similar in CLIP’s embedding space.
We demonstrate that our method results in natural and consistent stylizations
of objects for a diverse set of videos and target texts. We show how varying
the specificity of both the local and global target texts varies the
stylization and how augmenting the target texts with neutral prefixes can
result in more detailed stylizations. We also demonstrate that our global loss
focuses on the global semantics and the local losses on the fine-grained
details.
## 2 Related Work
Our work is related to video editing works and also to text-based stylization
works.
### 2.1 Video editing
Unlike images, editing or stylizing objects in videos requires the ability to
handle temporal consistency. One natural approach is to propagate the edits
from one frame to the next. Video Propagation Networks [10] use a bilateral
network to connect pixels in consecutive frames and an adaption network to
refine the pixels. Other approaches use optical flow to propagate edits made
on a few key-frames [28]. These approaches work well when there is a clear
correspondence between frames, but have difficulties, e.g., when the video
contains occlusions.
To address occlusion challenges, recent work has used deep learning
approaches, e.g. self-supervised methods for learning visual correspondence
from unlabeled videos [9, 29]. Instead, our work uses the representation
proposed by Neural Layered Atlases (NLA) [14], which decomposes a video into a
set of layered 2D atlases. Each atlas provides a unified representation of the
appearance of an object or background throughout the video. Similar to NLA,
Deformable Sprites [30] decomposes a video into a texture atlas that captures
an object’s motion across the entire video. Their method allows for temporally
consistent video edits, by applying edits to the decomposed atlas. In contrast
to NLA, they use optical flow to compute foreground objects instead of a
pretrained segmentation network to get the segmentation mask for an object in
the input video. However, both NLA and Deformable Sprites only allow for basic
manual editing. We use NLA’s atlas separation method for objects in videos,
but unlike NLA, we allow for text-driven stylization.
### 2.2 Text-based stylization
Our work bears similarities to recent image and 3D manipulation techniques
that edit style and appearances through natural language descriptions. These
descriptions are often embedded with the Contrastive Language Image
Pretraining (CLIP) [23] model, a multi-modal embedding model that learns an
image-text embedding space. Recent work used CLIP together with pretrained
generative networks for image editing and stylization [1, 21, 3, 5, 6, 8, 15].
For example, StyleCLIP [21], and StyleGAN-NADA [8] both use a pretrained
StyleGAN [12] and CLIP to perform image editing, either by using CLIP to
control a latent code or to adapt an image generator to a specific domain [8].
Other examples include using StyleGAN and CLIP for image stylization [5] or
Paint by Word [3] which uses CLIP paired with StyleGAN2 [13] and BigGAN [4] to
enable “painting” images in the style of a text prompt. Usually, pretrained
generators only work well for the specific domain they are trained on. In
contrast, our method does not require a pretrained generator. We train our own
generator on the set of video frames we wish to stylize.
Other examples of semantic text-guided manipulation in the context of 3D
objects include 3DStyleNet [31], a method for changing the geometric and
texture style of 3D objects, ClipMatrix [11] which uses text-prompts to create
digital 3D creatures, and methods for generating 3D voxels using CLIP [27].
Another line of recent work uses joint-embedding architectures [26, 25, 24,
19] for image generation, e.g., DALL-E [25] and its successor, DALL-E 2 [24],
which can also be used for stylizing images. DALL-E 2 uses a two-stage model,
where a CLIP image embedding is generated using a text prompt and a decoder is
then used to generate an image conditioned on the generated image embedding.
Training joint embedding architectures requires enormous datasets and many
training hours. Instead of training on a large dataset, we train on a set of
frames for a single video and use augmentations to extract many different
views of the input frames. As opposed to all the abovementioned techniques, we
work on videos.
Another line of work uses CLIP without relying on a pretrained generator, e.g.
given a natural language input, CLIPdraw [7] synthesizes novel drawings, and
CLIPstyler [17] stylizes images. Similarly, Texts2Mesh [18] does not rely on a
pretrained generator. In Text2Mesh, the CLIP embedding space is used to enable
text-driven editing of 3D meshes. Text2Mesh uses a multi-layer perceptron
(MLP) to apply a stylization to $(x,y,z)$-coordinates of an input mesh. The
neural optimization process of the MLP is guided by a semantic loss that
computes the similarity between multiple augmented 2D views embedded with CLIP
and a target text. Similarly, we do not rely on a pretrained generator.
Our work was developed concurrently to Text2Live [2], which shares many of the
same goals and methods as our work. Similarly to our method, Text2Live uses a
pretrained Neural Layered Atlases (NLA) model to separate a moving object in a
video from its background. Text2Live train a generator to apply text-driven
local edits to a single frame and use the NLA model to map the edits back to
the input video in a temporally consistent manner. In contrast to our
approach, Text2Live does not directly generate the edited output. Instead, it
generates an edit layer that is composited with the original input.
## 3 Method
We wish to apply natural and temporally consistent stylizations to objects in
videos using a natural language text prompt as guidance. To change the style
of an object to conform with a target text prompt in a temporally consistent
manner, we build on top of a Layered Neural Atlas method [14], which separates
the appearance of an object in a video from its background. We then use a pre-
trained text-image multimodal embedding of CLIP [23] in a set of objectives.
Minimizing this set of objectives aims at matching the style of a foreground
object in a video with that of a target text. The objectives include a global
and local objective. The global objective focuses on the global semantics by
maximizing the similarity between the global views and a target text that
relates to the underlying content. Instead, the local objective focuses on the
fine-grained details, by maximizing the similarity between the local views
with a target text that relates to local semantics of the stylization. We add
a sparsity loss, similar to a $L_{1}$-regularization term, that encourages the
predicted foreground color values to be minimal. Additionally, we add a
temporal triplet loss that encourages the embeddings of frames that are close
in time to also be close in CLIP’s embedding space. We begin by describing the
method of CLIP [23] and that of Neural Layered Atlases (NLA) on which our
method is based. We then describe the training and loss formulations used by
our method.
### 3.1 CLIP
CLIP is a multi-modal embedding method that trains an image encoder $E_{img}$
and a text encoder $E_{txt}$ to match between the embeddings of corresponding
image-text pairs using a contrastive loss formulation. This loss formulation
optimizes the similarity between corresponding image-text pair $T$ and $I$.
More specifically, $I$ and $T$ are first embedded:
$I_{emb}=E_{img}(I)\in\mathbb{R}^{512},\quad
T_{emb}=E_{txt}(T)\in\mathbb{R}^{512}$
The similarity between I and T is then measured by
$\operatorname{sim}(I_{emb},T_{emb})$ where
$\operatorname{sim}(a,b)=\frac{a\cdot b}{|a||b|}$, is the cosine similarity.
### 3.2 Neural Layered Atlases (NLA)
Figure 2: Our stylization pipeline. In the first step, we train the network
using the NLA procedure [14] to reconstruct input video frames. We then
finetune the editing atlas $A$ using our approach described in Fig. 3. In our
stylization pipeline, we create a set of cropped input video frames
$Q_{Crop}$. This set is passed through a stylization model to create a
foreground and background atlas. A set of stylized frames $Q_{Style}$ is
produced by $\alpha$-blending the predicted atlases. All weights of the MLPs
are frozen except for the editing atlas MLP $A$ which is finetuned. A closer
look at how our editing atlas is trained is given in Fig. 3. Figure 3:
Finetuning the editing atlas. As described in the stylization pipeline (Fig.
2), we use our stylization pipeline to get a set of stylized frames
$Q_{Style}$. To this end, we finetune our editing atlas, which is part of the
stylization model. We sample $n_{Global}$ global views $I^{Global}$ and
$n_{Local}$ local views $I^{Local}$. The set of images $I^{Global}$ are then
augmented using random perspectives and random background removal and used
together with the global target text $T_{Global}$ to compute a global loss
(Eq. 3). Three global augmented images are used to compute the temporal loss
(Eq. 4). Similarly, the $I^{Local}$ images are augmented and used together
with the local target text $T_{Local}$ to compute the local loss (Eq. 2).
Lastly, a sparsity loss (Eq. 5) is computed from the stylized frames
$Q_{Style}$.
Neural Layered Atlases (NLA) [14] decompose a video into a set of layered 2D
atlases. Each atlas provides a unified representation of the appearance of an
object or the background throughout the video. NLA use two mapping networks
$M_{f}$ and $M_{b}$, where each takes a pixel and time location $(x,y,t)$ in
the video as input and outputs the corresponding 2D $(u,v)$-coordinate in each
atlas:
$M_{f}(p)=(u_{f}^{p},u_{f}^{p}),\quad M_{b}(p)=(u_{b}^{p},u_{b}^{p})$
An atlas network $A$ takes the predicted 2D $(u,v)$-coordinate as input and
outputs the atlas’s RGB color at that location. Additionally, all pixel
coordinates are fed into the Alpha MLP network $M_{\alpha}$ which outputs the
opacity of each atlas at that location. The RGB color can then be
reconstructed at each pixel location by alpha-blending the predicted atlas
points according to the opacity value predicted by $M_{\alpha}$.
NLA enables consistent video editing. First, each atlas is discretized into an
image. A user can then manually apply edits using an editing program. These
edits are mapped back to the input video using the computed $(u,v)$-mapping.
To get the reconstructed color $c^{p}$ for pixel $p$, the color of the
predicted foreground color $c_{f}^{p}$, background color $c_{b}^{p}$ and
predicted opacity value $\alpha^{p}$ of the edited atlas are blended as
follows:
$c^{p}=(1-\alpha^{p})c_{b}^{p}+\alpha^{p}c_{f}^{p}$
### 3.3 Our Stylization Pipeline
Instead of having to manually apply edits to the discretized atlases as in
[14], we wish to apply edits automatically using a target text prompt.
Specifically, we are interested in modifying the RGB values such that they
conform with a target text. We focus on the Atlas MLP $A$, since $A$ makes the
RGB predictions. Instead of using a single atlas MLP $A$ for all atlases, we
create two copies of the atlas MLP after pre-training the NLA network, one
editing atlas MLP for the foreground object we want to stylize ($A$) and one
for all other atlases ($A_{b}$). We then freeze the weights of $M_{b}$,
$M_{f}$, $M_{\alpha}$, $A_{b}$ and finetune the weights of $A$. While the goal
in NLA is to optimize the model to produce the best reconstruction, we instead
want to create a stylization that conforms with the inputted target text.
Our stylization pipeline is illustrated in Fig. 2. The input is a set of raw
frames $Q_{Raw}$. Since we know the position of the object in the video from
the $\alpha$-map, we can compute a bounding box containing the whole object.
Once we have the bounding box, we crop each frame in $Q_{Raw}$ such that it
only contains the content within the bounding box plus a small margin. All the
cropped frames $Q_{Cropped}$ are passed through a pre-trained NLA model, where
all MLP weights have been frozen, except for the weights of the $A$ MLP. The
NLA method produces a set of stylized frames $Q_{Style}$.
To fine-tune the weights of $A$, we sample training batches in both time and
space. Our sampling method is illustrated in Fig. 3. First, we sample a set
$Q_{Sample}$ uniformly at random among all frames in the input video and pass
them through the stylization pipeline (Fig. 2) to create a set $Q_{Style}$.
For each frame in $Q_{Style}$ we sample $n_{Global}$ views $I^{Global}$ and a
set of $n_{Local}$ views $I^{Local}$. Each of the $n_{Global}$ views is
produced by sampling a crop with a size in the range $[0.9,1.0]$ of the
original frame size. Each of the $n_{Local}$ views is produced by sampling a
crop with a size in the range $[0.1,0.5]$ of the original frame size. To
ensure the local views contain the object we want to stylize, we use the
$\alpha$-map of the frame to determine the position of the object in the
frame. We then sample until we get $n_{Local}$ views where at least
$\frac{1}{3}$ of the sampled view is part of the object.
Once the local and global views have been sampled, we apply a random
perspective transformation and a random background removal which with some
probability $p$ removes the background (details in Appendix Sec. 6.2).
Additionally, each augmented frame is normalized with the same mean and
standard deviation as used to train the CLIP model [23].
Our objective function is composed of three main losses defined in CLIP’s
feature space: (1) $L_{\text{Local}}$ which focuses on local semantics, (2)
$L_{\text{Global }}$ which focuses on global semantics, and (3) $L_{Temp}$
which encourages temporal consistency. Additionally, we use the regularization
term $L_{\text{sparsity }}$ introduced in NLA [14], which encourages sparse
representations. In all loss terms, when we compute the similarity between a
text and each sampled view, we use the average embedding across all views:
$I_{emb}^{Local}=\frac{1}{n_{Local}}\sum_{i=1}^{n_{Local}}E_{img}\left(I_{i}^{\text{Local}}\right),I_{emb}^{Global}=\frac{1}{n_{Global}}\sum_{i=1}^{n_{Global}}E_{img}\left(I_{i}^{\text{Global}}\right)$
(1)
#### Local loss
$L_{\text{Local}}$ is applied to all views in $I^{\text{Local}}$. The goal is
to modify the image, such that the local details conform with a target text
$T_{\text{Local}}$:
$L_{\text{Local}}=1-\operatorname{sim}\left(I_{emb}^{Local},E_{txt}\left(T_{\text{Local}}\right)\right)$
(2)
where $\operatorname{sim}(a,b)=\frac{a\cdot b}{|a||b|}$ is the cosine
similarity, and $E_{txt}$ denote CLIP’s pre-trained text encoder. The local
views have a more zoomed-in view of the object we are stylizing. Additionally,
the local target text $T_{Local}$ contains local specific semantics, e.g.
“rough cactus texture.” Hereby, the local loss can focus on the texture and
fine-grained details of the stylization we apply to the input video.
#### Global loss
The global loss is applied to views in $I^{\text{Global}}$ that all include
the entire object being stylized. The intended goal is that the global loss
will preserve the overall context. In the target text $T_{\text{Global}}$, we
include words that describe the global context of the object we are trying to
stylize, e.g. “A swan made of cactus.” The global loss formulation is then
given by:
$L_{\text{Global}}=1-\operatorname{sim}\left(I_{emb}^{Global},E_{txt}\left(T_{\text{Global
}}\right)\right)$ (3)
#### Temporal loss
We use a triplet loss to include a temporal aspect and enforce that
consecutive frames should be more similar in CLIP’s embedding space than
frames that are further apart. To compute the temporal loss, we sample three
frames $t_{1},t_{2,},t_{3}$, where we have that $t_{1}<t_{2,}<t_{3}$ w.r.t.
the order of the frames in the input video. We then enforce that the
similarity between the sampled global views of $t_{1}$ and $t_{2}$ in the CLIP
embedding space should be greater than the similarity between $t_{1}$ and
$t_{3}$: Let
$I_{emb({t_{1}})}^{Global},{}I_{emb({t_{2}})}^{Global},I_{emb({t_{3}})}^{Global}$
denote the average embedded global views (computed in Eq. 1) for each of the
three frames. Then we compute the triplet loss $L_{Temp}$ as follows:
$\begin{split}\operatorname{Sim}_{t_{1}t_{3}-t_{1}t_{2}}&=\operatorname{sim}\left(I_{emb({t_{1}})}^{Global},I_{emb({t_{3}})}^{Global}\right)-\operatorname{sim}\left(I_{emb({t_{1}})}^{Global},I_{emb({t_{2}})}^{Global}\right)\\\
L_{Temp}&=\lambda_{Temp}\cdot\max\left(0,\operatorname{Sim}_{t_{1}t_{3}-t_{1}t_{2}}\right)\end{split}$
(4)
where $\lambda_{Temp}$ is a weighting of the temporal loss. If the frames are
further away from each other, the temporal loss should contribute less to the
overall loss. For this reason, we weigh the contribution of the temporal loss
by a Gaussian probability density function $g$ with a mean equal to zero and a
standard deviation of five. We compute the weight of a triplet by applying $g$
to the difference between $t_{1}$ and $t_{3}$:
$\lambda_{Temp}=g(t_{3}-t_{1})$
#### Sparsity loss
We use the same sparsity loss as in NLA [14]. Its intended function is to
encourage points that are mapped to the background atlas to have a zero value
in the foreground atlas, e.g. if a point $p$ is mapped to the background
atlases it should not contain information about the foreground atlas.
$L_{\text{sparsity }}=\left\|\left(1-\alpha^{P}\right)c_{f}^{P}\right\|$ (5)
where $c_{f}^{P}$ is the predicted color at $p$ for the foreground layer and
$\alpha^{P}$ is the opacity value at location $p$.
#### Full objective
The full loss term that is minimized is represented as:
$L=\lambda_{Sparsity}L_{\text{Sparsity }}+\lambda_{Temp}L_{\text{Temp
}}+\lambda_{Local}L_{\text{Local}}+\lambda_{Global}L_{\text{Global }}$ (6)
where $\lambda_{Local}$,$\lambda_{Global}$, $\lambda_{Temp}$,
$\lambda_{Sparsity}$ are hyperparameters used to control the weighting of each
loss term. As default $\lambda_{Local}=\lambda_{Global}=1$, while
$\lambda_{Temp}$ and and $\lambda_{Sparsity}$ vary depending on the input
video.
## 4 Experiments
We evaluate our method on a set of videos from the DAVIS dataset [22] across a
diverse set of target text prompts. Our goal is to perform consistent and
natural video editing. For this purpose, we present both a quantitative and
qualitative evaluation of our results and perform a careful ablation study of
each loss term in our objective function.
In Sec. 4.1 we demonstrate the capabilities of our method by showing various
stylizations for different videos. We also present a quantitative evaluation,
where we compare our method to an image baseline method applied to each frame
in the video. In Sec. 4.2 we demonstrate that the specificity of text prompts
influences the details of the results. In Sec. 4.3 we demonstrate how text
augmentation affects the results of the stylizations. In Sec. 4.4 we conduct a
series of ablations on our loss terms and demonstrate how the local and global
losses focus on different semantics. Finally, in Sec. 4.5 we illustrate some
of the limitations of our method.
| |
---|---|---
| |
“Swan” | “Shiny metal swan” | “Swan with wood bark skin”
| |
| |
“Boat” | “Shiny aluminum fishing boat” | “Boat made of copper”
| |
| |
“Dog” | “Dog with zebra fur” | “Golden dog”
Figure 4: Example results. Two representative frames from each edited video together with the global target text. | Q1 (Realism) | Q2 (Matching Text)
---|---|---
Blended Diffusion [1] | $2.22$ ($\pm 1.08$) | $2.10$ ($\pm 1.00$)
Ours | $\mathbf{3.47}$ ($\pm 1.10$) | $\mathbf{3.94}$ ($\pm 0.99$)
Table 1: Mean opinion scores (1-5) and standard deviation for Q1 and Q2.
### 4.1 Varied stylizations
In Fig. 4 we illustrate our video stylization method applied to three videos
and three texts. All local target texts used for these examples are similar to
the global target text, but without containing the information about the
underlying content, e.g., for the swan, the local target text is “metal skin.”
The results show that we can apply temporally consistent stylizations that
adhere to the target text specification. The swan example shows fine-grained
details of the target texts and also preserves the underlying content of the
swan. In the boat example, the stylization captures the details of the target
text. The boat has a texture similar to shiny aluminum and also has something
that looks like a fishing net at the end of the boat. The dog example shows
that our method can apply a realistic and consistent stylization to a video
containing occlusions.
We quantify the effectiveness of our method by comparing it to an image
baseline applied to each frame in a video input. As a baseline, we use a
pretrained Blended-diffusion (BF) [1] model with standard configurations. The
model takes as input an image, a ROI mask, and a target text. BF performs
local (region-based) edits based on a target text description and the ROI
mask. We conduct a user study to evaluate the perceived quality of the
stylized outputs generated by both the BF model and our method, and the degree
to which the outputs adhere to the global target text. Our user study
comprises $50$ users and $20$ stylized videos, each with a different target
text. For each video and target text combination, the users are asked to
assign a score ($1$-$5$) to two factors: (Q1) “How realistic is the video?,”
(Q2) “does the {object} in the video adhere to the text {content}. For Q1 we
make it clear to the user that “realistic” refers to the quality of the video
content. The results are shown in Table 1.
| | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
| | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
Figure 5: Target text specificity. Each example shows a representative frame
from the video. The experiment shows the specificity of a target affects the
stylization. Global target text prompts, row one: (a) “Armor,” (b) “Iron
armor,” (c) “Medieval iron armor,” (d) “Suit of shiny medieval iron armor,”
(e) “Full plate shiny medieval iron armor,” row two: (a) “Boat made of wood,”
(b) “Boat made of dark walnut wood (c) “Fishing boat made of wood,” (d) “Old
fishing boat made of wood,” (e) “Fishing boat made of wood planks.”
### 4.2 Prompt specificity
We demonstrate that varying the specificity of the target text prompt affects
the level of detail in the stylization. Our experiment is motivated by recent
work on prompt engineering [32] that shows how slight changes in the target
text can have a big impact on the CLIP similarity between a text and an image.
Fig. 5 shows an increasing level of detail for two videos and two target
texts. The target text specificity not only influences the level of detail, it
also makes it easier for CLIP to navigate in its embedding space. In the swan
example in column (a), we have “Swan with an armor,” which is a more ambiguous
target compared to the other swan examples with more detailed target texts and
stylizations. We hypothesize that this is because several stylizations can
satisfy the more simple target text, while a more specific target text narrows
down the set of possible directions in CLIP’s embedding space. The swan
examples in columns (d) and (e) indicate that CLIP has some understanding of
the different body parts of the swan. The “full plate” target text (d) covers
the entire head of the swan while the “suit” of armor in column (e) has a
clear cut around the head of the swan.
| | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
| | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
Figure 6: Prefix augmentations - varying number of prefixes to sample from in each iteration. Each figure shows a representative frame from each video and experiment configuration. The examples in row one use the texts: $T_{Global}$:“Origami swan with white paper skin,” $T_{Local}$: “Origami white paper skin,” and in row two: $T_{Global}$:“Dog with bengal tiger fur,” $T_{Local}$: “Bengal tiger fur.” (a) no prefixes, (b) $4$ local, no global, (c) $4$ global, no local (d) $4$ global & $4$ local, (e) $8$ global & $8$ local. The specific prefixes are described in Sec. 6.3. | | | |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
Figure 7: Ablation on each loss term \- All experiments were run with the same seed and with the texts: $T_{Global}$: “A swan with crocodile skin,” $T_{Local}$: “Crocodile skin.” Each figure shows a representative frame from a video, where we ablate one of our loss terms in Eq. 6. (a) All losses, (b) w/o local loss, (c) w/o global loss, (d) w/o temporal loss, (e) w/o sparsity loss. | |
---|---|---
(a) | (b) | (c)
Figure 8: Global and local semantics. A representative frame from each of the
edited video frames. The texts used are: (a) $T_{Global}$: “Swan with cactus
skin,” $T_{Local}$: “Cactus skin,” (b) $T_{Global}$: “Swan with cactus skin,”
$T_{Local}$: “Rough cactus skin,” (c) $T_{Global}$: “Swan made out of cactus,”
$T_{Local}$: “Catus skin.” (a) and (b) have the same global target text while
(a) and (c) have the same local target texts. In (b) the local details have
changed to a more rough texture. In (c) the global swan semantics are better
preserved.
### 4.3 Text augmentation
We add textual augmentation to our method to address some of the challenges
with prompt engineering. Inspired by Zhou et al. [32], we add neutral prefixes
to both the local and global target texts, e.g., “a photo of a {}.” We then
sample a new prefix each iteration for each of the target texts as a form of
regularization. Fig. 6 illustrates our text augmentation experiment. We
demonstrate that using prefixes increases the robustness and quality of the
results. In each experiment and each iteration, we sample one prefix among a
set of prefixes (details in appendix Sec. 6.3) for both the global and local
target texts. In this experiment, we vary the number of prefixes to sample
from and show that using an equal amount of prefixes for both the local and
global target texts produces better quality results than using no prefixes. In
both the swan and dog examples for columns (e) and (d) that use multiple
prefixes, we see that the stylizations are more detailed than the examples in
columns (a-c), e.g., in the dog example (d) and (e) the tiger fur has white
pigments around the belly area and also more natural tiger stripes.
### 4.4 Ablation study
We validate each term in our objective function (Eq. 6) with an ablation study
illustrated in Fig. 7. In (a), we see that all losses combined result in a
natural stylization that shows clear characteristics of a swan with crocodile
skin. In (b) we see that without the local loss the results lack fine-grain
details. It is still evident what the intended stylization was but the
crocodile texture is not as clear as in (a). The results in (c) show that
without the global loss the global semantics are less clear, e.g., the swan’s
neck is not stylized as detailed as the rest of the body. In (d), we see that
without the temporal loss, we get more edits outside the mask of the object.
In (e), we see that without the sparsity loss, the stylization is very noisy.
| |
---|---|---
(a) | (b) | (c)
Figure 9: Limitations. A representative frame from each of the edited video
frames. The texts used are: (a) $T_{Global}$: “Swan with cactus skin,”
$T_{Local}$: “Cactus skin,” (b) $T_{Global}$: “Boat made out of chocolate,”
$T_{Local}$: “Chocolate texture,” (c) $T_{Global}$: “Dog with zebra fur,”
$T_{Local}$: “Zebra fur.” In (a), the cactus texture is applied like photos of
cactus. In (b) the chocolate boat has a sailing ship printed at the end of the
boat. In (c), the dog has a face of a dog on its haunches.
In Fig 8 we illustrate how the local loss affects the fine-grained details and
the global loss affects the global semantics. We use (a) as a baseline and
then vary the local target texts in (b) and the global target texts in (c). In
(b) we see, how changing the local target text to include “rough” affects the
details of the cactus texture. In (c) we see how the global semantics of the
swan’s body become more realistic as an effect of changing the global target
text.
### 4.5 Limitations
Fig. 9 illustrates some of the limitations of our method. During training our
method occasionally starts to overfit and produce unintended stylizations. In
(a), we see that the body of the swan contains photo-like cacti instead of
natural cactus texture. In (b) and (c), we see how our model has used the
contexts of the global target text. In (b), a sailing ship has been added to
the end of the boat and in (c), a face of a dog has been added.
Our method is limited to text-prompts that do not entail features that cross
between the decomposed atlas layers. Some changes are best realized through a
shape change, e.g, “Swan with long hair.”, which is not currently possible in
our framework. Other limitations of our method include stylizations that focus
too much on some property of the target texts, e.g., we experienced that the
global target texts and similar variants “Swan with strawberry texture skin,”
gave noisy results, where the swan was colored red.
## 5 Conclusion
We considered the problem of developing intuitive and semantic control for
consistent editing and styling of objects in videos. This problem poses a
challenge in generating consistent content and style changes over time while
being able to produce fine-grained details and preserve the global semantics.
We proposed a method that uses CLIP [23] and the video object representation
[14] to stylize objects in videos by using both a global target text, to
control the global semantics of the stylization, and a local target text, to
control the fine-grained details. We demonstrated that the specificity and the
prefixes of the target texts can have a significant impact on the details
produced by our method’s stylization. In future work, it would be interesting
to investigate the limitations of CLIP. A model that is able to generate fine-
grained stylizations of videos and images could be leveraged to create data
augmentations in other learning settings. Another line of future work is to
extend our model to be able to generate shape changes or even new objects from
scratch.
### Acknowledgement
This research was supported by the Pioneer Centre for AI, DNRF grant number
P1. We would like to thank Ira Assent for the helpful discussions.
## References
* [1] Avrahami, O., Lischinski, D., Fried, O.: Blended diffusion for text-driven editing of natural images. CoRR abs/2111.14818 (2021), https://arxiv.org/abs/2111.14818
* [2] Bar-Tal, O., Ofri-Amar, D., Fridman, R., Kasten, Y., Dekel, T.: Text2live: Text-driven layered image and video editing (2022). https://doi.org/10.48550/ARXIV.2204.02491, https://arxiv.org/abs/2204.02491
* [3] Bau, D., Andonian, A., Cui, A., Park, Y., Jahanian, A., Oliva, A., Torralba, A.: Paint by word. CoRR abs/2103.10951 (2021), https://arxiv.org/abs/2103.10951
* [4] Brock, A., Donahue, J., Simonyan, K.: Large scale GAN training for high fidelity natural image synthesis. CoRR abs/1809.11096 (2018), http://arxiv.org/abs/1809.11096
* [5] Chefer, H., Benaim, S., Paiss, R., Wolf, L.: Image-based clip-guided essence transfer. CoRR abs/2110.12427 (2021), https://arxiv.org/abs/2110.12427
* [6] Crowson, K., Biderman, S., Kornis, D., Stander, D., Hallahan, E., Castricato, L., Raff, E.: Vqgan-clip: Open domain image generation and editing with natural language guidance (2022). https://doi.org/10.48550/ARXIV.2204.08583, https://arxiv.org/abs/2204.08583
* [7] Frans, K., Soros, L.B., Witkowski, O.: Clipdraw: Exploring text-to-drawing synthesis through language-image encoders (2021). https://doi.org/10.48550/ARXIV.2106.14843, https://arxiv.org/abs/2106.14843
* [8] Gal, R., Patashnik, O., Maron, H., Chechik, G., Cohen-Or, D.: Stylegan-nada: Clip-guided domain adaptation of image generators. CoRR abs/2108.00946 (2021), https://arxiv.org/abs/2108.00946
* [9] Jabri, A., Owens, A., Efros, A.: Space-time correspondence as a contrastive random walk. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H. (eds.) Advances in Neural Information Processing Systems. vol. 33, pp. 19545–19560. Curran Associates, Inc. (2020), https://proceedings.neurips.cc/paper/2020/file/e2ef524fbf3d9fe611d5a8e90fefdc9c-Paper.pdf
* [10] Jampani, V., Gadde, R., Gehler, P.V.: Video propagation networks. CoRR abs/1612.05478 (2016), http://arxiv.org/abs/1612.05478
* [11] Jetchev, N.: Clipmatrix: Text-controlled creation of 3d textured meshes. CoRR abs/2109.12922 (2021), https://arxiv.org/abs/2109.12922
* [12] Karras, T., Laine, S., Aila, T.: A style-based generator architecture for generative adversarial networks. CoRR abs/1812.04948 (2018), http://arxiv.org/abs/1812.04948
* [13] Karras, T., Laine, S., Aittala, M., Hellsten, J., Lehtinen, J., Aila, T.: Analyzing and improving the image quality of stylegan. CoRR abs/1912.04958 (2019), http://arxiv.org/abs/1912.04958
* [14] Kasten, Y., Ofri, D., Wang, O., Dekel, T.: Layered neural atlases for consistent video editing. CoRR abs/2109.11418 (2021), https://arxiv.org/abs/2109.11418
* [15] Kim, G., Ye, J.C.: Diffusionclip: Text-guided image manipulation using diffusion models. CoRR abs/2110.02711 (2021), https://arxiv.org/abs/2110.02711
* [16] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: Bengio, Y., LeCun, Y. (eds.) 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings (2015), http://arxiv.org/abs/1412.6980
* [17] Kwon, G., Ye, J.C.: Clipstyler: Image style transfer with a single text condition. CoRR abs/2112.00374 (2021), https://arxiv.org/abs/2112.00374
* [18] Michel, O., Bar-On, R., Liu, R., Benaim, S., Hanocka, R.: Text2mesh: Text-driven neural stylization for meshes. CoRR abs/2112.03221 (2021), https://arxiv.org/abs/2112.03221
* [19] Nichol, A., Dhariwal, P., Ramesh, A., Shyam, P., Mishkin, P., McGrew, B., Sutskever, I., Chen, M.: GLIDE: towards photorealistic image generation and editing with text-guided diffusion models. CoRR abs/2112.10741 (2021), https://arxiv.org/abs/2112.10741
* [20] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: Pytorch: An imperative style, high-performance deep learning library. In: Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran Associates, Inc. (2019), http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
* [21] Patashnik, O., Wu, Z., Shechtman, E., Cohen-Or, D., Lischinski, D.: Styleclip: Text-driven manipulation of stylegan imagery. CoRR abs/2103.17249 (2021), https://arxiv.org/abs/2103.17249
* [22] Pont-Tuset, J., Perazzi, F., Caelles, S., Arbelaez, P., Sorkine-Hornung, A., Gool, L.V.: The 2017 DAVIS challenge on video object segmentation. CoRR abs/1704.00675 (2017), http://arxiv.org/abs/1704.00675
* [23] Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., Krueger, G., Sutskever, I.: Learning transferable visual models from natural language supervision. CoRR abs/2103.00020 (2021), https://arxiv.org/abs/2103.00020
* [24] Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., Chen, M.: Hierarchical text-conditional image generation with clip latents (2022). https://doi.org/10.48550/ARXIV.2204.06125, https://arxiv.org/abs/2204.06125
* [25] Ramesh, A., Pavlov, M., Goh, G., Gray, S., Voss, C., Radford, A., Chen, M., Sutskever, I.: Zero-shot text-to-image generation. CoRR abs/2102.12092 (2021), https://arxiv.org/abs/2102.12092
* [26] Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E., Ghasemipour, S.K.S., Ayan, B.K., Mahdavi, S.S., Lopes, R.G., Salimans, T., Ho, J., Fleet, D.J., Norouzi, M.: Photorealistic text-to-image diffusion models with deep language understanding (2022). https://doi.org/10.48550/ARXIV.2205.11487, https://arxiv.org/abs/2205.11487
* [27] Sanghi, A., Chu, H., Lambourne, J.G., Wang, Y., Cheng, C.Y., Fumero, M., Malekshan, K.R.: Clip-forge: Towards zero-shot text-to-shape generation (2021). https://doi.org/10.48550/ARXIV.2110.02624, https://arxiv.org/abs/2110.02624
* [28] Texler, O., Futschik, D., Kucera, M., Jamriska, O., Sochorová, S., Chai, M., Tulyakov, S., Sýkora, D.: Interactive video stylization using few-shot patch-based training. CoRR abs/2004.14489 (2020), https://arxiv.org/abs/2004.14489
* [29] Wang, X., Jabri, A., Efros, A.A.: Learning correspondence from the cycle-consistency of time. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2019)
* [30] Ye, V., Li, Z., Tucker, R., Kanazawa, A., Snavely, N.: Deformable sprites for unsupervised video decomposition (2022). https://doi.org/10.48550/ARXIV.2204.07151, https://arxiv.org/abs/2204.07151
* [31] Yin, K., Gao, J., Shugrina, M., Khamis, S., Fidler, S.: 3dstylenet: Creating 3d shapes with geometric and texture style variations. CoRR abs/2108.12958 (2021), https://arxiv.org/abs/2108.12958
* [32] Zhou, K., Yang, J., Loy, C.C., Liu, Z.: Learning to prompt for vision-language models. CoRR abs/2109.01134 (2021), https://arxiv.org/abs/2109.01134
## 6 Appendix
### 6.1 Training details
We train on a single GPU (RTX $6000$). Our method is implemented using the
PyTorch framework [20] and will be made available. We use an ADAM optimizer
[16] with an initial learning rate of $1^{-4}$ and decay the learning rate by
a factor of $0.9$ every $200$ iterations. Our method takes about $40$ minutes
and $2000$ iterations with a batch size of three input frames, each of size
$432$x$768$, sampled from a video of maximally $70$ frames. High-quality
results usually appear after $10-20$ minutes.
All images are normalized and resized to $224$x$224$ before being passed to
the CLIP model. We normalization with mean $(0.48145466,0.4578275,0.40821073)$
and standard deviation $(0.26862954,0.26130258,0.27577711)$, which is the same
as CLIP was trained with. We use the pretrained “ViT-L/14” CLIP model loaded
from OpenAi’s GitHub page.
### 6.2 Image augmentation
* •
Random Crops: For the global views we crop the reconstructed frame with a
random scaling in the range: $[0.9,1.0]$ and for the local views we use a
scaling in the range $[0.1,0.5]$. For both global and local views we use a
random aspect ratio in the range $[0.8,1.2]$. The local crops have a chance of
not including the object we want to edit, e.g. a small crop from the bounding
box containing the Swan in Fig. 4 has a chance of only containing the
background. To combat this, we use the $\alpha$-map to locate the object, and
sample local crops until we get a crop that contains at least $\frac{1}{3}$ of
the object.
* •
Random Perspective Transformation: With probability $\frac{1}{2}$, we sample a
number $d$ uniformly at random in the range: $[0.1,0.5]$ and use $d$ as the
distortion scale.
* •
Random Background Removal: We locate the foreground object using the
$\alpha$-map. Then, with probability $\frac{1}{2}$ we color all background
pixels black.
### 6.3 Text augmentations
In our text augmentation we randomly sample one of the following prefixes in
each iteration:
1. 1.
“a photo of a {}”
2. 2.
“a {}”
3. 3.
“an image of a {}”
4. 4.
“the {}”
5. 5.
“image of a {}”
6. 6.
“image of the {}”
7. 7.
“photo of a {}”
8. 8.
“photo of the {}”
All prefixes are neutral and intended to not change the semantics of the
stylization. In the text augmentation experiments (Sec. 4.3), if we use four
prefixes, these are the first four in the list above.
|
11institutetext: IIIT Hyderabad, India22institutetext: 22email:
<EMAIL_ADDRESS>33email:
<EMAIL_ADDRESS>
# A framework for syntactic and semantic quality evaluation of ontologies
Vivek Iyer 1122 0000-0003-4451-8293 Lalit Mohan Sanagavarapu 1122
0000-0003-0745-1042 Y Raghu Reddy 1133 0000-0003-2280-5400
###### Abstract
The increasing focus on Web 3.0 is leading to automated creation and
enrichment of ontologies and other linked datasets. Alongside automation,
quality evaluation of enriched ontologies can impact software reliability and
reuse. Current quality evaluation approaches oftentimes seek to evaluate
ontologies in either syntactic (degree of following ontology development
guidelines) or semantic (degree of semantic validity of enriched
concepts/relations) aspects. This paper proposes an ontology quality
evaluation framework consisting of: (a) SynEvaluator and (b) SemValidator for
evaluating syntactic and semantic aspects of ontologies respectively.
SynEvaluator allows dynamic task-specific creation and updation of syntactic
rules at run-time without any need for programming. SemValidator uses Twitter-
based expertise of validators for semantic evaluation. The efficacy and
validity of the framework is shown empirically on multiple ontologies.
###### Keywords:
Ontology Quality Evaluation Syntactic Evaluation Semantic Validation
Crowdsourcing Twitter-based expertise
## 1 Introduction
The exponential increase in Internet users over the past decade has led to
generation of large volume of data. Web 3.0, otherwise commonly referred to as
Semantic web, seeks to represent internet data as knowledge through knowledge
graphs, ontologies and other knowledge systems [4]. These representations
enable knowledge integration, semantic ambiguity resolution, information
extraction, decision making, reasoning and many other use cases relevant to
the building of ‘intelligent’ software systems. Ontologies, in particular,
store domain-specific knowledge, and represent this knowledge through
concepts, relations, axioms and instances. They contain a formal structure and
achieve a certain level of rigor due to the presence of rules and constraints.
Ontologies are rarely static in nature. The range and the depth of the
knowledge stored are enriched over time. This impacts a wide variety of
software applications that utilize ontologies for reasoning, decision-making,
question-answering, etc. Ontology enrichment is thus a crucial step in the
ontology engineering process.
Traditionally, ontologies are created and managed by knowledge engineers and
domain experts resulting in high costs due to the expert human labour
involved. Automated or semi-automated approaches to ontology enrichment are
increasingly popular, driven by increased availability of domain-relevant
internet data and improvements in natural language processing and machine
learning models [36]. Research on ontology learning (both creation and
enrichment) snowballed in the last two decades [20], [36], [15], with
increased focus on fully automated Deep Learning based approaches [30]. Given
the variety in ontology enrichment approaches, it is important to evaluate the
quality of the enriched ontology.
Figure 1: Syntactic quality violation: No domain or range for property Figure
2: Semantic violation: Invalid enriched concept
Ontology evaluation approaches can broadly be divided into: manual, automated
and semi-automated approaches. In general, ontology evaluation happens on one
of two aspects: syntactic quality, or semantic quality. We define syntactic
quality of an ontology as a measure of its adherence to ontology development
guidelines or rules. For example, One such rule could necessitate presence of
both domain and range elements in properties. Examples of other rules or
guidelines could include explicit declaration of equivalent and inverse
properties, presence of annotations [34], following of unique naming
conventions [23], etc. Figure 1 shows, in Turtle syntax [3], a property
without a defined range element - thus violating the rule that necessitates
the presence of both domain and range elements. Semantic quality deals with
validity of enriched concepts, relations and instances. Figure 2 shows an
example ontology enriched with concepts extracted from a sentence to emphasize
the need for evaluation of enriched ontologies. The previously existing
concepts are shown in blue, the valid enriched concepts in green and the
invalid enriched ones in red. Ontology enrichment algorithms using Hearst
patterns [14] could mistakenly detect ‘antivirus’ as a type of ‘malware’. In
such cases, before creating a final ontology, the enriched ontology needs to
be validated for semantic quality, to accept or reject the enriched concepts,
properties and instances.
A variety of metric-based methods were proposed to evaluate various syntactic
quality-based characteristics of ontologies [9], [19], [32]. Publicly
available tools were proposed that allow users to evaluate syntactical quality
of ontologies using pre-defined metrics [18], [22] or rules [28], [31].
However, they offer limited customization and flexibility to the user for
creating task-specific rules for evaluation, even more so for non-programmers.
In regards to semantic evaluation, researchers have traditionally employed
domain experts [21], while in this decade, crowdsourced validators [17], [24]
are being used for semantic ontology validation. This paper proposes a
customizable and scalable framework that evaluates syntactic and semantic
aspects of ontology quality using SynEvaluator and SemValidator respectively.
* •
$SynEvaluator$: a tool that uses a rule-creation framework for allowing users
to non-programatically create rules during run-time and to set task-specific
priorities for these rules.
* •
$SemValidator$: a tool that uses crowdsourcing for validation of semantic
quality of enriched ontologies. In this paper, a Twitter-based expertise
estimation algorithm is used to weight validators’ decisions.
The source code for SynEvaluator and SemValidator is available on
GitHub111https://github.com/Remorax/SynEvaluator/,222https://github.com/Remorax/SemValidator/.
They are also deployed on Heroku as web-
applications333https://synevaluator.herokuapp.com/,444http://semvalidator.herokuapp.com/.
The rest of the paper is structured as follows: Section 2 describes related
work in ontology quality evaluation. The proposed framework constituting of
SynEvaluator and SemValidator is shown in section 3. Section 4 details the
experiments done to show the efficacy and accuracy of the framework, through
these tools. SynEvaluator is tested for its utility and accuracy for
implementing syntactical quality evaluation rules by comparing it against a
popular syntactic quality evaluation tool, OOPS! [27]. The efficacy of
$SemValidator$ is shown by conducting crowdsourced survey involving 28
validators on two popular ontologies, Stanford Pizza [8] and Information
Security ontology [10]. Accuracy of TweetExpert algorithm on responses to both
of these ontologies using multiple ML regression algorithms is shown. Finally,
section 5 summarizes the contributions and suggests possible future directions
of research.
## 2 Related Work
Ontology quality evaluation approaches can be broadly classified into a)
syntactic and b) semantic quality evaluation approaches. Syntactic evaluation
approaches primarily evaluate structural aspects of an ontology based on
ontology development guidelines, common pitfalls, structural metrics etc.
OntoClean [13], one of the earliest known works in this area, proposed a
methodology for validating adequacy of relationships in an ontology based on
notions drawn from philosophy such as essence, identity, and unity. Similarly,
OntoQA [32] proposed ontology evaluation on the basis of schema metrics and
instance metrics. They stated that ‘goodness’ or ‘validity’ of an ontology
varies between different users and domains. Gangemi et al. [11] proposed
structural, functional and usability-related measures using O2 and oQual, a
meta-ontology and an ontology model for ontology evaluation and validation.
Burton et al. [5] proposed an ontology evaluation model based on semiotic
theory. In order to apply the metrics proposed in these works, tools such as
S-OntoEval [7] drawn from semiotic theory and AktiveRank [1] that ranked
ontologies based on structural metrics like class match measure, density
measure etc. were proposed. However, the tools proposed in these articles are
either closed-source prototypes or theoretical frameworks and are not publicly
available.
There are also a few publicly available closed-source ontology (syntactic)
evaluation tools, such as OOPS! [28], DoORS [22] and OntoMetrics [18].
OntOlogy Pitfall Scanner (OOPS!) evaluates ontologies on the basis of
established ontology quality rules related to human understanding, logical
consistency, real world representation and modelling issues, and manually
assigned priorities. DoORS, evaluates ontologies based on metrics drawn from
Semiotic Theory while OntoMetrics uses metrics proposed in OntoQA, [11]. These
tools, however are not flexible or customizable, and evaluate ontologies using
a fixed set of rules or metrics. Users are unable to create/update customized
rules and set task-specific priorities, which can be crucial as requirements
and application scenarios vary. The available open source implementations [33]
require creation of new rules and priorities via programming which can be
daunting for non-programmers.
Semantic evaluation approaches focus on semantic validity of concepts and
relationships in an ontology. Traditionally, it has been formulated as a task
requiring simple accept/reject decisions from domain experts [2]. In the past
few years, a good number of crowdsourced ontology evaluation approaches have
been proposed. Hanika et al. [35] have developed the UComp Protégé plugin to
provide a platform for crowdsourced workers to validate classes, subclasses,
properties and instances. Kiptoo et al. [16] use crowdsourcing for axiom and
assertion verification in ontologies as well as for verification of subclass-
superclass relations by Amazon Mechanical Turks [24]. Pittet et al. used
crowdsourced workers to propose changes related to addition, deletion and
substitution errors [25]. Zhang et al. [37] used crowdsourced workers to
obtain written feedback (comments/suggestions/references) for making final
validation decisions. Requiring complex tasks (such as making data quality
decisions or requiring written feedback) from crowdsourced workers can be
expensive and unscalable as the size of ontology and/or number of workers
increases. Some approaches [35], [17], [25] used quality control mechanisms
like majority voting that are debatable. Noy et al. [24] addressed this by
using qualification questions and spam filtering techniques. While these
mechanisms eliminate spammers, it may not be applicable where large number of
workers have some degree of knowledge but only few are experts. Therefore, an
assessment of domain expertise on a continuous scale would be useful as a
quality control metric. Further, an integrated quality evaluation framework
that seeks to evaluate ontologies on both syntactic and semantic aspects, and
also addresses the above-mentioned problems would be useful for a holistic and
integrated approach to quality evaluation of enriched ontologies.
## 3 Proposed Framework
In this paper, an ontology evaluation framework that combines automated
syntactic evaluation and human-centric semantic evaluation is proposed.
SynEvaluator aims at increasing user flexibility by allowing customized rule
creation at runtime, as well as scalability (with respect to user base) by
proposing an approach that removes the need for programming. SemValidator
proposes a crowdsourcing based approach that uses the validators’ Twitter
profiles for quality control.
Figure 3: Work flow for semantic and syntactic quality evaluation of
ontologies.
The framework’s work flow is shown in Figure 3. The input to quality
evaluation process is an enriched ontology. The ontology may have been
enriched with concepts, relations and instances using some automated and semi-
automated algorithms. The ontology may contain syntactic quality errors (due
to violation of ontology development guidelines) and/or semantic errors (due
to wrongly enriched concepts). In the figure, for clarity, concepts and
relationships with potential syntactic violations are bordered in yellow, an
those that do not contain such violations are outlined in blue. Also, concepts
that potentially contain semantic violations are highlighted in yellow colour
and concepts that are semantically valid are highlighted in green.
An ontology engineer uploads an enriched ontology to SynEvaluator, and creates
syntactic quality evaluation rules. The rules created using a theoretical rule
creation framework are applied on the parsed ontology object through
SynEvaluator’s ontology evaluation module. This returns a list of detected
violations and the elements causing these violations. Using these elements as
suggestions, the ontology engineer can fix violations in a iterative manner.
The iterations may be repeated as needed. Then, the ontology engineer can
provide this ontology as input to SemValidator so that it can be validated for
semantic quality. As part of semantic validation, the ontology is provided to
crowdsourced validators who give their accept or reject validation decisions
for each of the enriched concepts, relations and instances. Simultaneously, an
estimate of the domain knowledge of each of these validators is calculated
from their Twitter profile using the TweetExpert algorithm. These scores
alluded to as ‘TweetExpert scores’, are used as a quality control mechanism to
ensure that the decisions of crowdsourced validators are given weightages
according to their knowledge of the ontology domain. Finally, the output of
this algorithm is used to take the final accept/reject decision for each
enriched element, resulting in an ontology with both good syntactic and
semantic quality.
### 3.1 Stage 1: SynEvaluator
In this section, the underlying terminology used in SynEvaluator and rule
creation framework is illustrated with examples. Further, the implementation
of SynEvaluator as a web application is detailed. Finally, the section ends
with a discussion on potential benefits and limitations of SynEvaluator.
#### 3.1.1 Defining the Rule Creation Framework
SynEvaluator allows users to create rules at run-time. The rules are
constructed from individual components like “Subjects", “Clauses" or “Operator
Expressions". The operational definitions of these and other relevant terms
are :
* •
Ontological Element: Refers to any element that forms a constituent part of an
ontology. It refers to any primary component (classes, individuals,
properties), their related elements (subclasses, domains, ranges), annotations
(labels, descriptions, comments) or attributes (ID, language, namespace).
* •
Rule: Refers to a sequence of one or more clauses, optionally connected by one
or more operator expressions that returns either one or more ontological
elements or a boolean value.
* •
Clause: Refers to a transformation applied on an ontological element(s) to
return either one or more ontological elements or a boolean value.
* •
Operator Expression: An expression used to compare and/or connect non-empty
sequences of clauses to return a boolean value.
* •
Subject: Refers to an ontological element (typically primary components such
as classes, individuals and properties), that is subjected to transformations
carried out through sequential clauses to form a rule.
Figure 4: Structure of supported expressions
Using these concepts, the expressions supported by SynEvaluator are formally
defined as shown in Figure 4. The notation used is similar to $RegEx$ notation
with $*$ denoting zero to infinity, $+$ denoting one to infinity, and $?$
denoting zero or one occurrences. A ‘Subject’ comprises the beginning and is
always the first keyword in a rule in out proposed framework. It goes through
a series of transformations as defined by sequences of clauses. Clauses can be
of of two types: a) Extractive Clauses and b) Functional Clauses. Extractive
Clauses consist of (Predicate, Object) pairs that use the Predicate to execute
a transformation on the return value from the previous clause using the Object
as argument. More specifically, object specifies the type of element
‘extracted’ by the predicate, and elements satisfying this (Predicate, Object)
pair are returned as output. Functional clauses, on the other hand, consist of
(Predicate, Function) pairs that involve executing a function of type
described by predicate on the return value from the previous clause. These
clauses typically check for existence of a certain functional property and
thus return a boolean value in response.
Currently, two kinds of functional properties are supported: i) ontological
(or structural) properties, that execute ontology-level functions (such as
uniqueness, validity, consistency etc) on ontological elements, and ii)
linguistic properties that linguistically analyze text (such as checking for
polysemes, conjunctions etc) returned from previous clauses. In case of
‘False’ value returned by a Functional Clause or an empty list (no matching
elements) returned by an Extractive Clause, that particular ontological
element is returned as an element containing a violation.
Extractive Clauses can also be chained together to form clause sequences.
Since functional clauses return a boolean value, they cannot be chained
further. Clause sequences can be compared through Operator Expressions.
Operator Expressions essentially consist of a ‘Predicate’ indicating operator
type, followed by an ‘Operator’ keyword. Operator Expressions comprise of two
main categories of operators, namely: (a) Logical Operators and (b)
Comparative Operators. Logical Operators like ‘And’, ‘Or’ and ‘Not’ are used
to create logical combinations of clause sequences. Comparative Operators like
‘Equality’, ‘Inverse’ and ‘Synonymy’ are used to compare return values.
Table 1: Keywords for different expression types in the proposed framework Subject | Extractive Clause | Functional Clause | Operator Expression
---|---|---|---
Predicate | Predicate | Predicate
Ontology Metadata | Has Related Element (1) | Has Ontological Property (1) | Uses Comparative Operator (1)
Ontological Element | Has Attribute (2) | Has Linguistic Property (2) | Uses Logical Operator (2)
Class | Object | Function | Operator
Instance | Domain (1) | ID Consistency (1) | Equality (1)
Property | Subclass (1) | Uniqueness (1) | Inverse (1)
Object Property | Disjoint Class (1) | Text Validity (1) | And (2)
Datatype Property | ID (2) | Contains Polysemes (2) | Or (2)
Symmetric Property | Language (2) | Contains Conjunctions (2) | Not (2)
Table 1 summarizes keywords supported by the proposed framework. Note that the
table lists 8 Subject keywords, while for Extractive Clauses, Functional
clauses and operator expressions it lists 2 predicates and 5 objects,
functions and operators respectively. This is due to differing syntax followed
by each expression type. Also, every predicate has a list of valid
Objects/Functions/Operators. This is shown in the table through bracketed
numbering. For example, the valid Predicates for ‘Has Attribute‘ are ‘ID‘ and
‘Language‘. The complete list of supported keywords is provided over
here555https://bit.ly/3zfdI8f.
#### 3.1.2 Examples of Created Rules
Figure 5 shows examples of 2 rules. In both examples 1 and 2, ‘Property’ is
the ‘Subject’ of the rule. ‘hasRelatedElement Domain’, ‘hasRelatedElement
Range’ and ‘hasOntologicalProperty Uniqueness’ are all clauses. In rule 1, the
‘hasRelatedElement Domain’ clause carries out a transformation that uses the
‘hasRelatedElement’ predicate to extract ‘Domain’ elements. A similar clause
is used for extracting ‘Range’ elements. These two clauses (or clause
sequences of length are combined using the operator expression
‘usesLogicalOperator And’. This rule thus necessitates non-null values for
both Domain and Range elements for each element of ‘Subject’, in this case,
‘Property’. Properties that do not contain both elements are therefore
returned as ontological elements containing violations.
Figure 5: Examples of rules implemented by SynEvaluator
Example 2 shows a rule where clauses have been chained together to constitute
a clause sequence of length 2. This sequence consists of the extractive clause
‘hasRelatedElement Domain’ followed by functional clause
‘hasOntologicalProperty Uniqueness’. The extractive clause essentially
extracts Domain element(s) of each Property. Then, the functional clause
applies ‘Uniqueness’ function on ontological elements returned by previous
clause with function type defined by Predicate “hasOntologicalProperty". In
case of non-existence of domain for a particular property or existence of
multiple domains, this rule would return that property as containing a
violation due to violation in first and second clauses respectively.
#### 3.1.3 Proposing SynEvaluator: the final web application
Figure 6: The SynEvaluator interface. Subjects are highlighted in blue,
Clauses in yellow, Operator Expressions in green and Rule Priorities in red.
Among Clauses, Extractive Clauses are shown in solid lines while Functional
Clauses are outlined with dotted lines.
The rule creation framework proposed above is used to create a web application
for use by the ontology engineer. The primary interface of this application,
SynEvaluator, is shown in Figure 6. It allows the users (ontology engineers)
to use dropdown menus to create functional and extractive clauses, operator
expressions and thus, rules, as well as set priorities for these rules. Users
can choose between ‘Low’, ‘Medium’ and ‘High’ priorities based on the task-
specific importance of the rule. Figure 8 shows, with the help of an activity
diagram, how a user could create an appropriate rule using SynEvaluator.
Finally, after uploading the enriched ontology and creating the rules,
SynEvaluator parses the ontology using its parsing module and executes the
created rules (Figure 3). Post evaluation, the user is presented with a list
of violating elements along with count and priority of each valid rule.
#### 3.1.4 Benefits and Limitations
The proposed framework makes it significantly easier for non-programmers to
create customised rules dynamically. Also, compared to previous quality
evaluation tools, due to the framework’s ability to reuse keywords to create
new rules, the developer effort required to hard-code rules is minimised.
Another major benefit is that the proposed framework can potentially be used
to query over entire OWL language. This can be done as any ontological
element/attribute can be extracted using extractive clauses and the
appropriate function executed on them. Lastly, due to functional clauses, it
is possible to execute ontological or ontology-level functions like normal
query languages and use linguistic analysis on text. This is particularly
useful in quality evaluation while applying appropriateness checks on IDs or
labels.
Figure 7: More examples of rules supported by SynEvaluator
In Figure 5 (Example 1), it can be seen that properties with missing domain or
range, a common quality violation, could be detected through a combination of
logical operators and extractive clauses. Another common violation is related
to the absence of annotations (comments, descriptions, etc.) for an
ontological element [34]. This can be converted into a rule, once again
through Extractive Clauses as shown in Figure 7 (Example 1). A third quality
violation would be when similar (or synonymous) classes are incorrectly
defined as equivalent classes [23]. This can be defined as in Figure 7
(Example 2), where the comparative operator ‘Dissimilarity’ is used to test
for semantic similarity through cosine similarity of label embeddings. As
mentioned previously, it is also possible to perform linguistic tests unlike
other query languages. A basic example of this is detection of conjunctions in
a label as shown in Figure 7(Example 3). This is useful to identify quality
violations where different concepts are merged in the same class using
conjunctions [28]. A more advanced example of a linguistic test would be a
rule detecting polysemous elements (Figure 7, Example 4). This is useful in
detecting violating elements that have labels denoting differing conceptual
ideas in differing senses. SynEvaluator implements this check through the use
of WordNet’s synsets to find out how many senses a word can have. Once these
rules are created, the ontology engineer can add domain/range elements;
annotations; remove synonymous equivalent classes and fix classes with
conjunctions and polysemes as appropriate. SynEvaluator can thus help in
fixing structural and linguistic quality violations.
The current version of SynEvaluator has a few limitations. It is currently
only possible to chain clauses together or use operators to compare chained
clauses. As a result, it is not possible to create rules with multiple lines.
One major consequence of this is that variable assignment is not supported,
and it is not possible to create a variable in one line and refer to it in
another, as part of the same rule. Aggregation operators, such as ‘Count‘ or
‘Sum‘, are currently not supported either. Finally, it is not possible to
create rules that require reasoning. The described limitations shall be
addressed in future iterations of SynEvaluator. In spite of the limitations,
the current framework (as shown in Experiments section) is still able to
support creation of the majority of quality evaluation rules.
Figure 8: Activity Diagram for creating a rule in SynEvaluator
### 3.2 Stage 2: SemValidator
SemValidator uses a crowdsourced approach to semantically evaluate ontologies.
The key feature of SemValidator is that it does not require ontology
engineers, domain experts or knowledge of OWL language for validation. This is
useful in a crowdsourced setting, where validators may have varying degrees of
expertise and knowledge. If further necessitates the use of appropriate
quality control mechanisms. To ensure quality, SemValidator uses TweetExpert
algorithm to calculate expertise score of a crowdsourced validator, which is
then used to weigh their decisions. This section describes the approach used
by TweetExpert and justifies the choices made. This is then followed by a
discussion on the assumptions made by TweetExpert and the feasibility of the
assumptions in the context of crowdsourcing. The section finally ends with a
description of the implementation of SemValidator and how it can be used by
crowdsourced validators.
#### 3.2.1 TweetExpert algorithm
TweetExpert algorithm takes Twitter usernames of validators as input. For each
of the validators, it calculates two scores: a) a ‘TweetSim’ score and b) a
‘FriendSim’ score. A ‘TweetSim’ score is intended to assess the similarity of
the validator’s tweets to the domain of the ontology while ‘FriendSim’
estimates the domain similarity of the validator’s friends (pages they are
following). To calculate the ‘TweetSim’ score, the validator’s ‘n’ most recent
tweets are extracted from their profile and their semantic similarity with the
domain keyword is computed using the Universal Sentence Encoder (USE) [6].
Here the ‘domain keyword’ is manually chosen as the word most relevant to the
domain of the ontology. For instance, “Pizza" for Stanford Pizza ontology [8]
and “Information Security" for the ISO 27001 based “Information Security"
ontology [10]. The current implementation uses only one keyword, but it is
possible to compute similarity scores with multiple keywords and average these
scores for greater accuracy. The similarities are sorted in decreasing order.
Out of n similarities, the top-K similarities are chosen. These top-K
similarities are then averaged to yield ‘TweetSim’ score. A similar approach
is used to calculate ‘FriendSim’ score. The ‘m’ most recent friends are
extracted to calculate their ‘TweetSim’ scores. After sorting in decreasing
order, the top-K most similar scores are averaged to yield ‘FriendSim’ score.
The reason behind extracting the most recent tweets and friends is to get a
better estimate of the validator’s current knowledge and interests. On the
other hand, a top-K average helps in both filtering out occasional out-of-
domain tweets from domain experts and smoothening out the effects of
coincidental in-domain outliers from non-experts. The value of K is thus
appropriately empirically chosen such that it is large enough to not include
in-domain outliers from laymen, but small enough to exclude any out-of-domain
tweets from experts. The pseudocode for TweetExpert is shown in Algorithm 1.
Result: TweetExpert score
Function _calculate_tweet_similarity(_$username$_)_:
user_tweets := extract_tweets(username)
tweet_similarities := calculate_USE_similarity(user_tweets, domain_name)
best_tweet_similarities := get_top_K_tweets(tweet_similarities)
tweet_similarity_score = average(best_tweet_similarities)
return tweet_similarity_score
Function _calculate_friend_similarity(_$username$_)_:
user_friends := extract_friends(username)
user_friends_scores := foreach _ $friend\in user\\_friends$_ do
calculate_tweet_similarity(friend)
best_friend_scores := get_top_K’_friends(user_friends_scores)
friend_similarity_score = average(best_friend_scores)
return friend_similarity_score
tweet_sim := calculate_tweet_relevance(username)
friend_sim := calculate_friend_relevance(username)
score := ML_predict_score(tweet_sim, friend_sim)
Algorithm 1 TweetExpert algorithm
Finally, the calculated ‘TweetSim’ and ‘FriendSim’ scores are input to a pre-
trained Machine Learning regression algorithm that predicts the final
$TweetExpert$ score using the two scores as feature vectors. The current
implementation uses Epsilon-Support Vector Regression (SVR), since it was
experimentally found to yield best results (shown in Experiments section).
However, the system uses strategy software design pattern [29] which enables
easy interchange of regression algorithms. The TweetExpert score is calculated
for each of the validators by repeating the process described above. The final
decision is taken using a weighted majority voting algorithm with the
TweetExpert scores being used as weights. The TweetExpert scores provide a way
to estimate a confidence value for the decisions input by each validator. This
is particularly crucial as a quality control metric in non-probabilistic
sampling techniques like crowdsourcing where number of validators can grow
exponentially in count and diversity.
#### 3.2.2 Assumptions
SemValidator makes reasonably grounded assumptions to establish efficacy and
suitability in a crowdsourced setting. For example, about 49% of crowdsourced
workers whose primary source of income is Amazon Mechanical Turk, a popular
crowdsourcing platform, are in the 18-29 age
range666https://pewrsr.ch/3vX5q2G. As of February 2021, about 42% of Americans
in the 18-29 age range use Twitter, with this age group being the most active
demographic on Twitter777https://bit.ly/3z5IDn9. It is also assumed that in
order for expertise estimation to work, Twitter users tweet reasonably
frequently. The average number of tweets per day per user, according to a 2016
study888https://bit.ly/3fSjWmA, is 4.422, which translates to over 1600 tweets
a year. This is typically sufficient since $K\sim 50-100$. Finally,
SemValidator assumes workers have public Twitter profiles, which would enable
tweets and friends of a worker to be extracted. This assumption is predicated
upon a recent 2019 survey999https://bit.ly/3pmMc3O that showed 87% of Twitter
users in USA have public accounts. Also, as part of future work, SemValidator
would allow login through Facebook and Linkedin as well and a similar
algorithm for expertise estimation could be used. This is expected to further
increase the applicability of this expertise-based approach for crowdsourcing.
#### 3.2.3 Proposing SemValidator: the final validation workbench
Figure 9: The main validation interface of SemValidator. The Stanford Pizza
ontology has been uploaded on SemValidator, with its concepts in blue and the
enriched concepts in green. One of the enriched concepts, “Tandoori Pizza" has
been selected, with options to “Accept" and “Reject" it.
The proposed workflow is used to develop a validation workbench for
crowdsourced workers that allows for accepting or rejecting enriched concepts,
relations and instances in an enriched ontology. The main validation interface
of this workbench, called SemValidator, is shown in Figure 9. This application
integrates Twitter authentication and uses the TweetExpert algorithm for
calculating validator expertise. SemValidator allows for two types of users:
(a) the administrator and (b) the validator. The administrator, typically the
ontology engineer, can upload/delete ontologies to be validated and also
access decisions made by validators. When the validator selects an ontology to
validate, the ontology is served using the WebVOWL ontology visualization
software. The enriched concepts and instances are highlighted in green, and on
selection, enable the validators to select accept/reject decisions
accordingly. Enriched relations, also in green, may also be accepted/rejected
independent of the concepts they relate. The validators’ decisions are
recorded by SemValidator and logged to a database. The administrator can
download this database after the crowdsourcing survey is completed, evaluate
validator expertise using their Twitter usernames and then make final
accept/reject decisions.
The syntactic and semantic evaluation aspects of the framework may be used
independently of each other. However, utilizing the framework in an integrated
manner as shown in Figure 3 is expected to give best results. This way,
syntactical violations can be detected easily and in an automated and
customizable manner, while semantic violations can now be detected more
accurately by crowdsourced validators. The resulting ontology has enhanced
syntactic and semantic ontological quality and is now fit for reuse.
## 4 Experiments
Stanford Pizza and ISO-IEC 27001 Information Security ontologies are evaluated
using the proposed framework to demonstrate ontology quality evaluation. Since
the focus of this work is on enriched ontologies, these are manually enriched
with concepts, properties and instances before quality evaluation. The RDF
triples used to enrich Pizza and Information Security ontologies are provided
over here101010https://bit.ly/3z5lGR1. Only 5 triples were chosen for this
iteration, considering that ontology enrichment is as an iterative process and
the count of triples per iteration is expected to be of this order. For Pizza,
the domain-specific webpages used for extraction consisted of culinary
articles111111https://bit.ly/3yYGGZP and food travel
blogs121212https://bit.ly/3iiEtCm. For Information Security they consisted of
informative articles and product pages by Cisco131313https://bit.ly/3wXRHZj,
Barracuda141414https://bit.ly/34PIUN9 and
SearchSecurity151515https://bit.ly/3wXRJQV. Please note that some relevant
data in this section is shown through external links due to space constraints.
### 4.1 Syntactic Quality Evaluation using SynEvaluator
This section attempts to evaluate the applicability and accuracy of
SynEvaluator in creating rules and detecting violations respectively. It is
hypothesized that SynEvaluator’s reusable theoretical framework for priority-
specific rule creation increases its applicability for diverse range of tasks,
but without compromising on violation-detection accuracy. To prove these
claims, SynEvaluator is compared against OOPS! [27], a popular, publicly
available SOTA tool that allows qualitative rule evaluation using rules.
Moreover, OOPS! compiles 41 most commonly observed pitfalls drawn from several
popular works on ontology quality evaluation [12], [23], [34] and the ones
described in the OOPS! catalogue [26] are chosen as rules for implementation.
This allows appropriate assessment of applicability for rule creation, while
also allowing accuracy evaluation by comparing SynEvaluator’s detected
violations to that of OOPS! Note that while SynEvaluator’s suitability for
rule creation is being assessed here by comparing with rule-based evaluation
approaches, it cannot be compared yet with other contemporary works focusing
on metric-based evaluation. This is being planned as part of future work by
introducing metric-to-rule conversion, which would allow metrics to be framed
as rules. One possible way this could be done is by adding conditions, such as
comparison operators, after metrics to form rules.
Table 2: No of pitfalls implemented by SynEvaluator (SE) Vs OOPS! SE OOPS! | | Implemented
---
| Not implemented
---
| Implemented
---
29 | 4
| Not implemented
---
1 | 7
SynEvaluator’s framework can be used to successfully formulate 30 of the 41
pitfalls compiled in the catalogue, compared to OOPS! which can automate 33.
As shown in Table 2, the majority of pitfalls (29 of 41) can be successfully
implemented by both OOPS! and SynEvaluator. 7 cannot be implemented by either
of them, and this involves rules that require human knowledge and reasoning
(such as overspecialization of ontology hierarchy, usage of wrong relations,
etc.). 1 of the pitfalls (involving linguistic polysemy detection) could be
implemented by SynEvaluator but not by OOPS!. There are 4 pitfalls implemented
by OOPS! which SynEvaluator cannot check. This includes rules that check for
undeclared disjoint concepts, undeclared transitive properties, equivalent
classes, etc. Supporting such rules involves a higher degree of ontological
reasoning that SynEvaluator is incapable of. Nevertheless, SynEvaluator’s
ability to implement vast majority of quality evaluation rules off-the-shelf
in a customizable manner increases confidence in its applicability for future
evaluation tasks that may involve more rules.
For comparing accuracy of violation detection, SynEvaluator’s reported
violations are compared with OOPS! [27]. Given the absence of ground truth and
the widespread popularity of OOPS!, it is assumed that OOPS! uses valid, non-
erroneous rule checks for the implemented pitfalls. It can be observed that
the violations reported by SynEvaluator match those of OOPS! in all pitfalls
except for P22. OOPS! mentions that these could be due to inconsistent naming
conventions, however SynEvaluator is unable to detect any such errors in the
ontologies. This may be a result of the incorrect naming checks currently
used. But for the other pitfalls, both tools returning the same number of
violations suggests that for the implemented framework, SynEvaluator is
accurate in evaluating rules and detecting pitfalls.
### 4.2 Semantic Quality Validation using SemValidator
SemValidator uses a crowdsourced survey to semantically validate the enriched
Pizza and Information Security ontologies. Crowdsourced validators were
invited by tweeting a description of the task along with the link to the
SemValidator application. Among the 31 validators that participated in the
survey, 3 had private Twitter accounts so their responses were discarded. The
28 validators with public accounts included 5 validators with a background in
Cybersecurity, 4 in Artificial Intelligence, 5 in Software Engineering, 4 in
Healthcare, 3 in Food domain (such as chefs and culinary experts), 3 in
Literature, 2 in Politics and 1 each in Fashion and Pure Science. These
diverse bunch of validators were asked to self-identify their domain of
expertise for statistical purposes before participating in the survey. To
preserve validator anonymity, the anonymized results of the survey are
provided here161616https://bit.ly/3pvzbFh. After completion of the survey for
both ontologies, ‘TweetSim’ and ‘FriendSim’ scores were calculated for all 28
validators using the TweetExpert algorithm, described previously. TweetExpert
took 1̃5 seconds to execute for our values of $K=20$ and $K^{\prime}=5$ per
validator profile. Finally, TweetExpert score was calculated by experimenting
with 4 standard regression algorithms: Linear Regression, Random Forest
Regression, K-Nearest Neighbours (KNN) Regression and Epsilon-Support Vector
Regression (SVR). A standard majority voting where all decisions have equal
weightage is considered as a naive baseline.
The input features for each algorithm were ‘TweetSim’ and ‘FriendSim’ scores
of a validator and the label was the ratio of correct answers (to total
answers) given by them. This was done to ensure validators with more correct
answers received higher ‘TweetExpert’ scores using similarity scores as
feature vectors. Since a gold standard for the correct answers did not exist,
CISOs were asked to manually validate the enriched triples from the
information security domain, while the authors themselves validated the
triples from the pizza domain. To determine the best among the 4 regression
algorithms for score prediction, 7-fold cross validation was carried out. The
28 responses (now reduced to feature vectors and labels) available for both
ontologies were divided into 7 folds of 4 responses each. 5 folds were used
for training, 1 for validation and 1 for testing. For each chosen test fold,
immediately succeeding fold was chosen as the validation set and all the other
folds were used for training.
Different sets of hyperparameters were considered for each regression
algorithm171717https://bit.ly/3wUa9SA. The best set of hyperparameters for
each algorithm was determined by selecting the algorithm with highest accuracy
on the validation set, calculated by determining the proportion of correctly
answered questions to the total number of questions. The predicted answer to a
question is determined by using weighted majority voting algorithm on the
predicted expertise scores in the validation set. The higher the number of
correct answers, the closer the predicted scores are to their actual value.
The same procedure was followed for calculating the accuracy on the test
folds. Accuracy obtained on test and validation sets for each of these
algorithms is shown in Table 3 for both ontologies. Epsilon Support Vector
Regression was found to yield best results on both datasets.
To determine whether training score prediction algorithms requires feature
vectors from the same domain, an SVR model is trained on a more generic
dataset, i.e the Pizza dataset and tested on Information Security, a niche
dataset. The model hyperparameters are chosen as the ones that performed best
on the Information Security dataset earlier. Varying percentages of the Pizza
dataset are used to construct training subsets, to observe variation in test
accuracy with subset size. To ensure randomness while choosing these subsets,
10 experimental trials were conducted where the dataset is shuffled before
each trial and trained on the first $x\%$ of samples, where $x$ is the
percentage of the training dataset chosen. The results are shown in Table 4.
The accuracy monotonically increases with increase in size of training subset.
The results suggest that a model pre-trained to predict the expertise score on
one domain could be reused multiple times in different domains, even if they
are niche.
Table 3: Mean Validation and Test Accuracy scores on Pizza and Security datasets Algorithm | Pizza | Security
---|---|---
Val | Test | Val | Test
Majority Voting | 71.43 | 71.43 | 51.43 | 51.43
Linear Regression | 85.71 | 83.57 | 68.57 | 71.43
Random Forest | 80.0 | 81.43 | 68.57 | 74.28
KNN | 85.71 | 80.0 | 77.14 | 77.14
SVR | 88.57 | 85.71 | 80.0 | 80.0
Table 4: Mean accuracy scores on test dataset (Information Security) with variation in % of training dataset (Pizza). Dataset% | 10% | 25% | 50% | 75% | 100%
---|---|---|---|---|---
Accuracy | 70% | 76% | 86% | 92% | 100%
Table 3 shows that the best-performing algorithm (TweetExpert with SVR
followed by weighted majority voting) significantly outperforms naive majority
voting. It can be observed that naive majority voting gives particularly bad
results for Information Security, a relatively niche domain, than Pizza, a
domain known more to laymen as well. When replacing naive majority voting with
TweetExpert + SVR, a drastic increase in performance in Information Security
(55.55%) vs Pizza (20%) can be observed. Even the worst performing regression
algorithm gives an increase of 38.8% and 14% respectively. As a result, it can
be inferred that estimating expertise using TweetExpert can be particularly
useful for quality control in niche domains.
## 5 Conclusion and Future Work
Ontology evaluation is a critical stage of any ontology engineering process.
In this paper, SynEvaluator and SemValidator are proposed for syntactic and
semantic evaluation of an enriched ontology respectively. Although the work
focuses on an enriched ontology, it can be easily extended for evaluation of
any generic ontology.
SynEvaluator automatically evaluates ontologies using customised rules created
dynamically at runtime, and improves on previous quality evaluation approaches
in a variety of ways. Firstly, it offers greater flexibility to the user in
terms of creating, updating and deleting custom rules as well as setting
priorities. Secondly, it proposes rule creation by the usage of a novel
theoretical framework that factors rules into sequences of ‘clauses’ and
‘operator expressions’. This facilitates creation of an interactive interface
that makes it easier for non-programmers to dynamically create rules. In
addition, chaining together independent keywords can facilitate creation of a
large number of rules without requiring additional developer programming.
SemValidator improves on previously proposed crowdsourced ontology validation
approaches by incorporating a Twitter-based quality control mechanism. The
TweetExpert algorithm is proposed for calculating the expertise score of a
validator using the tweets and friends extracted from their Twitter profile as
input.
The efficacy of SynEvalautor was shown by implementing rules to detect
pitfalls and the accuracy of the detected violations was compared with
publicly available OOPS! tool. Semantic quality evaluation using SemValidator
is performed on both Pizza and Information security ontologies with the help
of a crowdsourced survey of 28 validators. The experimental results showed a
significantly higher than naive majority voting.
The experimental results are encouranging but also can be used as an aid to
further extend the research work. For example, SynEvaluator can be expanded to
support additional, more complex operations such as using parantheses,
arithmetic operators, variable assignment etc. Although the TweetExpert
algorithm used by SemValidator currently calculates expertise using only
tweets and friends as feature vectors, it could be extended to use additional
information such as Twitter lists, followers, tweet metadata, etc.
## References
* [1] Alani, H., Brewster, C., Shadbolt, N.: Ranking ontologies with AKTiveRank. In: International Semantic Web Conference. pp. 1–15. Springer (2006)
* [2] Amardeilh, F., Laublet, P., Minel, J.L.: Document Annotation and Ontology Population from Linguistic Extractions. In: Proceedings of the 3rd International Conference on Knowledge Capture. pp. 161–168 (2005)
* [3] Beckett, D., Berners-Lee, T., Prud’hommeaux, E., Carothers, G.: Rdf 1.1 turtle. World Wide Web Consortium pp. 18–31 (2014)
* [4] Berners-Lee, T., Hendler, J., Lassila, O.: The Semantic Web. Scientific American 284(5), 34–43 (2001)
* [5] Burton-Jones, A., Storey, V.C., Sugumaran, V., Ahluwalia, P.: A semiotic metrics suite for assessing the quality of ontologies. Data & Knowledge Engineering 55(1), 84–102 (2005)
* [6] Cer, D., Yang, Y., Kong, S.y., Hua, N., Limtiaco, N., John, R.S., Constant, N., Guajardo-Céspedes, M., Yuan, S., Tar, C., et al.: Universal Sentence Encoder. arXiv preprint arXiv:1803.11175 (2018)
* [7] Dividino, R.Q., Romanelli, M., Sonntag, D., et al.: Semiotic-based Ontology Evaluation Tool (S-OntoEval). In: LREC (2008)
* [8] Drummond, N.: Stanford pizza ontology. https://protege.stanford.edu/ontologies/pizza/pizza.owl, accessed: 2021-03-28
* [9] Duque-Ramos, A., Fernández-Breis, J.T., Iniesta, M., Dumontier, M., Aranguren, M.E., Schulz, S., Aussenac-Gilles, N., Stevens, R.: Evaluation of the OQuaRE Framework for Ontology Quality. In: Expert Systems with Applications. vol. 40, pp. 2696–2703. Elsevier (2013)
* [10] Fenz, S., Goluch, G., Ekelhart, A., Riedl, B., Weippl, E.: Information security fortification by ontological mapping of the iso/iec 27001 standard. In: 13th Pacific Rim International Symposium on Dependable Computing (PRDC 2007). pp. 381–388. IEEE (2007)
* [11] Gangemi, A., Catenacci, C., Ciaramita, M., Lehmann, J.: A theoretical framework for ontology evaluation and validation. In: SWAP. vol. 166, p. 16 (2005)
* [12] Gómez-Pérez, A.: Evaluation of taxonomic knowledge in ontologies and knowledge bases (1999)
* [13] Guarino, N., Welty, C.: Evaluating ontological decisions with ontoclean. Communications of the ACM 45(2), 61–65 (2002)
* [14] Hearst, M.A.: Automatic Acquisition of Hyponyms from Large Text Corpora. In: Proceedings of the 14th conference on Computational linguistics. vol. 2, pp. 539–545. Association for Computational Linguistics (1992)
* [15] Iyer, V., Mohan, L., Bhatia, M., Reddy, Y.R.: A survey on ontology enrichment from text. In: Proceedings of the 16th International Conference on Natural Language Processing. pp. 95–104. NLP Association of India, International Institute of Information Technology, Hyderabad, India (Dec 2019), https://aclanthology.org/2019.icon-1.11
* [16] Kiptoo, C.C.: Ontology enhancement using crowdsourcing: a conceptual architecture. International Journal of Crowd Science (2020)
* [17] Kontokostas, D., Zaveri, A., Auer, S., Lehmann, J.: Triplecheckmate: A tool for crowdsourcing the quality assessment of linked data. In: International Conference on Knowledge Engineering and the Semantic Web. pp. 265–272. Springer (2013)
* [18] Lantow, B.: OntoMetrics: Putting Metrics into Use for Ontology Evaluation. In: KEOD. pp. 186–191 (2016)
* [19] Lozano-Tello, A., Gómez-Pérez, A.: Ontometric: A Method to Choose the Appropriate Ontology. In: Journal of Database Management. vol. 2, pp. 1–18. IDEA Group Publishing (2004)
* [20] Maedche, A., Staab, S.: Ontology Learning for the Semantic Web. Intelligent Systems 16(2), 72–79 (2001)
* [21] Makki, J., Alquier, A.M., Prince, V.: An nlp-based ontology population for a risk management generic structure. In: Proceedings of the 5th international conference on Soft computing as transdisciplinary science and technology. pp. 350–355 (2008)
* [22] McDaniel, M., Storey, V.C., Sugumaran, V.: Assessing the Quality of Domain Ontologies: Metrics and an Automated Ranking System. Data & Knowledge Engineering 115, 32–47 (2018)
* [23] Noy, N.F., McGuinness, D.L., et al.: Ontology development 101: A guide to creating your first ontology (2001)
* [24] Noy, N.F., Mortensen, J., Musen, M.A., Alexander, P.R.: Mechanical turk as an ontology engineer? using microtasks as a component of an ontology-engineering workflow. In: Proceedings of the 5th Annual ACM Web Science Conference. pp. 262–271 (2013)
* [25] Pittet, P., Barthélémy, J.: Exploiting users’ feedbacks: Towards a task-based evaluation of application ontologies throughout their lifecycle. In: International conference on knowledge engineering and ontology development. vol. 2 (2015)
* [26] Poveda, M.: Catalogue of common pitfalls. http://oops.linkeddata.es/catalogue.jsp, accessed: 2021-03-28
* [27] Poveda-Villalón, M., Gómez-Pérez, A., Suárez-Figueroa, M.C.: Oops!: A pitfall-based system for ontology diagnosis. In: Innovations, Developments, and Applications of Semantic Web and Information Systems, pp. 120–148. IGI Global (2018)
* [28] Poveda-Villalón, M., Suárez-Figueroa, M.C., Gómez-Pérez, A.: Validating Ontologies with OOPS! In: International Conference on Knowledge Engineering and Knowledge Management. pp. 267–281. Springer (2012)
* [29] Richard, E., RALPH, H., JOHNSON, R., et al.: Design patterns: Elements of reusable object-oriented software (1995)
* [30] Sanagavarapu, L.M., Iyer, V., Reddy, Y.R.: OntoEnricher: A Deep Learning Approach for Ontology Enrichment from Unstructured Text. arXiv preprint arXiv:2102.04081 (2021)
* [31] Schober, D., Tudose, I., Svatek, V., Boeker, M.: Ontocheck: verifying ontology naming conventions and metadata completeness in protégé 4. In: Journal of biomedical semantics. vol. 3, pp. 1–10. BioMed Central (2012)
* [32] Tartir, S., Arpinar, I.B., Sheth, A.P.: Ontological Evaluation and Validation. In: Theory and applications of ontology: Computer applications, pp. 115–130. Springer (2010)
* [33] Tibaut, A.: Ontology Evaluation. https://github.com/atibaut/ontology-evaluation (Online; accessed on 15-Mar-2021)
* [34] Vrandečić, D.: Ontology Evaluation. In: Handbook on Ontologies, pp. 293–313. Springer (2009)
* [35] Wohlgenannt, G., Sabou, M., Hanika, F.: Crowd-based ontology engineering with the ucomp protégé plugin. Semantic Web 7(4), 379–398 (2016)
* [36] Wong, W., Liu, W., Bennamoun, M.: Ontology Learning from Text: A Look Back and into the Future. ACM Computing Surveys (CSUR) 44(4), 1–36 (2012)
* [37] Zhang, Y., Saberi, M., Chang, E.: Semantic-based lightweight ontology learning framework: a case study of intrusion detection ontology. In: Proceedings of the international conference on web intelligence. pp. 1171–1177 (2017)
|
# Capacity Limitation and Optimization Strategy for Flexible Point-to-Multi-
Point Optical Networks
Ji Zhou, Haide Wang, Liangchuan Li, Weiping Liu, Changyuan Yu, and Zhaohui Li
Manuscript received; revised. This work was supported in part by the National
Natural Science Foundation of China (62371207, 62005102); Hong Kong Scholars
Program (XJ2021018). (Corresponding authors: Ji Zhou and Liangchuan Li),
(e-mail<EMAIL_ADDRESS>[email protected]).J. Zhou, H. Wang, and W.
Liu are with the Department of Electronic Engineering, College of Information
Science and Technology, Jinan University, Guangzhou 510632, China.Z. Xing and
L. Li are with Optical Research Department, Huawei Technologies Co Ltd,
Dongguan, 523808, China.C. Yu is with the Department of Electronic and
Information Engineering, The Hong Kong Polytechnic University, Hong Kong.Z. Li
is with the Guangdong Provincial Key Laboratory of Optoelectronic Information
Processing Chips and Systems, Sun Yat-sen University, Guangzhou 510275, China
###### Abstract
Point-to-multi-point (PtMP) optical networks become the main solutions for
network-edge applications such as passive optical networks and radio access
networks. Entropy-loading digital subcarrier multiplexing (DSCM) is the core
technology to achieve low latency and approach high capacity for flexible PtMP
optical networks. However, the high peak-to-average power ratio of the
entropy-loading DSCM signal limits the power budget and restricts the
capacity, which can be reduced effectively by clipping operation. In this
paper, we derive the theoretical capacity limitation of the flexible PtMP
optical networks based on the entropy-loading DSCM signal. Meanwhile, an
optimal clipping ratio for the clipping operation is acquired to approach the
highest capacity limitation. Based on an accurate clipping-noise model under
the optimal clipping ratio, we establish a three-dimensional look-up table for
bit-error ratio, spectral efficiency, and link loss. Based on the three-
dimensional look-up table, an optimization strategy is proposed to acquire
optimal spectral efficiencies for achieving a higher capacity of the flexible
PtMP optical networks.
###### Index Terms:
Flexible PtMP optical networks, theoretical capacity limitation, entropy-
loading DSCM, three-dimensional look-up table, optimization strategy.
## I Introduction
Point-to-multi-point (PtMP) optical networks with hub-to-leaves architecture
become the main solutions for the network-edge applications such as optical
access networks and optical metro networks [1, 2, 3]. Compared to traditional
point-to-point (PtP) optical networks, PtMP optical networks require less
optical transceivers, thus have lower capital expense (CapEx) and operating
expense (OpEx) costs [4, 5, 6]. In current commercial PtMP optical access
networks, only time-division multiple access (TDMA) has been widely applied,
which is statistical multiple access to make full use of bandwidth [7, 8, 9].
However, the TDMA-based PtMP optical networks naturally have a high latency,
which faces the challenge in the latency-sensitive scenarios [10, 11, 12].
Particularly, the low-latency requirement becomes stricter and stricter in the
future optical networks[13, 14, 15].
Owing to the development of coherent optical techniques, entropy-loading
digital subcarrier multiplexing (DSCM) has been commercially applied in the
800Gb/s long-haul optical communications [16, 17, 18]. Furthermore, coherent
entropy-loading DSCM can implement the revolutionary frequency-division
multiple access (FDMA) for the PtMP optical networks [19, 20, 21]. For the
leaves of the PtMP optical networks, the FDMA can provide the independent
frequency channel to achieve low latency, and the entropy loading allocates
maximal spectral efficiency to approach high capacity [22, 23, 24]. Meanwhile,
the leaves of FDMA can use a relatively low-bandwidth transceiver matching the
subcarrier bandwidth but that of TDMA needs a whole-bandwidth transceiver
matching the signal bandwidth. Unfortunately, the high peak-to-average-power-
ratio (PAPR) of the entropy-loading DSCM signal causes a low optical power
budget, thus limiting the capacity in the peak-power-constrained (PPC) PtMP
optical networks [25, 26, 27].
In this paper, the entropy-loading digital subcarriers with well-matched
spectral efficiencies are assigned to the leaves depending on their different
link losses, which can approach the highest possible capacity for the flexible
PtMP optical networks. Meanwhile, the clipping operation is used to reduce the
PAPR of the entropy-loading DSCM signal, thus increasing the optical power
budget and gaining more capacity. Furthermore, the theoretical capacity
limitation and optimization strategy will be investigated, which can guide the
optimal parameter setting for the flexible PtMP optical networks. The main
contributions of this paper are as follows:
* •
The theoretical capacity limitation is derived considering the clipping
operation for the flexible PtMP optical networks. Meanwhile, an optimal
clipping ratio is acquired based on the theoretical capacity limitation.
* •
We establish a three-dimensional look-up table for bit-error ratio (BER),
spectral efficiency, and link loss, and propose an optimization strategy to
approach the capacity limitation of the flexible PtMP optical networks.
The remainder of this paper is organized as follows. The capacity limitation
for flexible PtMP optical networks is derived in Section II. In Section III,
an optimization strategy for flexible PtMP optical networks is introduced in
detail. Finally, the paper is concluded in Section IV.
## II Capacity Limitation for Flexible PtMP Optical Networks
Figure 1 shows the schematic diagram of the DSCM-based flexible PtMP optical
networks. At the hub, the optical DSCM signal is generated and sent to the
leaves. The transmitted subcarriers are set to the same power. After the
optical distribution network (ODN), the leaves select and detect their
corresponding subcarriers by coherent optical detection. In the actual
situation, the optical signals to the different leaves pass the different
numbers of optical splitters and different lengths of fiber, leading to
different link losses. Based on the on-hand statistic of the deployed PON, the
maximum difference of the link losses among the leaves is probably larger than
10 dB [28]. Therefore, the received signal at the different leaves has
different power and signal-to-noise ratio (SNR). By considering the difference
of the leaves, we will investigate the theoretical capacity limitation for the
DSCM-based flexible PtMP optical networks, and simultaneously use the clipping
operation to increase the capacity.
Figure 1: The schematic diagram of the DSCM-based flexible PtMP optical
networks with different link losses.
At the hub, the DSCM signal is generated, of which the subcarriers are set to
the same average power $P_{\text{t}}$. After the ODN and fiber transmission,
the $i$-th subcarrier is selected and detected by the $i$-th leaf. The
received signal power of the $i$-th subcarrier can be defined as
$P_{\text{r-}i}=P_{\text{t}}/Loss_{i}=P_{\text{DSCM}}/(N\times Loss_{i})$ (1)
where ${P_{\text{DSCM}}}$ is the average power of the DSCM signal. $N$ is the
subcarrier number in the DSCM signal. $Loss_{i}$ is the link loss of the
$i$-th subcarrier.
Assuming that the devices for the leaves have good uniformity, all the
subcarriers suffer from the white noise with the same power. Therefore, the
SNR of the DSCM signal on $i$-th subcarrier is expressed as
$SNR_{i}=P_{\text{r-}i}/\sigma^{2}_{\text{n}}=P_{\text{DSCM}}/(N\times
Loss_{i}\times\sigma^{2}_{\text{n}})$ (2)
where $\sigma^{2}_{\text{n}}$ is the variance of the white noise on each
subcarrier. Obviously, the increase of ${P_{\text{DSCM}}}$ directly improves
the SNR. However, a high-PAPR DSCM signal has a low average power due to the
constrained peak power in the PPC PtMP optical networks. The clipping
operation can effectively reduce the PAPR of the DSCM signal to increase the
average power, which can be expressed as
$x_{\text{c}}=\left\\{\begin{array}[]{cc}A,&x~{}>~{}A\\\
x,&\left|x\right|\leq~{}A\\\ -A,&x<-A\end{array}\right.$ (3)
where $A$ is the clipping amplitude. The clipping ratio can be defined as
$20\times\text{log}_{10}(\eta)$ where $\eta$ is equal to
$A/\sqrt{P_{\text{DSCM}}}$. After the clipping operation, the frequency-domain
signal on each subcarrier is expressed as
$X_{\text{c}}=\alpha\times X+Noise_{\text{c}}$ (4)
where $X$ is the frequency-domain signal on each subcarrier before the
clipping operation. $Noise_{\text{c}}$ denotes the clipping noise. The
clipping attenuation $\alpha$ can be calculated by
$\alpha=\frac{E(x_{\text{c}}\cdot
x)}{E(x^{2})}=\frac{\int_{-\infty}^{\infty}x_{\text{c}}\cdot x\cdot
p(x)\text{d}x}{P_{\text{DSCM}}}=1-2Q(\eta)$ (5)
where $p(x)$ is the probability distribution function of the DSCM signal. Due
to the central limit theorem, the DSCM signal with enough subcarriers becomes
a Gaussian distribution with zero mean, which can be expressed as
$p(x)=\frac{1}{\sqrt{2\pi
P_{\text{DSCM}}}}e^{-\frac{x^{2}}{2P_{\text{DSCM}}}}.$ (6)
$Q(x)$ denotes the Q function, which can be defined as
$Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx.$ (7)
Figure 2: The simulation architecture of the flexible PtMP optical networks
based on the entropy-loading DSCM.
Based on the Eqs. (4) and (5), the average power $P_{\text{c}}$ of the
clipping noise $Noise_{\text{c}}$ can be calculated by
$\begin{split}P_{\text{c}}&=E(X_{\text{c}}^{2})-\alpha^{2}E(X^{2})\\\
&=\int_{-\infty}^{\infty}x_{\text{c}}^{2}\cdot
p_{\text{c}}(x)dx-\alpha^{2}P_{\text{DSCM}}\\\
&=2A^{2}Q(\eta)+\int_{-A}^{A}x^{2}p(x)dx-\alpha^{2}P_{\text{DSCM}}\\\
&=2P_{\text{DSCM}}\left\\{Q(\eta)[1+\eta^{2}-2Q(\eta)]-\frac{\eta}{\sqrt{2\pi}}e^{-\frac{\eta^{2}}{2}}\right\\}\\\
\end{split}$ (8)
where $p_{\text{c}}(x)$ is the probability distribution of the clipping DSCM
signal, which can be defined as
$p_{\text{c}}(x)=\left\\{\begin{array}[]{cc}p(x),&\left|x\right|\leq A\\\
Q(\eta)\delta(\left|x\right|-A),&\left|x\right|>A\end{array}\right.$ (9)
where $\delta(x)$ is the Dirac delta function with a unit impulse.
In the PPC PtMP optical networks, the signal peak is limited to constant value
$A_{\text{p}}$ due to the nonuse of expensive optical amplifier. The peak
amplitude $A$ of the clipping DSCM signal is matched to the $A_{\text{p}}$. We
define a matching coefficient $\beta$ as $A_{\text{p}}/A$ for the clipping
DSCM signal. The effective SNR on $i$-th subcarrier of the clipping DSCM
signal depends on the power of the clipping signal, clipping noise, and white
noise, which can be calculated by
$ESNR_{i}=\frac{\alpha^{2}\times\beta^{2}\times
P_{\text{DSCM}}}{\beta^{2}\times P_{\text{c}}+N\times
Loss_{i}\times\sigma_{\text{n}}^{2}}.$ (10)
Obviously, the $ESNR_{i}$ depends on the clipping ratio $\eta$ and the link
loss $Loss_{i}$. When the link loss of the $i$-th subcarrier is confirmed,
there is an optimal clipping ratio to maximize the effective SNR of the $i$-th
subcarrier. The optimal clipping ratios are different among the subcarriers
with different link losses. Considering the effective SNRs of all the
subcarriers, an optimal clipping ratio can be obtained to maximize the
capacity of the DSCM-based flexible P2MP network. The theoretical capacity
limitation of the DSCM-based flexible P2MP network can be defined as
$C=\sum_{i=1}^{N}C_{i}=B\sum_{i=1}^{N}\text{log}_{2}(1+ESNR_{i})$ (11)
where $B$ is the bandwidth of one subcarrier. The optimal clipping ratio to
maximize the capacity can be acquired by
$\eta_{\text{opt}}=\underset{\eta}{\arg\max}~{}B\sum_{i=1}^{N}\text{log}_{2}(1+ESNR_{i}).$
(12)
When the link losses of the leaves are measured, the theoretical results based
on Eq. (12) can get the optimal clipping ratio to achieve the highest capacity
for the DSCM-based flexible PtMP optical networks.
Figure 3: The effective SNR versus the clipping ratio for the subcarriers with
different link losses. The solid and dashed lines denote the simulation and
theoretical results, respectively.
Figure 2 shows the simulation architecture of the flexible PtMP optical
networks based on the entropy-loading DSCM. First of all, the SNR of each leaf
should be estimated to calculate the spectral efficiency for implementing
entropy loading. At the transmitter, the probabilistic-shaping 64-level
quadrature amplitude modulation (PS-64QAM) for each subcarrier is mapped
depending on the calculated spectral efficiency of each leaf. After the root-
raised-cosine-shaping (RRC) shaping and frequency upshift, the PS-64QAM-
modulated subcarriers are multiplexed to generate an entropy-loading DSCM
signal. The clipping operation is applied in the entropy-loading DSCM signal
to reduce the PAPR. After the normalization to the same peak amplitude
$A_{p}$, the different losses and the white noise with the same power are
added to the signals on the subcarriers for the different leaves. At the
receiver, the frequency downshift and matched filter are used to select the
subcarrier and recover the PS-64QAM signal for each leaf. Finally, the
recovered PS-64QAM signal is used to estimate the SNR and calculate the BER.
Figure 3 depicts the effective SNR versus clipping ratio for the subcarriers
with different link losses. The solid and dashed lines denote the simulation
and theoretical results, respectively. The subcarrier number $N$ is set to 8.
The losses of the leaves are set to the $\boldsymbol{Loss}$ of [1, 1.33, 1.74,
2.32, 3.05, 4.03, 5.25, 6.53]. The noise power $\sigma_{\text{n}}^{2}$ and the
peak value $A_{p}$ are set to 0.0237 and 2.579, respectively. The
theoretically effective SNR can be calculated by Eq. (10). Obviously, the
effective SNR obtained by the simulations coincides with the theoretically
effective SNR well. The effective SNR is improved with the decrease of
clipping ratio at the beginning owing to the increasing signal power and then
decreased with the decrease of clipping ratio due to the increasing clipping
noise. There is an optimal clipping ratio to maximize the effective SNR for
each subcarrier. However, the optimal clipping ratios are different among the
subcarriers with different losses.
Figure 4: The theoretical capacity limitation versus the clipping ratio for
the DSCM-based flexible PtMP optical networks when the symbol rate per
subcarrier is 8Gbaud.
The theoretical capacity limitation can be calculated by Eq. (11) for the
flexible PtMP network. Meanwhile, the optimal clipping ratio can be obtained
by Eq. (12) to achieve the highest capacity limitation. Fig. 4 shows the
theoretical capacity limitation versus clipping ratio for the DSCM-based
flexible PtMP optical networks when the symbol rate per subcarrier is 8Gbaud.
When the clipping operation is not used (i.e. a large clipping ratio is set
such as 13dB), the capacity limitation of the DSCM-based flexible PtMP optical
networks is approximately 263Gb/s. When the clipping operation is used and the
clipping ratio is set to the optimal value of $7$dB, the capacity limitation
of the DSCM-based flexible PtMP optical networks is approximately 361Gb/s. The
capacity limitation of the DSCM-based flexible PtMP optical networks with the
optimal clipping operation is approximately 98Gb/s higher than that of the
DSCM-based flexible PtMP optical networks without clipping operation.
## III Optimization Strategy for flexible PtMP Optical Networks
In this section, we will investigate the optimization strategy to increase the
capacity of the DSCM-based flexible PtMP optical networks. A three-dimensional
look-up table for BER, spectral efficiency, and link loss will be established
to acquire the spectral efficiencies for the subcarriers with different losses
under the same targeted BER.
Figure 5 depicts the BERs of the subcarriers for the DSCM-based flexible PtMP
optical networks without and with the clipping operation. When the clipping
operation is not employed, the spectral efficiencies of the 8 subcarriers are
set to the $\boldsymbol{SE}$ of [4.80, 4.40, 4.00, 3.60, 3.20, 2.80, 2.40,
2.00] bits/symbol depending on the $\boldsymbol{Loss}$ of [1, 1.33, 1.74,
2.32, 3.05, 4.03, 5.25, 6.53]. Under the $\boldsymbol{SE}$, the BER of each
subcarrier is on the forward error correction (FEC) limit with 7% overhead.
When the clipping operation is used and the clipping ratio is set to optimal
7dB, the BER performance of each subcarrier is much improved, particularly the
subcarrier with low spectral efficiency. To obtain a higher capacity, the
spectral efficiency of each subcarrier can be increased until the BER
approaches the 7% FEC limit. In this section, we will establish a three-
dimensional look-up table for the BER, spectral efficiency, and link loss.
When the link losses of the leaves and targeted BERs of the subcarriers are
confirmed, the corresponding spectral efficiency for each subcarrier can be
obtained from the three-dimensional look-up table to achieve the highest
capacity for the DSCM-based flexible PtMP optical networks.
Figure 5: The BERs of the subcarriers for the DSCM-based flexible PtMP optical
networks without and with the clipping operation.
For obtaining the three-dimensional look-up table, a theoretical BER should be
derived considering the white noise and clipping noise. The PS-64QAM signal
can be decomposed into independent in-phase and quadrature amplitudes, which
are PS 8-level pulse amplitude modulation (PS-8PAM) signals. Gray code is
applied to generate the PS-8PAM signals. Fig. 6 (a) shows the probability
density function for the in-phase PS-8PAM of the PS-64QAM in the clipping
entropy-loading DSCM signal. It demonstrates that the clipping noise does not
coincide with the Gaussian distribution. Figs. 6 (b)-(e) depict the
probability density functions of clipping noise on the amplitudes of 1, 3, 5,
and 7, respectively. Obviously, the distributions of the clipping noise on
different amplitudes are different. The piecewise power exponential model is
used to get the accurate fitting curves for the probability density function
of the clipping noise on the different amplitudes as the red lines in Figs. 6
(b)-(e) show. Based on the piecewise power exponential model, the theoretical
probability density function of the clipping noise can be defined as
$f_{Y_{\text{c}}}(y\mid k)=\left\\{\begin{array}[]{cc}A_{k,1}\times
e^{\frac{-\left|y-\mu_{k,1}\right|^{b_{k,1}}}{2\sigma_{k,1}^{2}}},{}&y\leq
D_{k}\\\ A_{k,2}\times
e^{\frac{-\left|y-\mu_{k,2}\right|^{b_{k,2}}}{2\sigma_{k,2}^{2}}},{}&y>D_{k}\end{array}\right.$
(13)
where $k$ is the amplitudes of [$-7$, $-5$, $-3$, $-1$, 1, 3, 5, 7] for the
in-phase amplitude of the PS-64QAM in the clipping entropy-loading DSCM
signal. $D_{k}$ is the value corresponding to the highest probability near the
amplitude of $k$. $A_{k,1}$, $\mu_{k,1}$, $b_{k,1}$ and $\sigma_{k,1}^{2}$ are
the fitting parameters of the signal at the left of $D_{k}$, while $A_{k,2}$,
$\mu_{k,2}$, $b_{k,2}$ and $\sigma_{k,2}^{2}$ are the those of the signal at
the right of $D_{k}$, respectively.
Figure 6: (a) Probability density function (PDF) of the in-phase PS-8PAM of
the PS-64QAM in the clipping entropy-loading DSCM signal. The Probability
density function of clipping noise on the amplitude of (b) 1, (c) 3, (d) 5,
and (e) 7, respectively. The red lines denote the fitting curves based on the
piecewise power exponential model.
Based on Eq. (13), the accurate fitting curves are used to acquire the fitting
parameters of $A_{k,1}$, $\mu_{k,1}$, $b_{k,1}$, $\sigma_{k,1}^{2}$,
$A_{k,2}$, $\mu_{k,2}$, $b_{k,2}$, $\sigma_{k,2}^{2}$, and $D_{k}$ for the
theoretical probability density function of the clipping noise. Meanwhile, to
make sure the integral value of $f_{Y_{\text{c}}}(y\mid k)$ is equal to $1$,
the $A_{k,i}$ should be updated by
$A_{k,i}=\frac{A_{k,i}}{\int_{-\infty}^{\infty}f_{Y_{\text{c}}}(y\mid
k)\text{d}y}$ (14)
where $i=1,2$ denotes the left and the right parts of the probability density
function, respectively.
The theoretical BER can be derived by the joint probability density function
based on the probability density functions of the white noise and clipping
noise. The white noise and clipping noise can be considered as two independent
continuous random variables $Y_{\text{n}}$ and $Y_{\text{c}}$. The white noise
coincides with Gaussian distribution, and its probability density functions
$f_{Y_{\text{n}}}$ is equal to
$1/(\sqrt{2\pi}\sigma_{\text{n}})e^{-y^{2}/(2\sigma_{\text{n}}^{2})}$. The
probability density functions of the clipping noise $f_{Y_{\text{c}}}$ is
equal to Eq. (13). The random variable of combined noise $Z$ is equal to
$Y_{\text{n}}+Y_{\text{c}}$, and its probability density functions $f_{Z}$ can
be calculated by
$f_{Z}(z)=\int_{-\infty}^{\infty}f_{Y_{\text{n}}}(y)f_{Y_{\text{c}}}(z-y)\text{d}y$
(15)
Substituting $f_{Y_{\text{n}}}$ and $f_{Y_{\text{c}}}$ into Eq. (15), the
probability density functions $f_{Z}$ can be expressed as
$\displaystyle f_{Z}(z\mid k)$
$\displaystyle=\frac{A_{k,1}}{\sqrt{2\pi}\sigma_{\text{n}}}\int_{z-D_{k}}^{+\infty}e^{-\frac{y^{2}}{2\sigma_{\text{n}}^{2}}}e^{-\frac{\left|z-y-\mu_{k,1}\right|^{b_{k,1}}}{2\sigma_{k,1}^{2}}}\text{d}y$
(16)
$\displaystyle+\frac{A_{k,2}}{\sqrt{2\pi}\sigma_{\text{n}}}\int_{-\infty}^{z-D_{k}}e^{-\frac{y^{2}}{2\sigma_{\text{n}}^{2}}}e^{-\frac{\left|z-y-\mu_{k,2}\right|^{b_{k,2}}}{2\sigma_{k,2}^{2}}}\text{d}y$
The BER of the PS-64QAM signal is similar to that of the decomposed PS-8PAM
signal. For the PS-8PAM signal, the probabilities of the amplitudes are
different, which can be defined as $\boldsymbol{Pr}=[Pr_{\pm 1},Pr_{\pm
3},Pr_{\pm 5},Pr_{\pm 7}]$. The theoretical error ratio of the first bit for
the PS-PAM8 signal can be calculated by
$\displaystyle E_{\text{b,~{}1}}$
$\displaystyle=2\left[Pr_{1}\int_{-\infty}^{-d}f_{Z}(z\mid
1)\text{d}z+Pr_{3}\int_{-\infty}^{-3d}f_{Z}(z\mid 3)\text{d}z\right.$ (17)
$\displaystyle\left.+Pr_{5}\int_{-\infty}^{-5d}f_{Z}(z\mid
5)\text{d}z+Pr_{7}\int_{-\infty}^{-7d}f_{Z}(z\mid 7)\text{d}z\right]$
where $d$ is the Euclidean distance. Similarly, the theoretical error ratio of
the second bit can be calculated by
$\displaystyle E_{\text{b},~{}2}$
$\displaystyle=2Pr_{1}\left[\int_{3d}^{\infty}f_{Z}(z\mid
1)\text{d}z+\int_{-\infty}^{-5d}f_{Z}(z\mid 1)\text{d}z\right]$ (18)
$\displaystyle+2Pr_{3}\left[\int_{d}^{\infty}f_{Z}(z\mid
3)\text{d}z+\int_{-\infty}^{-7d}f_{Z}(z\mid 3)\text{d}z\right]$
$\displaystyle+2Pr_{5}\int_{-9d}^{-d}f_{Z}(z\mid
5)\text{d}z+2Pr_{7}\int_{-11d}^{3d}f_{Z}(z\mid 7)\text{d}z.$
The theoretical error ratio of the third bit can be calculated by Eq. (19).
$\displaystyle E_{\text{b},~{}3}$
$\displaystyle=2\left\\{Pr_{1}\left[\int_{d}^{5d}f_{Z}(z\mid
1)\text{d}z+\int_{-7d}^{-3d}f_{Z}(z\mid
1)\text{d}z\right]+Pr_{3}\left[\int_{3d}^{\infty}f_{Z}(z\mid
3)\text{d}z+\int_{-5d}^{-d}f_{Z}(z\mid
3)\text{d}z+\int_{-\infty}^{-9d}f_{Z}(z\mid 3)\text{d}z\right]\right.$ (19)
$\displaystyle\left.+Pr_{5}\left[\int_{d}^{\infty}f_{Z}(z\mid
5)\text{d}z+\int_{-7d}^{-3d}f_{Z}(z\mid
5)\text{d}z+\int_{-\infty}^{-11d}f_{Z}(z\mid
5)\text{d}z\right]+Pr_{7}\left[\int_{-5d}^{-d}f_{Z}(z\mid
7)\text{d}z+\int_{-13d}^{-9d}f_{Z}(z\mid 7)\text{d}z\right]\right\\}$
As shown in Eqs. (17)-(19), the $d$ should be confirmed before calculating the
theoretical error ratios of the three bits for the PS-8PAM signal. The
relationship between $d$ and the average power of the PS-64QAM signal can be
defined as
$d=\sqrt{\frac{P_{\text{PS-64QAM}}}{4\sum_{i=0}^{3}(2i+1)^{2}\times
Pr_{2i+1}}}.$ (20)
where $P_{\text{PS-64QAM}}$ is the average power of the PS-64QAM signal on the
subcarrier, which can be calculated by $\alpha^{2}\times\beta^{2}\times
P_{\text{DSCM}}/(N\times Loss_{i})$. Finally, the theoretical BER of the
PS-64QAM signal on the subcarrier can be calculated by
$E_{\text{b}}=(E_{\text{b, 1}}+E_{\text{b, 2}}+E_{\text{b, 3}})/3.$ (21)
Figure 7: Three-dimensional look-up table for BER, spectral efficiency (SE),
and link loss under the optimal clipping ratio.
For the PS-64QAM signal, the spectral efficiency corresponds to the specific
probabilities $\boldsymbol{Pr}$ of the amplitudes. As shown in Eqs. (17)-(19),
the $E_{\text{b},~{}i}$ is related to the $\boldsymbol{Pr}$ (i.e., spectral
efficiency) and the $d$ where $i$ is from 1 to 3. Meanwhile, as Eq. (20)
shows, the $d$ is relevant to the $Loss_{i}$ and the $\boldsymbol{Pr}$ (i.e.,
spectral efficiency) under a given clipping ratio. In conclusion, the
theoretical BER of the PS-64QAM signal is determined by the $Loss_{i}$ and
spectral efficiency. Fig. 7 depicts the three-dimensional look-up table for
BER, spectral efficiency, and link loss under the optimal clipping ratio for
the flexible PtMP optical networks. In Fig. 4, the optimal clipping ratio can
be confirmed depending on the link losses of the leaves to achieve the highest
capacity limitation of the flexible PtMP optical networks. The three-
dimensional look-up table is established by Eq. (21) under the optimal
clipping ratio. As Fig. 5 shows, the BER of each subcarrier for the DSCM-based
flexible PtMP optical networks with the clipping operation has a large margin
compared to the 7% FEC limit. Therefore, when the targeted BER is set to 7%
FEC limit, the spectral efficiency can be further improved for each subcarrier
based on the three-dimensional look-up table.
Algorithm 1 Optimization strategy for the capacity of flexible point-to-multi-
point optical networks
1:Subcarrier number $N$; link losses $\boldsymbol{Loss}$; Noise variance
$\sigma^{2}_{\text{n}}$; Signal power $P_{\text{DSCM}}$; Targeted BER
$BER_{\text{T}}$; Maximum spectral efficiency $SE_{\text{max}}$; Spectral-
efficiency updating step size $\Delta SE$; Three-dimensional look-up tables
$\boldsymbol{LUT}(BER,~{}SE,~{}Loss)$.
2:Optimal clipping ratio $\eta_{\text{opt}}$ and optimal spectral efficiencies
$\boldsymbol{SE}$.
3:Calculate the effective SNRs $\boldsymbol{ESNR}$ of the subcarriers by Eq.
(10) based on subcarrier number $N$, link losses $\boldsymbol{Loss}$, noise
variance $\sigma^{2}_{\text{n}}$, and signal power $P_{\text{DSCM}}$.
4:Calculate the optimal clipping ratio $\eta_{\text{opt}}$ by Eq. (12) based
on the $\boldsymbol{ESNR}$.
5:Choose the corresponding three-dimensional look-up table $LUT(BER,SE,Loss)$
based on the optimal clipping ratio $\eta_{\text{opt}}$. $\triangleright$
Next, the iterative algorithm is used to find the optimal $\boldsymbol{SE}$
based on $BER_{\text{T}}$ and $\boldsymbol{Loss}$ in $LUT$.
6:Initialize: $\boldsymbol{SE}:=SE_{\text{max}}\times\text{ones}(1,N)$
7:for $i=1$ to $N$ do
8: Initialize: $BER_{i}$ by $(SE_{i},Loss_{i})$ in $LUT$
9: while $BER_{i}>BER_{\text{T}}$ do
10: Update $SE_{i}\leftarrow SE_{i}-\Delta SE$
11: Update $BER_{i}$ by $(SE_{i},Loss_{i})$ in the $LUT$
12: end while
13:end for
14:return Optimal clipping ratio $\eta_{\text{opt}}$ and optimal spectral
efficiencies $\boldsymbol{SE}$
Algorithm 1 shows the proposed optimization strategy for increasing the
capacity of the flexible PtMP optical networks. The input of the algorithm
includes the subcarrier number $N$, link losses $\boldsymbol{Loss}$, noise
variance $\sigma^{2}_{\text{n}}$, signal power $P_{\text{DSCM}}$, targeted BER
$BER_{\text{T}}$, maximum spectral efficiency $SE_{\text{max}}$, spectral-
efficiency updating step size $\Delta SE$, and three-dimensional look-up
tables $\boldsymbol{LUT}(BER,~{}SE,~{}Loss)$. The first step of the proposed
algorithm is to calculate the effective SNRs $\boldsymbol{ESNR}$ of the
subcarriers by Eq. (10) based on subcarrier number $N$, link losses
$\boldsymbol{Loss}$, noise variance $\sigma^{2}_{\text{n}}$, and signal power
$P_{\text{DSCM}}$. In the second step, the optimal clipping ratio
$\eta_{\text{opt}}$ is calculated by Eq. (12) based on the
$\boldsymbol{ESNR}$. In the third step, the corresponding three-dimensional
look-up table $LUT(BER,SE,Loss)$ is chosen based on the optimal clipping ratio
$\eta_{\text{opt}}$. The last step uses an iterative algorithm to find the
optimal $\boldsymbol{SE}$ based on $BER_{\text{T}}$ and $\boldsymbol{Loss}$
based on the $LUT$. In the iterative algorithm, the spectral efficiency of all
subcarriers is initialized as the maximum spectral efficiency
$SE_{\text{max}}$ (i.e., 6bit/symbol for 64QAM). The outer iterations traverse
all the subcarriers to find the optimized spectral efficiencies
$\boldsymbol{SE}$. In the inner iterations, the spectral efficiency $SE_{i}$
is reduced by the step size $\Delta SE$ each iteration until the BER updated
by $(SE_{i},Loss_{i})$ using the $LUT$ is lower than the targeted BER
$BER_{\text{T}}$. After outer and inner iterations, the optimal spectral
efficiencies $\boldsymbol{SE}$ are obtained for all the subcarriers of the
flexible PtMP optical network. Finally, the optimal clipping ratio
$\eta_{\text{opt}}$ and the optimal spectral efficiencies $\boldsymbol{SE}$
are returned.
Figure 8: The BERs of the subcarriers for the DSCM-based flexible PtMP optical
networks after improving spectral efficiency by using a conventional look-up
table (LUT) and a new look-up table.
Figure 8 depicts the BERs of the subcarriers for the DSCM-based flexible PtMP
optical networks after improving spectral efficiency by using a conventional
look-up table and a new look-up table. For the conventional look-up table, the
clipping noise is assumed to coincide with Gaussian distribution. Based on the
conventional look-up table, the spectral efficiencies for the 8 subcarriers
are set to [5.90, 5.72, 5.49, 5.19, 4.87, 4.54, 4.20, 3.90] bits/symbol,
respectively. However, the BERs of all the subcarriers cannot be under the
targeted 7% FEC limit. Thus, the assumption of Gaussian distribution is not
exact for the clipping noise. For the new look-up table, the clipping noise
coincides with the distribution based on Eq. (13). When the new look-up table
in Fig. 7 is used, the BERs of all the subcarriers are below and close to the
targeted 7% FEC limit. Therefore, the new look-up table is more precise than
the conventional look-up table.
When the clipping operation is not used, the spectral efficiencies for the 8
subcarriers are set to [4.80, 4.40, 4.00, 3.60, 3.20, 2.80, 2.40, 2.00]
bits/symbol, respectively. The corresponding capacity is approximately
217.6Gb/s (i.e. 203.3GB/s excluding the 7% FEC overhead) for the flexible PtMP
optical networks based on 8$\times$8Gbaud DSCM signal. When the new look-up
table is employed, the spectral efficiencies for the 8 subcarriers are set to
[5.65, 5.51, 5.27, 5.04, 4.75, 4.38, 4.08, 3.80] bits/symbol, respectively.
The corresponding capacity is approximately 307.84Gb/s (i.e. 287.7GB/s
excluding the 7% FEC overhead). Using the optimal clipping operation and new
look-up table, the capacity of the DSCM-based flexible PtMP optical networks
is approximately 90.4Gb/s higher than that without clipping operation, which
approaches the theoretical result in Section II. Thus, the optimal clipping
operation and new look-up table can be used to achieve 41% higher capacity.
## IV Conclusion
In this paper, we derive the theoretical capacity limitation versus the link
losses for the flexible PtMP optical networks based on the entropy-loading
DSCM signal. Meanwhile, the clipping operation with the optimal clipping ratio
reduces the PAPR of the entropy-loading DSCM signal to improve the effective
SNR for approaching the highest capacity limit. Based on the piecewise power
exponential model for the clipping noise, we establish a three-dimensional
look-up table for bit-error ratio, spectral efficiency, and link loss under
the optimal clipping ratio. Based on the three-dimensional look-up table, an
optimization strategy is proposed to acquire optimal spectral efficiencies for
achieving a higher capacity of the flexible PtMP optical networks.
## References
* [1] J. Zhang, G. Li, S. Xing, and N. Chi, “Flexible and adaptive coherent PON for next-generation optical access network,” _Optical Fiber Technology_ , vol. 75, p. 103190, 2023.
* [2] M. M. Hosseini, J. Pedro, N. Costa, A. Napoli, J. E. Prilepsky, and S. K. Turitsyn, “Multi-period Planning in Metro-Aggregation Networks using Point-to-Multipoint Transceivers,” in _IEEE Global Communications Conference_. IEEE, 2022, pp. 2921–2926.
* [3] M. M. Hosseini, J. Pedro, A. Napoli, N. Costa, J. E. Prilepsky, and S. K. Turitsyn, “Optimized design of filterless horseshoe networks exploiting point-to-multipoint coherent transceivers,” _Journal of Optical Communications and Networking_ , vol. 15, no. 9, pp. 569–578, 2023.
* [4] J. Bäck, P. Wright, J. Ambrose, A. Chase, M. Jary, F. Masoud, N. Sugden, G. Wardrop, A. Napoli, J. Pedro _et al._ , “CAPEX savings enabled by point-to-multipoint coherent pluggable optics using digital subcarrier multiplexing in metro aggregation networks,” in _European Conference on Optical Communications_. IEEE, 2020, pp. 1–4.
* [5] L. A. Campos, Z. Jia, H. Zhang, and M. Xu, “Coherent optics for access from P2P to P2MP,” _Journal of Optical Communications and Networking_ , vol. 15, no. 3, pp. A114–A123, 2023.
* [6] R. Li, Q. Lv, and Z. Zhu, “On the Network Planning of Wavelength Switched Optical Networks with P2MP Transceivers,” _Journal of Lightwave Technology_ , 2023.
* [7] E. Wong, “Next-generation broadband access networks and technologies,” _Journal of Lightwave Technology_ , vol. 30, no. 4, pp. 597–608, 2011.
* [8] F. J. Effenberger, “PON standardisation status and future prospects,” in _45th European Conference on Optical Communication (ECOC 2019)_. IET, 2019, pp. 1–3.
* [9] J.-i. Kani and D. van Veen, “Current TDM-PON Technologies,” in _Springer Handbook of Optical Networks_. Springer, 2020, pp. 849–870.
* [10] W. J. Shan, “The outlook for PON standardization: a tutorial,” _Journal of Lightwave Technology_ , vol. 38, no. 1, pp. 31–42, 2019.
* [11] J.-i. Kani, J. Terada, K.-I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” _Journal of Lightwave Technology_ , vol. 35, no. 3, pp. 527–534, 2017.
* [12] D. Schaber, P. Schulte, S. Calabrò, G. Caruso, and M. Kuschnerov, “PON Downstream Scheme Supporting Simultaneously Different ONU Categories,” _IEEE Photonics Technology Letters_ , vol. 35, no. 21, pp. 1171–1174, 2023\.
* [13] I. Dias, L. Ruan, C. Ranaweera, and E. Wong, “From 5G to beyond: Passive optical network and multi-access edge computing integration for latency-sensitive applications,” _Optical Fiber Technology_ , vol. 75, p. 103191, 2023.
* [14] D. Larrabeiti, L. M. Contreras, G. Otero, J. A. Hernández, and J. P. Fernandez-Palacios, “Toward end-to-end latency management of 5G network slicing and fronthaul traffic,” _Optical Fiber Technology_ , vol. 76, p. 103220, 2023.
* [15] S. Bidkar, K. Christodoulopoulos, T. Pfeiffer, and R. Bonk, “Evaluating Bandwidth Efficiency and Latency of Scheduling Schemes for 5G Fronthaul over TDM-PON,” in _2022 European Conference on Optical Communication (ECOC)_. IEEE, 2022, pp. 1–4.
* [16] H. Sun, M. Torbatian, M. Karimi, R. Maher, S. Thomson, M. Tehrani, Y. Gao, A. Kumpera, G. Soliman, A. Kakkar _et al._ , “800G DSP ASIC design using probabilistic shaping and digital sub-carrier multiplexing,” _Journal of Lightwave Technology_ , vol. 38, no. 17, pp. 4744–4756, 2020\.
* [17] D. Welch, A. Napoli, J. Bäck, S. Buggaveeti, C. Castro, A. Chase, X. Chen, V. Dominic, T. Duthel, T. A. Eriksson _et al._ , “Digital subcarrier multiplexing: enabling software-configurable optical networks,” _Journal of Lightwave Technology_ , vol. 41, no. 4, pp. 1175–1191, 2022.
* [18] D. Che and W. Shieh, “Approaching the capacity of colored-SNR optical channels by multicarrier entropy loading,” _Journal of Lightwave Technology_ , vol. 36, no. 1, pp. 68–78, 2018.
* [19] D. Welch, A. Napoli, J. Bäck, W. Sande, J. Pedro, F. Masoud, C. Fludger, T. Duthel, H. Sun, S. J. Hand _et al._ , “Point-to-multipoint optical networks using coherent digital subcarriers,” _Journal of Lightwave Technology_ , vol. 39, no. 16, pp. 5232–5247, 2021.
* [20] Z. Xing, K. Zhang, X. Chen, Q. Feng, K. Zheng, Y. Zhao, Z. Dong, J. Zhou, T. Gui, Z. Ye _et al._ , “First Real-time Demonstration of 200G TFDMA Coherent PON using Ultra-simple ONUs,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2023, pp. Th4C–4.
* [21] H. Wang, J. Zhou, Z. Xing, Q. Feng, K. Zhang, K. Zheng, X. Chen, T. Gui, L. Li, J. Zeng, J. Yang, W. Liu, C. Yu, and Z. Li, “Fast-Convergence Digital Signal Processing for Coherent PON Using Digital SCM,” _Journal of Lightwave Technology_ , vol. 41, no. 14, pp. 4635–4643, 2023.
* [22] Y. Fan, M. Fu, H. Jiang, X. Liu, Q. Liu, Y. Xu, L. Yi, W. Hu, and Q. Zhuge, “Point-to-Multipoint Coherent Architecture with Joint Resource Allocation for B5G/6G Fronthaul,” _IEEE Wireless Communications_ , vol. 29, no. 2, pp. 100–106, 2022.
* [23] D. Welch, A. Napoli, J. Bäck, N. Swenson, W. Sande, J. Pedro, F. Masoud, A. Chase, C. Fludger, H. Sun _et al._ , “Digital subcarriers: A universal technology for next generation optical networks,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2022, pp. Tu3H–1.
* [24] H. Wang, J. Zhou, J. Wei, W. Mo, Y. Feng, W. Liu, C. Yu, and Z. Li, “Record Capacity-Reach of C band IM/DD Optical Systems over Dispersion-Uncompensated Links,” in _CLEO: Science and Innovations_. Optica Publishing Group, 2022, pp. SM2J–6.
* [25] J. Zhou, L. Li, J. He, X. Lu, Y. Bo, G. Wang, Y. Huang, G. Liu, Y. Lu, S. Gao _et al._ , “Clipping discrete multi-tone for peak-power-constraint IM/DD optical systems,” _Science China Information Sciences_ , vol. 66, no. 5, p. 152302, 2023.
* [26] B. M. Oliveira, M. S. Neves, F. P. Guiomar, M. C. Medeiros, and P. P. Monteiro, “ML-based Optimization of Geometric Constellation Shaping for Unamplified Coherent Optical Systems,” in _2023 23rd International Conference on Transparent Optical Networks (ICTON)_. IEEE, 2023, pp. 1–4.
* [27] J. Zhou, J. He, X. Lu, G. Wang, Y. Bo, G. Liu, Y. Huang, L. Li, C. Yang, H. Wang _et al._ , “100G fine-granularity flexible-rate passive optical networks based on discrete multi-tone with PAPR optimization,” _Journal of Optical Communications and Networking_ , vol. 14, no. 11, pp. 944–950, 2022.
* [28] P. Parolari, A. Gatto, C. Neumeyr, and P. Boffi, “Flexible transmitters based on directly modulated VCSELs for next-generation 50G passive optical networks,” _Journal of Optical Communications and Networking_ , vol. 12, no. 10, pp. D78–D85, 2020.
|
# Data Leakage in Tabular Federated Learning
Mark Vero, Mislav Balunović, Dimitar I. Dimitrov, Martin Vechev
ETH Zurich, Zurich, Switzerland
<EMAIL_ADDRESS>{mislav.balunovic,
dimitar.iliev.dimitrov<EMAIL_ADDRESS>
###### Abstract
While federated learning (FL) promises to preserve privacy in distributed
training of deep learning models, recent work in the image and NLP domains
showed that training updates leak private data of participating clients. At
the same time, most high-stakes applications of FL (e.g., legal and financial)
use tabular data. Compared to the NLP and image domains, reconstruction of
tabular data poses several unique challenges: (i) categorical features
introduce a significantly more difficult mixed discrete-continuous
optimization problem, (ii) the mix of categorical and continuous features
causes high variance in the final reconstructions, and (iii) structured data
makes it difficult for the adversary to judge reconstruction quality. In this
work, we tackle these challenges and propose the first comprehensive
reconstruction attack on tabular data, called TabLeak. TabLeak is based on
three key ingredients: (i) a softmax structural prior, implicitly converting
the mixed discrete-continuous optimization problem into an easier fully
continuous one, (ii) a way to reduce the variance of our reconstructions
through a pooled ensembling scheme exploiting the structure of tabular data,
and (iii) an entropy measure which can successfully assess reconstruction
quality. Our experimental evaluation demonstrates the effectiveness of
TabLeak, reaching a state-of-the-art on four popular tabular datasets. For
instance, on the Adult dataset, we improve attack accuracy by 10% compared to
the baseline on the practically relevant batch size of 32 and further obtain
non-trivial reconstructions for batch sizes as large as 128. Our findings are
important as they show that performing FL on tabular data, which often poses
high privacy risks, is highly vulnerable.
## 1 Introduction
Federated Learning (McMahan et al., 2017) (FL) has emerged as the most
prominent approach to training machine learning models collaboratively without
requiring sensitive data of different parties to be sent to a single
centralized location. While prior work has examined privacy leakage in
federated learning in the context of computer vision (Zhu et al., 2019;
Geiping et al., 2020; Yin et al., 2021) and natural language processing
(Dimitrov et al., 2022a; Gupta et al., 2022; Deng et al., 2021), many
applications of FL rely on large tabular datasets that include highly
sensitive personal data such as financial information and health status
(Borisov et al., 2021; Rieke et al., 2020; Long et al., 2021). However, no
prior work has studied the issue of privacy leakage in the context of tabular
data, a cause of concern for public institutions which have recently launched
a competition111https://petsprizechallenges.com/ with a 1.6 mil. USD prize to
develop privacy-preserving FL solutions for fraud detection and infection risk
prediction, both being tabular datasets.
#### Key challenges
Leakage attacks often rely on solving optimization problems whose solutions
are the desired sensitive data points. Unlike other data types, tabular data
poses unique challenges to solving these problems because: (i) the
reconstruction is a solution to a mixed discrete-continuous optimization
problem, in contrast to other domains where the problem is fully continuous
(pixels for images and embeddings for text), (ii) there is high variance in
the final reconstructions because, uniquely to tabular data, discrete changes
in the categorical features significantly change the optimization trajectory,
and (iii) assessing the quality of reconstructions is harder compared to
images and text - e.g. determining whether a person with given reconstructed
characteristics exists is difficult. Together, these challenges imply that it
is difficult to make existing attacks work on tabular data.
#### This work
In this work, we propose the first comprehensive leakage attack on tabular
data in the FL setting, addressing the previously mentioned challenges. We
provide an overview of our approach in Fig. 1, showing the reconstruction of a
client’s private training data point $x=[\texttt{male, 18, white}]$, from the
corresponding training update $\nabla f$ received by the server. In Step 1, we
create $N$ separate optimization problems, each assigning different initial
values $z^{0}_{1},\dots z^{0}_{N}$ to the optimization variables, representing
our reconstruction of the client’s one-hot encoded data, $c(x)$. To address
the first challenge of tabular data leakage, we transform the mixed discrete-
continuous optimization problem into a fully continuous one, by passing our
current reconstructions $z_{1}^{t},\dots,z_{N}^{t}$ through a per-feature
softmax $\sigma$ at every step $t$. Using the softmaxed data $\sigma(z^{t})$,
we take a gradient step to minimize the reconstruction loss, which compares
the received client update $\nabla f$ with a simulated client update computed
on $\sigma(z^{t})$. In Step 2, we reduce the variance of the final
reconstruction by performing pooling over the $N$ different solutions
$z_{1},z_{2},...,z_{N}$, thus tackling the second challenge. In Step 3, we
address the challenge of assessing the fidelity of our reconstructions. We
rely on the observation that often when our proposed reconstructions
$z_{1},z_{2},...,z_{N}$ agree they also match the true client data, $c(x)$. We
measure the agreement using entropy. In the example above, we see that the
features sex and age produced a low entropy distribution. Therefore we assign
high confidence to these results (green arrows). In contrast, the
reconstruction of the feature race receives a low confidence rating (orange
arrow); rightfully so, as the reconstruction is incorrect.
We implemented our approach in an end-to-end attack called TabLeak and
evaluated it on several tabular datasets. Our attack is highly effective: it
can obtain non-trivial reconstructions for batch sizes as large as 128, and on
many practically relevant batch sizes such as 32, it improved reconstruction
accuracy by up to 10% compared to the baseline. Overall, our findings show
that FL is highly vulnerable when applied to tabular data.
#### Main contributions
Our main contributions are:
* •
Novel insights enabling efficient attacks on FL with tabular data: using
softmax to make the optimization problem fully continuous, ensembling to
reduce the variance, and entropy to assess the reconstructions.
* •
An implementation of our approach into an end-to-end tool called TabLeak.
* •
Extensive experimental evaluation, demonstrating effectiveness of TabLeak at
reconstructing sensitive client data on several popular tabular datasets.
Figure 1: Overview of TabLeak. Our approach transforms the optimization
problem into a fully continuous one by optimizing continuous versions of the
discrete features, obtained by applying softmax (Step 1, middle boxes),
resulting in $N$ candidate solutions (Step 1, bottom). Then, we pool together
an ensemble of $N$ different solutions $z_{1},z_{2},...,z_{N}$ obtained from
the optimization to reduce the variance of the reconstruction (Step 2).
Finally, we assess the quality of the reconstruction by computing the entropy
from the feature distributions in the ensemble (Step 3).
## 2 Background and Related Work
In this section, we provide the necessary technical background for our work,
introduce the notation used throughout the paper, and present the related work
in this field.
#### Federated Learning
Federated Learning (FL) is a training protocol developed to facilitate the
distributed training of a parametric model while preserving the privacy of the
data at source (McMahan et al., 2017). Formally, we have a parametric function
$f_{\theta}$, where $\theta\in\Theta$ are the (network) parameters and
$f_{\theta}:\mathcal{X}\rightarrow\mathcal{Y}$. Given a dataset as the union
of private datasets of clients $\mathcal{D}=\bigcup_{k=1}^{K}\mathcal{D}_{k}$,
we now wish to find a $\theta^{*}\in\Theta$ such that:
$\theta^{*}=\operatorname*{arg\,min}_{\theta\in\Theta}\;\frac{1}{N}\sum_{(x_{i},y_{i})\in\mathcal{D}}\,\mathcal{L}(f_{\theta}(x_{i}),y_{i}),$
(1)
in a distributed manner, i.e. without collecting the dataset $\mathcal{D}$ in
a central database. McMahan et al. (2017) propose two training algorithms:
FedSGD (a similar algorithm was also proposed by Shokri and Shmatikov (2015))
and FedAvg, that allow for the distributed training of $f_{\theta}$, while
keeping the data partitions $\mathcal{D}_{k}$ at client sources. The two
protocols differ in how the clients compute their local updates in each step
of training. In FedSGD, each client calculates the update gradient with
respect to a randomly selected batch of their own data and shares the
resulting gradient with the server. In FedAvg, the clients conduct a few
epochs of local training on their own data before sharing their resulting
parameters with the server. After the server has received the
gradients/parameters from the clients, it aggregates them, updates the model,
and broadcasts it to the clients. In each case, this process is repeated until
convergence, where FedAvg usually requires fewer rounds of communication.
#### Data Leakage Attacks
Although FL was designed with the goal of preserving the privacy of clients’
data, recent work has uncovered substantial vulnerabilities. Melis et al.
(2019) first presented how one can infer certain properties of the clients’
private data in FL. Later, Zhu et al. (2019) demonstrated that an honest but
curious server can use the current state of the model and the client gradients
to reconstruct the clients’ data, breaking the main privacy promise of
Federated Learning (FL). Under this threat model, there has been extensive
research on designing tailored attacks for images (Geiping et al., 2020; Geng
et al., 2021; Huang et al., 2021; Jin et al., 2021; Balunovic et al., 2022;
Yin et al., 2021; Zhao et al., 2020; Jeon et al., 2021; Dimitrov et al.,
2022b) and natural language (Deng et al., 2021; Dimitrov et al., 2022a; Gupta
et al., 2022). However, no prior work has comprehensively dealt with tabular
data, despite its significance in real-world high-stakes applications (Borisov
et al., 2021). Some works also consider a threat scenario where the malicious
server is allowed to change the model or the updates communicated to the
clients (Wen et al., 2022; Fowl et al., 2022); but in this work we focus on
the honest-but-curious setting.
In training with FedSGD, given the model $f_{\theta}$ at an iteration $t$ and
the gradient $\nabla_{\theta}\,\mathcal{L}(f_{\theta}(x),y)$ of some client,
we solve the following optimization problem to retrieve the client’s private
data:
$\hat{x},\,\hat{y}=\operatorname*{arg\,min}_{(x^{\prime},y^{\prime})\in\mathcal{X}\times\mathcal{Y}}\mathcal{E}(\nabla_{\theta}\,\mathcal{L}(f_{\theta}(x),y),\nabla_{\theta}\,\mathcal{L}(f_{\theta}(x^{\prime}),y^{\prime}))+\lambda\mathcal{R}(x^{\prime}).$
(2)
Where in Eq. 2 we denote the gradient matching loss as $\mathcal{E}$ and
$\mathcal{R}$ is an optional regularizer for the reconstruction. The work of
Zhu et al. (2019) has used the mean square error for $\mathcal{E}$, on which
Geiping et al. (2020) improved using the cosine similarity loss. Zhao et al.
(2020) first demonstrated that the private labels $y$ can be estimated before
solving Eq. 2, reducing the complexity of Eq. 2 and improving the attack
results. Their method was later extended to batches by Yin et al. (2021) and
refined by Geng et al. (2021). Eq. 2 is typically solved using continuous
optimization tools such as L-BFGS (Liu and Nocedal, 1989) and Adam (Kingma and
Ba, 2014). Although analytical approaches exist, they do not generalize to
batches with more than a single data point (Zhu and Blaschko, 2021).
Depending on the data domain, distinct tailored alterations to Eq. 2 have been
proposed in the literature, e.g., using the total variation regularizer for
images (Geiping et al., 2020) and exploiting pre-trained language models in
language tasks (Dimitrov et al., 2022a; Gupta et al., 2022). These mostly non-
transferable domain-specific solutions are necessary as each domain poses
unique challenges. Our work is first to identify and tackle the key challenges
to data leakage in the tabular domain.
#### Mixed Type Tabular Data
Mixed type tabular data is a data type commonly used in health, economic and
social sciences, which entail high-stakes privacy-critical applications
(Borisov et al., 2021). Here, data is collected in a table of feature columns,
mostly human-interpretable, e.g., age, nationality, and occupation of an
individual. We formalize tabular data as follows. Let $x\in\mathcal{X}$ be one
line of data, containing discrete or categorical features and continuous or
numerical features. Let $\mathcal{X}$ contain $K$ discrete feature columns and
$L$ continuous feature columns, i.e.
$\mathcal{X}=\mathcal{D}_{1}\times\mathcal{D}_{2}\times\dots\times\mathcal{D}_{K}\times\mathcal{U}_{1}\times\dots\times\mathcal{U}_{L}$,
where $\mathcal{D}_{i}\subset\mathbb{N}$ with cardinality
$|\mathcal{D}_{i}|=D_{i}$ and $\mathcal{U}_{i}\subset\mathbb{R}$. For the
purpose of deep neural network training, the categorical features are often
encoded in a numerical vector. We denote the encoded data batch or line as
$c(x)$, where we preserve the continuous features and encode the categorical
features by a one-hot encoding. The one-hot encoding of the $i$-th discrete
feature $c^{D}_{i}(x)$ is a vector of length $D_{i}$ that has a one at the
position marking the encoded category, while all other entries are zeros. We
obtain the represented category by taking the argmax of $c^{D}_{i}(x)$
(projection to obtain $x$). Using the described encoding, one line of data
$x\in\mathcal{X}$ translates to:
$c(x)=\left[c^{D}_{1}(x),\,c^{D}_{2}(x),\,\dots,\,c^{D}_{K}(x),\,x^{C}_{1},\,\dots,\,x^{C}_{L}\right]$,
containing $d\coloneqq L+\sum_{i=1}^{K}D_{i}$ entries.
## 3 Tabular Leakage
In this section, we briefly summarize the key challenges in tabular leakage
and present our solution to these challenges in the subsequent subsections and
our end-to-end attack.
#### Key Challenges
We now list the three key challenges that we address in our work: (i) the
presence of both categorical and continuous features in tabular data require
the attacker to solve a significantly harder mixed discrete-continuous
optimization problem (addressed in Sec. 3.1), (ii) the large distance in the
encodings of the categorical features introduces high variance in the leakage
problem (addressed in Sec. 3.2), and (iii) in contrast to images and text, it
is hard for an adversary to assess the quality of the reconstructed data in
the tabular domain, as most reconstructions may be projected to credible input
data points (we address this via an uncertainty quantification scheme in Sec.
3.3).
### 3.1 The Softmax Structural Prior
We now discuss our solution to challenge (i) – we introduce the softmax
structural prior, which turns the hard mixed discrete-continuous optimization
problem into a fully continuous one. This drastically reduces its complexity,
while still facilitating the recovery of correct discrete structures.
To start, notice that the recovery of one-hot encodings can be enforced by
ensuring that all entries of the recovered vector are either zero or one, and
exactly one of the entries equals to one. However, these constraints enforce
integer properties, i.e. they are non-differentiable and can not be used in
combination with the powerful continuous optimization tools used for gradient
leakage attacks. Relaxing the integer constraint by allowing the reconstructed
entries to take real values in $[0,1]$, we are still left with a constrained
optimization problem not well suited for popular continuous optimization
tools, such as Adam (Kingma and Ba, 2014). Therefore, we are looking for a
method that can implicitly enforce the constraints introduced above.
Let $z\in\mathbb{R}^{d}$ be our approximate intermediate solution for the true
one-hot encoded data $c(x)$ at some optimization step. Then, we are looking
for a differentiable function $\sigma:\mathbb{R}^{D_{i}}\rightarrow[0,1]$,
such that:
$\sum_{j=1}^{D_{i}}\sigma(z_{i}^{D})[j]=1\qquad\text{and}\qquad\sigma(z_{i}^{D})[j]\in[0,1]\quad\forall
j\in\mathcal{D}_{i}.$ (3)
Notice that the two conditions in Eq. 3 can be fulfilled by applying a softmax
to $z_{i}^{D}$, i.e. define:
$\sigma(z_{i}^{D})[j]\coloneqq\frac{\text{exp}(z^{D}_{i}[j])}{\sum_{k=1}^{D_{i}}\text{exp}(z^{D}_{i}[k])}\qquad\forall
j\in\mathcal{D}_{i}.$ (4)
Note that it is easy to show that Eq. 4 fulfills both conditions in Eq. 3 and
that it is differentiable. Putting this together, in each round of
optimization we will have the following approximation of the true data point:
$c(x)\approx\sigma(z)=\left[\sigma(z^{D}_{1}),\,\dots,\,\sigma(z^{D}_{K}),\,z^{C}_{1},\,\dots,\,z^{C}_{L}\right]$.
In order to preserve notational simplicity, we write $\sigma(z)$ to mean the
application of softmax to each group of entries representing a given
categorical variable separately.
### 3.2 Pooled Ensembling
As mentioned earlier, the mix of categorical and continuous features
introduces further variance in the difficult reconstruction problem which
already has multiple local minima and high sensitivity to initialization (Zhu
and Blaschko, 2021) (challenge (ii)). Concretely, as the one-hot encodings of
the categorical features are orthogonal to each other, a change in the encoded
category can drastically change the optimization trajectory. We alleviate this
problem by adapting an established method of variance reduction in noisy
processes (Hastie et al., 2009), i.e. we run independent optimization
processes with different initializations and ensemble their results through
feature-wise pooling.
Note that the features in tabular data are tied to a certain position in the
recovered data vector, thereby we can combine independent reconstructions to
obtain an improved and more robust final estimate of the true data by applying
feature-wise pooling. Formally, we run $N$ independent rounds of optimization
with $i.i.d.$ initializations recovering potentially different reconstructions
$\left\\{\sigma(z_{j})\right\\}_{j=1}^{N}$. Then, we obtain a final estimate
of the true encoded data, denoted as $\sigma^{D}_{i}(\hat{z})$, by pooling
them:
$\sigma^{D}_{i}(\hat{z})=\text{pool}\left(\left\\{\sigma(z^{D}_{ji})\right\\}_{j=1}^{N}\right)\quad\forall
i\in[K]\quad\text{and}\quad\hat{z}_{i}^{C}=\text{pool}\left(\left\\{(z^{C}_{ji})\right\\}_{j=1}^{N}\right)\quad\forall
i\in[L].$ (5)
Where the $\text{pool}(\cdot)$ operation can be any permutation invariant
mapping that maps to the same structure as its inputs. In our attack, we use
median pooling for both continuous and categorical features.
Figure 2: Maximum similarity matching of a sample from the collection of
reconstructions to the best-loss sample $\hat{x}^{best}$.
Notice that because a batch-gradient is invariant to permutations of the
datapoints in the corresponding batch, when reconstructing from such a
gradient we may retrieve the batch-points in a different order at every
optimization instance. Hence, we need to reorder each batch such that their
lines match to each other, and only then we can conduct the pooling. We
reorder by first selecting the sample that produced the best reconstruction
loss at the end of optimization $\hat{z}^{best}$, with projection
$\hat{x}^{best}$. Then, we match the lines of every other sample in the
collection with respect to $\hat{x}^{best}$. Concretely, we calculate the
similarity (described in detail in Sec. 4) between each pair of lines of
$\hat{x}^{best}$ and another sample $\hat{x}_{i}$ in the collection and find
the maximum similarity reordering of the lines with the help of bipartite
matching solved by the Hungarian algorithm (Kuhn, 1955). This process is
depicted in Fig. 2. Repeating this for each sample, we reorder the entire
collection with respect to the best-loss sample, effectively reversing the
permutation differences in the independent reconstructions. Therefore, after
this process we can directly apply feature-wise pooling for each line over the
collection.
### 3.3 Entropy-based Uncertainty Estimation
We now address challenge (iii) above. To recap, it is significantly harder for
an adversary to asses the quality of an obtained reconstruction when it comes
to tabular data, as almost any reconstruction may constitute a credible
datapoint when projected back to mixed discrete-continuous space. Note that
this challenge does not arise as prominently in the image (or text) domain,
because by looking at a picture one can easily judge if it is just noise or an
actual image. To address this issue, we propose to estimate the reconstruction
uncertainty by looking at the level of agreement over a certain feature for
different reconstructions. Concretely, given a collection of reconstructions
as in Sec. 3.2, we can observe the distribution of each feature over the
reconstructions. Intuitively, if this distribution is "peaky", i.e.
concentrates the mass heavily on a certain value, then we can assume that the
feature has been reconstructed correctly, whereas if there is high
disagreement between the reconstructed samples, we can assume that this
feature’s recovered final value should not be trusted. We can quantify this by
measuring the entropy of the feature distributions induced by the recovered
samples.
#### Categorical Features
Let
$p(\hat{x}_{i}^{D})_{m}\coloneqq\frac{1}{N}\,\text{Count}_{j}(\hat{x}_{ji}^{C}=m)$
be the relative frequency of projected reconstructions of the $i$-th discrete
feature of value $m$ in the ensemble. Then, we can calculate the normalized
entropy of the feature as
$\bar{H}^{D}_{i}=\frac{-1}{\log\,|\mathcal{D}_{i}|}\,\sum_{m=1}^{D_{i}}p(\hat{x}_{i}^{D})_{m}\,\log\,p(\hat{x}_{i}^{D})_{m}$.
Note that the normalization allows for comparing features with supports of
different size, i.e. it ensures that $\bar{H}^{D}_{i}\in[0,1]$, as $0\leq
H(k)\leq\log\,|\mathcal{K}|$ for any discrete random variable
$k\in\mathcal{K}$ of finite support.
#### Continuous Features
In case of the continuous features, we calculate the entropy assuming that
errors of the reconstructed samples follow a Gaussian distribution. As such,
we first estimate the sample variance $\hat{\sigma}^{2}_{i}$ for the $i$-th
continuous feature and then plug it in to calculate the entropy of the
corresponding Gaussian:
$H^{C}_{i}=\frac{1}{2}+\frac{1}{2}\,\log\,2\pi\hat{\sigma}^{2}_{i}$. As this
approach is universal over all continuous features, it is enough to simply
scale the features themselves to make their entropy comparable. For example,
this can be achieved by working only with standardized features.
Note that as the categorical and the continuous features are fundamentally
different from an information theoretic perspective, we have no robust means
to combine them in a way that would allow for equal treatment. Therefore, when
assessing the credibility of recovered features, we will always distinguish
between categorical and continuous features.
### 3.4 Combined Attack
Algorithm 1 Our combined attack against training by FedSGD
1:function SingleInversion(Neural Network: $f_{\theta}$, Client Gradient:
$\nabla_{\theta}\,\mathcal{L}(f_{\theta}(c(x)),y)$, Reconstructed Labels:
$\hat{y}$, Initial Reconstruction: $z_{i}^{0}$, Iterations: $T$, N. of
Discrete Features: $K$)
2: for $t$ in $1,2,\dots,T$ do
3: for $k$ in $1,2,\dots,K$ do
4: $\sigma(z_{ik}^{D})\leftarrow$ softmax($z_{ik}^{D}$)
5: end for
6: $z_{i}^{t+1}\leftarrow
z_{i}^{t}-\eta\,\nabla_{z}\mathcal{E}_{CS}(\nabla_{\theta}\,\mathcal{L}(f_{\theta}(c(x)),y),\nabla_{\theta}\,\mathcal{L}(f_{\theta}(\sigma(z_{i}^{t})),\hat{y}))$
7: end for
8: return $z_{i}^{T}$
9:end function
10:
11:function TabLeak(Neural Network: $f_{\theta}$, Client Gradient:
$\nabla_{\theta}\,\mathcal{L}(f_{\theta}(c(x)),y)$, Reconstructed Labels:
$\hat{y}$, Ensemble Size: $N$, Iterations: $T$, N. of Discrete Features: $K$)
12: $\left\\{z_{i}^{0}\right\\}_{i=1}^{N}\sim\mathcal{U}_{[0,1]^{d}}$
13: for $i$ in $1,2,\dots,N$ do
14: $z_{i}^{T}\leftarrow$ SingleInversion($f_{\theta}$,
$\nabla_{\theta}\,\mathcal{L}(f_{\theta}(c(x)),y)$, $\hat{y}$, $z_{i}^{0}$,
$T$, $K$)
15: end for
16:
$\hat{z}^{best}\leftarrow\operatorname*{arg\,min}_{z_{j}^{T}}\,\mathcal{E}_{CS}(\nabla_{\theta}\,\mathcal{L}(f_{\theta}(c(x)),y),\nabla_{\theta}\,\mathcal{L}(f_{\theta}(\sigma(z_{j}^{T})),\hat{y}))$
17: $\sigma(\hat{z})\leftarrow$
MatchAndPool($\left\\{\sigma(z_{i}^{T})\right\\}_{i=1}^{N},\sigma(\hat{z}^{best})$)
18: $\bar{H}^{D},H^{C}\leftarrow$
CalculateEntropy($\left\\{\sigma(z_{i}^{T})\right\\}_{i=1}^{N}$)
19: $\hat{x}\leftarrow$ Project($\sigma(\hat{z})$)
20: return $\hat{x}$, $\bar{H}^{D}$, $H^{C}$
21:end function
Now we provide the description of our end-to-end attack, TabLeak. Following
Geiping et al. (2020), we use the cosine similarity loss as our reconstruction
loss, defined as:
$\mathcal{E}_{CS}(\nabla_{\theta_{t}}\,\mathcal{L}(f_{\theta_{t}}(c(x)),y),\nabla_{\theta_{t}}\,\mathcal{L}(f_{\theta_{t}}(\sigma(z)),\hat{y})),\quad\text{with}\quad\mathcal{E}_{CS}(l,g)\coloneqq
1-\frac{\langle l,\,g\rangle}{\|l\|_{2}\,\|g\|_{2}},$ (6)
where $(x,y)$ are the true data, $\hat{y}$ are the labels reconstructed
beforehand, and we optimize for $z$. Our algorithm is shown in Alg. 1. First,
we reconstruct the labels using the label reconstruction method of Geng et al.
(2021) and provide them as an input to our attack. Then, we initialize $N$
independent dummy samples for an ensemble of size $N$ (12). Starting from each
initial sample we optimize independently (13-15) via the SingleInversion
function. In each optimization step, we apply the softmax structural prior of
Sec. 3.1, and let the optimizer differentiate through it (4). After the
optimization processes have converged or have reached the maximum number of
allowed iterations $T$, we identify the sample $\hat{z}^{best}$ producing the
best reconstruction loss (16). Using this sample, we match and median pool to
obtain the final encoded reconstruction $\sigma(\hat{z})$ in 17 as described
in Sec. 3.2. Finally, we return the projected reconstruction $\hat{x}$ and the
corresponding feature-entropies $\bar{H}^{D}$ and $H^{C}$, quantifying the
uncertainty in the reconstruction.
## 4 Experimental evaluation
In this section, we first detail the evaluation metric we used to assess the
obtained reconstructions. We then briefly explain our experimental setup.
Next, we evaluate our attack in various settings against baseline methods,
establishing a new state-of-the-art. Finally, we demonstrate the effectiveness
of our entropy-based uncertainty quantification method.
#### Evaluation Metric
As no prior work on tabular data leakage exists, we propose our metric for
measuring the accuracy of tabular reconstruction, inspired by the 0-1 loss,
allowing the joint treatment of categorical and continuous features. For a
reconstruction $\hat{x}$, we define the accuracy metric as:
$\text{accuracy}(x,\,\hat{x})\coloneqq\frac{1}{K+L}\left(\sum_{i=1}^{K}\mathbb{I}\\{x^{D}_{i}=\hat{x}^{D}_{i}\\}+\sum_{i=1}^{L}\mathbb{I}\\{\hat{x}^{C}_{i}\in[x^{C}_{i}-\epsilon_{i},\,x^{C}_{i}+\epsilon_{i}]\\}\right),$
(7)
where $x$ is the ground truth and $\\{\epsilon_{i}\\}_{i=1}^{L}$ are constants
determining how close the reconstructed continuous features have to be to the
original value in order to be considered successfully leaked. We provide more
details on our metric in App. A and experiments with additional metrics in
Sec. C.3.
#### Baselines
We consider two main baselines: (i) Random Baseline does not use the gradient
updates and simply randomly samples reconstructions from the per-feature
marginals of the input dataset. Due to the structure of tabular datasets, we
can easily estimate the marginal distribution of each feature. For the
categorical features this can be done by simple counting, and for the
continuous features we do it by defining a binning scheme with $100$ equally
spaced bins between the lower and upper bounds of the feature. Although this
baseline is usually not realizable in practice (as it assumes prior knowledge
of the marginals), it helps us calibrate our metric as performing below this
baseline signals that there is no information being extracted from the client
updates. Note that because both the selection of a batch and the random
baseline represent sampling from the (approximate) data generating
distribution, the random baseline monotonously approaches perfect accuracy
with increasing batch size, (ii) Cosine Baseline is based on the work of
Geiping et al. (2020), who established a strong attack for images. We transfer
their attack to tabular data by removing the total variation prior used for
images. Note that in the case of most competitive attacks on image and text,
when removing the domain specific elements, they reduce to this baseline,
therefore it is a reasonable choice for benchmarking a new domain.
#### Experimental Setup
For all attacks, we use the Adam optimizer (Kingma and Ba, 2014) with learning
rate $0.06$ for $1\,500$ iterations and without a learning rate schedule to
perform the optimization in Alg. 1. In line with Geiping et al. (2020), we
modify the update step of the optimizer by reducing the update gradient to its
element-wise sign. The neural network we attack is a fully connected neural
network with two hidden layers of $100$ neurons each. We conducted our
experiments on four popular mixed-type tabular binary classification datasets,
the Adult census dataset (Dua and Graff, 2017), the German Credit dataset (Dua
and Graff, 2017), the Lawschool Admission dataset (Wightman, 2017), and the
Health Heritage dataset from Kaggle222Source: https://www.kaggle.com/c/hhp.
Due to the space constraints, here we report only our results on the Adult
dataset, and refer the reader to App. D for full results on all four datasets.
Finally, for all reported numbers below, we attack a neural network at
initialization and estimate the mean and standard deviation of each reported
metric on $50$ different batches. For experiments with varying network sizes
and attacks against provable defenses, please see App. C. For further details
on the experimental setup of each experiment, we refer the reader to App. B
#### General Results against FedSGD
In Tab. 1 we present the results of our strong attack TabLeak against FedSGD
training, together with two ablation experiments, each time removing either
the pooling (no pooling) or the softmax component (no softmax). We compare our
results to the baselines introduced above, on batch sizes $8,16,32,64$, and
$128$, once assuming knowledge of the true labels (top) and once using labels
reconstructed by the method of Geng et al. (2021) (bottom). Notice that the
noisy label reconstruction only influences the results for lower batch sizes,
and manifests itself mostly in higher variance in the results. It is also
worth to note that for batch size $8$ (and lower, see App. D) all attacks can
recover almost all the data, exposing a trivial vulnerability of FL on tabular
data. In case of larger batch sizes, even up to $128$, TabLeak can recover a
significant portion of the client’s private data, well above random guessing,
while the baseline Cosine attack fails to do so, demonstrating the necessity
of a domain tailored attack. In a later paragraph, we show how we can further
improve our reconstruction on this batch size and extract subsets of features
with $>90\%$ accuracy using the entropy. Further, the results on the ablation
attacks demonstrate the effectiveness of each attack component, both providing
a non-trivial improvement over the baseline attack that is preserved when
combined in our strongest attack. Demonstrating generalization beyond Adult,
we include our results on the German Credit, Lawschool Admissions, and Health
Heritage datasets in Sec. D.1, where we also outperform the Cosine baseline
attack by at least $10\%$ on batch size $32$ on each dataset.
Table 1: The mean inversion accuracy [%] and standard deviation of different
methods over varying batch sizes with given true labels (True $y$) and with
reconstructed labels (Rec. $\hat{y}$) on the Adult dataset.
Label | Batch | TabLeak | TabLeak | TabLeak | Cosine | Random
---|---|---|---|---|---|---
| Size | | (no pooling) | (no softmax) | |
True $y$ | $8$ | $\mathbf{95.1\pm 9.2}$ | $93.9\pm 10.2$ | $92.9\pm 6.5$ | $91.1\pm 7.3$ | $53.9\pm 4.4$
$16$ | $\mathbf{89.5\pm 7.6}$ | $84.5\pm 9.9$ | $80.5\pm 4.3$ | $75.0\pm 5.2$ | $55.1\pm 3.9$
$32$ | $\mathbf{77.6\pm 4.8}$ | $72.4\pm 4.6$ | $70.8\pm 3.2$ | $66.6\pm 3.5$ | $58.0\pm 2.9$
$64$ | $\mathbf{71.2\pm 2.8}$ | $66.2\pm 2.8$ | $66.9\pm 2.7$ | $62.5\pm 3.1$ | $59.0\pm 3.2$
$128$ | $\mathbf{68.8\pm 1.3}$ | $64.1\pm 1.4$ | $64.0\pm 2.1$ | $59.5\pm 2.1$ | $61.2\pm 3.1$
Rec. $\hat{y}$ | $8$ | $\mathbf{86.9\pm 11.6}$ | $84.6\pm 13.4$ | $85.8\pm 9.9$ | $83.3\pm 9.7$ | $53.9\pm 4.4$
$16$ | $\mathbf{82.4\pm 8.4}$ | $78.3\pm 9.0$ | $77.7\pm 4.1$ | $73.0\pm 3.5$ | $55.1\pm 3.9$
$32$ | $\mathbf{75.3\pm 4.8}$ | $70.6\pm 4.3$ | $70.2\pm 3.2$ | $66.3\pm 3.4$ | $58.0\pm 2.9$
$64$ | $\mathbf{70.4\pm 3.2}$ | $65.9\pm 3.6$ | $66.8\pm 2.6$ | $63.1\pm 3.2$ | $59.0\pm 3.2$
$128$ | $\mathbf{68.7\pm 1.3}$ | $64.4\pm 1.5$ | $63.8\pm 2.1$ | $59.5\pm 2.1$ | $61.2\pm 3.1$
#### Categorical vs. Continuous Features
An interesting effect of having mixed type features in the data is that the
reconstruction success clearly differs by feature type. As we can observe in
Fig. 3, the continuous features produce an up to $30\%$ lower accuracy than
the categorical features for the same batch size. We suggest that this is due
to the nature of categorical features and how they are encoded. While trying
to match the gradients by optimizing the reconstruction, having the correct
categorical features will have a much greater effect on the gradient
alignment, as when encoded, they take up the majority of the data vector.
Also, when reconstructing a one-hot encoded categorical feature, we only have
to be able to retrieve the location of the maximum in a vector of length
$D_{i}$, whereas for the successful reconstruction of a continuous feature we
have to retrieve its value correctly up to a small error. Therefore,
especially when the optimization process is aware of the structure of the
encoding scheme (e.g., by using the softmax structural prior), categorical
features are much easier to reconstruct. This poses a critical privacy risk in
tabular federated learning, as sensitive features are often categorical, e.g.,
gender or race.
Figure 3: The inversion accuracy on the Adult dataset over varying batch size
separated for discrete (D) and continuous (C) features.
#### Federated Averaging
In training with FedAvg (McMahan et al., 2017) participating clients conduct
local training of several updates before communicating their new parameters to
the server. Note that the more local updates are conducted by the clients, the
harder a reconstruction attack becomes, making leakage attacks against FedAvg
more challenging. Although this training method is of significantly higher
practical importance than FedSGD, most prior work does not evaluate against
it. Building upon the work of Dimitrov et al. (2022b) (for details please see
App. B and the work of Dimitrov et al. (2022b)), we evaluate our combined
attack and the cosine baseline in the setting of Federated Averaging. We
present our results of retrieving a client dataset of size $32$ over varying
number of local batches and epochs on the Adult dataset in Tab. 2, while
assuming full knowledge of the true labels. We observe that while our combined
attack significantly outperforms the random baseline of $58.0\%$ accuracy even
up to $40$ local updates, the baseline attack fails to consistently do so
whenever the local training is longer than one epoch. As FedAvg with tabular
data is of high practical relevance, our results which highlight its
vulnerability are concerning. We show further details for the experimental
setup and results on other datasets in App. B and App. D, respectively.
Table 2: Mean and standard deviation of the inversion accuracy [%] on FedAvg
with local dataset sizes of $32$ on the Adult dataset. The accuracy of the
random baseline for $32$ datapoints is $58.0\pm 2.9$.
| TabLeak | Cosine
---|---|---
#batches | 1 epoch | 5 epochs | 10 epochs | 1 epoch | 5 epochs | 10 epochs
1 | $\mathbf{77.4\pm 4.5}$ | $\mathbf{71.1\pm 2.9}$ | $\mathbf{67.6\pm 3.7}$ | $65.2\pm 2.7$ | $56.1\pm 4.1$ | $53.2\pm 4.2$
2 | $\mathbf{75.7\pm 5.0}$ | $\mathbf{71.7\pm 3.9}$ | $\mathbf{67.7\pm 4.2}$ | $64.8\pm 3.3$ | $56.4\pm 4.8$ | $56.2\pm 4.8$
4 | $\mathbf{75.9\pm 4.4}$ | $\mathbf{71.0\pm 3.2}$ | $\mathbf{67.4\pm 3.4}$ | $64.8\pm 3.4$ | $58.7\pm 4.6$ | $56.6\pm 5.0$
Table 3: The mean accuracy [%] and entropies with the corresponding standard
deviations over batch sizes of the categorical (top) and continuous (bottom)
features.
| Batch Size
---|---
| 8 | 16 | 32 | 64 | 128
Acc. | $98.5\pm 5.6$ | $97.2\pm 4.3$ | $91.0\pm 4.4$ | $83.2\pm 3.6$ | $78.5\pm 1.8$
$\bar{H}^{D}$ | $0.15\pm 0.13$ | $0.26\pm 0.11$ | $0.40\pm 0.06$ | $0.48\pm 0.04$ | $0.53\pm 0.03$
Acc. | $90.9\pm 14.7$ | $78.8\pm 13.5$ | $59.2\pm 6.9$ | $55.1\pm 3.0$ | $55.7\pm 2.0$
$H^{C}$ | $-1.11\pm 0.95$ | $-0.11\pm 0.63$ | $0.77\pm 0.30$ | $1.21\pm 0.19$ | $1.48\pm 0.10$
#### Assessing Reconstructions via Entropy
Table 4: The mean accuracy [%] and the share of data [%] in each entropy
bucket for batch size 128 on the Adult dataset.
Entropy | Categorical Features
---|---
Bucket | Accuracy [%] | Data [%]
0.0-0.2 | $95.7$ | $8.1$
0.2-0.4 | $90.5$ | $23.4$
0.4-0.6 | $79.8$ | $27.7$
0.6-0.8 | $69.8$ | $29.2$
0.8-1.0 | $61.2$ | $11.6$
Overall | $78.5$ | $100$
Random | $73.8$ | $100$
We now investigate how an adversary can use the entropy (introduced in Sec.
3.3) to assess the quality of their reconstructions. In Tab. 3 we show the
mean and standard deviation of the accuracy and the entropy of both the
discrete and the continuous features over increasing batch sizes after
reconstructing with TabLeak (ensemble size $30$). We observe an increase in
the mean entropy over the increasing batch sizes, corresponding to accuracy
decrease in the reconstructed batches. Hence, an attacker can understand the
global effectiveness of their attack by looking at the retrieved entropies,
without having to compare their results to the ground truth.
We now look at a single batch of size $128$ and put each categorical feature
into a bucket based on their reconstruction entropy after attacking with
TabLeak (ensemble size $30$). In Tab. 4 we present our results, showing that
features falling into lower entropy buckets (0.0-0.2 and 0.2-0.4) inside a
batch are significantly more accurately reconstructed ($>90\%$) than the
overall batch ($78.5$%). Note that this bucketing can be done without the
knowledge of the ground-truth, yet the adversary can concretely identify the
high-fidelity features in their noisy reconstruction. This shows that even for
reconstructions of large batches that seem to contain little-to-no information
(close to random baseline), an adversary can still extract subsets of the data
with high accuracy. Tables containing both feature types on all four datasets
can be found in Sec. D.4, providing analogous conclusions.
## 5 Conclusion
In this work we presented TabLeak, the first data leakage attack on tabular
data in the setting of federated learning (FL), obtaining state-of-the-art
results against both popular FL training protocols in the tabular domain. As
tabular data is ubiquitous in privacy critical high-stakes applications, our
results raise important concerns regarding practical systems currently using
FL. Therefore, we advocate for further research on advancing defenses
necessary to mitigate such privacy leaks.
## 6 Ethics Statement
As tabular data is often used in high-stakes applications and may contain
sensitive data of natural or legal persons, confidential treatment is
critical. This work presents an attack algorithm in the tabular data domain
that enables an FL server to steal the private data of its clients in
industry-relevant scenarios, deeming such applications potentially unsafe.
We believe that exposing vulnerabilities of both recently proposed and widely
adopted systems, where privacy is a concern, can benefit the development of
adequate safety mechanisms against malicious actors. In particular, this view
is shared by the governmental institutions of the United States of America and
the United Kingdom that jointly supported the launch of a competition
(https://petsprizechallenges.com/) aimed at advancing the privacy of FL in the
tabular domain, encouraging the participation of both teams developing
defenses and attacks. Also, as our experiments in Sec. C.1 show, existing
techniques can help mitigate the privacy threat, hence we encourage
practitioners to make use of them.
## References
* Abadi et al. (2016) M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang, “Deep learning with differential privacy,” in _Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security_ , ser. CCS ’16. New York, NY, USA: Association for Computing Machinery, 2016, p. 308–318. [Online]. Available: https://doi.org/10.1145/2976749.2978318
* Balunovic et al. (2022) M. Balunovic, D. I. Dimitrov, R. Staab, and M. Vechev, “Bayesian framework for gradient leakage,” in _International Conference on Learning Representations_ , 2022. [Online]. Available: https://openreview.net/forum?id=f2lrIbGx3x7
* Borisov et al. (2021) V. Borisov, T. Leemann, K. Seßler, J. Haug, M. Pawelczyk, and G. Kasneci, “Deep neural networks and tabular data: A survey,” _CoRR_ , vol. abs/2110.01889, 2021. [Online]. Available: https://arxiv.org/abs/2110.01889
* Deng et al. (2021) J. Deng, Y. Wang, J. Li, C. Wang, C. Shang, H. Liu, S. Rajasekaran, and C. Ding, “Tag: Gradient attack on transformer-based language models,” in _EMNLP (Findings)_ , 2021, pp. 3600–3610. [Online]. Available: https://aclanthology.org/2021.findings-emnlp.305
* Dimitrov et al. (2022a) D. I. Dimitrov, M. Balunović, N. Jovanović, and M. Vechev, “Lamp: Extracting text from gradients with language model priors,” 2022.
* Dimitrov et al. (2022b) D. I. Dimitrov, M. Balunović, N. Konstantinov, and M. Vechev, “Data leakage in federated averaging,” 2022. [Online]. Available: https://arxiv.org/abs/2206.12395
* Dua and Graff (2017) D. Dua and C. Graff, “UCI machine learning repository,” 2017. [Online]. Available: http://archive.ics.uci.edu/ml
* Fowl et al. (2022) L. H. Fowl, J. Geiping, W. Czaja, M. Goldblum, and T. Goldstein, “Robbing the fed: Directly obtaining private data in federated learning with modified models,” in _International Conference on Learning Representations_ , 2022\. [Online]. Available: https://openreview.net/forum?id=fwzUgo0FM9v
* Geiping et al. (2020) J. Geiping, H. Bauermeister, H. Dröge, and M. Moeller, “Inverting gradients-how easy is it to break privacy in federated learning?” pp. 16 937–16 947, 2020.
* Geng et al. (2021) J. Geng, Y. Mou, F. Li, Q. Li, O. Beyan, S. Decker, and C. Rong, “Towards general deep leakage in federated learning,” 2021.
* Gupta et al. (2022) S. Gupta, Y. Huang, Z. Zhong, T. Gao, K. Li, and D. Chen, “Recovering private text in federated learning of language models,” 2022. [Online]. Available: https://arxiv.org/abs/2205.08514
* Hastie et al. (2009) T. Hastie, R. Tibshirani, and J. Friedman, _The Elements of Statistical Learning: Data Mining, Inference, and Prediction_ , ser. Springer series in statistics. Springer, 2009. [Online]. Available: https://books.google.ch/books?id=eBSgoAEACAAJ
* Huang et al. (2021) Y. Huang, S. Gupta, Z. Song, K. Li, and S. Arora, “Evaluating gradient inversion attacks and defenses in federated learning,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 7232–7241, 2021.
* Jeon et al. (2021) J. Jeon, K. Lee, S. Oh, J. Ok _et al._ , “Gradient inversion with generative image prior,” pp. 29 898–29 908, 2021.
* Jin et al. (2021) X. Jin, P.-Y. Chen, C.-Y. Hsu, C.-M. Yu, and T. Chen, “Cafe: Catastrophic data leakage in vertical federated learning,” pp. 994–1006, 2021.
* Kingma and Ba (2014) D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _International Conference on Learning Representations_ , 12 2014.
* Kuhn (1955) H. W. Kuhn, “The hungarian method for the assignment problem,” _Naval Research Logistics Quarterly_ , vol. 2, no. 1-2, pp. 83–97, 1955. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800020109
* Liu and Nocedal (1989) D. C. Liu and J. Nocedal, “On the limited memory bfgs method for large scale optimization,” _Mathematical Programming_ , vol. 45, pp. 503–528, 1989.
* Long et al. (2021) G. Long, Y. Tan, J. Jiang, and C. Zhang, “Federated learning for open banking,” 2021. [Online]. Available: https://arxiv.org/abs/2108.10749
* McMahan et al. (2017) B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” pp. 1273–1282, 2017.
* Melis et al. (2019) L. Melis, C. Song, E. De Cristofaro, and V. Shmatikov, “Exploiting unintended feature leakage in collaborative learning,” in _2019 IEEE symposium on security and privacy (SP)_. IEEE, 2019, pp. 691–706.
* Rieke et al. (2020) N. Rieke, J. Hancox, W. Li, F. Milletarì, H. R. Roth, S. Albarqouni, S. Bakas, M. N. Galtier, B. A. Landman, K. Maier-Hein, S. Ourselin, M. Sheller, R. M. Summers, A. Trask, D. Xu, M. Baust, and M. J. Cardoso, “The future of digital health with federated learning,” _npj Digital Medicine_ , vol. 3, no. 1, sep 2020. [Online]. Available: https://doi.org/10.1038%2Fs41746-020-00323-1
* Shokri and Shmatikov (2015) R. Shokri and V. Shmatikov, “Privacy-preserving deep learning,” in _Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security_ , ser. CCS ’15. New York, NY, USA: Association for Computing Machinery, 2015, p. 1310–1321. [Online]. Available: https://doi.org/10.1145/2810103.2813687
* Wen et al. (2022) Y. Wen, J. A. Geiping, L. Fowl, M. Goldblum, and T. Goldstein, “Fishing for user data in large-batch federated learning via gradient magnification,” in _Proceedings of the 39th International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, Eds., vol. 162. PMLR, 17–23 Jul 2022, pp. 23 668–23 684. [Online]. Available: https://proceedings.mlr.press/v162/wen22a.html
* Wightman (2017) F. L. Wightman, “LSAC national longitudinal bar passage study,” 2017.
* Yin et al. (2021) H. Yin, A. Mallya, A. Vahdat, J. M. Alvarez, J. Kautz, and P. Molchanov, “See through gradients: Image batch recovery via gradinversion,” pp. 16 337–16 346, 2021.
* Zhao et al. (2020) B. Zhao, K. R. Mopuri, and H. Bilen, “idlg: Improved deep leakage from gradients,” 2020. [Online]. Available: https://arxiv.org/abs/2001.02610
* Zhu and Blaschko (2021) J. Zhu and M. B. Blaschko, “R-{gap}: Recursive gradient attack on privacy,” in _International Conference on Learning Representations_ , 2021\. [Online]. Available: https://openreview.net/forum?id=RSU17UoKfJF
* Zhu et al. (2019) L. Zhu, Z. Liu, and S. Han, “Deep leakage from gradients,” 2019.
## Appendix A Accuracy Metric
To ease the understanding, we start by repeating our accuracy metric here,
where we measure the reconstruction accuracy between the retrieved sample
$\hat{x}$ and the ground truth $x$ as:
$\text{accuracy}(x,\,\hat{x})\coloneqq\frac{1}{K+L}\left(\sum_{i=1}^{K}\mathbb{I}\\{x^{D}_{i}=\hat{x}^{D}_{i}\\}+\sum_{i=1}^{L}\mathbb{I}\\{\hat{x}^{C}_{i}\in[x^{C}_{i}-\epsilon_{i},\,x^{C}_{i}+\epsilon_{i}]\\}\right).$
(8)
Note that the binary treatment of continuous features in our accuracy metric
enables the combined measurement of the accuracy on both the discrete and the
continuous features. From an intuitive point of view, this measure closely
resembles how one would judge the correctness of numerical guesses. For
example, guessing the age of a $25$ year old, one would deem the guess good if
it is within $3$ to $4$ years of the true value, but the guesses $65$ and $87$
would be both qualitatively incorrect. In order to facilitate scalability of
our experiments, we chose the $\left\\{\epsilon_{i}\right\\}_{i=1}^{L}$ error-
tolerance-bounds based on the global standard deviation if the given
continuous feature $\sigma^{C}_{i}$ and multiplied it by a constant,
concretely, we used $\epsilon_{i}=0.319\,\sigma^{C}_{i}$ for all our
experiments. Note that $Pr[\mu-0.319\,\sigma<x<\mu+0.319\,\sigma]\approx 0.25$
for a Gaussian random variable $x$ with mean $\mu$ and variance $\sigma^{2}$.
For our metric this means that assuming Gaussian zero-mean error in the
reconstruction around the true value, we accept our reconstruction as privacy
leakage as long as we fall into the $25\%$ error-probability range around the
correct value. In Tab. 5 we list the tolerance bounds $\epsilon_{i}$ for the
continuous features of the Adult dataset produced by this method. We would
like to remark here, that we fixed our metric parameters before conducting any
experiments, and did not adjust them based on any obtained results. Note also
that in App. C we provide results where the continuous feature reconstruction
accuracy is measured using the commonly used regression metric of root mean
squared error (RMSE), where TabLeak also achieves the best results, signaling
that the success of our method is independent of our chosen metric.
Table 5: Resulting tolerance bounds on the Adult dataset when using
$\epsilon_{i}=0.319\,\sigma_{i}^{C}$, as used by us for our experiments.
feature | age | fnlwgt | education-num | capital-gain | capital-loss | hours-per-week
---|---|---|---|---|---|---
tolerance | 4.2 | 33699 | 0.8 | 2395 | 129 | 3.8
## Appendix B Further Experimental Details
Here we give an extended description to our experimental details provided in
Sec. 4. For all attacks, we use the Adam optimizer (Kingma and Ba, 2014) with
learning rate $0.06$ for $1\,500$ iterations and without a learning rate
schedule. We chose the learning rate based on our experiments on the baseline
attack where it performed best. In line with Geiping et al. (2020), we modify
the update step of the optimizer by reducing the update gradient to its
element-wise sign. We attack a fully connected neural network with two hidden
layers of $100$ neurons each at initialization. However, we provide a network-
size ablation in Fig. 8, where we evaluate our attack against the baseline
method for $5$ different network architectures. For each reported metric we
conduct $50$ independent runs on 50 different batches to estimate their
statistics. For all FedSGD experiments we clamp the continuous features to
their valid ranges before measuring the reconstruction accuracy, both for our
attacks and the baseline methods. We ran each of our experiments on single
cores of Intel(R) Xeon(R) CPU E5-2690 v4 @ 2.60GHz.
#### Federated Averaging Experiments
For experiments on attacking the FedAvg training algorithm, we fix the
clients’ local dataset size at $32$ and conduct an attack after local training
with learning rate $0.01$ on the initialized network described above. We use
the FedAvg attack-framework of Dimitrov et al. (2022b), where for each local
training epoch we initialize an independent mini-dataset matching the size of
the client dataset, and simulate the local training of the client. At each
reconstruction update, we use the mean squared error between the different
epoch data means ($D_{\text{inv}}=\ell_{2}$ and $g=\text{mean}$ in Dimitrov et
al. (2022b)) as the permutation invariant epoch prior required by the
framework, ensuring the consistency of the reconstructed dataset. For the full
technical details, please refer to the manuscript of Dimitrov et al. (2022b).
For choosing the prior parameter $\lambda_{\text{inv}}$, we conduct line-
search on each setup and attack method pair individually on the parameters
$[0.0,0.5,0.1,0.05,0.01,0.005,0.001]$, and pick the ones providing the best
results. Further, to reduce computational overhead, we reduce the ensemble
size of TabLeak from $30$ to $15$ for these experiments on all datasets.
## Appendix C Further Experiments
In this subsection, we present three further experiments:
* •
Results of attacking neural networks defended using differentially private
noisy gradients in Sec. C.1.
* •
Ablation study on the impacts of the neural network’s size on the
reconstruction difficulty in Sec. C.2.
* •
Measuring the Root Mean Squared Error (RMSE) of the reconstruction of
continuous features in Sec. C.3.
### C.1 Attack against Gaussian DP
Differential privacy (DP) has recently gained popularity, as a way to prevent
privacy violations in FL (Abadi et al., 2016; Zhu et al., 2019). Unlike,
empirical defenses which are often broken by specifically crafted adversaries
(Balunovic et al., 2022), DP provides guarantees on the amount of data leaked
by a FL model, in terms of the magnitude of random noise the clients add to
their gradients prior to sharing them with the server (Abadi et al., 2016; Zhu
et al., 2019). Naturally, DP methods balance privacy concerns with the
accuracy of the produced model, since bigger noise results in worse models
that are more private. In this subsection, we evaluate TabLeak, and the Cosine
baseline against DP defended gradient updates, where zero-mean Gaussian noise
is added with standard deviations $0.001$, $0.01$, and $0.1$ to the client
gradients. We present our results on the Adult, German Credit, Lawschool
Admissions, and Health Heritage datasets in Fig. 4, Fig. 5, Fig. 6, and Fig.
7, respectively. Although both methods are affected by the defense, our method
consistently produces better reconstructions than the baseline method.
However, for high noise level ($\sigma=0.1$) and larger batch size both
attacks break, advocating for the use of DP defenses in tabular FL to prevent
the high vulnerability exposed by this work.
(a) $\sigma=0.001$
(b) $\sigma=0.01$
(c) $\sigma=0.1$
Figure 4: Mean and standard deviation accuracy [%] curves over batch size at
varying Gaussian noise level $\sigma$ added to the client gradients for
differential privacy on the Adult dataset.
(a) $\sigma=0.001$
(b) $\sigma=0.01$
(c) $\sigma=0.1$
Figure 5: Mean and standard deviation accuracy [%] curves over batch size at
varying Gaussian noise level $\sigma$ added to the client gradients for
differential privacy on the German Credit dataset.
(a) $\sigma=0.001$
(b) $\sigma=0.01$
(c) $\sigma=0.1$
Figure 6: Mean and standard deviation accuracy [%] curves over batch size at
varying Gaussian noise level $\sigma$ added to the client gradients for
differential privacy on the Lawschool Admissions dataset.
(a) $\sigma=0.001$
(b) $\sigma=0.01$
(c) $\sigma=0.1$
Figure 7: Mean and standard deviation accuracy [%] curves over batch size at
varying Gaussian noise level $\sigma$ added to the client gradients for
differential privacy on the Health Heritage dataset.
### C.2 Varying Network Size
To understand the effect the choice of the network has on the obtained
reconstruction results, we defined $4$ additional fully connected networks,
two smaller, and two bigger ones to evaluate TabLeak on. Concretely, we
examined the following five architectures for our attack:
1. 1.
a single hidden layer with $50$ neurons,
2. 2.
a single hidden layer with $100$ neurons,
3. 3.
two hidden layers with $100$ neurons each (network used in main body),
4. 4.
three hidden layers with $200$ neurons each,
5. 5.
and three hidden layers with $400$ neurons each.
We attack the above networks, aiming to reconstruct a batch of size $32$. We
plot the accuracy of TabLeak and the cosine baseline as a function of the
number of parameters in the network in Fig. 8 for all four datasets. We can
observe that with increasing number of parameters in the network, the
reconstruction accuracy significantly increases on all datasets, and rather
surprsingly, allowing for near perfect reconstruction of a batch as large as
$32$ in some cases. Observe that on both ends of the presented parameter scale
the differences between the methods degrade, i.e. they either both converge to
near-perfect reconstruction (large networks) or to random guessing (small
networks). Therefore, the choice of our network for conducting the experiments
was instructive in examining the differences between the methods.
(a) Adult
(b) German Credit
(c) Lawschool Admissions
(d) Health Heritage
Figure 8: Mean attack accuracy curves with standard deviation for batch size
$32$ over varying network size (measured in number of parameters, #Params, log
scale) on all four datasets. We mark the network we used for our other
experiments with a dashed vertical line.
### C.3 Continuous Feature Reconstruction Measured by RMSE
In order to examine the potential influence of our choice of reconstruction
metric on the obtained results, we further measured the reconstruction quality
of continuous features on the widely used Root Mean Squared Error (RMSE)
metric as well. Concretely, we calculate the RMSE between the $L$ continuous
features of our reconstruction $\hat{x}^{C}$ and the ground truth $x$ in a
batch of size $n$ as:
$\text{RMSE}(x^{C},\hat{x}^{C})=\frac{1}{L}\sum_{i=1}^{L}\sqrt{\frac{1}{n}\sum_{j=1}^{n}(x_{ij}^{C}-\hat{x}_{ij}^{C})^{2}}.$
(9)
As our results in Fig. 9 demonstrate, TabLeak achieves significantly lower
RMSE than the Cosine baseline on large batch sizes, for all four datasets
examined. This indicates that the strong results obtained by TabLeak in the
rest of the paper are not a consequence of our evaluation metric.
(a) Adult
(b) German Credit
(c) Lawschool Admissions
(d) Health Heritage
Figure 9: The mean and standard deviation of the Root Mean Square Error (RMSE)
of the reconstructions of the continuous features on all four datasets over
batch sizes.
## Appendix D All Main Results
In this subsection, we include all the results presented in the main part of
this paper for the Adult dataset alongside with the corresponding additional
results on the German Credit, Lawschool Admissions, and the Health Heritage
datasets.
### D.1 Full FedSGD Results on all Datasets
In Tab. 6, Tab. 7, Tab. 8, and Tab. 9 we provide the full attack results of
our method compared to the Cosine and random baselines on the Adult, German
Credit, Lawschool Admissions, and Health Heritage datasets, respectively.
Looking at the results for all datasets, we can confirm the observations made
in Sec. 4, i.e. (i) the lower batch sizes are vulnerable to any non-trivial
attack, (ii) not knowing the ground truth labels does not significantly
disadvantage the attacker for larger batch sizes, and (iii) TabLeak provides a
strong improvement over the baselines for practically relevant batch sizes
over all datasets examined.
Table 6: The mean inversion accuracy [%] and standard deviation of different
methods over varying batch sizes with given true labels (top) and with
reconstructed labels (bottom) on the Adult dataset.
Label | Batch | TabLeak | TabLeak | TabLeak | Cosine | Random
---|---|---|---|---|---|---
| Size | | (no pooling) | (no softmax) | |
True $y$ | $1$ | $99.4\pm 2.8$ | $99.1\pm 4.4$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $43.3\pm 11.8$
$2$ | $99.2\pm 5.5$ | $99.1\pm 6.5$ | $\mathbf{99.9\pm 1.0}$ | $97.6\pm 6.9$ | $47.1\pm 7.9$
$4$ | $98.0\pm 4.5$ | $96.6\pm 7.5$ | $\mathbf{98.9\pm 4.0}$ | $96.4\pm 7.2$ | $49.8\pm 4.9$
$8$ | $\mathbf{95.1\pm 9.2}$ | $93.9\pm 10.2$ | $92.9\pm 6.5$ | $91.1\pm 7.3$ | $53.9\pm 4.4$
$16$ | $\mathbf{89.5\pm 7.6}$ | $84.5\pm 9.9$ | $80.5\pm 4.3$ | $75.0\pm 5.2$ | $55.1\pm 3.9$
$32$ | $\mathbf{77.6\pm 4.8}$ | $72.4\pm 4.6$ | $70.8\pm 3.2$ | $66.6\pm 3.5$ | $58.0\pm 2.9$
$64$ | $\mathbf{71.2\pm 2.8}$ | $66.2\pm 2.8$ | $66.9\pm 2.7$ | $62.5\pm 3.1$ | $59.0\pm 3.2$
$128$ | $\mathbf{68.8\pm 1.3}$ | $64.1\pm 1.4$ | $64.0\pm 2.1$ | $59.5\pm 2.1$ | $61.2\pm 3.1$
Rec. $\hat{y}$ | $1$ | $99.4\pm 2.8$ | $99.3\pm 3.6$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $43.3\pm 11.8$
$2$ | $98.2\pm 8.9$ | $98.1\pm 9.1$ | $\mathbf{98.6\pm 7.5}$ | $95.9\pm 11.5$ | $47.1\pm 7.9$
$4$ | $89.5\pm 13.8$ | $88.0\pm 15.2$ | $\mathbf{90.0\pm 13.0}$ | $87.9\pm 13.7$ | $49.8\pm 4.9$
$8$ | $\mathbf{86.9\pm 11.6}$ | $84.6\pm 13.4$ | $85.8\pm 9.9$ | $83.3\pm 9.7$ | $53.9\pm 4.4$
$16$ | $\mathbf{82.4\pm 8.4}$ | $78.3\pm 9.0$ | $77.7\pm 4.1$ | $73.0\pm 3.5$ | $55.1\pm 3.9$
$32$ | $\mathbf{75.3\pm 4.8}$ | $70.6\pm 4.3$ | $70.2\pm 3.2$ | $66.3\pm 3.4$ | $58.0\pm 2.9$
$64$ | $\mathbf{70.4\pm 3.2}$ | $65.9\pm 3.6$ | $66.8\pm 2.6$ | $63.1\pm 3.2$ | $59.0\pm 3.2$
$128$ | $\mathbf{68.7\pm 1.3}$ | $64.4\pm 1.5$ | $63.8\pm 2.1$ | $59.5\pm 2.1$ | $61.2\pm 3.1$
Table 7: The mean inversion accuracy [%] and standard deviation of different
methods over varying batch sizes with given true labels (top) and with
reconstructed labels (bottom) on the German Credit dataset.
Label | Batch | TabLeak | TabLeak | TabLeak | Cosine | Random
---|---|---|---|---|---|---
| Size | | (no pooling) | (no softmax) | |
True $y$ | $1$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $43.9\pm 9.8$
$2$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $99.9\pm 0.7$ | $98.0\pm 7.1$ | $45.1\pm 6.6$
$4$ | $\mathbf{99.9\pm 0.4}$ | $99.2\pm 3.6$ | $99.5\pm 1.2$ | $97.8\pm 6.0$ | $50.3\pm 4.5$
$8$ | $99.7\pm 1.1$ | $99.1\pm 2.2$ | $\mathbf{98.2\pm 2.5}$ | $96.1\pm 5.2$ | $51.8\pm 3.2$
$16$ | $\mathbf{95.9\pm 3.4}$ | $94.0\pm 4.3$ | $84.1\pm 3.4$ | $79.3\pm 4.4$ | $54.5\pm 3.0$
$32$ | $\mathbf{83.6\pm 2.9}$ | $79.4\pm 3.1$ | $72.1\pm 1.9$ | $69.7\pm 2.2$ | $56.8\pm 2.2$
$64$ | $\mathbf{73.0\pm 1.3}$ | $70.8\pm 1.4$ | $68.9\pm 1.4$ | $66.6\pm 1.8$ | $59.4\pm 1.9$
$128$ | $\mathbf{71.3\pm 0.8}$ | $69.1\pm 0.8$ | $67.4\pm 1.5$ | $64.5\pm 1.5$ | $61.0\pm 2.1$
Rec. $\hat{y}$ | $1$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $43.9\pm 9.8$
$2$ | $\mathbf{100.0\pm 0.0}$ | $99.5\pm 3.5$ | $99.9\pm 0.7$ | $98.8\pm 5.2$ | $45.1\pm 6.6$
$4$ | $\mathbf{99.6\pm 2.6}$ | $99.5\pm 2.9$ | $99.2\pm 3.0$ | $97.4\pm 6.4$ | $50.3\pm 4.5$
$8$ | $\mathbf{97.2\pm 6.1}$ | $96.8\pm 6.8$ | $96.0\pm 6.2$ | $94.8\pm 6.5$ | $51.8\pm 3.2$
$16$ | $\mathbf{91.7\pm 6.5}$ | $90.0\pm 7.3$ | $82.3\pm 4.6$ | $77.9\pm 4.6$ | $54.5\pm 3.0$
$32$ | $\mathbf{81.5\pm 3.4}$ | $77.6\pm 2.8$ | $71.5\pm 2.0$ | $69.1\pm 2.1$ | $56.8\pm 2.2$
$64$ | $\mathbf{72.9\pm 1.4}$ | $70.5\pm 1.4$ | $68.6\pm 1.3$ | $66.5\pm 1.7$ | $59.4\pm 1.9$
$128$ | $\mathbf{71.1\pm 0.9}$ | $69.1\pm 0.7$ | $67.1\pm 1.6$ | $64.4\pm 1.6$ | $61.0\pm 2.1$
Table 8: The mean inversion accuracy [%] and standard deviation of different
methods over varying batch sizes with given true labels (top) and with
reconstructed labels (bottom) on the Lawschool Admissions dataset.
Label | Batch | TabLeak | TabLeak | TabLeak | Cosine | Random
---|---|---|---|---|---|---
| Size | | (no pooling) | (no softmax) | |
True $y$ | $1$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $38.9\pm 14.6$
$2$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $99.9\pm 1.0$ | $96.3\pm 10.4$ | $38.4\pm 11.5$
$4$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $99.7\pm 1.2$ | $97.6\pm 6.9$ | $43.2\pm 7.2$
$8$ | $98.7\pm 3.8$ | $\mathbf{98.8\pm 3.7}$ | $96.0\pm 5.0$ | $94.5\pm 5.8$ | $49.4\pm 4.6$
$16$ | $\mathbf{94.8\pm 5.6}$ | $93.5\pm 6.5$ | $81.1\pm 4.5$ | $77.3\pm 5.5$ | $53.0\pm 3.1$
$32$ | $\mathbf{84.8\pm 3.9}$ | $82.4\pm 4.1$ | $73.3\pm 2.8$ | $71.0\pm 2.8$ | $57.6\pm 2.3$
$64$ | $\mathbf{78.2\pm 2.0}$ | $76.6\pm 2.0$ | $73.0\pm 2.1$ | $71.7\pm 2.2$ | $60.4\pm 2.2$
$128$ | $\mathbf{77.3\pm 1.2}$ | $76.0\pm 1.1$ | $73.7\pm 2.6$ | $71.8\pm 2.7$ | $63.4\pm 1.5$
Rec. $\hat{y}$ | $1$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $\mathbf{100.0\pm 0.0}$ | $38.9\pm 14.6$
$2$ | $99.1\pm 6.0$ | $\mathbf{99.3\pm 5.0}$ | $98.7\pm 7.1$ | $95.9\pm 12.0$ | $38.4\pm 11.5$
$4$ | $\mathbf{99.6\pm 3.0}$ | $99.0\pm 4.9$ | $98.7\pm 6.3$ | $96.8\pm 8.5$ | $43.2\pm 7.2$
$8$ | $\mathbf{95.9\pm 7.8}$ | $95.3\pm 8.3$ | $93.4\pm 7.2$ | $91.9\pm 7.9$ | $49.4\pm 4.6$
$16$ | $\mathbf{91.2\pm 7.3}$ | $89.1\pm 8.3$ | $80.5\pm 4.7$ | $77.4\pm 5.4$ | $53.0\pm 3.1$
$32$ | $\mathbf{83.2\pm 4.1}$ | $80.9\pm 4.3$ | $72.7\pm 2.2$ | $71.0\pm 2.0$ | $57.6\pm 2.3$
$64$ | $\mathbf{77.2\pm 2.4}$ | $76.0\pm 2.2$ | $72.7\pm 2.1$ | $71.5\pm 2.4$ | $60.4\pm 2.2$
$128$ | $\mathbf{77.1\pm 1.2}$ | $75.9\pm 1.3$ | $73.9\pm 2.7$ | $71.8\pm 2.8$ | $63.4\pm 1.5$
Table 9: The mean inversion accuracy [%] and standard deviation of different
methods over varying batch sizes with given true labels (top) and with
reconstructed labels (bottom) on the Health Heritage dataset.
Label | Batch | TabLeak | TabLeak | TabLeak | Cosine | Random
---|---|---|---|---|---|---
| Size | | (no pooling) | (no softmax) | |
True $y$ | $1$ | $\mathbf{99.8\pm 1.6}$ | $\mathbf{99.8\pm 1.6}$ | $\mathbf{99.8\pm 1.6}$ | $\mathbf{99.8\pm 1.6}$ | $34.8\pm 13.1$
$2$ | $97.7\pm 8.3$ | $97.2\pm 10.2$ | $\mathbf{98.6\pm 3.3}$ | $97.9\pm 5.6$ | $36.9\pm 9.8$
$4$ | $\mathbf{98.2\pm 6.5}$ | $96.1\pm 9.8$ | $97.8\pm 4.2$ | $95.6\pm 8.1$ | $37.0\pm 5.3$
$8$ | $\mathbf{96.0\pm 8.2}$ | $94.2\pm 10.5$ | $89.2\pm 9.1$ | $86.2\pm 9.0$ | $39.2\pm 3.8$
$16$ | $\mathbf{86.1\pm 8.8}$ | $80.6\pm 9.9$ | $67.8\pm 4.8$ | $63.6\pm 5.5$ | $41.4\pm 3.7$
$32$ | $\mathbf{70.0\pm 4.5}$ | $64.7\pm 3.9$ | $61.4\pm 4.0$ | $57.7\pm 4.1$ | $43.4\pm 2.8$
$64$ | $\mathbf{64.7\pm 2.8}$ | $59.6\pm 2.7$ | $61.5\pm 4.3$ | $57.4\pm 4.7$ | $45.0\pm 3.7$
$128$ | $\mathbf{63.0\pm 1.4}$ | $57.9\pm 1.6$ | $59.9\pm 5.0$ | $55.6\pm 4.8$ | $46.8\pm 3.2$
Rec. $\hat{y}$ | $1$ | $99.8\pm 1.6$ | $\mathbf{99.9\pm 0.8}$ | $99.8\pm 1.6$ | $99.6\pm 2.5$ | $34.8\pm 13.1$
$2$ | $\mathbf{95.4\pm 13.6}$ | $94.8\pm 15.1$ | $95.2\pm 13.6$ | $92.5\pm 16.9$ | $36.9\pm 9.8$
$4$ | $\mathbf{86.6\pm 20.2}$ | $84.7\pm 22.0$ | $84.7\pm 20.8$ | $83.5\pm 20.7$ | $37.0\pm 5.3$
$8$ | $\mathbf{82.4\pm 15.6}$ | $80.5\pm 16.3$ | $77.3\pm 13.3$ | $74.5\pm 13.8$ | $39.2\pm 3.8$
$16$ | $\mathbf{75.9\pm 12.4}$ | $71.4\pm 11.4$ | $64.8\pm 7.6$ | $60.9\pm 6.3$ | $41.4\pm 3.7$
$32$ | $\mathbf{64.8\pm 5.7}$ | $60.8\pm 4.7$ | $59.8\pm 3.8$ | $56.9\pm 4.0$ | $43.4\pm 2.8$
$64$ | $\mathbf{62.6\pm 2.6}$ | $59.6\pm 2.6$ | $60.9\pm 4.0$ | $57.7\pm 4.7$ | $45.0\pm 3.7$
$128$ | $\mathbf{62.7\pm 1.6}$ | $59.2\pm 1.6$ | $59.6\pm 5.1$ | $55.7\pm 5.0$ | $46.8\pm 3.2$
### D.2 Categorical vs. Continuous Features on all Datasets
In Fig. 10, we compare the reconstruction accuracy of the continuous and the
discrete features on all four datasets. We confirm our observations, shown in
Fig. 3 in the main text, that a strong dichotomy between continuous and
discrete feature reconstruction accuracy exists on all 4 datasets.
(a) Adult
(b) German Credit
(c) Lawschool Admissions
(d) Health Heritage
Figure 10: Mean reconstruction accuracy curves with corresponding standard
deviations over varying batch size, separately for the discrete and the
continuous features on all four datasets.
### D.3 Federated Averaging Results on all Datasets
In Tab. 10, Tab. 11, Tab. 12, and Tab. 13 we present our results on attacking
the clients in FedAvg training on the Adult, German Credit, Lawschool
Submissions, and Health Heritage datasets, respectively. We described the
details of the experiment in App. B above. Confirming our conclusions drawn in
the main part of this manuscript, we observe that TabLeak achieves non-trivial
reconstruction accuracy over all settings and even for large numbers of
updates, while the baseline attack often fails to outperform random guessing,
when the number of local updates is increased.
Table 10: Mean and standard deviation of the inversion accuracy [%] with local
dataset size of $32$ in FedAvg training on the Adult dataset. The accuracy of
the random baseline for $32$ datapoints is $58.0\pm 2.9$.
| TabLeak | Cosine
---|---|---
#batches | 1 epoch | 5 epochs | 10 epochs | 1 epoch | 5 epochs | 10 epochs
1 | $\mathbf{77.4\pm 4.5}$ | $\mathbf{71.1\pm 2.9}$ | $\mathbf{67.6\pm 3.7}$ | $65.2\pm 2.7$ | $56.1\pm 4.1$ | $53.2\pm 4.2$
2 | $\mathbf{75.7\pm 5.0}$ | $\mathbf{71.7\pm 3.9}$ | $\mathbf{67.7\pm 4.2}$ | $64.8\pm 3.3$ | $56.4\pm 4.8$ | $56.2\pm 4.8$
4 | $\mathbf{75.9\pm 4.4}$ | $\mathbf{71.0\pm 3.2}$ | $\mathbf{67.4\pm 3.4}$ | $64.8\pm 3.4$ | $58.7\pm 4.6$ | $56.6\pm 5.0$
Table 11: Mean and standard deviation of the inversion accuracy [%] with local
dataset size of $32$ in FedAvg training on the German Credit dataset. The
accuracy of the random baseline for $32$ datapoints is $56.9\pm 2.1$.
| TabLeak | Cosine
---|---|---
#batches | 1 epoch | 5 epochs | 10 epochs | 1 epoch | 5 epochs | 10 epochs
1 | $\mathbf{95.2\pm 3.8}$ | $\mathbf{87.9\pm 6.2}$ | $\mathbf{83.4\pm 4.6}$ | $78.2\pm 4.6$ | $65.4\pm 6.2$ | $62.5\pm 6.1$
2 | $\mathbf{95.5\pm 3.9}$ | $\mathbf{88.2\pm 5.2}$ | $\mathbf{84.0\pm 6.6}$ | $78.3\pm 5.8$ | $68.8\pm 6.6$ | $63.4\pm 4.8$
4 | $\mathbf{95.6\pm 3.6}$ | $\mathbf{85.5\pm 6.0}$ | $\mathbf{81.0\pm 6.1}$ | $79.2\pm 4.9$ | $67.4\pm 4.8$ | $62.6\pm 6.5$
Table 12: Mean and standard deviation of the inversion accuracy [%] with local
dataset size of $32$ in FedAvg training on the Lawschool Admissions dataset.
The accuracy of the random baseline for $32$ datapoints is $57.8\pm 2.3$.
| TabLeak | Cosine
---|---|---
#batches | 1 epoch | 5 epochs | 10 epochs | 1 epoch | 5 epochs | 10 epochs
1 | $\mathbf{85.6\pm 3.8}$ | $\mathbf{83.3\pm 2.9}$ | $\mathbf{80.7\pm 4.1}$ | $72.2\pm 2.6$ | $68.1\pm 3.1$ | $65.2\pm 2.8$
2 | $\mathbf{86.0\pm 3.8}$ | $\mathbf{83.0\pm 3.2}$ | $\mathbf{79.8\pm 3.5}$ | $72.5\pm 1.9$ | $68.3\pm 4.4$ | $66.2\pm 2.8$
4 | $\mathbf{85.8\pm 3.5}$ | $\mathbf{81.7\pm 3.8}$ | $\mathbf{79.3\pm 4.3}$ | $72.5\pm 2.4$ | $69.4\pm 3.9$ | $67.9\pm 3.8$
Table 13: Mean and standard deviation of the inversion accuracy [%] with local
dataset size of $32$ in FedAvg training on the Health Heritage dataset. The
accuracy of the random baseline for $32$ datapoints is $43.4\pm 3.5$.
| TabLeak | Cosine
---|---|---
#batches | 1 epoch | 5 epochs | 10 epochs | 1 epoch | 5 epochs | 10 epochs
1 | $\mathbf{68.5\pm 5.0}$ | $\mathbf{62.2\pm 3.5}$ | $\mathbf{57.4\pm 3.0}$ | $53.8\pm 5.5$ | $41.4\pm 3.6$ | $41.1\pm 3.4$
2 | $\mathbf{68.1\pm 4.9}$ | $\mathbf{62.4\pm 4.1}$ | $\mathbf{57.0\pm 2.8}$ | $52.4\pm 5.7$ | $43.4\pm 4.28$ | $44.4\pm 4.3$
4 | $\mathbf{67.3\pm 5.8}$ | $\mathbf{62.0\pm 3.5}$ | $\mathbf{57.0\pm 3.0}$ | $52.5\pm 6.6$ | $43.4\pm 5.7$ | $44.8\pm 4.4$
### D.4 Full Results on Entropy on all Datasets
In Tab. 14, Tab. 15, Tab. 16, and Tab. 17 we provide the mean and standard
deviation of the reconstruction accuracy and the entropy of the continuous and
the categorical features over increasing batch size for attacking with TabLeak
on the four datasets. In support of Sec. 4, we can observe on all datasets a
trend of increasing entropy over decreasing reconstruction accuracy as the
batch size is increased; and as such providing a signal to the attacker about
their overall reconstruction success.
To generalize our results on the local information contained in the entropy,
we show the mean reconstruction accuracy of both the discrete and the
continuous features with respect to bucketing them based on their entropy in a
batch of size $128$ in Tab. 18, Tab. 19, Tab. 20, and Tab. 21 for all four
datasets, respectively. We can see that with the help of this bucketing, we
can identify subsets of the reconstructed features that have been retrieved
with a (sometimes significantly e.g., up to 24%) higher accuracy than the
overall batch.
Table 14: The mean accuracy [%] and entropies with the corresponding standard
deviations over batch sizes of the categorical and the continuous features on
the Adult dataset.
| Discrete | Continuous
---|---|---
| Accuracy | Entropy | Accuracy | Entropy
1 | $100.0\pm 0.0$ | $0.01\pm 0.04$ | $98.7\pm 6.5$ | $-3.1\pm 0.26$
2 | $99.5\pm 3.5$ | $0.01\pm 0.06$ | $98.8\pm 8.2$ | $-2.82\pm 0.57$
4 | $99.5\pm 1.4$ | $0.08\pm 0.11$ | $95.8\pm 9.9$ | $-1.89\pm 0.99$
8 | $98.5\pm 5.6$ | $0.15\pm 0.13$ | $90.9\pm 14.7$ | $-1.11\pm 0.95$
16 | $97.2\pm 4.3$ | $0.26\pm 0.11$ | $78.8\pm 13.5$ | $-0.11\pm 0.63$
32 | $91.0\pm 4.4$ | $0.40\pm 0.06$ | $59.2\pm 6.9$ | $0.77\pm 0.30$
64 | $83.2\pm 3.6$ | $0.48\pm 0.04$ | $55.1\pm 3.0$ | $1.21\pm 0.19$
128 | $78.5\pm 1.8$ | $0.53\pm 0.03$ | $55.7\pm 2.0$ | $1.48\pm 0.10$
Table 15: The mean accuracy [%] and entropies with the corresponding standard
deviations over batch sizes of the categorical and the continuous features on
the German Credit dataset.
| Discrete | Continuous
---|---|---
| Accuracy | Entropy | Accuracy | Entropy
1 | $100.0\pm 0.0$ | $0.00\pm 0.01$ | $100.0\pm 0.0$ | $-3.10\pm 0.18$
2 | $100.0\pm 0.0$ | $0.03\pm 0.05$ | $100.0\pm 0.0$ | $-2.41\pm 0.97$
4 | $100.0\pm 0.0$ | $0.07\pm 0.05$ | $99.8\pm 1.1$ | $-1.66\pm 0.80$
8 | $100.0\pm 0.0$ | $0.10\pm 0.07$ | $99.1\pm 3.1$ | $-1.38\pm 0.54$
16 | $99.5\pm 1.4$ | $0.25\pm 0.07$ | $89.1\pm 8.1$ | $-0.35\pm 0.22$
32 | $93.0\pm 2.1$ | $0.43\pm 0.04$ | $66.0\pm 4.9$ | $0.60\pm 0.13$
64 | $81.9\pm 1.8$ | $0.56\pm 0.02$ | $57.5\pm 2.2$ | $1.08\pm 0.06$
128 | $78.2\pm 1.1$ | $0.59\pm 0.02$ | $58.4\pm 1.7$ | $1.30\pm 0.05$
Table 16: The mean accuracy [%] and entropies with the corresponding standard
deviations over batch sizes of the categorical and the continuous features on
the Lawschool Admissions dataset.
| Discrete | Continuous
---|---|---
| Accuracy | Entropy | Accuracy | Entropy
1 | $100.0\pm 0.0$ | $0.01\pm 0.03$ | $100.0\pm 0.0$ | $-3.28\pm 0.29$
2 | $100.0\pm 0.0$ | $0.02\pm 0.05$ | $100.0\pm 0.0$ | $-2.85\pm 0.87$
4 | $100.0\pm 0.0$ | $0.03\pm 0.04$ | $100.0\pm 0.0$ | $-2.45\pm 0.78$
8 | $99.8\pm 1.1$ | $0.11\pm 0.10$ | $96.4\pm 11.1$ | $-1.77\pm 0.62$
16 | $98.1\pm 2.8$ | $0.24\pm 0.11$ | $87.1\pm 13.4$ | $-0.65\pm 0.49$
32 | $93.4\pm 3.0$ | $0.41\pm 0.08$ | $65.1\pm 8.0$ | $0.18\pm 0.20$
64 | $87.0\pm 2.4$ | $0.55\pm 0.05$ | $57.7\pm 4.8$ | $0.78\pm 0.12$
128 | $83.5\pm 1.5$ | $0.60\pm 0.03$ | $62.6\pm 3.4$ | $1.07\pm 0.11$
Table 17: The mean accuracy [%] and entropies with the corresponding standard
deviations over batch sizes of the categorical and the continuous features on
the Health Heritage dataset.
| Discrete | Continuous
---|---|---
| Accuracy | Entropy | Accuracy | Entropy
1 | $100.0\pm 0.0$ | $0.02\pm 0.05$ | $99.6\pm 2.5$ | $-2.97\pm 0.33$
2 | $100.0\pm 0.0$ | $0.05\pm 0.09$ | $96.5\pm 12.9$ | $-2.55\pm 0.80$
4 | $99.7\pm 1.8$ | $0.08\pm 0.10$ | $97.5\pm 9.0$ | $-1.71\pm 0.79$
8 | $99.1\pm 3.7$ | $0.13\pm 0.11$ | $94.3\pm 11.3$ | $-1.06\pm 0.64$
16 | $96.6\pm 7.5$ | $0.26\pm 0.10$ | $80.2\pm 11.2$ | $-0.11\pm 0.42$
32 | $85.1\pm 6.4$ | $0.43\pm 0.06$ | $61.7\pm 3.8$ | $0.72\pm 0.23$
64 | $73.1\pm 4.7$ | $0.52\pm 0.03$ | $59.6\pm 2.5$ | $1.13\pm 0.20$
128 | $66.1\pm 2.4$ | $0.57\pm 0.02$ | $60.7\pm 1.6$ | $1.44\pm 0.13$
Table 18: The mean accuracy [%] and the share of data [%] in each entropy bucket for batch size 128 on the Adult dataset. Entropy | Categorical Features | Entropy | Continuous Features
---|---|---|---
Bucket | Accuracy [%] | Data [%] | Bucket | Accuracy [%] | Data [%]
0.0-0.2 | $95.7$ | $8.1$ | $\infty$-0.72 | 72.7 | 1.2
0.2-0.4 | $90.5$ | $23.4$ | 0.72-1.16 | 65.5 | 13.6
0.4-0.6 | $79.8$ | $27.7$ | 1.16-1.6 | 56.4 | 50
0.6-0.8 | $69.8$ | $29.2$ | 1.6-2.04 | 51.1 | 32.4
0.8-1.0 | $61.2$ | $11.6$ | 2.04-$\infty$ | 41.8 | 2.9
Overall | $78.5$ | $100$ | Overall | 55.7 | $100$
Random | $73.8$ | $100$ | Random | 44.4 | $100$
Table 19: The mean accuracy [%] and the share of data [%] in each entropy bucket for batch size 128 on the German Credit dataset. Entropy | Categorical Features | Entropy | Continuous Features
---|---|---|---
Bucket | Accuracy [%] | Data [%] | Bucket | Accuracy [%] | Data [%]
0.0-0.2 | $98.1$ | $7.4$ | $\infty$-0.72 | 55.7 | 1.2
0.2-0.4 | $92.5$ | $15.3$ | 0.72-1.16 | 62.3 | 28.1
0.4-0.6 | $83.3$ | $22.2$ | 1.16-1.6 | 57.7 | 56.4
0.6-0.8 | $71.6$ | $33.7$ | 1.6-2.04 | 53.4 | 13.2
0.8-1.0 | $66.0$ | $21.3$ | 2.04-$\infty$ | 48.2 | 0.2
Overall | $78.2$ | $100$ | Overall | $58.4$ | $100$
Random | $73.5$ | $100$ | Random | $37.8$ | $100$
Table 20: The mean accuracy [%] and the share of data [%] in each entropy bucket for batch size 128 on the Lawschool Admissions dataset. Entropy | Categorical Features | Entropy | Continuous Features
---|---|---|---
Bucket | Accuracy [%] | Data [%] | Bucket | Accuracy [%] | Data [%]
0.0-0.2 | $95.5$ | $3.4$ | $\infty$-0.72 | 69.5 | 20.7
0.2-0.4 | $92.1$ | $14.2$ | 0.72-1.16 | 63.3 | 35.1
0.4-0.6 | $88.0$ | $32.7$ | 1.16-1.6 | 60.1 | 32.9
0.6-0.8 | $81.2$ | $30.4$ | 1.6-2.04 | 55.5 | 10.8
0.8-1.0 | $70.7$ | $19.3$ | 2.04-$\infty$ | 54.1 | 0.5
Overall | $83.5$ | $100$ | Overall | $62.6$ | $100$
Random | $81.1$ | $100$ | Random | $19.1$ | $100$
Table 21: The mean accuracy [%] and the share of data [%] in each entropy bucket for batch size 128 on the Health Heritage dataset. Entropy | Categorical Features | Entropy | Continuous Features
---|---|---|---
Bucket | Accuracy [%] | Data [%] | Bucket | Accuracy [%] | Data [%]
0.0-0.2 | $90.7$ | $6.2$ | $\infty$-0.72 | 69.1 | 1.1
0.2-0.4 | $84.7$ | $22.1$ | 0.72-1.16 | 65.6 | 17.0
0.4-0.6 | $70.5$ | $21.2$ | 1.16-1.6 | 61.8 | 52.9
0.6-0.8 | $54.8$ | $32.3$ | 1.6-2.04 | 55.9 | 26.5
0.8-1.0 | $50.3$ | $18.4$ | 2.04-$\infty$ | 49.4 | 2.5
Overall | $66.1$ | $100$ | Overall | $60.7$ | $100$
Random | $69.8$ | $100$ | Random | $34.2$ | $100$
|
# What will it take to generate fairness-preserving explanations?
Jessica Dai Sohini Upadhyay Stephen H. Bach Himabindu Lakkaraju
###### Abstract
In situations where explanations of black-box models may be useful, the
fairness of the black-box is also often a relevant concern. However, the link
between the fairness of the black-box model and the behavior of explanations
for the black-box is unclear. We focus on explanations applied to tabular
datasets, suggesting that explanations do not necessarily preserve the
fairness properties of the black-box algorithm. In other words, explanation
algorithms can ignore or obscure critical relevant properties, creating
incorrect or misleading explanations. More broadly, we propose future research
directions for evaluating and generating explanations such that they are
informative and relevant from a fairness perspective.
Machine Learning, ICML
## 1 Introduction & Motivation
While fairness and explainability are both generally considered core
components of “responsible” machine learning, surprisingly little work has
explored the two principles in tandem. However, especially in light of common
goals of generating explanations for a black-box models, it is critical that
the explanation itself can be reliably trusted to illustrate important
fairness properties of the black-box. For example, Suresh et al. (2021)’s
framework for characterizing stakeholders in explainable machine learning
provides objectives such as debugging or improving the model, ensuring
regulatory compliance, informing downstream actions, justifying actions based
on algorithm output, and contesting a decision; and specific tasks like
assessing the reliability of a prediction; detecting mistaken or
discriminatory behavior; and understanding the influence of different inputs.
Prior work in this area has outlined similar goals for explanations (Bhatt et
al., 2020a). For obvious reasons, if fairness is a concern related to the
model more broadly, it is also a critical consideration for these tasks and
objectives in the context of explanations. Furthermore, while of course
calculating particular fairness desiderata for the underlying black-box
directly might surface unfairness, the stakeholders who are using an
explainable ML algorithm may not have access to the information needed for
such an analysis; as a result, we might hope that explanations themselves
contain sufficient and accurate information for any stakeholder to confidently
make claims and downstream decisions based on the explanation. This is
especially important given that end-users of explanations may be vulnerable to
overtrusting or being manipulated by explanations (Lakkaraju & Bastani, 2020).
However, current methods for evaluating explanations are designed to be almost
entirely application-agnostic, and therefore do not consider any criteria
related to fairness. While terminology varies across the literature, commonly
used evaluation metrics for explanations include fidelity, the extent to which
a surrogate model generated by an explanation algorithm produces predictions
similar to the black-box, and stability, the extent to which explanations
generated for similar (but non-identical) inputs are similar to one another
(Bhatt et al., 2020b; Yeh et al., ).
A growing portion of the literature points to dangers in focusing solely on
these targets when designing explanation algorithms. Slack et al. (2020b) and
Zhang et al. (2019), for instance, highlight the high degree of inconsistency
of explanations generated by perturbation-based methods under certain
parameter settings—in other words, multiple explanations generated for the
same input may result in wildly different explanations. Kumar et al. (2020)
and Hancox-Li & Kumar (2021), meanwhile, investigate SHAP (Lundberg & Lee,
2017), and find problems from both technical and philosophical perspectives.
Under the framework of fairness specifically, Slack et al. (2020a) and Aïvodji
et al. (2019) illustrate specific ways in which either a black-box algorithm
or an explanation, respectively, may be adversarially constructed such that
the explanation, while having high fidelity (or achieving other desirable
metrics), misleadingly suggests that the black-box model is fair when in
reality it is not. However, adversarial construction may not be necessary for
misleading explanations to occur.
For some baseline intuition as to how fairness and explanations may interact,
consider the following. Explanations are often intended to provide a
digestible approximation of the black-box algorithm’s decision boundary,
whether locally (in the neighborhood of a particular input) or globally (for
all possible inputs). Additionally, fairness concerns arise when there is a
meaningful difference in how two or more demographic groups are distributed or
labelled in the training data, which leads to a meaningful disparity in how
the black-box machine learning algorithm performs on the two groups by
whatever metric one may choose (Corbett-Davies & Goel, 2018). The demographic
information may or may not be used by the black-box. In the case that it is
not used, the explanation will not explicitly encode information about the
sensitive attribute, and an end-user relying on the explanation alone will
have little information about the fairness of the black-box. In the case that
it is used, such as when fairness-constrained learning algorithms or
postprocessing methods are applied, then the black-box may learn a decision
boundary such that the boundaries are different when conditioned on group
membership. However, explanation methods are not designed to approximate such
boundaries. Furthermore, the explanation itself will include information about
the sensitive attribute, such as in the form of a feature importance score; it
is not immediately clear what the proper interpretation of that score should
be. Finally, in either case, the known issue of isolating feature attributions
when features may be correlated with one another (Kumar et al., 2020) is
especially relevant when considering fairness applications.
### 1.1 A simple example
Consider the following scenario where the black-box takes in three features:
group, x0, and x1, where group $\in\\{0,1\\}$, x0 and x1 are continuous, and
x0 is correlated with group membership. For some reason or another (perhaps by
applying a fairness intervention in the training process), the black-box’s
learned decision boundaries are different when conditioned on group
membership: specifically, the black-box predicts 1 for group 0 when x1 $>6$,
and for group 1 when x1 $>5$. Figure 1(a) illustrates this black-box decision
boundary.
(a) Decision boundary of the black-box
(b) Decision boundary learned by LIME
Figure 1: The true decision boundaries of the black-box vs the decision
boundary learned by LIME. The left cluster is the minority group, comprising
27% of the total population.
In the case that both groups constitute $50\%$ of the population, an
explanation method that optimizes for fidelity as measured by performance on
sampled neighbors will approximate the decision boundary at around x1
$>5.5$—an explanation that is simply incorrect for both groups based on what
we know about the black-box. If one group is a minority of the population,
however, the explanation’s approximated decision boundary will be closer to
the majority group’s decision boundary, meaning overall better explanations
for the majority group and overall worse explanations for the minority group.
This is illustrated in Figure 1(b), which visualizes the decision boundary
learned by LIME: note that the learned boundary is much closer to x1 $>5$, the
majority group’s boundary, over all data points, not just the points
corresponding to the majority group. Notably, this is a problem that seems to
arise whenever group-conditional decision boundaries are meaningfully
distinct. The explanations generated here, therefore, may be both misleading
and incorrect.
## 2 Our Framework
We propose a two-part framework for further work in this area: first,
determining what constitutes a mismatch in fairness properties; and second,
generating fairness-preserving explanations.
### 2.1 Diagnosing Fairness Mismatch
First, we provide an initial attempt at outlining what metrics or diagnostic
tests may be useful in detecting a mismatch in fairness; these also serve,
therefore, as potential criteria or definitions for what a fairness-preserving
vs fairness-obscuring explanation may look like. These metrics are broadly
motivated by the principle that if the model is fair, the explanations should
not raise false alarms; similarly, if the model is unfair, the explanations
should not suggest that it is innocuous. In this section we attempt to
pinpoint what exactly it means for an explanation to “raise a false alarm” or
suggest that the model “is innocuous.”
Group fairness. There are a variety of metrics through which models can be
audited or monitored for group fairness: demographic parity focuses primarily
on group-wise outcomes, while other metrics such as equalized odds, equal
opportunity, or predictive parity, reflect some combination of the group-
conditional confusion matrices (Verma & Rubin, 2018).
Let $\mathcal{M}$ represent a metric of group fairness which takes in the
predictions of some model (and potentially information about the true labels);
$f$ represent the black-box; $E_{f}$ be the surrogate model from an
explanation for $f$; and $E_{f}(\vec{x})$ represent evaluating the surrogate
model on some input $\vec{x}$. Then, group fairness is preserved when:
$|\mathcal{M}(f(\vec{x}))-\mathcal{M}(E_{f}(\vec{x}))|\leq\epsilon$
In other words, when, if substituting the black-box model with the
explanation’s surrogate model, the predictions generated result in similar
values of the fairness metric $\mathcal{M}$.
Two obvious issues arise with this initial proposition. First, while this is
straightforward for global explanation methods, many of the most popular
explanation methods like LIME (Ribeiro et al., 2016) or SHAP (Lundberg & Lee,
2017) are local explanation methods, designed to explain specific points: that
is, there is no notion of a global surrogate model from which group fairness
metrics can easily be calculated. Second, only the demographic parity metric
does not require information about the ground-truth labelling of data points;
all other metrics require this information. The question then becomes how to
determine the set of points $x$ on which $\mathcal{M}$ will be calculated for
local explanations. One potential approach is to use the sampled points in the
local neighborhood generated by the explanation method, and calculate
$\mathcal{M}$ on the neighborhood for each of the points in the dataset. Of
course, this approach means that no ground-truth labelling is available for
this set of sampled points, and thus the only metric that can be verified to
match or mismatch in this way is demographic parity.
Counterfactual fairness. In classification, counterfactual fairness and
individual fairness have similar motivations: identifying how the prediction
for a particular input $x$ would change if only the group membership of $x$
was changed (Dwork et al., 2012). Though there is debate about the extent to
which counterfactual or individual fairness is distinct (if not orthogonal)
from group fairness (Lahoti et al., 2019; Binns, 2020), a “fairness-
preserving” explanation should nevertheless capture the counterfactual
behavior of the black-box model. To that end, let $x^{\prime}$ represent the
input $x$ with a changed value for group membership, and $E_{f(x)}$ illustrate
explanations generated for input $x$. Then, counterfactual fairness is
preserved when:
$E_{f(x)}-E_{f(x^{\prime})}\approx f(x)-f(x^{\prime})$
In other words, when the difference between the explanation generated for $x$
and the explanation generated for $x^{\prime}$ follows the difference between
the model’s behavior on $x$ and $x^{\prime}$. This abstraction also raises
open questions about how exactly the similarity should be determined.
Sensitive attribute. The treatment of the sensitive attribute in cases where
it is included in the inputs to the black-box model is also worth additional
attention. In this case, unlike group and counterfactual fairness, we do not
propose a particular normative value of how the sensitive attribute ought to
be treated by the explanation algorithm in relation to the black-box. For
example, the feature importance for the sensitive attribute being 0 does not
necessarily imply that the black-box is not discriminatory: the influence of
the sensitive attribute may have been attributed to another, correlated
feature. Moreover, many algorithms for fair machine learning explicitly use
the sensitive attribute in order to achieve some measure of fairness, such as
the method proposed in Hardt et al. (2016). In this sense, a feature
importance of 0 might even be alarming rather than reassuring. As a result,
future work in this area may include methods which give more meaningful ways
to interpret the influence of the sensitive attribute.
Additional considerations. Finally, of note here is the distinction between
evaluating an explanation algorithm itself for how well it preserves fairness
properties in general, and evaluating a given, specific explanation for
whether it is preserving relevant fairness properties once the explanation for
a particular input or model has been generated. These are different tasks—the
first, for example, may be useful for a model developer or engineer in the
process of choosing an explanation method, while the second may be more
relevant to auditing processes once a black-box model (and a corresponding
explanation algorithm) has been deployed. Additional work distinguishing what
approaches or metrics might be comparatively useful in either situation is
warranted; in particular, all of these proposed metrics require a
comparatively high amount of information and access to the black-box, and may
be better-suited towards the first task (evaluating algorithms in general)
rather than the second (auditing individual explanations).
### 2.2 Generating Fairness-Preserving Explanations
As discussed above, algorithms for finding explanations typically focus on
optimizing for metrics such as fidelity and sensitivity. One approach for
generating a fairness-preserving explanation can be similar to early
approaches to fair machine learning algorithms: adding a penalty term in the
objective function for the extent to which the explanation is fairness-
preserving (Kamishima et al., 2012; Zafar et al., 2017). For example, the
original LIME objective function (Ribeiro et al., 2016) is as follows:
$\xi(x)=\arg\min_{g\in G}\mathcal{L}(f,g,\pi_{x})+\Omega(g)$
where $\xi(x)$ is the optimal explanation for input $x$ to model $f$, $G$ is
the class of sparse linear models, $\mathcal{L}$ is a measure of fidelity,
$\pi_{x}$ is a local region around $x$, and $\Omega$ is a measure of
complexity. A modified objective function including a term such as $\psi(f,g)$
measuring the preservation of fairness properties described in Section 2.1,
fits naturally:
$\xi_{\textit{fair}}(x)=\arg\min_{g\in
G}\mathcal{L}(f,g,\pi_{x})+\lambda_{1}\Omega(g)+\lambda_{2}\psi(f,g)$
Here, $\lambda_{1}$ and $\lambda_{2}$ are tuning parameters for the complexity
$\Omega$ and fairness-preservation term $\psi$, respectively.
Figure 2 illustrates the results of using this modified, fairness-preserving
objective function when finding the explanation $\xi$. Here, the dataset used
was COMPAS; the black-box was a three-layer deep neural net; and $\psi$ is
derived from the group fairness equation in Section 2.1. Specifically,
$\psi=|DP(f(x))-DP(E_{f}(x))|$, where $DP$ is the demographic parity metric:
$P(Y=1|S=1)-P(Y=1|S=0)$. In this experiment, the number of perturbations used
to generate the LIME explanation was varied to show the asymptotic fairness
mismatch, as a greater number of perturbations generally results in a higher-
certainty explanation. The fairness mismatch plotted on the y-axis is
calculated exactly in the same way as $\psi$ explained above. Our introduction
of this approach is meant more as a provocation to start the conversation
rather than a full-fledged proposal or argument that this method is
necessarily ideal; however, the results are promising and warrant further
investigation in this direction.
Figure 2: Original vs fairness-preserving LIME algorithms: number of
perturbations used for LIME vs average fairness mismatch over explanations for
all points in the dataset.
## 3 Discussion & Conclusion
In this work, we have given some intuition and preliminary results as to why
it is important to probe the fairness of explanations: not just because of the
often high-stakes and consequential goals for which explanations are used, but
because existing explanation methods focusing on metrics like fidelity may
result in misleading and incorrect explanations even in the absence of an
adversarial actor constructing explicitly discriminatory black-boxes, or
designing explanation methods that explicitly hide discrimination.
Furthermore, fairness can also be viewed as a specific lens on performance for
the model overall. In fact, the phenomenon illustrated in Figure 1 can be
considered to be a performance issue—strictly incorrect decision boundaries,
though the minority group’s decision boundary is much more incorrect—that can
be detected by testing for fairness mismatch as proposed in Section 2. Of
course, the exact behavior in this scenario may be the consequence of LIME’s
choice to focus on sparse linear models, and choosing a more complex
interpretable model class (such as shallow decision trees) may alleviate the
issue.
Nevertheless, an action like this is only made possible by the first step of
diagnosing the fairness mismatch between the black-box and the explanation’s
surrogate model. Thinking about explanations in this context, therefore, also
raises broader questions about the extent to which explanations are in fact
capturing what we want; or, alternatively, ways in which the limitations of
particular explanations or explanation methods may be communicated clearly to
stakeholders and end-users.
In this work, we suggested a framework for evaluating the fairness-preserving
properties of explanations, and proposed one generic approach for producing
fairness-preserving explanations. However, this extended abstract is also
meant to argue for the consideration of evaluation metrics for explanations
more broadly: while fairness was the first angle we considered, there are
undoubtedly additional necessary properties of the model—even privacy, for
example—that explanations should preserve.
## Acknowledgements
We would like to thank the anonymous reviewers for their insightful feedback.
This work is supported in part by the NSF award #IIS-2008461, and Google. The
views expressed are those of the authors and do not reflect the official
policy or position of the funding agencies.
## References
* Aïvodji et al. (2019) Aïvodji, U., Arai, H., Fortineau, O., Gambs, S., Hara, S., and Tapp, A. Fairwashing: the risk of rationalization. In _International Conference on Machine Learning_ , pp. 161–170. PMLR, 2019.
* Bhatt et al. (2020a) Bhatt, U., Andrus, M., Weller, A., and Xiang, A. Machine learning explainability for external stakeholders. _arXiv preprint arXiv:2007.05408_ , 2020a.
* Bhatt et al. (2020b) Bhatt, U., Weller, A., and Moura, J. M. Evaluating and aggregating feature-based model explanations. _arXiv preprint arXiv:2005.00631_ , 2020b.
* Binns (2020) Binns, R. On the apparent conflict between individual and group fairness. In _Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency_ , pp. 514–524, 2020.
* Corbett-Davies & Goel (2018) Corbett-Davies, S. and Goel, S. The measure and mismeasure of fairness: A critical review of fair machine learning. _arXiv preprint arXiv:1808.00023_ , 2018.
* Dwork et al. (2012) Dwork, C., Hardt, M., Pitassi, T., Reingold, O., and Zemel, R. Fairness through awareness. In _Proceedings of the 3rd innovations in theoretical computer science conference_ , pp. 214–226, 2012.
* Hancox-Li & Kumar (2021) Hancox-Li, L. and Kumar, I. E. Epistemic values in feature importance methods: Lessons from feminist epistemology. In _Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency_ , pp. 817–826, 2021.
* Hardt et al. (2016) Hardt, M., Price, E., and Srebro, N. Equality of opportunity in supervised learning. _arXiv preprint arXiv:1610.02413_ , 2016.
* Kamishima et al. (2012) Kamishima, T., Akaho, S., Asoh, H., and Sakuma, J. Fairness-aware classifier with prejudice remover regularizer. In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , pp. 35–50. Springer, 2012.
* Kumar et al. (2020) Kumar, I. E., Venkatasubramanian, S., Scheidegger, C., and Friedler, S. Problems with shapley-value-based explanations as feature importance measures. In _International Conference on Machine Learning_ , pp. 5491–5500. PMLR, 2020.
* Lahoti et al. (2019) Lahoti, P., Gummadi, K. P., and Weikum, G. ifair: Learning individually fair data representations for algorithmic decision making. In _2019 IEEE 35th International Conference on Data Engineering (ICDE)_ , pp. 1334–1345. IEEE, 2019.
* Lakkaraju & Bastani (2020) Lakkaraju, H. and Bastani, O. “how do i fool you?” manipulating user trust via misleading black box explanations. In _Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society_ , pp. 79–85, 2020.
* Lundberg & Lee (2017) Lundberg, S. and Lee, S.-I. A unified approach to interpreting model predictions. _arXiv preprint arXiv:1705.07874_ , 2017.
* Ribeiro et al. (2016) Ribeiro, M. T., Singh, S., and Guestrin, C. “why should i trust you?” explaining the predictions of any classifier. In _Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining_ , pp. 1135–1144, 2016.
* Slack et al. (2020a) Slack, D., Hilgard, S., Jia, E., Singh, S., and Lakkaraju, H. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. In _Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society_ , pp. 180–186, 2020a.
* Slack et al. (2020b) Slack, D., Hilgard, S., Singh, S., and Lakkaraju, H. How much should i trust you? modeling uncertainty of black box explanations. _arXiv preprint arXiv:2008.05030_ , 2020b.
* Suresh et al. (2021) Suresh, H., Gomez, S. R., Nam, K. K., and Satyanarayan, A. Beyond expertise and roles: A framework to characterize the stakeholders of interpretable machine learning and their needs. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ , pp. 1–16, 2021.
* Verma & Rubin (2018) Verma, S. and Rubin, J. Fairness definitions explained. In _2018 ieee/acm international workshop on software fairness (fairware)_ , pp. 1–7. IEEE, 2018.
* (19) Yeh, C.-K., Hsieh, C.-Y., Suggala, A. S., Inouye, D. I., and Ravikumar, P. On the (in) fidelity and sensitivity of explanations.
* Zafar et al. (2017) Zafar, M. B., Valera, I., Rogriguez, M. G., and Gummadi, K. P. Fairness constraints: Mechanisms for fair classification. In _Artificial Intelligence and Statistics_ , pp. 962–970. PMLR, 2017.
* Zhang et al. (2019) Zhang, Y., Song, K., Sun, Y., Tan, S., and Udell, M. ” why should you trust my explanation?” understanding uncertainty in lime explanations. _arXiv preprint arXiv:1904.12991_ , 2019.
|
# Predicting Customer Lifetime Value in Free-to-Play Games
Paolo Burelli
IT University of Copenhagen - Tactile Games
<EMAIL_ADDRESS>
###### Abstract
As game companies increasingly embrace a service-oriented business model, the
need for predictive models of players becomes more pressing. Multiple
activities, such as user acquisition, live game operations or game design need
to be supported with information about the choices made by the players and the
choices they could make in the future. This is especially true in the context
of free-to-play games, where the absence of a pay wall and the erratic nature
of the players’ playing and spending behavior make predictions about the
revenue and allocation of budget and resources extremely challenging.
In this chapter we will present an overview of customer lifetime value
modeling across different fields, we will introduce the challenges specific to
free-to-play games across different platforms and genres and we will discuss
the state-of-the-art solutions with practical examples and references to
existing implementations.
## 1 Introduction
Customer lifetime value (CLV or LTV) refers broadly to the revenue that a
company can attribute to one or more customer over the length of their
relationship with the company [55]. The process of predicting the lifetime
value consists in producing one or more monetary values that correspond to the
sum of all the different types of revenues that a specific customer, or a
specific cohort, will generate in the future. The purposes of this prediction
are manifold: for example, having an early estimation of a customer’s
potential value allows more accurate budgeting for future investment;
moreover, monitoring the remaining potential revenue from an established
customer could permit preemptive actions in case of decreased engagement.
Predicting customer lifetime value is a complex challenge and, to date, there
is no single established practice. Furthermore, due to its wide potential
impact in different business aspects, the problem is being researched in
different communities using a plethora of different techniques, varying from
parametric statistical models to deep learning [28, 70]. One of the initial
primary drivers of research in CLV was direct marketing [4] with a focus on
its use in marketing decision problems, such as acquisition or the trade-off
between acquisition and retention costs [6]. Research expanded progressively
to other fields of marketing and customer relationship management, especially
thanks to the pervasiveness of digital technology and the possibility to have
more accurate tracking of the customer relationship [37].
The increased quality and quantity of customer data available for analysis
widened the areas of application of CLV related principles, and helped
accelerating the development of new models for CLV prediction. Initial
analytical models were developed on assumptions about constant and uniform
characteristics of the user behavior (e.g. transaction frequency, margin of
profit). Over time, models have evolved to take into account uncertainty and
variations of user behavior [63] and to be able to draw information from a
wide spectrum of variables [69].
Models have traditionally been based, on transactional data, meaning that the
variables describing the user behavior are based on different characteristics
of the customers transactions with the company (e.g. recency, frequency,
monetary (RFM) [63]). However, with the advent of e-commerce and similar
technologies, the available information about the customer behavior became
increasingly richer, since it started to be possible to track not only the
purchases, but also which objects were observed, how often would the customer
visit the store and many other non-strictly transactional details.
One of the application area with probably the richest amount of user data is
video games; however, until recently, due to its distribution and revenue
models, the need for customer lifetime value prediction was relatively minor.
Traditionally, computer games distribution has relied on a premium pricing
model, in which a game is developed, released and sold for a price. Following
this pricing scheme, the monetization of the customer happens before the
player starts playing the game and its post-purchase behavior does not affect
the customer lifetime value directly [3].
Two main aspects radically changed the relationship between the game
developers and the players/customers: digital distribution platforms and the
free-to-play business model. Digital distribution platforms, such as
Steam111Valve Corporation. Steam. https://store.steampowered.com 2018. or the
App Store222Apple Incorporated. App Store. https://www.apple.com/lae/ios/app-
store 2018, along with game developers’ own on-line services allow tracking of
user behavior beyond a single game, making it more meaningful and necessary to
predict the customer lifetime value across multiple games, either being
purchased on the same store or being linked to the same on-line account.
Free-to-play (F2P/Freemium) games follow a different revenue model than the
traditional one: games are freely available, at no cost, to the players and
the revenue comes from in-game advertisement and the sale of in-game items.
These two revenue streams, contrarily to the classic game revenue model, begin
only after the players/customers have been acquired and vary dramatically
between players. This kind of relationship between the company and the players
resembles in part the relationship between a store and its customer, e.g.
monetary transactions happen at irregular intervals, the transaction value is
not constant and there is no explicit customer churn event.
However, differently from physical or on-line stores, the users are not only
customers but also players, and the data available about their behavior
includes a wide range of features that goes beyond transactions or goods
browsing, such as players progression within the game or players’ skill level.
Within the field of game analytics, the analysis of these features plays a
major role in evaluating the quality and the potential success of a game [19];
furthermore, the behavior has been linked with major business related metrics
such as retention [53] or customer lifetime value [70].
This abundance of accurate and high frequency data, covering different aspects
of the customer experience make free-to-play-games an ideal application area
for research in user modeling and prediction of future customer behavior. In
turn, the development of such models would see an immediate application in the
industry, as they would allow a better optimization of important processes
such as user acquisition and customer relationship management [45].
This, as well as the ever-expanding share of the game industry that leverages
this business model, highlights customer life time value prediction as a key
challenge in games data science and games research. For these reasons, in this
chapter, we will make an attempt to give an overview of the research and
applications of CLV modeling including an in-depth analysis of the works that
have been focused on the F2P field. In the next section, we will outline a
number of foundational concepts that will serve as a basis to the rest of the
chapter. In section 3, we will describe different ways in which CLV can be
used inside and outside of the game industry. In section 4, we will give an
overview of different models used to predict CLV with a more in-depth focus on
models employed in the game industry. In section 5 we will describe a number
of software packages that can be used to perform CLV prediction, and finally,
in section 6, we will outline a number of current and future directions within
the field.
## 2 Definitions and terminology
Pfeifer et al. [55], in their 2004 article, describe a problem present across
the different research works on customer lifetime value: incoherent
terminology. The reasons behind this problem are probably manifold, e.g. the
different research communities; however, in this section we will attempt to
synthesize a number of concepts and terms that are common across the different
fields involved and are foundational to the study of customer lifetime value
prediction in games and beyond.
First of all, it is important to define what is customer lifetime value;
Pfeifer et al. [55] summarize it as: “the present value of the future cash
flows attributed to the customer relationship”. As the authors point out, this
definition makes a couple of important assumptions: first, the value is
associated to the cash flow, which means it is not limited to the inbound flow
of revenue generated by a customer but also to the costs attributed to that
customer; however, the costs considered in this definition are only the ones
directly attributable to a given customer. Second, the term value expresses a
valuation at the current point in time of some future revenues and costs, this
implies some form of discount based on the time in which the future cash flows
are predicted.
In their definition, Pfeifer et al. do not attempt to encompass all possible
definitions of CLV within the literature; instead, they propose one possible
explicit definition of CLV that is terminologically correct and coherent. This
means that other research works on predicting CLV do not necessarily adhere to
the aforementioned definition. Berger and Nasr [4], for instance, do not
consider acquisition cost as part of the lifetime value, considering instead
CLV to be the “maximum profitable acquisition cost” [37]. Sifa et al. [69], on
the other hand, do not take into consideration the present value and use no
factor to discount the future predicted revenue.
One further aspect of this definition that is not shared by the different
research works is the time frame considered for the calculation. Gupta et al.
[28], in their review, find that a number of researchers employ an arbitrary
time horizon for the future prediction [56], while others user an infinite
horizon [25]. While it might appear that employing a fixed time horizon is
just a way to simplify the modeling process, both approaches have their own
merits. Using an infinite horizon does permits to avoid making any assumption
about the maximum length of the customer relationship with the company,
therefore; it allows a more accurate prediction of Pfeifer’s definition of
CLV; however, often for budgeting or other purposes, it is important to know
the return on investment within a given period of time (e.g. 1 year), making a
fixed horizon CLV prediction extremely useful.
Another consideration relative to the temporal aspect of the customer
relationship that has to be taken into account when studying CLV prediction is
whether the relationship is exclusive or it allows the customer to use other
competitors’ services; these two different type of relationship are often
labeled as either lost-for-good or always-a-share [20]. Identifying which type
of relationship is being modeled for CLV prediction is extremely important as
it affects the temporal length of the relationship and the definition of churn
(i.e. when a customer stops using the service).
This is especially true in free-to-play-games where there is no visible churn
event; in this context, the definition of the user state at any given moment
(active or inactive) depends on an ad-hoc formula based one some synthesis of
the players’ actions [61]. Many CLV prediction models include or are built on
top of a churn prediction model, and both the formula and the choice of model
will depend on whether it is possible for the customer to return after a
period of inactivity [54] or whether his or her relationship with the company
is considered finished [27].
Finally, it is important to specify that all of the works cited and described
in this chapter do not consider the effects of the competition with other
services on the customer lifetime value. As pointed out by Gupta et al. [28],
this is the case for most current modeling approaches because of the lack of
data about the competitors.
## 3 Applications of CLV prediction
While marketing serves as the primary application area of customer lifetime
value prediction, a number of other activities, such as customer relationship
management or live game operations, are being increasingly driven by data and
different key performance indicators (KPIs) such as CLV. Schmittlein et al.
[63], in one of the earliest works on customer lifetime value estimation,
motivate their research by stating that:
> ”…the issue is important in at least three settings: monitoring the size and
> growth rate of a firm’s ongoing customer base, evaluating a new product’s
> success based on the pattern of trial and repeat purchases, and targeting a
> subgroup of customers for advertising and promotions.”
They envision that, by knowing which customer are active and what purchases
they make, a manager is able to do more accurate budgeting, a better product
evaluation and target the customers more accurately with re-engagement
initiatives. Similarly to Schmittlein et al., a number of other research works
have tackled the above problems; Table 1 gives and overview of the main
activities for which CLV prediction has been employed.
Each application area includes a number of different activities (e.g. special
offers or advertisement) united by a similar usage of CLV prediction. In the
case of budgeting and finance, CLV predictions is used to predict revenue and
plan budgets for the different activities within the company [63, 18], finding
the balance between acquisition and retention investments [6] or to make a
financial estimation of the company [29]. The second activity, Product
Development, is mentioned only by Schmittlein et al. [63]; however, as we will
describe later in this section, CLV is an important KPI in the evaluation of
the game quality in F2P games and it can be very useful to drive the game
design process.
### 3.1 Customer Acquisition
Customer Acquisition encompasses all activities aimed at getting new customers
to start using the service or buying the products offered. As mentioned by
Berger et al. [4, 5], having an estimation of the customers’ lifetime value
can help deciding more accurately how much to spend on a given promotional
campaign and how much margin for profitability there could be in a new market
segment; Dwyer [20] further develops this concepts by identifying in CLV the
ceiling of the customer acquisition spend. Having an early indication of
whether the costs of acquisition are matched by the CLVs, can help deciding
whether a certain promotional initiative should be continued or stopped [45,
69, 70].
### 3.2 Customer Retention
Customer retention and segmentation initiatives are often intertwined and not
completely distinguishable; in this article, we describe customer
retention/loyalty activities as any activity explicitly aimed at prolonging
the lifetime of a customer, while, with customer targeting/segmentation, we
identify all activities aimed at identifying different homogeneous groups
within the customer base that can be targeted with a custom user experience.
Such a customer experience could be aimed at prolonging the customer lifetime;
however, for the purpose of this categorization, we define this as a secondary
objective.
Application area | Article
---|---
Budgeting and Finance | [63, 6, 18, 29]
Product Development | [63]
Customer Acquisition | [20, 4, 5, 45, 69, 70]
Customer Retention/Loyalty | [4, 50, 56, 44, 59, 58]
| [48, 67, 11, 61, 70]
Customer Targeting/Segmentation | [50, 56, 75, 74, 35, 30, 67, 42, 40]
Other | [1]
Table 1: Applications of CLV prediction in different business activities
Latchowetz et al., in their study on the impact of season ticket holders on
the NBA franchises revenue [44], argue that a customer should not be evaluated
solely on the revenue that she or he generates within a season. At the time of
their estimation, an average season ticket holder had a lifetime value of more
than eighty thousand US dollars; therefore, the authors advice the
entertainment industry to acknowledge the importance of using customer
lifetime value and developing long term retention strategies for their
customers.
Berger and Nasr [4] discuss how knowing the customer lifetime value can help
determining the effects of adopting different marketing strategies for
retention and acquisition and how to balance the spending between the two
activities. The authors point out also that any acquisition strategy might
impact also the customers’ retention; therefore, it is hard to see the two
activities independently and both their budgets must by aligned to the
predicted customer lifetime value. Mulhern extends further Berger and Nasr’s
idea about the strategic use of CLV prediction by seeing it as a primary
indicator to driving resources allocation in the marketing mix [50].
Rosset at al.[59, 58] extend the idea of using CLV prediction to drive
retention activities and expand it by investigating how to estimate the impact
of retention efforts to the CLV. They demonstrate an application of their
approach in the context of a retention campaign for a group of potential
churners. They show how, by having a model able to predict CLV in different
configurations of the service, it is possible to compare the current estimated
lifetime value of a group of potential churners with different projected
lifetime values given a number of different incentives to improve the cohort’s
retention; thus, taking a more informed decision on the correct initiative to
be taken.
The predictions used for these decisions, as any other prediction of future
events, have some degree for uncertainty in the form of prediction error or
bias; taking this uncertainty into account can be also important in decision
making. Malhouse and Blattberg [48] study the impact of uncertainty in the
management of the customer relationship helping to understand not only whether
engaging into a retention activity can be profitable but also how often it
could be.
### 3.3 Customer Segmentation
Most of the research works presented in this section make the assumption that
a higher retention (i.e. a longer lifetime for the customers) will have a
positive impact on the company’s revenue in the form of an increased CLV;
however, while this is generally true, Reinartz and Kumar [56] found that this
assumption does not always howl and different customer segments have different
spending patterns, which means that, in a number of cases, prolonging the
customer’s lifetime would no yield any more revenue. For this reason, they
suggest that, the company should not necessarily pursuit long-term customer
relationship, but it should customize the relationship based on prediction on
future customer lifetime value and retention.
Segmenting the target audience and personalizing marketing activities is an
established practice [17]; however, the predominant guiding factors used to
define the segments and the most appropriate actions have, for long time, been
based on demographics, questionnaires and interviews [47]. The advent of large
scale data collection and analytics, in many industries, has dramatically
changed many of the common practices in marketing research, allowing
practitioners to access more detailed information from wider audiences at
lowers costs.
Mulhern [50], for example, proposes customer lifetime value as a possible
segmentation criterion beside usage volume and brand loyalty: he suggests that
one simple segmentation could divide the users in customers worth keeping, as
their CLV is profitable, and customers that are not worth the retention cost.
Verhoef and Donkers [75] propose a similar segmentation to design personalized
insurance offers.
Venkatesan and Kumar [74] pose the question whether using predicted CLV for
segmentation con yield better results than more established KPIs such as
previous-period customer revenue or past customer value. To compare the
effectiveness of the different metrics, they rank the customers based on each
metric and segment the customers based on the raking; the results show the
segmentation based on CLV to be superior to the other segmentation approaches.
Similar strategies to the ones mentioned so far have been applied to a number
of different industries such as telecommunications [35], banking [30],
retailing [67, 42] or beauty [40]. In the last few years, a similar trend has
emerged in the gaming industry and, especially in the mobile game and free-to-
play industry, all the aforementioned marketing practices are established and
widely undertaken [66].
### 3.4 Free-To-Play Games
Figure 1: Three screen grabs from the iOS version of Bee Brilliant Blast by
Tactile Games depicting, from left to right, the phone’s store interface, a
typical puzzle level and the in-game store interface, in which the customer
can purchase different amounts of in-game currency.
Free-To-Play video games are games than can be played without an upfront
payment and potentially completely free of charge; players can, within the
game, purchase some virtual goods and pay to enable some features. While this
type of games was originally mostly distributed through on-line social media
platforms, the business model is currently vastly diffused on many different
platforms, such as mobile phones, home consoles and personal computers [2].
Figure 1 shows an example of a mobile free-to-play game; as it is possible to
see on the first screen grab to the left, the game is freely available on the
phone’s store, however, the store specifies that the game contains the
possibility to perform in-app purchases. The last image to the right shows the
in-game store that allows the customer to purchase different packs of in-game
currency, which can be used throughout the game to unlock different features.
Using an intermediary currency, while a very common practice, is not defining
characteristic of free-to-play games, and some games allow customer to enable
directly in-game features directly using a real currency.
The second main source of revenue in free-to-play games comes from
advertisement: a large portion of free-to-play game display to the players ads
in different formats (e.g. videos or interstitials). These ads are provided by
external services that act as mediators between the advertiser and the
publisher. The game (i.e. the publisher) received ads from the ad provider
through an API and shows them to the player, the ad provider will in return
pay the game developer depending on the number of ads being displayed, the
number of times the ads are clicked or the number of times the application
being advertised is being installed.
Both aforementioned revenue streams – in-app purchase and advertisement – have
a non-contractual nature, as the player can freely choose if and when to make
a purchase or click on and advertisement campaign; furthermore, a conversion
rate (i.e. the percentage of players making an in-app purchase) well below 10%
is considered normal in the industry [51], making it especially challenging to
predict the customers’ lifetime value. At the same activities such as customer
acquisition and retention rely on accurate CLV predictions for accurate
targeting and budgeting.
As of November 2018, there are around 816000 games available only on the Apple
App Store333https://www.pocketgamer.biz/metrics/app-store/categories/;
therefore, targeting the right customers and accurately measure their
potential impact in the company’s revenue is particularly important in
customer acquisition in the free-to-play games market, given the extremely
high number of games competing for the same users. A significant part of the
customer acquisition efforts in the industry is executed through performance
marketing campaigns: in these campaigns, he advertiser competes with other
advertisers to show ads the potential new customers, often times through some
form of auctioning system. Customer lifetime value predictions can be used as
a benchmark to evaluate the profit margin of these campaigns as both the
potential revenue coming from the new customers and their cost of acquisition
vary greatly depending on the segmentation chosen for the campaign, the market
context and the quality of the advertisement creative content [70]. Even
beyond performance marketing, an accurate CLV prediction is very important to
be able to budget expensive and higher risk marketing efforts such as
television advertisement.
Furthermore, the F2P games market is characterized by very low retention
number – e.g. the best android games have a day 30 retention rate below
5%444https://www.forbes.com/sites/johnkoetsier/2017/10/20/the-very-best-
android-game-has-just-4-5-user-retention-after-30-days/ – making any effort to
improve such low number a priority for most companies. Within this context,
CLV prediction can be used, as previously described, to target different
segments of users with special offers; However, it is also possible to
personalize the game experience at large by adapting, for instance, the flow
of the game progression or by providing custom events. Harrison and Roberts
show an example of such adaptation aimed to improving retention in a digital
version of Scrabble [32].
Given the importance of CLV prediction in the free-to-play and other
industries, a number of algorithms have been developed over time , in the next
two section we will describe the most significant ones and analyses a number
of software packages that can be used for CL prediction.
## 4 Models
Berger and Nasr are the first researchers that attempted a categorization of
CLV prediction models [4]. In this early review of the field, all methods
mentioned are aimed at calculating an average customer lifetime value for the
entirety of the company’s customer base or a given cohort. In a latter review
[43], Kumar et al. group these approaches under the label ”average LTV
approach”, while Jain and Singh [37] use the term ”basic structural model”. In
this chapter we adopt the definition used by Kumar et al. and we include a
number of other practices commonly used in the free-to-play industry that
approach LTV prediction in a similar fashion.
The methods for CLV prediction discussed in this article are dividend in two
further categories: customer history models and machine learning models. The
first category contains all methods that attempt to predict either a single
customer’s or a cohort’s lifetime value based on some mathematical projection
if the past behavior of the given customer or cohort. The second category
includes all algorithms which make use of some learning algorithm to build a
computational model of the customer behavior, which is then used to make
predictions.
### 4.1 Average Models
The aim of the models discussed in this section is to give an estimation of
the future discounted cash flow for a group of customers or for the whole user
base. The common aspect between these models is that they make a number of
assumptions: they assume a constant (or otherwise known) margin of profit over
time, they do not consider the stochastic nature of the purchase behavior of
the customers and they assume that the customer behavior is uniform across the
estimated cohort.
The most basic model proposed by Berger and Nasr [4] assumes that the customer
retention rate and the costs of retention are constant over time and both
costs and revenues happen periodically at a constant rate, within these
conditions the formula for CLV is as follows:
$CLV=GC*\sum_{\\_}{i=0}^{n}\frac{r^{i}}{(1+d)^{i}}-M*\sum_{\\_}{i=0}^{n}\frac{r^{i-1}}{(1+d)^{i-0.5}}$
(1)
Where CG is the customer’s expected gross contribution margin per period, M is
the promotion costs per customer per period, n is the prediction horizon
expressed in number of periods, r is the retention between one period and the
next, and d is the discount rate.
Further iterations of this model presented by Berger and Nasr [4], by Jain and
Singh [37] and by Kumar et al. [43], allow more sophisticated ways to express
non constant retention rates and margins of profit and to include acquisition
costs in the calculation.
In their article about the relationship between the length of service and CLV,
Rosset et al. [58] formulate an extended average model that employs a Kaplan-
Meier [39] estimator to calculate the duration of the relationship between the
company and the average customer .
A similar approach is currently among the basic models used in the free-to-
play industry for CLV prediction [65]: after selecting a specific function to
express the retention curve of a given cohort, the function is fit on the
training data and the resulting customer lifetime value is calculated as
follows:
$CLV=\sum_{\\_}{i=0}^{n}ARPDAU*ret(i)$ (2)
Where $ARPDAU$ stands for average revenue per daily active user and $ret(i)$
is the value of the chosen retention function at the $i^{th}$ day. This method
for CLV prediction, while quite simplistic and not necessarily extremely
accurate, has the advantage of being easily readable and based on established
business KPIs such as APRDAU and retention.
Another example of simplistic but widely adopted model is described by Runge
[60]: instead of basing the projected revenue on retention and APRDAU, the
method fits a function to approximate the average monetization curve, which is
then used to project the customer lifetime value for a player based the amount
of money the she spent at the present day. If we consider $rev(n)$ as the
revenue produced by a customer up the $n^{th}$ day of the customer
relationship and $mon(n)$ as the average fraction of the revenue that is
produced by day $n$ the formula for CLV prediction is as follows:
$CLV=rev(n)/mon(n)$ (3)
The major advantage of this model is its simplicity and, similarly to the
retention-based model, its readability. However, with such a model it is not
possible to predict the CLV for customers that have not yet produced any
revenue – i.e. the projection would always result 0; moreover, in a context
such as free-to-play games, in which the amount of paying user is very small,
the variably of the behavior between different paying user has a large impact
on the prediction, and such variability is completely disregarded by the
model.
### 4.2 Customer History Models
The aforementioned methods based on retention and monetization curves are
partly based on historical customer data as, in both methods, a function
representing the average retention or monetization behaviors is fit on some
past customer data and then used for prediction. These methods however do not
allow to use any particular user’s track record to personalize the prediction,
hence the average nature of the methods.
Based on the customer categorization by Jackson [36] that divides the
customers in two types: always-a-share and lost-for-good, Dwyer [20] argues
that an average retention model is not sufficient to model CLV for customers
belonging to the always-a-share category. For this type of customers, he
proposed a migration model based on the customers’ recency of purchase.
Always-a-share customers differ from lost-for-good customers in that the
latter have a long-term commitment with the company and switching to a
competitor is costly, while the first ones can have simultaneous relationships
with multiple companies competing for the same product/service. Typical
examples of lost-for-good customers are bank customers or tenants in a rental
property. Whether free-to-play game players can be considered belonging to the
first or second category is an open question as there is no monetary barrier
that stops a player to switch to another game; however, to a certain extent,
the longer the time a player invests in a game, the less likely is that the
player will be stopping playing that game.
In his customer migration model, Dwyer uses purchase recency to estimate the
customer’s probability of performing a purchase in the next period and the
potential value of such a purchase. In the training phase, based on past
customers’ data, the method builds a model of purchase probabilities and
values that depend on the length of the period of inactivity. These recency
cells are then used as a lookup table to predict, depending on each customer’s
past purchase behavior data, her purchases in the next time unit. Dwyer’s
model allows to leverage the customer’s past behavior for a personalized CLV
prediction and considers the probabilistic nature of the customers purchase
behavior.
#### 4.2.1 Recency, Frequency and Monetary Value
The main limitation of Dwyer’s methods is that the probability of a purchase
in a given period is based only in the recency of the last purchase, which is
not necessarily true for all businesses. Especially in free-to-play game,
different types of purchase will affect differently the future purchases, for
example, a player purchasing a large package of virtual currency recently
might be less likely to make purchase as the amount of resources acquired will
allow her to play longer without a need for any help within the game.
The concept of recency used by Dwyer is an established metric in direct
marketing and alongside purchase frequency and average purchase monetary value
– commonly known together as RFM – have been long used in the field to predict
customer behavior [28]. Hughes describes a method for customer quality
estimation based on these three variables: the current customer based of the
company is sorted along these three variables each categories into 5
quantiles, creating therefore $5*5*5$ groups, these groups are then used to
score the different customers and target them with specific offers [34].
Inspired by Hughes’s work, Shih and Liu propose a method based on RFM and CLV
clustering to rank the customer according to their profitability [68]. As a
first step the method relies on a group of expert evaluation to identify the
relative importance of the recency, frequency and monetary variables using
analytical hierarchical processing. The customers are than clustered based on
the RFM space and the resulting clusters are ranked through a simple weighted
sum of the three normalized variables.
The aforementioned methods are not designed to produce a numerical predictor
of CLV but only to score the customers by their potential profitability.
Furthermore, these methods are limited in predicting the customer behavior
only for one future time period [26]. Finally, these methods disregard the
fact that customers’ pas behavior is, often times, the result of past company
activities.
Fader et al. describe a model for CLV prediction, using RFM as input variables
based on the e Pareto/NBD framework [63], which overcomes some of the
aforementioned limitations [26].
#### 4.2.2 Pareto/NBD
The Pareto/NBD model [63] aims at predicting individual customer’s purchase
behavior based on their past purchase patterns. More specifically, for each
customer, based on the recency ($t$) and frequency ($X$) of the purchases and
the length of the relationship between the customer and the company ($T$), the
model estimates the expected future number of purchases. The model is based on
five assumptions:
* •
while active, a customer makes purchases following a Poisson process with rate
$\lambda$;
* •
the purchase rate $\lambda$ differs between customers and is distributed
according to a gamma distribution across the customer base;
* •
the duration of the active period of the customer is exponentially distributed
with a death rate $\mu$;
* •
the death rate $\mu$ differs between customers and is distributed according to
a gamma distribution across the customer base;
* •
the death rate and the purchase rates are independent.
Based in these assumptions, the authors find that the customers “deaths” for a
sample follow a Pareto distribution of the second kind [38], while the number
of purchases made by an active customer follow a negative binomial
distribution [22] – hence the name Pareto/NBD. The two distributions are
controlled by four parameters ($r$ and $\alpha$ for NBD and $s$ and $\beta$
fro Pareto);
Schmittlein suggests two approaches to estimate the parameters based on the
customers’ past behavior: maximum likelihood and fitting observed moments; in
case a model in needed without prior data, Schmittlein discusses also the
possibility to handpick the parameters based on management judgments. Once the
parameters have been identified, each customer’s future number of purchases is
predicted using the two aforementioned distributions and the customer’s $X$,
$t$ and $T$.
One of the main issues with the Pareto/NBD is it’s computational complexity
[25] as the estimation requires multiple evaluations of the Gauss
Hypergeometric Function; this problem is mitigated by the modified BG/NBD
model by Fader et al. [25], which models the customer activity using a beta-
geometric model that is easier to implement efficiently.
Both aforementioned models are able to predict the future number of purchases
for a given customer and they can, for instance, be used to count expected
active users ate a give point in time, however these methods do not model in
any way the value of each purchase, therefore, they can’t directly predict the
customer lifetime value.
Reinartz and Kumar, in their studies on profitable lifetime duration [56, 57],
employ Schmittlein’s model to be able to calculate the number of time periods
in which a customer will perform a purchase. The authors transform the
continuous probability that a customer is active into a dichotomous measure of
whether the customer is active or inactive it a given point in time. Given a
probability threshold, it is possible to identify a customer’s future date of
churn, which, combined with the first date of the customer relationship, gives
an expected customer relationship duration. This duration, expressed as number
of periods $n$, is used to calculate that customer’s lifetime value using the
formula defined by Berger and Nasr [4] and described in Equation 1.
Schmittlein and Peterson[64] propose an extension of the original Pareto/NBD
in which the future monetary value of each transaction is samples from a
normal distributed around the mean monetary value of a cohort. However, Fader
et al. [26] observe that, in the data they analyzed, there are large
differences between mean, mode and median, indicating that the distribution of
the monetary values is highly skewed. Therefore, the authors propose to model
the average transaction value using an adapted version of the gamma-gamma
model proposed by Colombo and Jiang [15].
One of the main strengths and, at the same, the main limitations of the
Pareto/NBD or BG/NBD models is that they rely on a small number of purely
transactional data. This is a strength in that it makes the model general
enough to be easily applied in different context; however, the resulting model
will always risk to be sub-optimal as it disregards potentially relevant
information. Sing et al. address this limitation and propose an estimation
framework that can flexibly incorporate multiple statistical distributions and
consider a number of covariates such as age or gender [72].
Glady et al. [27] address another important limitation of the aforementioned
models: they all assume independence between the frequency of transactions and
the profit per transaction. The authors of the study demonstrate that such
assumption does not hold in multiple real-world data sets and propose a
modified Pareto/Dependent model that performs better than Pareto/NBD in such
circumstances.
### 4.3 Markov Chain Models
An alternative approach to customer lifetime value prediction is proposed for
the first time by Pfeifer and Carraway [54]: the authors suggest to model the
customer relationship as a Markov Chain Model (MCM), in which the different
states of the model represent different conditions of the relationship between
the customer and the company in terms of transactions and customer activity,
and the transition probabilities between states represent the probability of a
customer to move from one condition to the other one – e.g. for a customer to
make a purchase or to churn.
Figure 2: Simple customer relationship Markov Chain Model. States r1 to r4
represent different value of recency from 1 to 4.
Figure 2 shows an example of a simple Markov Chain Model representing the
possible states of the relationship between a customer and the company and the
valid transitions between the states. In the depicted case, the states are
based on the recency of the last purchase (e.g. r1 corresponds to recency
equal to 1) and there are only 5 valid stages before churn. However, as in the
examples reported by Pfeifer and Carraway, in a real-world scenario, the
states are based on a multitude of factors and there are many more valid
transitions each with their own probability.
While Pfeifer and Carraway show how it is possible to calculate CLV using a
MCM representation of the customer relationship with a given set of states and
transaction probabilities, Etzion et al. focus on the process of learning the
states and the transition from past customer data [24]. The process requires
first to identify the variables determining the states of the model, to define
their ranges and discretize them.
At this point the transition probabilities between the states can be
calculated through the following three steps:
1. 1.
initialize a transition matrix – i.e. a square matrix that contains the
probability of transitioning between one state and another one – with zeros in
all cells;
2. 2.
for each customer performing a transition between a state $i$ and a state $j$,
the matrix cells identified by $ij$ is incremented.
3. 3.
at the end of the processing, each line of the matrix is normalized between 0
and 1 using a min-max normalization.
The resulting matrix can be used, as suggested by Pfeifer and Carraway [54] to
calculate CLV for a customer in any given state.
Ching et al. [12] employ the same approach to build multiple transition
matrices that describe the customer behaviors in different market conditions –
e.g. with or without a promotion – finding that the transition matrices differ
and so does the CLV and the customer retention. The authors further showcase
how to use stochastic dynamic programming to estimate the optimal promotion
strategy that optimizes CLV given the previously found transition matrices and
a set of constraint on the promotions budget.
### 4.4 Supervised Learning Models
The general principle at the core of all supervised learning algorithms is to
learn a function between some example input and output data – e.g. past
customer behavior and their recorded lifetime value. This example data is
usually called training data, and the function resulting from this training
phase – the model – is used to predict the target variable – e.g. CLV – on
newly collected data. This procedure is not dissimilar to some of the MCM
based approaches previously described [24]; however, the models we present in
this section, differ from MCM models in that they trade explanatory power for
the possibility to capture more complex behaviors using some form of
computational/black-box model.
This means that the resulting models are less useful to inform the business
about specific customers’ preferences, but they can potentially be more
accurate and be deployed, for instance, for marketing automation. It is
important to note that this is not a hard distinction and many models
described in this and the previous sections include a mix of explanatory and
black-box models.
An example of a model incorporating different learning algorithms is the one
proposed by Haenlein et al. [30]. The authors describe an approach to CLV
prediction that uses a combination of CART analysis [8] and MCM: first, the
customers of a retail bank are divided into age groups and, for each age
group, a regression tree model is built to predict the profit for a customer
within that group; second, the transition probabilities between the groups are
modeled as a Markov Chain so that the model is able to follow the transitions
of the customers throughout their lifetime. The resulting lifetime value is
calculated as a discounted sum of each of the CLV predicted in the possible
customer states weighted by the transition probabilities.
Cheng at al. [11] approach CLV prediction in a similar way but they extend the
model by building a different Markov Chain for each period the customer
relationship; each chain includes a number of states depending of the activity
level of the customers in that specific period. The total CLV for a customer
is similarly calculated by summing the predicted CLVs over the different
states and periods weighted by the transition probabilities and the survival
probability of each period. The models used to predict the profit contribution
of a customer and a given state is modeled using an artificial neural network,
while the survival probability in each period is calculated using a logistic
regression model.
Within the context of free-to-play games, supervised learning methods have
been employed in a number of studies. Compared to the previously mentioned
approaches, supervised learning offers two key advantages that make them
particularly interesting within this industry: the ability to leverage a wide
variety of data about the customer behavior and the ability to make
predictions about CLV for players that are not yet customers – i.e. that have
not yet made a single purchase.
The first aspect is important as in the freemium games industry, customers are
also players and the data describing the gaming behavior of players/customers
is much richer and at a much higher frequency compared to their purchase
behavior data. Furthermore, there are a number of studies that show how past
in-game behavior is a predictor of future in-game behavior [46], of player
preferences [9] and of the probability of making a purchase within the game
[31].
The second aspect is partially related to the first one: since the predictions
can be based also on non-purchase related data, it is potentially possible to
perform predictions also for players that have not yet made a purchase and
that might never be making one. As described in Section 3, most of free-to-
play players do not make any in-game purchase and generate little to no
revenue; it is therefore important for a model that predicts CLV in such
context to be able to discern which customers will be making a purchase before
the first one is made.
In one of the first studies on the application of supervised learning for CLV
prediction in freemium games, Lange et al. present a comparison of the
performance of a series of linear and non-linear models in the prediction of
CLV [45]. The dataset used for the study contains in-game events from the
first 7 days of gameplay of about 38 million players; the input variables
include data about player spending, gameplay, game progression, social
interactions, success metrics and game settings preferences. The target
variable is the amount of revenue generated by each player after 180 days.
The results of the study show that an ensemble of two artificial neural
networks using the absolute-differences technique over-performed the other
methods in terms of mean square error in a 10-fold cross-validated comparison.
A similar study is presented by Sifa et al. [69] in their article on purchase
behavior prediction in mobile free-to-play games. Similarly to Lange et al.,
the authors present a comparative study of multiple supervised learning
algorithms; however, the objective of the prediction is slightly different as
the authors argue that CLV prediction is a combination of three prediction
tasks: predicting whether player will ever make a purchase, predict the number
of purchases and predict the value of each purchase. The results of the study
show that, for the classification task, the models having the best performance
are tree-based models – e.g. random forest and decision tree classifiers – and
that it is extremely valuable to re-sample the dataset in to avoid as the
players performing purchases are extremely under represented. For this
purpose, the authors employ the SMOTE-NC [10] method to over-sample the
“converted” users data points.
In a latter article, Sifa et al. present a study that attempts to predict CLV
in one single step [70], more in line with the approach presented by Lange et
al. [45]. The study evaluates the performance of a deep neural network in
predicting day 360 cumulative undiscounted revenue per player, and it compares
the results against a number of algorithms such as random forest and linear
regression. The results of the comparison show that the deep neural network
out-performs the other regression algorithms in terms or normalized root mean
square error.
Furthermore, the authors demonstrate how to adapt SMOTE-NC to re-sample the
dataset and reduce the imbalance between customers that made a purchase and
customers that did not. This process requires a number of adjustments of the
standard implementation of SMOTE-NC, as the original algorithms is design for
classification problems, while a direct prediction of CLV is a regression
problem.
## 5 Software Packages
In Section 4, we gave an overview of the current state-of-the art in CLV
prediction mostly from an academic perspective, listing and explaining a
number of scientific articles describing the main approaches to solve the
prediction problem. However, most of the algorithms presented so far have been
translated, to different degrees, in software packages that can be applied to
easily perform CLV predictions on new datasets.
In this article, we identify three main packages that can be either used
directly to predict CLV or that can be used to construct a model for CLV
prediction. The software packages are Lifetimes [16] and BTYD [21] that
implement some of the algorithms presented in Section 4.2.2, and scikit-learn
[49] a popular machine learning package for the Python programming language
that can be used to implement the models presented in Section 4.4. Alternative
packages can be used to develop supervised learning models – e.g. Keras [14]
or XGBoost [13] – however, most of them are compatible with scikit-learn,
therefore we chose this package as the representative in this article.
To build and operate with Markov Chain Models, there are a large number of
different alternatives for both the Python and the R language – e.g. the DTMC
pack [73] and PyMC3 [62] – however, these packages are designed to offer
functionalities that go far beyond the code needed to implement the models
presented in Section 4.3, which can be easily implemented using standard
algebra functionalities. For this reason, we chose not cover any MCM package
in this Section.
### 5.1 Lifetimes/BTYD
The Lifetimes [16] and BTYD [21] packages are both software implementations of
the Pareto/NBD, BG/NBD and Gamma-Gamma models; they allow to learn the
parameters of the various distributions from past purchases data and to
predict future purchases numbers and the purchases values for new customers.
In this section we will give sample of how to train and produce predictions
with Lifetimes; however, the procedure is very similar for the BTYD software
package.
The main components of the Lifetimes package are the classes BetaGeoFitter and
GammaGammaFitter. The first class (BetaGeoFitter) contains all the logic
necessary to fit a BG/NBD model from transactional data in the format of
purchase recency and frequency, and current age of the customer in terms of
customer relationship duration. It is also possible to alternatively fit a
Pareto/NBD model using the ParetoNBDFitter class instead.
Given a dataset with correctly formatted inputs, the code to fit the
BetaGeoFitter looks as follows:
⬇
from lifetimes import BetaGeoFitter
bgf = BetaGeoFitter(penalizer_coef=0.0)
bgf.fit(data[’frequency’], data[’recency’], data[’T’])
The package provides a function named summarydatafromtransactiondata that
allows to produce correctly formatted data for the algorithm from a list of
transactions.
The second class (GammaGammaFitter) provides all the methods necessary to fit
a Gamma-Gamma model and can be used, in combination with the BetaGeoFitter, to
predict the customer lifetime value. The Gamma-Gamma model is also trained on
transactional data containing the frequency and the total monetary value of
the purchases made by each customer.
The code to fit the Gamma-Gamma model looks as follows:
⬇
from lifetimes import GammaGammaFitter
returning_customers_summary = data[data[’frequency’]>0]
ggf = GammaGammaFitter(penalizer_coef = 0)
ggf.fit(returning_customers_summary[’frequency’],
returning_customers_summary[’monetary_value’])
And the code to perform CLV predictions on new data using the previously fit
models looks like this:
⬇
ggf.customer_lifetime_value(
bgf,
new_data[’frequency’],
new_data[’recency’],
new_data[’T’],
new_data[’monetary_value’],
time=180,
discount_rate=0.01
The time and discount rate depend on the parameters of the specific prediction
to be performed.
Further documentation and tutorials about the two packages can be found on
their project web
pages555https://github.com/CamDavidsonPilon/lifetimes666https://cran.r-project.org/web/packages/BTYD/index.html,
and both packages are currently available for download and install
respectively on the Python Package Index777https://pypi.org and The
Comprehensive R Archive Network888https://cran.r-project.org.
### 5.2 scikit-learn
Scikit-learn [49] is a library for the Python programming language that
provides functionalists to perform supervised and unsupervised learning tasks.
The supervised learning part of the library implements a wide variety of
algorithms ranging from linear models to tree-based methods for both
classification and regression. In this section, we will show how to use
scikit-learn to build a regression models based on the Random Forest algorithm
similar to the one presented in [45] and [70].
Random Forest [7] is an ensemble algorithms that combines a number of tree
predictors trained on different portions of the data, the output of the model
is generally either the mode or the mean of the outputs of the different tree
predictors depending on whether the task is a classification or a regression
task. Scikit-learn implements this class of algorithms though the
RandomForestRegressor and RandomForestClassifier classes; for the purpose of
this article we will show how to use the RandomForestRegressor to predict CLV
through regression.
Given a dataset similar to the one used in [70] with one line per player, the
code to fit the RandomForestRegressor looks as follows:
⬇
from sklearn.ensemble import RandomForestRegressor
X = data[’number_of_sessions’, ’number_of_rounds’,
’number_of_days’, ’number_of_purchases’,
’total_purchase_amount’]
y = data[”day_360_CLV”]
model = RandomForestRegressor(n_estimators=100)
model.fit(X, y)
For readability, the number of features selected as inputs is reduced in
comparison to [70]; furthermore, the number of trees in the ensemble has been
chosen without any specific motivation as no precise description is included
in the article. Both the size of the ensemble and maximum depth the trees, as
well as the other parameters of the ensemble, are problem specific and should
be selected either algorithmically or through extensive experimentation.
The code necessary to perform CLV prediction using the previously fit model
looks as follows:
⬇
X = new_data[’number_of_sessions’, ’number_of_rounds’,
’number_of_days’, ’number_of_purchases’,
’total_purchase_amount’]
y = model.predict(X)
The resulting array $y$ will contain the predicted customer lifetime values.
The sci-kit learn library includes also a number of functionalities to
estimate the predictions error and to perform different forms of validation
and testing – e.g. k-folds cross-validation. More information can be found on
the project’s documentation page999https://scikit-
learn.org/stable/documentation.html.
## 6 Conclusions and Future Directions
In this chapter we have described the applications of customer lifetime value
prediction in the free-to-play games and other industries and we attempted to
give a comprehensive description of the different methods that are currently
used to predict CLV. We have identified a number of activities that can
benefit from an accurate prediction ranging from customer acquisition to
market segmentation and we have described how the need for such prediction is
even more important in free-to-play games due to the characteristics of the
marker and the customer relationship – e.g. incredibly high competition in
user acquisition and very low number of paying customers compared to the
player base.
In Section 4, we have identifies within the literature four main group of
methods for the prediction: average based, Pareto/NBD and derivatives, Markov
Chain Models and Supervised Learning Models. The methods that are currently
dominant within the free-to-play games industry are either average based or
some form of Pareto/NBD. However, we can see an emerging trend in recent years
of more studies being published on the application of different supervised
learning algorithms to CLV prediction.
These methods have a number of advantages over classical statistical methods
in this context as they make no assumption of the distributions of the input
data and easily allow multiple co-variates to be included in the model.
Furthermore, in an industry with such a low rate of conversion to paying
users, being able to predict revenue from a player that has not yet made any
purchase has the potential to be very useful, for instance, for early
estimation of the profitability of a player acquisition campaign.
Based on the analysis presented and in light of the recent improvements within
the field of machine learning, we close the article by proposing a number of
possible interesting future directions for customer lifetime value prediction
research.
##### Deep Learning
One group of supervised learning algorithms that has only minimally been
explored in one of the most recent articles [70] is Deep Neural Networks. The
work by Sifa et al. don’t give an in-depth description of the deep multi-layer
perceptron used; however, in the absence of other information, we can assume
that the model was based on a standard fully-connected feed-forward neural
network with multiple hidden layers. Given the results achieved with a
relatively simple architecture, it would be interesting to test more advanced
deep learning techniques such as auto-encoders for feature extraction [76] or
deep convolutional neural networks [41].
##### Time Series
All of the models presented in this chapter act on some form of summary data
representing the behavior of customers/players up to a given point in time.
While some of these summary representations do include some representation of
the temporal aspect of the behaviors – e.g. the recency and frequency features
in RFM based models – these cannot capture any particular sequences of
purchases or any sequence of in-game events that is connected to the
probability that a player will perform a purchase or not. One possible
approach to leverage this kind of information is to treat player behavior data
as time series and perform some form of time-series regression or
classification to predict CLV or a purchase event. Within aforementioned deep
learning field algorithms such as Long-short term memory networks [33] or the
more recent temporal convolution networks [23].
##### Transfer Learning and Lifelong Learning
Game developers in the mobile free-to-play market are challenged with the
increasing need to manage a multitude of games live at the same time. These
games have often very long lifetimes and, throughout their lifetimes, the game
is adjusted and evolved and the player base changes dues to the game evolution
and the changes in the customer acquisition initiatives. This adds a new
dimension of the problem of CLV prediction as models need to be constantly
updated and new models need to be built quickly for new version of the games
and for new games. Within this scenario, research in cross-game player
behavior analysis [52], transfer learning [77] and life-long machine learning
[71], and their application to CLV prediction will become increasingly
important.
## References
* [1] Harsha Aeron, Tarun Bhaskar, Ramasubramanian Sundararajan, Ashwani Kumar, and Janakiraman Moorthy. A metric for customer lifetime value of credit card customers. Journal of Database Marketing & Customer Strategy Management, 15(3):153–168, jun 2008.
* [2] Kati Alha, Elina Koskinen, Janne Paavilainen, Juho Hamari, and Jani Kinnunen. Free-to-Play Games : Professionals’ Perspectives. Nordic DiGRA, pages 1–14, 2014.
* [3] Charles Baden-Fuller and Stefan Haefliger. Business Models and Technological Innovation. Long Range Planning, 46(6):419–426, 2013.
* [4] Paul D. Berger and Nada I. Nasr. Customer lifetime value: Marketing models and applications. Journal of Interactive Marketing, 12(1):17–30, 1998.
* [5] Paul D. Berger, Bruce Weinberg, and Richard C. Hanna. Customer lifetime value determination and strategic implications for a cruise-ship company. Journal of Database Marketing & Customer Strategy Management, 11(1):40–52, sep 2003.
* [6] Robert C Blattberg and John Deighton. Manage Marketing by the Customer Euqity. Harvard business review, 74(4):136, 1996.
* [7] Leo Breiman. Random forests. Machine Learning, 2001.
* [8] Leo Breiman. Classification And Regression Trees. Routledge, oct 2017.
* [9] Paolo Burelli and Georgios N. Yannakakis. Adaptive Virtual Camera Control Trough Player Modelling. User Modelling and User-Adapted Interaction, 2015.
* [10] N. V. Chawla, K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer. SMOTE: Synthetic Minority Over-sampling Technique. Journal of Artificial Intelligence Research, 16:321–357, jun 2002\.
* [11] C.-J. Cheng, S.W. Chiu, C.-B. Cheng, and J.-Y. Wu. Customer lifetime value prediction by a Markov chain based data mining model: Application to an auto repair and maintenance company in Taiwan. Scientia Iranica, 19(3):849–855, jun 2012.
* [12] W. K. Ching, M. K. Ng, K. K. Wong, and E. Altman. Customer lifetime value: Stochastic optimization approach. Journal of the Operational Research Society, 55(8):860–868, 2004\.
* [13] Hyunsu Cho. XGBoost Python Package, 2018.
* [14] Francois Chollet. Keras: Deep Learning for humans, 2018.
* [15] Richard Colombo and Weina Jiang. A stochastic RFM model. Journal of Interactive Marketing, 13(3):2–12, 1999.
* [16] Cam Davidson-Pilon. Lifetimes: Measure customer lifetime value in Python, 2018.
* [17] Peter R. Dickson and James L. Ginter. Market Segmentation, Product Differentiation, and Marketing Strategy. Journal of Marketing, 1987.
* [18] Bas Donkers, Peter C. Verhoef, and Martijn de Jong. Predicting Customer Lifetime Value in Multi-Service Industries. Technical report, Erasmus Research Institute of Management (ERIM), 2003\.
* [19] Anders Drachen, Magy Seif El-Nasr, and Alessandro Canossa. Game Analytics – The Basics. In Game Analytics, pages 13–40. 2013.
* [20] F Robert Dwyer. Customer Lifetime Valuation to Support Marketing Decision Making. Journal of Direct Marketing, 11(4):6–13, 1997.
* [21] Lukasz Dziurzynski, Edward Wadsworth, and Daniel McCarthy. BTYD: Implementing Buy ’Til You Die Models, 2014.
* [22] A. S. C. Ehrenberg. Repeat Buying. North-Holland, 1972.
* [23] Maha Elbayad, Laurent Besacier, and Jakob Verbeek. Pervasive Attention: 2D Convolutional Neural Networks for Sequence-to-Sequence Prediction. In Conference on Computational Natural Language Learning, Brussels, Belgium, 2018.
* [24] O. Etzion, A. Fisher, and S. Wasserkrug. e-CLV: a modelling approach for customer lifetime evaluation in e-commerce domains, with an application and case study for online auctions. In IEEE International Conference on e-Technology, e-Commerce and e-Service, 2004. EEE ’04. 2004, pages 149–156. IEEE, 2004.
* [25] Peter S Fader, Bruce G S Hardie, Ka Lok Lee, and Peter S Fader. Counting Your Customers the Easy Way : An Alternative to the Pareto / NBD Model. Marketing Science, 24(2):275–284, 2005.
* [26] Peter S. Fader, Bruce G.S. Hardie, and Ka Lok Lee. RFM and CLV: Using Iso-Value Curves for Customer Base Analysis. Journal of Marketing Research, 42(4):415–430, nov 2005.
* [27] Nicolas Glady, Bart Baesens, and Christophe Croux. A Modified Pareto / NBD Approach for Predicting Customer Lifetime Value. Expert Systems with Applications, 36(2, Part 1):1–26, 2009.
* [28] Sunil Gupta, Dominique M. Hanssens, Bruce G. S. Hardie, Wiliam Kahn, V. Kumar, Nathaniel Lin, N. Ravishanker, S. Sriram, Nalini Ravishanker S. Sriram, V. Kumar, and Sunil Gupta. Modeling Customer Lifetime Value. Journal of Service Research, 9(2):139–155, 2006.
* [29] Sunil Gupta and Donald R. Lehmann. Customer Lifetime Value and Firm Valuation. Journal of Relationship Marketing, 5(2-3):87–110, oct 2006.
* [30] Michael Haenlein, Andreas M. Kaplan, and Anemone J. Beeser. A Model to Determine Customer Lifetime Value in a Retail Banking Context. European Management Journal, 25(3):221–234, jun 2007.
* [31] Nicolai Hanner and Ruediger Zarnekow. Purchasing behavior in free to play games: Concepts and empirical validation. In Proceedings of the Annual Hawaii International Conference on System Sciences, volume 2015-March, pages 3326–3335, 2015.
* [32] Brent Harrison and David L. Roberts. Analytics-Driven Dynamic Game Adaption for Player Retention in a 2-Dimensional Adventure Game. IEEE Conference on Computatonal Intelligence and Games, CIG, (Aiide):23–29, 2013.
* [33] Sepp Hochreiter and Jürgen Schmidhuber. Long Short-Term Memory. Neural Computation, 9(8):1735–1780, nov 1997.
* [34] Arthur M. Hughes. Strategic Database Marketing: The Masterplan for Starting and Managing a Profitable Customer-Based Marketing Program. McGraw-Hill, 2000.
* [35] Hyunseok Hwang, Taesoo Jung, and Euiho Suh. An LTV model and customer segmentation based on customer value: a case study on the wireless telecommunication industry. Expert Systems with Applications, 26(2):181–188, feb 2004.
* [36] Barbara Jackson. Winning and keeping industrial customers. Lexington Books, 1985.
* [37] Dipak Jain and Siddhartha S. Singh. Customer lifetime value research in marketing: A review and future directions. Journal of Interactive Marketing, 16(2):34–46, jan 2002.
* [38] Norman L. Johnson and Samuel Kotz. Continuous Univariate Distributions, volume 1. John Wiley & Sons, 1970.
* [39] E. L. Kaplan and Paul Meier. Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 1958.
* [40] Mahboubeh Khajvand, Kiyana Zolfaghar, Sarah Ashoori, and Somayeh Alizadeh. Estimating customer lifetime value based on RFM analysis of customer purchase behavior: Case study. Procedia Computer Science, 3(January):57–63, 2011.
* [41] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097–1105, Lake Tahoe, NV, USA, 2012. Neural Information Processing Systems Foundation.
* [42] V. Kumar. A Customer Lifetime Value-Based Approach to Marketing in the Multichannel, Multimedia Retailing Environment. Journal of Interactive Marketing, 24(2):71–85, may 2010.
* [43] V. Kumar, Girish Ramani, and Timothy Bohling. Customer lifetime value approaches and best practice applications. Journal of Interactive Marketing, 18(3):60–72, 2004.
* [44] Tony Lachowetz, Mark McDonald, William Sutton, and John Clark. The National Basketball Association: Application of Customer Lifetime Value. Sport Marketing Quarterly, 10(3):181, 2001.
* [45] Sascha Lange, Michael Lenz, and Martin Riedmiller. Case Study : Behavioral Prediction of Future Revenues in Freemium Games. In Workshop New Challenges in Neural Computation, pages 26–33, 2014\.
* [46] Tobias Mahlmann, Anders Drachen, Julian Togelius, Alessandro Canossa, and Georgios N. Yannakakis. Predicting player behavior in Tomb Raider: Underworld. In IEEE Conference on Computational Intelligence and Games, pages 178–185, Copenhagen, Denmark, 2010. IEEE.
* [47] Naresh K. Malhotra. Review of marketing research, 2008.
* [48] Edward C. Malthouse and Robert C. Blattberg. Can we predict customer lifetime value? Journal of Interactive Marketing, 19(1):2–16, jan 2005.
* [49] Andreas Mueller. scikit-learn: A set of python modules for machine learning and data mining, 2018.
* [50] Francis Mulhern. Customer Profitability Analysis: Measurement, Concentration and Research Directions. Journal of Interactive Marketing, 13(1):25–40, 1999.
* [51] David B. Nieborg. Crushing Candy: The Free-to-Play Game in Its Connective Commodity Form. Social Media and Society, 1(2), 2015.
* [52] Héctor Perez Martínez, Maurizio Garbarino, Georgios N. Yannakakis, Hector Perez Martinez, Maurizio Garbarino, and Georgios N. Yannakakis. Generic physiological features as predictors of player experience. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), volume 6974 LNCS, pages 267–276, Memphis, 2011.
* [53] Africa Perianez, Alain Saas, Anna Guitart, and Colin Magne. Churn Prediction in Mobile Social Games: Towards a Complete Assessment Using Survival Ensembles. In 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pages 564–573, 2016.
* [54] Phillip E. Pfeifer and Robert L. Carraway. Modeling customer relationships as Markov chains. Journal of Interactive Marketing, 14(2):43–55, jan 2000.
* [55] Phillip E Pfeifer, Mark E Haskins, and Robert M Conroy. Customer Lifetime Value, Customer Profitability, and the Treatment of Acquisition Spending. Journal of Managerial Issues, 17(1):25, 2004.
* [56] Werner J. Reinartz and V. Kumar. On the Profitability of Long-Life Customers in a Noncontractual Setting: An Empirical Investigation and Implications for Marketing. Journal of Marketing, 64(4):17–35, oct 2000.
* [57] Werner J. Reinartz and V. Kumar. The Impact of Customer Relationship Characteristics on Profitable Lifetime Duration. Journal of Marketing, 67(1):77–99, jan 2003.
* [58] Saharon Rosset, Einat Neumann, U R I Eick, and Nurit Vatnik. Customer Lifetime Value Models for Decision Support. Data Mining and Knowledge Discovery, 7(3):321–339, 2003.
* [59] Saharon Rosset, Einat Neumann, Uri Eick, Nurit Vatnik, and Yizhak Idan. Customer lifetime value modeling and its use for customer retention planning. Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD ’02, page 332, 2002.
* [60] Julian Runge. The Golden Curve: Determining Player Value in Freemium Apps - https://gameanalytics.com/blog/golden-curve-determining-player-value-freemium-apps.html (Accessed: 2018-11-08), 2014.
* [61] Julian Runge, Peng Gao, Florent Garcin, and Boi Faltings. Churn prediction for high-value players in casual social games. In 2014 IEEE Conference on Computational Intelligence and Games, pages 1–8. IEEE, aug 2014.
* [62] John Salvatier, Thomas V Wiecki, and Christopher Fonnesbeck. Probabilistic programming in Python using {PyMC}3. {PeerJ} Computer Science, 2:e55, apr 2016.
* [63] David C. Schmittlein, Donald G. Morrison, and Richard Colombo. Counting Your Customers : Who Are They and What Will They Do Next? Management Science, 33(1):1–24, 1987.
* [64] David C Schmittlein and Robert A Peterson. Customer Base Analysis : An Industrial Purchase Process Application. Marketing Science, 1994.
* [65] Eric Benjamin Seufert. Two Methods for Modeling LTV with a Spreadsheet - https://www.slideshare.net/EricSeufert/ltv-spreadsheet-models-eric-seufert (Accessed: 2018-11-08), 2013.
* [66] Venkatesh Shankar and Sridhar Balasubramanian. Mobile Marketing: A Synthesis and Prognosis. Journal of Interactive Marketing, 23(2):118–129, 2009.
* [67] Cc Shen and Hm Chuang. A study on the applications of data mining techniques to enhance customer lifetime value. WSEAS Transactions on Information Science and Applications, 6(2):319–328, 2009.
* [68] Ya-Yueh Shih and Chung-Yuan Liu. A method for customer lifetime value ranking — Combining the analytic hierarchy process and clustering analysis. Journal of Database Marketing & Customer Strategy Management, 11(2):159–172, dec 2003.
* [69] Rafet Sifa, Fabian Hadiji, Julian Runge, Anders Drachen, Kristian Kersting, and Christian Bauckhage. Predicting Purchase Decisions in Mobile Free-to-Play Games. In Proceedings, The Eleventh AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment (AIIDE-15), pages 79–85, 2015\.
* [70] Rafet Sifa, Julian Runge, Christian Bauckhage, and Daniel Klapper. Customer Lifetime Value Prediction in Non-Contractual Freemium Settings: Chasing High-Value Users Using Deep Neural Networks and SMOTE. In Data, Text and Web Mining for Business Analytics, volume 9, pages 923–932, 2018.
* [71] Daniel L Silver, Qiang Yang, and Lianghao Li. Lifelong Machine Learning Systems : Beyond Learning Algorithms. AAAI Spring Symposium Series, (Solomonoff 1989):49–55, 2013.
* [72] Siddharth S. Singh, Sharad Borle, and Dipak C. Jain. A Generalized Framework for Estimating Customer Lifetime Value When Customer Lifetimes are Not Observed. SSRN Electronic Journal, pages 1–32, 2007.
* [73] Giorgio Alfredo Spedicato. Discrete Time Markov Chains with R. The R Journal, 2017.
* [74] Rajkumar Venkatesan and V Kumar. A Customer Lifetime Value Framework for Customer Selection and Resource Allocation Strategy. Journal of Marketing, 68(4):106–125, oct 2004.
* [75] Peter C. Verhoef and Bas Donkers. Predicting Customer Potential Value: an Application in the Insurance Industry. Technical Report 0, Erasmus Research Institute of Management, 2001.
* [76] Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning - ICML ’08, number April, pages 1096–1103, New York, New York, USA, 2008. ACM Press.
* [77] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in Neural Information Processing Systems, volume 27, pages 1–9, 2014.
|
# Privacy-Preserving Representations are not Enough: Recovering Scene Content
from Camera Poses
Kunal Chelani1 Torsten Sattler2 Fredrik Kahl1 Zuzana Kukelova3
1 Chalmers University of Technology
2 Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical
University in Prague
3 Visual Recognition Group, Faculty of Electrical Engineering, Czech Technical
University in Prague
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Visual localization is the task of estimating the camera pose from which a
given image was taken and is central to several 3D computer vision
applications. With the rapid growth in the popularity of AR/VR/MR devices and
cloud-based applications, privacy issues are becoming a very important aspect
of the localization process. Existing work on privacy-preserving localization
aims to defend against an attacker who has access to a cloud-based service. In
this paper, we show that an attacker can learn about details of a scene
without any access by simply querying a localization service. The attack is
based on the observation that modern visual localization algorithms are robust
to variations in appearance and geometry. While this is in general a desired
property, it also leads to algorithms localizing objects that are similar
enough to those present in a scene. An attacker can thus query a server with a
large enough set of images of objects, _e.g_., obtained from the Internet, and
some of them will be localized. The attacker can thus learn about object
placements from the camera poses returned by the service (which is the minimal
information returned by such a service). In this paper, we develop a proof-of-
concept version of this attack and demonstrate its practical feasibility. The
attack does not place any requirements on the localization algorithm used, and
thus also applies to privacy-preserving representations. Current work on
privacy-preserving representations alone is thus insufficient.
## 1 Introduction
Visual localisation refers to the problem of estimating the camera pose of a
given image in a known scene. It is a core problem in several 3D computer
vision applications, including self-driving cars [17, 18] and other autonomous
robots [50], and Augmented Reality [25, 23, 5].
Figure 1: In the context of privacy-preserving localization, we show that it
is possible to learn about the content of a scene using camera poses returned
by a localization service, without any direct access to the scene
representation. (1st column) Examples of images from the scene, used to build
the scene representation. The images are shown for illustrative purposes and
are not available to an attacker trying to learn about the scene. (2nd column)
The attacker queries the service with images of objects, _e.g_., downloaded
from the Internet. (3rd & 4th column) Using the camera poses for the query
image returned by the localization service, the attacker is able to identify
the types of objects present in the scene and to accurately place them in the
scene. We show the estimated object poses overlaid over the ground truth
structure of the scene (which is not accessible to the attacker). The attacker
is able to faithfully recover the placement of objects. Overall, our results
demonstrate that simple feedback such as camera poses is already sufficient to
potentially reveal private details.
A popular approach for Augmented/Mixed/Virtual Reality (XR) applications is to
use a client-server mechanism for localization: the user device (client) sends
image data to a cloud-based system (server) that computes and returns the
camera pose [25, 46, 23]. Examples of such services include Google’s Visual
Positioning System [29], Microsoft’s Azure Spatial Anchors [24], and Niantic’s
Lightship [39]. Cloud-based localization services are popular for multiple
reasons - first, performing localization on the server reduces storage
requirements and the computational load, and thus energy consumption, which is
important for client devices such as mobile phones and headsets; second, it
enables using robust mapping and localization algorithms that are too
expensive for mobile devices; third, in the context of collaborative mapping,
_e.g_., for the AR cloud or autonomous driving, maintaining a single scene
representation in a centralized place is far easier than keeping multiple
copies on various mobile devices up-to-date.
Naturally, sending user data to a server, _e.g_., in the form of images to be
localized or 3D maps recorded by users that will be used for localization,
raises privacy concerns [41, 42, 9]. Work on privacy-preserving localization
aims to resolve these concerns by ensuring that private details cannot be
recovered from the data sent [42, 14, 26] to or stored on the server [41, 36,
28, 52, 11, 15, 11].
Existing work focuses on scenarios where an attacker gains access to the
localization service or can eavesdrop on the communication between client and
server. In this work, we demonstrate that it is possible for an attacker to
learn about the content of a scene stored on a localization server without
direct access to the server. We show that a localization service will reveal
scene-related information through estimated camera poses, _i.e_., through its
normal operation process. The attack is based on two recent developments: (1)
modern visual localization algorithms are designed to be robust against
changes such as illumination and seasonal variations [44]. This is an
essential property for cloud-based localization services in order to operate
robustly and reliably. However, since these algorithms are robust to (slight)
variations in appearance and geometry, they will also localize images showing
objects that are similar (but not necessarily identical) to those objects
present in the scene. (2) massive amounts of images depicting objects in
different variations are readily available on the Internet. Taken together,
both developments allow an attacker to repeatedly query the server with images
and to recover the positions of the objects in the scene based on the camera
poses returned by the server (_cf_. Fig. 1). In this paper, we demonstrate the
feasibility of this attack by developing a proof-of-concept implementation of
the attack.
In summary, we make the following contributions: (1) we identify a new line of
attack in the context of privacy-preserving visual localization based on the
camera poses returned by a cloud-based server. (2) we show the feasibility of
the attack through a proof-of-concept implementation of the attack. Through
experiments, we explore the performance of our implementation as well as the
trade-off between localization robustness and potential defenses against the
attack. (3) the attack is agnostic to the underlying localization algorithm
and thus applicable even if the localization system is otherwise perfectly
privacy-preserving. This paper thus proposes a new research direction for
privacy-preserving localization, where the aim for the localization service is
to correctly identify whether a query image was taken in the concerned scene
or not, in order to prevent leaking information through camera poses.
## 2 Related Work
Visual localization. Most state-of-the-art visual localization algorithms are
based on establishing 2D-3D matches between a query image and a 3D model of
the scene. These correspondences are then used for camera pose estimation. The
3D model can either be stored explicitly [33, 31, 32, 20, 21, 43, 19, 27],
_e.g_., in the form of a Structure-from-Motion (SfM) point cloud, or
implicitly in the form of the weights of a machine learning model [3, 2, 1,
38, 45, 6]. In the former case, local feature descriptors are associated with
3D points of the model. It has been shown that this information is sufficient
to recover detailed images from the 3D map [28, 40], although sparsifying
these models [4, 51] might effectively make them privacy-preserving [7].
Approaches based on implicit representations map image pixels or patches to 3D
points by training scene coordinate regression models [38, 3]. Recently, it
was claimed that such approaches are inherently privacy-preserving [11].
However, feature-based methods currently scale better to large scenes and are
able to better handle condition changes [44], such as illumination or seasonal
changes, between the query image and the database images used to build the the
scene representation. The resulting robustness is highly important in many
applications of visual localization, including AR and robotics. The robustness
is a direct consequence of recent advances in local features [10, 13, 30] and
effective feature matchers [32, 53, 43, 48]. In this paper, we show that
robustness to changing conditions enables an attacker to learn about the
content of the scene: robustness to changing conditions not only bridges the
gap between (small) variations in scene appearance and geometry observed in
images depicting the same place, but also leads to correspondences between
images depicting similar but not identical objects, _e.g_., different chairs.
In turn, these correspondences can be used to localize the object in the
scene, which is the basis for the attack described in this work. Note that the
properties we exploit are inherent to robust localization algorithms and are
not restricted to feature-based methods. Ultimately, any robust localization
system needs to handle variations in shape and appearance.
Privacy-preserving visual localization. Existing work on privacy-preserving
localization focuses on two points of attack: (1) ensuring that data sent to a
localization service does not reveal private information. (2) ensuring that
data stored on a localization service does not reveal private information. For
the former case, it has been shown that images can be recovered from local
features [49, 12, 9]. Work on privacy-preserving queries to a localization
server thus mostly aims at developing features that prevent image recovery
[14, 26] or on obfuscating the feature geometry [42, 16]. Similarly, work on
privacy-preserving scene representation aims to obfuscate the geometry [41,
37] (although scene geometry can be recovered under certain conditions [7]),
splitting the maps over multiple server for increased data security [15],
using implicit representations [11], or storing raw geometry without any
feature descriptors [52].
This paper presents a new line of attack that complements existing work.
Previous work considers a scenario where the attacker gains access to the
service. In contrast, we show that it is possible to recover scene content
from the very basic information provided by any localization service, namely
the camera poses estimated for query images. As such, the attack is still
feasible even if the data send to and stored on the server is completely
privacy-preserving. Our work thus shows that existing privacy-preserving
localization approaches are not sufficient to ensure user privacy.
## 3 Recovering Scenes from Camera Poses
Any localization system returns the camera poses of localized query images. At
the same time, modern localization algorithms aim to be robust to shape and
appearance variations in order to be robust to changes in viewing conditions.
This feature, however, opens up the possibility that not only genuine queries,
but also images of objects that are similar to the ones present in the scene
can be localized. The camera poses of the localized images can then in turn be
used to infer the positions of certain objects in the scene, potentially
revealing more information about the scene than the cloud-based service / a
user would like to disclose.
Naturally, an attacker does not know which objects are present in the scene
and thus which images to use for their queries. The Internet is a source of a
theoretically unlimited number of images, videos, and 3D models of objects of
different types and appearances. This naturally leads to an idea of a
potential attack, where an attacker just downloads such images and videos,
bombards the server with localization requests, and uses poses of localized
images to reveal detailed scene structure.
In the following sections, we investigate this new type of attack, and we try
to answer several questions: Can an attacker with access to images and videos
of objects similar to those present in the scene easily learn about the
presence/absence of different objects and their placement in the scene just
from the poses returned by a localization service? What are the challenges of
such an attack, and are these challenges easily solvable? Can cloud-based
services easily prevent such attacks? To this end, we present a proof-of-
concept implementation of the attack.111We only aim to show feasibility. We
believe that better attack algorithms are certainly possible. Later, Sec. 6
discusses an approach to potentially mitigate the attack and why its
effectiveness is limited.
### 3.1 Formalization
We assume a localization server $\mathcal{L}$ that is responsible for
localizing images in a scene $\mathcal{S}$. $\mathcal{L}$ tries to align each
query image it receives with the scene representation as best as possible. If
an image can be localized, the server returns a 6-dof camera pose
$[\mathbf{R}|\mathbf{t}]$. We assume that the scale of the translation
component $\mathbf{t}$ is in known.
An adversary $\mathcal{A}$ is querying $\mathcal{L}$ with many images of
different objects, where each image contains only one dominant object to avoid
confusion about which object from the image was localized in the scene.
$\mathcal{A}$, using the poses returned by $\mathcal{L}$, wants to learn about
the presence/absence of objects in the scene $\mathcal{S}$, and wants to infer
their (approximate) positions. As such, $\mathcal{A}$ tries to construct an
(approximate) ”copy” of the scene $\mathcal{S}$ or at least its layout.
In this setting $\mathcal{A}$ needs to deal with two challenges:
1. 1.
$\mathcal{A}$ queries $\mathcal{L}$ with images of objects that, in general,
differ geometrically from the actual objects in the scene. In the best case,
the pose returned by the server provides the best-possible approximate
alignment between the query and actual object. In general, the returned poses
will be noisy and can be quite inaccurate if only a part of the object,
_e.g_., a chair’s leg, is aligned. Creating an accurate ”copy” of the scene
from such poses is a challenging problem.
2. 2.
$\mathcal{A}$ has, in general, no a-priori information about the type of the
scene and which objects are visible in it. Since $\mathcal{L}$ can also return
poses for objects that are not in the scene, $\mathcal{A}$ needs to have a
mechanism for deciding the presence/absence of an object based on the returned
poses. Naturally, having to deal with noisy and inaccurate poses makes the
decision process harder.
In general, it is not possible to overcome these challenges by using a single
image of each object. A single camera pose returned by $\mathcal{L}$, without
additional information, does not provide enough data for deciding about the
presence/absence of the object in the scene and the quality of the pose.
However, given the large amount of images available on the Internet, and in
particular the availability of videos, $\mathcal{A}$ can use several images of
the same object taken from different viewpoints. Jointly reasoning about all
of the corresponding poses obtained for these images can then be used to
decide the presence and position of the object.
Figure 2: Object alignment example: 1. A 3D model $\mathcal{M}$ of an object
and corresponding camera poses $\mathbf{P}_{o}$ in the attacker’s local
coordinate system, built from a sequence of object images. 2. The server scene
with a similar object. 3. The noisy poses returned by the server for the
queried object images. 4. Sequences of local and server-provided poses aligned
to approximately place the object in the scene.
### 3.2 3D Object Placement
Assuming that the attacker knows that an object is present, they still need to
predict its position and orientation in the scene based on the pose estimates
provided by the server. To this end, the attacker can use that multiple images
of the same object taken from different viewpoints are available. These images
can be used by $\mathcal{A}$ to build a local 3D model $\mathcal{M}$, _e.g_.,
using SfM [34] and MVS [35], and to compute the poses $\mathbf{P}_{o}$ of
these images w.r.t. this model. In turn, $\mathcal{L}$ provides a set
$\hat{\mathbf{P}}_{o}$ of poses for (a subset of) these images in the
coordinate system of the scene model $\mathcal{S}$. The problem of placing the
object in the copy of the scene $\mathcal{S}$ thus reduces to the problem of
aligning both sets of poses (_cf_. Fig. 2). The camera poses
$\hat{\mathbf{P}}_{o}$ provided by $\mathcal{L}$ can be very noisy and can
contain outliers. Thus, the alignment process needs to be robust.
Algorithm 1 Best single camera based alignment between sets of poses
Input
$\mathbf{P}_{o}=\\{[\mathbf{R}_{i}|\mathbf{t}_{i}]\\}$,$\mathbf{\hat{P}}_{o}=\\{[\mathbf{\hat{R}}_{i}|\mathbf{\hat{t}}_{i}]\\}$,
$\delta_{r}$,$\delta_{t}$
Output $\textbf{R}_{best},\textbf{t}_{best},\epsilon$
1:procedure Get-Best-Alignment
2: $\texttt{N}\leftarrow{|\mathbf{P}_{o}|}$
3: $\texttt{Inliers\\_best}\leftarrow\phi$
4: for i = 1 to N do
5: $\mathbf{R}_{est}\leftarrow\mathbf{\hat{R}}^{\top}_{i}\mathbf{R}_{i}$
6:
$\mathbf{t}_{est}\leftarrow\mathbf{\hat{R}}^{\top}_{i}(\mathbf{t}_{i}-\mathbf{\hat{t}}_{i})$
7: $\texttt{Inliers}\leftarrow\phi$
8: for j = 1 to N do
9:
$\Delta_{r}\leftarrow\angle(\mathbf{R}_{j}\mathbf{R}^{\top}_{est}\mathbf{\hat{R}}_{j}^{\top})$
10:
$\Delta_{t}\leftarrow||\mathbf{\hat{R}}^{\top}_{j}\mathbf{\hat{t}}_{j}-\mathbf{R}_{est}\mathbf{R}^{\top}_{j}\mathbf{{t}}_{j}+\mathbf{t}_{est})||$
11: if $\Delta_{r}<\delta_{r}$ and $\Delta_{t}<\delta_{t}$ then
12: $\texttt{Inliers}\leftarrow\texttt{Inliers}\cup\\{\texttt{j}\\}$
13: if $|\texttt{Inliers}|>|\texttt{Inliers\\_best}|$ then
14: $\texttt{Inliers\\_best}\leftarrow\texttt{Inliers}$
15: $\epsilon\leftarrow|\texttt{Inliers\\_best}|/\texttt{N}$
16: $\mathbf{R}_{best},\mathbf{t}_{best}\leftarrow$ Average(Inliers_best)
As mentioned above, for simplicity we assume that the scale of the 3D model
stored by $\mathcal{L}$ is known.222In the context of user-generated maps,
captured by devices with IMUs such as mobile phones or dedicated XR headsets,
it seems realistic to assume that the scale of the maps is provided in meters.
Similarly, the scale of the local model $\mathcal{M}$ can be (approximately)
recovered using the known size of the object. In this case, the two poses, in
the coordinate systems of $\mathcal{M}$ and $\mathcal{S}$, for a single image
already provide an alignment hypothesis, _i.e_., the relative pose between
them. As outlined in Alg. 1, we evaluate all hypotheses. The input to Alg. 1
are the two sets of poses, $\mathbf{P}_{o}$ and $\hat{\mathbf{P}}_{o}$, and
two error thresholds - $\delta_{r}$ for rotation and $\delta_{t}$ for
translation. For each pair of corresponding camera poses - local and server-
provided, a relative transformation is computed (line 5-6). One set of poses
is transformed using this estimated transformation, and errors for rotation
and translation between corresponding pairs are computed (Lines 9-10). Using
the two thresholds, we determine which other pose pairs are inliers to the
pose hypothesis (Lines 11-12). The transformation with the largest number of
inliers is selected (Lines 13-14) and refined by averaging the relative poses
of all inliers.
Obviously, not knowing the scale of the scene model $\mathcal{S}$ is
insufficient to prevent the attacker from placing the object in the scene as
the scale and relative transformation can be recovered from two pairs of
poses. Additionally, there are ways to further robustify the alignment
process. _E.g_., if images of multiple very similar instance of an object and
the corresponding 3D models are available, it seems reasonable to assume that
images of different instances taken from similar viewpoints will also result
in similar pose estimates by $\mathcal{L}$. These estimates can then be used
to average out noise in the poses. Similarly, the relation between different
objects, _e.g_., a monitor standing on a desk, can be used to stabilize the
process of placing objects in the scene. However, we do not investigate such
advanced strategies in this paper.
### 3.3 Deciding the Presence/Absence of an Object
We assume that $\mathcal{L}$ is running a localization algorithm that is
robust to shape and appearance variations and that is aligning query images to
the scene as best as it can. At the same time, $\mathcal{L}$ can also return
poses for objects that are not in the scene, as well poses for objects that
are not even from the same categories or similar to objects present in the
scene. Deciding if an object is present or not in a scene based on the poses
returned for its images by the localization server is therefore a challenging
problem.
Figure 3: Example images from IKEA-Scenes (left) and one of the objects of
corresponding scenes in IKEA-Objects (right).
For an attacker $\mathcal{A}$ trying to recover scene information via camera
poses, it is impossible to determine which type of objects are present using
just a single camera pose returned for one query image of each of the objects.
To overcome this challenge, $\mathcal{A}$ can employ several possible
techniques; _e.g_., they can use statistics about object co-occurrence to
select the set of queries and associated camera poses having a high
probability of their spatial distribution. Another simple solution is to use
multiple images of the same object taken from different viewpoints or to
cluster query images into groups depicting similar objects that are assumed to
be matched with the same object in the scene $\mathcal{S}$. $\mathcal{A}$ can
then use different images from these groups to query $\mathcal{L}$ and decide
on the presence/absence of the object based on the consistency of returned
poses. Even though the returned poses can be noisy and can contain outlier
poses, in general, it is expected that a reasonably large subset of images
depicting the same object from different viewpoints or depicting objects from
the same group will show consistency of returned poses if a similar object is
present in $\mathcal{S}$. On the other hand, poses obtained for images of an
object that is absent can be expected to exhibit a much higher variance.
In this paper, we discuss and evaluate another strategy for presence/absence
decision that allows us to show the completeness of the attack and present its
proof-of-concept implementation. We assume that the attacker $\mathcal{A}$
learns certain statistics for each object or category from a curated training
data that comprises of scenes with known presence/absence of these objects or
object categories. This can be done for different types of localization
schemes over huge amounts of 3D data. The attacker can then use these learned
statistics to infer the presence/absence of objects when attacking an unknown
scene $\mathcal{S}$.
For experimental results in the later sections, we use the inlier-ratio
$\mathbf{\epsilon}$ obtained from the object positioning step (Line 15 in Alg.
1) as this statistic. We can assume that for each object (or a class of
objects) $o$, $\mathcal{A}$ has inlier-ratios $\mathbf{\epsilon}^{+}_{o}$ and
$\mathbf{\epsilon}^{-}_{o}$ that are trained on scenes with known presence or
absence of $o$. _E.g_., $\mathbf{\epsilon}^{+}_{o}$ and
$\mathbf{\epsilon}^{-}_{o}$ can be computed as the medians of
$\mathbf{\epsilon}_{o}$ over all ”present(+)/absent(-)” scenes. Based on these
statistics, the presence or absence of $o$ in the unknown scene $\mathcal{S}$
can be decided by comparing the distances of
$\mathbf{\epsilon}^{\mathcal{S}}_{o}$ to $\mathbf{\epsilon}^{+}_{o}$ and
$\mathbf{\epsilon}^{-}_{o}$. We provide concreteness to this idea when
assessing its effectiveness over a real world dataset in Section 5.2.
Figure 4: Qualitative results for aligning objects in different scenes of the
IKEA-Scenes dataset. We evaluate three combinations of local features and
matchers. Aligned objects are color-coded green to red along the gravity
direction to make their orientation better visible.
## 4 Datasets
We use multiple datasets for our experiments:
IKEA-Scenes and IKEA-Objects \- We captured image sequences of seven different
inspiration-rooms at an IKEA furniture store (_cf_. Fig.3). 1,000-2,500 images
were captured for each room, depending on its size. 4-10 objects from each
room were selected, and a separate sequence of images was captured for each of
them in the inventory section of the store, where the surrounding environment
was different from that of the inspiration rooms. Note that the two instances
of each object have the same model, but in many cases differ in color and
size. Presence/absence of additional objects such as cushions on a sofa, or a
computer on a desk can additionally change the overall appearance of the two
instances. In total, the dataset comprises 38 object instances covered by
100-200 images each. While capturing the dataset, we tried to only have a
single object occupying a large part of each image. However, this was not
always possible and no post processing has been applied to mask out objects.
We call the inspiration-room data IKEA-Scenes and the data from the inventory
section IKEA-Objects.
ScanNet-Office-Scene \- To show that the objects do not need to be of the
exact same model for the proposed attack to work, we consider a generic office
scene - scene0040 from the ScanNet [8] dataset.
Office-Objects \- We collected image sequences of 5 common office room objects
- a door, a whiteboard, an office chair, a desk with computer, and a
bookshelf. These images are used as queries by the attacker.
RIO10 \- RIO10 [47] is a localization benchmark dataset which we use to
evaluate the effectiveness of a potential defence strategy that a localization
server might employ.
We manually scale all local 3D models constructed by the attacker to roughly
metric scale.
## 5 Experimental Evaluation
This section presents a series of experiments that show the practical
feasibility of the attack introduced in Sec. 3. First, we show via qualitative
results that the method proposed in Sec. 3.2 allows the attacker to place the
3D models of relevant objects close to the actual corresponding objects in the
scene. We then explain and evaluate a simple implementation of the method
described in Sec. 3.3 that the attacker can use to decide the presence/absence
of objects.
For querying the localization server, we use images from the datasets
described above. To implement the server, we use HLoc [31, 32] (with default
thresholds and parameters), a state-of-the-art visual localization approach.
HLoc uses feature descriptors to establish 2D-3D matches between features
extracted from the query image and 3D scene points. The resulting
correspondences are then used for pose estimation. We demonstrate the reliance
of the attack on the robustness of the localization process by evaluating
three different local image features and matchers: Superpoint [10] features
with the SuperGlue[32] (most robust), R2D2 [30] with Nearest Neighbor (NN)
matching, and SIFT [22] with NN matching (least robust).
### 5.1 3D Object Placement
Figure 5: (a) Example images from ScanNet-Office-Scene and corresponding
objects in Office-Objects. (b) Qualitative results for aligning generic office
objects in ScanNet [8] scene0040, using Superpoint+Superglue and R2D2+NN.
We qualitatively evaluate the accuracy of the 3D object placements obtained
using the approach from Sec. 3.2 for the IKEA-Scenes and ScanNet-Office-Scene
datasets. We use qualitative results rather than quantitative metrics since it
is hard to quantify when a placement is realistic enough. _E.g_., consider the
predicted positions of the oven in the 3rd row of Fig. 4. The first two
predictions are far enough from the ground truth position that a metric such
as the IoU of the 3D bounding boxes of the objects will discard them as wrong.
Yet, the estimated positions are close enough to the ground truth to provide
the attacker with a good layout of the scene.
Fig. 4 shows results for placing selected items from the IKEA-Objects dataset
in 4 different scenes from the IKEA-Scenes dataset. Fig. 5 shows results for
placing objects from the Office-Objects dataset in the ScanNet-Office-Scene
dataset. As can be seen, using a robust localization process based on
Superpoint features and the Superglue matcher or R2D2 features allows the
attacker to place the objects close to their ground truth positions. In
particular, the results from Fig. 5 show that the alignment also works well
when the queried object is not the same model of different color/size but also
a very different one in terms of shape and overall appearance. The results
clearly demonstrate the practical feasibility of the placement strategy.
We used slightly different values for the error thresholds required by the
positioning algorithm based on the object size and obtained poses. Such an
approach is feasible if a human supervises the attack. Code and data is
available at https://github.com/kunalchelani/ObjectPositioningFromPoses.
### 5.2 Deciding the Presence/Absence of an Object
Scene | Superpoint+Superglue | R2D2+NN | SIFT + NN
---|---|---|---
$10^{\circ},0.25m$ | $30^{\circ},0.5m$ | $60^{\circ},2m$ | $10^{\circ},0.25m$ | $30^{\circ},0.5m$ | $60^{\circ},2m$ | $10^{\circ},0.25m$ | $30^{\circ},0.5m$ | $60^{\circ},2m$
Precision | Recall | P | R | P | R | P | R | P | R | P | R | P | R | P | R | P | R
Scene1 | 0.6 | 0.85 | 0.75 | 0.85 | 0.67 | 0.85 | 0.57 | 0.57 | 0.36 | 0.57 | 0.28 | 0.57 | 0.33 | 0.57 | 0.45 | 0.71 | 0.33 | 0.43
Scene2 | 0.36 | 0.4 | 0.36 | 0.5 | 0.37 | 0.6 | 0.34 | 0.4 | 0.3 | 0.3 | 0.35 | 0.6 | 0.33 | 0.4 | 0.26 | 0.5 | 0.28 | 0.6
Scene3 | 0.55 | 0.71 | 0.36 | 0.57 | 0.25 | 0.43 | 0.31 | 0.71 | 0.47 | 1 | 0.41 | 1.0 | 0.3 | 0.42 | 0.5 | 0.42 | 0.44 | 1.0
Scene4 | 0.17 | 0.4 | 0.23 | 0.6 | 0.14 | 0.4 | 0.34 | 0.6 | 0.28 | 0.4 | 0.2 | 0.4 | 0.15 | 0.4 | 0.15 | 0.4 | 0.17 | 0.4
Scene5 | 0.33 | 0.6 | 0.4 | 0.8 | 0.44 | 0.8 | 0.5 | 0.6 | 0.34 | 0.4 | 0.5 | 0.6 | 0.22 | 0.4 | 0.25 | 0.4 | 0.33 | 0.4
Scene6 | 0.25 | 0.6 | 0.28 | 0.6 | 0.22 | 0.4 | 0.22 | 0.4 | 0.3 | 0.6 | 0.33 | 0.8 | 0.14 | 0.2 | 0.2 | 0.2 | 0.25 | 0.6
Scene7 | 0.5 | 0.5 | 0.5 | 0.33 | 0.33 | 0.5 | 0.6 | 0.5 | 0.5 | 0.5 | 0.38 | 0.5 | 0 | 0 | 0.14 | 0.17 | 0 | 0
Table 1: Precision (P) and recall (R) of our method to determine the presence
of objects for the IKEA-scenes and IKEA-Objects datasets.
Figure 6: Effectiveness of a potential approach to prevent the proposed attack
based on not providing poses for queries containing only a few objects. Only
objects contributing at least 10% of the inliers found on the object with the
most inliers are considered. As can be seen, finding a suitable threshold for
the minimum number of visible objects can be difficult.
In Sec. 3.3, we suggested strategies which an attacker can use to decide
whether an object is present or not in a scene $\mathcal{S}$. for each object.
Concretely, using a set of training scenes, the attacker has learned
representative values $\epsilon^{+}$ and $\epsilon^{-}$ for the inlier-ratio
returned by Alg. 1 for cases where the object is present(+) respectively
absent(-). When deciding the presence of an object $\mathbf{o}$ in a scene
$\mathcal{S}$, the attacker uses the inlier ratio ($\epsilon$) from Alg. 1 to
make their decision. The object $\mathbf{o}$ is considered to be present in
the scene if $|\epsilon-\epsilon^{+}|<|\epsilon-\epsilon^{-}|$ and otherwise
considered as absent.
We use the IKEA-Scenes and IKEA-Objects dataset for this experiment. When
deciding the presence/absence of an object in a scene, the other 6 scenes are
used as training scenes. Many of the objects from IKEA-Objects are only
present in one of the scenes from IKEA-Scenes. In these cases, no reference
value for $\epsilon^{+}$ is available for these scenes. In such cases, the
object is considered as present if $\epsilon>\epsilon^{-}$. This strategy is
motivated by the assumption that correctly placing an object that is present
results in a higher inlier-ratio than placing objects that are not present.
Tab. 1 shows precision and recall of this strategy. Since the computation of
the inlier-ratio $\epsilon$ depends upon the error thresholds, we present the
results for three different sets of thresholds. The results show that for most
scenes, it is possible to obtain a precision/recall of approx. 0.4/0.6, which,
_e.g_., translates to 3 out of 5 present, and around 29 out of 33 absent
objects from IKEA-objects being correctly classified. The average precision
using random guessing in these scenes is 0.19. This, together with the quality
of the placement, clearly validates the feasibility of the proposed attack.
## 6 Preventing the Attack?
A natural way to prevent the presented attack is to try to distinguish between
genuine and malicious queries. By not sending poses for query images deemed as
(potentially) malicious, the localization service effectively prevents the
attacker from using pose estimates to learn about the scene.
One potential classification strategy is based on the fact that the attacker
sends images focusing on a single object. In this case, we expect that most of
the 3D points from the inlier 2D-3D matches found by HLoc lie on a single 3D
object. We thus count the number of 3D objects that contribute at least a
certain fraction of inliers (X% of the inliers of the object contributing the
largest number of inliers). If the number is too small, the query image is
considered to be malicious and is rejected.
Fig. 6 shows results for three different objects used to attack three
different scenes of the RIO10 dataset [47]. Here, we use the instance-level
labels provided by the dataset, which include background classes such as floor
and walls, to define objects. As can be seen, rejecting the majority of
malicious queries while retaining genuine queries can be challenging. The
reason is that even while focusing on a single object, other objects might be
partially visible in the queries, _e.g_., part of a desk for monitors,
different pillows on a couch, books on a shelf, _etc_. In addition, genuine
queries might focus on small parts of the scene or even individual objects.
Thus, finding a suitable threshold on the minimum number of visible objects
can be hard. Furthermore, note that this defense strategy requires the service
to have knowledge about the objects in the scene, either extracted from the
queries or the scene representation. This requirement creates a potential
privacy risk if an attacker is able to gain access to the service.
## 7 Conclusions and Future work
In this paper, we have considered the problem of privacy-preserving
localization. Prior work aims to defend attacks for the case where the
attacker gains access to a cloud-based localization service. In contrast, we
show that it is possible for an attacker to recover information about the
scene by using the service as intended: by querying the server with images of
different objects, an attacker is able to determine which objects are present
and to estimate their position in the scene. The attack is based on the
minimum amount of information that a localization service needs to provide to
its users, _i.e_., camera poses for query images, and exploits that modern
localization systems are robust to changing conditions. Experiments with our
proof-of-concept implementation show the practical feasibility of the attack.
The attack is applicable even if the localization algorithm used by the server
is otherwise perfectly privacy-preserving.
Our results show that existing privacy-preserving approaches are not
sufficient to ensure user privacy, creating the need for further research. In
particular, first experiments show that preventing the attack proposed in this
paper without reducing localization performance and creating other angles of
attack is a non-trivial task and interesting direction for future work.
Acknowledgements. This work was supported by the EU Horizon 2020 project
RICAIP (grant agreement No. 857306), the European Regional Development Fund
under project IMPACT (No. CZ.02.1.01/0.0/0.0/15_003/0000468), the Czech
Science Foundation (GAČR) JUNIOR STAR Grant No. 22-23183M, Chalmers AI
Research Center (CHAIR), WASP and SSF.
## References
* [1] Eric Brachmann and Carsten Rother. Learning Less is More - 6D Camera Localization via 3D Surface Regression. In CVPR, 2018.
* [2] Eric Brachmann and Carsten Rother. Expert sample consensus applied to camera re-localization. In ICCV, 2019.
* [3] Eric Brachmann and Carsten Rother. Visual camera re-localization from RGB and RGB-D images using DSAC. arXiv:2002.12324, 2020.
* [4] Federico Camposeco, Andrea Cohen, Marc Pollefeys, and Torsten Sattler. Hybrid Scene Compression for Visual Localization. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
* [5] R. O. Castle, G. Klein, and D. W. Murray. Video-rate localization in multiple maps for wearable augmented reality. In ISWC, 2008.
* [6] T. Cavallari, L. Bertinetto, J. Mukhoti, P. Torr, and S. Golodetz. Let’s take this online: Adapting scene coordinate regression network predictions for online rgb-d camera relocalisation. In 3DV, 2019.
* [7] Kunal Chelani, Fredrik Kahl, and Torsten Sattler. How Privacy-Preserving Are Line Clouds? Recovering Scene Details From 3D Lines. In (CVPR), pages 15668–15678, June 2021.
* [8] Angela Dai, Angel X. Chang, Manolis Savva, Maciej Halber, Thomas Funkhouser, and Matthias Nießner. Scannet: Richly-annotated 3d reconstructions of indoor scenes. In CVPR, 2017.
* [9] Deeksha Dangwal, Vincent T. Lee, Hyo Jin Kim, Tianwei Shen, Meghan Cowan, Rajvi Shah, Caroline Trippel, Brandon Reagen, Timothy Sherwood, Vasileios Balntas, Armin Alaghi, and Eddy Ilg. Analysis and mitigations of reverse engineering attacks on local feature descriptors. BMVC, 2021.
* [10] Daniel DeTone, Tomasz Malisiewicz, and Andrew Rabinovich. Superpoint: Self-supervised interest point detection and description, 2018\.
* [11] Tien Do, Ondrej Miksik, Joseph DeGol, Hyun Soo Park, and Sudipta N. Sinha. Learning to detect scene landmarks for camera localization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2022.
* [12] Alexey Dosovitskiy and Thomas Brox. Inverting visual representations with convolutional networks. In CVPR 2016, pages 4829–4837, 06 2016.
* [13] Mihai Dusmanu, Ignacio Rocco, Tomas Pajdla, Marc Pollefeys, Josef Sivic, Akihiko Torii, and Torsten Sattler. D2-Net: A trainable CNN for joint detection and description of local features. In CVPR, 2019.
* [14] Mihai Dusmanu, Johannes L. Schönberger, Sudipta N. Sinha, and Marc Pollefeys. Privacy-preserving visual feature descriptors through adversarial affine subspace embedding. CVPR, 2021.
* [15] Marcel Geppert, Viktor Larsson, Johannes L. Schönberger, and Marc Pollefeys. Privacy preserving partial localization. In CVPR, 2022.
* [16] Marcel Geppert, Viktor Larsson, Pablo Speciale, Johannes L. Schönberger, and Marc Pollefeys. Privacy Preserving Structure-from-Motion. In ECCV, 2020.
* [17] Marcel Geppert, Peidong Liu, Zhaopeng Cui, Marc Pollefeys, and Torsten Sattler. Efficient 2D-3D Matching for Multi-Camera Visual Localization. In ICRA, 2019.
* [18] L. Heng, B. Choi, Z. Cui, M. Geppert, S. Hu, B. Kuan, P. Liu, R. Nguyen, Y. C. Yeo, A. Geiger, G. H. Lee, M. Pollefeys, and T. Sattler. Project AutoVision: Localization and 3D Scene Perception for an Autonomous Vehicle with a Multi-Camera System. In ICRA, 2019.
* [19] Martin Humenberger, Yohann Cabon, Nicolas Guerin, Julien Morat, Jérôme Revaud, Philippe Rerole, Noé Pion, Cesar de Souza, Vincent Leroy, and Gabriela Csurka. Robust Image Retrieval-based Visual Localization using Kapture. arXiv:2007.13867, 2020.
* [20] A. Irschara, C. Zach, J.-M. Frahm, and H. Bischof. From Structure-from-Motion Point Clouds to Fast Location Recognition. In CVPR, 2009.
* [21] Y. Li, N. Snavely, D. Huttenlocher, and P. Fua. Worldwide Pose Estimation Using 3D Point Clouds. In ECCV, 2012.
* [22] D. Lowe. Distinctive Image Features from Scale-Invariant Keypoints. IJCV, 60(2), 2004.
* [23] Simon Lynen, Bernhard Zeisl, Dror Aiger, Michael Bosse, Joel Hesch, Marc Pollefeys, Roland Siegwart, and Torsten Sattler. Large-scale, real-time visual–inertial localization revisited. IJRR, 39(9):1061–1084, 2020.
* [24] Microsoft. Spatial Anchors, 2020.
* [25] S. Middelberg, T. Sattler, O. Untzelmann, and L. Kobbelt. Scalable 6-DOF Localization on Mobile Devices. In ECCV, 2014.
* [26] Tony Ng, Hyo Jin Kim, Vincent T. Lee, Daniel DeTone, Tsun-Yi Yang, Tianwei Shen, Eddy Ilg, Vassileios Balntas, Krystian Mikolajczyk, and Chris Sweeney. Ninjadesc: Content-concealing visual descriptors via adversarial learning. CVPR, 2022.
* [27] Vojtech Panek, Zuzana Kukelova, and Torsten Sattler. MeshLoc: Mesh-Based Visual Localization. In ECCV, 2022.
* [28] Francesco Pittaluga, Sanjeev J Koppal, Sing Bing Kang, and Sudipta N Sinha. Revealing scenes by inverting structure from motion reconstructions. In CVPR, pages 145–154, 2019.
* [29] Tilman Reinhardt. Using Global Localization to Improve Navigation, 2019.
* [30] Jerome Revaud, Philippe Weinzaepfel, César Roberto de Souza, and Martin Humenberger. R2D2: repeatable and reliable detector and descriptor. In NeurIPS, 2019.
* [31] Paul-Edouard Sarlin, Cesar Cadena, Roland Siegwart, and Marcin Dymczyk. From coarse to fine: Robust hierarchical localization at large scale. In CVPR, 2019.
* [32] Paul-Edouard Sarlin, Daniel DeTone, Tomasz Malisiewicz, and Andrew Rabinovich. SuperGlue: Learning feature matching with graph neural networks. In CVPR, 2020.
* [33] T. Sattler, B. Leibe, and L. Kobbelt. Efficient & Effective Prioritized Matching for Large-Scale Image-Based Localization. PAMI, 39(9):1744–1756, 2017.
* [34] Johannes L. Schönberger and Jan-Michael Frahm. Structure-From-Motion Revisited. In CVPR, June 2016.
* [35] Johannes Lutz Schönberger, Enliang Zheng, Marc Pollefeys, and Jan-Michael Frahm. Pixelwise View Selection for Unstructured Multi-View Stereo. In European Conference on Computer Vision (ECCV), 2016.
* [36] Mikiya Shibuya, Shinya Sumikura, and Ken Sakurada. Privacy Preserving Visual SLAM. In ECCV, 2020.
* [37] Mikiya Shibuya, Shinya Sumikura, and Ken Sakurada. Privacy preserving visual SLAM. In (ECCV), 2020.
* [38] Jamie Shotton, Ben Glocker, Christopher Zach, Shahram Izadi, Antonio Criminisi, and Andrew Fitzgibbon. Scene Coordinate Regression Forests for Camera Relocalization in RGB-D Images. In CVPR, 2013.
* [39] Tory Smith. Niantic lightship. 2022\.
* [40] Zhenbo Song, Wayne Chen, Dylan Campbell, and Hongdong Li. Deep Novel View Synthesis from Colored 3D Point Clouds. In ECCV, 2020.
* [41] Pablo Speciale, Sing Bing Kang, Marc Pollefeys, Johannes Schönberger, and Sudipta Sinha. Privacy preserving image-based localization. In CVPR. IEEE, June 2019.
* [42] Pablo Speciale, Johannes L. Schonberger, Sudipta N. Sinha, and Marc Pollefeys. Privacy Preserving Image Queries for Camera Localization. In The IEEE International Conference on Computer Vision (ICCV), 2019\.
* [43] Jiaming Sun, Zehong Shen, Yuang Wang, Hujun Bao, and Xiaowei Zhou. LoFTR: Detector-free local feature matching with transformers. CVPR, 2021.
* [44] Carl Toft, Will Maddern, Akihiko Torii, Lars Hammarstrand, Erik Stenborg, Daniel Safari, Masatoshi Okutomi, Marc Pollefeys, Josef Sivic, Tomas Pajdla, Fredrik Kahl, and Torsten Sattler. Long-term visual localization revisited. PAMI, 2020.
* [45] J. Valentin, A. Dai, M. Niessner, P. Kohli, P. Torr, S. Izadi, and C. Keskin. Learning to Navigate the Energy Landscape. In International Conference on 3D Vision (3DV), 2016.
* [46] Jonathan Ventura, Clemens Arth, Gerhard Reitmayr, and Dieter Schmalstieg. Global Localization from Monocular SLAM on a Mobile Phone. IEEE Transactions on Visualization and Computer Graphics, 20(4):531–539, 2014.
* [47] Johanna Wald, Torsten Sattler, Stuart Golodetz, Tommaso Cavallari, and Federico Tombari. Beyond controlled environments: 3d camera re-localization in changing indoor scenes. In European Conference on Computer Vision (ECCV), 2020.
* [48] Qing Wang, Jiaming Zhang, Kailun Yang, Kunyu Peng, and Rainer Stiefelhagen. Matchformer: Interleaving attention in transformers for feature matching, 2022.
* [49] P. Weinzaepfel, H. Jégou, and P. Pérez. Reconstructing an image from its local descriptors. In CVPR, 2011.
* [50] A. Wendel, A. Irschara, and H. Bischof. Natural landmark-based monocular localization for mavs. In ICRA, 2011.
* [51] Luwei Yang, Rakesh Shrestha, Wenbo Li, Shuaicheng Liu, Guofeng Zhang, Zhaopeng Cui, and Ping Tan. Scenesqueezer: Learning to compress scene for camera relocalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 8259–8268, June 2022.
* [52] Qunjie Zhou, Sérgio Agostinho, Aljoša Ošep, and Laura Leal-Taixé. Is geometry enough for matching in visual localization? In ECCV, 2022.
* [53] Qunjie Zhou, Torsten Sattler, and Laura Leal-Taixe. Patch2pix: Epipolar-guided pixel-level correspondences. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.
In Section A, we present additional qualitative results for alignments of
objects from the IKEA-Objects set in all scenes from the IKEA-Scenes dataset.
Section B presents qualitative results for object alignments in some of the
RIO10 scenes, corresponding to the quantitative results shown in Figure 6 of
the main paper.
## Appendix A Qualitative results - Ikea-Scenes and Ikea-Objects
Qualitative alignment results. In Figures 7-27, we present alignment results
for four selected objects present in each scene from the the IKEA-Scenes
dataset. We include results for poses obtained by using 1) Superpoint [10]
features with Superglue [32]-based matching and 2) R2D2 [30] features with
Nearest Neighbor matching within the Hloc [31, 32] pipeline. We selected
objects of varying sizes, shapes, categories, textures _etc_. as to show the
feasibility of the attack. As can be seen, it is often possible to quite
accurately place objects in a scene based on camera poses estimated by a
visual localization system.
Camera poses. The same figures also show the set of camera poses returned by
the server (blue) and the subset of poses (green) selected as inliers by our
alignment method (see Algorithm 1 and Section 3.1 in the main paper). Note
that these poses are obtained by localizing a sequence of query images sampled
from a video. Hence, temporally close frames can be expected to show spatial
coherence in their pose estimates. However, due to the difference in
appearance between objects that are present in the scenes and the objects that
are used for the attack, as well as due to viewpoint changes, there can be
many outlier poses. Still, Algorithm 1 (of the main paper) is able to identify
subsets of camera poses that allow to appropriately position the 3D models of
the objects, demonstrating the robustness of the approach.
Failure cases. For each scene, failure cases of the alignment step are
highlighted using red colored boxes. Alignments that result in positioning the
attacking object such that it is either too far from the corresponding object
in the scene or its ”up-direction” is very different from that of the
corresponding object are considered as failure cases. Only visual inspection
has been used to decide whether a case is considered a success or failure. As
can be seen from visualizations, the 3D models of the attacking objects that
result in failures are often quite different (in terms of appearance) from the
corresponding objects in the scene. Some failure cases such as the Sofa
Linanas in Figure 21 can be attributed to this difference while many other
cases such as Sofa Soderhamn in Figure 18 and Stool Kyrre in Figure 21 show
the success in such difficult cases. Other reasons for failure are a low
number of matches from texture-less or weakly textured objects. Chair Odger in
Figure 18 and Chair Froset in Figure 9 (using Superpoint features [10] with
the Superglue matcher [32]), are examples of such cases. Scene02, Scene04, and
Scene06, shown in Figures 10-12, 16-18, 22-24 respectively, are complex rooms
(_e.g_., an open concept kitchen in Scene04) composed of very similar looking
objects, and hence challenging cases for such an attack. This results in more
failure cases. Still, as shown in the figures, the attack succeeds in many
cases, which shows its feasibility.
## Appendix B RIO10 example alignment results
In Section 6 of the main paper, we discuss a potential strategy to defend
against the attack introduced in our work. Yet, Figure 6 of the main paper
shows that this strategy causes the localization process to not only reject
malicious queries, but also to reject genuine query images. In Figure 6 of the
main paper, we consider 3 different objects attacking 3 different scenes of
the RIO10 dataset. To visualize these scenarios, Figure 28 shows qualitative
results for object alignments in scenes corresponding to those plots. These
results further emphasize that the attack does not require images of the exact
same objects as present in the scene, but can also be carried out using images
of similar instances from the same class of objects. Furthermore, these
alignments also do not change significantly when the HLoc[31] based server
uses stricter inlier thresholds (5,2 or 1 pixel as compared to the default 12
pixels) for the RANSAC based localization process. This counters another
simple defence strategy of using stricter inlier thresholds in such a
localization pipeline to avoid localization of malicious queries. Figure 29
shows the qualitative results of object alignment for varying inlier
thresholds. As can be seen, the alignment does not get worse (in fact,
improves) upon using stricter thresholds for the three cases considered here.
Note that for these examples, although the number of inliers decreases as the
threshold become stricter, the solutions of poses with least sum of
reprojection errors are still able to position the attack-object
appropriately. A more rigorous experimentation against this defence would
definitely be more insightful but that would require an appropriate
quantitative measure of the quality of alignment and this is left as future
work.
## Appendix C Additional results for object presence classification
For the task object presence classification, we present precision and recall
results for each of the objects in IKEA-Objects in Table 2.
Figure 7: Scene01 of IKEA-Scenes with selected objects in focus. Figure 8: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene01 of IKEA-Scenes. Figure 9: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene01 of IKEA-Scenes. Failure case: The aligned model for Chair Froset when using Superpoint+Superglue is a bit far from the actual object and also oriented incorrectly. Figure 10: Scene02 of IKEA-Scenes with selected objects in focus. Figure 11: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene02 of IKEA-Scenes. Failure case: The aligned model for Table Lisabo when using Superpoint+Superglue is a bit far from the actual object and also oriented incorrectly. Figure 12: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene02 of IKEA-Scenes. Failure cases: The aligned models of Lamp Misterhult when using R2D2+NN or Superpoint+Superglue are very far from the actual object. This can be attributed to the similar wooden appearance of the lamp and several wooden objects in the scene, which leads to incorrect matches. Figure 13: Scene03 of IKEA-Scenes with selected objects in focus. Figure 14: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene03 of IKEA-Scenes. Failure case: The aligned model for Bed table Klipsk when using Superpoint+Superglue is close to the actual object, but incorrectly oriented. Figure 15: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene03 of IKEA-Scenes. Failure case: The aligned model for Chair Linneback when using R2D2+NN is close to the actual object, but incorrectly oriented. Figure 16: Scene04 of IKEA-Scenes with selected objects in focus. Figure 17: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene04 of IKEA-Scenes. Figure 18: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene04 of IKEA-Scenes. Failure case: The aligned model for Chair Odger when using Superpoint+Superglue is close to the actual object, but incorrectly oriented. Figure 19: Scene05 of IKEA-Scenes with selected objects in focus. Figure 20: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene05 of IKEA-Scenes. Figure 21: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene05 of IKEA-Scenes. Failure cases: The aligned model for Sofa Linanas when using Superpoint+Superglue or R2D2+NN is incorrectly oriented and also far from the actual object. Figure 22: Scene06 of IKEA-Scenes with selected objects in focus. Figure 23: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene06 of IKEA-Scenes. Figure 24: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene06 of IKEA-Scenes. Failure cases: The aligned model for Chair Strandmon when using Superpoint+Superglue or R2D2+NN is incorrectly aligned and also far from the actual object. Figure 25: Scene07 of IKEA-Scenes with selected objects in focus. Figure 26: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene07 of IKEA-Scenes. Failure case: The aligned model for Organizer Tjena when using Superpoint+Superglue is far from the actual object and also incorrectly oriented. Figure 27: Qualitative results for aligning corresponding objects from IKEA-Objects using poses for localizing them in Scene07 of IKEA-Scenes. Figure 28: Alignment results for querying scenes from the RIO10 dataset [47] with objects from the Office-Objects and IKEA-Objects datasets (_cf_. Fig. 6 in the main paper for quantitative results). The textured mesh corresponds to the scene and the point cloud corresponds to the 3D model of the object used for the attack. The alignment was produced using the poses obtained from the server, using Superpoint+Superglue for matching. As can be seen, it is possible to position the objects with reasonable accuracy, despite differences in appearance and geometry. Figure 29: Impact of changing the inlier threshold for RANSAC based localization on alignment - qualitative results. The final aligned object is not impacted drastically in most cases by a stricter RANSAC threshold. In fact, in several cases, it results in a better alignment of the attacking object with the corresponding object in the scene. The textured mesh corresponds to the scene and the point cloud corresponds to the 3D model of the object used for the attack. The alignment was produced using the poses obtained from the server, using Superpoint+Superglue for matching Object Name | Superpoint + Superglue | R2D2 + NN
---|---|---
$10^{\circ}$,0.25m | $30^{\circ}$, 0.5m | $60^{\circ}$, 2m | $10^{\circ}$,0.25m | $30^{\circ}$, 0.5m | $60^{\circ}$, 2m
P | R | P | R | P | R | P | R | P | R | P | R
bookshelf | 1 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 0.5 | 0.5
chair_agam | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
chair_froset | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
chair_gaming | 0.33 | 1 | 0.5 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0.5 | 1
chair_linneback | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1
chair_odger | 0.33 | 0.5 | 0.33 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 1 | 0.5 | 0.5 | 0.5
chair_poang_small | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
chair_storsele | 0.33 | 1 | 0.33 | 1 | 1 | 1 | 0.5 | 1 | 0.25 | 1 | 0.5 | 1
chair_strandmon | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1
chair_vedbo | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1
cupboard_hauga | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
cupboard_kallax | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0.25 | 1
cycle | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1
klipsk_bed_table | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.5 | 1
lamp_ceiling_agunarryd | 1 | 1 | 1 | 1 | 1 | 1 | 0.33 | 1 | 0 | 0 | 0.33 | 1
lamp_ceiling_appleviken | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0
lamp_ceiling_mojna | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
lamp_ceiling_nymane | 0 | 0 | 1 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 0 | 0 | 0.25 | 0.5
lamp_ceiling_ranarp | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
lamp_evedal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
lamp_fancy | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
lamp_navlinge | 0.33 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.25 | 0.5 | 0.33 | 0.5 | 0.25 | 0.5
lamp_star | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1
lamp_table_misterhult | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0
lamp_table_nymane | 0.5 | 0.5 | 1 | 0.5 | 0.33 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5
lamp_table_tertial | 0 | 0 | 0.5 | 1 | 1 | 1 | 0.5 | 1 | 0 | 0 | 0.33 | 1
organizer_kvarnik | 0.5 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0.5 | 1
oven | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 0.5 | 0.5
sofa_landskrona | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1
sofa_linanas | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sofa_soderhamn | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
stool_kyrre | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
stool_marius | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
strainer | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
table_corner_gladom | 1 | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1
table_lisabo_square | 1 | 0.5 | 0.5 | 0.5 | 0.25 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5
vas_gradvis | 0.25 | 1 | 1 | 1 | 0.5 | 1 | 0 | 0 | 0 | 0 | 0.5 | 1
wall_hanging_crescent | 0.33 | 0.5 | 0.25 | 0.5 | 0.33 | 0.5 | 0 | 0 | 1 | 0.5 | 0.5 | 0.5
Table 2: Object wise precision and recall results for IKEA-Objects when
attacking IKEA-Scenes
|
# The cosmology dependence of galaxy clustering and lensing from a hybrid
$N$-body–perturbation theory model
Nickolas Kokron1,2 , Joseph DeRose3,4, Shi-Fan Chen3, Martin White3,5, Risa H.
Wechsler1,2
1 Kavli Institute for Particle Astrophysics and Cosmology and Department of
Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
2 Kavli Institute for Particle Astrophysics and Cosmology, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
3 Department of Physics, University of California, Berkeley, 366 LeConte Hall,
Berkeley, CA 94720, USA
4 Santa Cruz Institute for Particle Physics, University of California, Santa
Cruz, CA 95064, USA
5 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 93720,
USA
Contact e-mail<EMAIL_ADDRESS>
###### Abstract
We implement a model for the two-point statistics of biased tracers that
combines dark matter dynamics from $N$-body simulations with an analytic
Lagrangian bias expansion. Using Aemulus, a suite of $N$-body simulations
built for emulation of cosmological observables, we emulate the cosmology
dependence of these nonlinear spectra from redshifts $z=0$ to $z=2$. We
quantify the accuracy of our emulation procedure, which is sub-per cent at
$k=1\,h{\rm Mpc}^{-1}$ for the redshifts probed by upcoming surveys and
improves at higher redshifts. We demonstrate its ability to describe the
statistics of complex tracer samples, including those with assembly bias and
baryonic effects, reliably fitting the clustering and lensing statistics of
such samples at redshift $z\simeq 0.4$ to scales of $k_{\rm max}\approx
0.6\,h\mathrm{Mpc}^{-1}$. We show that the emulator can be used for unbiased
cosmological parameter inference in simulated joint clustering and
galaxy–galaxy lensing analyses with data drawn from an independent $N$-body
simulation. These results indicate that our emulator is a promising tool that
can be readily applied to the analysis of current and upcoming datasets from
galaxy surveys.
###### keywords:
cosmology: theory – large-scale structure of Universe – methods: statistical –
methods: computational
††pubyear: 2021††pagerange: The cosmology dependence of galaxy clustering and
lensing from a hybrid $N$-body–perturbation theory model–LABEL:lastpage
## 1 Introduction
We are entering a golden era for studying the large-scale structure of the
Universe. Over the next decade, ambitious imaging surveys will map out large
swathes of the sky to unprecedented depths, imaging billions of galaxies and
their shapes (Ivezić et al., 2019; Laureijs et al., 2011; Doré et al., 2015,
2019), enabling studies of weak gravitational lensing by the intervening
distribution of matter (Bartelmann & Schneider, 2001; Mandelbaum, 2018). Weak
lensing has only recently begun to contribute competitive cosmological
constraints on dark matter and dark energy (Abbott et al., 2018; Heymans et
al., 2020), but is one of the most promising future directions to pursue.
Meanwhile, spectroscopic surveys will observe tens of millions of radial
positions of galaxies (Takada et al., 2014; Aghamousa et al., 2016), enabling
unparalleled understanding of the spatial distribution of galaxies in our
Universe. The cross-correlation between positions and lensing, galaxy–galaxy
lensing, is and will continue to be a key driver of cosmological constraints
from galaxy surveys.
The quality and quantity of these upcoming datasets imposes a significant
challenge in their analysis. Even now, models for summary statistics such as
correlation functions and power spectra are inadequate across the full range
of scales probed by such surveys (Krause et al., 2017; Nishimichi et al.,
2020). Either a large amount of the data must be discarded, or mitigation
schemes must be developed to prevent contamination from scales where the
models are insufficiently calibrated or constrained (MacCrann et al., 2020;
Park et al., 2020). Models for clustering and lensing must be substantially
improved if we are to extract the maximal information about the Universe we
live in, from surveys that are already ongoing or planned. To date, two
separate approaches have been developed to build models for the observables of
cosmic surveys: analytically, through perturbative techniques, or numerically,
using non-linear $N$-body simulations.
Perturbation theory provides a systematic, analytic way to compute $N$-point
summary statistics to systematically higher precision and smaller scales
(Bernardeau et al., 2002). Below the nonlinear scale the effects of these
nonlinearities can be tamed and parametrized within the framework of effective
theories (Baumann et al., 2012; Carrasco et al., 2012; Vlah et al., 2015).
This increased precision, however, comes at the cost of very large
inaccuracies beyond the nonlinear scale at which the self-gravitating dark
matter fluid ceases to be perturbative (Blas et al., 2014; McQuinn & White,
2016). In addition, perturbative frameworks provide a rigorous, first-
principles approach to include physics beyond the standard $\Lambda$CDM model
in large-scale structure observables such as neutrinos, baryonic effects and
more exotic early-universe scenarios (Lewandowski et al., 2015; Senatore &
Zaldarriaga, 2017; Aviles & Banerjee, 2020; Chen et al., 2020c; Laguë et al.,
2020; Ivanov et al., 2020; 2020arXiv200612420D; Aviles et al., 2020).
Understanding the domain of applicability of perturbation theory is still an
active field of research (Baldauf et al., 2016b; Nishimichi et al., 2020; Chen
et al., 2020a).
The other approach, simulation-based modelling, involves numerically solving
the equations of motion for an initial distribution of matter (Hockney &
Eastwood, 1988; Bagla, 2005; Kuhlen et al., 2012). The resulting catalogs can
be analysed in a way analogous to data to obtain predictions of cosmological
observables across a wide range of scales at the cosmological parameters of
the simulation.
However, a limiting factor in simulation-based analyses is that $N$-body
simulations require significant computational resources for a single
realization. Thus, standard inference procedures such as Markov Chain Monte
Carlo (MCMC) become prohibitively expensive when using models derived from
simulations.
In order to ameliorate the issues with simulation-based inference, recent
developments in statistical learning have popularized so-called emulators as
models (Heitmann et al., 2010, 2009; Lawrence et al., 2010). Emulators combine
a set of simulations that representatively sample cosmological parameter space
with sophisticated regression techniques to ‘fill in the blanks’ across
parameter space. Once trained, an emulator provides rapid evaluations of a
model which can be seamlessly integrated in analysis pipelines. For example,
recent emulators for the nonlinear matter power spectrum (Knabenhans et al.,
2019) have runtimes with negligible overhead compared to the underlying
Boltzmann codes used for linear predictions.
While galaxy surveys observe luminous tracers of the underlying dark matter
density distribution, most suites of $N$-body simulations used to construct
emulators deal only with the dark matter component. Thus, emulators for galaxy
survey observables are presented with the additional challenge of capturing
the relationship between the galaxy distribution and the underlying dark
matter. Understanding the details of this relationship, known as the
galaxy–halo connection, is an active field of research (see e.g. Wechsler &
Tinker, 2018, for a recent review). Even for well-studied samples of galaxies,
there are no consensus models to describe this relationship. For any given
model of the galaxy–halo connection, an entirely new emulator has to be
trained (Kwan et al., 2015; Wibking et al., 2019; Zhai et al., 2019;
McLaughlin et al., 2021). Emulation of models with a large number of free
parameters is also a challenging task, with techniques such as Gaussian
processes scaling as $\mathcal{O}(N^{3})$ with $N$ training points and a
substantially larger set of training data being required as one increases the
dimensionality of the model. The simplest forms of galaxy–halo connections
such as halo occupation distributions have five free parameters (Zheng et al.,
2005), and it is expected that for more complex selections of galaxy samples
the number will grow considerably (Guo et al., 2019; Yuan et al., 2018; Favole
et al., 2020; Zu, 2020).
In comparison, modern perturbation theory approaches to galaxy clustering
operate at the field level via so-called bias expansions, which encode the
response of small-scale galaxy physics (e.g. the galaxy–halo connection) to
large-scale structure via a series of bias coefficients (see e.g. Desjacques
et al. 2018 for a recent review). A key advantage of bias models is that while
their dependence on parameters is simple and analytic, they should describe
the statistics of a broad range of galaxy (and halo) samples as long as they
are formed by processes that respect the symmetries of the underlying
processes of structure and galaxy formation, namely rotational and Galilean
invariance and the equivalence principle. Indeed, it was recently shown that
the bias expansion can be directly derived by generating all possible
dynamical terms and eliminating combinations not allowed by these symmetries
(Fujita & Vlah, 2020).
The challenges in using bias models come, instead, from the aforementioned
limitations of perturbation theory models themselves. Similarly to
perturbation theories for the clustering of dark matter, bias models are not
expected to hold across all scales. Instead, they are expected to be valid at
scales larger than or comparable to the Lagrangian size of haloes. This regime
is where one is insensitive to the internal structure of haloes (McDonald &
Roy, 2009; Fujita et al., 2020; Lazeyras & Schmidt, 2019; Vlah et al., 2016).
It is worth noting, however, that the nonlinear and halo scales are not
identical and scale differently with redshift — at higher redshifts
perturbative models may be more limited by the larger Lagrangian radii of
(typically more luminous or massive) samples than dynamical nonlinearities,
and vice versa at lower redshifts. This distinction is particularly apparent
in the Lagrangian basis (Matsubara, 2008; Vlah et al., 2016), in which galaxy
clustering due to dynamics and biasing are explicitly disentangled. Recently
Modi et al. (2020) suggested a way to combine the generality of bias
expansion-based models with $N$-body simulations in a manner that is
particularly suited for emulation, particularly in the regime where dynamics
become nonlinear on scales larger than the halo scales of interest. Since
higher-order Lagrangian biases have been found in simulations to be small for
low and intermediate mass haloes (Abidi & Baldauf, 2018; Lazeyras & Schmidt,
2018), this scheme keeps the dynamical nonlinearities from $N$-body
simulations to all orders while including Lagrangian bias only up to second
order.
In the remainder of this work we concern ourselves with the construction of an
emulator for the halo–halo and halo–matter correlations with analytic
dependence on bias parameters, extending the method presented in Modi et al.
(2020) to a generic cosmological parameter dependence which can then be
readily used for cosmological clustering analyses. The structure is as
follows: in section 2 we briefly review the Lagrangian description of galaxy
bias. In section 3 we describe the hybrid technique which combines
displacements obtained from $N$-body simulations with Lagrangian bias. Section
4 describes the Aemulus suite of simulations (DeRose et al., 2019b), which we
use to build the training data for the emulator. The measurements of the
‘basis spectra’ of the hybrid Lagrangian bias model, and their emulation, are
outlined in section 5. Section 6 concerns itself with assessing the
performance of the emulator. Specifically, sub-section 6.1 addresses the scale
and redshift-dependent error for each of the ten basis functions that span the
model. Subsection 6.2 assesses how well the model describes the statistics of
complicated galaxy samples, including those possessing concentration and spin
secondary biases, as well as the effect of baryons at small scales. Our final
test, subsection 6.3, pits the emulator against a series of increasingly
complex simulated likelihood analyses, in order to assess potential biases in
inferred cosmological parameters using our emulator and their origin.
## 2 Lagrangian bias expansion
In the Lagrangian approach to bias formulated in Matsubara (2008), the
observed clustering of galaxies is obtained through first weighting fluid
elements by a local functional $F[\delta(\textbf{q})]$ at their initial
(Lagrangian) positions q and then advecting these weights to their observed
positions via fluid trajectories $\textbf{x}=\textbf{q}+\mathbf{\Psi}$, where
$\mathbf{\Psi}(\textbf{q},t)$ is the Lagrangian displacement. As discussed in
the introduction, the bias functional $F$ is obtained by summing up all scalar
terms allowed by Galilean invariance and the equivalence principle up to a
given order in the initial conditions; up to quadratic order we have (Vlah et
al., 2016)
$\displaystyle F(\bm{q})\approx\,1+$ $\displaystyle
b_{1}\delta_{L}(\bm{q})+\frac{b_{2}}{2!}(\delta_{L}^{2}(\bm{q})-\langle\delta_{L}^{2}\rangle)\,+$
(1) $\displaystyle b_{s^{2}}(s_{L}^{2}(\bm{q})-\langle
s_{L}^{2}\rangle)+\,b_{\nabla^{2}}\nabla^{2}\delta_{L}(\bm{q})+\,\epsilon(\bm{q}),$
where $s^{2}=s_{ij}s_{ij}$ is the tidal shear tensor. The bias expansion is
local above the halo scale and the initial fields in the above functional are
to be interpreted as smoothed; any ‘nonlocal’ effects as we approach this
scale, as well as dependences on smoothing, are parametrized to lowest order
by the derivative bias $b_{\nabla^{2}}$. Modes below the halo scale,
uncorrelated with the large scales of interest, are represented by the
stochastic noise $\epsilon$.
From the weighting $F(\textbf{q})$, the observed clustering is given via
number conservation to be
$1+\delta_{\alpha}(\textbf{x},z)=\int
d^{3}q\,\delta^{D}(\textbf{x}-\textbf{q}-\mathbf{\Psi}(\textbf{q},z))F(\bm{q}),$
(2)
where the Lagrangian displacement $\mathbf{\Psi}$ denotes the movement of the
fluid element relative to its initial position. At any given order, the
Lagrangian galaxy overdensity above can be mapped onto e.g. the Eulerian basis
of McDonald & Roy (2009) by Taylor expanding $\mathbf{\Psi}$. However, keeping
the nonlinear mapping in the integral above will generate a tower of Eulerian
bias parameters even if only a few of the Lagrangian bias parameters are
nonzero (see e.g. Abidi & Baldauf, 2018). We will treat the bias values,
$b_{\alpha}$, as free parameters. Ab initio predictions of the $b_{\alpha}$
for general tracer populations is a harder problem, and a current active area
of research.
## 3 Lagrangian bias and simulations
Recently, it has been proposed that one can combine the fully resolved dark
matter dynamics of an $N$-body simulation with the analytic perturbative bias
techniques we outlined in the previous section (Modi et al., 2020). The use of
dynamics from an $N$-body simulation means this hybrid model circumvents the
need for perturbative calculations related to the equations of motion of the
dark matter fluid itself. Additionally, $N$-body simulations are relatively
inexpensive (compared to hydrodynamical simulations) and well-controlled,
well-defined limits for observables exist so that convergence of measured
quantities can be assessed systematically (e.g. Power et al., 2016; Mansfield
& Avestruz, 2020; Joyce et al., 2020). As such, this hybrid model combines two
techniques with solid theoretical foundations, ensuring robustness of its
predictions. We will briefly describe the technique and how one implements it
below, but refer the reader to Modi et al. (2020) for a more complete
discussion.
When creating initial conditions of an $N$-body simulation, one starts from a
noiseless linear cosmological density field, $\delta_{L}(\bm{x})$.
Traditionally, this density is only used to sample initial displacements which
impart a cosmological signal on a set of _pre-_ initial conditions. First-
order displacements using the Zeldovich approximation,
$\Psi(\bm{q})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot\bm{q}}\frac{i\bm{k}}{k^{2}}\delta_{L}(\bm{k}),$
(3)
result in so-called 1LPT initial conditions. However, higher order initial
conditions (Crocce et al., 2006; Garrison et al., 2016; Michaux et al., 2020)
are now ubiquitous in modern simulations.
The noiseless initial density field can also be used to construct the
different component fields of the Lagrangian bias expansion of the initial
conditions:
${O}_{L}\supset{1,\delta_{L},\delta_{L}^{2},s_{L}^{2},\nabla^{2}\delta_{L},\cdots},$
(4)
where the subscript $L$ indicates these are the Lagrangian fields. Advecting
$N$-body particles weighted by $\mathcal{O}_{L}$ to a specific snapshot
results in bias-weighted fields,
$\delta_{\mathcal{O}_{L}}(\textbf{x})\equiv\int d^{3}\textbf{q}\
\mathcal{O}_{L}(\textbf{q})\
\delta_{D}(\textbf{x}-\textbf{q}-\Psi(\textbf{q})),$ (5)
which trace the non-linear dark matter distribution. In Fig. 1 (middle panel)
we show an example of the different bias-weighted fields produced by this
procedure. These fields are similar to the ‘Eulerian-shifted’ operator basis
of Schmittfull et al. (2019). A notable difference is that in our case the
displacements are fully resummed, while the Eulerian-shifted basis of
Schmittfull et al. (2019) only resums the Zeldovich displacement (1LPT).
Higher order displacements ($n$LPT) are Taylor-expanded up to third order as
part of their bias expansion. The difference is because our aim in this paper
is to attempt to model scales beyond the reach of standard one-loop
perturbation theory, whereas the goal of Schmittfull et al. (2019) was to
validate one-loop perturbation theory at the field level (see also Taruya et
al. 2018).
The power spectrum of any combination of tracers can then generically be
written as ($X,\,Y\equiv{\delta_{\mathcal{O}_{L}}}$)
$P^{ab}(k)=\sum_{X,Y}b_{X}^{a}b^{b}_{Y}P_{XY}(k)+P_{SN},$ (6)
where $P_{XY}$ is the cross-power spectrum at a fixed cosmology between the
different fields at a given redshift. For example, the unweighted spectrum,
$P_{11}$, is the non-linear matter power spectrum.
This Lagrangian bias model can handle cross-correlations of arbitrary tracers.
However, we also note that given a set of bias parameters for a single tracer
sample $\alpha$, $\\{b_{X}^{\alpha},\,X\in\mathcal{O}_{L}\\}$, one can also
self-consistently predict the tracer–matter cross-correlation by taking the
second sample to have $b_{Y}^{m}=0$ except for $Y=1$. In this case there are
only $P_{X1}$ terms. The tracer–matter cross-correlation is the primary cosmic
contribution to the signal of galaxy–galaxy lensing, one of the key
cosmological observables of current and upcoming galaxy surveys (Prat et al.,
2018; Yoo et al., 2006; Wibking et al., 2020; Mandelbaum, 2018). The
tracer–matter cross-correlation is also the primary contribution to the cross-
correlation between galaxy positions and lensing of the cosmic microwave
background (CMB), one of the most powerful and complementary statistics that
is measured between galaxy and CMB surveys (Bianchini et al., 2015; Pullen et
al., 2016; DiPompeo et al., 2017; Peacock & Bilicki, 2018; Omori et al., 2019;
Singh et al., 2019; Krolewski et al., 2020). For notational convenience,
throughout the remainder of this paper we will refer to the tracer–tracer
correlation as $P^{hh}(k)$ and the tracer–matter correlation as $P^{hm}(k)$.
This hybrid approach of combining $N$-body simulations with Lagrangian bias
can fit the power spectrum of tracers to significantly smaller scales than
standard Lagrangian perturbation theory (Modi et al., 2020). While the
dependence on the Lagrangian bias parameters $b_{X}$ is analytic in this
model, one still requires an $N$-body simulation to measure the basis spectra.
An $N$-body simulation at a given point of cosmological parameter space then
provides a measurement of the basis spectra at that point. With $N$-body
simulations that sufficiently sample parameter space one can estimate the
cosmological dependence of these basis functions across the entire space. This
is precisely the goal of this work.
Figure 1: Visualization of the methodology implemented in this paper, from the
advection process to the measurements of the basis spectra. Our emulation
scheme approximates the cosmology and redshift dependence of each spectrum in
the ten panels in the lower part of the figure. The top panel has each
Lagrangian field scaled to have equal variance, in order to highlight the
qualitative differences between the fields. The middle panel shows the bias
weighted-fields that result from the advection process. Different weights
highlight qualitatively different aspect of the matter density. The cross-
spectra of these fields give the spectra shown in the lower panel.
## 4 The Aemulus simulations
In order to properly emulate the cosmology dependence of the basis spectra
$P_{XY}(k)$, the underlying suite of $N$-body simulations used for
measurements of observables must be constructed carefully.
The Aemulus suite of $N$-body simulations (DeRose et al., 2019b) has been
purpose-built for precise emulation of cosmological observables measured in
galaxy surveys. The suite is composed of a set of 75 simulations that span 47
points in the $w$CDM parameter space allowed by a combination of modern CMB,
BAO and type Ia supernova experiments.
Each Aemulus box has a size $L_{\rm box}=1050\,h^{-1}$Mpc with $N=1400^{3}$
particles, corresponding to a mass resolution of $3.51\times
10^{10}\left(\frac{\Omega_{m}}{0.3}\right)h^{-1}M_{\odot}$. The Aemulus
simulations have undergone rigorous convergence and validation tests for
several observables. There are 10 particle snapshots ranging from $0<z<3$,
allowing for measurements the redshift-dependence of the non-linear basis
spectra.
Aemulus’ halo mass function emulator has sufficient accuracy to remain valid,
for the defined cosmological parameter space, through the Rubin Observatory’s
Y1 LSST survey (McClintock et al., 2019), while the galaxy correlation
function can predict the clustering of massive galaxy samples, such as those
observed by DESI, to within 1 per cent down to scales of $r\approx
1\,h^{-1}$Mpc (Zhai et al., 2019).
Thus, Aemulus represents an appropriate setting to construct an emulator for
the Lagrangian bias basis spectra described in section 3. The only missing
component is that the initial conditions code used in Aemulus, 2LPTIC (Crocce
et al., 2012), does not output the noiseless linear density fields. We patched
the code to read out this field and re-generated the initial conditions.
Figure 2: Ratio of the measured basis spectra compared to LPT predictions for
one of the cosmologies in the Aemulus test set. The mean of the five
independent boxes in the test set is shown, and the shaded band represents one
standard deviation as inferred from the boxes. The dashed vertical line at
$k=0.1h{\rm Mpc}^{-1}$ shows the point where we revert to predictions of LPT.
As discussed in the text, we find some small multiplicative differences at
large scales for most basis spectra, that are larger for the basis spectra
built from higher powers of the density field. This is most likely due to
discrepancies in growth factors obtained between linear theory and $N$-body
simulations.
## 5 Emulating the basis spectra
### 5.1 Measuring basis spectra
We now describe in detail our implementation of the hybrid Lagrangian biasing
scheme described in Section 3. Schematically, the process of obtaining
measurements of the basis spectra from an $N$-body box can be broken down into
four steps:
1. 1.
Compute the Lagrangian bias fields: given the noiseless density field
$\delta_{L}$ one constructs the other weight fields $\mathcal{O}_{L}$ by
applying the appropriate transformations.
2. 2.
Advect particles to a given snapshot: every particle ID can be associated with
a grid cell $\\{i,j,k\\}$ in the fields $\mathcal{O}_{L}$. Every particle in a
snapshot receives a weight
$\left(\frac{D(z)}{D(z_{0})}\right)^{n}\times\mathcal{O}_{L}[i,j,k]$, where
$\left(\frac{D(z)}{D(z_{0})}\right)$ is the ratio of growth factors between
the snapshot and initial conditions, and $n$ is the number of powers in the
linear density field that make up $\mathcal{O}_{L}$.
3. 3.
Paint the weighted particles to a grid, to form the late-time bias fields.
4. 4.
Measure the basis spectra: the painted bias fields are cross-correlated with
each other to measure the basis spectra $P_{XY}$ for that given cosmology and
redshift.
This procedure imposes some additional storage requirements. While a particle
catalog normally has seven entries for every particle, $(ID,\,\bm{x},\bm{v})$,
each bias field weight will add an additional entry. Naively saving component
weights at every snapshot will lead to a 57 per cent increase in catalog size.
However, the time evolution of the weights is determined entirely by the
linear growth function and can be determined on the fly. Thus, the fractional
increase in catalog size will only be of order $\sim(1/7)(N_{b}/N_{z})$, where
$N_{b}$ is the number of bias-weighted fields computed and $N_{z}$ is the
number of snapshots used. For the second order basis of
$\mathcal{O}=\\{1,\delta_{L},\delta_{L}^{2},s_{L}^{2},\nabla^{2}\delta_{L}\\}$
this represents a fractional increase in catalog size of 6 per cent. Even if
the weights are not stored, all of the steps outlined above can be carried out
on the fly when needed.
In Fig. 1 we show a comparison between the predictions of one-loop Lagrangian
perturbation theory and the basis spectra averaged across five Aemulus boxes
with the same cosmology, from the test suite. For all basis spectra we recover
the LPT result at large scales to within a few per cent. While one would
expect the agreement at large scales to be exact, it is well known that
$N$-body simulations struggle to correctly recover linear growth at large
scales (Heitmann et al., 2010; Schneider et al., 2016; Garrison et al., 2016)
due to transients from the grid that particles are initialized on, and the
discrete nature of the kick-drift-kick operators used in time-stepping. This
discrepancy is also present in Aemulus, as can be seen in fig. 13 of DeRose et
al. (2019b). The Aemulus simulations have a 1 per cent mismatch in growth at
large scales, which is redshift independent at the largest scales. Differences
in growth between linear theory and the simulations would then be amplified
for the basis spectra built from multiple fields. In Appendix C we explore the
$k\to 0$ differences between LPT and our emulator, present prescriptions for
enforcing consistency and discuss the small impact they have on parameter
inference.
At small scales, we see that non-linear structure formation imbues significant
differences between LPT and the simulations. At the highest redshift shown,
$z=2$, the agreement for the three spectra that dominate the signal ($\langle
1,1\rangle,\,\langle 1,\delta\rangle,\mathrm{and}\langle\delta,\delta\rangle$)
is close throughout all scales probed in our simulation. Thus, for the scales
under consideration, we find no need to extend the emulator to $z>2$.
Above $z=1$, the Aemulus simulations only have snapshots at $z=2$ and $z=3$,
and thus any attempt to emulate redshift evolution between these snapshots is
too poorly sampled for the emulator to achieve our desired performance. For
$z\geq 2$ the emulator reverts to predictions from velocileptors (Chen et al.,
2020b), a public code to predict LPT power spectra and correlation functions
to one loop order. This agrees quite well with most basis spectra given Fig.
1. When reverting to LPT at $z>2$, our implementation includes an additional
free parameter. This parameter corresponds to the $k^{2}$ counterterm for
matter that takes into account the effects of small-scale physics not captured
by perturbation theory (Vlah et al., 2015). We note that there are no specific
impediments to measuring basis spectra, or the emulation scheme adopted, at
higher redshifts. Given simulations that are sufficiently well sampled in
time, out to the furthest bin one wishes to include, the techniques described
here should apply.
The LPT predictions shown in Fig. 1 are a limit of a more complete theory that
includes redshift-space distortions (Chen et al., 2020a, b) . The agreement
between $N$-body simulations and this subset of LPT at large scales implies
the bias parameters in the full theory and our hybrid model are equivalent; a
set of bias parameters obtained from fitting the emulator to a sample can then
be used in tandem with RSD measurements analysed purely with perturbation
theory at a slightly more restrictive $k_{\mathrm{max}}$. Since the RSD
measurements are done in 3D, rather than projection, one can achieve small
measurement errors at more restrictive $k_{\rm max}$ making this combination
an efficient one, e.g. for testing general relativity (Alam et al., 2017;
Zhang et al., 2020).
We note that we omit results for the basis spectra $\langle
X,\nabla^{2}\delta\rangle$. The initial weight field $\nabla^{2}\delta_{L}$
has a large amount of power at very small scales, making its Fourier transform
unwieldy due to the presence of an explicit smoothing scale of $k\sim L_{\rm
grid}^{-1}$. As a result, we find the basis spectra as measured through the
advection procedure have a cosmology-dependent amplitude mismatch when
compared to LPT predictions at large scales. Therefore we adopt the
approximation $\langle X,\nabla^{2}\delta\rangle\approx-k^{2}\langle
X,1\rangle$ in the actual emulation scheme. Since these higher derivative bias
contributions most closely correspond to the effects of baryonic physics and
finite-size effects for haloes, we check that the approximation performs
similarly in Section 6.2. Specifically, in Fig. 8 we explicitly show the
differences between the measured $P_{1\nabla^{2}}$ and the approximation
employed. We also note the approximation lowers the complexity of the
emulation scheme, reducing the full set of basis functions at second order to
be emulated from 15 to 10.
### 5.2 Principal components of non-linear spectra
Once the basis spectra have been measured across all boxes, the emulator is
built by adopting a suitable interpolation scheme between the different
spectra. While other emulators using the Aemulus simulations have been
constructed using Gaussian processes (GPs), we adopt a different approach
here, similar to that used in the Euclid emulator (Knabenhans et al., 2019),
using a combination of principal component analysis and polynomial chaos
expansions (PCE) (Xiu, 2010).
We prefer PCE to GP emulation for a few practical reasons. GPs are more
difficult to train, requiring explicit choices for kernels and tuning of real
valued hyper–parameters. Additionally, the run time for evaluating a trained
GP scales with the amount of data used for training, while the run-time of a
PCE model evaluation scales only with the order of the PCE. Furthermore, the
polynomial nature of PCEs means that they have fast, analytic gradients,
making them easy to integrate with sampling techniques such as Hamiltonian
Monte Carlo (Hoffman & Gelman, 2011), although we have not done so in this
work. GPs may still be preferred when the model being emulated is highly
complex, but, as we show in the following sections, we are able to attain a
nearly optimal emulator performance with the simpler and faster PCE scheme.
To begin, we compute one-loop LPT predictions for each basis spectrum at every
cosmology and redshift in the Aemulus training design, which we will refer to
as $P_{\rm XY}^{\rm LPT}(k,\mathbf{\Omega})$, where $\mathbf{\Omega}$ denotes
the cosmology and redshift in question. To do this we make use of the
velocileptors code (Chen et al., 2020b).
We then compute the ratio between the LPT predictions and the measured basis
spectra, $P_{XY}^{\rm NL}(k,\mathbf{\Omega})$, from each snapshot. These
ratios are thus consistent with unity at small wavenumbers, and while they
deviate significantly from unity at high $k$ they have significantly less
dynamic range than the basis spectra. In order to de-noise these ratios, we
apply a Savitsky–Golay (Savitzky & Golay, 1964) filter of order three using an
11-point window in $k$. Doing so dramatically reduces the amount of noise in
the spectra, and is a simple alternative to reduce noise at high $k$, where
techniques such as fixed amplitude, paired phase simulations do little to
reduce variance (Angulo & Pontzen, 2016; Villaescusa-Navarro et al., 2018;
Chuang et al., 2019). As a final preprocessing step, we also take the base-10
logarithm of these smoothed ratios in order to further decrease the dynamic
range. This yields the quantity that we emulate, which we call
$\Gamma^{XY}(k,\mathbf{\Omega})$,
Figure 3: The first two principal components of the log-ratios between
$N$-body and LPT spectra, $\Gamma^{XY}$, for each basis spectrum. The
principal components are very smooth compared to the raw basis spectrum
measurements from the simulations. Two principal components are sufficient to
explain greater than 99 per cent of the variance in all spectra as a function
of redshift and cosmology.
$\Gamma^{XY}(k,\mathbf{\Omega})\equiv\log_{10}\left(\frac{P_{XY}^{\rm
NL}(k,\mathbf{\Omega})}{P_{XY}^{\rm LPT}(k,\mathbf{\Omega})}\right)$ (7)
After these pre-processing steps, we proceed by constructing a principal
component basis for these spectra. At this point we restrict ourselves to
$0.1<k<1$ and $0<z<2$, however we note that in principle there are no issues
extending to broader scales and redshifts if simulations allow for it.
Let $\mathbf{X}_{XY}$ be the $N\times M$ array containing $\Gamma^{XY}$, where
$N=N_{\rm cosmo}\times N_{z}$, $N_{\rm cosmo}$ is the number of cosmologies in
our training set, $N_{z}$ is the number of redshift outputs per cosmology in
our training set and $M$ is the number of $k$ values under consideration. Then
a basis of principal components can be constructed by computing the
eigenvectors of the covariance matrix of $\mathbf{X}_{XY}$:
$\displaystyle\mathbf{C}_{XY}$ $\displaystyle=\mathbf{X}_{XY}^{\rm
T}\mathbf{X}_{XY},$ (8)
$\displaystyle=\mathbf{W}_{XY}\mathbf{\Lambda}_{XY}\mathbf{W}_{XY}^{\rm T},$
where the rows of $\mathbf{W}_{XY}$ are the eigenvectors, i.e., the principal
components, in question and $\mathbf{\Lambda}_{XY}$ is a diagonal matrix of
the eigenvalues, which are equal to the variance of the data described by each
eigenvector. In all cases, greater than 99 per cent of the variance in each
basis spectrum is described by the first two principal components, shown in
Figure 3. We thus disregard all other principal components for the duration of
this work. Given the results discussed in Section 6, we deem this to be
sufficient. Having computed the principal components, we then determine the
projection of them onto each measured $\Gamma^{XY}$ via:
$\displaystyle\mathbf{A}_{XY}=\mathbf{X}_{XY}\mathbf{W}_{XY},$ (9)
where $\mathbf{A}_{XY}$ is an $N\times 2$ matrix containing the principle
component coefficients $\alpha^{XY}_{i}(\mathbf{\Omega})$ for each cosmology
and redshift in our training set. It is the dependence of these coefficients
on cosmology and redshift that we build a surrogate model for using polynomial
chaos expansions (Wiener, 1938).
### 5.3 Emulating cosmology dependence with polynomial chaos
With our principal components in hand, every point in cosmological parameter
space sampled by the training set has coefficients for the approximation
$\displaystyle\Gamma^{XY}(k,\mathbf{\Omega})\approx\sum_{i}\alpha_{i}^{XY}(\mathbf{\Omega})\mathrm{PC}_{i}^{XY}(k).$
(10)
The problem of emulating the cosmology dependence of the $\Gamma^{XY}$
functions is now reduced to that of figuring out the cosmology dependence of
the PC coefficients $\alpha_{i}(\mathbf{\Omega})$. A polynomial chaos
expansion (PCE) (of order $N$) of this dependence is the decomposition of the
$\alpha_{i}$ onto a basis of products of orthogonal polynomials
$\Phi_{\mathbf{i}}(\mathbf{\Omega})$ organized by a multi-index $\mathbf{i}$
(Xiu, 2010):
$\displaystyle\alpha(\mathbf{\Omega})=\sum_{|\mathbf{i}|\leq
N}c_{\mathbf{i}}\Phi_{\mathbf{i}}(\mathbf{\Omega}).$ (11)
Each component of the multi-index $\mathbf{i}=(i_{1},\cdots,i_{d})$, denotes
the order of the polynomial for that cosmological parameter, e.g.,
$\displaystyle\Phi_{\mathbf{i}}(\mathbf{\Omega})=\phi_{i_{1}}(\Omega_{1})\cdots\phi_{i_{d}}(\Omega_{d}),$
(12)
and so $\phi_{i_{d}}(\Omega_{d})$ is a univariate orthogonal polynomial of
order $i_{d}$.
While this is in principle a decomposition into a combinatorially large space
of coefficients $c_{\mathbf{i}}$, it is known to be a sparse representation
(Blatman & Sudret, 2008, 2011), and there exist many algorithms (and numerical
libraries) optimized to perform regression over this space and obtain values
for the coefficients. We use the package Chaospy (Feinberg & Langtangen, 2015;
Feinberg et al., 2018) to perform the decomposition and subsequent regression.
Note that since the parameter dependence of the principal components is given
by a combination of polynomials, our model in principle has an analytic
dependence on cosmology, redshift, and bias. Since the coefficients are
determined via regression, a PCE emulator does not recover the input data
exactly. However, the tests conducted in section 6 indicate that this drawback
is not an issue.
In total, the hyperparameters in the model are:
1. 1.
The number of principal components used, $N_{\mathrm{PC}}$.
2. 2.
The maximum order of the multi-index $|\mathbf{i}|$. In practice we separately
optimize over the maximum polynomial order of each individual parameter
$i_{d}$, with $i_{d}\leq 4$.
As mentioned previously, we restrict ourselves to $N_{\mathrm{PC}}=2$, as this
is sufficient to capture over 99 per cent of the variance in each basis
spectrum. To optimize over the polynomial orders $i_{d}$, we run a simple grid
search across the aforementioned values for the seven $w$CDM parameters
$\mathbf{\Omega}=(\Omega_{b}h^{2},\Omega_{c}h^{2},\sigma_{8},H_{0},n_{s},N_{\mathrm{eff}},w)$
and evaluate our results on the Aemulus test suite. We select the set of
orders that minimizes global error across all test boxes and snapshots. We
describe the tests of this optimized emulator below.
Figure 4: Coefficients of $\mathrm{PC}_{1}^{XY}(k)$ for the first three basis
spectra as a function of $\sigma_{8}$, colored by redshift. The coefficients
vary smoothly for all redshifts as $\sigma_{8}$ is varied. It is the
dependence of these coefficients that we emulate via PCE as a function of
cosmology and redshift. The panels look similar for the remaining basis
spectra. Figure 5: Emulation residuals for basis spectra. _Lower left
triangle:_ the fractional error obtained for each basis spectrum when compared
to the measurements averaged from each set of boxes in the test suite. _Upper
right triangle:_ the relative size of the emulator residuals compared to the
total halo–halo spectrum measured for a fiducial halo sample. In each panel,
the dark blue curves are the mean residuals across all redshifts and test
boxes, the red curves report the median residual error across the test suite
as a function of redshift, and the black curves report the expected sample
variance at the volume of an Aemulus training box.
## 6 Results
### 6.1 Analysis of emulator residuals
A crucial step in producing viable emulators of cosmological observables is
characterizing the accuracy of the emulation scheme. We use the Aemulus set of
test boxes to assess the performance of the scheme described in the previous
section. The test boxes span seven points in cosmological parameter space,
each with five independent realizations of that cosmology. We use the average
of five basis spectra at each test cosmology as reference quantities to
understand the errors induced in the emulation procedure as a function of
scale, across parameter space.
We report the accuracy of our optimized PCE emulator for the basis spectra
over the range $0.1\leq k\leq 1.0\,h\,{\rm Mpc}^{-1}$ in the lower left panel
of Fig. 5. Across most redshift bins in the test suite and for most basis
spectra we achieve better than 1 per cent accuracy in the test set. At $z=0$
we observe worse performance, however this can be attributed to numerical
difficulties in computing the LPT spectra at $z=0$ at small scales, as can be
seen in Fig. 1. As there is little cosmological information in the very low
redshift universe, we do not consider this to be a significant issue. Indeed,
our additional validation tests support that the model has sufficient accuracy
to analyse current survey data.
Adopting a fiducial set of bias parameters corresponding to a halo sample of
$12.5\leq\log_{10}\left(\frac{M}{h^{-1}M_{\odot}}\right)\leq 13$, we compute
the emulator residuals for each basis spectrum relative to the _total_
$P^{hh}(k)$. The results are shown in the upper right triangle of Fig. 5. The
individual basis spectrum error rarely exceeds a permille of the total power.
This implies that the slightly larger errors for cubic basis spectra shown in
Fig. 5 are sub-leading relative to the total signal we expect to model.
### 6.2 Fitting assembly bias and baryons
Beyond samples of fixed halo mass, the general bias expansion in Eq. 1 should
also be able to describe the clustering statistics of more complex tracer
populations. It is well known that haloes of a fixed mass bin exhibit
different clustering properties depending on whether they are sub-selected on
certain properties. This effect, originally discovered in the context of
assembly history, and generally known as assembly bias or secondary bias, has
been observed for selections on concentration, occupation, local environment,
spin, and other secondary halo properties (Wechsler et al., 2002; Gao et al.,
2005; Wechsler et al., 2006; Dalal et al., 2008; Mao et al., 2018; Salcedo et
al., 2018; Mansfield & Kravtsov, 2020).
Figure 6: Emulator predictions at fixed cosmology for halo samples exhibiting
concentration (top panels) and spin (bottom panels) assembly bias. Central
panels show the signal from the halo sample with no selection on a secondary
parameter. The left and right panels show samples split on the lowest and
highest quartiles of the relevant secondary bias parameter, respectively.
Shaded bands show the regions where residuals are within 2 per cent and 1 per
cent respectively, while the dashed envelope shows the expected cosmic
variance for a sample with $V\approx 5.8(h^{-1}{\rm Gpc})^{3}$. The spectra
are measured at $z=0.7$ and the fit is performed with the data vector out to
$k_{\rm max}=0.6\,h{\rm Mpc}^{-1}$.
As a test of our model, we construct halo catalogs with different amounts of
concentration and spin secondary bias, splitting the sample by quartile. The
magnitude of the effect varies differently as a function of mass for each
secondary bias parameter. Thus, we adopt separate halo mass bins for each
parameter, in a regime where we have both reliable estimates of the secondary
quantities and know that the secondary bias effect is not drastic, following
fig. 4 of Sato-Polito et al. (2019). The mass range
$12\leq\log_{10}\left(\frac{M}{h^{-1}M_{\odot}}\right)\leq 12.5$ was used to
build samples contaminated with concentration bias, and
$12.5\leq\log_{10}\left(\frac{M}{h^{-1}M_{\odot}}\right)\leq 13$ for spin
bias. We consider the highest and lowest quartile samples in both
concentration and spin, as well as a sample with no secondary bias, sub-
sampled to the same number density as the samples contaminated with secondary
bias. We additionally do not subtract the shot-noise contribution from
measured spectra, and opt instead to include it in our covariance matrix as
detailed in Eqn. 14.
Using the emulator for the basis spectra evaluated at the cosmology of these
test boxes, we jointly fit the halo–halo and halo–matter spectra
$\\{P_{hh},P_{hm}\\}$ with five parameters:
$b_{i}=\\{b_{1},b_{2},b_{s^{2}},b_{\nabla^{2}},\bar{n}^{-1}\\}$. We minimize
the $\chi^{2}$ between the mean of five simulations assuming a disconnected
covariance for the observables as described in Eq. 14, with
$V=5\times(1.05\,h^{-1}{\rm Gpc})^{3}$ each. The resulting fits are shown in
Fig. 6.
We fit the spectra to a maximum scale of $k_{\rm max}=0.6\,h\,{\rm Mpc}^{-1}$.
For most panels, we see that the hybrid $N$-body/Lagrangian bias model can
jointly describe the clustering and lensing spectra to within 1 per cent down
to scales even smaller than employed for the model fit. At large scales, the
lowest spin assembly bias bin seems to be systematically higher by at most 10
per cent. Changing the $k_{\rm max}$ of the fit down to $0.2\,h\,{\rm
Mpc}^{-1}$ does not qualitatively alleviate the large-scale discrepancies. We
observe similar behavior if the average of the basis spectra from this
cosmology are used instead of the emulator, implying this is not an issue of
the emulator and could perhaps be attributed to large-scale noise. Another
possibility is that a second-order Lagrangian bias model is unable to fully
capture the effects of spin secondary bias, but we leave this investigation to
future work.
In Fig. 7 we show the reduced $\chi^{2}$ for the fits to the samples split on
concentration. We see that the goodness of fit degrades significantly past
$k\simeq 0.6h\,{\rm Mpc}^{-1}$ for some subsamples. The fits to smaller
$k_{\rm max}$ have $\chi^{2}/{\rm d.o.f.}\lesssim 1.5$. Note that in these
tests we use the emulator at a volume that is significantly larger than the
boxes it was trained on, and the covariance matrices do not have any
contributions due to the emulator uncertainty. If we instead use the mean
basis spectra the $\chi^{2}/{\rm d.o.f.}$ cross the $\chi^{2}/{\rm d.o.f.}\sim
1$ threshold at $k_{\rm max}\sim 0.6h\,{\rm Mpc}^{-1}$ and grow significantly
afterwards, signalling a potential breakdown of the applicability of this
Lagrangian bias model to these samples.
Figure 7: The goodness of fit $\chi^{2}/{\rm d.o.f.}$ from increasing $k_{\rm
max}$ for the halo sample selected on concentration quartiles, using the
emulator as a model. Note the significant degradation of the goodness of fit
for the subsample split on the lowest quartile after $k_{\rm max}=0.6$.
Baryonic physics is known to impact the statistics of biased tracers at the
scales we are considering (White, 2004; Zhan & Knox, 2004; Chisari et al.,
2019; van Daalen et al., 2020). In our model, the $\langle
1,\nabla^{2}\delta\rangle$ basis spectrum should have the scale dependence
required to capture the first-order impacts of baryons (Lewandowski et al.,
2015). In order to test this, we produce mock ‘baryonified’ spectra using the
fitting function of van Daalen et al. (2020), which is obtained from analysis
of a comprehensive suite of hydrodynamic simulations. We compare the fitting
function to two parametrizations for the impact of baryons:
1. 1.
Including terms that scale as the basis functions $b_{\nabla^{2}}\langle
1,\nabla^{2}\delta\rangle$ and
$b_{1}b_{\nabla^{2}}\langle\delta,\nabla^{2}\delta\rangle$.
2. 2.
Same as above, but substituting the basis functions with the approximation
$\langle X,\nabla^{2}\delta\rangle\simeq-k^{2}\langle X,1\rangle$.
The results of this test are shown in Figure 8. While the baryonic suppression
factors presented by the two parametrizations differ, in the bottom panel we
see that both capture the effects of baryons to within 1 per cent out to
$k\approx 0.8\,h\,\mathrm{Mpc}^{-1}$, whereas not including the contributions
leads to errors larger than 1 per cent at $k\approx
0.2\,h\,\mathrm{Mpc}^{-1}$.
Additionally, our framework can simultaneously treat the effects of finite
halo size and baryonic physics. As both are captured by the same basis
spectra, this corresponds to treating the halo tracer as having one set of
$b_{\nabla}^{2}$ and the matter tracer in the $P^{hm}$ correlation as having a
separate higher derivative coefficient $b^{\prime}_{\nabla^{2}}$, while
keeping all other bias parameters equal to zero.
Figure 8: Higher derivative bias terms and their comparison to the baryonic
physics fitting function of van Daalen et al. (2020). The top panel shows the
fitting function, the basis spectrum as measured in the $N$-body simulations
and the approximation we employ in the text. In the lower panel we show
residuals between the different treatments and the fitting function. The blue
curve in the lower panel shows the difference between the unprocessed dark
matter power spectrum and the fitting function. The green curve is the
approximation to the higher derivative fitting functions that is implemented
in our analyses.
### 6.3 Recovering input cosmology
In this section we present an increasingly complex series of tests to ensure
our emulator can be used for cosmological inference, i.e., to demonstrate that
it can recover input cosmological parameters in an unbiased way. The general
structure of the analyses we run is as follows.
The input data-vectors will be the joint halo–halo and halo–matter power
spectra $\mathbf{d}=\\{P_{hh}(k),P_{hm}(k)\\}$. We assume a Gaussian
likelihood in the residuals between $\mathbf{d}$ and the emulator prediction
at a cosmology $\mathbf{x}(\mathbf{\Omega})$:
$\log\mathcal{L}(d|\mathbf{\Omega})\propto-(\mathbf{d}-\mathbf{x}(\mathbf{\Omega}))^{T}\mathbf{C}^{-1}(\mathbf{d}-\mathbf{x}(\mathbf{\Omega})).$
(13)
We adopt a baseline covariance matrix that includes only dependence on the
two-point functions of the tracer density field, known as the disconnected
contribution (Li et al., 2019). The result is a block-diagonal matrix with
format
$\mathbf{C}(k,k^{\prime})\equiv\frac{2\pi^{2}\delta_{k,k^{\prime}}}{k^{2}\Delta
kV}\times\begin{cases}2P_{hh}^{2}(k),&\text{ for }hh\times hh\\\
2P_{hh}(k)P_{hm}(k),&\text{ for }hh\times hm\\\
\bigg{[}P_{hh}(k)P_{mm}(k)&\text{ for }hm\times hm\\\
+P_{hm}^{2}(k)\bigg{]},\par\end{cases}$ (14)
for each sub-block. We use non-linear power spectra and $P_{hh}$ includes the
shot-noise contribution. At the smaller scales we probe in the resulting
analyses, the purely disconnected approximation is known to fail and off-
diagonal (connected) components become increasingly important (Meiksin &
White, 1999; Scoccimarro et al., 1999; Cooray & Hu, 2001; Mohammed et al.,
2016; Lacasa, 2018). The intent of this paper is not to conclusively quantify
the information content available at small scales. Rather, we would like to
ensure that the emulator is an unbiased model when pushing to such small
scales. Therefore, we consider the form of the covariance in Eqn. 14 to be a
sufficient baseline to carry out our analyses. We assess its performance in
more detail in Appendix A. As we will discuss in more detail in section 6.3.2,
the approximation of taking only the disconnected contribution neglects two
forms of error: that arising from the connected contribution, and model error
from the emulator itself. We discuss the contribution to the covariance from
emulator error in Appendix A, and find that in the regime under which our
tests are carried out, its inclusion is important in achieving unbiased
constraints.
We sample the posterior distributions of the model parameters via Markov Chain
Monte Carlo (MCMC), using emcee (Goodman & Weare, 2010; Foreman-Mackey et al.,
2013). Chains are run with either $N=64$ or $N=128$ walkers across 8000 (4000)
steps respectively. We checked that these values ensure converged chains for
the simulated likelihood analyses we run; the posteriors are not altered
significantly by doubling the length or number of walkers. We adopt wide
uniform priors on the bias parameters,
$b_{i}\sim U(-5,5),$ (15)
and uniform priors surrounding the boundaries of the Aemulus training suite,
specified in Table 1.
Parameter | Range
---|---
$\Omega_{b}h^{2}$ | [0.0207 , 0.0237]
$\Omega_{c}h^{2}$ | [0.101 , 0.132]
$w_{0}$ | [-1.399 , -0.566]
$n_{s}$ | [0.928 , 0.997]
$\sigma_{8}$ | [0.575 , 0.964]
$H_{0}$ | [61.69 , 74.77]
$N_{\mathrm{eff}}$ | [2.62 , 4.28]
Table 1: Boundaries of the cosmological parameters of simulations spanned by
the Aemulus training suite. These are the values used as flat priors for
cosmological parameters.
#### 6.3.1 Synthetic Data
As a first test of the emulator, we perform a simulated likelihood analysis on
a noiseless data vector drawn from the emulator itself. We fit the basis
spectra to a halo sample of mass $12\leq\log_{10}M_{h}/M_{\odot}\leq 12.5$
from one of Aemulus’ test boxes. The cosmology and best-fitting bias values
are used as inputs to the emulator to produce a mock noiseless data-vector. As
the data in this test is not a random draw from a distribution, the exact
format of the covariance matrix does not matter. However, we use the block-
diagonal disconnected covariance of Eqn. 14 with
$V=(1050\,h^{-1}\mathrm{Mpc})^{3}$ so as to replicate an analysis on an
individual Aemulus test box.
The results of this first mock analysis are shown in in Fig. 9. The three-
parameter analysis constrains all cosmological and bias parameters in an
unbiased fashion, indicating that there are no issues in fitting the emulator
to itself at this volume. We also conduct seven-parameter analysis for $w$CDM
parameters. The results returns unbiased posteriors relative to the true input
values, however it is hard to constrain all $w$CDM parameters using a single
halo sample at the volume of a single Aemulus box. For this reason, several of
the cosmological parameters simply saturate the priors and remain
unconstrained.
#### 6.3.2 Halo samples from the test suite
Figure 9: Cosmological parameter inference using the emulator where the data
are a noiseless draw from itself. We vary the subset of parameters
$\omega_{c},\,\sigma_{8},$ and $H_{0}$, using a Gaussian likelihood and purely
disconnected covariance with volume $V=(1.05)^{3}(h^{-1}\mathrm{Gpc})^{3}$.
The fiducial values used to generate the data vector are shown in the dashed
lines. The bias parameter posteriors are equally unbiased and Gaussian, but
omitted from the figure for aesthetic purposes. Figure 10: Cosmological
parameter inference using the emulator fit to the mean of five realizations at
the seven Aemulus test cosmologies. We vary the cosmological parameters
$\omega_{c},\,\sigma_{8},$ and $H_{0}$, using a disconected covariance with
volume $V=(1.05)^{3}(h^{-1}\mathrm{Gpc})^{3}$, including a contribution
arising from correlated emulator residuals. The contours are shown in the
space of differences relative to the true cosmology of each box.
A subsequent test we perform is inference on halo catalogs drawn from the
Aemulus test suite. We refer to Aemulus I (DeRose et al., 2019b) for details
on how the halo finding procedure was done. This fiducial halo sample contains
the mass bin $13\leq\log_{10}\left(\frac{M_{h}}{h^{-1}M_{\odot}}\right)\leq
13.5$ at $z=0.4$.
We run a suite of chains for this halo sample, to assess emulator performance
in terms of inferring cosmological parameters. We measure the halo–halo and
halo–matter power spectra for each independent test box across the seven
different test cosmologies. The data vector is averaged over the five
independent realizations from the test suite. This set of chains allows us to
assess the emulator biases such that they are less susceptible to projection
effects. We can also study the cosmology dependence of the emulator in this
way and the interplay between the bias coefficients of our model and
cosmological parameters.
Using solely the purely disconnected covariance matrix in Eqn. 14 leads to
strong biases in inferred cosmological parameters, despite all residuals being
smaller than 1 per cent as a function of scale. This can be understood by the
fact that sample variance at small scales will eventually become smaller than
the $1-2$ per cent emulator error observed in Fig. 5 (see also Fig. 13).
However, the aforementioned figure allows us to estimate the emulator
uncertainty as a function of scale. This can then be included as a separate
contribution to the covariance matrix. We detail how this is done in Appendix
A 111All contour plots shown from this point forward will include the effects
of emulator error unless stated otherwise..
The result of the test is shown in Fig. 10, with the full set of contours
shown in Fig. 17. The cosmological parameters inferred scatter around the best
fits for $\omega_{c}$ and $\sigma_{8}$, whereas they recover $H_{0}$ to within
one standard deviation for most cosmologies but biased slightly high. However,
we note these tests are conservative, as they neglect the contribution to the
covariance matrix arising from shape noise, the lensing equivalent of shot-
noise that would contribute to the $hm\,\times\,hm$ term of the covariance
matrix. Given the conservative nature of this test we deem the emulator
performance to be sufficient and continue with the final and most stringent
test we consider in this work.
#### 6.3.3 A redMaGiC sample from an independent simulation
$\log M_{\mathrm{min}}$ | $\sigma_{\log M}$ | $f_{c}$ | $\log M_{0}$ | $\log M_{1}^{{}^{\prime}}$ | $\alpha$
---|---|---|---|---|---
12.1 | 0.4 | 0.13 | 11.45 | 13.73 | 1.48
Table 2: HOD parameters used to populate the redMaGiC sample described in
section 6.3.3.
So far, we have reported tests performed on samples that originate either from
the emulator itself or from the same suite of simulations used to construct
it. It is also important that the model is useful for inference on spectra
measured from tracer samples generated by independent methods, both in how
halo samples are defined and the underlying $N$-body simulation used. For
example, in Modi et al. (2020) it was shown that this hybrid Lagrangian bias
model can successfully fit galaxy power spectra produced from a halo
occupation distribution (HOD; see e.g. Zheng et al. 2005).
We perform a final test: a simulated likelihood analysis with spectra produced
from populating an independent $N$-body simulation with an HOD that matches
the density and clustering properties of redMaGiC galaxies (Rozo et al.,
2016). redMaGiC galaxies are the primary photometric Luminous Red Galaxy
sample used in current and future weak lensing surveys (Elvin-Poole et al.,
2018).
The HOD parametrization we adopt is an extension of the model presented in
Zheng et al. (2007), allowing for the central occupation at high mass to be
less than unity
$\displaystyle\langle
N_{\mathrm{cen}}(M)\rangle=\frac{f_{c}}{2}\left[1+\mathrm{erf}\left(\frac{\log
M-\log M_{\mathrm{min}}}{\sigma_{\log M}}\right)\right],$ (16)
$\displaystyle\langle
N_{\mathrm{sat}}(M)\rangle=\frac{1}{2}\left[1+\left(\frac{\log M-\log
M_{\mathrm{min}}}{\sigma_{\log
M}}\right)\right]\left(\frac{M-M_{0}}{M_{1}^{{}^{\prime}}}\right)^{\alpha}.$
(17)
The HOD parameters corresponding to the redMaGiC samples used can be found in
Table 2, and are derived from a redMaGiC sample selected from simulations
similar to those presented in DeRose et al. (2019a).
We paint redMaGiC galaxies onto halo catalogs measured from the UNIT
simulations (Chuang et al., 2019) at $z\approx 0.59$, a redshift different
from the Aemulus snapshots. A UNIT realization boasts a comparable volume to
Aemulus of $V=1\,(h^{-1}\mathrm{Gpc})^{3}$ at a significantly higher number of
particles, $N=(4096)^{3}$. Every UNIT simulation has two realizations with
opposite phases and fixed amplitudes. Averaging two-point statistics measured
from these paired–fixed realizations leads to very high sample variance
suppression at large scales, comparable to averaging $\sim 150$ simulations of
the same volume.
The cosmological parameter constraints corresponding to this test are shown in
Fig. 11. The emulator recovers the input cosmology of UNIT within its $68$ per
cent contours. Although this test is idealized, the constraints inferred are
promising if they translate even moderately well to a realistic analysis: a
2.5 per cent constraint on $\omega_{c}$, a 0.5 per cent constraint on
$\sigma_{8}$ and a 1.6 per cent constraint of $H_{0}$. In a realistic lensing
analysis one would expect these quantities to be degraded due to the inclusion
of shape noise and only having access to two-dimensional lensing maps instead
of the 3D matter field. Nevertheless, even a 100% degradation of these
constraints due to the aforementioned complications would still result in
highly competitive measurements of these parameters. Note we adopt no priors
beyond the (moderately informative) priors set by the boundaries of the
Aemulus suite.
Figure 11: Cosmological parameter constraints from the redMaGiC sample
constructed from the UNIT simulations. The true cosmological parameters and
the best-fit bias parameters assuming the true cosmology are shown in the
dashed lines. All parameters are recovered to well within the one-sigma
errors.
The simulated likelihood analysis performed on this sample additionally allow
us to quantify both model and emulator errors in a space that is closer to
observations that will be carried out in the near future. As redMaGiC galaxies
are commonly used as lens samples in galaxy–galaxy lensing analyses, we can
translate the $P^{hh},P^{hm}$ residuals to those in the observables
$C_{\ell}^{gg},C_{\ell}^{g\kappa}$. We assume a redshift distribution $n(z)$
for redMaGiC galaxies consistent with data (Elvin-Poole et al., 2018) and
fiducial parametrizations for the source sample that are consistent with those
that will be achieved in future imaging surveys (Mandelbaum et al., 2018). For
a redMaGiC sample spanning $z=[0.45,0.6]$ we present the results in Fig. 12.
The harmonic space observables are calculated assuming the Limber
approximation, with the additional approximation that the residuals between 3D
power spectra do not evolve as a function of redshift. The residuals stay
within one per cent out to $\ell\approx 1000$. If we instead use residuals
from fitting the emulator at fixed cosmology to the same sample out to $k_{\rm
max}=1.0\,h{\rm Mpc}^{-1}$ the residuals remain within ten per cent out to
$\ell_{\rm max}=2000$, at the cost of worse performance at large scales. This
indicates that the combined emulator and model error remain well under control
for the analysis of current galaxy–galaxy lensing datasets.
Figure 12: Residuals of the emulator fit to the redMaGiC sample in the space
of a projected analysis. Residuals are shown for the range $\ell\in[50,2000]$.
The redshift distributions of this analysis are consistent with those of
current and upcoming surveys. The dashed envelope corresponds to the sample
variance contribution in the absence of noise with sky coverage consistent
with upcoming surveys and angular binning of $\Delta\ell=50$. That is,
shot/shape noise will only increase the size of this envelope. The light gray
and dark gray bands correspond to 2 and 1 per cent error bands, respectively.
## 7 Conclusions
In this work we have built an emulator to study the cosmology dependence of
the model of Modi et al. (2020) for the two-point statistics of biased
tracers. The model combines $N$-body simulations with a symmetries-based bias
expansion to provide accurate predictions beyond the regime of validity of
standard perturbative approaches.
Specifically, we built an emulator for the cosmology and redshift dependence
of the ten non-linear basis functions that span this model. We use
measurements from the Aemulus suite of simulations, which has been designed to
enable the construction of emulators that satisfy the modelling requirements
of upcoming cosmic surveys. The model and emulation techniques used are
general; there are no limitations to extending the range of validity given the
availability of an improved suite of simulations.
We find that:
1. 1.
The emulator recovers each basis spectrum to $\lesssim$ 1 per cent accuracy
across a wide range of scales, $0.1<k/\left(h^{-1}{\rm Mpc}\right)\leq 1.0$,
and redshifts, $0\leq z\leq 2$.
2. 2.
The Lagrangian bias model is capable of capturing the clustering and lensing
statistics of samples imbued with non-trivial amounts of secondary bias and
contamination from baryonic physics.
3. 3.
The test set used to validate the emulator can also be used to calibrate its
‘theoretical uncertainty’. This allows us to include contributions to the
covariance matrix of an analysis related to model error, which cannot be
neglected when pushing to small scales.
4. 4.
The emulator, as constructed, can recover unbiased cosmological parameters
from realistic simulated likelihood analyses.
These findings indicate that our emulator is a robust tool that can be readily
applied to analyses of current and even upcoming datasets. The code will be
made publicly available github and can be integrated with modern sampling
packages such as Cobaya (Torrado & Lewis, 2020). We also point out a few
further directions to be investigated as a result of this work.
First, while the simulations used here are sufficient to obtain per cent level
emulator accuracy, improved simulations will be important for maximizing the
applicability of this model. The biggest immediate limitation of this emulator
is the extent of the cosmological parameter space that it is trained on. We
plan on running simulations over a broader parameter space, including massive
neutrinos, in the near future. Another limiting factor in the current emulator
construction is our ability to match the basis spectra measured from our
simulations to their perturbation theory analogs at low $k$. Running larger
simulation volumes, or implementing a method for sample variance mitigation
such as that presented in Chartier et al. (2020), would ameliorate this issue
by reducing noise in the $N$-body measurements. This will allow them to be
matched more easily to the perturbation theory predictions at scales that are
still safely within perturbative reach. Mismatches in the linear growth
predictions from $N$-body simulations also limit the accuracy of the large
scale matching. Simulations with more stringent time-stepping criteria would
reduce these inaccuracies, at the cost of increased run-time. For this reason,
methods that explicitly enforce linear growth on large scales may be worth
exploring in the future (Feng et al., 2016; Howlett et al., 2015). Finally,
the accuracy of the model for redshift evolution of the basis spectra in the
current emulator is limited by the number of snapshots saved in the Aemulus
suite. For this reason, saving snapshots with finer redshift resolution out to
higher redshifts will be a priority when running future simulations to upgrade
the current emulator.
While in this paper we have restricted ourselves to predictions of survey
observables in Fourier space, one could use this same field-level approach to
measure configuration-space correlation statistics instead. The model employed
should also be able to describe the statistics of biased tracers at the field
level, beyond two-point statistics. This includes both field-level
characterizations of the Lagrangian bias model similarly to what was
investigated in Schmittfull et al. (2019) and higher order functions such as
the bispectrum or the collapsed tri-spectra that form the connected component
of covariance matrices.
The field-level approach to bias modelling described in Schmittfull et al.
(2019) was recently extended to redshift space (Schmittfull et al., 2020). For
our emulator to be used to describe the statistics of 3D galaxy clustering in
spectroscopic galaxy surveys, it would need to be extended to redshift space
in a similar manner. Alternatively, we note that the bias parameters in this
model are equivalent to those of the Lagrangian perturbation theory of Chen et
al. (2020a, b). This suggests one could perform a joint analysis that combines
perturbation theory for describing the redshift-space clustering, where the 3D
nature of the measurements allow tight constraints even on quasi-linear
scales, and an emulator for describing projected statistics, which need to
extend to smaller scales in order to beat down sample variance. In addition to
providing a large dynamic range and sensitivity to both metric potentials, the
combination of measurements would help to break bias parameter degeneracies
and thus improve cosmological constraints.
The second release of the Aemulus suite, Aemulus-$\nu$, will include two-fluid
simulations that capture the effects of massive neutrinos on the matter
density field. The techniques described in this paper can be translated to
this new set of simulations to construct an emulator that can be used to
constrain the sum of neutrino masses, one of the key science drivers of
ongoing and future cosmological surveys.
We leave these extensions to future work.
## Acknowledgements
We thank Simone Ferraro and Anže Slosar for helpful comments on a draft of the
paper and Sean McLaughlin for many helpful discussions. We are grateful to the
Aemulus collaboration for making the simulation suite used here publicly
available. This work was supported in part by U.S. Department of Energy
contracts to SLAC (DE-AC02-76SF00515) and by Stanford University. N.K. thanks
the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF
Cybertraining Grant #1829740, the Brinson Foundation, and the Moore
Foundation. S.C. is supported by the National Science Foundation Graduate
Research Fellowship (Grant No. DGE 1106400) and by the UC Berkeley Theoretical
Astrophysics Center Astronomy and Astrophysics Graduate Fellowship. M.W. is
supported by the U.S. Department of Energy and the NSF. This research has made
use of NASA’s Astrophysics Data System and the arXiv preprint server.
Some of the computing for this project was performed on the Sherlock cluster.
We would like to thank Stanford University and the Stanford Research Computing
Center for providing computational resources and support that contributed to
these research results.
Calculations and figures in this work have been made using nbodykit (Hand et
al., 2018), GetDist (Lewis, 2019), and the SciPy Stack (Harris et al., 2020;
Virtanen et al., 2020; Hunter, 2007).
## Data Availability
The data underlying this article are available in the Aemulus Project’s
website.
## References
* Abbott et al. (2018) Abbott T., et al., 2018, Phys. Rev. D, 98, 043526
* Abidi & Baldauf (2018) Abidi M. M., Baldauf T., 2018, JCAP, 07, 029
* Aghamousa et al. (2016) Aghamousa A., et al., 2016, arXiv e-prints
* Alam et al. (2017) Alam S., Miyatake H., More S., Ho S., Mandelbaum R., 2017, Mon. Not. Roy. Astron. Soc., 465, 4853
* Angulo & Pontzen (2016) Angulo R. E., Pontzen A., 2016, Mon. Not. Roy. Astron. Soc., 462, L1
* Aviles & Banerjee (2020) Aviles A., Banerjee A., 2020, J. Cosmology Astropart. Phys., 2020, 034
* Aviles et al. (2020) Aviles A., Valogiannis G., Rodriguez-Meza M. A., Cervantes-Cota J. L., Li B., Bean R., 2020, arXiv e-prints, p. arXiv:2012.05077
* Bagla (2005) Bagla J. S., 2005, Curr. Sci., 88, 1088
* Baldauf et al. (2016a) Baldauf T., Mirbabayi M., Simonović M., Zaldarriaga M., 2016a, arXiv e-prints
* Baldauf et al. (2016b) Baldauf T., Schaan E., Zaldarriaga M., 2016b, JCAP, 03, 007
* Bartelmann & Schneider (2001) Bartelmann M., Schneider P., 2001, Phys. Rept., 340, 291
* Baumann et al. (2012) Baumann D., Nicolis A., Senatore L., Zaldarriaga M., 2012, J. Cosmology Astropart. Phys., 2012, 051
* Bernardeau et al. (2002) Bernardeau F., Colombi S., Gaztanaga E., Scoccimarro R., 2002, Phys. Rept., 367, 1
* Bianchini et al. (2015) Bianchini F., et al., 2015, The Astrophysical Journal, 802, 64
* Blas et al. (2014) Blas D., Garny M., Konstandin T., 2014, J. Cosmology Astropart. Phys., 2014, 010
* Blatman & Sudret (2008) Blatman G., Sudret B., 2008, Comptes Rendus Mecanique, 336, 518
* Blatman & Sudret (2011) Blatman G., Sudret B., 2011, Journal of Computational Physics, 230, 2345
* Carrasco et al. (2012) Carrasco J. J. M., Hertzberg M. P., Senatore L., 2012, JHEP, 09, 082
* Chartier et al. (2020) Chartier N., Wandelt B., Akrami Y., Villaescusa-Navarro F., 2020, arXiv e-prints, p. arXiv:2009.08970
* Chen et al. (2020a) Chen S.-F., Vlah Z., Castorina E., White M., 2020a, Redshift-Space Distortions in Lagrangian Perturbation Theory (arXiv:2012.04636)
* Chen et al. (2020b) Chen S.-F., Vlah Z., White M., 2020b, JCAP, 07, 062
* Chen et al. (2020c) Chen S.-F., Vlah Z., White M., 2020c, JCAP, 11, 035
* Chisari et al. (2019) Chisari N. E., et al., 2019, Open J. Astrophys., 2, 4
* Chuang et al. (2019) Chuang C.-H., et al., 2019, Mon. Not. Roy. Astron. Soc., 487, 48
* Chudaykin et al. (2020) Chudaykin A., Ivanov M. M., Simonović M., 2020, arXiv e-prints
* Cooray & Hu (2001) Cooray A., Hu W., 2001, The Astrophysical Journal, 554, 56–66
* Crocce et al. (2006) Crocce M., Pueblas S., Scoccimarro R., 2006, Mon. Not. Roy. Astron. Soc., 373, 369
* Crocce et al. (2012) Crocce M., Pueblas S., Scoccimarro R., 2012, 2LPTIC: 2nd-order Lagrangian Perturbation Theory Initial Conditions (ascl:1201.005)
* Dalal et al. (2008) Dalal N., White M., Bond J. R., Shirokov A., 2008, Astrophys. J., 687, 12
* DeRose et al. (2019a) DeRose J., et al., 2019a, arXiv e-prints, p. arXiv:1901.02401
* DeRose et al. (2019b) DeRose J., et al., 2019b, Astrophys. J., 875, 69
* Desjacques et al. (2018) Desjacques V., Jeong D., Schmidt F., 2018, Phys. Rept., 733, 1
* DiPompeo et al. (2017) DiPompeo M. A., Hickox R. C., Eftekharzadeh S., Myers A. D., 2017, MNRAS, 469, 4630
* Doré et al. (2015) Doré O., et al., 2015, Cosmology with the SPHEREX All-Sky Spectral Survey (arXiv:1412.4872)
* Doré et al. (2019) Doré O., et al., 2019, WFIRST: The Essential Cosmology Space Observatory for the Coming Decade (arXiv:1904.01174)
* Elvin-Poole et al. (2018) Elvin-Poole J., et al., 2018, Phys. Rev. D, 98, 042006
* Favole et al. (2020) Favole G., et al., 2020, Mon. Not. Roy. Astron. Soc., 497, 5432
* Feinberg & Langtangen (2015) Feinberg J., Langtangen H. P., 2015, Journal of Computational Science, 11, 46
* Feinberg et al. (2018) Feinberg J., Eck V. G., Langtangen H. P., 2018, SIAM Journal on Scientific Computing, 40, A199
* Feng et al. (2016) Feng Y., Chu M.-Y., Seljak U., McDonald P., 2016, MNRAS, 463, 2273
* Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
* Fujita & Vlah (2020) Fujita T., Vlah Z., 2020, J. Cosmology Astropart. Phys., 2020, 059
* Fujita et al. (2020) Fujita T., Mauerhofer V., Senatore L., Vlah Z., Angulo R., 2020, JCAP, 01, 009
* Gao et al. (2005) Gao L., Springel V., White S. D., 2005, Mon. Not. Roy. Astron. Soc., 363, L66
* Garrison et al. (2016) Garrison L. H., Eisenstein D. J., Ferrer D., Metchnik M. V., Pinto P. A., 2016, Mon. Not. Roy. Astron. Soc., 461, 4125
* Goodman & Weare (2010) Goodman J., Weare J., 2010, Communications in Applied Mathematics and Computational Science, 5, 65
* Guo et al. (2019) Guo H., et al., 2019, Astrophys. J., 871, 147
* Hand et al. (2018) Hand N., Feng Y., Beutler F., Li Y., Modi C., Seljak U., Slepian Z., 2018, The Astronomical Journal, 156, 160
* Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357–362
* Heitmann et al. (2009) Heitmann K., Higdon D., White M., Habib S., Williams B. J., Wagner C., 2009, Astrophys. J., 705, 156
* Heitmann et al. (2010) Heitmann K., White M., Wagner C., Habib S., Higdon D., 2010, Astrophys. J., 715, 104
* Heymans et al. (2020) Heymans C., et al., 2020, arXiv e-prints, p. arXiv:2007.15632
* Hockney & Eastwood (1988) Hockney R., Eastwood J., 1988, Computer Simulation Using Particles. CRC Press, https://books.google.com/books?id=nTOFkmnCQuIC
* Hoffman & Gelman (2011) Hoffman M. D., Gelman A., 2011, arXiv e-prints, p. arXiv:1111.4246
* Howlett et al. (2015) Howlett C., Manera M., Percival W. J., 2015, Astronomy and Computing, 12, 109
* Hunter (2007) Hunter J. D., 2007, Computing in Science Engineering, 9, 90
* Ivanov et al. (2020) Ivanov M. M., McDonough E., Hill J. C., Simonović M., Toomey M. W., Alexander S., Zaldarriaga M., 2020, Phys. Rev. D, 102, 103502
* Ivezić et al. (2019) Ivezić v., et al., 2019, Astrophys. J., 873, 111
* Joyce et al. (2020) Joyce M., Garrison L., Eisenstein D., 2020, Mon. Not. Roy. Astron. Soc.
* Knabenhans et al. (2019) Knabenhans M., et al., 2019, Mon. Not. Roy. Astron. Soc., 484, 5509
* Krause et al. (2017) Krause E., et al., 2017, arXiv e-prints
* Krolewski et al. (2020) Krolewski A., Ferraro S., Schlafly E. F., White M., 2020, J. Cosmology Astropart. Phys., 2020, 047
* Kuhlen et al. (2012) Kuhlen M., Vogelsberger M., Angulo R., 2012, Phys. Dark Univ., 1, 50
* Kwan et al. (2015) Kwan J., Heitmann K., Habib S., Padmanabhan N., Finkel H., Lawrence E., Frontiere N., Pope A., 2015, Astrophys. J., 810, 35
* Lacasa (2018) Lacasa F., 2018, Astronomy & Astrophysics, 615, A1
* Laguë et al. (2020) Laguë A., Bond J. R., Hložek R., Marsh D. J., Söding L., 2020, arXiv e-prints
* Laureijs et al. (2011) Laureijs R., et al., 2011, Euclid Definition Study Report (arXiv:1110.3193)
* Lawrence et al. (2010) Lawrence E., Heitmann K., White M., Higdon D., Wagner C., Habib S., Williams B., 2010, The Astrophysical Journal, 713, 1322–1331
* Lazeyras & Schmidt (2018) Lazeyras T., Schmidt F., 2018, J. Cosmology Astropart. Phys., 2018, 008
* Lazeyras & Schmidt (2019) Lazeyras T., Schmidt F., 2019, JCAP, 11, 041
* Lewandowski et al. (2015) Lewandowski M., Perko A., Senatore L., 2015, JCAP, 05, 019
* Lewis (2019) Lewis A., 2019, GetDist: a Python package for analysing Monte Carlo samples (arXiv:1910.13970)
* Li et al. (2019) Li Y., Singh S., Yu B., Feng Y., Seljak U., 2019, Journal of Cosmology and Astroparticle Physics, 2019, 016–016
* MacCrann et al. (2020) MacCrann N., Blazek J., Jain B., Krause E., 2020, Mon. Not. Roy. Astron. Soc., 491, 5498
* Mandelbaum (2018) Mandelbaum R., 2018, Ann. Rev. Astron. Astrophys., 56, 393
* Mandelbaum et al. (2018) Mandelbaum R., et al., 2018, arXiv e-prints
* Mansfield & Avestruz (2020) Mansfield P., Avestruz C., 2020, Mon. Not. Roy. Astron. Soc.
* Mansfield & Kravtsov (2020) Mansfield P., Kravtsov A. V., 2020, Mon. Not. Roy. Astron. Soc., 493, 4763
* Mao et al. (2018) Mao Y.-Y., Zentner A. R., Wechsler R. H., 2018, Mon. Not. Roy. Astron. Soc., 474, 5143
* Matsubara (2008) Matsubara T., 2008, Phys. Rev. D, 78, 083519
* McClintock et al. (2019) McClintock T., et al., 2019, Astrophys. J., 872, 53
* McDonald & Roy (2009) McDonald P., Roy A., 2009, Journal of Cosmology and Astroparticle Physics, 2009, 020–020
* McLaughlin et al. (2021) McLaughlin S., Wechsler R. H., Banerjee A., DeRose J., Mao Y.-Y., Tinker J. L., Zhai Z., 2021, arXiv e-prints, p. To appear
* McQuinn & White (2016) McQuinn M., White M., 2016, J. Cosmology Astropart. Phys., 2016, 043
* Meiksin & White (1999) Meiksin A., White M., 1999, Monthly Notices of the Royal Astronomical Society, 308, 1179–1184
* Michaux et al. (2020) Michaux M., Hahn O., Rampf C., Angulo R. E., 2020, Mon. Not. Roy. Astron. Soc., 500, 663
* Modi et al. (2017) Modi C., White M., Vlah Z., 2017, J. Cosmology Astropart. Phys., 2017, 009
* Modi et al. (2020) Modi C., Chen S.-F., White M., 2020, Mon. Not. Roy. Astron. Soc., 492, 5754
* Mohammed et al. (2016) Mohammed I., Seljak U., Vlah Z., 2016, Monthly Notices of the Royal Astronomical Society, 466, 780–797
* Nishimichi et al. (2020) Nishimichi T., D’Amico G., Ivanov M. M., Senatore L., Simonović M., Takada M., Zaldarriaga M., Zhang P., 2020, arXiv e-prints
* Omori et al. (2019) Omori Y., et al., 2019, Physical Review D, 100
* Park et al. (2020) Park Y., Rozo E., Krause E., 2020, arXiv e-prints
* Peacock & Bilicki (2018) Peacock J. A., Bilicki M., 2018, MNRAS, 481, 1133
* Power et al. (2016) Power C., Robotham A. S. G., Obreschkow D., Hobbs A., Lewis G. F., 2016, Monthly Notices of the Royal Astronomical Society, 462, 474–489
* Prat et al. (2018) Prat J., et al., 2018, Physical Review D, 98
* Pullen et al. (2016) Pullen A. R., Alam S., He S., Ho S., 2016, MNRAS, 460, 4098
* Rozo et al. (2016) Rozo E., et al., 2016, Mon. Not. Roy. Astron. Soc., 461, 1431
* Salcedo et al. (2018) Salcedo A. N., Maller A. H., Berlind A. A., Sinha M., McBride C. K., Behroozi P. S., Wechsler R. H., Weinberg D. H., 2018, Monthly Notices of the Royal Astronomical Society, 475, 4411–4423
* Sato-Polito et al. (2019) Sato-Polito G., Montero-Dorta A. D., Abramo L. R., Prada F., Klypin A., 2019, Mon. Not. Roy. Astron. Soc., 487, 1570
* Savitzky & Golay (1964) Savitzky A., Golay M. J. E., 1964, Analytical Chemistry, 36, 1627
* Schmittfull et al. (2019) Schmittfull M., Simonović M., Assassi V., Zaldarriaga M., 2019, Physical Review D, 100
* Schmittfull et al. (2020) Schmittfull M., Simonović M., Ivanov M. M., Philcox O. H. E., Zaldarriaga M., 2020, Modeling Galaxies in Redshift Space at the Field Level (arXiv:2012.03334)
* Schneider et al. (2016) Schneider A., et al., 2016, Journal of Cosmology and Astroparticle Physics, 2016, 047–047
* Scoccimarro et al. (1999) Scoccimarro R., Zaldarriaga M., Hui L., 1999, The Astrophysical Journal, 527, 1–15
* Senatore & Zaldarriaga (2017) Senatore L., Zaldarriaga M., 2017, arXiv e-prints, p. arXiv:1707.04698
* Singh et al. (2019) Singh S., Mandelbaum R., Seljak U., Rodríguez-Torres S., Slosar A., 2019, Monthly Notices of the Royal Astronomical Society, 491, 51–68
* Takada et al. (2014) Takada M., et al., 2014, PASJ, 66, R1
* Taruya et al. (2018) Taruya A., Nishimichi T., Jeong D., 2018, Phys. Rev. D, 98, 103532
* Torrado & Lewis (2020) Torrado J., Lewis A., 2020, Cobaya: Code for Bayesian Analysis of hierarchical physical models (arXiv:2005.05290)
* Villaescusa-Navarro et al. (2018) Villaescusa-Navarro F., et al., 2018, Astrophys. J., 867, 137
* Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
* Vlah et al. (2015) Vlah Z., White M., Aviles A., 2015, J. Cosmology Astropart. Phys., 2015, 014
* Vlah et al. (2016) Vlah Z., Castorina E., White M., 2016, Journal of Cosmology and Astroparticle Physics, 2016, 007–007
* Wechsler & Tinker (2018) Wechsler R. H., Tinker J. L., 2018, Ann. Rev. Astron. Astrophys., 56, 435
* Wechsler et al. (2002) Wechsler R. H., Bullock J. S., Primack J. R., Kravtsov A. V., Dekel A., 2002, Astrophys. J., 568, 52
* Wechsler et al. (2006) Wechsler R. H., Zentner A. R., Bullock J. S., Kravtsov A. V., 2006, Astrophys. J., 652, 71
* White (2004) White M. J., 2004, Astropart. Phys., 22, 211
* Wibking et al. (2019) Wibking B. D., et al., 2019, Mon. Not. Roy. Astron. Soc., 484, 989
* Wibking et al. (2020) Wibking B. D., Weinberg D. H., Salcedo A. N., Wu H.-Y., Singh S., Rodríguez-Torres S., Garrison L. H., Eisenstein D. J., 2020, Mon. Not. Roy. Astron. Soc., 492, 2872
* Wiener (1938) Wiener N., 1938, American Journal of Mathematics, 60, 897
* Xiu (2010) Xiu D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, USA
* Yoo et al. (2006) Yoo J., Tinker J. L., Weinberg D. H., Zheng Z., Katz N., Dave R., 2006, The Astrophysical Journal, 652, 26–42
* Yuan et al. (2018) Yuan S., Eisenstein D. J., Garrison L. H., 2018, Mon. Not. Roy. Astron. Soc., 478, 2019
* Zhai et al. (2019) Zhai Z., et al., 2019, Astrophys. J., 874, 95
* Zhan & Knox (2004) Zhan H., Knox L., 2004, Astrophys. J. Lett., 616, L75
* Zhang et al. (2020) Zhang Y., et al., 2020, Mon. Not. Roy. Astron. Soc.
* Zheng et al. (2005) Zheng Z., et al., 2005, Astrophys. J., 633, 791
* Zheng et al. (2007) Zheng Z., Coil A. L., Zehavi I., 2007, Astrophys. J., 667, 760
* Zu (2020) Zu Y., 2020, arXiv e-prints
* van Daalen et al. (2020) van Daalen M. P., McCarthy I. G., Schaye J., 2020, Mon. Not. Roy. Astron. Soc., 491, 2424
## Appendix A Including emulator error in the covariance matrix
As seen in Fig. 5, there is a scale-dependent error associated with our
emulation scheme. This error is small, on the order of $\sim$ 1 per cent, and
within the accuracy requirements for the next generation of surveys. However,
at the smallest scales we would like to test this model, $k\simeq
0.6\,h\,\mathrm{Mpc}^{-1}$, it will often be larger than the combined cosmic
variance and shot noise (and absence of shape noise) of our tests. In this
regime, the combination of using the average of only five boxes as our data,
the approximate disconnected form of the covariance in Eqn. 14 and failing to
include model uncertainty in an analysis could then lead to biased inference
on cosmological parameters (Baldauf et al., 2016a; Chudaykin et al., 2020).
Since the Aemulus test suite is composed of 35 simulations, at seven distinct
points of cosmological parameter space, we can use the emulator residuals at
these points to construct a model for the theoretical uncertainty. In this
appendix we discuss our procedure to construct this model and study its impact
when employed in inference.
Let $\bar{P}_{XY}(k,\Omega_{i})$ be the mean basis spectrum measured from five
Aemulus boxes at the cosmology $\Omega_{i}$. For a given box, we define
normalized emulator residuals as
$\hat{r}^{XY}(k)=\frac{\hat{P}_{XY}(k,\Omega_{i})-P^{\mathrm{Emu}}_{XY}(k,\Omega_{i}))}{\bar{P}_{XY}(k,\Omega_{i})},$
(18)
where $P^{\mathrm{Emu}}_{XY}$ is the emulator prediction at the same
cosmology. Normalized this way, we assume the residuals are cosmology
independent. At each redshift we have 35 sets of residuals. With these
measurements we can build an estimate of the residual correlation matrix
$\mathrm{Corr}^{\mathrm{Emu}}(k,k^{\prime})=\frac{\mathrm{Cov}[\hat{r}^{XY}(k),\hat{r}^{XY}(k^{\prime})]}{\sqrt{\mathrm{Cov}(k,k)\mathrm{Cov}(k^{\prime},k^{\prime})}}$
(19)
which captures how correlated the emulator residuals are across the test set
as a function of scale. The quantities in the numerator and denominator of
Eqn. 19 are the same, but we apply the shorthand
$\mathrm{Cov}(k,k)\equiv\mathrm{Cov}[\hat{r}^{XY}(k),\hat{r}^{XY}(k)]$ to not
overload the expression. We proceed to define an emulator floor,
$f_{\mathrm{Emu}}$, specifying what fraction of the signal is of the order
emulator error. From Fig. 5, the dominant source of uncertainty will come from
the error in the $P_{11}$ spectrum. This implies $f_{\mathrm{Emu}}\simeq 0.01$
at small scales for redshifts $z>0$. We then estimate that the emulator error
will scale as
$\mathrm{Cov}^{\mathrm{Err}}(k,k^{\prime})=(f_{\mathrm{Emu}}P_{hh,hm}(k))^{2}\times\mathrm{Corr}^{\mathrm{Emu}}(k,k^{\prime}),$
(20)
where $P_{hh,hm}$ is used depending on whether we are including this
contribution to the block corresponding to the halo–halo correlation or the
halo-matter correlation. We then add this contribution in quadrature to Eq. 14
$\mathrm{Cov}(k,k^{\prime})=\mathrm{Cov}^{G}(k,k^{\prime})+\mathrm{Cov}^{\mathrm{Err}}(k,k^{\prime}).$
(21)
We run chains with the covariance in Eq. 21, as well as chains including only
the diagonal contribution due to uncertainty, which we will call the ‘floor’
covariance. The contours for cosmological parameters are shown in Fig. 15.
While this is clearly an approximate treatment, we observe that including this
contribution helps prevent significant biases in cosmological parameter
inference due to the noisy input data and very low noise assumed in the fit.
Figure 13: Comparison of our model for emulator uncertainty compared to the
disconnected component of the covariance matrix. The left panel corresponds to
$P_{hh}P_{hh}$ contribution and the right panel to $P_{hm}P_{hm}$. Figure 14:
Correlation matrix of emulator residuals described in Appendix A. We see at
small scales, past $k\simeq 0.4h{\rm Mpc}^{-1}$, the emulator residuals are
significantly correlated. Figure 15: UNIT contours with the different
covariance forms discussed in Appendix A. Chains are run with the standard
scale cuts of $k_{\rm max}=0.6\,h^{-1}{\rm Mpc}$.
## Appendix B Subsets of the bias model
A common critique of EFT-based models is that they are over-parametrized, and
can fit to any signal due to the large number of free parameters. For
perturbative Lagrangian bias models, this question has been previously
explored in the context of CMB lensing cross-correlations. In Modi et al.
(2017), it was shown that significant biases are obtained in $\sigma_{8}$ in
these analyses if one uses a simplified model with linear galaxy bias and non-
linear matter power spectra. To address whether this holds for our model, we
run a series of tests of the emulator, with differing subsets of the bias
parameters set to zero. The full set we adopt is
1. 1.
‘All $b_{i}$’s ’, the full bias parametrization.
2. 2.
‘$b_{1}$ only’, where $b_{2}=b_{s^{2}}=b_{\nabla^{2}}=0$.
3. 3.
‘$b_{1},\,b_{\nabla^{2}}$’, where $b_{2}=b_{s^{2}}=0$.
4. 4.
‘No $b_{s^{2}}$’, where $b_{s^{2}}=0$.
5. 5.
‘No $b_{2}$’, where $b_{2}=0$.
Figure 16: Posteriors for varying subsets of the bias model in Eqn. 1, for two
different scale cut configurations.
A contribution due to shot-noise is included in all of the chains. All chains
in Fig. 16 are run with the same data vector and covariance matrices, and the
$k_{\rm max}$ cuts highlighted in each row. We observe significant biases for
every subset of bias parameters, except for the complete parameterization
which recovers the input cosmological parameters as previously discussed in
section 6.3.3. This implies, at least in this simplified analysis, that the
full set of bias parameters is required to achieve unbiased inference with
this model.
To check the scale-dependence of the importance of the full parameterization,
the second row of Fig. 16 repeats this test limiting ourselves to
$k_{\mathrm{max}}=0.4h{\rm Mpc}^{-1}$. The full bias model and the subset
including only linear, quadratic and higher derivative biases perform
comparatively well.
## Appendix C The $k\to 0$ limit of the emulator
In this appendix we investigate the impact of not correctly recovering large-
scale linear growth in $N$-body simulations on the emulator, as highlighted in
§1. We implement two different forms of enforcing consistency with linear
theory at large scales:
* •
Strictly reverting to LPT at $k<k_{\rm min}$. This introduces a ‘kink’ in the
basis spectra predicted by the emulator.
* •
Extrapolating the principal component predictions out to $k<k_{\rm min}$, but
with a filter to enforce linear growth.
The filter is applied to the $\Gamma^{XY}(k)$ that we use to build the
emulator,
$\displaystyle\Gamma^{XY}(k,{\bf\Omega)}\to F(k)\Gamma^{XY}(k,{\bf\Omega}).$
(22)
With this filtering approach, we recover LPT at large scales by construction
without the discontinuity introduced by simply forcing LPT after some
transition. The functional form we adopted for $F(k)$ is
$\displaystyle
F(k)=\frac{1}{2}\left[1+\tanh\left(\alpha\frac{k-k_{*}}{k_{*}}\right)\right].$
(23)
This quantity asymptotes to 0 at large scales, ensuring the $\Gamma^{XY}$ are
0, and thus the ratios are consistent with unity. Fiducial values adopted are
$k_{*}=0.125$ and $\alpha=2.5$ but the impact is similar for other values.
Since the samples we use to test the emulator are also derived from boxes with
incorrect growth, for all figures in this paper we adopt a ‘fiducial model’
where we use the $\Gamma^{XY}$ with no corrections at large-scales. The
emulator then has large-scale growth compatible with the boxes.
If we perform a simulated likelihood analysis with the other variants that
enforce LPT at large scales we see small shifts in some cosmological
parameters away from their true values. The shifts in parameters are all less
than one $\sigma$, and one must keep in mind that the noise levels in our
analysis are quite stringent (for example, we have no shape noise in the
simulated lensing constraint). When phrased in terms of the uncertainties in
parameters obtained by recent analyses (Heymans et al., 2020), these shifts
are less than $(1/4)\,\sigma$.
Figure 17: The same chains as Fig. 10 but showing all parameters varied.
|
# Vortex solution in elliptic coordinates
Wladimir Lyra New Mexico State University, Department of Astronomy, PO Box
30001 MSC 4500, Las Cruces, NM 88001, USA
###### Abstract
Vortices (flows with closed elliptic streamlines) are exact nonlinear
solutions to the compressible Euler equation. In this contribution, we use
differential geometry to derive the transformations between Cartesian and
elliptic coordinates, and show that in elliptic coordinates a constant
vorticity flow reduces to $\dot{\mu}=0$ and $\dot{\nu}={\rm const}$ along the
streamline $\mu_{0}$ that matches the vortex eccentricity.
## 1 Introduction
Vortices are important for planet formation, theorized as favorable locations
for dust trapping (Barge & Sommeria, 1995). Crescent-shaped asymmetries have
been observed in sub-mm images of protoplanetary disks (van der Marel et al.,
2013), although their unambiguous identification vortices has been elusive. A
patch of constant vorticity follows the solution ${\bm{u}}=\Omega
y/\chi\hat{{\bm{x}}},-x\chi\hat{{\bm{y}}}$, where $(x,y)$ are the Cartesian
coordinates, $\Omega$ is a constant and $\chi=x/y>1$ is the vortex aspect
ratio. Given the elliptic streamlines, a solution in terms of elliptic
coordinates $(\mu,\nu)$ is desirable.
## 2 Elliptical coordinates
The orthogonal elliptical coordinate system is
$\displaystyle x$ $\displaystyle=$ $\displaystyle f\cosh\mu\cos\nu,$ (1)
$\displaystyle y$ $\displaystyle=$ $\displaystyle f\sinh\mu\sin\nu,$ (2)
where $f=a\epsilon$ is the focal length, $a$ the semi-major axis, and
$\epsilon$ the eccentricity. Constant $\mu$ define ellipses, constant $\nu$
define hyperbolae. The coordinates describe confocal ellipses: the focal
distance is constant, so changing $\mu$ changes not only the semimajor axis
but also the eccentricity.
### 2.1 Metric
The metric of this system is
$\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle\frac{\partial{x^{\alpha}}}{\partial{q^{i}}}\frac{\partial{x^{\beta}}}{\partial{q^{j}}}g_{\alpha\beta},$
(3) $\displaystyle=$ $\displaystyle
f^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)\,\delta_{ij}.$
where ${\bm{x}}=(x,y)$ and ${\bm{q}}=(\mu,\nu)$ are Cartesian and elliptic
coordinates; $g_{\alpha\beta}=\delta_{\alpha\beta}$ is the metric of Cartesian
space. From this transformation, it follows that the scale factors are equal
$\displaystyle h_{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{g_{\mu\mu}}=f\sqrt{\sinh^{2}\mu+\sin^{2}\nu},$ (4)
$\displaystyle h_{\nu}$ $\displaystyle=$
$\displaystyle\sqrt{g_{\nu\nu}}=f\sqrt{\sinh^{2}\mu+\sin^{2}\nu}.$ (5)
We hereafter use $h=h_{\mu}=h_{\nu}$. We also use the equivalent definition
$h=\frac{f}{\sqrt{2}}\sqrt{\cosh\,2\mu-\cos\,2\nu}.$ (6)
The derivatives with respect to the coordinates are
$\displaystyle\partial_{\mu}h$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h}\,\sinh\,2\mu,$ (7)
$\displaystyle\partial_{\nu}h$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h}\,\sin\,2\nu.$ (8)
We calculate the Christoffel symbols in non-coordinate basis
$\displaystyle\Gamma_{\hat{\alpha}\hat{\beta}\hat{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(c_{\hat{\alpha}\hat{\beta}\hat{\gamma}}+c_{\hat{\alpha}\hat{\gamma}\hat{\beta}}-c_{\hat{\beta}\hat{\gamma}\hat{\alpha}}\right),$
(9) $\displaystyle\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\gamma}}$
$\displaystyle=$ $\displaystyle
g^{\hat{\alpha}\hat{\zeta}}\Gamma_{\hat{\zeta}\hat{\beta}\hat{\gamma}},$ (10)
where
$c_{\hat{\beta}\hat{\gamma}\hat{\alpha}}=g_{\hat{\alpha}\hat{\zeta}}{c_{\hat{\beta}\hat{\gamma}}}^{\hat{\zeta}}$
are the connection coefficients, given by
$[e_{\hat{\beta}},e_{\hat{\gamma}}]={c_{\hat{\beta}\hat{\gamma}}}^{\hat{\alpha}}\partial_{\hat{\alpha}}.$
(11)
Given that $e_{\hat{\mu}}=h^{-1}\partial_{\mu}$ and
$e_{\hat{\nu}}=h^{-1}\partial_{\nu}$, we have
$\displaystyle[e_{\hat{\mu}},e_{\hat{\nu}}]$ $\displaystyle=$
$\displaystyle\frac{1}{h}\left[\frac{\partial{}}{\partial{\mu}}\left(\frac{1}{h}\frac{\partial{}}{\partial{\nu}}\right)-\frac{\partial{}}{\partial{\nu}}\left(\frac{1}{h}\frac{\partial{}}{\partial{\mu}}\right)\right]$
(12) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{4}}\left(\sin\,2\nu\frac{\partial{}}{\partial{\mu}}-\sinh\,2\mu\frac{\partial{}}{\partial{\nu}}\right)$
$\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left(\sin\,2\nu\,\partial_{\hat{\mu}}-\sinh\,2\mu\,\partial_{\hat{\nu}}\right)$
$\displaystyle=$ $\displaystyle-[e_{\hat{\nu}},e_{\hat{\mu}}].$
The connection coefficients are thus
$\displaystyle{c_{{\hat{\mu}}{\hat{\nu}}}}^{\hat{\mu}}=-{c_{{\hat{\nu}}{\hat{\mu}}}}^{\hat{\mu}}$
$\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\frac{f^{2}}{2h^{3}}\sin\,2\nu,$ (13)
$\displaystyle{c_{{\hat{\mu}}{\hat{\nu}}}}^{\hat{\nu}}=-{c_{{\hat{\nu}}{\hat{\mu}}}}^{\hat{\nu}}$
$\displaystyle=$ $\displaystyle-\frac{f^{2}}{2h^{3}}\sinh\,2\mu.$ (14)
And the Christoffel symbols are
$\displaystyle\Gamma^{\hat{\mu}}_{{\hat{\nu}}{\hat{\mu}}}=-\Gamma^{\hat{\nu}}_{{\hat{\mu}}{\hat{\mu}}}$
$\displaystyle=$ $\displaystyle\frac{f^{2}}{2h^{3}}\sin\,2\nu,$ (15)
$\displaystyle\Gamma^{\hat{\nu}}_{{\hat{\mu}}{\hat{\nu}}}=-\Gamma^{\hat{\mu}}_{{\hat{\nu}}{\hat{\nu}}}$
$\displaystyle=$ $\displaystyle\frac{f^{2}}{2h^{3}}\sinh\,2\mu.$ (16)
The elliptic and Cartesian unit vectors $\hat{e}_{i}$ and $\hat{x}_{i}$
transform according to
$\hat{e}_{i}=\frac{1}{h_{i}}\frac{\partial x_{j}}{\partial e_{i}}\hat{x_{j}},$
(17)
i.e.,
$\displaystyle\left[\begin{array}[]{c}\hat{\mu}\\\ \hat{\nu}\\\
\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}h_{\mu}^{-1}\partial_{\mu}x&{\color[rgb]{1,1,1}..}&h_{\mu}^{-1}\partial_{\mu}y\\\
h_{\nu}^{-1}\partial_{\nu}x&{\color[rgb]{1,1,1}..}&h_{\nu}^{-1}\partial_{\nu}y\\\
\end{array}\right]\left[\begin{array}[]{c}\hat{x}\\\ \hat{y}\\\
\end{array}\right]$ (24) $\displaystyle=$
$\displaystyle\frac{f}{h(\mu,\nu)}\left[\begin{array}[]{ccc}{\color[rgb]{1,1,1}-}\sinh\mu\cos\nu&{\color[rgb]{1,1,1}..}&\cosh\mu\sin\nu\\\
-\cosh\mu\sin\nu&{\color[rgb]{1,1,1}..}&\sinh\mu\cos\nu\\\
\end{array}\right]\left[\begin{array}[]{c}\hat{x}\\\ \hat{y}\\\
\end{array}\right]$ (29)
This can be written compactly as
$\hat{e}_{i}=E_{ij}\hat{x}_{j},$ (30)
where $E_{ij}$ is the elliptic rotation matrix. Its inverse is
$\displaystyle E^{-1}$ $\displaystyle=$
$\displaystyle\frac{f}{h}\left[\begin{array}[]{ccc}\sinh\mu\cos\nu&{\color[rgb]{1,1,1}..}&-\cosh\mu\sin\nu\\\
\cosh\mu\sin\nu&{\color[rgb]{1,1,1}..}&{\color[rgb]{1,1,1}-}\sinh\mu\cos\nu\\\
\end{array}\right].$ (33)
The velocity is
$u_{i}=h_{i}\dot{q}_{i}\hat{q}_{i},$ (34)
which means
$\displaystyle{\bm{u}}$ $\displaystyle=$
$\displaystyle\dot{x}\hat{x}+\dot{y}\hat{y},$ (35) $\displaystyle=$
$\displaystyle h_{\mu}\dot{\mu}\hat{\mu}+h_{\nu}\dot{\nu}\hat{\nu}.$
We can also get the velocity by the rotation matrix
$u_{{\hat{e}}_{i}}=E_{ij}u_{c_{j}},$ (36)
i.e.,
$\displaystyle u_{\hat{\mu}}=E_{11}u_{x}+E_{12}u_{y},$ (37) $\displaystyle
u_{\hat{\nu}}=E_{21}u_{x}+E_{22}u_{y}.$ (38)
Yielding the variation of the coordinate bases
$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle
fh^{-2}\left({\color[rgb]{1,1,1}-}\sinh\mu\cos\nu\,u_{x}+\cosh\mu\sin\nu\,u_{y}\right),$
(39) $\displaystyle\dot{\nu}$ $\displaystyle=$ $\displaystyle
fh^{-2}\left(-\cosh\mu\sin\nu\,u_{x}+\sinh\mu\cos\nu\,u_{y}\right).$ (40)
## 3 Vortex motion
Consider a vortex in Cartesian coordinates
$\displaystyle u_{x}$ $\displaystyle=$ $\displaystyle-\varOmega y\chi$ (41)
$\displaystyle u_{y}$ $\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\varOmega x/\chi$ (42)
We seek to transform this into elliptic coordinates. The vortex motion occurs
on ellipses of constant eccentricity, whereas the elliptic coordinate system
defines confocal ellipses of different eccentricity. An elliptic coordinate
system based on constant eccentricity (Chang & Oishi, 2010), although matching
the flow geometry, is not orthogonal, which complicates analysis (Lyra & Lin,
2013). If the streamlines matched the eccentricities of the confocal ellipses,
the velocity would everywhere reduce to $\dot{\mu}=0$ and $\dot{\nu}={\rm
const}$. However, that is not the case, as one can verify that this is not
divergenceless. In fact, there is only one streamline that obeys $\dot{\mu}=0$
and $\dot{\nu}={\rm const}$, which is the streamline of eccentricity matching
the eccentricity of the vortex. This is the particular ellipse $\mu_{0}$,
given by $\tanh\mu_{0}=\chi^{-1}$. We write the velocities as
$\displaystyle u_{x}$ $\displaystyle=$
$\displaystyle-\varOmega\,f\,\cosh\mu\sin\nu,$ (43) $\displaystyle u_{y}$
$\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\varOmega\,f\,\sinh\mu\cos\nu.$ (44)
We transform these into elliptical coordinates by the rotation matrix
$u_{{\hat{e}}_{i}}=E_{ij}u_{c_{j}}$ (45)
yielding
$\displaystyle u_{\hat{\mu}}$ $\displaystyle=$
$\displaystyle-\Omega\frac{f^{2}}{2h}\left(\frac{\cosh\,2\mu-\cosh\,2\mu_{0}}{\sinh\,2\mu_{0}}\right)\sin\,2\nu,$
(46) $\displaystyle u_{\hat{\nu}}$ $\displaystyle=$
$\displaystyle\Omega\frac{f^{2}}{2h}\left(\frac{\sinh\,2\mu}{\sinh\,2\mu_{0}}\right)(\cosh\,2\mu_{0}-\cos\,2\nu).$
(47)
The divergence is
$\displaystyle{\bm{\nabla}}\cdot{A}$ $\displaystyle=$ $\displaystyle
u^{\hat{\alpha}}_{;\hat{\alpha}}=u^{\hat{\alpha}}_{,\hat{\alpha}}+\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\alpha}}u^{\hat{\beta}},$
(49) $\displaystyle=$ $\displaystyle
u^{\hat{\alpha}}_{,\hat{\alpha}}+\frac{f^{2}}{2h^{3}}\left(u^{\hat{\mu}}\sin\,2\nu+u^{\hat{\nu}}\sinh\,2\mu\right),$
or, abandoning the co-variant formulation,
${\bm{\nabla}}\cdot{A}=\frac{1}{h^{2}}\left(\frac{\partial{hu_{\hat{\mu}}}}{\partial{\mu}}+\frac{\partial{hu_{\hat{\nu}}}}{\partial{\nu}}\right).$
(51)
we conclude that the flow is divergenceless.
Eq. (46) and Eq. (47) may seem daunting at first, but following the motion at
the ellipse of $\mu$=$\mu_{0}$ simplifies it considerably. For $\mu=\mu_{0}$,
Eq. (46) cancels. For Eq. (47), the factor in parentheses becomes unity; the
next term, given Eq. (6), is $2h/f^{2}$. Thus, for $\mu$=$\mu_{0}$, the motion
is $u_{\hat{\mu}}$=0, $u_{\hat{\nu}}$=$\varOmega\,h_{0}$. Comparing with Eq.
(35) yields
$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (52)
$\displaystyle\dot{\nu}$ $\displaystyle=$ $\displaystyle\varOmega.$ (53)
For the particular $\mu=\mu_{0}$ ellipse, the motion has constant $\mu$: a
closed elliptic streamline. The angle $\nu$ rotates uniformly. Notice that
this does not mean that the velocity itself is uniform, since $h$ depends on
$\nu$. The explicit dependency of $u_{\nu}$ on $\nu$ is
$\displaystyle u_{\nu}^{2}$ $\displaystyle=$
$\displaystyle\frac{\varOmega^{2}\,f^{2}}{2}\left(\cosh\,2\mu-\cos\,2v\right)$
(54) $\displaystyle=$
$\displaystyle\varOmega^{2}\,f^{2}\left(\sinh^{2}\mu+\sin^{2}\,v\right).$
## 4 Energy conservation
That the kinetic energy $K=u_{\nu}^{2}/2$ depends on $\nu$, a function of
time, may seem strange at first. We show that this happens because the
velocity change is compensated by a change in pressure ($p$), conserving the
total energy. The energy equation is
$\frac{\partial E}{\partial
t}=-{\bm{\nabla}}\cdot{\left[{\bm{u}}\left(E+p\right)\right]}+{\bm{F}}_{b}\cdot{\bm{u}}$
(55)
where ${\bm{F}}_{b}$ is a body force. The total energy is $E=K+\varepsilon$,
where and $\varepsilon=k_{B}T$ is the internal energy ($k_{B}$ is Boltzmann’s
constant and $T$ is the temperature). In the absence of a body force and for
constant temperature, this reduces to
$\frac{\partial}{\partial
t}\left(\frac{u^{2}}{2}\right)=-\left({\bm{u}}\cdot{\bm{\nabla}}\right)\left(\frac{u^{2}}{2}+p/\rho\right)$
(56)
therefore
$\frac{d}{dt}\left(\frac{u^{2}}{2}\right)=-\left({\bm{u}}\cdot{\bm{\nabla}}\right)p/\rho$
(57)
The enthalpy is found by Euler’s equation
$\displaystyle\partial_{x}p/\rho$ $\displaystyle=$ $\displaystyle-
u_{x}\partial_{x}u_{y}=\varOmega^{2}x$ (58) $\displaystyle\partial_{y}p/\rho$
$\displaystyle=$ $\displaystyle-u_{y}\partial_{y}u_{x}=\varOmega^{2}y$ (59)
Taking the $x$-derivative above and the $y$ below, we find
$\nabla^{2}{p/\rho}=\varOmega^{2}$; therefore
$p/\rho=\frac{1}{2}\varOmega^{2}\left(x^{2}+y^{2}\right)+C$ (60)
which is an intriguing result: an incompressible elliptical vortex produces an
axis-symmetric pressure distribution. Transforming into elliptic coordinates
and eliminating the constant
$p/\rho=\frac{1}{2}\Omega^{2}f^{2}(\cosh^{2}\mu\cos^{2}\nu+\sin^{2}\mu\sin^{2}\nu).$
(61)
Along the $\mu_{0}$ streamline, the advection reduces to the $\nu$-term
$\displaystyle-\left({\bm{u}}\cdot{\bm{\nabla}}\right)P/\rho$ $\displaystyle=$
$\displaystyle-u_{\nu}h^{-1}\partial_{\nu}p/\rho$ (62) $\displaystyle=$
$\displaystyle\frac{1}{2}\varOmega^{3}f^{2}\sin\,2\nu,$
whereas the time derivative of the kinetic energy is
$\displaystyle\frac{d}{dt}\left(\frac{u^{2}}{2}\right)$ $\displaystyle=$
$\displaystyle\Omega^{2}h\frac{dh}{dt}$ (63) $\displaystyle=$
$\displaystyle\frac{1}{2}\Omega^{3}f^{2}\sin\,2\nu.$
That the two variations match amounts to conservation of energy: along the
ellipse, the material slows down or speeds up in order to match the pressure
variation.
## 5 Euler equation in elliptical coordinates
We consider now the force balance. We use the transformations here derived to
write the Euler equation in elliptic coordinates
$\partial_{t}{\bm{u}}+\left({\bm{u}}\cdot{\bm{\nabla}}\right){\bm{u}}=-{\bm{\nabla}}{p/\rho}.$
(64)
Using covariant derivatives, this reads
$\partial_{t}u^{\hat{k}}+u^{\hat{p}}\partial_{\hat{p}}u^{\hat{k}}+\Gamma^{\hat{k}}_{\hat{m}\hat{n}}u^{\hat{m}}u^{\hat{n}}=-\partial_{\hat{k}}{p/\rho}.$
(65)
For $\mu$, the correction due to the Christoffel symbols is
$\displaystyle\Gamma^{\hat{\mu}}_{{\hat{m}}{\hat{n}}}u^{{\hat{m}}}u^{{\hat{n}}}$
$\displaystyle=$ $\displaystyle\Gamma^{{\hat{\mu}}}_{{\hat{\nu}}{\hat{\mu}}},$
(66) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[u^{{\hat{\nu}}}u^{{\hat{\mu}}}\,\sin\,2\nu-(u^{{\hat{\nu}}})^{2}\sinh\,2\mu\right].$
The same procedure for $\nu$ yields
$\displaystyle\Gamma^{{\hat{\nu}}}_{{\hat{m}}{\hat{n}}}u^{{\hat{m}}}u^{{\hat{\nu}}}$
$\displaystyle=$
$\displaystyle\Gamma^{\hat{\nu}}_{{\hat{\nu}}{\hat{\mu}}}u^{\hat{\mu}}u^{\hat{\nu}}+\Gamma^{\hat{\mu}}_{{\hat{\mu}}{\hat{\mu}}}(u^{{\hat{\mu}}})^{2},$
(67) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[u^{\hat{\nu}}u^{\hat{\mu}}\,\sinh\,2\mu-(u^{\hat{\mu}})^{2}\,\sin\,2\nu\right].$
Abandoning the co-variant notation
$\displaystyle\partial_{t}\,u_{\mu}$ $\displaystyle=$
$\displaystyle-\left(h^{-1}u_{\mu}\partial_{\mu}+h^{-1}u_{\nu}\partial_{\nu}\right)u_{\mu}-h^{-1}\partial_{\mu}{p/\rho}-\frac{f^{2}}{2h^{3}}\left(u_{\nu}u_{\mu}\,\sin\,2\nu-
u_{\nu}^{2}\sinh\,2\mu\right),$ (68) $\displaystyle\partial_{t}\,u_{\nu}$
$\displaystyle=$
$\displaystyle-\left(h^{-1}u_{\mu}\partial_{\mu}+h^{-1}u_{\nu}\partial_{\nu}\right)u_{\nu}-h^{-1}\partial_{\nu}{p/\rho}-\frac{f^{2}}{2h^{3}}\left(u_{\nu}u_{\mu}\,\sinh\,2\mu-
u_{\mu}^{2}\,\sin\,2\nu\right).$ (69)
This differs from the usual equations by the presence of the extra force
$\displaystyle{\bm{F}}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[\left(u_{\nu}u_{\mu}\,\sin\,2\nu-
u_{\nu}^{2}\sinh\,2\mu\right)\hat{\mu}+\left(u_{\nu}u_{\mu}\,\sinh\,2\mu-
u_{\mu}^{2}\,\sin\,2\nu\right)\hat{\nu}\right]$ (70)
Contracting this force with the velocity yields ${\bm{F}}\cdot{\bm{u}}=0$,
which shows that this force is inertial. For the vortical flow, again
following the $\mu_{0}$ streamline where $\dot{\mu}=0$ and
$\dot{\nu}=\varOmega$, these reduce to
$\displaystyle-\frac{\varOmega^{2}f^{2}}{2}\sinh\,2\mu_{0}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}{p/\rho}$ (71)
$\displaystyle\left(u_{\nu}\partial_{\nu}\right)u_{\nu}$ $\displaystyle=$
$\displaystyle-\partial_{\nu}{p/\rho}$ (72)
I.e, a constant centrifugal force that balances the normal pressure gradient,
and inertia in the tangential direction exchanging kinetic energy with the
pressure field. Fig 1 sketches the forces.
Figure 1: Force balance in an elliptic vortex streamline (solid line). Dotted
circles represent the pressure contours. The velocity (black arrow) is tangent
to the streamline, in the $\hat{\nu}$ direction. The pressure gradient (blue
arrow) is broken down in its $\hat{\mu}$ and $\hat{\nu}$ components (brown
arrows). The $\hat{\mu}$ component is balanced by the centrifugal force (red
arrow); the $\hat{\nu}$ component is balanced by advection.
## References
* Barge & Sommeria (1995) Barge, P., & Sommeria, J. 1995, A&A, 295, L1
* Chang & Oishi (2010) Chang, P., & Oishi, J. S. 2010, ApJ, 721, 1593, doi: 10.1088/0004-637X/721/2/1593
* Lyra & Lin (2013) Lyra, W., & Lin, M.-K. 2013, ApJ, 775, 17, doi: 10.1088/0004-637X/775/1/17
* van der Marel et al. (2013) van der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2013, Science, 340, 1199, doi: 10.1126/science.1236770
|
# Sharp Signal Detection under Ferromagnetic Ising Models
Sohom Bhattacharya Sohom Bhattacharya
Department of Statistics, Stanford University, CA, USA,
<EMAIL_ADDRESS>, Rajarshi Mukherjee Rajarshi Mukherjee
Department of Biostatistics, Harvard University, MA, USA,
<EMAIL_ADDRESS>and Gourab Ray Gourab Ray
Department of Mathematics and Statistics, University of Victoria, Vancouver,
BC, Canada,
<EMAIL_ADDRESS>
###### Abstract.
In this paper we study the effect of dependence on detecting a class of
structured signals in Ferromagnetic Ising models. Natural examples of our
class include Ising Models on lattices, and Mean-Field type Ising Models such
as dense Erdős-Rényi, and dense random regular graphs. Our results not only
provide sharp constants of detection in each of these cases and thereby
pinpoint the precise relationship of the detection problem with the underlying
dependence, but also demonstrate how to be agnostic over the strength of
dependence present in the respective models.
## 1\. Introduction
††2010 Mathematics Subject Classification: 62G10, 62G20, 60C20††Keywords and
phrases: Ising Model, Signal Detection, Structured Sparsity, Sharp Constants
Let $\mathbf{X}=(X_{1},\ldots,X_{n})^{\top}\in\\{\pm 1\\}^{n}$ be a random
vector with the joint distribution of $\mathbf{X}$ given by an Ising model
defined as:
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x}):=\frac{1}{Z(\beta,\mathbf{Q},\mathbf{\boldsymbol{\mu}})}\exp{\left(\frac{\beta}{2}\mathbf{x}^{\top}\mathbf{Q}\mathbf{x}+\boldsymbol{\mu}^{\top}\mathbf{x}\right)},\qquad\forall\mathbf{x}\in\\{\pm
1\\}^{n},$ (1)
where $\mathbf{Q}$ is an $n\times n$ symmetric and hollow matrix,
$\boldsymbol{\mu}:=(\mu_{1},\ldots,\mu_{n})^{\top}\in\mathbb{R}^{n}$ is an
unknown parameter vector to be referred to as the external magnetization
vector, $\beta\in\mathbb{R}$ is a real number usually referred to as the
“inverse temperature”, and $Z(\beta,\mathbf{Q},\mathbf{\boldsymbol{\mu}})$ is
a normalizing constant. It is clear that the pair $(\beta,\mathbf{Q})$
characterizes the dependence among the coordinates of $\mathbf{X}$, and
$X_{i}$’s are independent if $\beta\mathbf{Q}=\mathbf{0}_{n\times n}$. The
matrix $\mathbf{Q}$ will usually be associated with a certain sequence of
simple labeled graphs $\mathbb{G}_{n}=(V_{n},E_{n})$ with vertex set
$V_{n}=\\{1,\dots,n\\}$ and edge set $E_{n}\subseteq V_{n}\times V_{n}$ and
corresponding $\mathbf{Q}=|V_{n}|G_{n}/2|E_{n}|$, where $G_{n}$ is the
adjacency matrix of $\mathbb{G}_{n}$. Note that we do not absorb $\beta$ in
the matrix $\mathbf{Q}$. This is because we want to understand the effect of
the nuisance parameter $\beta$ on the inference about $\boldsymbol{\mu}$.
We are interested in testing against a collection of alternatives defined by a
class of subsets $\mathcal{C}_{s}$ of $\mathbb{R}_{+}^{n}$ each of which is of
size $s$. More precisely, given any class of subsets
$\mathcal{C}_{s}\subset\\{S\subset\mathbb{R}_{+}^{n}:|S|=s\\}$ of
$\mathbb{R}_{+}^{n}$ having size $s$ each, we consider testing the following
hypotheses
$H_{0}:\boldsymbol{\mu}=\mathbf{0}\quad{\rm vs}\quad
H_{1}:\boldsymbol{\mu}\in\Xi(\mathcal{C}_{s},A),$ (2)
where
${\Xi}(\mathcal{C}_{s},A):=\left\\{\begin{array}[]{c}\boldsymbol{\mu}\in\mathbb{R}_{+}^{n}:\mathrm{supp}(\boldsymbol{\mu})\in\mathcal{C}_{s}{\rm\
and\ }\min\limits_{i\in{\rm supp}(\boldsymbol{\mu})}\mu_{i}\geq
A\end{array}\right\\},$
and
${\rm supp}(\boldsymbol{\mu}):=\\{i\in\\{1,\ldots,n\\}:\mu_{i}\neq 0\\}.$
Thus the class of alternatives $\Xi(\mathcal{C}_{s},A)$ puts non-zero signals
on one of candidate sets in $\mathcal{C}_{s}$ where each signal set has size
$s$. Of primary interest here is to explore the effect of $(\beta,\mathbf{Q})$
in testing (2) when $\mathcal{C}_{s}$ has low complexity in a suitable sense.
In this regard, previously, [6, 4] studied the detection of block-sparse and
thick shaped signals on lattices while [1] considered general class of signals
of combinatorial nature. However these papers crucially assume independence
between the outcomes and thereby correspond to $\beta=0$ in (1) in our
context. Following up on this line of research, several other papers have also
considered detection of signals over lattices and networks (see e.g. [26, 50,
7, 45, 49, 13, 35] and references therein). However, in overwhelming majority
of the literature, the underlying networks only describe the nature of signals
– such as rectangles or thick clusters in lattices [6]. A fundamental question
however remains – “how does dependence characterized by a network modulate the
behavior of such detection problems ?” Only recently, [26] explored the effect
of dependence 111Effect of dependence in signal detection for Gaussian
outcomes has also been explored for detecting unstructured arbitrary sparse
signals [31, 32]. on such structured detection problems for stationary
Gaussian processes – with examples including linear lattices studied through
the lens of Gaussian auto-regressive observation schemes. Dependence
structures beyond Gaussian random variables are often more challenging to
analyze (due to possible lack of closed form expressions of resulting
distributions) and allow for interesting and different behavior of such
testing problems – see e.g. [42]. One of the motivations of this paper is to
fill this gap in the literature and show how dependent binary outcomes can
substantially change the results for detecting certain classes structured
signals. One of the motivations of this paper is to pinpoint the precise
effect of dependence on the behavior of such testing problems. In particular,
[42, 18] demonstrate how dependence might have a subtle effect on the minimax
separation rate of sparse testing problems. In this paper we crystallize the
effect of such dependence by going beyond optimal rates and characterizing
sharp asymptotic constant for minimax separation while testing against
suitably structured hypotheses $\mathcal{C}_{s}$.
To describe our results in the context of the model-problem pair (1)-(2) we
adopt a standard minimax framework as follows. Let a statistical test for
$H_{0}$ versus $H_{1}$ be a measurable $\\{0,1\\}$ valued function of the data
$\mathbf{X}$, with $1$ denoting rejecting the null hypothesis $H_{0}$ and $0$
denoting not rejecting $H_{0}$. The worst case risk of a test $T:\\{\pm
1\\}^{\Lambda_{n}(d)}\to\\{0,1\\}$ for testing (2) is defined as
$\displaystyle\mathrm{Risk}(T,{\Xi}(\mathcal{C}_{s},A),\beta,\mathbf{Q})$
$\displaystyle:=\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(T(\mathbf{X})=1\right)+\sup_{\boldsymbol{\mu}\in{\Xi}(\mathcal{C}_{s},A)}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(T(\mathbf{X})=0\right).$
(3)
We say that a sequence of tests $T_{n}$ corresponding to a sequence of model-
problem pair (1) and (2), to be asymptotically powerful (respectively
asymptotically not powerful) against $\Xi(\mathcal{C}_{s},A)$ if
$\limsup\limits_{n\rightarrow\infty}\mathrm{Risk}(T_{n},\Xi(\mathcal{C}_{s},A),\beta,\mathbf{Q})=0\text{
(respectively
}\liminf\limits_{n\rightarrow\infty}\mathrm{Risk}(T_{n},\Xi(\mathcal{C}_{s},A),\beta,\mathbf{Q})>0).$
(4)
The goal of the current paper is to characterize how for some low complexity
class $\mathcal{C}_{s}$, the sparsity $s$, and strength $A$ of the signal
jointly determine if there is an asymptotically powerful test, and how the
behavior changes with $(\beta,\mathbf{Q})$.
With the above framework, our main results can be summarized as follows.
1. (i)
For some classical mean-field type models we show that detecting low
complexity sets (see Theorem 1 for exact definition) has same constant of
detection for low and critical dependence and has a larger constant of minimax
separation (i.e. strictly more information theoretic hardness) for higher
dependence. Our examples naturally include the complete graph (Theorem 1),
dense Erdős-Rényi and dense random regular graphs(Theorem 3).
2. (ii)
For detecting thick rectangular signals in Ising models over lattices of
general dimensions, we present the sharp minimax separation constants (Theorem
6) for low dependence and high dependence (under a “pure phase” defined in
Section 3). In contrast to dense regular graphs, the problem has a strict
monotone increasing nature of the constant as one approaches criticality from
the low dependence direction (Lemma 7). The exact monotonic nature of the
constant in the high dependence case is not clear and is left as future
research direction.
3. (iii)
We further demonstrate how the sharp optimal tests can be obtained adaptively
over the dependence strength $\beta$ (Theorem 2 and Theorem 8).
The rest of this paper is organized as follows. In Section 2 we present our
results for detecting low complexity type signals in dense regular type
graphs. Subsequently, Section 3 considers detecting thick rectangular signals
over lattices. Finally all the proofs and associated technical lemmas are
collected in Section 6.
### 1.1. Notation
$[n]$, for $\mathbf{v}=(v_{1},\ldots,v_{d})^{T}\in\mathbb{R}^{d}$ define
$\bar{\mathbf{v}}=\frac{1}{d}\sum_{i=1}^{d}v_{i}$, for any set $A$ define
$\mathbbm{1}(\cdot\in A)$ as the indicator function for the set. For any set
$S\in\mathcal{S}$, let $\boldsymbol{\mu}_{S}(A)$ denote the vector with
$\mu_{i}=A\mathbbm{1}_{i\in S}$. Let $\mathbf{1}$ denote the vector of all
$1$s. We also let $\mathbbm{1}$ to denote generic indicator functions.
The results in this paper are mostly asymptotic (in $n$) in nature and thus
requires some standard asymptotic notations. If $a_{n}$ and $b_{n}$ are two
sequences of real numbers then $a_{n}\gg b_{n}$ (and $a_{n}\ll b_{n}$) implies
that ${a_{n}}/{b_{n}}\rightarrow\infty$ (and ${a_{n}}/{b_{n}}\rightarrow 0$)
as $n\rightarrow\infty$, respectively. Similarly $a_{n}\gtrsim b_{n}$ (and
$a_{n}\lesssim b_{n}$) implies that
$\liminf_{n\rightarrow\infty}{{a_{n}}/{b_{n}}}=C$ for some $C\in(0,\infty]$
(and $\limsup_{n\rightarrow\infty}{{a_{n}}/{b_{n}}}=C$ for some
$C\in[0,\infty)$). Alternatively, $a_{n}=o(b_{n})$ will also imply $a_{n}\ll
b_{n}$ and $a_{n}=O(b_{n})$ will imply that $\limsup_{n\rightarrow\infty}\
a_{n}/b_{n}=C$ for some $C\in[0,\infty)$). If $C>0$ then we write
$a_{n}=\Theta(b_{n})$. If $a_{n}/b_{n}\rightarrow 1$, then we say $a_{n}\sim
b_{n}$. For any $a,b\in\mathbb{Z}$, $a\leq b$, let
$[a,b]:=\\{a,a+1,\ldots,b\\}$.
## 2\. Mean-Field type Interactions
In this section, we collect our results on the testing problem (2) for some
specific examples of mean-field type models [8] such as Ising models on the
complete graph, and dense Erdős-Rényi and random regular graphs. However, for
the precise statement the upper bounds of our results we first define a class
of signals $\mathcal{C}_{s}$. This is captured by a notion of complexity
defined through the following weighted Hamming type of metric (see e.g. – [4,
6]) on $2^{[n]}$. Mathematically, for any two subsets $S_{1},S_{2}\subset[n]$
we let $\gamma(S_{1},S_{2}):=\sqrt{2}\left(1-\frac{|S_{1}\cap
S_{2}|}{\sqrt{|S_{1}||S_{2}|}}\right)$ denote their distance. Subsequently,
for any $\varepsilon>0$ we let
$|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon)|$ denote the
$\varepsilon$-covering number of $\mathcal{C}_{s}$ w.r.t. $\gamma$. Our main
result in terms of detecting signals in $\Xi(\mathcal{C}_{s},A)$ pertains to
classes of signals with suitably low complexity defined through the asymptotic
behavior of $|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon)|$. In this
regard we first provide a complete picture for all temperature regimes in the
mean-field Curie-Weiss model followed by demonstrating how similar results
might be obtained in high temperature regimes for other dense regular type
graphs.
### 2.1. Complete Graph
We begin by stating and discussing our results for the Curie-Weiss model.
###### Theorem 1.
Consider testing (2) in the model (1) with
$\mathbf{Q}_{ij}=\frac{\mathbf{1}(i\neq j)}{n}$ correspond to the complete
graph.
1. i.
Assume that $\mathcal{C}_{s}$ satisfies the following condition w.r.t. the
metric $\gamma$ for some sequence $\varepsilon_{n}\rightarrow 0$:
$\Theta(\log n)=\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|\ll
s\ll n/\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|.$
Then the following hold.
1. _a_)
For $\beta<1$, $s\ll\frac{n}{\log n}$, there exists a sequence of
asymptotically powerful test if
$\liminf_{n\rightarrow\infty}\sqrt{s}\tanh(A)(\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|)^{-1/2}>\sqrt{2}.$
(5)
2. _b_)
For $\beta=1$, the same conclusion is valid for $s\ll\frac{\sqrt{n}}{\log n}$.
3. _c_)
For $\beta>1$ and $s\ll\frac{n}{\log n}$, there exists a sequence of
asymptotically powerful test if
$\liminf_{n\rightarrow\infty}\sqrt{s}\tanh(A)(\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|)^{-1/2}>\sqrt{2}\cosh(\beta
m),$ (6)
where $m$ is the unique positive root of the equation $m=\tanh(\beta m)$.
2. ii.
Assume that, there exists a subset
$\tilde{\mathcal{C}}_{s}\subset\mathcal{C}_{s}$ of disjoint sets such that
$\Theta(\log n)=\log|\tilde{\mathcal{C}}_{s}|\ll s\ll
n/\log|\tilde{\mathcal{C}}_{s}|.$
Then the following hold.
1. _a_)
For $\beta<1$ and $s\ll\frac{n}{\log n}$, all tests are asymptotically
powerless if
$\liminf_{n\rightarrow\infty}\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}<\sqrt{2}.$
(7)
2. _b_)
For $\beta=1$, the same conclusion is valid for $s\ll\frac{\sqrt{n}}{\log n}$.
3. _c_)
For $\beta>1$ and $s\ll\frac{n}{\log n}$, no tests are asymptotically powerful
if
$\liminf_{n\rightarrow\infty}\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}<\sqrt{2}\cosh(\beta
m),$ (8)
A few remarks are in order regarding the conditions, involved optimal
procedures, and implications of Theorem 1. We first note that the upper and
lower bounds are sharp as long as there exists
$\tilde{C}_{s},\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ such that
$\log|\tilde{C}_{s}|\sim\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|$
for some $\varepsilon_{n}\rightarrow 0$. In particular, the sharp constants
match for testing a class of signals whose size is dominated (on log-scale) by
the size of a subclass of disjoint sets. Next we note that the conditions on
$s$ posited in the various parts of the theorem are actually optimal. Indeed,
as was discussed in [18], for $\beta=1$ and $s\gg\sqrt{n}$ the rate of
detection is much faster (requiring only $s\tanh(A)\gg n^{1/4}$) and the
problem might not offer a phase transition at the level of sharp constants.
Next we note that the optimal test above is based on a suitably calibrated
scanning procedure. However, the scanning procedure is somewhat different
depending on the regime of dependence. In high and critical dependence
($0\leq\beta\leq 1$) the procedure can be described as follows. For
$S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ we first define
$Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$ and our scan test rejects for large values
of
$Z_{\max}:=\max\limits_{S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}$.
In contrast, for the high dependence regime ($\beta>1$) we perform a somewhat
randomized scan test by first generating $W_{n}\sim
N(\bar{\mathbf{X}},1/(n\beta))$ and subsequently for $W_{n}>0$, rejecting
$H_{0}$ for large values of $Z_{\max}-m\sqrt{s}$ and when $W_{n}\leq 0$
rejecting $H_{0}$ for large values of $Z_{\max}+m\sqrt{s}$ 222$m:=m(\beta)$ is
the unique positive root of $m=\tanh(\beta m)$. The fact that these sequence
of tests are indeed sharp optimal is thereafter demonstrated by precise
analyses of the Type II errors (through a mean-variance control) and a
matching lower bound calculation obtained through a truncated second moment
approach. Both the analyses of the tests and the proofs for matching lower
bounds rely on the moderation deviation behavior of
$Z_{S},S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ – which are
obtained through Lemma 1. Finally, a direct analysis of the sharp constants of
detection, $\sqrt{2}$ for $\beta\leq 1$ and $\sqrt{2\cosh^{2}(\beta
m(\beta))}$ for $\beta>1$, reveals that the problem becomes harder as one
moves away from the critical dependence $\beta=1$. In particular, the sharp
constant of detection can be succinctly defined through by
$\sqrt{2\cosh^{2}(\beta m(\beta))}$ which we display below in Figure 1 to
demonstrate this phenomenon.
Figure 1. Behavior of $\sqrt{2}\cosh(m(\beta))$
To describe our next result, we note that the proof of the upper bound in
Theorem 1 assumes the knowledge of $\beta>0$. Our next result therefore
pertains to showing that the sharp rates obtained above can actually be
obtained adaptively over the knowledge of the inverse temperature $\beta>0$.
To this end we begin by noting that using a consistent of $\beta>0$ might seem
hopeless to begin with since consistent estimation of $\beta$ is not possible
when $\beta\in[0,1)$. However, when $\beta\leq 1$, our optimal test does not
depend on the specific knowledge of $\beta$. Therefore our idea can be
described as follows: (1) construct a consistent test to decide whether
$\beta\leq 1$ or $\beta>1$; (2) if the test rejects in favor of $\beta>1$ then
use a pseudo-likelihood estimator of $\beta$ under the working model of
$\boldsymbol{\mu}=0$ 333this is important since joint estimation of
$\beta,\boldsymbol{\mu}$ can be significantly harder information theoretically
and if the test return in favor of $\beta\leq 1$ then construct the $\beta$
independent optimal test in Theorem 1.
###### Theorem 2.
Theorem 1 holds for unknown $\beta>0$ if $\|\boldsymbol{\mu}\|_{\infty}=O(1)$.
The proof of Theorem 2, which is deferred to Section 6, requires the
assumption on $\boldsymbol{\mu}$ in terms of its maximal element. Although
this requirement can be relaxed to $\|\boldsymbol{\mu}\|_{\infty}=o(n/s)$, our
arguments were unable to get rid of it completely. We keep further
explorations in this regard for future research.
### 2.2. Dense Regular Graphs
The results in the Curie-Weiss model in the last section provides insight on
the possible behavior of this testing problem under mean-field type models
[8]. Here we demonstrate that this intuition of similar behavior to the Curie-
Weiss model with regard to this inferential problem is indeed true for some
specific examples of mean-field type models such as dense Erdős-Rényi and
random regular graphs, which is our first result in this direction. In
particular, we let $\mathbb{G}_{n}=(V_{n},E_{n})\sim\mathcal{G}_{n}(n,p)$
denote an Erdős-Rényi random graph with edges $E_{n}$ formed by joining pairs
of vertices $i,j\in V_{n}=\\{1,\ldots,n\\}$ independently with probability
$p\in(0,1)$. In a similar vein we let
$\mathbb{G}_{n}=(V_{n},E_{n})\sim\mathcal{G}_{n}(n,d)$ denote an randomly
drawn graph from the collection of all $d$-regular graphs on
$V_{n}=\\{1,\ldots,n\\}$.
###### Theorem 3.
The same conclusion of Theorem 1 hold for any $\beta\geq 0$ when either (i)
$\mathbf{Q}=\frac{G_{n}}{np}$ with $G_{n}$ being the adjacency matrix of
$\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,p)$ with $p=\Theta(1)$; or (ii)
$\mathbf{Q}=\frac{G_{n}}{d}$ with $G_{n}$ being the adjacency matrix of
$\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,d)$ with $d=\Theta(n)$.
A few remarks in order regarding the statement and proof of Theorem 3. First,
the result should be understood as a high probability statement w.r.t. the
randomness of the underlying Erdős-Rényi random graph. In particular, we prove
that same results as in Theorem 1 hold with probability converging to $1$
under the Erdős-Rényi measure on $\mathbb{G}_{n}$. In this regard, the
requirement of $p=\Theta(1)$ is mostly used for the $\beta>1$ case and can be
relaxed for $\beta\leq 1$ regime. However, to keep our discussions consistent
over values of $\beta$ we only consider the $p=\Theta(1)$ case. Finally, the
proof of the theorem mainly operates through careful comparison of suitable
event probabilities and partition functions under Ising models over Erdős-
Rényi and complete graphs respectively – the proofs of which can be found in
Section 6.4. We also show that the result above can be obtain without the
knowledge of $\beta$.
###### Theorem 4.
Theorem 3 holds for unknown $\beta>0$ if $\|\boldsymbol{\mu}\|_{\infty}=O(1)$.
The requirement of $\|\boldsymbol{\mu}\|_{\infty}=O(1)$ can be relaxed to
$\|\boldsymbol{\mu}\|_{\infty}=o(n/s)$. However, at this moment we have not
been able to relax this completely.
## 3\. Short Range Interactions on Lattices
To describe the detection problem for nearest neighbor interaction type
models, it is convenient to represent the points $i=1,\dots,n$ to be vertices
of $d$-dimensional hyper-cubic lattice and the underlying graph (i.e. the
graph corresponding to $\mathbf{Q}$) to be the nearest neighbor (in sense of
Euclidean distance) graph on these vertices. More precisely, given positive
integers $n,d$, we consider a growing sequence of lattice boxes of dimension
$d$ defined as
$\\{\Lambda_{n,d}\\}_{n\geq
1}=\\{[-n^{1/d},n^{1/d}]^{d}\cap\mathbb{Z}^{d}\\}_{n\geq 1}$
where $\mathbb{Z}^{d}$ denotes the d-dimensional integer lattice.
Subsequently, we consider a family of random variables defined on the vertices
of $\Lambda_{n,d}$ as $\mathbf{X}\in\\{-1,+1\\}^{\Lambda_{n,d}}$ with the
following probability mass function (p.m.f.)
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x})=\frac{1}{Z(\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{\boldsymbol{\mu}})}\exp{\left(\frac{\beta}{2}\mathbf{x}^{\top}\mathbf{Q}(\Lambda_{n,d})\mathbf{x}+\boldsymbol{\mu}^{\top}\mathbf{x}\right)},\qquad\forall\mathbf{x}\in\\{\pm
1\\}^{\Lambda_{n,d}},$ (9)
where as usual
$\mathbf{Q}(\Lambda_{n,d})=(\mathbf{Q}(\Lambda_{n,d})_{ij})_{i,j\in\Lambda_{n,d}}$
is a symmetric and hollow array (i.e. $\mathbf{Q}(\Lambda_{n,d})_{ii}=0$ for
all $i\in\Lambda_{n,d}$) with elements indexed by pairs of vertices in
$\Lambda_{n,d}$ (organized in some pre-fixed lexicographic order),
$\boldsymbol{\mu}:=(\mu_{i}:i\in\Lambda_{n,d})^{\top}\in\mathbb{R}^{\Lambda_{n,d}}$
referred to as the external magnetization vector indexed by vertices of
$\Lambda_{n,d}$ , $\beta>0$ is the “inverse temperature”, and
$Z(\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{\boldsymbol{\mu}})$ is a
normalizing constant. Note that in this notation $i,j\in\Lambda_{n,d}$ are
vertices of the $d$-dimensional integer lattice and hence correspond to
$d$-dimensional vector with integer coordinates. Therefore, using this
notation, by nearest neighbor graph will correspond to the matrix
$\mathbf{Q}_{ij}=\mathbf{Q}_{ij}(\Lambda_{n,d})=\mathbf{1}(0<\|i-j\|_{1}=1)$.
Our results in this model will be derived for any dependence other than a
critical dependence $\beta_{c}(d)$ to be defined next. The case of critical
dependence for lattices remains open even in terms of optimal rate for the
minimax separation and therefore we do not pursue the issue of optimal
constants in this regime. To describe a notion of critical temperature,
consider $\mathcal{Q}(d)$ be the sequence of matrices
$\\{\mathbf{Q}(\Lambda_{n,d})\\}_{n\geq 1}$ and define
$\displaystyle\beta_{c}(d)=\beta_{c}(\mathcal{Q}(d))=\inf\left\\{\beta>0:\lim_{h\downarrow
0}\lim_{n\rightarrow\infty}\mathbb{E}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}(h)}\left(\frac{1}{n}\sum_{i\in\Lambda_{n,d}}X_{i}\right)>0\right\\},$
where we let $\boldsymbol{\mu}(h)$ denote the vector in
$\mathbb{R}^{|\Lambda_{n,d}|}$ with all coordinates equal to $h$. The
existence and equivalence of the above notions (such as ones including
uniqueness of infinite volume measure) of critical temperature in nearest
neighbor Ising Model is a topic of classical statistical physics and we refer
the interested reader to the excellent expositions in [27, 20] for more
details. This value of $\beta_{c}(d)$ (which is known to be strictly positive
for any fixed $d\geq 1$) is referred to as the critical inverse temperature in
dimension $d$ and the behavior of the system of observations $\mathbf{X}$
changes once $\beta$ exceeds this threshold. For $d=1$, it is known from the
first work in this area [34] that $\beta_{c}(1)=+\infty$ and consequently the
Ising model in 1-dimension is said to have no phase transitions. The seminal
work of [43] provides a formula for $\beta_{c}(2)$ and obtaining an analytical
formula for $\beta_{c}(d)$ for $d\geq 3$ remains open. Consequently, results
only pertain to the existence of a strictly non-zero finite $\beta_{c}(d)$
which governs the macroscopic behavior of the system of observations
$X_{i},i\in\Lambda_{n,d}$ as $n\rightarrow\infty$. In particular, the average
magnetization $n^{-1}\sum X_{i}$ converges to $0$ in probability for
$\beta<\beta_{c}(d)$ and to an equal mixture of two delta-dirac random
variables $m_{+}(\beta)$ and $m_{-}(\beta)=-m_{+}(\beta)$, for
$\beta>\beta_{c}(d)$ ([38]). This motivates defining Ising models in pure
phases as follows.
Let $\Lambda^{\mathsf{w}}_{n,d}$ denote the graph obtained by identifying the
vertices not in $\Lambda_{n,d}$ into a single vertex $\partial\Lambda_{n,d}$
and then erasing all the self loops. Let
$\mathbf{Q}(\Lambda^{\mathsf{w}}_{n,d})$ denote the adjacency matrix
corresponding to nearest neighbour interaction in this modified graph. We
denote
$\displaystyle\mathbb{P}^{+}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x})=\mathbb{P}_{\beta,\mathbf{Q}(\Lambda^{\mathsf{w}}_{n,d}),\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x}|\partial\Lambda_{n,d}=+1),$
$\displaystyle\mathbb{P}^{-}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x})=\mathbb{P}_{\beta,\mathbf{Q}(\Lambda^{\mathsf{w}}_{n,d}),\boldsymbol{\mu}}(\mathbf{X}=\mathbf{x}|\partial\Lambda_{n,d}=-1),$
to be Ising Models (see (1)) with $+$ and $-$ boundary conditions
respectively. On the other hand
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$ is referred to as the Ising
model with _free boundary condition_. It is well known [27], that for
$0\leq\beta<\beta_{c}(d)$ the asymptotic properties of the models
$\mathbb{P}^{+}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$,
$\mathbb{P}^{-}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$, and
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$ are similar (i.e. they have
all the same infinite volume $n\rightarrow\infty$ weak limit). However, for
$\beta>\beta_{c}(d)$, the model
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$ behaves asymptotically as the
mixture of $\mathbb{P}^{+}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$ and
$\mathbb{P}^{-}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$. We only present our
result for the measure $\mathbb{P}^{+}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$ in
such cases. Although we believe that a similar result might hold for both
negative boundary condition (i.e.
$\mathbb{P}^{-}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$) as well as free boundary
condition (i.e. the original model
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}$) we do not yet have access to
a rigorous argument in this regard. In the rest of this section, we therefore
shall use the superscript “$\rm bc$” in our probability, expectation, and
variance operators (e.g. $\mathbb{P}^{\rm
bc}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}$, $\mathbb{E}^{\rm
bc}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}$, and
$\mathrm{Var}^{\rm bc}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}$)
where $\mathrm{bc}\in\\{\mathsf{f},+\\}$ and stands for “boundary condition”
referring to either the free boundary condition (when $\rm bc=\mathsf{f}$) or
plus boundary condition (when $\rm bc=+$).
To present our results in this case, we begin with describing the structure of
our alternatives. Since our model has an inherent geometry given by the
lattice structure in $d$-dimensions, it is natural to consider signals which
can be described by such geometry. Similar to one of the emblematic cases
considered in [4, 6, 49, 13, 35], here we discuss testing against block sparse
alternatives of size $s$ define by $\Xi(\mathcal{C}_{s},A)$ with
$\displaystyle\mathcal{C}_{s}=\mathcal{C}_{s,\rm
rect}:=\left\\{\prod\limits_{j=1}^{d}[a_{j}:b_{j}]\cap\Lambda_{n}(d):\
b_{j}-a_{j}=\lceil s^{1/d}\rceil\right\\}.$ (10)
Although we only present the results for sub-cube detection with equal size of
the sides , one can extend the results to detection of thick clusters (see [6]
for details) with modifications of the arguments presented here. The
development and analysis of multi-scale tests similar to those explored in [4,
49, 35] is also important. However we keep this for future research to keep
our discussions in this paper focused on understanding the main driving
principles behind the sharp constants of detection under Ising dependence.
This a class of alternatives is known to require
$\tanh(A)\asymp\sqrt{\frac{\log{n}}{s}}$ for consistent detection (see e.g.
[4, 5, 6] for independent outcomes case and [26, 18] for dependent models) and
allows a sharp transition at a level of multiplicative constants. Indeed, such
sharp constants of phase transition has been derived for either independent
outcome models [4, 5, 6] and for dependent Gaussian outcomes [26]. For our
problem, the derivation of the sharp optimal constant of detection is somewhat
subtle and we first discuss a way to formalize this asymptotic constant below.
We begin by noting that it is reasonable to believe that a sequence of optimal
test can be obtained by scanning over a suitable subclass of the potentially
signal rectangles. We define the test first to gain intuition about the sharp
constant of detection. To put ourselves in the context of notation similar to
Section 2, we use $\mathcal{C}_{s,\rm rect}$ to denote the class of all thick
rectangles of volume $s$ and for any $S\in\mathcal{C}_{s,\rm rect}$
$\displaystyle Z_{S}=\frac{1}{\sqrt{s}}\sum_{i\in
S}\left(X_{i}-\mathbb{E}^{\rm
bc}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{0}}(X_{i})\right),$
We would like to scan over a suitable subclass $\tilde{\mathcal{C}}_{s,\rm
rect}$ which captures the essential complexity of $\mathcal{C}_{s,\rm rect}$
both in terms of theoretical and computational aspects. From a theoretical
perspective, we shall require a precise understanding of the moderate
deviation behavior of $Z_{S}$ and which in turn crucially relies on
understanding the asymptotic behavior of the variance of $Z_{S}$. In
particular, it is reasonable to believe that in the “pure phase” (which is the
case for free boundary high temperature and low temperature plus boundary
case) $Z_{S}$ should asymptotically behave like a Gaussian random variable and
thus its moderate deviation exponent is characterized by its variance (See
[24, Theorem V.7.2] for a result of this nature). To justify that there is a
valid candidate for the limit of
$\mathrm{Var}^{+}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{0}}(Z_{S})$ we have
the following result which is one of the main components of this section.
###### Theorem 5.
Suppose
$\mathrm{dist}(S,\partial\Lambda_{n,d}):=\inf\limits_{j\in\partial\Lambda_{n,d},i\in
S}\|i-j\|_{1}\gg\log^{2}{n}$. Then there exists $\chi^{\mathsf{f}}(\beta)$
(for $\beta<\beta_{c}(d)$) and $\chi^{+}(\beta)$ (for $\beta>\beta_{c}(d)$)
such that
$\displaystyle\lim_{s\rightarrow\infty}\mathrm{Var}^{\mathrm{b}c}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{0}}(Z_{S})$
$\displaystyle=\chi^{\mathrm{b}c}(\beta),\quad\text{for}\quad\beta<\beta_{c}(d),\mathrm{bc}\in\\{\mathsf{f},+\\};$
(11)
$\displaystyle\lim_{s\rightarrow\infty}\mathrm{Var}^{+}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\mathbf{0}}(Z_{S})$
$\displaystyle=\chi^{+}(\beta),\quad\text{for}\quad\beta>\beta_{c}(d).$ (12)
Armed with Theorem 5 we are now ready to state the main sharp detection
threshold for detecting rectangles over lattices.
###### Theorem 6.
Suppose $\mathbf{X}\sim\mathbb{P}^{\rm
bc}_{\beta,\mathbf{Q}(\Lambda_{n,d}),\boldsymbol{\mu}}$ for $\rm
bc\in\\{\mathsf{f},+\\}$and consider testing (2) against
$\mathcal{C}_{s}=\mathcal{C}_{s,\rm rect}$. Then the following hold with $\rm
bc\in\\{\mathsf{f},+\\}$ for $\beta<\beta_{c}(d)$ and $\rm bc=+$ for
$\beta>\beta_{c}(d)$.
1. (1)
A test based on $\max_{S\in\tilde{\mathcal{C}}_{s,\rm rect}}Z_{S}$ is
asymptotically powerful if $s\gg(\log{n})^{d}$
$\liminf\tanh(A)\sqrt{s/\log{(n/s)}}>\sqrt{2\chi^{\rm bc}(\beta)}.$
2. (2)
All tests are asymptotically powerless if
$\limsup\tanh(A)\sqrt{s/\log{(n/s)}}<\sqrt{2\chi^{\rm bc}(\beta)}$.
A few remarks are in order regarding the results in Theorem 6. First, we have
not tried optimize the requirement $s\gg(\log{n})^{d}$ in our upper bound
above. Indeed, as noted in [18] one needs $s\gg\log{n}$ for any successful
detection and hence our results on sharp constants matches the requirement on
$s$ up to log-factors. Moreover, our next verifies that the susceptibility is
indeed an increasing function of $0\leq\beta<\beta_{c}(d)$.
###### Proposition 7.
$\chi(\beta)$ is differentiable and strictly monotonically increasing for
$\beta\in(0,\beta_{c}(d))$.
It is worth noting that this monotonicity without the strictness can be seen
as a consequence of the Edwards-Sokal coupling and the monotonicity in $p$ of
the FK-Ising model ([30]). Finally, we demonstrate that the results above can
be obained without the knowledge of $\beta\neq\beta_{c}(d)$.
###### Theorem 8.
Theorem 6 continues to hold for unknown $\beta\neq\beta_{c}(d)$ and $\rm
bc=\mathsf{f}$.
Although this completes the picture for non-critical temperature in nearest
neighbor Ising models over lattices, the behavior of the testing problem at
$\beta=\beta_{c}(d)$ remains open. At this moment we believe that the blessing
of getting a better rate and/or constant at criticality continues to hold for
lattices as well.
## 4\. Discussions
In this paper we have considered sharp constants of detecting structured
signals under Ising dependence. Although we have derived sharp phase
transitions in some popular classes of of both mean-field type Ising models
(at all temperatures) and nearest-neighbor Ising models on lattices (at all
non-critical temperatures) several related directions remain open. As an
immediate interesting question pertains to the Ising model on lattices and
figuring out the exact detection thresholds at the critical temperature to
complete the narrative of precise benefit of critical dependence in this
model. As was discussed in [41] this might require new ideas. Moreover, even
for non-critical temperatures it remains to explore the multi-scale procedures
for adaptive testing of thick clusters for Ising models over lattices (see
e.g. [4, 49, 35] and references therein). Moreover distributional
approximation for the test statistics used here is also a crucial direction
for the sake of improved practical applicability of our result. We keep these
questions as future research directions.
## 5\. Acknowledgements
GR is supported by NSERC 50311-57400.
## References
* Addario-Berry et al. [2010] Louigi Addario-Berry, Nicolas Broutin, Luc Devroye, Gábor Lugosi, et al. On combinatorial testing problems. _The Annals of Statistics_ , 38(5):3063–3092, 2010.
* Aizenman et al. [1987] Michael Aizenman, David J Barsky, and Roberto Fernández. The phase transition in a general class of ising-type models is sharp. _Journal of Statistical Physics_ , 47(3-4):343–374, 1987.
* Aizenman et al. [2015] Michael Aizenman, Hugo Duminil-Copin, and Vladas Sidoravicius. Random currents and continuity of ising model’s spontaneous magnetization. _Communications in Mathematical Physics_ , 334(2):719–742, 2015.
* Arias-Castro et al. [2005] Ery Arias-Castro, David L Donoho, and Xiaoming Huo. Near-optimal detection of geometric objects by fast multiscale methods. _IEEE Transactions on Information Theory_ , 51(7):2402–2425, 2005.
* Arias-Castro et al. [2008] Ery Arias-Castro, Emmanuel J Candès, Hannes Helgason, and Ofer Zeitouni. Searching for a trail of evidence in a maze. _The Annals of Statistics_ , pages 1726–1757, 2008.
* Arias-Castro et al. [2011] Ery Arias-Castro, Emmanuel J Candes, and Arnaud Durand. Detection of an anomalous cluster in a network. _The Annals of Statistics_ , pages 278–304, 2011.
* Arias-Castro et al. [2018] Ery Arias-Castro, Rui M Castro, Ervin Tánczos, and Meng Wang. Distribution-free detection of structured anomalies: Permutation and rank-based scans. _Journal of the American Statistical Association_ , 113(522):789–801, 2018.
* Basak and Mukherjee [2015] Anirban Basak and Sumit Mukherjee. Universality of the mean-field for the potts model. _Probability Theory and Related Fields_ , pages 1–44, 2015.
* Besag [1974] Julian Besag. Spatial interaction and the statistical analysis of lattice systems. _Journal of the Royal Statistical Society: Series B (Methodological)_ , 36(2):192–225, 1974.
* Bhattacharya and Mukherjee [2018a] Bhaswar B Bhattacharya and Sumit Mukherjee. Inference in ising models. _Bernoulli_ , 24(1):493–525, 2018a.
* Bhattacharya and Mukherjee [2018b] Bhaswar B Bhattacharya and Sumit Mukherjee. Inference in ising models. _Bernoulli_ , 24(1):493–525, 2018b.
* Bodineau [2003] Thierry Bodineau. Slab percolation for the ising model. _arXiv preprint math/0309300_ , 2003.
* Butucea and Ingster [2013] Cristina Butucea and Yuri I Ingster. Detection of a sparse submatrix of a high-dimensional noisy matrix. _Bernoulli_ , 19(5B):2652–2688, 2013.
* Chatterjee [2007a] Sourav Chatterjee. Estimation in spin glasses: A first step. _The Annals of Statistics_ , pages 1931–1946, 2007a.
* Chatterjee [2007b] Sourav Chatterjee. Estimation in spin glasses: A first step. _The Annals of Statistics_ , pages 1931–1946, 2007b.
* Daskalakis et al. [2019] Constantinos Daskalakis, Nishanth Dikkala, and Gautam Kamath. Testing ising models. _IEEE Transactions on Information Theory_ , 2019.
* Deb and Mukherjee [2020] Nabarun Deb and Sumit Mukherjee. Fluctuations in mean-field ising models. _arXiv preprint arXiv:2005.00710_ , 2020.
* Deb et al. [2020] Nabarun Deb, Rajarshi Mukherjee, Sumit Mukherjee, and Ming Yuan. Detecting structured signals in ising models. _arXiv preprint arXiv:2012.05784_ , 2020.
* Duminil-Copin [2016] Hugo Duminil-Copin. Random currents expansion of the ising model. _arXiv preprint arXiv:1607.06933_ , 2016.
* Duminil-Copin [2017] Hugo Duminil-Copin. Lectures on the ising and potts models on the hypercubic lattice. _arXiv preprint arXiv:1707.00520_ , 2017.
* Duminil-Copin and Tassion [2016] Hugo Duminil-Copin and Vincent Tassion. A new proof of the sharpness of the phase transition for bernoulli percolation and the ising model. _Communications in Mathematical Physics_ , 343(2):725–745, 2016.
* Duminil-Copin et al. [2017] Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion. Sharp phase transition for the random-cluster and potts models via decision trees. _arXiv preprint arXiv:1705.03104_ , 2017.
* Duminil-Copin et al. [2018] Hugo Duminil-Copin, Subhajit Goswami, and Aran Raoufi. Exponential decay of truncated correlations for the ising model in any dimension for all but the critical temperature. _arXiv preprint arXiv:1808.00439_ , 2018.
* Ellis [2006] Richard S Ellis. _Entropy, large deviations, and statistical mechanics_ , volume 1431\. Taylor & Francis, 2006.
* Ellis and Newman [1978] Richard S. Ellis and Charles M. Newman. The statistics of curie-weiss models. _Journal of Statistical Physics_ , 19(2):149–161, 1978.
* Enikeeva et al. [2020] Farida Enikeeva, Axel Munk, Markus Pohlmann, and Frank Werner. Bump detection in the presence of dependency: Does it ease or does it load? _Bernoulli_ , 26(4):3280–3310, 2020.
* Friedli and Velenik [2017] Sacha Friedli and Yvan Velenik. _Statistical mechanics of lattice systems: a concrete mathematical introduction_. Cambridge University Press, 2017.
* Ghosal and Mukherjee [2020] Promit Ghosal and Sumit Mukherjee. Joint estimation of parameters in ising model. _The Annals of Statistics_ , 48(2):785–810, 2020\.
* Grimmett [1999] Geoffrey Grimmett. _Percolation_ , volume 321 of _Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_. Springer-Verlag, Berlin, second edition, 1999. ISBN 3-540-64902-6. doi: 10.1007/978-3-662-03981-6. URL https://doi.org/10.1007/978-3-662-03981-6.
* Grimmett [2006] Geoffrey R Grimmett. _The random-cluster model_ , volume 333. Springer Science & Business Media, 2006.
* Hall and Jin [2008] Peter Hall and Jiashun Jin. Properties of higher criticism under strong dependence. _The Annals of Statistics_ , pages 381–402, 2008.
* Hall and Jin [2010] Peter Hall and Jiashun Jin. Innovated higher criticism for detecting sparse signals in correlated noise. _The Annals of Statistics_ , 38(3):1686–1732, 2010.
* Ingster et al. [2010] Yuri I Ingster, Alexandre B Tsybakov, and Nicolas Verzelen. Detection boundary in sparse regression. _Electronic Journal of Statistics_ , 4:1476–1526, 2010\.
* Ising [1925] Ernst Ising. Beitrag zur theorie des ferromagnetismus. _Zeitschrift für Physik A Hadrons and Nuclei_ , 31(1):253–258, 1925.
* König et al. [2020] Claudia König, Axel Munk, and Frank Werner. Multidimensional multiscale scanning in exponential families: Limit theory and statistical consequences. _The Annals of Statistics_ , 48(2):655–678, 2020\.
* Lebowitz [1972] Joel L Lebowitz. Bounds on the correlations and analyticity properties of ferromagnetic ising spin systems. _Communications in Mathematical Physics_ , 28(4):313–321, 1972.
* Lebowitz [1974] Joel L Lebowitz. Ghs and other inequalities. _Communications in Mathematical Physics_ , 35(2):87–92, 1974.
* Lebowitz [1977] Joel L Lebowitz. Coexistence of phases in ising ferromagnets. _Journal of Statistical Physics_ , 16(6):463–476, 1977.
* Liggett et al. [1997] Thomas M Liggett, Roberto H Schonmann, and Alan M Stacey. Domination by product measures. _The Annals of Probability_ , 25(1):71–95, 1997\.
* Martin-Löf [1973] Anders Martin-Löf. Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the ising model at low temperature. _Communications in Mathematical Physics_ , 32(1):75–92, 1973.
* Mukherjee and Ray [2019] Rajarshi Mukherjee and Gourab Ray. On testing for parameters in ising models. _arXiv preprint arXiv:1906.00456_ , 2019.
* Mukherjee et al. [2016] Rajarshi Mukherjee, Sumit Mukherjee, and Ming Yuan. Global testing against sparse alternatives under ising models. _arXiv preprint arXiv:1611.08293_ , 2016.
* Onsager [1944] Lars Onsager. Crystal statistics. i. a two-dimensional model with an order-disorder transition. _Physical Review_ , 65(3-4):117, 1944.
* Pisztora [1996] Agoston Pisztora. Surface order large deviations for Ising, Potts and percolation models. _Probab. Theory Related Fields_ , 104(4):427–466, 1996. ISSN 0178-8051. doi: 10.1007/BF01198161. URL https://doi.org/10.1007/BF01198161.
* Sharpnack et al. [2015] James Sharpnack, Alessandro Rinaldo, and Aarti Singh. Detecting anomalous activity on networks with the graph fourier scan statistic. _IEEE Transactions on Signal Processing_ , 64(2):364–379, 2015.
* Simon [1980] Barry Simon. Correlation inequalities and the decay of correlations in ferromagnets. _Communications in Mathematical Physics_ , 77(2):111–126, 1980.
* Tikhomirov and Youssef [2019] Konstantin Tikhomirov and Pierre Youssef. The spectral gap of dense random regular graphs. _The Annals of Probability_ , 47(1):362–419, 2019\.
* Vu [2005] Van H Vu. Spectral norm of random matrices. In _Proceedings of the thirty-seventh annual ACM symposium on Theory of computing_ , pages 423–430, 2005.
* Walther [2010] Guenther Walther. Optimal and fast detection of spatial clusters with scan statistics. _The Annals of Statistics_ , 38(2):1010–1033, 2010.
* Zou et al. [2017] Shaofeng Zou, Yingbin Liang, and H Vincent Poor. Nonparametric detection of geometric structures over networks. _IEEE Transactions on Signal Processing_ , 65(19):5034–5046, 2017.
## 6\. Proofs of Main Results
We divide the proofs our main results according to Sections 2 and 3.
### 6.1. Proof of Results in Section 2.1
#### 6.1.1. Proof of Theorem 1
##### Proof of Theorem 1i.
We divide our proof according to the various parts of the theorem.
##### Proof of Theorem 1i.i._a_)
Recall from the discussion following the statement of Theorem 1 that the
claimed optimal test is given the by the scan test which can be described as
follows. For $S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ we
first define $Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$ and our scan test rejects for
large values of
$Z_{\max}:=\max\limits_{S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}$.
The cut-off for the test is decided by the moderate deviation behavior of
$Z_{S}$’s given in Lemma 1 – which implies that for any $\delta>0$, the test
given by
$T_{n}(\delta)=\mathbf{1}\left\\{Z_{\max}>\sqrt{2(1+\delta)\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\right\\}$
has Type I error converging to $0$.
Turning to the Type II error, consider any
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\in\Xi(\mathcal{C}_{s},A)$ and
note that by monotonicity arguments (i.e. stochastic increasing nature of the
distribution of $Z_{\max}$ as a function of coordinates of $\boldsymbol{\mu}$)
it is enough to restrict to the case where
$A=\Theta(\sqrt{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}})$.
Thereafter, note that by GHS inequality(cf. 15) one has
$\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i\in\tilde{S}^{\star}}X_{i})=O(s)$
for $\beta<1$ since $s\ll\frac{n}{\log{n}}$ by appealing to [18, Lemma 9(a)].
As a result,
$\frac{\sum_{\i\in\tilde{S^{\star}}}(X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i}))}{\sqrt{s}}=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)$.
Therefore, as usual it is enough to show that there exists $\delta>0$ such
that
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
To show this end, first let $S^{\star}\in\mathcal{C}_{s}$ be such that the
signal lies on $S^{\star}$ i.e. for all $i\in S^{\star}$ one has
$\boldsymbol{\mu}_{i}\geq A$. Note that by monotonicity arguments it is enough
to consider $\boldsymbol{\mu}_{i}=A$. By definition of covering, we can find a
$\tilde{S}^{\star}\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ such
that $\gamma\left(\tilde{S}^{\star},S^{\star}\right)\leq\varepsilon_{n}$ i.e.
$|\tilde{S}^{\star}\cap S^{\star}|\geq s(1-\varepsilon_{n}/\sqrt{2})$.
Thereafter note that by Lemma 20 we have that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\geq
A|\tilde{S}^{\star}\cap S^{\star}|-A^{2}\sum_{i\in\tilde{S}^{\star}\cap
S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j}),$
However, by [18, Lemma 9 (a)] we have that
$\displaystyle A^{2}\sum_{i\in\tilde{S}^{\star}\cap S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j})$
$\displaystyle\lesssim\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}{s}\left(\frac{|\tilde{S}^{\star}\cap
S^{\star}||S^{\star}|}{n}+|\tilde{S}^{\star}\cap S^{\star}|\right)$
$\displaystyle\leq\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|s}}{{n}}+\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll
As.$
Consequently, we immediately have that there exists $\epsilon>0$ such that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)$
$\displaystyle\geq
As(1+o(1))\geq\sqrt{2(1+\epsilon)s\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$
Therefore, we can conclude that for any $\delta<\epsilon$ we have
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
This completes the proof of the upper bound for $0<\beta<1$.
##### Proof of Theorem 1i.i._b_)
An optimal test is given the by the scan test which can be described similar
to the case $\beta<1$. For
$S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$, define
$Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$ and reject for large values of
$Z_{\max}:=\max\limits_{S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}$.
Lemma 1 implies that for any $\delta>0$, the test given by
$T_{n}(\delta)=\mathbf{1}\left\\{Z_{\max}>\sqrt{2(1+\delta)\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\right\\}$
has Type I error converging to $0$ whenever $s\ll\sqrt{n}/\log{n}$.
Turning to the Type II error, consider any
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\in\Xi(\mathcal{C}_{s},A)$ and
note that by monotonicity arguments it is enough to restrict to the case where
$A=\Theta(\sqrt{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}})$.
Thereafter, note that by GHS inequality(cf. 15) one has
$\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i\in\tilde{S}^{\star}}X_{i})=O(s)$
even for $\beta=1$ since $s\ll\frac{\sqrt{n}}{\log{n}}$ by appealing to [18,
Lemma 9(c)]. As a result,
$\frac{\sum_{\i\in\tilde{S^{\star}}}(X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i}))}{\sqrt{s}}=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)$.
Therefore, as usual it is enough to show that there exists $\delta>0$ such
that
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
To show this end, first let $S^{\star}\in\mathcal{C}_{s}$ be such that the
signal lies on $S^{\star}$ i.e. for all $i\in S^{\star}$ one has
$\boldsymbol{\mu}_{i}\geq A$. Note that by monotonicity arguments it is enough
to consider $\boldsymbol{\mu}_{i}=A$. By definition of covering, we can find a
$\tilde{S}^{\star}\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ such
that $\gamma\left(\tilde{S}^{\star},S^{\star}\right)\leq\varepsilon_{n}$ i.e.
$|\tilde{S}^{\star}\cap S^{\star}|\geq s(1-\varepsilon_{n}/\sqrt{2})$.
Thereafter note that by Lemma 20 we have that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\geq
A|\tilde{S}^{\star}\cap S^{\star}|-A^{2}\sum_{i\in\tilde{S}^{\star}\cap
S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j}),$
However, by [18, Lemma 9 (c)] we have that
$\displaystyle A^{2}\sum_{i\in\tilde{S}^{\star}\cap S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j})$
$\displaystyle\lesssim\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}{s}\left(\frac{|\tilde{S}^{\star}\cap
S^{\star}||S^{\star}|}{\sqrt{n}}+|\tilde{S}^{\star}\cap S^{\star}|\right)$
$\displaystyle\leq\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|s}}{\sqrt{n}}+\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll
As.$
Consequently, we immediately have that there exists $\epsilon>0$ such that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)$
$\displaystyle\geq
As(1+o(1))\geq\sqrt{2(1+\epsilon)s\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$
Therefore, we can conclude that for any $\delta<\epsilon$ we have
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
This completes the proof of the upper bound for $\beta=1$.
##### Proof of Theorem 1i.i._c_)
We use a randomized scan test here described as follows: Given data
$X_{i},i\in[n]$, generate a random variable $W_{n}\sim
N(\bar{\mathbf{X}},1/(n\beta))$. If $W_{n}>0$, we reject the null hypotheses
if
$Z_{\max}:=\max\limits_{S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}>m\sqrt{s}+t_{n}(\delta)$.
If $W_{n}\leq 0$ we reject if $Z_{\max}>-m\sqrt{s}+t_{n}(\delta)$, where
$m:=m(\beta)$ is the unique positive root of $m=\tanh(\beta m)$ and
$t_{n}(\delta)=\sqrt{2(1+\delta)(1-m^{2})\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$.
It turns out that our analysis of this test works whenever
$\sum_{i=1}^{n}\mu_{i}\ll n$. This is however not an issue since the test
based on conditionally centered $\sum_{i=1}^{n}X_{i}$ works as soon as
$\sum_{i=1}^{n}\mu_{i}\gg\sqrt{n}$ [42] – and hence simple Bonferroni
correction (i.e. the test rejects when either the randomized test described
above or the the conditionally centered sum based test of [42]) yields our
desired result. Hence we focus here on the case when
$\sqrt{n}\ll\sum_{i=1}^{n}\mu_{i}\ll n^{3/4}$.
By the moderate deviation behavior of $Z_{S}$’s given in Lemma 1, the Type-I
error converges to $0$.
Turning to the Type II error, consider any
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\in\Xi(\mathcal{C}_{s},A)$.
Now, let $S^{*}$ be the rectangle with non-zero signal with the true signal
set under the alternative. We choose $\delta$ with
$\tanh(A)=\sqrt{2(1+\delta)^{2}(1-m^{2})^{-1}\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}/s}$.
By definition of covering, we can find a
$\tilde{S}^{\star}\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ such
that $\gamma\left(\tilde{S}^{\star},S^{\star}\right)\leq\varepsilon_{n}$ i.e.
$|\tilde{S}^{\star}\cap S^{\star}|\geq s(1-\varepsilon_{n}/\sqrt{2})$. We want
to control
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(A_{n,\tilde{S}^{\star}}\cup
B_{n,\tilde{S}^{\star}})$ where
$\displaystyle A_{n,S}$
$\displaystyle=\\{Z_{S}>m\sqrt{s}+t_{n}(\delta),W_{n}>0\\},$ $\displaystyle
B_{n,S}$ $\displaystyle=\\{Z_{S}>-m\sqrt{s}+t_{n}(\delta),W_{n}\leq 0\\}.$
(13)
We have to show
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(A_{n,\tilde{S}^{\star}}\cup
B_{n,\tilde{S}^{\star}})\rightarrow 1.$ (14)
We plan to show
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(A_{n,\tilde{S}^{\star}})-\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(W_{n}>0)\rightarrow
0$ and
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(B_{n,\tilde{S}^{\star}})-\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(W_{n}\leq
0)\rightarrow 0$. We prove of the first limit and the proof of the second one
is similar.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(A_{n,\tilde{S}^{\star}})=\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\mathbf{1}(W_{n}>0)\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left[\bar{X}_{S}>m(\beta)+t_{n}(\delta)/\sqrt{s}|W_{n}\right]\right)$
$\displaystyle\leq\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\mathbf{1}(W_{n}>0)\underbrace{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left[\frac{\sum_{i\in
S}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum\limits_{i\in
S}X_{i}|W_{n})}{\sqrt{s}}>\sqrt{s}(m-\tanh(\beta
W_{n}+A))+t_{n}(\delta)|W_{n}\right]}_{G_{n}(W_{n})}\right)$
We claim that
$G_{n}(W_{n})\mathbf{1}(W_{n}>0)-\mathbf{1}(W_{n}>0)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}0$.
This will imply
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(A_{n})-\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(W_{n}>0)\rightarrow
0$ by DCT. To show the in probability convergence of
$G_{n}(W_{n})\mathbf{1}(W_{n}>0)$ we note that for any $\epsilon>0$
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\mathbf{1}(W_{n}>0)(1-G_{n}(W_{n}))>\epsilon)$
$\displaystyle\leq\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(G_{n}(W_{n})<1-\epsilon,W_{n}>0).$
We therefore need to understand $G_{n}(W_{n})$ on the event $W_{n}>0$. By
Lemma 4,
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(|W_{n}-m|>\zeta_{n},W_{n}>0)\rightarrow
0$ for some slowly decreasing $\zeta_{n}$. More specifically, this follows
from 4 since this lemma not only implies concentration of $W_{n}$ around
$m_{n}$ in $\sqrt{n}$-scale and but also that
$m_{n}-m=\Theta(\frac{1}{n}\sum_{i=1}^{n}\mu_{i})=o(n^{-1/4})$ by the property
of $\boldsymbol{\mu}$. Next fix $\eta>0$ such that if
$W_{n}>0,|W_{n}-m|\leq\zeta_{n}$ for some decreasing
$\frac{1}{\sqrt{n}}\ll\zeta_{n}\rightarrow\ll\frac{1}{n^{1/4}}$,
$\operatorname{sech}^{2}(\beta
W_{n}+A)\leq(1+\eta)^{2}\operatorname{sech}^{2}(\beta m)$ and
$\sqrt{s}(m-\tanh(\beta W_{n}+A))\leq-(1-\eta)\operatorname{sech}^{2}(\beta
m)A\sqrt{s}$. Hence, on the event $W_{n}>0,|W_{n}-m|\leq\zeta_{n}$ we have
$\displaystyle\sqrt{\frac{s}{\operatorname{sech}^{2}(\beta
W_{n}+A)}}(m-\tanh(\beta
W_{n}+A))+\frac{t_{n}(\delta)}{\operatorname{sech}^{2}(\beta W_{n}+A)}$
$\displaystyle\leq-\sqrt{2\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\left[\frac{(1-\eta)(1+\delta)}{1+\eta}-1\right]$
Choosing $\eta$ small enough we have the last display
$\Theta(-\sqrt{\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|})$
and the result follows by Chebyshev’s Inequality. The same proof goes through
for $W_{n}\leq 0$ by appealing to Lemma 4.
##### Proof of Theorem 1ii.
We divide our proof according to the various parts of the theorem.
##### Proof of Theorem 1ii.ii._a_)
We follow the path of truncated second moment method with respect to a
suitable prior over $\tilde{\mathcal{C}}_{s}\subset\mathcal{C}_{s}$. Owing to
the exchangeability of the Curie-Weiss model it is natural to consider the
uniform prior $\pi$ (say) over all $\boldsymbol{\mu}\in\Xi(\mathcal{C}_{s},A)$
such $\mathrm{supp}(\boldsymbol{\mu})\in\tilde{C}_{s}$ and
$\boldsymbol{\mu}_{i}=A$ for $i\in\mathrm{supp}(\boldsymbol{\mu})$. The
likelihood ratio $L_{\pi}$ (say) corresponding to this prior can be written as
$\displaystyle
L_{\pi}=\frac{1}{|\tilde{C}_{s}|}\sum_{S\in\tilde{C}_{s}}\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\mathbf{0})}\exp\left(A\sum_{i\in
S}X_{i}\right),$
where $\boldsymbol{\mu}_{S}(A)$ is the vector with support $S$ and entries
equal to $A$ on its support. Now recall the definition of $Z_{S}$ from the
proof of Theorem 1i.i._a_) define an event
$\Omega_{n}(S,\delta)=\\{Z_{S}\leq\tilde{t}_{n}(\delta)\\}$ – where for any
$\delta>0$ we define
$\tilde{t}_{n}(\delta)=\sqrt{2(1+\delta)\log|\tilde{C}_{s}|}$. Subsequently,
we let
$\displaystyle\tilde{L}_{\pi}(\delta):=\frac{1}{|\tilde{C}_{s}|}\sum_{S\in\tilde{C}_{s}}\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\mathbf{0})}\exp\left(A\sum_{i\in
S}X_{i}\right)\mathbf{1}(\mathbf{X}\in\Omega_{n}(S,\delta))$ (15)
denote a truncated likelihood ratio at level $\delta>0$. It is thereafter
enough the verify that there exists $\delta>0$ such that the following three
claims hold [33]
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(L_{\pi}\neq\tilde{L}_{\pi}(\delta))$
$\displaystyle\rightarrow 0;$ (16)
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(\tilde{L}_{\pi}(\delta))$
$\displaystyle\rightarrow 1;$ (17)
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(\tilde{L}_{\pi}^{2}(\delta))$
$\displaystyle\leq 1+o(1).$ (18)
We now show them in sequence. To show (16) note that
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(L_{\pi}\neq\tilde{L}_{\pi}(\delta))=\sum_{S\in\tilde{C}_{s}}\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(\frac{1}{\sqrt{s}}\sum_{i\in
S}X_{i}>\tilde{t}_{n}(\delta)\right)\rightarrow 0.$
The convergence to $0$ in the display above follows from Lemma 1 for any
$\delta>0$ – by the same verbatim argument that showed the Type I error
convergence to $0$ in the proof of Theorem 1i.i._a_).
Next we turn to (17). To verify this, we first note that by a simple change of
measure
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(\tilde{L}_{\pi}(\delta))=\frac{1}{|\tilde{\mathcal{C}}_{s}|}\sum_{S\in\tilde{C}_{s}}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}\left(\frac{1}{\sqrt{s}}\sum_{i\in
S}X_{i}\leq\tilde{t}_{n}(\delta)\right).$
To analyze the R.H.S of the display above note that
$\frac{1}{\sqrt{s}}\sum_{i\in
S}(X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)})$ is tight
since $\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(\sum_{i\in
S}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i\in
S}X_{i})=O(s)$ by G.H.S inequality. Also, by arguments similar to the control
of Type II error in the proof of Theorem 1i.i._a_) and the fact that
$\limsup_{n\rightarrow\infty}\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}<\sqrt{2}$,
we have that for any $\delta>0$ that
$\tilde{t}_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(\sum_{i\in
S}X_{i})\rightarrow\infty$ uniformly in $S\in\tilde{\mathcal{C}}_{s}$.
Therefore by Chebyshev’s Inequality we conclude that
$1-\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(\tilde{L}_{\pi}(\delta))\rightarrow
0$ for any $\delta>0$.
Finally we shall make a choice of $\delta>0$ while verifying (18). First note
that since $\tilde{\mathcal{C}}_{s}$ consists of disjoint sets we have
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}\left(\tilde{L}_{\pi}^{2}(\delta)\right)$
$\displaystyle=\frac{1}{|\tilde{\mathcal{C}}_{s}|^{2}}\sum_{S\in\tilde{\mathcal{C}}_{s}}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A))}{Z^{2}(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))Z(\beta,\mathbf{Q},\mathbf{0})}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}(\Omega_{n}(S,\delta))$
$\displaystyle+\frac{1}{|\tilde{\mathcal{C}}_{s}|^{2}}\sum_{S_{1}\neq
S_{2}\in\tilde{\mathcal{C}}_{s}}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}}(A))Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{2}}(A))Z(\beta,\mathbf{Q},\mathbf{0})}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A)}(\Omega_{n}(S_{1},\delta)\cap\Omega_{n}(S_{2},\delta))$
$\displaystyle=T_{1}+T_{2}\quad\text{(say),}$ (19)
We will first show that $T_{2}\leq 1+o(1)$. To see this note that
$\displaystyle T_{2}\leq\frac{1}{|\tilde{\mathcal{C}}_{s}|^{2}}\sum_{S_{1}\neq
S_{2}\in\tilde{\mathcal{C}}_{s}}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}}(A))Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{2}}(A))Z(\beta,\mathbf{Q},\mathbf{0})}=1+o(1),$
by the proof for the control of the second term in equation (9) of [18,
Theorem 3] along with the correlation bounds presented in [18, Lemma 9(a)]. By
symmetry, the value of $T_{1}$ equals
$T_{1}=\frac{1}{|\tilde{\mathcal{C}}_{s}|}\frac{Z^{2}_{n}(\beta,0)}{Z^{2}_{n}(\beta,s,A)}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A))}{Z^{2}(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))Z(\beta,\mathbf{Q},\mathbf{0})}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}(\Omega_{n}(S,\delta)).$
(20)
Let $\varepsilon>0$ small enough such that
$\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}=2(1-\varepsilon)$. Set
$\lambda>0$ such that
$\tanh(2A-\lambda/\sqrt{s})=\tilde{t}_{n}(\delta)/\sqrt{s}$. Note,
$\cosh(2A)\leq(1-\frac{(t_{n}(\delta)+\lambda)^{2}}{s})^{-1/2}$. Therefore,
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}\Big{(}Z_{S}\leq\tilde{t}_{n}(\delta)\Big{)}$
$\displaystyle\leq
e^{\lambda\tilde{t}_{n}(\delta)}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}\Big{(}\exp{(-\frac{\lambda}{\sqrt{s}}\sum\limits_{i\in
S}X_{i})}\Big{)}$
$\displaystyle=(1+o(1))e^{\lambda\tilde{t}_{n}(\delta)}\Big{(}\frac{\cosh(2A-\frac{\lambda}{\sqrt{s}})}{\cosh(2A)}\Big{)}^{s}$
$\displaystyle\leq(1+o(1))e^{\lambda\tilde{t}_{n}(\delta)}e^{\frac{\tilde{t}^{2}_{n}(\delta)}{2}}\frac{1}{\cosh^{s}(2A)}$
$\displaystyle\leq(1+o(1))\exp\left(\lambda\tilde{t}_{n}(\delta)+\frac{\tilde{t}^{2}_{n}(\delta)}{2}-\frac{(\lambda+t_{n}(\delta))^{2}}{2}\right)$
(21) $\displaystyle=(1+o(1))\exp(-\frac{\lambda^{2}}{2}).$ (22)
where the first equality is by Lemma 3. Choose $\eta>0$ small enough such
that,
$2\sqrt{\frac{1-\varepsilon}{1+\delta}}(1-\eta)\frac{\tilde{t}_{n}(\delta)}{\sqrt{s}}\leq
2\sqrt{(1-\varepsilon)}(1-\eta)\tanh(A)\leq\tanh(2A)\leq\frac{\lambda+\tilde{t}_{n}(\delta)}{\sqrt{s}}.$
Hence,
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}\left(Z_{S}\leq\tilde{t}_{n}(\delta)\Big{)}\leq(1+o(1))\exp\Big{(}-(2\sqrt{\frac{1-\varepsilon}{1+\delta}}(1-\eta)-1)^{2}\frac{\tilde{t}^{2}_{n}(\delta)}{2}\right).$
(23)
By (20),
$\displaystyle T_{1}$
$\displaystyle=(1+o(1))\exp\Big{(}s\\{\log\cosh(2A)-2\log\cosh(A)\\}-\log|\tilde{\mathcal{C}}_{s}|-(2\sqrt{\frac{1-\varepsilon}{1+\delta}}(1-\eta)-1)^{2}\frac{\tilde{t}^{2}_{n}(\delta)}{2}\Big{)}$
$\displaystyle\leq(1+o(1))\exp\Big{(}sA^{2}-\log|\tilde{\mathcal{C}}_{s}|-(2\sqrt{\frac{1-\varepsilon}{1+\delta}}(1-\eta)-1)^{2}\frac{\tilde{t}^{2}_{n}(\delta)}{2}\Big{)}=o(1),$
(24)
for small enough $\eta>0$. This completes the proof of (18), finishing the
proof of lower bound.
##### Proof of Theorem 1ii.ii._b_)
The proof is verbatim same as that for $\beta<1$ in part Theorem 1ii.ii._a_)
and the only change comes from applying Lemma 1 which is applicable now for
$s\ll\sqrt{n}/\log{n}$ – a condition that is part of the theorem’s assumption.
##### Proof of Theorem 1ii.ii._c_)
To prove the lower bound, fix $\varepsilon>0$ such that
$\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}=2(1-\varepsilon)(1-m^{2})^{-1}$.
Defining
$L_{\pi}=\frac{1}{|\tilde{C}_{s}|}\sum_{S\in\tilde{C}_{s}}\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\mathbf{0})}\exp\left(A\sum_{i\in
S}X_{i}\right),$ (25)
it suffices to show $L_{\pi}\rightarrow 1$ in probability. Defining
$\Omega_{n}(S,\delta)=\\{(A_{n,S}\cup B_{n,S})^{c}\\}$ – where for any
$\delta>0$ we define
$\tilde{t}_{n}(\delta)=\sqrt{2(1+\delta)\log|\tilde{C}_{s}|}$. Subsequently,
we let
$\displaystyle\tilde{L}_{\pi}(\delta):=\frac{1}{|\tilde{C}_{s}|}\sum_{S\in\tilde{C}_{s}}\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\mathbf{0})}\exp\left(A\sum_{i\in
S}X_{i}\right)\mathbf{1}(\mathbf{X}\in\Omega_{n}(S,\delta)),$ (26)
and it is enough to show (16),(17),(18). The proof of type I error in Theorem
1i.i._c_) yields $\mathbb{P}(L_{\pi}\neq\tilde{L}_{\pi})\rightarrow 0$. The
proof of (17) follows similar to the Type II error of in Theorem 1i.i._c_). To
prove (18), recall that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}\left(\tilde{L}_{\pi}^{2}(\delta)\right)$
(27)
$\displaystyle=\frac{1}{|\tilde{\mathcal{C}}_{s}|^{2}}\sum_{S\in\tilde{\mathcal{C}}_{s}}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A))}{Z^{2}(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))Z(\beta,\mathbf{Q},\mathbf{0})}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}(\Omega_{n}(S,\delta))$
$\displaystyle+\frac{1}{|\tilde{\mathcal{C}}_{s}|^{2}}\sum_{S_{1}\neq
S_{2}\in\tilde{\mathcal{C}}_{s}}\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}}(A))Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{2}}(A))Z(\beta,\mathbf{Q},\mathbf{0})}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A)}(\Omega_{n}(S_{1},\delta)\cap\Omega_{n}(S_{2},\delta))$
$\displaystyle=T_{1}+T_{2},$ (28)
Here, $T_{2}\leq 2+o(1)$, because
$\frac{Z^{2}(\beta,\mathbf{Q},\mathbf{0})Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}\cup
S_{2}}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{1}}(A))Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S_{2}}(A))Z(\beta,\mathbf{Q},\mathbf{0})}=(2+o(1))\frac{\cosh^{2s}(\beta
m+A)+\cosh^{2s}(\beta m-A)}{(\cosh^{s}(\beta m+A)+\cosh^{s}(\beta
m-A))^{2}}=2+o(1),$
where we have used $\frac{\cosh^{s}(\beta m-A)}{\cosh^{s}(\beta m+A)}=o(1)$.
To bound $T_{1}$, choose $\eta>0$ small to be specified later. Set
$\lambda=\frac{\tilde{t}_{n}(\delta)\sqrt{s}(1-\eta)}{1-m^{2}}$.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(2A)}(Z_{S}\leq\sqrt{s}+\tilde{t}_{n}(\delta)|W_{n}>0)$
$\displaystyle\leq\exp\Big{(}{\lambda(m+\tilde{t}_{n}(\delta)/\sqrt{s})+s[\log\cosh(\beta
m+2A-\frac{\lambda}{s})-\log\cosh(\beta m+2A)]}\Big{)}$
$\displaystyle\leq\exp\Big{(}\lambda(m-\tanh(\beta
m+2A)+\frac{\tilde{t}_{n}(\delta)}{\sqrt{s}})+\frac{\lambda^{2}}{2s}\operatorname{sech}^{2}(\beta
m)+o(\log|\tilde{C}_{s}|)\Big{)}$
$\displaystyle\leq\exp\Big{(}-\frac{\lambda\tilde{t}_{n}(\delta)}{\sqrt{s}}(1-\eta)+\frac{\lambda^{2}}{2s}\operatorname{sech}^{2}(\beta
m)+o(\log|\tilde{C}_{s}|)\Big{)}$
$\displaystyle=\exp\Big{(}-(1+\varepsilon)(1-\eta)^{2}\log|\tilde{C}_{s}|+o(\log|\tilde{C}_{s}|)\Big{)},$
where the third inequality is by $\tanh(\beta m+2A)\geq
m+2A\operatorname{sech}^{2}(\beta m)$ and small $\eta>0$. Similar bound holds
for $W_{n}<0$. Therefore,
$\displaystyle T_{1}$ $\displaystyle\leq C\exp\Big{(}s[\log\cosh(\beta
m+2A)-2\log\cosh(\beta m+A)+\log\cosh(\beta
m)]-\log|\tilde{C}_{s}|-(1+\varepsilon)(1-\eta)^{2}\log|\tilde{C}_{s}|\Big{)}$
$\displaystyle\leq
C\exp\Big{(}sA^{2}-\log|\tilde{C}_{s}|-(1+\varepsilon)(1-\eta)^{2}\log|\tilde{C}_{s}|\Big{)}=o(1),$
by small $\varepsilon,\eta>0$ and since $\tanh(A)\sim A$. Therefore,
$\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(\tilde{L}^{2}_{\pi})\leq 2+o(1)$.
Hence,
$\liminf_{n\rightarrow\infty}\inf_{T}\mathrm{Risk}(T,{\Xi}(\mathcal{C}_{s},A),\beta,\mathbf{Q})\geq
1-\limsup_{n\rightarrow\infty}\frac{1}{2}\sqrt{\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}(L^{2}_{\pi})-1}>0.$
Consequently, no test is asymptotically powerful.
#### 6.1.2. Proof of Theorem 2
To obtain an adaptive test, we first test the hypothesis
$H_{0,\beta}:\beta\leq 1$ vs $H_{1,\beta}:\beta>1$. This is obtained through
the test
$T_{n}^{(1)}=\mathbbm{1}\left(|\bar{\mathbf{X}}|\geq\frac{1}{\log{n}}\right).$
Subsequently, if $T_{n}^{(1)}=0$ we simply use the test
$T_{n}^{(2)}=\mathbbm{1}\left(Z_{\max}>\sqrt{2(1+\delta)\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\right)$
for $\delta>0$ as in the proof of Theorem 1i.i._a_). In contrast, if
$T_{n}^{(1)}=1$ we estimate $\beta$ assuming the working model
“$\mathbf{X}\stackrel{{\scriptstyle\text{working}}}{{\sim}}\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}$”
using the Pseudo-likelihood method [9, 15, 10, 28] and with $\hat{\beta}_{\rm
working}$ denote this estimator we consider rejecting $H_{0}$ using
$\displaystyle T_{n}^{(3)}$
$\displaystyle=\mathbbm{1}(Z_{\max}>m(\hat{\beta}_{\rm
working})+\sqrt{2(1+\delta)(1-m^{2}(\hat{\beta}_{\rm
working}))\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|},W_{n}\geq
0)$ $\displaystyle+\mathbbm{1}(Z_{\max}>-m(\hat{\beta}_{\rm
working})+\sqrt{2(1+\delta)(1-m^{2}(\hat{\beta}_{\rm
working}))\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|},W_{n}<0)$
where $\delta>0$ can be chosen as in the proof of Theorem Theorem 1i.i._c_)
and $m(\beta)$ is the unique positive root of $m=\tanh(\beta m)$. Our final
$\beta$-adaptive test is thereafter given by
$T_{n}=(1-T_{n}^{(1)})T_{n}^{(2)}+T_{n}^{(1)}T_{n}^{(3)}$. We note that in
this part of the proof we do not perform the Bonferroni correction of the test
$T_{n}^{(3)}$ with a test based on conditionally centered version of
$\sum_{i=1}^{n}X_{i}$ since here $\sum_{i=1}^{n}\mu_{i}\ll s\ll n$. Therefore
as it is clear from the proof of Theorem 1i.i._c_), it suffices to only
consider the randomized scan test described through $T_{n}^{(3)}$.
We first show that uniformly over all
$\boldsymbol{\mu}\in\\{\mathbf{0}\\}\cup\Xi(\mathcal{C}_{s},A)$ one has that
$|\bar{\mathbf{X}}|=o_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(\frac{1}{\log{n}})$
for $\beta\leq 1$ and $|\bar{\mathbf{X}}|\gg\frac{1}{\log{n}}$ with
probability converging to $1$ for $\beta>1$. For the first claim, first let
$\beta\leq 1$. Then from Lemma 5 we have that
$|\bar{\mathbf{X}}|=o_{\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}}(\frac{1}{\log{n}})$
for $\beta\leq 1$. Now note that by Lemma 20 we have that there exists
constants $C_{1},C_{2}>0$ such that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}\right)\leq
C_{1}\sum_{i=1}^{n}\tanh(\mu_{i})+C_{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\tanh(\mu_{j})\left(\mathbf{Q}_{i,j}+\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\right)$
Now suppose $\beta<1$. Then we have from [18] that
$0\leq\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\lesssim
1/n$. Hence for $\beta<1$ we have
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}\right)\lesssim\sum_{i=1}^{n}\tanh(\mu_{i})\sum_{i=1}^{n}\sum_{j=1}^{n}\tanh(\mu_{j})\left(\mathbf{Q}_{i,j}+\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\right)\lesssim\sum_{j=1}^{n}\tanh(\mu_{j})\lesssim
s$
since $\|\boldsymbol{\mu}\|_{0}\leq s$. Hence under any $\boldsymbol{\mu}$ and
$\beta<1$ one has
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(1)}=1)$
$\displaystyle=\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}>\frac{n}{\log{n}}\right)+\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}<-\frac{n}{\log{n}}\right)$
$\displaystyle\leq\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}>\frac{n}{\log{n}}\right)+\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(\sum_{i=1}^{n}X_{i}<-\frac{n}{\log{n}}\right)\quad\text{by
monotonicity of}\ \sum_{i=1}^{n}X_{i}$
$\displaystyle=\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\frac{\sum_{i=1}^{n}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}>\frac{n}{\log{n}\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}-\frac{\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}\right)+o(1)$
$\displaystyle\leq\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\frac{\sum_{i=1}^{n}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}>C_{1}\frac{\sqrt{n}}{\log{n}}-C_{2}\frac{s}{\sqrt{n}}\right)+o(1)$
where in the last line we have used the fact that by GHS inequality (Lemma 15)
and Lemma 5 we have that
$n\lesssim\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i=1}^{n}X_{i})\lesssim
n$ for $\beta<1$. Now note that by our assumptions
$s/\sqrt{n}\ll\sqrt{n}/\log{n}$ since $s\ll n/\log{n}$ and hence the first
term of the display above goes to $0$ uniformly in $\boldsymbol{\mu}$ for
$\beta<1$.
Turning to $\beta=1$, we have from [18] that
$0\leq\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\lesssim
1/\sqrt{n}$. Hence for $\beta=1$ we have
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})\lesssim\sum_{i=1}^{n}\tanh(\mu_{i})\sum_{i=1}^{n}\sum_{j=1}^{n}\tanh(\mu_{j})\left(\mathbf{Q}_{i,j}+\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\right)\lesssim\sum_{j=1}^{n}\tanh(\mu_{j})\lesssim
s\sqrt{n}$
since $\|\boldsymbol{\mu}\|_{0}\leq s$. Hence under any $\boldsymbol{\mu}$ and
$\beta=1$ one has
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(1)}=1)$
$\displaystyle=\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}>\frac{n}{\log{n}}\right)+\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}<-\frac{n}{\log{n}}\right)$
$\displaystyle\leq\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i=1}^{n}X_{i}>\frac{n}{\log{n}}\right)+\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(\sum_{i=1}^{n}X_{i}<-\frac{n}{\log{n}}\right)\quad\text{by
monotonicity of}\ \sum_{i=1}^{n}X_{i}$
$\displaystyle=\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\frac{\sum_{i=1}^{n}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}>\frac{n}{\log{n}\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}-\frac{\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}\right)+o(1)$
$\displaystyle\leq\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\frac{\sum_{i=1}^{n}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}{\sqrt{\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})}}>C_{1}\frac{n^{1/4}}{\log{n}}-C_{2}\frac{s}{n^{1/4}}\right)+o(1)$
where in the last line we have used the fact that by GHS inequality (Lemma 15)
and Lemma 5 we have that
$n^{3/4}\lesssim\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i=1}^{n}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i=1}^{n}X_{i})\lesssim
n^{3/4}$ for $\beta<1$. Now note that by our assumptions $s/n^{1/4}\ll
n^{1/4}/\log{n}$ since $s\ll\sqrt{n}/\log{n}$ and hence the first term of the
display above goes to $0$ uniformly in $\boldsymbol{\mu}$ for $\beta=1$ as
well. This shows that for any $\boldsymbol{\mu}$ one has that $T_{n}^{(1)}$
has Type I error converging to $0$ for testing $H_{0,\beta}$ vs $H_{1,\beta}$.
Next we show that uniformly over all
$\boldsymbol{\mu}\in\\{\mathbf{0}\\}\cup\Xi(\mathcal{C}_{s},A)$
$|\bar{\mathbf{X}}|\gg\frac{1}{\log{n}}$ with probability converging to $1$
for $\beta>1$. The claim is trivial for $\boldsymbol{\mu}=\mathbf{0}$ by [25].
In particular, it also follows that
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0)\leq e^{-Cn}$ for some
$C>0$ depending on $\beta>1$. Then for any $\boldsymbol{\mu}$ with
$\|\boldsymbol{\mu}\|_{\infty}\leq M$ for some $M>0$ one has
$\displaystyle\sup_{\boldsymbol{\mu}:\|\boldsymbol{\mu}\|_{\infty}\leq
M}\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0)$
$\displaystyle=\sup_{\boldsymbol{\mu}:\|\boldsymbol{\mu}\|_{\infty}\leq
M}\frac{Z(\beta,\mathbf{Q},\mathbf{0})}{Z(\beta,\mathbf{Q},\boldsymbol{\mu})}\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}\left(\mathbbm{1}(T_{n}^{(1)}=0)e^{\sum_{i=1}^{n}\mu_{i}X_{i}}\right)$
$\displaystyle=\sup_{\boldsymbol{\mu}:\|\boldsymbol{\mu}\|_{\infty}\leq
M}e^{2s\|\boldsymbol{\mu}\|_{\infty}}\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0)\leq
e^{2sM-Cn}\rightarrow 0$
since $s\ll n$. This completes the proof for $\beta>1$ for the consistency of
testing $H_{0,\beta}$ vs $H_{1,\beta}$ using $T_{n}^{(1)}$ under any
$\boldsymbol{\mu}\in\mathbb{R}_{+}^{n}$ s.t. $\|\boldsymbol{\mu}\|_{0}\leq
s\ll n$ and $\|\boldsymbol{\mu}\|_{\infty}=O(1)$. We next turn to the
consistency of the test
$T_{n}=(1-T_{n}^{(1)})T_{n}^{(2)}+T_{n}^{(1)}T_{n}^{(3)}$ for testing the
actual hypotheses (2) of interest.
First consider the case $\beta\leq 1$. Let us first consider the Type I error
of the test $T_{n}$.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}=1)=\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0,T_{n}^{(2)}=1)+\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1,T_{n}^{(3)}=1).$
By consistency of the test $T-n^{(1)}$ for testing $H_{0,\beta}$ vs
$H_{1,\beta}$ one has that for $\beta\leq 1$ one has
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1)\rightarrow 0$. Hence
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1,T_{n}^{(3)}=1)\rightarrow
0$ as well. Hence it is enough to show that
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(2)}=1)\rightarrow 0$ under
$\beta\leq 1$. However this follows by arguments verbatim to the proof of Type
I errors in Theorem 1i.i._a_) and Theorem 1i.i._b_) since the test
$T_{n}^{(2)}$ is free of $\beta$. This completes the control of Type I error
for $\beta\leq 1$. Turning to the Type II error, once again note that for any
$\boldsymbol{\mu}\in\mathcal{C}_{s}$
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}=0)=\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0,T_{n}^{(2)}=0)+\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1,T_{n}^{(3)}=0).$
Once again by consistency of the test $T-n^{(1)}$ for testing $H_{0,\beta}$ vs
$H_{1,\beta}$ (uniformly against any $\boldsymbol{\mu}$ in our class having
$\|\boldsymbol{\mu}\|_{\infty}=O(1)$) one has that for $\beta\leq 1$ one has
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(1)}=1)\rightarrow 0$
uniformly in such $\boldsymbol{\mu}$’s. Hence
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1,T_{n}^{(3)}=1)\rightarrow
0$ uniformly as well and is enough to show that
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(2)}=0)\rightarrow 0$
uniformly in $\boldsymbol{\mu}$ under $\beta\leq 1$. This once again follows
by arguments verbatim to the proof of Type II errors in Theorem 1i.i._a_) and
Theorem 1i.i._b_) since the test $T_{n}^{(2)}$ is free of $\beta$.
We next turn to the final case of analyzing the test $T_{n}$ under $\beta>1$.
Let us first consider the Type I error of the test $T_{n}$.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}=1)=\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0,T_{n}^{(2)}=1)+\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=1,T_{n}^{(3)}=1).$
By consistency of the test $T_{n}^{(1)}$ for testing $H_{0,\beta}$ vs
$H_{1,\beta}$ one has that for $\beta>1$ one has
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0)\rightarrow 0$. Hence
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(1)}=0,T_{n}^{(2)}=1)\rightarrow
0$ as well. Hence it is enough to show that
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(T_{n}^{(3)}=1)\rightarrow 0$ under
$\beta>1$. Now note that the only dependence of the test $T_{n}^{(3)}$ on
$\hat{\beta}_{\rm working}$ is in its cutoff and the proof of the Type I error
in Theorem 1i.i._c_) shows that it is enough to show that $m(\hat{\beta}_{\rm
working})\stackrel{{\scriptstyle\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}}}{{\rightarrow}}m(\beta)$
which follows from Lemma 19. In particular, Lemma 19 is applicable since
$\liminf_{n\rightarrow\infty}\frac{1}{n}\log\frac{1}{2^{n}}Z(\beta,\mathbf{Q}^{\rm
CW},\mathbf{0})>0,$ (29)
using [11, (7.9)]. Hence, by Lemma 19, is applicable. Turning to the Type II
error of our test $T_{n}$ once again by consistency of the test $T_{n}^{(1)}$
for testing $H_{0,\beta}$ vs $H_{1,\beta}$ (uniformly against any
$\boldsymbol{\mu}$ in our class having $\|\boldsymbol{\mu}\|_{\infty}=O(1)$)
one has that for $\beta>1$ one has
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(1)}=0)\rightarrow 0$
uniformly in such $\boldsymbol{\mu}$’s. Hence
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(1)}=0,T_{n}^{(2)}=0)\rightarrow
0$ uniformly as well and is enough to show that
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(T_{n}^{(2)}=0)\rightarrow 0$
uniformly in $\boldsymbol{\mu}$ under $\beta>1$. Once again from the proof of
the Type II error in Theorem 1i.i._c_) it is enough to show that
$m(\hat{\beta}_{\rm
working})\stackrel{{\scriptstyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}}{{\rightarrow}}m(\beta)$
uniformly in $\boldsymbol{\mu}$ in our class which follows from Lemma 19.
### 6.2. Technical Lemmas for Proofs of Theorems in Section 2.1
###### Lemma 1.
Suppose $\mathbf{X}\sim\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}}$ with $\mathbf{Q}^{\rm CW}_{i,j}=\mathbf{1}(i\neq j)/n$
and let $Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$. Define
$t_{n}(\delta):=\sqrt{2(1+\delta)(1-m^{2})\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$
for some $\delta>0$, where $m:=m(\beta)$ is the unique positive root of
$m=\tanh(\beta m)$.
1. (a)
If $\beta<1,s\ll\frac{n}{\log n}$, $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(Z_{S}>t_{n}(\delta))\leq(1+o(1))e^{-t^{2}_{n}(\delta)}$.
2. (b)
If $\beta=1,s\ll\frac{\sqrt{n}}{\log n}$, then same conclusion as (a) holds.
3. (c)
If $\beta>1,s\ll\frac{n}{\log n}$,
$P_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(Z_{S}>m\sqrt{s}+t_{n}(\delta))\leq(1+o(1))\exp\left(-\frac{t^{2}_{n}(\delta)}{2(1-m^{2})}\right)$
(30)
Further,
$P_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(Z_{S}>-m\sqrt{s}+t_{n}(\delta),W_{n}<0)\leq(1+o(1))\exp\left(-(1+o(1))\frac{t^{2}_{n}(\delta)}{2(1-m^{2})}\right)$
(31)
###### Proof.
1. (a)
Pick $\lambda>0$ such that
$\tanh\left(\frac{\lambda}{\sqrt{s}}\right)=\frac{t_{n}(\delta)}{\sqrt{s}}$.
So, $\lambda\geq t_{n}(\delta)$, and
$\cosh\left(\frac{\lambda}{\sqrt{s}}\right)=\left(1-\frac{t^{2}_{n}(\delta)}{s}\right)^{-1/2}.$
Also,
$s\tanh\left(\frac{\lambda}{\sqrt{s}}\right)=\sqrt{st_{n}(\delta)}=O(\sqrt{s\log|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|})\ll\sqrt{n}$.
Hence,
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW}\mathbf{0}}(Z_{S}>t_{n}(\delta)$ $\displaystyle\leq e^{-\lambda
t_{n}(\delta)}\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(\frac{\lambda}{\sqrt{s}}))}{Z(\beta,\mathbf{Q}^{\rm
CW},\mathbf{0})}$ $\displaystyle=(1+o(1))e^{-\lambda
t_{n}(\delta)}\cosh^{s}\left(\frac{\lambda}{\sqrt{s}}\right)\leq(1+o(1))e^{-t^{2}_{n}(\delta)},$
where the equality is due to Lemma 2.
2. (b)
When $\beta=1$, we pick same $\lambda$. Now,
$s\tanh\left(\frac{\lambda}{\sqrt{s}}\right)\ll n^{1/4}$ and the bound is
obtained using Lemma 2 again and the $o(1)$ term does not depend on the choice
of $S$.
3. (c)
When $\beta>1$, pick $\lambda=\frac{t_{n}(\delta)\sqrt{s}}{1-m^{2}}$. Here,
$s\tanh(\lambda/s)\ll\sqrt{n}$ and $\lambda/s\rightarrow 0$.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(Z_{S}>m\sqrt{s}+t_{n}(\delta))$
$\displaystyle\leq\exp\left(-\frac{\lambda t}{\sqrt{s}}-\lambda
m\right)\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}\exp\left(\frac{\lambda}{s}\sum_{i\in S}X_{i}\right)$
$\displaystyle=\exp\left(-\frac{\lambda t}{\sqrt{s}}-\lambda
m\right)\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(\frac{\lambda}{s}))}{Z(\beta,\mathbf{Q}^{\rm
CW},\mathbf{0})}$ $\displaystyle\leq(1+o(1))\exp\left(-\frac{\lambda
t}{\sqrt{s}}-\lambda m\right)\left(\frac{\cosh(\beta m+\lambda/s)}{\cosh(\beta
m)}\right)^{s}$ $\displaystyle=(1+o(1))\exp\left(-\frac{\lambda
t}{\sqrt{s}}-\lambda m+s\log\cosh(\beta m+\lambda/s)-s\log\cosh(\beta
m)\right)$ $\displaystyle\leq(1+o(1))\exp\left(-\frac{\lambda
t}{\sqrt{s}}\underbrace{-\lambda m(\beta)+\frac{s\lambda}{s}\tanh(\beta
m(\beta))}_{=0}+\frac{s\lambda^{2}}{2s^{2}}\operatorname{sech}^{2}(\beta
m(\beta))\right)$ $\displaystyle=(1+o(1))\exp\left(-\frac{\lambda
t}{\sqrt{s}}+\frac{\lambda^{2}}{2s}\operatorname{sech}^{2}(\beta
m(\beta))\right)$
$\displaystyle=(1+o(1))\exp\left(-\frac{t^{2}}{2(1-m^{2})}\right),$
where the second inequality is due to Lemma 2 and the third inequality occurs
since the third term of the Taylor expansion is negative.
When $W_{n}<0$, note that,
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(Z_{S}>-m\sqrt{s}+t_{n}(\delta),W_{n}<0)$
$\displaystyle\leq\exp\left(-\frac{\lambda t}{\sqrt{s}}+\lambda
m\right)\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}\exp\left(\frac{\lambda}{s}\sum_{i\in
S}X_{i}\mathbbm{1}_{W_{n}\leq 0}\right)$
$\displaystyle\leq(1+o(1))\exp\left(-\frac{\lambda
t_{n}(\delta)}{\sqrt{s}}+\lambda m\right)\left(\frac{\cosh(\beta
m-\lambda/s)}{\cosh(\beta m)}\right)^{s}$
Previously, the third term in Taylor expansion was $<0$, here we simply bound
it by $O\left(\frac{t^{3}_{n}(\delta)}{s}\right)=o(t^{2}_{n}(\delta))$.
∎
The following Lemma provides estimates for ratios of normalizing constants in
the Curie-Weiss model.
###### Lemma 2.
For $s\in[n]$ and $A,\beta>0$ let $Z_{n}(\beta,s,A)=Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))$ be the partition function of the Curie-Weiss
model $\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\boldsymbol{\mu}_{S}(A)}$ with
$\mathbf{Q}^{\rm CW}_{i,j}=\mathbbm{1}(i\neq j)/n$ Then the following
conclusions hold for any $C>0$:
1. (a)
If $\beta<1,s\ll\frac{n}{\log n}$ and $s\tanh(A)\ll\sqrt{n}$, then we have
$\displaystyle\frac{Z_{n}(\beta,s,A)}{Z_{n}(\beta,0,0)}=(1+o(1))\cosh(A)^{s}.$
(32)
2. (b)
If $\beta>1,s\ll\frac{n}{\log n}$ and $s\tanh(A)\ll\sqrt{n}$ we have
$\mathbb{E}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}\left(\exp(A\sum_{i\in
S}X_{i})|W_{n}>0\right)=(1+o(1))\frac{\cosh(\beta m+A)^{s}}{\cosh(\beta
m)^{s}},$ $\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}\left(\exp(A\sum_{i\in
S}X_{i})|W_{n}<0\right)=(1+o(1))\frac{\cosh(\beta m-A)^{s}}{\cosh(\beta
m)^{s}},$
where $W_{n}$ is the auxilliary variable for the Curie-Weiss model (cf. [42,
lemma 3]). Therefore,
$\displaystyle\frac{Z_{n}(\beta,s,A)}{Z_{n}(\beta,0,0)}=(1+o(1))\frac{1}{2}\left(\frac{\cosh(\beta
m-A)^{s}}{\cosh(\beta m)^{s}}+\frac{\cosh(\beta m+A)^{s}}{\cosh(\beta
m)^{s}}\right)$ (33)
where $m$ is the unique positive root of the equation $m=\tanh(\beta m)$.
3. (c)
If $\beta=1$ and $s\ll\frac{n}{\log n}$, then for $s\tanh(A)\ll n^{1/4}$ we
have (32) holds.
###### Proof.
We begin with the representation used in [42, lemma 3]. Define a random
variable $W_{n}$ which given $\mathbf{X}$ has a distribution
$N(\bar{\mathbf{X}},1/n)$. Then under $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}$, given $W_{n}=w_{n}$, each $X_{i}$’s are i.i.d with
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(X_{i}=x_{i}|W_{n}=w_{n})=\frac{e^{\beta w_{n}x_{i}}}{e^{\beta
w_{n}}+e^{-\beta w_{n}}}.$
Therefore, for any $\mu_{i}\in\mathbb{R}$ one has
$\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(\exp(\mu_{i}X_{i})|W_{n})=\frac{\cosh(\mu_{i}+\beta
W_{n})}{\cosh(\beta W_{n})}$. Therefore for any
$\boldsymbol{\mu}\in\mathbb{R}^{n}$
$\displaystyle\frac{Z_{n}(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu})}{Z_{n}(\beta,\mathbf{Q}^{\rm CW},\mathbf{0})}$
$\displaystyle=\frac{\sum_{\mathbf{x}\in\\{\pm
1\\}^{n}}\exp\left(\frac{\beta}{n}\sum_{i<j}x_{i}x_{j}+\sum_{i=1}^{n}\mu_{i}x_{i}\right)}{\sum_{\mathbf{x}\in\\{\pm
1\\}^{n}}\exp\left(\frac{\beta}{n}\sum_{i<j}x_{i}x_{j}\right)}=\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}\left(\exp(\sum_{i=1}^{n}\mu_{i}X_{i})\right)$
$\displaystyle=\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}\left[\exp\left\\{\sum_{i=1}^{n}\left(\log\cosh(\mu_{i}+\beta
W_{n})-\log\cosh(\beta W_{n})\right)\right\\}\right].$ (34)
Consequently we have
$\displaystyle\frac{Z_{n}(\beta,s,A)}{Z_{n}(\beta,0,0)}=\frac{Z_{n}(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}{Z_{n}(\beta,\mathbf{Q}^{\rm
CW},\mathbf{0})}=\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(e^{T_{n}}),\quad T_{n}:=s[\log\cosh(\beta
W_{n}+A)-\log\cosh(\beta W_{n})].$ (35)
Also use [42, Lemma 3] to note that $W_{n}$ marginally has a density
proportional to $e^{-nf(w)}$, with $f(w):=\beta w^{2}/2-\log\cosh(\beta w)$.
With this we are ready to prove the lemma.
1. (a)
To begin note that $|\log\cosh(\beta W_{n})|\leq\frac{\beta^{2}}{2}W_{n}^{2}$.
Also, a two term Taylor’s series expansion in $W_{n}$ gives
$\displaystyle\Big{|}\log\cos(\beta W_{n}+A)-\log\cosh(A)-\beta
W_{n}\tanh(A)\Big{|}\leq$ $\displaystyle\frac{\beta^{2}}{2}W_{n}^{2}.$
Combining these two and setting $B:=\tanh(A)$ we have
$|T_{n}-s\log\cosh(A)|\leq\beta|W_{n}|sB+\beta^{2}sW_{n}^{2},$
the RHS of which converges to $0$ in probability, as
$\sqrt{n}W_{n}=O_{\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}}(1)$ by
part (a) of Lemma 5. Thus to prove (32), using uniform integrability it
suffices to show that
$\displaystyle\limsup_{n\rightarrow\infty}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2T_{n}-2s\log\cosh(A)}<\infty.$ (36)
To this effect, again using the above Taylor’s expansion gives
$T_{n}-s\log\cosh(A)\leq\beta s|W_{n}|B+s\beta^{2}W_{n}^{2},$
using which the RHS (36) can be bounded as follows:
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2T_{n}-2s\log\cosh(A)}\leq\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2\beta sB|W_{n}|+2s\beta^{2}W_{n}^{2}}\leq
2\mathbb{E}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}e^{2\beta
sBW_{n}+2s\beta^{2}W_{n}^{2}}$
Setting $\lambda_{1}:=\beta(1-\beta),\lambda_{2}:=\beta$ we have
$\lambda_{1}\frac{w^{2}}{2}\leq f(w)\leq\lambda_{2}\frac{w^{2}}{2},$
which readily gives
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}e^{2\beta
sW_{n}B+s\beta^{2}W_{n}^{2}}\leq$ $\displaystyle
2\frac{\int_{\mathbb{R}}e^{2\beta
sBz+2s\beta^{2}z^{2}-n\lambda_{1}z^{2}/2}dz}{\int_{\mathbb{R}}e^{-n\lambda_{2}z^{2}/2}dz}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{n\lambda_{2}}{n\lambda_{1}-4\beta^{2}s}}\text{exp}\Big{\\{}\frac{2\beta^{2}s^{2}B^{2}}{n\lambda_{1}-4\beta^{2}s}\Big{\\}},$
the RHS of which converges to $1$. Thus (36) holds, and so the proof is
complete.
2. (b)
To begin, setting $C_{n}:=\int_{\mathbb{R}}e^{-nf(w)}dw$ we have
$\displaystyle C_{n}\mathbb{E}e^{T_{n}}=$
$\displaystyle\int_{-\infty}^{0}\prod_{i=1}^{n}e^{\log\cosh(\beta
w+\mu_{i})-\log\cosh(\beta
w)}e^{-nf(w)}dw+\int_{0}^{\infty}\prod_{i=1}^{n}e^{\log\cosh(\beta
w+\mu_{i})-\log\cosh(\beta w)}e^{-nf(w)}dw$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\prod_{i=1}^{n}e^{\log\cosh(-\beta
w+\mu_{i})-\log\cosh(-\beta
w)}e^{-nf(w)}dw+\int_{0}^{\infty}\prod_{i=1}^{n}e^{\log\cosh(\beta
w+\mu_{i})-\log\cosh(\beta w)}e^{-nf(w)}dw$ $\displaystyle\leq$ $\displaystyle
2\int_{0}^{\infty}\prod_{i=1}^{n}e^{\log\cosh(\beta w+\mu_{i})-\log\cosh(\beta
w)}e^{-nf(w)}dw,$
where the last inequality uses the fact that
$\displaystyle\log\cosh(\beta w+\mu_{i})-\log\cosh(\beta w)=$
$\displaystyle\int_{0}^{\mu_{i}}\tanh(\beta w+z)dz$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\mu_{i}}\tanh(-\beta w+z)dz=\log\cosh(-\beta
w+\mu_{i})-\log\cosh(-\beta w),$
as $\tanh$ is monotone increasing. This shows that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}e^{T_{n}}$
$\displaystyle=\frac{1}{2}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(e^{T_{n}}|W_{n}>0)+\frac{1}{2}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(e^{T_{n}}|W_{n}<0).$
We will now show that,
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(e^{T_{n}}|W_{n}>0)=(1+o(1))\frac{\cosh(\beta
m+A)^{s}}{\cosh(\beta m)^{s}}.$ (37)
A similar argument takes care of the second term. For this, use part (b) of
Lemma 5 to note that $\Big{(}\sqrt{n}(W_{n}-m)|W_{n}>0\Big{)}=O_{p}(1)$. For
$W_{n}>0$, a Taylor’s series expansion gives
$\displaystyle\Big{|}\log\cosh(\beta W_{n}+A)-\log\cosh(\beta
m+A)-\beta(W_{n}-m)\tanh(\beta m+A)\Big{|}$ $\displaystyle\leq$
$\displaystyle\frac{\beta^{2}}{2}(W_{n}-m)^{2}$
$\displaystyle\Big{|}\log\cosh(\beta W_{n})-\log\cosh(\beta
m)-\beta(W_{n}-m)\tanh(\beta m)\Big{|}$ $\displaystyle\leq$
$\displaystyle\frac{\beta^{2}}{2}(W_{n}-m)^{2},$
which on taking a difference gives
$\Big{|}T_{n}-s(\log\cosh(\beta m+A)-\log\cosh(\beta m))\Big{|}\leq
s\tilde{B}|W_{n}-m|+\beta^{2}s(W_{n}-m)^{2},$
where $\tilde{B}:=|\tanh(\beta m+A)-\tanh(\beta m)|$. The RHS in the display
above converges to $0$ in probability, conditioned on the event $T_{n}>0$. As
before, (37) will follow from this via uniform integrability if we can show
that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2T_{n}-2s\log\cosh(\beta m+A)+2s\log\cosh(\beta m)}<\infty.$
(38)
For showing (38), again use the above Taylor’s expansion to note that
$T_{n}-s\log\cosh(\beta m+A)+s\log\cosh(\beta m)\leq
s\tilde{B}|W_{n}-m|+\beta^{2}s(W_{n}-m)^{2}.$
We now claim that on $[0,\infty)$ the function $f(w)$ satisfies
$\displaystyle\frac{\lambda_{2}}{2}(w-m)^{2}\leq
f(w)-f(m)\leq\frac{\lambda_{1}}{2}(w-m)^{2}$ (39)
for some positive constants $\lambda_{1}>\lambda_{2}$. Given the claim, a
similar calculation as in part (a) shows that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(e^{T_{n}}|W_{n}>0)\leq$ $\displaystyle
2\frac{\int_{\mathbb{R}}e^{2\beta
s\tilde{B}z+2s\beta^{2}z^{2}-n\lambda_{1}z^{2}/2}dz}{\int_{-m}^{\infty}e^{-n\lambda_{2}z^{2}/2}dz}$
$\displaystyle=$
$\displaystyle(1+o(1))\sqrt{\frac{n\lambda_{2}}{n\lambda_{1}-4\beta^{2}s}}\text{exp}\Big{\\{}\frac{2\beta^{2}s^{2}\tilde{B}^{2}}{n\lambda_{1}-4\beta^{2}s}\Big{\\}},$
which converges to $1$ as before, thus verifying (38) and hence completing the
proof of the Lemma.
It thus remains to prove (39). To this effect, note that the function $f(.)$
on $[0,\infty)$ has a unique global minima at $z=m$, and
$f^{\prime\prime}(m)>0$. Thus the function $F:[0,\infty)\mapsto\mathbb{R}$
defined by
$F(w)=\frac{f(w)-f(m)}{2(w-m)^{2}},z\neq m,\quad F(m):=f^{\prime\prime}(m)$
is continuous and strictly positive on $[0,\infty)$, and
$\lim_{t\rightarrow\infty}F(t)=\beta_{2}>0$. Thus setting
$\lambda_{2}:=\inf_{t\in[0,\infty)}F(t)$,
$\lambda_{1}:=\sup_{t\in[0,\infty)}F(t)$, it follows that $\lambda_{2}>0$ and
$\lambda_{1}<\infty$, and so the proof of (39) is complete.
3. (c)
We now use a four term Taylor expansion to get
$\displaystyle\Big{|}\log\cosh(W_{n}+A)-\log\cosh(A)-\frac{W_{n}^{2}}{2}\text{sech}^{2}(A)|$
$\displaystyle\leq$ $\displaystyle
B|W_{n}|+\frac{1}{2}B|W_{n}|^{3}+\frac{1}{4}|W_{n}|^{4}$
$\displaystyle\Big{|}\log\cosh(W_{n})-\frac{W_{n}^{2}}{2}\Big{|}$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}|W_{n}|^{4}$
This gives
$|T_{n}-s\log\cosh(A)|\leq
sB|W_{n}|+sB^{2}\frac{W_{n}^{2}}{2}+\frac{1}{2}sB|W_{n}|^{3}+\frac{1}{2}sW_{n}^{4}$
The RHS above converges to $0$ on noting that $sB\ll n^{1/4}$, and
$W_{n}=O_{p}(n^{-1/4})$ (which follows by part (c) of Lemma 5).
As before, using uniform integrability it suffices to show that
$\displaystyle\limsup_{n\rightarrow\infty}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2T_{n}-2s\log\cosh(A)}<\infty.$ (40)
To this end, use the bound $x-\log 2\leq\log\cosh(x)\leq x$ for $x\geq 0$ to
conclude
$T_{n}-s\log\cosh(A)\leq 2s\log 2.$
Also, since
$\lim_{z\rightarrow 0}\frac{f(z)}{z^{4}}=\frac{1}{12},$
there exists $\delta\in(0,1)$ and $0<\lambda_{2}\leq\lambda_{1}<\infty$ such
that for $|z|\leq\delta$ we have
$\lambda_{1}z^{4}\leq f(z)\leq\lambda_{2}z^{4},$
which gives
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}e^{2W_{n}-2s\log\cosh(A)}\leq$ $\displaystyle e^{2s\log
2}\mathbb{P}(|T_{n}|>\delta)+\frac{\int_{-\delta}^{\delta}e^{5sB|z|-n\lambda_{1}z^{4}}dz}{\int_{-\delta}^{\delta}e^{-n\lambda_{2}z^{4}}dz},$
(41)
where the last line uses the bound
$\max\Big{(}sB^{2}z^{2},sB|z|^{3}+sz^{4}\Big{)}\leq sB|z|$
for all $|z|\leq\delta$. We now bound each of the terms in the RHS of (41).
The denominator in the RHS of (41) by a change of variable equals
$n^{-1/4}\int_{-\delta n^{1/4}}^{\delta
n^{1/4}}e^{-\lambda_{2}t^{4}}dt=(1+o(1))n^{-1/4}\int_{-\infty}^{\infty}e^{-\lambda_{2}t^{4}}dt.$
For estimating the numerator of RHS, set
$t_{n}:=\Big{(}\frac{10sB}{n\lambda_{1}}\Big{)}^{1/3}$ and note that for
$t>t_{n}$ we have $5sB|z|\leq\frac{1}{2}n\lambda_{1}z^{4}$. This gives
$\displaystyle\int_{-\delta}^{\delta}e^{5sB|z|-n\lambda_{1}z^{4}}dz\leq$
$\displaystyle
2\int_{0}^{t_{n}}e^{5sBz}dz+2\int_{t_{n}}^{\delta}e^{-\frac{n\lambda_{1}}{2}t^{4}}dt$
$\displaystyle\leq$ $\displaystyle
2t_{n}e^{5sBt_{n}}+2\int_{0}^{\infty}e^{-\frac{n\lambda_{1}}{2}t^{4}}dt$
$\displaystyle=$ $\displaystyle
2t_{n}e^{5sBt_{n}}+2n^{-1/4}\int_{0}^{\infty}e^{-\lambda_{1}t^{4}/2}dt.$
The first term above is asymptotically $2t_{n}\ll n^{-1/4}$, where we use the
bound $sB\ll n^{1/4}$. Thus the numerator in the second term in the RHS of
(41) is $O(n^{-1/4})$. Proceeding to estimate the first term, use Lemma 5 to
note that $\mathbb{P}(|W_{n}|>\delta)$ decays exponentially in $n$, whereas
$e^{2s\log 2}\leq e^{\frac{2n\log 2}{\log n}}$ is sub exponential. It thus
follows that the RHS of (41) is bounded, and so the proof of part (c) is
complete.
∎
###### Lemma 3.
Suppose $\mathbf{X}\sim\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}$ with $\mathbf{Q}^{\rm CW}_{i,j}=\mathbf{1}(i\neq
j)/n$. If $\lambda\in\mathbb{R}$, $\beta<1$, $s\ll n/\log n$,
$B:=s\max\\{\tanh(\lambda+A),\tanh(A)\\}\ll\sqrt{n}$, then
$\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}\big{(}\exp{(\lambda\sum\limits_{i\in
S}X_{i})}\big{)}=(1+o(1))\Big{(}\frac{\cosh(A+\lambda)}{\cosh(A)}\Big{)}^{s}.$
(42)
###### Proof.
Observe that,
$\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}\big{(}\exp{(\lambda\sum\limits_{i\in
S}X_{i})}\big{)}=\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A+\lambda))}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}=\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A+\lambda))/Z(\beta,\mathbf{Q},\mathbf{0})}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))/Z(\beta,\mathbf{Q},\mathbf{0})}.$
Now, Lemma 2 completes the proof. ∎
###### Lemma 4.
Suppose $\mathbf{X}\sim\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}}$ with $\mathbf{Q}^{\rm CW}_{i,j}=\mathbf{1}(i\neq j)/n$.
When $\beta>1$, let $m$ be the unique positive root of $m=\tanh(\beta m)$. Fix
$\boldsymbol{\mu}\in\Xi(s,A)$. Then the following hold uniformly in
$\boldsymbol{\mu}\in\mathbb{R}_{+}^{n}$ such that
$\sum_{\i=1}^{n}\boldsymbol{\mu}_{i}\ll n$.
1. (a)
Fix any sequence $\kappa_{n}\rightarrow 0$. Then then there exists a sequence
$\zeta_{n}\rightarrow 0$ (depending only on the pre-fixed sequence
$\kappa_{n}$) such that,
$\sup_{\boldsymbol{\mu}:\sum_{i=1}^{n}\mu_{i}\leq\kappa_{n}n}\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}}\left(|W_{n}-m|>\zeta_{n},W_{n}\geq 0\right)\rightarrow
0$
2. (b)
Fix any sequence $\kappa_{n}\rightarrow 0$. Then then there exists a sequence
$\zeta_{n}\rightarrow 0$ (depending only on the pre-fixed sequence
$\kappa_{n}$) such that,
$\sup_{\boldsymbol{\mu}:\sum_{i=1}^{n}\mu_{i}\leq\kappa_{n}n}\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}}\left(|W_{n}+m|>\zeta_{n},W_{n}\leq 0\right)\rightarrow
0$
###### Proof.
For the proof of the case $W_{n}\geq 0$ we first note that by [24] there
exists $C>0$
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(|W_{n}-m|>{\zeta_{n}},W_{n}\geq
0\right)\leq\exp\left(-Cn\zeta_{n}^{2}\right)$ and hence for
$\sum_{i=1}^{n}\mu_{i}\ll n$ we have with
$\mathcal{E}_{n}=\\{|W_{n}-m|>\frac{\zeta_{n}}{\sqrt{n}},W_{n}\geq 0\\}$ and
any sequence $\kappa_{n}\rightarrow 0$ that
$\displaystyle\sup_{\boldsymbol{\mu}:\sum_{i=1}^{n}\mu_{i}\leq\kappa_{n}n}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\mathcal{E}_{n}\right)$
$\displaystyle=\sup_{\boldsymbol{\mu}:\sum_{i=1}^{n}\mu_{i}\leq\kappa_{n}n}\frac{Z(\beta,\mathbf{Q},\mathbf{0})}{Z(\beta,\mathbf{Q},\boldsymbol{\mu})}\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}\left(\mathbbm{1}(\mathcal{E}_{n})e^{\sum_{i=1}^{n}\mu_{i}X_{i}}\right)$
$\displaystyle=\sup_{\boldsymbol{\mu}:\sum_{i=1}^{n}\mu_{i}\leq\kappa_{n}n}e^{2\sum_{i=1}^{n}\mu_{i}}\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(\mathcal{E}_{n})\leq
e^{2\sum_{i=1}^{n}\boldsymbol{\mu}_{i}-Cn\zeta_{n}^{2}}\rightarrow 0$
whenever $n\zeta_{n}^{2}\gg\kappa_{n}n$. This completes the proof of the
lemma. For the case $W_{n}\leq 0$ [24] there exists $C>0$
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}\left(|W_{n}+m|>{\zeta_{n}},W_{n}\geq
0\right)\leq\exp\left(-Cn\zeta_{n}^{2}\right)$ and the rest of the proof is
similar.
∎
The next lemma follows from [25, Theorems 1-3].
###### Lemma 5.
Suppose $\mathbf{X}\sim\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}}$ with $\mathbf{Q}^{\rm CW}_{i,j}=\mathbbm{1}(i\neq
j)/n$.
1. (a)
If $\beta\in(0,1)$ then under $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}$ we have
$\sqrt{n}\bar{\mathbf{X}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\frac{1}{1-\beta}\right).$
2. (b)
If $\beta>1$ then under $\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}$ we
have
$\sqrt{n}(\bar{\mathbf{X}}-m|\bar{\mathbf{X}}>0)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\frac{1-m^{2}}{1-\beta(1-m^{2})}\right),$
where $m$ is the unique positive root of the equation $t=\tanh(\beta t)$.
3. (c)
If $\beta=1$ then under $\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}$ we
have
$n^{1/4}\bar{\mathbf{X}}\stackrel{{\scriptstyle d}}{{\rightarrow}}Y,$
where $Y$ is a random variable with density proportional to $e^{-y^{4}/12}$.
### 6.3. Proof of Results in Section 2.2
#### 6.3.1. Proof of Theorem 3
We will divide the proof into three parts based on high $(\beta<1)$, critical
$(\beta=1)$, and low $(\beta>1)$ temperature. Throughout we will use
$\mathbf{Q}$ for either scaled adjacency matrix of of the graph $G(n,p)$ (with
$p=\Theta(1)$) or random regular graph on $n$ vertices each having degree
$d=\Theta(n)$.
##### Proof for upper bound in High Temperature $(0\leq\beta<1)$ Regime:
We first focus on the upper bound. As before, the optimal test is given the by
the scan test which rejects for large values of $Z_{\max}:=\max\limits_{S\in
S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}$ (recall that
for $S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ we defined
$Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$). The cut-off for the test is decided by
the moderate deviation behavior of $Z_{S}$’s given in Lemma 10 – which implies
that for any $\delta>0$, the test given by
$T_{n}(\delta)=\mathbf{1}\left\\{Z_{\max}>\sqrt{2(1+\delta)\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\right\\}$
has Type I error converging to $0$.
Turning to the Type II error, consider any
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\in\Xi(\mathcal{C}_{s},A)$.
First note that by GHS inequality
$\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i\in\tilde{S}^{\star}}X_{i})=O(s)$.
As a result,
$\frac{\sum_{i\in\tilde{S^{\star}}}(X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i}))}{\sqrt{s}}=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)$.
Therefore, as usual it is enough to show that there exists $\delta>0$ such
that
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
To show this end, first let $S^{\star}\in\mathcal{C}_{s}$ be such that the
signal lies on $S^{\star}$ i.e. for all $i\in S^{\star}$ one has
$\boldsymbol{\mu}_{i}\geq A$. Note that by monotonicity arguments it is enough
to consider $\boldsymbol{\mu}_{i}=A$. By definition of covering, we can find a
$\tilde{S}^{\star}\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ such
that $\gamma\left(\tilde{S}^{\star},S^{\star}\right)\leq\varepsilon_{n}$ i.e.
$|\tilde{S}^{\star}\cap S^{\star}|\geq s(1-\varepsilon_{n}/\sqrt{2})$.
Thereafter note that by Lemma 20 we have that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\geq
A|\tilde{S}^{\star}\cap S^{\star}|-A^{2}\sum_{i\in\tilde{S}^{\star}\cap
S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j}),$
(43)
However, by [18, Lemma 9 (a) and Theorem 7] we have that
$\displaystyle A^{2}\sum_{i\in\tilde{S}^{\star}\cap S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j})$
$\displaystyle\lesssim\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}{s}\left(\frac{|\tilde{S}^{\star}\cap
S^{\star}||S^{\star}|}{n}+|\tilde{S}^{\star}\cap S^{\star}|\right)$
$\displaystyle\leq\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|s}}{{n}}+\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll
As,$
since $\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll s\ll
n/\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}$. Consequently,
we immediately have that there exists $\epsilon>0$ such that
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)$
$\displaystyle\geq
As(1+o(1))\geq\sqrt{2(1+\epsilon)s\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$
Therefore, we can conclude that for any $\delta<\epsilon$ we have
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
This completes the proof of the upper bound for $0<\beta<1$.
##### Proof for upper bound at Critical Temperature $(\beta=1)$ Regime:
Similar to $\beta<1$ regime, the optimal test is given the by the scan test
which rejects for large values of $Z_{\max}:=\max\limits_{S\in
S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})}Z_{S}$ (recall that
for $S\in\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})$ we defined
$Z_{S}=\sum_{i\in S}X_{i}/\sqrt{s}$). The cut-off for the test is decided by
the moderate deviation behavior of $Z_{S}$’s given in Lemma 10 – which implies
that for any $\delta>0$, the test given by
$T_{n}(\delta)=\mathbf{1}\left\\{Z_{\max}>\sqrt{2(1+\delta)\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}\right\\}$
has Type I error converging to $0$.
Turning to the Type II error, we again consider any
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\in\Xi(\mathcal{C}_{s},A)$.
First note that by GHS inequality
$\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in\tilde{S}^{\star}}X_{i})\leq\mathrm{Var}_{\beta,\mathbf{Q},\mathbf{0}}(\sum_{i\in\tilde{S}^{\star}}X_{i})=O(s)$
even for $\beta=1$ since $s\ll\frac{\sqrt{n}}{\log{n}}$ by appealing to [18,
Lemma 9(c)]. Hence, it is enough to show that there exists $\delta>0$ such
that
$t_{n}(\delta)-\frac{1}{\sqrt{s}}\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\left(\sum_{i\in\tilde{S}^{\star}}X_{i}\right)\rightarrow-\infty$.
As before, let $S^{\star}\in\mathcal{C}_{s}$ be such that the signal lies on
$S^{\star}$ i.e. for all $i\in S^{\star}$ one has $\boldsymbol{\mu}_{i}\geq
A$. By monotonicity arguments it is enough to consider
$\boldsymbol{\mu}_{i}=A\mathbbm{1}_{i\in S^{\star}}$. Following (43), it is
enough to lower bound $A^{2}\sum_{i\in\tilde{S}^{\star}\cap
S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j})$.
By [18, Lemma 9 (c) and Theorem 7], we have that
$\displaystyle A^{2}\sum_{i\in\tilde{S}^{\star}\cap S^{\star}}\sum_{j\in
S}\mathrm{Cov}_{\beta,\mathbf{Q},\tilde{\mathbf{\eta}}_{S^{\star}}(A)}(X_{i},X_{j})$
$\displaystyle\lesssim\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}{s}\left(\frac{|\tilde{S}^{\star}\cap
S^{\star}||S^{\star}|}{\sqrt{n}}+|\tilde{S}^{\star}\cap S^{\star}|\right)$
$\displaystyle\leq\frac{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|s}}{\sqrt{n}}+\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll
As,$
since $\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}\ll
s\ll\frac{n}{\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$.
This completes the proof of the upper bound for $\beta=1$.
##### Proof for upper bound in Low Temperature $(\beta>1)$ Regime:
To prove the upper bound on detection threshold, we will use the same
randomized test as described in Theorem 1i.i._c_). Let the rejection region of
the test be denoted by $\Omega_{n}$. Using [17, Theorem 1.6], we obtain
$\log\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(\Omega_{n})\leq
C_{\delta}\Big{(}\log\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(\Omega_{n})+\|\mathbf{Q}\|^{2}_{F}+\sum_{i=1}^{n}(R_{i}-1)^{2}\Big{)}.$
Now, $\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}(\Omega_{n})=o(1)$ by
Theorem 1i.i._c_). The error term in RHS is $O(1)$ by [17, Section 1.2].
Hence, type-I error of the proposed test converges to $0$.
For the type-II error, fix $\eta>0$ and
$\tanh(A)=\sqrt{2(1+\delta)^{2}(1-m^{2})^{-1}\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}/s}$.
To get the desired cut-off of the test, take $\delta>0$ to be chosen later and
set
$t_{n}(\delta)=\sqrt{2(1+\delta)(1-m^{2})\log{|\mathcal{N}(\mathcal{C}_{s},\gamma,\varepsilon_{n})|}}$.
Assume $\boldsymbol{\mu}$ be the true signal with support $S^{\star}$ and we
scan over $S$ such that $\gamma(S,S^{\star})\leq\varepsilon_{n}=o(1)$. We will
show that $\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}>ms+t\sqrt{s}|\bar{\mathbf{X}}\geq 0)\rightarrow 1$. Note that,
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}>ms+t\sqrt{s}|\bar{\mathbf{X}}\geq 0)$
$\displaystyle=\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\frac{\sum_{i\in
S}X_{i}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}|\bar{\mathbf{X}}>0)}{\sqrt{Var_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}|\bar{\mathbf{X}}>0)}}>\frac{ms+t\sqrt{s}-\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}|\bar{\mathbf{X}}>0)}{\sqrt{Var_{\beta,\mathbf{Q},\boldsymbol{\mu}}(\sum_{i\in
S}X_{i}|\bar{X}>0)}}|\bar{\mathbf{X}}\geq 0)$
Since LHS is $O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)$, it is
enough to show the RHS (henceforth denoted by $T_{n}$) converges to $-\infty$.
Defining $\alpha_{n}=\sqrt{\frac{\log n}{n}}$, we obtain the following
estimates from [18, Lemma 9] for some absolute constants $c_{i}>0$, $i\in[3]$:
$\displaystyle\max_{i,j}\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j}|\bar{\mathbf{X}}\geq
0)\leq\frac{c_{1}}{n}$ (44)
$\displaystyle\max_{i}|\mathrm{Var}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i}|\bar{X}\geq
0)-\operatorname{sech}^{2}(\beta m+\mu_{i})|\leq c_{2}\alpha_{n}$ (45)
$\displaystyle\max_{i}|\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i}|\bar{X}\geq
0)-\tanh(\beta m+\mu_{i})|\leq c_{3}\alpha_{n}.$ (46)
All the above statements hold with high probability with respect to the
randomness of $\mathbb{G}_{n}$.
Using the bounds, we obtain
$\displaystyle T_{n}$ $\displaystyle\leq\frac{ms-
t_{n}(\delta)\sqrt{s}-\sum_{i\in S}\tanh(\beta
m+\mu_{i})+sc_{3}\alpha_{n}}{\sqrt{\sum_{i\in S}\operatorname{sech}^{2}(\beta
m+\mu_{i})+sc_{2}\alpha_{n}-\frac{cs^{2}}{n}}}\leq\frac{ms-
t_{n}(\delta)\sqrt{s}-s\tanh(\beta
m+A)+sc_{3}\alpha_{n}}{\sqrt{s\operatorname{sech}^{2}(\beta
m+A)+sc_{2}\alpha_{n}-\frac{cs^{2}}{n}}}$
$\displaystyle\leq\frac{1}{C\sqrt{s}}\left(ms-
t_{n}(\delta)\sqrt{s}-s\big{(}m-(1+o(1)A\operatorname{sech}^{2}(\beta
m)\big{)}+sc_{3}\alpha_{n}\right)$
$\displaystyle\leq\frac{1}{C}\left(-t_{n}(\delta)+As(1+o(1))(1-m^{2})+o(As)\right),$
since $\alpha_{n}=o(A)$. By choosing $\delta$ based on $\eta$, the final bound
converges to $\infty$. The same calculation holds for
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}\big{(}\sum_{i\in
S}X_{i}>-ms+t\sqrt{s}|\bar{\mathbf{X}}<0\big{)}$ yielding Type-II error
converging to $0$.
##### Proof for lower bound in $\beta>0$ Regime:
To prove the lower bound, fix $\beta>$ and $\varepsilon>0$ such that
$\sqrt{s}\tanh(A)(\log|\tilde{\mathcal{C}}_{s}|)^{-1/2}=2(1-\varepsilon)c_{\beta}$
where $c_{\beta}=1$ if $\beta\leq 1$, and $(1-m^{2})^{-1}$ otherwise. Suppose
there exists a test with rejection region $\Omega_{n}$ which can test $H_{0}$
vs $H_{1}$. Hence,
$\mathbb{P}_{\beta,\mathbf{Q},\mathbf{0}}(\Omega_{n})\rightarrow 0$. Using
Lemma 6, we have $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(\Omega_{n})\rightarrow 0$. However, since every test not
asymptotically powerful for Curie-Weiss model in this regime of signal,
implying that, $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(\Omega^{c}_{n})\geq\eta$, for some $\eta>0$ for
all large enough $n$. This implies, using Lemma 6, there exists $\nu>0$ such
that $\mathbb{P}_{\beta,\mathbf{Q},\mu_{S}(A)}(\Omega_{n})\geq\nu$. Hence, the
type-I and type-II errors cannot converge to $0$ simultaneously completing the
proof of the lower bound.
#### 6.3.2. Proof of Theorem 4
To obtain an adaptive test, we apply the same procedure as described by the
proof of Theorem 2. The proof of type-I error convergence when
$\boldsymbol{\mu}=0$ follows from concentration of $\bar{\mathbf{X}}$ which is
immediate by [17, Theorem 1.1, Theorem 1.2]. For general $\boldsymbol{\mu}$,
we only require the fact
$\mathrm{Cov}_{\beta,\mathbf{Q},\boldsymbol{\mu}}(X_{i},X_{j})\lesssim 1/n$
which follows from [18, Lemma 9]. Finally we also can obtain consistent
estimator of $\beta$ thanks to [11, Corollary 3.1, Corollary 3.2] and Lemma
19. This completes our proof exactly as Theorem 2.
### 6.4. Technical Lemmas for Proofs of Theorems in Section 2.2
###### Lemma 6.
Fix $\beta>0$. Let $\mathbf{Q}^{\rm CW}$ be the scaled adjacency matrix of the
complete graph and $\mathbf{Q}$ is either scaled adjecency matrix of of the
graph $G(n,p)$ (with $p=\Theta(1)$) or random regular graph on $n$ vertices
each having degree $d=\Theta(n)$. Fix any $\delta>0$. For any event
$\mathcal{E}_{n}$: the following holds with probability $\geq 1-\delta$: If
$\exists\eta>0$ such that $\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mu_{S}(A)}(\mathcal{E}_{n})\geq\eta$, then $\exists\nu>0$ such that
$\mathbb{P}_{\beta,\mathbf{Q},\mu_{S}(A)}(\mathcal{E}_{n})>\nu$.
###### Proof.
Defining $\mathcal{A}_{n}:=G_{n}-11^{T}/n+I/n$, note that
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(\mathcal{E}_{n})=\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\frac{\beta}{2}\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{\sigma\in\mathcal{E}_{n}}.$
For the auxiliary variable $W_{n}$, define a vector
$\tilde{T}_{n}=\tilde{T}_{n}(W_{n})$ such that $\tilde{T}_{n,i}=\tanh(\beta
W_{n}+A\mathbbm{1}_{i\in S})$. Observe that,
$\sigma^{\top}\mathcal{A}_{n}\sigma=(\sigma^{\top}-\tilde{T}_{n})\mathcal{A}_{n}(\sigma-\tilde{T}_{n})+2\tilde{T}^{\top}_{n}\mathcal{A}_{n}(\sigma-\tilde{T}_{n})+\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}=:Y_{n}+\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}.$
Define the good set $J_{n,\varepsilon}:=\\{m-\varepsilon\leq|W_{n}|\leq
m+\varepsilon\\}$. Then,
$\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(\mathcal{E}_{n})\geq\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}\left(\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J_{n,\varepsilon}}\right)$
(47)
The ratio of partition function is $\geq 1/C_{u}$ for some $C_{u}>0$, w.p.
$\geq 1-\delta$ by Lemma 8.
To analyse the expectation above, define $A_{M}=\\{\sigma:|Y_{n}|\leq M\\}$
for some $M>0$ and observe that,
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\frac{\beta}{2}\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J_{n,\varepsilon}}\mathbbm{1}_{\sigma\in\mathcal{E}_{n}}$
$\displaystyle\geq\mathbb{E}\left(\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}\Big{(}e^{\frac{\beta}{2}\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J_{n,\varepsilon}}\mathbbm{1}_{\sigma\in\mathcal{E}_{n}}\mathbbm{1}_{A_{M}}|W_{n}\Big{)}\right)$
$\displaystyle\geq
e^{-\frac{\beta}{2}M}\mathbb{E}\left(e^{\frac{\beta}{2}\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(J_{n,\varepsilon}\cap\mathcal{E}_{n}\cap
A_{M}|W_{n})\right)$
The proof of Lemma 8 yields $\exists\theta>0$ and $\delta>0$ small enough such
that
$\inf_{W_{n}}\exp(\frac{\beta}{2}\tilde{T}_{n}(W_{n})^{T}\mathcal{A}_{n}\tilde{T}_{n}(W_{n}))\geq
e^{-\theta}$ with probability $\geq 1-\delta$. Therefore,
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\frac{\beta}{2}\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J_{n,\varepsilon}}\mathbbm{1}_{\sigma\in\mathcal{E}_{n}}\geq
e^{-\frac{\beta}{2}M-\theta}\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(J_{n,\varepsilon}\cap\mathcal{E}_{n}\cap
A_{M}).$
By Hanson-Wright inequality(cf. Lemma 21), pick $M$ large enough such that
$\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(A_{M}|W_{n})\geq 1-\eta/4$. Hence,
$\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\boldsymbol{\mu}_{S}(A)}(A_{M})\geq
1-\eta/4$ By picking $\varepsilon>0$ small enough, we can also get
$\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(J_{n,\varepsilon})\geq 1-\eta/4$. Hence,
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\frac{\beta}{2}\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J_{n,\varepsilon}}\geq\frac{\eta}{2}e^{-\frac{\beta}{2}M-\theta}=:\nu>0.$
This concludes the proof of this lemma. ∎
###### Lemma 7.
There exists a constant $C_{\varepsilon}>0$ such that
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(J^{c}_{n,\varepsilon})\leq e^{-Cn},$
where $\mathbf{Q}^{\rm CW}$ corresponds to the coupling matrix of a Curie-
Weiss model on $n$ vertices.
###### Proof.
$\displaystyle\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(J^{c}_{n,\varepsilon})$
$\displaystyle=\frac{Z(\beta,\mathbf{Q}^{\rm
CW},\mathbf{0})}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}\mathbb{E}_{\beta,\mathbf{Q},\mathbf{0}}\left(e^{A\sum_{i\in
S}{X_{i}}}\mathbbm{1}_{J^{c}_{n,\varepsilon}}\right)$ $\displaystyle\leq
e^{As}\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\mathbf{0}}(J^{c}_{n,\varepsilon})\leq e^{As-Cn},$
for some $C>0$ depending on $\varepsilon>0$ (where the proof of
$\mathbb{P}_{\beta,\mathbf{Q}^{\rm CW},\mathbf{0}}(J^{c}_{n,\varepsilon})\leq
e^{-Cn}$ follows from [24]). The proof follows from the assumption that $As\ll
n$. ∎
###### Lemma 8.
Let $\mathbf{Q}^{\rm CW}_{i,j}$ denote the coupling matrix of Curie-Weiss
model and $\mathbf{Q}$ be the scaled adjacency matrix of either the graph
$G(n,p)$ (with $p=\Theta(1)$) or random regular graph on $n$ vertices each
having degree $d=\Theta(n)$. Then for any $\delta>0$ the following holds with
probability $\geq 1-\delta$: for any $\beta>0,\boldsymbol{\mu}_{S}(A)$:
$\displaystyle
C_{l}\leq\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}\leq C_{u}\quad,$
for constants $C_{l},C_{u}$ depending on $\beta,\delta$.
###### Proof.
Define $\mathcal{A}_{n}:=G-\mathbf{1}\mathbf{1}^{T}+I/n$. For the auxiliary
variable $W_{n}$, define a vector $\tilde{T}_{n}=\tilde{T}_{n}(W_{n})$ such
that $\tilde{T}_{n,i}=\tanh(\beta W_{n}+A\mathbbm{1}_{i\in S})$. Observe that,
$\mathbf{X}^{\top}\mathcal{A}_{n}\mathbf{X}=(\mathbf{X}-\tilde{T}_{n})^{\top}\mathcal{A}_{n}(\mathbf{X}-\tilde{T}_{n})+2\tilde{T}^{\top}_{n}\mathcal{A}_{n}(\mathbf{X}-\tilde{W}_{n})+\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}=:Y_{n}+\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}.$
Define the good set $J_{n,\varepsilon}:=\\{m-\varepsilon\leq|W_{n}|\leq
m+\varepsilon\\}$, where $m$ is the unique positive root of $m=\tanh(\beta
m)$. For the upper bound, note that,
$\displaystyle\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}$
$\displaystyle=\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}\Big{(}e^{\frac{\beta}{2}\mathbf{X}^{T}\mathcal{A}_{n}\mathbf{X}}\Big{)}$
$\displaystyle=\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\mathbf{X}^{\top}\mathcal{A}_{n}\mathbf{X}}\mathbbm{1}_{J^{c}_{n,\varepsilon}}+\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\mathbf{X}^{\top}\mathcal{A}_{n}\mathbf{X}}\mathbbm{1}_{J_{n,\varepsilon}},$
To bound the first summand, note that for any $\delta>0$ one has that there
exists a sequence $\epsilon_{n}(\delta)\rightarrow 0$ such that with
probability $1-\epsilon_{n}(\delta)$ one has that $\|\mathcal{A}_{n}\|_{\rm
op}\leq n^{-1/2+\delta}$ for either $G(n,p)$ [48, Theorem 1.1] or for random
regular graph [47, Theorem A]. As a result, for any $\delta>0$ one has that
with probability with probability $1-\epsilon_{n}(\delta)$,
$\sup\limits_{\sigma\in\\{\pm\\}^{n}}e^{\mathbf{X}^{T}\mathcal{A}_{n}\mathbf{X}}\leq
e^{n^{1/2+\delta}}$. Subsequently, with probability larger than
$1-\epsilon_{n}(\delta)$ the following hold
$\displaystyle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\sigma^{\top}\mathcal{A}_{n}\sigma}\mathbbm{1}_{J^{c}_{n,\varepsilon}}\leq
e^{n^{1/2+\delta}}\mathbb{P}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(J^{c}_{n,\varepsilon}).$
This inequality and Lemma 7 yields that the first summand is $o(1)$. To
analyze $\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}e^{\mathbf{X}^{\top}\mathcal{A}_{n}\mathbf{X}}\mathbbm{1}_{J_{n,\varepsilon}}$
we plan to invoke Hanson-Wright inequality (cf. Lemma 21). To this end, we
choose $\varepsilon>0$ small enough such that
$\lim\sup\max_{i}s_{\tilde{T}_{n,i}}\lambda_{1}(\beta\mathcal{A}_{n})<1,$
where $s$ in Lemma 21. The choice of $\varepsilon$ is possible since $A=o(1)$.
Now, we apply 21 on $\mathbb{E}(e^{\frac{\beta}{2}Y_{n}}|W_{n})$ and the error
term in Lemma 21 is $O(1)$ as shown in [17, Section 1.2]. Hence, the proof
concludes once we observe that $\log\mathbb{E}^{\rm
CW}e^{\tilde{T}^{\top}_{n}\mathcal{A}\tilde{T}_{n}}=O(1)$ uniformly over $S$.
To see this, note that it is enough to show that for any $\delta>0$ there
exists a $C_{\delta}>0$ such that
$\sup_{W_{n}}\exp(\tilde{T}_{n}(W_{n})^{T}\mathcal{A}_{n}\tilde{T}_{n}(W_{n}))\leq
C_{\delta}$ with probability $\geq 1-\delta$ under either Erdős-Rényi
randomness or random regular graph. To that end, note that with $S(\mu)$
denoting the support of $\boldsymbol{\mu}$ one has
$\displaystyle\tilde{T}_{n}(W_{n})^{T}\mathcal{A}_{n}\tilde{T}_{n}(W_{n})$
$\displaystyle=T_{1}+2T_{2}+T_{3}$
where
$\displaystyle T_{1}$ $\displaystyle=\frac{1}{\tilde{d}}\sum_{(i,j)\in
S(\boldsymbol{\mu})^{c}\times S(\boldsymbol{\mu})^{c}}\tanh^{2}(\beta
W_{n})\left(G_{i,j}-\frac{\tilde{d}}{n}\right);$ $\displaystyle T_{2}$
$\displaystyle=\frac{1}{\tilde{d}}\sum_{(i,j)\in S(\boldsymbol{\mu})\times
S(\boldsymbol{\mu})^{c}}\tanh(\beta W_{n}+\mu_{i})\tanh(\beta
W_{n})\left(G_{i,j}-\frac{\tilde{d}}{n}\right);$ $\displaystyle T_{3}$
$\displaystyle=\frac{1}{\tilde{d}}\sum_{(i,j)\in S(\boldsymbol{\mu})\times
S(\boldsymbol{\mu})}\tanh(\beta W_{n}+\mu_{i})\tanh(\beta
W_{n}+\mu_{j})\left(G_{i,j}-\frac{\tilde{d}}{n}\right).$
Above $\tilde{d}=np$ when $\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,p)$ and
$\tilde{d}=d$ when $\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,d)$. In particular, in
that case
$\displaystyle T_{1}$ $\displaystyle=\tanh^{2}(\beta
W_{n})\frac{1}{\tilde{d}}\sum_{(i,j)\in S(\boldsymbol{\mu})^{c}\times
S(\boldsymbol{\mu})^{c}}\left(G_{i,j}-\frac{\tilde{d}}{n}\right)$
$\displaystyle=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)O_{\mathbb{P}_{G_{n}}}\left(\frac{(n-s)}{\tilde{d}}\right)\quad\text{by
Lemma \ref{lem:random_regular_covariance}}.$
Similarly,
$\displaystyle T_{2}$ $\displaystyle=\tanh(\beta W_{n})\tanh(\beta
W_{n}+A)\frac{1}{\tilde{d}}\sum_{(i,j)\in S(\boldsymbol{\mu})\times
s(\boldsymbol{\mu})^{c}}\left(G_{i,j}-\frac{\tilde{d}}{n}\right)$
$\displaystyle=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)O_{\mathbb{P}_{G_{n}}}\left(\frac{\sqrt{(n-s)s}}{\tilde{d}}\right)\quad\text{by
Lemma \ref{lem:random_regular_covariance}},$
and
$\displaystyle T_{3}$ $\displaystyle=\tanh^{2}(\beta
W_{n}+A)\frac{1}{\tilde{d}}\sum_{(i,j)\in S(\boldsymbol{\mu})\times
s(\boldsymbol{\mu})}\left(G_{i,j}-\frac{\tilde{d}}{n}\right)$
$\displaystyle=O_{\mathbb{P}_{\beta,\mathbf{Q},\boldsymbol{\mu}}}(1)O_{\mathbb{P}_{G_{n}}}\left(\frac{\sqrt{(n-s)s}}{\tilde{d}}\right)\quad\text{by
Lemma \ref{lem:random_regular_covariance}},$
These estimates immediately verifies claim that $T_{j}$’s $O_{\mathbb{P}}(1)$
whenever $\tilde{d}=\Theta(n)$ – which is guaranteed by either $p=\Theta(1)$
when $\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,p)$ or $d=\Theta(n)$ when
$\mathbb{G}_{n}\sim\mathcal{G}_{n}(n,d)$.
For the lower bound on ratio of partition functions we begin by noting that by
convexity of $\mathbf{Q}\rightarrow\log{Z(\beta,\mathbf{Q},\boldsymbol{\mu})}$
for every $\boldsymbol{\mu}$,444for any differentiable convex function
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ one has $f(y)-f(x)\geq\langle\nabla
f(x),y-x\rangle$
$\displaystyle\exp\left(\beta\langle\mathbb{E}_{\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A)}(\hat{\Sigma}),-\Delta_{\mathbf{Q}}\rangle_{\rm
TR}\right)\leq\frac{Z(\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A))}{Z(\beta,\mathbf{Q}^{\rm
CW},\boldsymbol{\mu}_{S}(A))}\leq\exp\left(-\beta\langle\mathbb{E}_{\beta,\mathbf{Q},\boldsymbol{\mu}_{S}(A)}(\hat{\Sigma}),\Delta_{\mathbf{Q}}\rangle_{\rm
TR}\right)$
where $\hat{\Sigma}_{i,j}=X_{i}X_{j}$,
|
# Fingerprinting Robot Movements via Acoustic Side Channel
Ryan Shah University of StrathclydeUnited Kingdom<EMAIL_ADDRESS>,
Mujeeb Ahmed University of StrathclydeUnited Kingdom
<EMAIL_ADDRESS>and Shishir Nagaraja Newcastle UniversityUnited
Kingdom<EMAIL_ADDRESS>
###### Abstract.
In this paper, we present an acoustic side channel attack which makes use of
smartphone microphones recording a robot in operation to exploit acoustic
properties of the sound to fingerprint a robot’s movements. In this work we
consider the possibility of an insider adversary who is within physical
proximity of a robotic system (such as a technician or robot operator),
equipped with only their smartphone microphone. Through the acoustic side-
channel, we demonstrate that it is indeed possible to fingerprint not only
individual robot movements within 3D space, but also patterns of movements
which could lead to inferring the purpose of the movements (i.e. surgical
procedures which a surgical robot is undertaking) and hence, resulting in
potential privacy violations. Upon evaluation, we find that individual robot
movements can be fingerprinted with around 75% accuracy, decreasing slightly
with more fine-grained movement meta-data such as distance and speed.
Furthermore, workflows could be reconstructed with around 62% accuracy as a
whole, with more complex movements such as pick-and-place or packing
reconstructed with near perfect accuracy. As well as this, in some
environments such as surgical settings, audio may be recorded and transmitted
over VoIP, such as for education/teaching purposes or in remote telemedicine.
The question here is, can the same attack be successful even when VoIP
communication is employed, and how does packet loss impact the captured audio
and the success of the attack? Using the same characteristics of acoustic
sound for plain audio captured by the smartphone, the attack was 90% accurate
in fingerprinting VoIP samples on average across baseline movements, which is
around 15% higher than the baseline without the VoIP codec employed. This is
an interesting result as it opens up new research questions regarding
anonymous communications to protect robotic systems from acoustic side channel
attacks via VoIP communication networks.
robot, security, privacy, acoustic, side channel, attack, passive, deep
learning, neural network, voip
††ccs: Security and privacy Systems security††ccs: Security and privacy Side-
channel analysis and countermeasures
## 1\. Introduction
The prominence of teleoperated robotic systems has seen a recent rise in a
variety of application areas, such as industrial (Quarta et al., 2017) and
surgical environments (Ahn et al., 2015; Tewari et al., 2002), with promises
of higher levels of accuracy and precision. Given that many of these systems
are becoming increasingly connected, they are vulnerable to an expanded threat
landscape in the cyber domain. Attacks from this angle are primarily active
attacks such as tampering with the integrity of messages in-flight or
hijacking the robot controller directly (Bonaci et al., 2015). However, little
attention has been paid to the capabilities of a passive attacker and the
damage potential of stealthier attacks. Specifically, passive attacks such
side channel attacks which exploit information leakages without the need to
change the normal behaviour of the system, can result in huge losses that stem
from the compromise of operational confidentiality. Side channel attacks in
the cyber domain have the potential to compromise the operational
confidentiality of organisations that own such systems (Shah et al., 2022),
yet those targeting robots in the physical domain are still to be explored.
In this paper, we aim to investigate whether an adversary can exploit
information leakages from the acoustic side channel, by capturing audible
emanations from a robotic system during normal operations, to mount an attack
that targets operational confidentiality. In this context, we look at two
possible threats posed by an insider attacker. First, a malicious robot
operator or technician on the ground could use a recording device, such as a
smartphone, near the robot to record entire workflows or individual movements.
By fingerprinting this leaked information, they could sell this on to
competing organisations for a malicious advantage. While it can be argued that
an attacker may not be able to get close enough to the robot to place the
recording device, many robotic systems now employ sensors to aid safety
mechanisms to prevent harm to nearby humans or environmental changes. This can
allow the attacker windows of opportunity to place the recording device near
the robot or be near enough to capture meaningful acoustic emanations. A
second possible threat comes from a telemonitoring perspective. While
telemonitoring is less common in industrial settings, in surgical settings the
use of medical recording devices, such as medical data recorders or
intraoperative video recorders, are used for post-surgical review or teaching
(alongside patient consent) to learn from suboptimal scenarios and improve
performance (Saun et al., 2019; Vodafone, 2019; Al-Jabir et al., 2020). While
privacy laws and medicolegal requirements govern the use of such devices, data
from them is not typically required as evidence in court so long as patient
confidentiality is maintained (Dalen et al., 2019). However, acoustic
emanations captured by such recordings could reveal the operations the robot
is carrying out, and ultimately piece together surgical procedures. In
combination with other metadata, such as patient admission and exit times,
this could compromise patient confidentiality.
In this attack, we recorded the acoustic emanations, through a smartphone
recorder, for individual robot movements, as well as recording entire
workflows corresponding to typical warehousing operations such as picking and
placing objects from one place to another. Using the collected data, we
extract a set of acoustic characteristics which are used as input to an
artificial neural network (ANN). We found that baseline movements (of minimum
speed and distance) can be fingerprinted with at least ~75% accuracy as a
baseline. The speed and distance of movements are not as successfully
fingerprinted in this attack, compared to the radio frequency side channel.
Entire warehousing workflows with ~64% accuracy. Ultimately, it is clear that
a passive insider adversary has the potential not only to reveal what a robot
is doing but take the resulting liabilities of such an attack to an extreme
that impacts even the organisations that employ them. As well as this, in
certain robotics environments, such as in surgical settings, procedures may be
streamed and/or recorded for viewing, education or research (Muensterer et
al., 2014; Kulkarni et al., 2020; Hosseini et al., 2013). Therefore, it is
important to question how VoIP impacts the audio samples for movements and
workflows and, ultimately, the success of the attack. Using the Opus codec – a
common choice for most modern VoIP applications – the attack was 90% accurate
for computing movement fingerprints for baseline speed and distance, which is
nearly 15% more accurate than the baseline without the Opus codec employed,
presenting new research questions regarding side channel attacks via VoIP
communication networks which target robotic systems.
The remainder of this paper is as follows. In Section 2 we provide background
on teleoperated robots and acoustic emanations, to which we then describe the
threat model. We then outline the attack and our findings in Section 3, and
provide an in-depth discussion in Section 5. In Section 6 we discuss related
work and conclude in Section 7.
## 2\. Background
### 2.1. Teleoperated Robots
The use of robotics has seen an increase in installations in a variety of
application areas (Reuters, 2019) and play pivotal roles bringing benefits to
quality of service, efficiency and precision, among others. Among them,
teleoperated robots are most prominent in many industrial (Aschenbrenner et
al., 2015; Avila et al., 2020; Grabowski et al., 2021; Li et al., 2017; Bartoš
et al., 2021) and surgical (Sung and Gill, 2001; Tewari et al., 2002;
Hannaford et al., 2012) environments, and share a common system architecture.
This type of system makes use of a human operator (i.e. a specially-trained
surgeon) who operates the controller (i.e. surgeon’s console or teach pendant)
which translates human movements or inputs into those which the robot can
interpret. These input (console and other sources of information) and output
(actuators) devices are linked together via an electronic control system (ECS)
and typically connected to the organisation’s network in which the robot
operates. An overview of a typical teleoperated robot architecture can be seen
in Figure 1.
Figure 1. Teleoperated Robot Architecture
### 2.2. Acoustic Characteristics
While the robot moves, its electromechanical components will emit (audible)
sound, which when captured could be used to mount an information leakage
attack. The first step to determining an appropriate attack strategy is to
understand the different characteristics of acoustic emanations, and what may
be most useful from an attack perspective.
#### 2.2.1. Root Mean Square Energy
Root Mean Square (RMS) energy (Bachu et al., 2008) is a measure of the
amplitude based on all samples in a frame of audio, and can be thought of as
an indicator of loudness of the audio signal (Panagiotakis and Tziritas,
2005). This may be useful in the context of this attack given that
combinations of movements (i.e. simultaneous movements along 2 or more axes)
may emanate a louder sound given the use of more stepper motors, for example.
As well as this, as the robot axes pass the microphone, the sound may be
louder and thus this feature may help provide further information to the
discrimination between movements in different positions.
#### 2.2.2. Zero-Crossing Rate
Zero-Crossing Rate (ZCR) is a measure of the number of times a signal crosses
the horizontal time axis and can help identify pitch variations in monophonic
tones (sound emitted from one location) (Panagiotakis and Tziritas, 2005).
Given the robot is stationary in this case, the ZCR may be a useful feature
candidate.
#### 2.2.3. Spectral Centroid
The spectral centroid provides information corresponding to frequency bands
contain most of the energy, wherein lower centroid (energy) values is likened
to duller sounds and higher centroid values for brighter sounds (Le et al.,
2011). In a robotic system, smaller movement distances and speeds will
naturally require less energy and appear more dull sounding to the human ear,
whereas faster and longer movements have better tonality, and may ultimately
provide useful for distinguishing between different movements of the same
source.
#### 2.2.4. Spectral Bandwidth
Spectral bandwidth is defined as the full width of band of light (wavelength
interval) at half the peak maximum (Klapuri and Davy, 2007; Atahan et al.,
2021). Acoustic signals oscillate about a point, and the bandwidth for each
time interval in a signal is the sum of maximum deviation on both sides of
this point. The point of the centroid of the signal may vary for different
robot movements and thus may be an important feature for fingerprinting.
#### 2.2.5. Spectral Rolloff
Spectral rolloff is the fraction of frequency bins under a cutoff point where
the total energy of the spectrum is contained, and can help distinguish
between noisy sounds and more harmonic sounds (below roll-off point) (Kos et
al., 2013). This feature may provide useful to this attack as it can roll off
frequencies that may fall outside of the useful range of frequencies where the
energy of the acoustic energy of movements is contained.
#### 2.2.6. Spectral Contrast
Spectral contrast is the measure of energy of frequencies in windows of time
(Jiang et al., 2002) and can help identify strong spectral peaks to reflect
the distribution between harmonic and non-harmonic components of the acoustic
emanations. As a robot moves, the frequency contents may have energy that
changes with time and capturing the spectral contrast can help measure this
energy variation.
#### 2.2.7. Chroma Feature
Chroma feature, sometimes referred to as a chromagram, profiles a sound into
12 pitch class profiles (Müller, 2015). In music analysis, the attempt is to
capture the harmonic and melodic characteristics of a song where pitches can
be categorised to one of the scales in the equally-tempered set of the notes
$\\{C,C\\#,D,D\\#,E,F,F\\#,G,G\\#,A,A\\#,B\\}$ (Cho and Bello, 2013; Paulus et
al., 2010). While recorded robot movements are not akin to songs that are
analysed in this fashion, the pitch of sound may correlate with the speed and
distance of movement and may provide useful as a mid-level feature for
fingerprinting movements.
#### 2.2.8. Mel-Cepstrum Frequency Coefficients
The Mel scale is a scale of pitches that is felt to be equal in distance from
one another. For example, in audible acoustics listened by a human,
differences in frequency content can be observed if the source of acoustic
emanations are in the same distance and atmosphere (Greenwood, 1997; Martinez
et al., 2012). The short-term power spectrum of acoustic emanations can be
represented by the Mel frequency cepstral (MFC) and a combination of
coefficients (MFCCs) make up the MFC. The MFC equally distributes frequency
bands to approximate human auditory response. If variations in robot movements
can be inferred from audible sound, then looking at the mel frequency
coefficients (the list of amplitudes of the spectrum in the mel scale) will
provide useful information to the attack.
### 2.3. Threat Model
Many previous attacks focus on an active attacker, which can involve the
tampering of messages (Bonaci et al., 2015) or replaying attacks between the
robot or controller (McClean et al., 2013). In this work, the primary attacker
is a passive insider, such as a malicious technician or operator. Being an
insider close to the robot would allow them to record the acoustic emanations
during the robot’s normal operations using a smartphone, which they may have
on them and use covertly (Peng and Choi, 2013). As well as this, it is also
possible than an insider attacker is able to covertly plant a microphone which
could transmit recorded audio to the attacker remotely or be retrieved at a
later time. In either case, if an attacker is able to mount an information
leakage attack to fingerprint robot movement patterns from acoustic
emanations, this could lead to the revelation of daily workflows (i.e. in a
warehouse) and ultimately compromise the operational confidentiality of the
organisation. For example, this information could be given to competitors to
gain an advantage or use it maliciously.
A second possible threat comes from a telemonitoring perspective. While
telemonitoring is less common in industrial settings, in surgical settings the
use of medical recording devices, such as medical data recorders or
intraoperative video recorders, are used for post-surgical review or teaching
(alongside patient consent) to learn from suboptimal scenarios and improve
performance (Saun et al., 2019; Vodafone, 2019; Al-Jabir et al., 2020). While
privacy laws and medicolegal requirements govern the use of such devices, data
from them is not typically required as evidence in court so long as patient
confidentiality is maintained (Dalen et al., 2019). However, acoustic
emanations captured by such recordings could reveal the operations the robot
is carrying out, and ultimately piece together surgical procedures. In
combination with other metadata, such as patient admission and exit times,
this could compromise patient confidentiality.
Ultimately, reviewing the nature of acoustic emanations in robotic systems, as
well as the proposed threat model, the aim is to investigate whether an
advesary will be able to record the acoustic emanations from a robot during
its normal operation, and make use of distinct features present across the
recorded audio to fingerprint robot movements and workflows. Several
hypothetical factors will come into play which could influence the potential
success of this attack. First, the type of operations being carried out by the
robot can vary in terms of speed and distance of movement, and so the attack
should be robust enough to fingerprint between these parameters. Second, the
distance at which the microphone is situated away from the robot, naturally
due to the Doppler effect (Doppler, 1903) wherein sounds soften with distance,
will also have an impact on the success of the attack and should be
investigated. Finally, given that in some cases VoIP technology will be
employed, such as for recording purposes or to livestream medical procedures
with surgical robots, the impact of VoIP on the attack should be evaluated.
### 2.4. Hypotheses and Goals
In this work we aim to investigate whether an advesary will be able to
effectively record the acoustic emanations from a robot during its normal
operation, and make use of distinct features present across the recorded audio
to fingerprint robot movements and workflows. We hypothesise that several
factors will come into play which could influence the potential success of
this attack. First, the type of operations being carried out by the robot can
vary in terms of speed and distance of movement, and so the attack should be
robust enough to fingerprint between these parameters. Second, we also
hypothesise that the distance at which the microphone is situated away from
the robot, naturally due to the Doppler effect (Doppler, 1903) wherein sounds
soften with distance, will also have an impact on the success of the attack
and should be investigated. Ultimately, the following research questions are
proposed:
1. ($R_{1}$)
Can an attacker fingerprint individual robot movements on each axes, as well
as permutations of them?
2. ($R_{2}$)
How is movement fingerprinting affected by:
1. (i)
The speed and distance of movements?
2. (ii)
The distance the recording device (i.e. smartphone) is away from the robot?
3. ($R_{3}$)
Can entire robot workflows be reconstructed from acoustic emanations?
4. ($R_{4}$)
How do VoIP codecs influence the success of the attack?
## 3\. Attack Methodology
In this paper, we investigate an acoustic side-channel attack which exploits
audio emanations from a robot during its operation. Specifically, the aim of
this attack is to fingerprint a robots movements from acoustic characteristics
alone, recorded by smartphone devices in a passive manner. For subsequent
discussion, we aim to answer the following questions:
1. ($R_{1}$)
Can an attacker fingerprint individual robot movements on each axes, as well
as permutations of them?
2. ($R_{2}$)
How is movement fingerprinting affected by:
1. (i)
The speed and distance of movements?
2. (ii)
The distance the smartphone or microphone is away from the robot?
3. ($R_{3}$)
Can an attacker recover information about the objects a robot is handling,
such as its weight?
Figure 2. Robot Environment for Acoustic Side Channel
### 3.1. Robot Environment
The context of this study surrounds modern teleoperated surgical robots, whos
typical architecture can be viewed (at a high level) as a pairing between the
robotic system itself and its controller (surgeon’s console). For this work,
we use uFactory’s uARM Swift Pro which runs on an Arduino Mega 2560 with
MicroPython installed. The controller is emulated on a Windows 10 laptop which
uses the uARM Python (3.8.X) SDK to enable controller instructions to be
written in Python which are then translated into Gcode that is understood by
the robot. An overview of the robot environment used in this study is depicted
in Figure 2. For capturing the acoustic emanations which arise when the robot
operates, we position the robot in the center of a table with the
smartphone/microphone placed in several distances away (30cm to 1m) from the
robot as shown in Figure 2.
### 3.2. Experiment Parameters
With the robot setup for evaluating our acoustic side-channel attack for
fingerprinting the robot’s movements, we now outline the parameters of our
study. Specifically, we will discuss the speed and distance of the movement
operation being carried out, the type of smartphone/micrphone, the distance
the smartphone microphone is away from the robot and finally,
Speed and Distance. In addition to capturing the acoustic emanations which
arise during operation along the X, Y and Z axes, and combined movement
operations, it is important to evaluate more fine-grained movements. To this,
we programmed robot movements with varying distances (in millimetres) as well
as varying speeds of movement (mm/s). This is because in realistic cases, a
surgical robot for example would not move in each direction with constant
distance and speed. Therefore, it is vital to understand whether an adversary
can also fingerprint meta-information as well as just the movements
themselves.
Microphone Distance. In terms of recording the acoustic emanations during
robot operation, it is important to evaluate the impact of distance the
microphone is away from the robot. In a real situation, it is highly unlikely
that an adversary would be very close to or in front of the robot, especially
in cases like surgical robots where it could not only be dangerous to stand
too close but being close enough may trigger potential safety features
implemented to prevent injury. For this study, given the size of our uARM
robot ($150mm\times 140mm\times 281mm$) and the volume of sound which is given
off during its operation, we cannot investigate large distances as would be
granted with a large surgical robot, for example. However, given this
limitation, we recorded sounds at distances ranging from 30cm to 1m.
VoIP. The final parameter for this study is to evaluate the impact VoIP has on
the success of the attack. For this study, the codec employed by the majority
of VoIP applications is Opus (Valin et al., 2012, 2016). The first step is to
observe how the codec performs, but also how packet loss will also affect
audio quality and the success of the attack.
Figure 3. Depiction of Common Warehousing Workflows Our dataset contains
common warehousing workflows such as pushing, pulling, packing and moving
objects
### 3.3. Movement Dataset
After determining the appropriate acoustic features to extract from the
captured sounds, the next step was to create the dataset. In this dataset,
there are 2 subsets. Within both subsets, there are samples pertaining to both
individual and permutations of movements with varying speeds and distances of
movement, the microphone distance, and robotic warehousing workflows (Figure
3). These workflows include those such as pick-and-place, push and pull
operations, which were replicated from those found in existing industrial
robot datasets such as the Forward Dynamics Dataset Using KUKA LWR and Baxter
(Polydoros and Nalpantidis, 2016) for pick and place and the Inverse Dynamics
Dataset Using KUKA (Rueckert et al., 2017) for push/pull. For these workflows,
movements were slightly perturbated to account for a small degree of entropy
that may be present in real-world operations (i.e. those that may arise due to
drift in equipment calibration or wear-and-tear). In contrast to the first
subset, the second subset contains the same samples but are passed through the
Opus codec to evaluate the impact of VoIP on recorded audio in this attack.
Specifically, while all samples are passed through the Opus codec, they are
further split by packet loss. Packet loss has been shown to negatively impact
call quality in VoIP communications (Ortega et al., 2018; Laghari et al.,
2020), as they induce impact in the form of dropped calls or parts of speech,
slow rate of speech (latency) or noise/interference. Because of this, these
further subsets are divided by packet loss values of 1%, 5%, 10%, 25% and 50%.
As a whole, the first subset contains 27.2K samples for individual movements
and 658 samples for warehousing workflows, with each using 20% of the total
samples for validation and another 20% for testing. The second contains the
same amount of samples for each of the packet losses evaluated.
Dataset Pre-Processing. The features in the dataset, as listed above, are
computed using the librosa (McFee et al., 2015) Python library. For each
feature, the mean value of each feature across each signal sample is taken and
computed from a Short-Time Fourier Transform (STFT) with a Hann window and FFT
length of 8192. For the MFCCs, 14 coefficients were used. Typically, 8–13 are
used with the zeroth excluded given it only represents the average log-energy
of the input signal (Rao and Vuppala, 2014). However, given this is a new
problem to be explored, this is also kept to later examine its importance for
fingerprinting.
### 3.4. Neural Network
Before an evaluation can take place, an important step is constructing an
appropriate neural network architecture for fingerprinting movements and
ensuring a successful attack. To create the neural network, a sequential model
was used where neurons are grouped in a linear fashion. This was created using
the Keras API (Chollet et al., 2015). The parameters and structure for the
layers in the neural network were evaluated on the dataset using a cross-
validated grid search to find the most optimal number of neurons, layers,
acttion function and dropouts if necessary. The input for the maximum number
of neurons to be tested was calculated using the formula proposed by Demuth et
al. (Demuth et al., 2014) with an alpha branching factor of $2$. Using the
grid search with 3 cross validations, the most optimal neural network
architecture for this feature set consists of 5 layers. First, the input layer
containing 21 neurons for each of the input features. Next, there are 4 hidden
layers. The first is a Dense layer with 290 neurons and uses the ReLU
activation function (Eckle and Schmidt-Hieber, 2019). The next hidden layer is
a Dropout layer which is used to randomly set input units to 0 at a rate of
$0.05$ at each step during training to prevent overfitting. The next layer is
another Dense layer of 350 neurons with ReLU activation, followed by another
Dropout with a rate of $0.05$ to prevent overfitting. Finally, the last layer
is a Dense output layer of 7 neurons, one for each of the movement classes,
and uses the SoftMax activation function (Dunne and Campbell, 1997) to have
the output in the range of $[0,1]$ for use as predicted probabilities. Sparse
categorical cross-entropy is used as labels are integers and not one-hot
encoded, for which categorical cross-entropy would be used (Zhang and Sabuncu,
2018). The optimiser used is Adam with a learning rate of $0.001$. This
learning rate was chosen as others, such as those with higher learning rates,
resulted in lowered accuracy scores. The model was fitted with a batch size of
32 and was run for 1000 epochs.
Choice of Activation and Optimisation Functions. The ReLU activation function
was chosen over other activation functions, as the reduced likelihood of
vanishing gradient allows for a constant gradient resulting in faster
learning. Further, the sparsity of representations are shown to be more
beneficial than dense representations, as seen in other activations such as
sigmoids (Krizhevsky et al., 2012; Li and Yuan, 2017; Agarap, 2018). The
softmax activation function, combined with categorical cross-entropy (Zhang
and Sabuncu, 2018) for the loss function, was chosen due to the simple fact
that this is a multi-class classification problem. Simply, a sample can belong
to one of the 7 classes, with each class corresponding to one of the robot
movements. As well as this, the Adam optimiser was an ideal candidate. It is
an extension to the Stochastic Gradient Descent (SGD) method, based on
adaptive estimation of first- and second-order moments (Kingma and Ba, 2014).
Specifically, it allows for the updating of network weights iteratively based
on the training data, and fits best with the weighted sample sets in
opposition to other tried methods such as standard SGD, RMSProp and SGD +
Nesterov Momentum.
## 4\. Evaluation
After setting up the robot environment and capturing the acoustic emanations
during various stages of operations, the next step is to evaluate the success
of the attack. As per the research questions listed above, the evaluation of
this attack and related results will be set out in that order.
### 4.1. Individual Movement Fingerprints
The first research question ($R_{1}$) aims to investigate whether an attacker
can infer individual movements (on each axis) and permutations of these
movements from the recorded audio. To compare this against other parameters,
this experiment is considered as a baseline where the speed and distance of
movement are the lowest possible values (1mm and 12.5mm/s respectively), and
no VoIP codec used. As seen in Table 1, an average accuracy of around 75% can
be observed across all movements, with the YZ movement having the highest
precision among the movements. In comparison with the RF side channel, there
is a clear drop in accuracy of around 20% but the acoustic side channel
outperforms traffic analysis by around 10%. Interestingly, Y-involved
movements are better recovered than other movements overall, which was not the
case in the RF side channel (albeit a higher accuracy). This may be due to the
Y-axis moving across the microphone range. Looking at the Z-involved
movements, these are among the lowest. This may be due to the Z axis involving
a vertical movement only and not moving nearer the microphone for better
recording.
Movement | Precision | Recall
---|---|---
X | 76% | 81%
Y | 77% | 78%
Z | 61% | 71%
XY | 78% | 80%
XZ | 68% | 65%
YZ | 85% | 78%
XYZ | 72% | 67%
Accuracy | 75%
Table 1. Baseline Classification Results As a whole, the baseline accuracy is
75% which is fairly good inference accuracy for an attacker. Z-based movements
show the lowest precision and recall for fingerprinting, perhaps due to
vertical motion and no horizontal spread across the recording device
### 4.2. Impact of Movement Distance
For the next research question ($R_{2}$), the evaluation will look into how
the distance and speed ($R_{2(i)}$) of robot movements, and the distance of
the recording device ($R_{2(ii)}$), impact the success of fingerprinting
movements from the acoustic side channel. First, as a robot moves, there is
likely to be more sound that can be recovered as the distance of movement
increases. As seen in Table 2, an increase by a single distance unit increases
the model accuracy by 1%, improving Y-involved movement precision by around
10%. furthermore, the Z movement also gains a slight increase in precision.
Unfortunately, this results in lowered accuracy for the other movements. This
increase in distance results in the sound of movement being held for longer
and may either provide useful for distinguishing variance between movements or
even reduce this variance. To explore this, larger distances of movements are
explored. At 5mm, there is a drop in accuracy of around 4%, with X-involved
movements having much higher accuracy. At 10mm, the accuracy of the model
overall decreases significantly to 57%. Y-involved movements in this case are
much poorly fingerprinted, yet X-involved movements have a further increase in
precision. For the Z movement at this stage, there is unfortunately a further
drop in precision but the recall remains relatively similar. At 25mm, the
accuracy starts to improve by 7% with the X movement having similar precision
and recall to 10mm, and most other movements have an increase in both
precision and recall. Finally, at 50mm, the accuracy nears that of the
baseline and 2mm, however X-involved movement accuracy is significantly
improved.
D = Distance (mm), P = Precision, R = Recall
---
| D=2 | D=5 | D=10 | D=25 | D=50 |
P | R | P | R | P | R | P | R | P | R |
X | 69% | 71% | 77% | 87% | 85% | 58% | 83% | 70% | 86% | 84% |
Y | 88% | 77% | 77% | 79% | 80% | 54% | 66% | 43% | 90% | 88% |
Z | 65% | 83% | 64% | 79% | 51% | 81% | 64% | 71% | 83% | 66% |
XY | 68% | 60% | 67% | 63% | 63% | 53% | 57% | 65% | 79% | 81% |
XZ | 62% | 57% | 83% | 47% | 67% | 49% | 60% | 61% | 64% | 79% |
YZ | 94% | 94% | 66% | 83% | 37% | 54% | 55% | 57% | 56% | 59% |
XYZ | 69% | 81% | 76% | 68% | 45% | 52% | 63% | 84% | 64% | 58% |
Accuracy | 76% | 72% | 57% | 64% | 74% |
Table 2. Classification Results With Distance Parameter At a slight increase
in distance, the accuracy remains similar to the baseline, but further
increases in distances lead to a reduction in fingerprinting accuracy.
Notably, unlike the baseline, X-involved movement are better fingerprinted at
distance
### 4.3. Impact of Movement Speed
After looking at movement distance, the next parameter for robot movements is
the speed at which the robot is moving along each of the axes ($R_{2(i)}$). As
seen in Table 3, the speed parameter is less accurately fingerprinted by the
attack compared to the distance parameter by at least 10% on average.
Interestingly, a similar pattern is observed rgarding X-involved movements,
with accuracy increasing with speed, except from the XYZ movement. While there
are slight drops in accuracy, the precision and recall across most movements
remains similar as speed increases. This is interesting, as the initial
hypothesis was that a higher speed would result in higher pitched acoustic
emanations, however the results seem to contradict this. In any case, perhaps
the perceptual characteristics for human audio, while a clear pitch change is
present listening to the robot in the lab, the feature algorithms regarding
pitch (i.e. chroma feature) may not pick up on this for robot sounds.
S = Speed (mm/s), P = Precision, R = Recall |
---|---
| S=25 | S=50 | S=75 | S=100 |
P | R | P | R | P | R | P | R |
X | 57% | 81% | 54% | 74% | 78% | 53% | 72% | 81% |
Y | 79% | 76% | 61% | 42% | 59% | 45% | 72% | 69% |
Z | 50% | 56% | 52% | 58% | 62% | 84% | 77% | 75% |
XY | 73% | 72% | 46% | 40% | 67% | 57% | 57% | 70% |
XZ | 79% | 57% | 75% | 79% | 57% | 60% | 60% | 56% |
YZ | 67% | 59% | 66% | 45% | 53% | 65% | 66% | 69% |
XYZ | 65% | 63% | 51% | 66% | 54% | 57% | 62% | 47% |
Accuracy | 66% | 58% | 60% | 66% |
Table 3. Classification Results With Speed Parameter The speed parameter
performs worse than the distance parameter in the acoustic side channel, a
similar pattern as seen with the radio frequency side channel
### 4.4. Microphone Distance
While observing more fine-grained information leakage is useful to an
attacker, one problem that may impact the success of the attack is the
distance the recording device is away from the robot – in this case, the
smartphone. Naturally, due to the Doppler effect, the intensity of sound (i.e.
loudness) decreases over distances, and one would hypothesise that because of
this the accuracy may be significantly impacted as the distance of recording
increases. In this experiment, two other microphone distances (50cm and 100cm)
are also tested in addition to the baseline recorded at 30cm. While these are
not large recording distances, given the small scale of the robot used for the
evaluation of the attack, these are relatively suitable candidates to be
tested. As seen in Table 4, as the distance the microphone is away the robot
is increased, the accuracy of the attack compared to the baseline decreases by
around 10% at each recording distance step. Notably, this is much more
significant for Z-based movements which were previously described to have
poorer fingerprinting accuracy due to the limited range of motion that does
not cross the recording device (remains stationary and moves vertically). In
this case, a point a future work may be to evaluate the impact on position of
the smartphone around the robot, aside from facing in front. Collectively,
inference from multiple angles may provide better fingerprinting accuracy in
all cases.
MD = Microphone Distance (cm), P = Precision, R = Recall
---
| MD=30 | MD=50 | MD=100
| P | R | P | R | P | R
X | 76% | 81% | 57% | 79% | 75% | 76%
Y | 77% | 78% | 67% | 74% | 67% | 68%
Z | 61% | 71% | 48% | 66% | 45% | 63%
XY | 78% | 80% | 88% | 91% | 64% | 68%
XZ | 68% | 65% | 61% | 52% | 51% | 40%
YZ | 85% | 78% | 83% | 54% | 47% | 35%
XYZ | 72% | 67% | 52% | 39% | 33% | 33%
Accuracy | 75% | 65% | 54%
Table 4. Classification Results With Microphone Distance As the microphone
distance increases away from the robot being recorded, on average the accuracy
decreases around 10% at each step compared to the baseline - more
significantly for Z-based movements
### 4.5. Workflow Recovery
The next step in the evaluation looks at whether entire warehousing workflows
can be reconstructed through the acoustic side channel attack. While a pattern
matching approach can be successful using individual movement fingerprints,
the ability to reconstruct entire workflows may be useful from an auditing
perspective, for example, where offsets in normal movement signals can be
flagged and investigated further. As seen in Table 5, the explored warehousing
workflows can be recovered on average with around 62% accuracy. Notably, the
pick-and-place and packing workflows are recovered with much higher success
than the push and pull workflows. Simply, the former have much more variation
in the pattern of movements and thus the variance helps with fingerprinting.
In the case of push and pull movements, they are highly similar and it can be
hypothesised that only the direction of movement away from the microphone
(i.e. pull is a reverse of push) provides at least some degree of accuracy
between the two.
Workflow | Precision | Recall
---|---|---
Push | 37% | 16%
Pull | 31% | 59%
Pick-and-Place | 100% | 96%
Packing | 97% | 100%
Accuracy | 64%
Table 5. Workflow Reconstruction Results Common warehousing workflows can be
reconstructed in their entirety are better recovered through the acoustic side
channel if they are more complex and varied. Push and pull operations are less
accurate due to the fact they are very similar movements
### 4.6. Impact of VoIP
In certain robotics environments, such as in surgical settings, procedures may
be streamed and/or recorded for viewing, education or research (Muensterer et
al., 2014; Kulkarni et al., 2020; Hosseini et al., 2013). Therefore, it is
important to question how VoIP impacts the audio samples for movements and
workflows and, ultimately, the success of the attack. In many modern VoIP
applications, the Opus codec is the preferred choice (Valin et al., 2012,
2016) given its standardisation and rank of higher quality compared to other
audio formats for a variety of bitrates. To explore this,the open-source
nature of Opus allows for easy implementation to encode and decode the audio
samples and, during decoding, investigate various packet losses. In VoIP
applications, Packet Loss Concealment (PLC) is used as a decoder feature for
receiving data from an unreliable source, which masks the effects of packet
loss in VoIP communications. In realistic settings, packets may arrive late,
be dropped or be corrupted, which may result in not only a lowered audio
quality but in the worst case, dropped parts of the audio or the entire audio
sample entirely. Given that in VoIP applications, a 1% packet loss is
considered an acceptable rate for VoIP to minimise impact on call quality
(James et al., 2004; Amirzade Dana et al., 2020), however in the event of
network failures or availability attacks this may be higher. For completeness,
5 packet losses of 1%, 5%, 10%, 25% and 50% are evaluated. Furthermore, as it
was shown that constant bitrate quality does not perform as well as variable
bitrate quality (Rämö and Toukomaa, 2011), samples are encoded and decoded
with variable bitrate. This experiment used the same model as the previous
experiments, but with a batch size of 256 and 100 epochs of training. As seen
in Table 6, the results for the baseline speed and distance of movement
(12.5mm/s and 1mm respectively) under various packet losses via the Opus codec
can be seen. Interestingly, at low packet loss, the classification accuracy is
around 90% and increases by around 15% compared to the baseline without VoIP
employed. Further, X movements are more accurately fingerprinted across all
packet losses compared to the baseline without VoIP. As the packet loss
reaches more undesirable amounts of 25% and 50%, the accuracy slightly
decreases but the accuracy still remains much higher than the baseline without
VoIP. This may be due to the PLC algorithm switching between CELT or SILK mode
and variable bit rate. Specifically, frames that are deemed important are re-
encoded at a lower bitrate and allows for partial recovery for improtant lost
packets. This may be targetting the movement audio within the sample thus
leading to higher variance among classes.
L = Loss (%), P = Precision, R = Recall |
---|---
| L=1 | L=5 | L=10 | L=25 | L=50 |
P | R | P | R | P | R | P | R | P | R |
X | 99% | 99% | 99% | 100% | 100% | 100% | 100% | 100% | 99% | 100% |
Y | 90% | 94% | 87% | 96% | 90% | 92% | 82% | 97% | 86% | 97% |
Z | 86% | 72% | 88% | 68% | 91% | 73% | 86% | 72% | 90% | 74% |
XY | 88% | 91% | 91% | 88% | 88% | 91% | 94% | 81% | 90% | 81% |
XZ | 89% | 93% | 82% | 83% | 82% | 82% | 80% | 85% | 80% | 85% |
YZ | 86% | 89% | 87% | 88% | 85% | 89% | 91% | 80% | 91% | 85% |
XYZ | 93% | 98% | 94% | 97% | 92% | 96% | 89% | 97% | 90% | 97% |
Accuracy | 90% | 90% | 90% | 88% | 89% |
Table 6. Classification Results (Baseline) With Opus Codec and Packet Loss
Interestingly, the precision and recall remains relatively similar across
packet losses, with a slightly drop in accuracy for undesirable large packet
losses. Notably, there is an increase in accuracy of around 15% compared to
the baseline without the Opus codec employed
## 5\. Discussion
The acoustic side channel attack we propose showcases the potential for
successfully compromising the operational confidentiality of organisations in
which robotic systems under attack are deployed. While many active attacks
have shown to result in potentially devastating consequences, the capabilities
of a passive insider attacker are truly underestimated. In this section, a
discussion on the proposed attack is provided.
### 5.1. Influence of Noise
During the recording of acoustic samples for robot movements, there is likely
some degree of background noise that should be accounted for. Given the
recordings were made in a computer lab, background noise effects may include
the likes of light chatter, keyboard tapping and rolling chairs, among others.
While relatively good accuracy is observed even with the background noise, it
is important to also look into techniques to eliminate such noise to determine
whether this results in better fingerprinting accuracy.
In human audio, sound recordings contain the relative signal of the
oscillations due to density and pressure of air in the ear. In digital audio,
sound waves are encoded in digital form as numerical samples in a continuous
sequence. The recordings taken in this attack are recorded at a sampling rate
of 44.1KHz with 16-bit depth and thus there are $65,536$ possible values the
signal can take in the sequence.
Figure 4. FFT of Acoustic Signal Peaks can be observed at 60Hz corresponding
to electric hum, with other peaks at 150Hz and 200Hz (among others) which may
correlate with robot movement Figure 5. FFT of Acoustic Signal (Filtered) The
amplitude at points correlating with electric hum or those outwidth the human
hearing range are set to 0 (filtered out)
As shown in Figure 4, the amplitude of the frequency content of the acoustic
signal can be observed using the Fast Fourier Transform (FFT). In this attack,
we make use of techniques originally applied to human acoustics, but given
that the robot movements produce sound that is audible to the human ear as
well. Looking at the frequency content, notable amplitude is not found past
1KHz, so this is zoomed in further to 250Hz. There is a notable spike around
60Hz, which is the frequency standard common to alternating current and is an
effect known as electric hum due to electrical noise getting into an acoustic
recording medium. The next largest peaks can be observed at around 150Hz and
200Hz which may correspond with the robot movements. As a first step to noise
reduction/filtering, one technique is amplitude filtering, where the
amplitudes of FFT values to be filtered can be set to 0Hz, to which the
original signal can be recreated using an inverse FFT. In this experiment, the
electric hum, as well as frequencies outwidth the human hearing range of
~20Hz–20KHz are filtered by dropping the amplitude of these ranges. A
depiction of the amplitude drop can be seen in Figure 5. Looking at Table 7,
the accuracy of baseline movement fingerprints can be observed with amplitude
filtering in place. While the accuracy overall decreases by 1% compared to the
baseline without amplitude filtering, the precision for Y and XY movements
increase. This may be due to unfortunate noise events present in these samples
that the filter has rectified. However, there is still a reduction in overall
accuracy, which may mean that electric hum and other peaks may not be the best
indicators of noise to remove when recording a robotics system. In this case,
as a point of future work other noise reduction techniques that have shown to
be successful in other areas, such as stationary or non-stationary spectral
gating (Neumann and Schuller, 1991; Inouye et al., 2014) which reduces noise
in time-domain signals by estimating noise thresholds for the frequency bands
in a signal to gate (mask) noise below the threshold, are worth exploring in
the hope the attack accuracy may increase.
Movement | Precision | Recall
---|---|---
X | 72% | 81%
Y | 75% | 73%
Z | 68% | 72%
XY | 86% | 71%
XZ | 69% | 68%
YZ | 76% | 83%
XYZ | 72% | 70%
Accuracy | 74%
Table 7. Amplitude Filtering Classification Results While the accuracy is
slightly reduced compared to the baseline with no filtering, the precision for
some movements increases further, with better recall seen in most cases
### 5.2. Other VoIP Codecs
Opus is the primary choice for many VoIP applications due to its royalty free
and open source nature, alongside the benefits of higher quality and low-
bandwidth streaming, in comparison with other codecs such as Speex (Valin,
2016) or SILK (Vos et al., 2010) (Opus’ predecessor). While it may be
interesting to evaluate other codecs, Opus is the main choice for the majority
of modern applications, such as Zoom, Teams and Discord (Rajaratnam et al.,
2018; Castro, 2020) and is taking over previously dominating codecs.
### 5.3. Defences
While the attack is successful, and even more so when the attack targets VoIP
communiations, a natural question pertains to countermeasures and defences
against the acoustic side channel attack. In this work, acoustic emanations
result in unintentional information leakages about robot behaviours and can
ultimately lead to the compromise of operational confidentiality.
One defence that could be considered is to make use of vibration- or sound-
reduction mechanisms to hinder the effect of the attack. As seen in Section
4.4, as the microphone distance increases the accuracy of fingerprinting also
decreases. While this is due to the Doppler effect that is naturally at play
with regard to sound intensity (i.e. loudness), a reduction in this from other
means may result in the same outcome of reduced success of fingerprinting.
Techniques in this space include the likes of using vibration isolation pads
(Desai and Patil, [n.d.]) or damping to reduce vibration (Gravagne et al.,
2001; Khan and Li, 2020) for the robot as a whole. In the case of noise
reduction for robot components such as stepper motors, potential defences
include using a clean damper (Ma et al., 2019) or higher resolution stepper
motors.
Another potential defence is to make use of a masking noise, to interfere with
attack inference by distorting the signal related to information leakage in
the acoustic side channel (Backes et al., 2010; Anand and Saxena, 2016; Kim et
al., 2015). Adding a masking signal has shown success, but two challenges need
to be addressed. First, the mask must be similar to the signal requiring
masking to ensure difficult separation. Second, the masking noise should not
cause any degrading effect on usability of the robotic system. For example, if
the masking noise is to cover up other sound such as those used for
emergencies or other operator feedback, then this will be much less than ideal
and potentially lead to catastrophic liabilities.
## 6\. Related Work
While acoustic side channel attacks have not been explored for robotic
systems, enhancing the novelty of this work, there has been previous research
in the area of acoustic side channels. In a similar respect to robotics, the
exploration of information leakage in the acoustic side channel has been
explored for 3D printers (Backes et al., 2010) – some of which making use of
smartphones to carry out the attack (Song et al., 2016; Bilal, 2017) – and
additive manufacturing systems (Chhetri et al., 2017). However, many of these
attacks solely focus on IP theft. The acoustic side channel attack presented
in this work focus solely on the movement of the robot arm and the compromise
of operational confidentiality, which when looking at the bigger picture is
much more valuable to an attacker. Furthermore, the reconstruction of G-code
is an unnecessary extra step as movements which correspond to these can be
inferred from individual movement fingerprinting under the assumption the
robot is operated by an Arduino. Furthermore, while the robot in this work is
operated by an Arduino, the focus is on reconstructing movements from the
acoustic emanations, irrespective of the microcontroller used and thus applies
to robotic systems in general and not those restricted to being operated by an
Arduino.
## 7\. Conclusion
In conclusion, it is clear that even acoustic emanations provide a high level
of accuracy for fingerprinting movements and showcases a highly important
passive side channel attack in the physical domain, which can be carried out
with a fairly cheap smartphone. While more fine-grained movements and entire
workflows in warehousing settings can be inferred, our contributions
demonstrate that the recent usage of VoIP technologies also leave potential
for information leakage through these communication channels, with the result
leaving movement fingeprints to be more accurately reconstructed. This is an
interesting result as it opens up new research questions regarding anonymous
communications to protect robotic systems from acoustic side channel attacks
via VoIP communication networks.
###### Acknowledgements.
The authors are grateful for the support by the Engineering and Physical
Sciences Research Council (11288S170484-102) and the support of the National
Measurement System of the UK Department of Business, Energy & Industrial
Strategy, which funded this work as part of NPL’s Data Science program.
## References
* (1)
* Agarap (2018) Abien Fred Agarap. 2018\. Deep learning using rectified linear units (relu). _arXiv preprint arXiv:1803.08375_ (2018).
* Ahn et al. (2015) Ho Seok Ahn, Min Ho Lee, and Bruce A MacDonald. 2015. Healthcare robot systems for a hospital environment: CareBot and ReceptionBot. In _2015 24th IEEE International Symposium on Robot and Human Interactive Communication (RO-MAN)_. IEEE, 571–576.
* Al-Jabir et al. (2020) A. Al-Jabir, A. Kerwan, M. Nicola, Z. Alsafi, M. Khan, C. Sohrabi, N. O’Neill, C. Iosifidis, M. Griffin, G. Mathew, and R. Agha. 2020\. Impact of the Coronavirus (COVID-19) pandemic on surgical practice - Part 1. _Int J Surg_ 79 (Jul 2020), 168–179.
* Amirzade Dana et al. (2020) Parvaneh Amirzade Dana, Zahra Esmaeilbeig, and Mohammad-Reza Sadeghi. 2020. Reliability enhancement and packet loss recovery of any steganographic method in voice over IP. _Wireless Networks_ 26, 8 (2020), 5817–5823.
* Anand and Saxena (2016) S Abhishek Anand and Nitesh Saxena. 2016. A sound for a sound: Mitigating acoustic side channel attacks on password keystrokes with active sounds. In _International Conference on Financial Cryptography and Data Security_. Springer, 346–364.
* Aschenbrenner et al. (2015) Doris Aschenbrenner, Michael Fritscher, Felix Sittner, Markus Krauß, and Klaus Schilling. 2015\. Teleoperation of an industrial robot in an active production line. _IFAC-PapersOnLine_ 48, 10 (2015), 159–164.
* Atahan et al. (2021) Yunus Atahan, Ahmet Elbir, Abdullah Enes Keskin, Osman Kiraz, Bulent Kirval, and Nizamettin Aydin. 2021. Music Genre Classification Using Acoustic Features and Autoencoders. In _2021 Innovations in Intelligent Systems and Applications Conference (ASYU)_. IEEE, 1–5.
* Avila et al. (2020) Jose Luis Ordoñez Avila, Hector Jimenez, Tania Marquez, Carlos Muñoz, Alberto Max Carrazco, Maria Elena Perdomo, David Miselem, and David Nolasco. 2020. Study Case: Teleoperated Voice Picking Robots prototype as a logistic solution in Honduras. In _2020 5th International Conference on Control and Robotics Engineering (ICCRE)_. IEEE, 19–24.
* Bachu et al. (2008) RG Bachu, S Kopparthi, B Adapa, and BD Barkana. 2008\. Separation of voiced and unvoiced using zero crossing rate and energy of the speech signal. In _American Society for Engineering Education (ASEE) zone conference proceedings_. American Society for Engineering Education, 1–7.
* Backes et al. (2010) Michael Backes, Markus Dürmuth, Sebastian Gerling, Manfred Pinkal, Caroline Sporleder, et al. 2010\. Acoustic $\\{$Side-Channel$\\}$ Attacks on Printers. In _19th USENIX Security Symposium (USENIX Security 10)_.
* Bartoš et al. (2021) Michal Bartoš, Vladimír Bulej, Martin Bohušík, Ján Stanček, Vitalii Ivanov, and Peter Macek. 2021\. An overview of robot applications in automotive industry. _Transportation Research Procedia_ 55 (2021), 837–844.
* Bilal (2017) Muhammad Bilal. 2017\. A review of internet of things architecture, technologies and analysis smartphone-based attacks against 3D printers. _arXiv preprint arXiv:1708.04560_ (2017).
* Bonaci et al. (2015) Tamara Bonaci, Jeffrey Herron, Tariq Yusuf, Junjie Yan, Tadayoshi Kohno, and Howard Jay Chizeck. 2015. To make a robot secure: An experimental analysis of cyber security threats against teleoperated surgical robots. _arXiv preprint arXiv:1504.04339_ (2015).
* Castro (2020) Rodolfo Castro. 2020\. Is your company’s network ready for Microsoft teams.
* Chhetri et al. (2017) Sujit Rokka Chhetri, Arquimedes Canedo, and Mohammad Abdullah Al Faruque. 2017. Confidentiality breach through acoustic side-channel in cyber-physical additive manufacturing systems. _ACM Transactions on Cyber-Physical Systems_ 2, 1 (2017), 1–25.
* Cho and Bello (2013) Taemin Cho and Juan P Bello. 2013. On the relative importance of individual components of chord recognition systems. _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ 22, 2 (2013), 477–492.
* Chollet et al. (2015) François Chollet et al. 2015\. Keras. https://keras.io.
* Dalen et al. (2019) ASHM Dalen, J Legemaate, WS Schlack, DA Legemate, and MP Schijven. 2019. Legal perspectives on black box recording devices in the operating environment. _Journal of British Surgery_ 106, 11 (2019), 1433–1441.
* Demuth et al. (2014) Howard B Demuth, Mark H Beale, Orlando De Jess, and Martin T Hagan. 2014. _Neural network design_. Martin Hagan.
* Desai and Patil ([n.d.]) Tanvi D Desai and SR Patil. [n.d.]. Experimental and Numerical Analysis of Vibration Isolation Materials on Vibration Reduction within Plazma Torch. ([n. d.]).
* Doppler (1903) Christian Doppler. 1903\. _Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels: Versuch einer das Bradley’sche Aberrations-Theorem als integrirenden Theil in sich schliessenden allgemeineren Theorie_. K. Böhm Gesellschaft der Wissenschaften.
* Dunne and Campbell (1997) Rob A Dunne and Norm A Campbell. 1997. On the pairing of the softmax activation and cross-entropy penalty functions and the derivation of the softmax activation function. In _Proc. 8th Aust. Conf. on the Neural Networks, Melbourne_ , Vol. 181. Citeseer, 185\.
* Eckle and Schmidt-Hieber (2019) Konstantin Eckle and Johannes Schmidt-Hieber. 2019. A comparison of deep networks with ReLU activation function and linear spline-type methods. _Neural Networks_ 110 (2019), 232–242.
* Grabowski et al. (2021) Andrzej Grabowski, Jarosław Jankowski, and Mieszko Wodzyński. 2021. Teleoperated mobile robot with two arms: the influence of a human-machine interface, VR training and operator age. _International Journal of Human-Computer Studies_ 156 (2021), 102707\.
* Gravagne et al. (2001) Ian A Gravagne, Christopher D Rahn, and Ian D Walker. 2001\. Good vibrations: a vibration damping setpoint controller for continuum robots. In _Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No. 01CH37164)_ , Vol. 4. IEEE, 3877–3884.
* Greenwood (1997) Donald D Greenwood. 1997\. The Mel Scale’s disqualifying bias and a consistency of pitch-difference equisections in 1956 with equal cochlear distances and equal frequency ratios. _Hearing research_ 103, 1-2 (1997), 199–224.
* Hannaford et al. (2012) Blake Hannaford, Jacob Rosen, Diana W Friedman, Hawkeye King, Phillip Roan, Lei Cheng, Daniel Glozman, Ji Ma, Sina Nia Kosari, and Lee White. 2012\. Raven-II: an open platform for surgical robotics research. _IEEE Transactions on Biomedical Engineering_ 60, 4 (2012), 954–959.
* Hosseini et al. (2013) Azamossadat Hosseini, Hamid Moghaddasi, Samad Sajadi, and Mozhgan Karimi. 2013. Telesurgery information management systems in university hospitals of Tehran. _Archives of Advances in Biosciences_ 4, 4 (2013).
* Inouye et al. (2014) Joshua M Inouye, Silvia S Blemker, and David I Inouye. 2014\. Towards undistorted and noise-free speech in an MRI scanner: correlation subtraction followed by spectral noise gating. _The Journal of the Acoustical Society of America_ 135, 3 (2014), 1019–1022.
* James et al. (2004) Jim H James, Bing Chen, and Laurie Garrison. 2004. Implementing VoIP: a voice transmission performance progress report. _IEEE Communications Magazine_ 42, 7 (2004), 36–41.
* Jiang et al. (2002) Dan-Ning Jiang, Lie Lu, Hong-Jiang Zhang, Jian-Hua Tao, and Lian-Hong Cai. 2002. Music type classification by spectral contrast feature. In _Proceedings. IEEE International Conference on Multimedia and Expo_ , Vol. 1. IEEE, 113–116.
* Khan and Li (2020) Ameer Hamza Khan and Shuai Li. 2020. Sliding mode control with PID sliding surface for active vibration damping of pneumatically actuated soft robots. _IEEE Access_ 8 (2020), 88793–88800.
* Kim et al. (2015) Younghyun Kim, Woo Suk Lee, Vijay Raghunathan, Niraj K Jha, and Anand Raghunathan. 2015. Vibration-based secure side channel for medical devices. In _2015 52nd ACM/EDAC/IEEE Design Automation Conference (DAC)_. IEEE, 1–6.
* Kingma and Ba (2014) Diederik P Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_ (2014).
* Klapuri and Davy (2007) Anssi Klapuri and Manuel Davy. 2007. Signal processing methods for music transcription. (2007).
* Kos et al. (2013) Marko Kos, Zdravko Kačič, and Damjan Vlaj. 2013\. Acoustic classification and segmentation using modified spectral roll-off and variance-based features. _Digital Signal Processing_ 23, 2 (2013), 659–674.
* Krizhevsky et al. (2012) Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. 2012\. Imagenet classification with deep convolutional neural networks. In _Advances in neural information processing systems_. 1097–1105.
* Kulkarni et al. (2020) Sushmita Kulkarni, Dattaprasad A Torse, and Deepak Kulkarni. 2020. A Cloud based Medical Transcription using Speech Recognition Technologies. _International Research Journal of Engineering and Technology (IRJET)_ 7, 5 (2020), 6160–6163.
* Laghari et al. (2020) Asif Ali Laghari, Rashid Ali Laghari, Asif Ali Wagan, and Aamir Iqbal Umrani. 2020. Effect of packet loss and reorder on quality of audio streaming. _EAI Endorsed Transactions on Scalable Information Systems_ 7, 25 (2020).
* Le et al. (2011) Phu Ngoc Le, Eliathamby Ambikairajah, Julien Epps, Vidhyasaharan Sethu, and Eric HC Choi. 2011\. Investigation of spectral centroid features for cognitive load classification. _Speech Communication_ 53, 4 (2011), 540–551.
* Li et al. (2017) Chunxu Li, Chenguang Yang, Jian Wan, Andy SK Annamalai, and Angelo Cangelosi. 2017. Teleoperation control of Baxter robot using Kalman filter-based sensor fusion. _Systems Science & Control Engineering_ 5, 1 (2017), 156–167.
* Li and Yuan (2017) Yuanzhi Li and Yang Yuan. 2017. Convergence analysis of two-layer neural networks with relu activation. In _Advances in neural information processing systems_. 597–607.
* Ma et al. (2019) Xinbo Ma, Pak Kin Wong, and Jing Zhao. 2019. Practical multi-objective control for automotive semi-active suspension system with nonlinear hydraulic adjustable damper. _Mechanical Systems and Signal Processing_ 117 (2019), 667–688.
* Martinez et al. (2012) Jorge Martinez, Hector Perez, Enrique Escamilla, and Masahisa Mabo Suzuki. 2012. Speaker recognition using Mel frequency Cepstral Coefficients (MFCC) and Vector quantization (VQ) techniques. In _CONIELECOMP 2012, 22nd International Conference on Electrical Communications and Computers_. IEEE, 248–251.
* McClean et al. (2013) Jarrod McClean, Christopher Stull, Charles Farrar, and David Mascarenas. 2013. A preliminary cyber-physical security assessment of the robot operating system (ros). In _Unmanned Systems Technology XV_ , Vol. 8741. International Society for Optics and Photonics, 874110.
* McFee et al. (2015) Brian McFee, Colin Raffel, Dawen Liang, Daniel P Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto. 2015. librosa: Audio and music signal analysis in python. In _Proceedings of the 14th python in science conference_ , Vol. 8. Citeseer, 18–25.
* Muensterer et al. (2014) Oliver J Muensterer, Martin Lacher, Christoph Zoeller, Matthew Bronstein, and Joachim Kübler. 2014. Google Glass in pediatric surgery: an exploratory study. _International journal of surgery_ 12, 4 (2014), 281–289.
* Müller (2015) Meinard Müller. 2015\. Fundamentals of Music Processing. (2015).
* Neumann and Schuller (1991) Ingrid Neumann and Gerd Schuller. 1991. Spectral and temporal gating mechanisms enhance the clutter rejection in the echolocating bat, Rhinolophus rouxi. _Journal of comparative physiology A_ 169, 1 (1991), 109–116.
* Ortega et al. (2018) Martín Ortega Ortega, Gustavo Chafla Altamirano, and Mara Falconí Abad. 2018. Evaluation of the voice quality and QoS in real calls using different voice over IP codecs. In _2018 IEEE Colombian Conference on Communications and Computing (COLCOM)_. IEEE, 1–6.
* Panagiotakis and Tziritas (2005) Costas Panagiotakis and Georgios Tziritas. 2005. A speech/music discriminator based on RMS and zero-crossings. _IEEE Transactions on multimedia_ 7, 1 (2005), 155–166.
* Paulus et al. (2010) Jouni Paulus, Meinard Müller, and Anssi Klapuri. 2010\. State of the Art Report: Audio-Based Music Structure Analysis.. In _Ismir_. Utrecht, 625–636.
* Peng and Choi (2013) Yinni Peng and Susanne YP Choi. 2013. Mobile phone use among migrant factory workers in south China: Technologies of power and resistance. _The China Quarterly_ 215 (2013), 553–571.
* Polydoros and Nalpantidis (2016) Athanasios S Polydoros and Lazaros Nalpantidis. 2016. A reservoir computing approach for learning forward dynamics of industrial manipulators. In _2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 612–618.
* Quarta et al. (2017) Davide Quarta, Marcello Pogliani, Mario Polino, Federico Maggi, Andrea Maria Zanchettin, and Stefano Zanero. 2017. An experimental security analysis of an industrial robot controller. In _2017 IEEE Symposium on Security and Privacy (SP)_. IEEE, 268–286.
* Rajaratnam et al. (2018) Krishan Rajaratnam, Kunal Shah, and Jugal Kalita. 2018\. Isolated and ensemble audio preprocessing methods for detecting adversarial examples against automatic speech recognition. _arXiv preprint arXiv:1809.04397_ (2018).
* Rämö and Toukomaa (2011) Anssi Rämö and Henri Toukomaa. 2011. Voice quality characterization of IETF Opus codec. In _Twelfth Annual Conference of the International Speech Communication Association_.
* Rao and Vuppala (2014) K Sreenivasa Rao and Anil Kumar Vuppala. 2014. _Speech processing in mobile environments_. Springer.
* Reuters (2019) Reuters. 2019. U.S. companies put record number of robots to work in 2018\. https://www.reuters.com/article/us-usa-economy-robots/u-s-companies-put-record-number-of-robots-to-work-in-2018-idUSKCN1QH0K0.
* Rueckert et al. (2017) Elmar Rueckert, Moritz Nakatenus, Samuele Tosatto, and Jan Peters. 2017. Learning inverse dynamics models in o (n) time with lstm networks. In _2017 IEEE-RAS 17th International Conference on Humanoid Robotics (Humanoids)_. IEEE, 811–816.
* Saun et al. (2019) Tomas J Saun, Kevin J Zuo, and Teodor P Grantcharov. 2019\. Video technologies for recording open surgery: a systematic review. _Surgical innovation_ 26, 5 (2019), 599–612.
* Shah et al. (2022) Ryan Shah, Chuadhry Mujeeb Ahmed, and Shishir Nagaraja. 2022\. Can You Still See Me?: Reconstructing Robot Operations Over End-to-End Encrypted Channels. _arXiv preprint arXiv:2205.08426_ (2022).
* Song et al. (2016) Chen Song, Feng Lin, Zhongjie Ba, Kui Ren, Chi Zhou, and Wenyao Xu. 2016\. My smartphone knows what you print: Exploring smartphone-based side-channel attacks against 3d printers. In _Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security_. 895–907.
* Sung and Gill (2001) Gyung Tak Sung and Inderbir S Gill. 2001. Robotic laparoscopic surgery: a comparison of the da Vinci and Zeus systems. _Urology_ 58, 6 (2001), 893–898.
* Tewari et al. (2002) Ashutosh Tewari, James Peabody, Richard Sarle, Guruswami Balakrishnan, Ashok Hemal, Alok Shrivastava, and Mani Menon. 2002\. Technique of da Vinci robot-assisted anatomic radical prostatectomy. _Urology_ 60, 4 (2002), 569–572.
* Valin (2016) Jean-Marc Valin. 2016\. Speex: A free codec for free speech. _arXiv preprint arXiv:1602.08668_ (2016).
* Valin et al. (2016) Jean-Marc Valin, Gregory Maxwell, Timothy B Terriberry, and Koen Vos. 2016. High-quality, low-delay music coding in the opus codec. _arXiv preprint arXiv:1602.04845_ (2016).
* Valin et al. (2012) Jean-Marc Valin, Koen Vos, and Timothy Terriberry. 2012\. _Definition of the Opus audio codec_. Technical Report.
* Vodafone (2019) Vodafone. 2019. 5GDig: The winners – from Skype for surgeons to AR. https://www.vodafone.com/news/technology/5gdig-winners-2019-supponor-ar-surgeonmate.
* Vos et al. (2010) Koen Vos, Soeren Jensen, and Karsten Soerensen. 2010. SILK speech codec. _IETF draft_ 30 (2010).
* Zhang and Sabuncu (2018) Zhilu Zhang and Mert Sabuncu. 2018. Generalized cross entropy loss for training deep neural networks with noisy labels. In _Advances in neural information processing systems_. 8778–8788.
|
# The 6Li$(p,\gamma)^{7}$Be reaction rate in the light of the new LUNA data
S. B. Dubovichenko Fesenkov Astrophysical Institute ”NCSRT” ASA MDASI RK,
050020 Almaty, Kazakhstan al-Farabi Kazakh National University, 050040
Almaty, Kazakhstan A. S. Tkachenko Fesenkov Astrophysical Institute ”NCSRT”
ASA MDASI RK, 050020 Almaty, Kazakhstan R. Ya. Kezerashvili New York City
College of Technology, City University of New York, Brooklyn, 11201 New York,
USA Graduate School and University Center, City University of New York, 10016
New York, USA N. A. Burkova al-Farabi Kazakh National University, 050040
Almaty, Kazakhstan A. V. Dzhazairov-Kakhramanov Fesenkov Astrophysical
Institute ”NCSRT” ASA MDASI RK, 050020 Almaty, Kazakhstan
###### Abstract
We present new calculations of the astrophysical $S-$factor and reaction rate
for the 6Li$(p,\gamma)^{7}$Be reaction at energies of 10 keV to 5 MeV in the
framework of a modified potential cluster model with forbidden states,
including low lying resonances. The astrophysical $S(E)-$factor is compared
with the available experimental data and calculations done within different
models. The results for the $S-$factor are in good agreement with the data set
(for $E<0.3$ MeV) and calculations (for $E<0.6$ MeV) of LUNA collaboration
(Phys. Rev. C 102, 052802, 2020). The recommended extrapolated zero value
$S(0)$ turned out to be 101 eV $\cdot$ b. Using the theoretical total cross-
sections, the 6Li$(p,\gamma)^{7}$Be capture reaction rate is calculated at
temperatures ranging from 0.01 to 10 $T_{9}$ and compared with NACRE and NACRE
II. Analytical expressions for the $S-$factor and reaction rate are given, and
the effect of low-lying resonances on the reaction rate is estimated. We
suggest to update the NACRE and NACRE II databases in light of the new LUNA
data and present calculations.
low and astrophysical energies, $p^{6}$Li system, thermonuclear reaction rate,
potential cluster model
††preprint: APS/123-QED
## I Introduction
The radiative 6Li$(p,\gamma)^{7}$Be capture reaction is of great interest in
nuclear astrophysics [1, 2]. Since 1955 the 6Li$(p,\gamma)^{7}$Be reaction at
low energies has been studied by several experimental groups [3, 4, 5, 6, 7,
8, 9, 10, 11]. Measurements of the astrophysical $S-$factor of this reaction
were limited to the energy range of 35 keV to 1.2 MeV. The astrophysical
$S-$factor and reaction rate were studied in the framework of different
theoretical approaches and methods [12, 13, 14, 15, 16, 17, 18, 11]. A
detailed review of the theoretical and experimental current status is given in
Ref. [9].
In 2020, new experimental data were obtained in the Laboratory for Underground
Nuclear Astrophysics (LUNA) [9], and it excluded the possibility of resonance
mentioned in [8]. It seems challenging to consider this reaction in the
astrophysical energy range, for which experimental data are available. In
particular, our interest is the re-examination of $S-$factor and has two foci:
i. to consider 6Li$(p,\gamma)^{7}$Be within the framework of the modified
potential cluster model with the classification of bound and scattering states
according to Young’s orbital diagrams [28]; ii. to describe the $S-$factor
using all available experimental data and to obtain the reaction rate. While
assessing the reliability of MPCM, it is reasonable to extend the energy
interval up to 5 MeV to estimate the role of resonances in this energy range.
Ten years ago, we studied this reaction, but limited ourselves to an energy
range up to 1 MeV. Moreover, we did not take into account resonances, nor did
we consider the reaction rate [14, 15].
In this paper, we investigate the energy dependence of the astrophysical
$S-$factor of $\ {}^{6}$Li$(p,\gamma)^{7}$Be reaction at energies of 10 keV to
5 MeV. By considering several resonances, including a wide resonance at
$E_{x}=9.9$ MeV, the reaction rate in the temperature range of 0.01 to 10
$T_{9}$ is calculated. We demonstrate that it is possible to correctly convey
the available experimental data based on potentials that are consistent with
the energies of bound states and their asymptotic constants. For the
scattering potentials, we use the parameters consistent with the resonance
spectrum of the final nucleus.
The results obtained for the reaction rate are approximated by curves of a
particular type to simplify their use in applied research. These results apply
to problems in nuclear astrophysics related to light atomic nuclei and ultra-
low energies.
This work is organized as follows. Secs. II and III present the theoretical
framework and fundamentals of the modified potential cluster model (MPCM),
constructing principles of discrete states potentials and the description of
the $p^{6}$Li channel in the continuous spectrum in the MPCM. Cross-section
for the radiative capture processes and discussion of wave functions
asymptotics are given in Sec. IV. Classification of cluster states and bound
states potentials are given in Sec. V, and Sec. VI contains the description of
the $p^{6}$Li channel in the continuous spectrum. Sec. VII is devoted to the
astrophysical $S-$factor and the 6Li$(p,\gamma)^{7}$Be reaction rate.
Appendices A and B include the description of the finite-difference method
that we are using in present calculations and a table of numerical values of
the $p^{6}$Li reaction rate in the temperature range of 0.001 $T_{9}$ to 10
$T_{9}$, respectively. We outline conclusions in Sec. VIII.
## II Theoretical framework
Charged-particle induced reactions represent one of the main inputs in stellar
evolution. There are a number of theoretical methods used for description of
nuclear reactions at stellar energies that are based on fundamental principles
of quantum mechanics [20].
Since its first application in 1963 [21], the potential model approach has a
special place among models for description of low energy reactions. However,
over the course of over a half century, this model has been significantly
modified and improved. Below we present the fundamentals of the modified
potential cluster model (MPCM), where the Young diagrams are used for the
classification of orbital states and construction of potentials ([28] and
Refs. herein).
The basic features of the MPCM approach are as follows:
1. 1.
The MPCM is a two-particle model that accounts for the internal
characteristics of clusters: their sizes, charges, masses, quadrupole and
magnetic momenta, which are used to calculate the reaction total cross-
sections or other characteristics of the final nucleus.
2. 2.
The classification of cluster states is performed according to Young’s orbital
diagrams, leading to the concept of forbidden states in some partial waves
[28]. The Pauli principle is implemented via exclusion of the forbidden states
(FSs), manifesting in proper node behavior of the radial wave function (WF).
Forbidden states that lead to low-lying bound states are not physically
realized due to the orthogonality of corresponding functions and allowed state
functions.
3. 3.
The Gaussian type inter-cluster interaction potentials are constructed, taking
into account these forbidden states in certain partial waves. For each partial
wave with specified quantum numbers, the potential is constructed with two
parameters, assuming it depends explicitly on Young’s orbital diagrams.
4. 4.
Potentials of the bound states (BSs) are constructed based on asymptotic
constants (AC) and binding energies. Potentials of the scattering processes
are constructed based on the spectra of the final nucleus or the scattering
phase shifts of the particles of the input channel. Parameters of the
potentials are fixed or variable within the AC error intervals and vary within
the energy or width errors of resonant or excited states.
5. 5.
The radial WFs of the allowed states of the continuous and discrete spectra
are tailored appropriately using correct asymptotics.
In light of the new experimental LUNA data [9], we reexamine the
6Li$(p,\gamma)^{7}$Be reaction $S-$factor and reaction rate within the
framework of the MPCM.
### Classification of cluster states
The total wave functions (WF) have the form of an antisymmetrized product of
completely antisymmetric internal wave functions of clusters
$\Psi(1,\dots,A_{1})=\Psi(\mathbf{R_{1}})$ and
$\Psi(A_{1}+1,\dots,A)=\Psi(\mathbf{\ R_{2}})$, multiplied by the
corresponding wave function $\Phi(\mathbf{R})$ of relative motion [22, 23, 24]
$\Psi=\hat{A}\\{\Psi(\mathbf{R_{1}})\Psi(\mathbf{R_{2}})\Phi(\mathbf{R})\\}.$
(1)
In Eq. (1) $\hat{A}$ is the antisymmetrization operator that permutes nucleons
from the clusters $A_{1}$ and $A_{2}$, $\mathbf{\ R_{1}}$ and $\mathbf{R_{2}}$
are the center-of-mass radius-vectors of the clusters, and
$\mathbf{R=R}_{1}\mathbf{-R}_{2}$ is the relative motion coordinate.
The wave functions (1) are characterized by specific quantum numbers,
including $JLS$ — total momentum, orbital quantum momentum, and spin,
respectively — and Young’s diagrams $\\{f\\}$, which determine the orbital
part of WF permutation symmetry of the relative motion of the clusters.
In the general case, the possible Young’s orbital diagram $\\{f\\}_{L}$ of
some nucleus $A(\\{f\\})$, consisting of two parts
$A_{1}(\\{f_{1}\\})+A_{2}(\\{f_{2}\\})$, is the direct outer product of
Young’s orbital diagrams $\\{f\\}_{L}=\\{f_{1}\\}_{L}\times\\{f_{2}\\}_{L}$
and is determined by Littlewood’s theorem [23, 24]. According to Elliot’s
theorem, each Young’s diagram is associated with a certain orbital angular
momentum or their combination.
Spin-isospin diagrams are the direct inner product of the spin and isospin
Young diagrams of a nucleus consisting of $A$ nucleons
$\\{f\\}_{ST}=\\{f\\}_{S}\otimes\\{f\\}_{T}$. For a system with no more than
eight particles such diagrams are provided in Table C of Ref. [25]. A detailed
procedure for defining the corresponding momenta can be found in the classical
monograph [26]. Let us note that in Ref. [26] the definition for inner and
outer products is reverses.
The total Young’s diagram of the nucleus is defined as the direct inner
product of the orbital and spin-isospin diagram
$\\{f\\}=\\{f\\}_{L}\otimes\\{f\\}_{ST}$. The total wave function of the
system under antisymmetrization does not vanish identically, only if it
contains an antisymmetric component $\\{1^{N}\\}$, where $N$ is the number of
nucleons. In this case the conjugates $\\{f\\}_{L}$ and $\\{f\\}_{ST}$ are
multiplied. Therefore, the diagrams $\\{f\\}_{L}$ conjugated to $\\{f\\}_{ST}$
are allowed in this channel. All other orbital symmetries are forbidden since
they lead to zero total wave function of the particle system after
antisymmetrization.
## III The potentials construction within the MPCM
Let us describe in more detail the procedure for constructing the intercluster
partial potentials. Below we define the criteria and outline the sequence for
finding parameters for the potentials and indicating their errors as well as
ambiguities.
### III.1 Discrete states
For the bound states of two clusters, the interaction potentials within the
framework of the MPCM are constructed based on the requirement imposed to
describe the main observable characteristics of such a nucleus. In this case,
the potential parameters are fixed. It should be noted that this requirement
is an idealized scenario that exists in the nucleus since it assumes that the
ground state (GS) is a two-body single channel with probability closed to
unity. First, we find the parameters of the bound state potentials. For the GS
with a given number of bound allowed and forbidden states in the partial wave,
these parameters are fixed unambiguously in terms of the binding energy, the
radius of the nucleus, and the AC. When constructing the partial interaction
potentials in the MPCM, it is assumed that interactions depend not only on the
orbital angular momentum $L$ but also on the total spin $S$ and the total
angular momentum $J$ of the system and also depend on Young’s orbital
diagrams. As in earlier work [28], we use Gaussian interaction potentials,
which depend on the quantum numbers $JLS$, and Young’s diagrams $\\{f\\}_{L}$.
Therefore, for different $JLS$, we have different values of the parameters of
the partial potentials.
The accuracy of determining the parameters of the BS potential is connected
directly with the accuracy of the AC. The potential does not contain any other
ambiguities, since according to Young’s diagrams, the classification of states
makes it possible to unambiguously fix the number of bound forbidden and
allowed states in a given partial wave. The number of bound states ultimately
determines the depth of the potential, while the width depends entirely on the
value of the AC. If one fixes two parameters of the potential using two
particular quantities — the binding energy and the AC — the error of the
binding energy is seen to be much less than that of the AC.
It should be noted that any calculations of the charge radius reflect the
errors of the underlying model. In any model, the magnitude of such a radius
depends on the integral of the model wave functions, thereby compounding
sources of error. At the same time, the values of AC are determined from the
asymptotic behavior of the model WFs at one point and contain significantly
less error. The potentials of the BSs are constructed to obtain the best
agreement with the values of the AC extracted independently from the
experimental data. For more details, see Ref. [27].
### III.2 Continuum states
For the potentials of the continuous spectrum, the intercluster potential of
the nonresonant scattering process for a given number of allowed and forbidden
BSs in the considered partial wave is also constructed quite unambiguously
based on the scattering phase shifts. The accuracy of the potential parameters
sometimes as high as 20–30%, is associated with the precision of the extracted
scattering phase shifts from experimental data. For the 6Li$(p,\gamma)^{7}$Be
reaction, the potential is unambiguous since the classification, according to
Young’s diagrams, makes it possible to fix the number of bound states. This
completely determines the potential depth, and its width is determined by the
shape of the scattering phase shifts.
When constructing the nonresonant scattering potential based on the data for
the nuclear spectra, it is difficult to estimate the accuracy of the
parameters even for a given number of BSs. However, one can expect that it
will not exceed the error discussed above. This potential should lead to a
scattering phase shift close to zero or rise to a smoothly decreasing phase
shift at low energies, since there are no resonance levels in the spectra of
the nucleus.
In resonance scattering, when a relatively narrow resonance is present in the
partial wave at low energies for a given number of BSs, the potential is
constructed completely unambiguously. The accuracy of determining the
parameters of the interaction potentials is determined by the following
factors. The depth of the potential depends on the resonance energy $E_{x}$
and the number of BSs. The width is determined by the accuracy of the
experimental values of the level width $\Gamma$.
The error of the parameters, approximately 5–10%, usually does not exceed the
error of the energy level width. This also applies to the construction of the
partial potential from the resonant scattering phase shifts and the
determination of its parameters from the spectral resonance of the nucleus
[19, 28].
## IV Cross-section and WFs asymptotics
To calculate the total cross-sections of radiative capture processes, we use
the well-known formula for the transitions of $NJ$ multipolarity [19, 28]
$\begin{gathered}\sigma(NJ,J_{f})=\frac{8\pi Ke^{2}}{\hbar^{2}\
k^{3}}\frac{\mu}{(2S_{1}+1)(2S_{2}+1)}\frac{J+1}{J[(2J+1)!!]^{2}}A_{J}^{2}(NJ,K)\times\\\
\times\sum\limits_{L_{i},J_{i}}P_{J}^{2}(NJ,J_{f},J_{i})I_{J}^{2}(k,J_{f},J_{i})\end{gathered}$
(2)
where the matrix elements of orbital $EJ(L)$-transitions have the following
form $(S=S_{i}=S_{f})$
$P_{J}^{2}(EJ,J_{f},J_{i})=\delta_{S_{i}S_{f}}[(2J+1)(2L_{i}+1)(2J_{i}+1)(2J_{f}+1)](L_{i}0J0|L_{f}0)^{2}\begin{Bmatrix}L_{i}&S&J_{i}\\\
J_{f}&J&L_{f}\end{Bmatrix},$ (3)
$A_{J}(EJ,K)=K^{J}\mu^{J}\left(\frac{Z_{1}}{m_{1}^{J}}+(-1)^{J}\frac{Z_{2}}{m_{2}^{J}}\right),\
\ \ \ I_{J}(k,J_{f},J_{i})=\langle\chi_{f}|r^{J}|\chi_{i}\rangle,$ (4)
and the matrix elements of the magnetic $M1(S)$-transition are written as
follows $(S=S_{i}=S_{f},L=L_{i}=L_{f})$
$P_{1}^{2}(M1,J_{f},J_{i})=\delta_{S_{i}S_{f}}\delta_{L_{i}L_{f}}[S(S+1)(2S+1)(2J_{i}+1)(2J_{f}+1)]\begin{Bmatrix}S&L&J_{i}\\\
J_{f}&1&S\end{Bmatrix},$ (5) $A_{1}(M1,K)=\frac{\hbar
K}{m_{0}c}\sqrt{3}\left(\muup_{1}\frac{m_{2}}{m_{1}+m_{2}}-\muup_{2}\frac{m_{1}}{m_{1}+m_{2}}\right),\
\ \ \ I_{J}(k,J_{f},J_{i})=\langle\chi_{f}|r^{J-1}|\chi_{i}\rangle,\ J=1.$ (6)
In Eqs. (2)–(6) $K=E_{\gamma}/\hbar c$ is the wave number of the emitted
photon with energy $E_{\gamma}$, $m_{1}$, $m_{2}$ and $\muup_{1}$, $\muup_{2}$
are the masses and magnetic momenta of the clusters, respectively, and $\mu$
is the reduced mass of the system. Namely, in the present calculations for the
reaction 6Li$(p,\gamma)^{7}$Be, $m_{1}\equiv m_{p}=1.00727646677$ amu,
$m_{2}\equiv m_{{}^{6}\text{Li}}=6.0151232$ amu [29] and
$\muup_{1}\equiv\muup_{p}=2.792847\muup_{0},$
$\muup_{2}\equiv\muup_{{}^{6}\text{Li}}=0.822\muup_{0}$ [30, 31], where
$\muup_{0}$ is nuclear magneton, $\hbar^{2}/m_{0}=41.4686$ MeV$\cdot$fm2,
where $m_{0}=931.494$ MeV is the atomic mass unit (amu).
The point-like Coulomb potential is of the form
$V_{\text{Coul}}\text{(MeV)}=1.439975Z_{1}Z_{2}/r$, where $r$ is the relative
distance between the particles of the channel in fm, and $Z_{1}$ and $Z_{2}$
are the charges in units of the elementary charge. The Coulomb parameter
$\eta=\mu Z_{1}Z_{2}e^{2}/k\hbar^{2}$ is represented in the form
$\eta=3.44476\cdot 10^{-2}\ Z_{1}Z_{2}\mu/k$, where $k$ is the wavenumber in
fm-1 and is determined by the energy $E_{c.m.}$ of the interacting particles,
$k^{2}=2\mu E_{c.m.}/\hbar^{2}$.
In our calculations, we use the dimensionless AC denoted as $C_{\text{w}}$
[32]
$\chi_{L}(r)=\sqrt{2k_{0}}\ C_{\text{w}}\ W_{-\eta L+1/2}(2k_{0}r).$ (7)
A dimensional asymptotic constant $C$ is related to the asymptotic
normalization coefficient (ANC) $A_{NC}$ by the expression [27]
$A_{NC}=\sqrt{S_{F}}C,$ (8)
where $S_{F}$ is the spectroscopic factor and $C$ is the dimensional AC that
can be represented using the asymptotics of the WF
$\chi_{L}(r)\xrightarrow[r\rightarrow R]{}CW_{-\eta
L+1/2}\left(2k_{0}r\right).$ (9)
In Eq. (9) $R$ is the large distance where the nuclear potential vanishes and
$\chi_{L}(r)$ is the wave function of the bound state obtained from the
solution of the radial Schr$\ddot{\text{o}}$dinger equation and normalized to
unity. The Whittaker function $W_{-\eta L+1/2}$ of the bound state determines
the asymptotic behaviour of the WF. The wave number $k_{0}$ is related to the
channel binding energy $E_{b}$.
For a continuous spectrum, the function $\chi_{i}$ found numerically is
matched to asymptotics $u_{L}(R)$ of the form
$N_{L}u_{L}(r)\xrightarrow[r\rightarrow
R]{}F_{L}(kr)+\tan(\deltaup_{S,L}^{J})G_{L}(kr).$ (10)
Here $F_{L}$ and $G_{L}$ are Coulomb regular and irregular functions [33].
They are the solutions of the Schr$\ddot{\text{o}}$dinger equation with the
Coulomb potential. $\deltaup_{S,L}^{J}$ are the scattering phase shifts
depending on the $JLS$ momenta of the system and $N_{L}$ is the normalizing
constant of the numerical radial function $u_{L}(R)$ for the continuum.
## V Cluster states classification and the BS potentials
Consider the classification of the BSs of the $p^{6}$Li system according to
Young’s diagrams. In this case there is only one Young’s orbital diagram for
the 6Li nucleus $\\{42\\}$. It is believed that the system’s potentials are
dependent on the diagrams or combinations of these diagrams in various states.
Thus, if the orbital diagram $\\{42\\}$ allowed in the 2H4He-cluster channel
is accepted for the 6Li nucleus, then the $p^{6}$Li system with spin $S=1/2$
contains a forbidden level with diagram $\\{52\\}$ and orbital momenta of
$L=0,2$, and the allowed states with configurations $\\{43\\}$ for $L=1,3$ and
$\\{421\\}$ for $L=1,2$. Hence, the $p^{6}$Li potentials must have a forbidden
state related to $\\{52\\}$ in the $S$ wave. The allowed bound state
corresponds to the $P$ wave with the two Young’s diagrams $\\{43\\}$ and
$\\{421\\}$. In the quartet spin channel $S=3/2$ of the system, only one
diagram $\\{421\\}$ is allowed for $L=1,2$ [28]. Since there are two allowed
diagrams $\\{43\\}$ and $\\{421\\}$ in the doublet spin state of the $p^{6}$Li
system, the scattering states turn out to be mixed in orbital symmetries. At
the same time, only one allowed diagram $\\{43\\}$ usually corresponds to the
doublet ground state of the 7Be nucleus in the $p^{6}$Li channel with
$J^{\pi}=3/2^{-}$ and $L=1$.
Here, the $p^{6}$Li system is completely analogous to the $p^{2}$H channel in
the 3He nucleus. In the latter case the doublet state is also mixed according
to Young’s diagrams $\\{3\\}$ and $\\{21\\}$ [24]. Therefore, the potentials
constructed based on the elastic scattering phase shifts of the $p^{6}$Li
system or the level spectra of the 7Be nucleus cannot be used to describe the
GS of the 7Be nucleus in the $p^{6}$Li channel. Pure in orbital symmetry with
Young’s diagram $\\{43\\}$, the ${}^{2}P_{3/2}$ potential of the ground state
of 7Be reproduces the binding energy of the GS of the nucleus consistent with
the $p^{6}$Li system and its asymptotic constant.
The scattering potentials are constructed based on the spectra of the 7Be
nucleus from Ref. [30]; it has no major difference from the newer compilation
[34].
The orbital state’s classification of the $p^{6}$Li system is shown in Table
1.
Table 1: The classification of the orbital states of the $p^{6}$Li system [14, 19]. The following notations are used: $S$ and $L$ are spin and orbital angular momentum of the system, respectively, $\\{f\\}_{S}$, $\\{f\\}_{T}$ and $\\{f\\}_{ST}$ for isospin $T=1/2$, and $\\{f\\}_{L}$ are spin, isospin and spin-isospin, and possible orbital Young’s diagrams, and $\\{f\\}_{\text{AS}}$, $\\{f\\}_{\text{FS}}$ are Young’s diagrams of allowed and forbidden orbital states. $S$ | $\\{f\\}_{S}$ | $\\{f\\}_{T}$ | $\\{f\\}_{ST}$ | $\\{f\\}_{L}$ | $L$ | $\\{f\\}_{\text{AS}}$ | $\\{f\\}_{\text{FS}}$
---|---|---|---|---|---|---|---
$1/2$ | {43} | {43} | {7}+{61}+{52}+{511}+{43}+ | {52} | $0,2$ | — | {52}
+{421}+{331}+{4111}+ | {43} | $1,3$ | {43} | —
+{322}+{3211}+{2221} | {421} | $1,2$ | {421} | —
$3/2$ | {52} | {43} | {61}+{52}+{511}+ | {52} | $0,2$ | — | {52}
+{43}+2{431}+{331}+ | {43} | $1,3$ | — | {43}
+{322}+{3211} | {421} | $1,2$ | {421} | —
We use a Gaussian potential that depends on the momenta of the system [28] and
Young’s diagrams
$V(r,JLS,\\{f_{L}\\})=-V_{0}(JLS,\\{f_{L}\\})\exp(-\alpha_{JLS,\\{f_{L}\\}}r^{2}).$
(11)
In Eq. (11) $V_{0}$ is the potential depth and $\alpha$ is related to the
potential width. The choice of parameters for the bound and scattering states
is discussed in detail in Sec. VI.
A compilation of the AC data for the ground and first excited state in the
$p^{6}$Li channel of 7Be is presented in Table 2, with
$C_{\text{w}}^{2}=A_{NC}^{2}/2k_{0}S_{F}$. Here $\sqrt{2k_{0}}=0.983$ fm-1/2
for the GS and $\sqrt{2k_{0}}=0.963$ fm-1/2 for the FES.
Table 2: AC data for the ground and the first excited states in $p^{6}$Li channel of 7Be. BS | Reference | $A_{NC}$, fm-1/2 | $S_{F}$ | $C_{\text{w}}$
---|---|---|---|---
GS | Nollett and Wiringa [35], 2011 | 2.85(3) | 1 | 2.90(3)
| Huang et al. [13], 2010 | 2.01 | 0.66 – 1 | 2.28(24)
| Timofeyuk [36], 2013 | 1.80 | 0.46 – 0.87 | 2.32(37)
| Burtebayev et al. [37], 2013 | 1.77(8) | 0.55 – 0.81 | 2.23(31)
| Gnech and Marcucci [18], 2019 | 2.654 | 1.003 | 2.65
| Kiss et al. [11], 2021 | 2.19(9) | 0.98(30) | 2.35(43)
FES | Huang et al. [13], 2010 | 1.91 | 0.66 – 1.02 | 2.20(24)
| Timofeyuk [36], 2013 | 1.91 | 0.62 – 1.21 | 2.17(36)
| Burtebayev et al. [37], 2013 | 1.95(9) | 0.85 – 1.03 | 2.10(20)
| Gnech and Marcucci [18], 2019 | 2.528 | 1.131 | 2.53
| Kiss et al. [11], 2021 | 2.18(6) | 1.08(32) | 2.26(39)
Summarizing the data in Table 2, we conclude that all cited data on the AC are
overlapped. While constructing the corresponding GS and FES potentials, we
used the average values indicated in bold font in Table 2. Just at the end of
our calculation story, a publication by Kiss et al. [11] appeared, and it
happened that these latest experimental results turned to be within the
defined intervals for ANC given above.
We use GS and FES in the form of only doublet ${}^{2}P_{3/2}$ and
${}^{2}P_{1/2}$ states, but we take the experimental data on ANC from [37]
since it is assumed that these states result in the observed ANC values. We do
not consider these states as a mix of doublet and quartet states, for
instance, ${}^{2+4}P_{3/2}$, is a prime example, as the quartet channel is not
allowed for the orbital Young’s diagram $\\{43\\}$ in Table 1.
All AC values are used here as a framework to obtain the parameters of the
$p^{6}$Li interaction BSs potentials. These potentials correspond to the
lower, upper, and average values of AC and accurately reproduce the binding
energies [30] of the bound states. The parameters of the potentials are
presented in Table 3.
Table 3: Parameters of $p^{6}$Li system bound state potentials and bound states characteristics. $E_{x}$, $E_{b}$, and $V_{0}$ are provided in MeV. The $R_{\text{ch}}$ and $R_{\text{m}}$ are given in fm. $\\#$ | BS | $E_{x}$ | $J^{\pi}$ | $E_{b}$ | ${}^{2S+1}L_{J}$ | $V_{0}$ | $\alpha$, fm-2 | $C_{\text{w}}$ | $R_{\text{ch}}$ | $R_{\text{m}}$
---|---|---|---|---|---|---|---|---|---|---
1 | GS | 0 | 3/2- | –5.60580 | ${}^{2}P_{3/2}$ | $100.750920$ | $0.25$ | $1.75(1)$ | $2.49$ | $2.51$
2 | GS | 0 | 3/2- | –5.60580 | ${}^{2}P_{3/2}$ | $74.504070$ | $0.17$ | $2.26(1)$ | $2.58$ | $2.58$
3 | GS | 0 | 3/2- | –5.60580 | ${}^{2}P_{3/2}$ | $60.998575$ | $0.13$ | $2.74(1)$ | $2.64$ | $2.61$
4 | FES | 0.4291 | 1/2- | –5.17670 | ${}^{2}P_{1/2}$ | $99.473500$ | $0.25$ | $1.68(1)$ | $2.52$ | $2.54$
5 | FES | 0.4291 | 1/2- | –5.17670 | ${}^{2}P_{1/2}$ | $73.333835$ | $0.17$ | $2.16(1)$ | $2.59$ | $2.59$
6 | FES | 0.4291 | 1/2- | –5.17670 | ${}^{2}P_{1/2}$ | $59.898120$ | $0.13$ | $2.61(1)$ | $2.65$ | $2.62$
## VI $\boldmath p^{6}\text{Li}$ channel in the continuous spectrum
We usually assume that the scattering potentials can lead to the FSs [28], and
if the FSs are absent, the potential depth can be set to zero. The present
case refers to the scattering $P$ potentials without the FS, while the $S$ and
$D$ potentials have the bound forbidden state and, even at zero phase shifts,
must have a nonzero depth.
In the presence of one BS, the phase shift starts at $180^{\circ}$ [24], as
shown in Fig. 1 for the $S$ scattering phase shifts. Furthermore, we consider
$E2$ transitions from resonant $F$ scattering states with phase shifts shown
in Fig. 2. The parameters of potentials for all scattering processes for
transitions to GS and FES are given in Tables 4 and 5, respectively.
Table 4: The spectrum of 7Be levels [30] and scattering states in the $p^{6}$Li channel for the capture to the ${}^{2}P_{3/2}$ GS at a binding energy of 5.6058 MeV, along with $P^{2}_{J}$ from expressions (3) and (5). $E_{x}$, $E_{\text{res}}$, $\Gamma_{c.m.}$ and $V_{0}$ are provided in MeV. # | $E_{x}$, | $J^{\pi}$ | $E_{\text{res}}$, | $\Gamma_{c.m.}$, | GS transition: | $P^{2}_{J}$ | $V_{0}$ | $\alpha$, | $E_{\text{res}}$ | $\Gamma_{c.m.}$
---|---|---|---|---|---|---|---|---|---|---
exp. | exp. | exp. | $\left[{}^{2S+1}L_{J}\right]_{i}\rightarrow\left[{}^{2S+1}L_{J}\right]_{f}$ | fm-2 | theory | theory
1 | No res. | $5/2^{+}$ | — | — | $E1:\ ^{2}D_{5/2}\rightarrow\ ^{2}P_{3/2}$ | $36/5$ | $58.0$ | $0.4$ | — | —
2 | No res. | $3/2^{+}$ | — | — | $E1:\ ^{2}D_{3/2}\rightarrow\ ^{2}P_{3/2}$ | $4/5$ | $58.0$ | $0.4$ | — | —
3 | No res. | $1/2^{+}$ | — | — | $E1:\ ^{2}D_{5/2}\rightarrow\ ^{2}P_{3/2}$ | $4$ | $58.0$ | $0.4$ | — | —
4 | No res. | $1/2^{-}$ | — | — | $M1:\ ^{2}P_{1/2}\rightarrow\ ^{2}P_{3/2}$ | $4/3$ | $0.0$ | $1.0$ | — | —
5 | $7.2(1)$ | $5/2^{-}$ | $1.59(10)$ | $0.40(5)$ | $E2:\ ^{2}F_{5/2}\rightarrow\ ^{2}P_{3/2}$ | $12/7$ | $111.60$ | $0.1$ | $1.60(1)$ | $0.62(1)$
6 | $9.29(31)$ | $7/2^{-}$ | $3.68(31)$ | $1.93(96)$ | $E2:\ ^{2}F_{7/2}\rightarrow\ ^{2}P_{3/2}$ | $72/7$ | $44.34$ | $0.05$ | $3.68(1)$ | $1.50(1)$
7 | $9.9$ | $3/2^{-}$ | $4.3$ | $1.8$ | $M1:\ ^{2}P_{3/2}\rightarrow\ ^{2}P_{3/2}$ | $5/3$ | $432.0$ | $1.5$ | $4.30(1)$ | $1.80(2)$
Table 5: The spectrum of 7Be energy levels [30] and scattering states in the $p^{6}$Li channel for the proton capture to the ${}^{2}P_{1/2}$ FES at a binding energy of 5.1767 MeV, along with $P^{2}_{J}$ from expressions (3) and (5). $E_{\text{x}}$, $E_{\text{res}}$, $\Gamma_{c.m.}$ and $V_{0}$ are provided in MeV. # | $E_{\text{x}}$, | $J^{\pi}$ | $E_{\text{res}}$, | $\Gamma_{c.m.}$, | FES transition: | $P^{2}_{J}$ | $V_{0}$ | $\alpha$, | $E_{\text{res}}$ | $\Gamma_{c.m.}$
---|---|---|---|---|---|---|---|---|---|---
exp. | exp. | exp. | $\left[{}^{2S+1}L_{J}\right]_{i}\rightarrow\left[{}^{2S+1}L_{J}\right]_{f}$ | fm-2 | theory | theory
1 | No res. | $3/2^{+}$ | — | — | $E1:\ ^{2}D_{3/2}\rightarrow\ ^{2}P_{1/2}$ | $4$ | $58.0$ | $0.4$ | — | —
2 | No res. | $1/2^{+}$ | — | — | $E1:\ ^{2}S_{1/2}\rightarrow\ ^{2}P_{1/2}$ | $2$ | $58.0$ | $0.4$ | — | —
3 | No res. | $1/2^{-}$ | — | — | $M1:\ ^{2}P_{1/2}\rightarrow\ ^{2}P_{1/2}$ | $1/6$ | $0.0$ | $1.0$ | — | —
4 | $7.2(1)$ | $5/2^{-}$ | $1.59(10)$ | $0.40(5)$ | $E2:\ ^{2}F_{5/2}\rightarrow\ ^{2}P_{1/2}$ | $6$ | $111.6$ | $0.1$ | $1.60(1)$ | $0.62(1)$
5 | $9.9$ | $3/2^{-}$ | $4.3$ | $1.8$ | $M1:\ ^{2}P_{3/2}\rightarrow\ ^{2}P_{1/2}$ | $4/3$ | $432.0$ | $1.5$ | $4.30(1)$ | $1.80(2)$
Figure 1: Doublet and quartet $S$ phase shifts of elastic $p^{6}$Li scattering
at low energies. The ${}^{2}S$ and ${}^{4}S$ phase shifts are taken from Ref.
[38] and shown by $\medblackcircle$ and $\medblacktriangleup$, respectively.
Results from [14, 15] obtained according to [38] are shown by the dashed-
dotted curves. Results of the present work are shown by the solid curve.
In addition, we consider the resonance at an excitation energy of 9.9 MeV [30]
according to Fig. 3 (4.3 MeV above the threshold) in a ${}^{2}P_{3/2}$
scattering state of width 1.8 MeV in c.m. Considering such an $M1$ transition
to the ${}^{2}P_{3/2}$ ground state or an $M1$ transition from the
${}^{2}P_{1/2}$ scattering state to the ${}^{2}P_{1/2}$ FES are possible due
to the presence of different Young’s diagrams in the bound and scattering
states. Recall that the BSs have the diagram $\\{43\\}$, and the scattering
states are mixed according to the two diagrams $\\{43\\}+\\{421\\}$ [15].
Table 4 shows possible transitions to the 7Be nucleus GS from various
$p^{6}$Li scattering states with ${}^{2S+1}L_{J}$. The possible transitions to
the FES from different scattering states with ${}^{2S+1}L_{J}$ are shown in
Table 5. The resonance energies and widths are obtained with the corresponding
parameters of the scattering potentials. For the $P_{1/2}$ scattering wave,
zero-depth potentials are used since the scattering $P$ waves do not contain
forbidden BSs. For the ${}^{2}D$ wave potentials, ${}^{2}S$ wave parameters
are used for $L=2$.
Figure 2: $P$ and $F$ phase shifts of elastic $p+^{6}$Li scattering obtained
for scattering potentials with the parameters from Tables 4 and 5. Figure 3:
Schematics of the energy spectrum of 7Be. The energy are given in MeV and
figure is not drown to scale [30]. a the above-threshold resonance at 6.73 MeV
with a width of 1.2 MeV refers to the 4He3He channel that is not considered in
the present work.
Resonant phase shifts of elastic scattering $p+^{6}$Li are shown in Fig. 2.
The above-threshold resonance at 6.73 MeV with a width of 1.2 MeV indicated in
Fig. 3 refers to the 4He3He channel [30] and is not considered in our previous
works [14, 15]. Note again that in the MPCM we used, Young’s orbital diagram
{43} is forbidden in the quartet state, as shown in Table 1, and this
particular diagram corresponds to the GS of the 7Li nucleus. Therefore, in the
GS there is only a doublet ${}^{2}P_{3/2}$ state (without impurity of
${}^{4}P_{3/2}$), which is allowed for the diagram {43}. Thus, our model [14,
15] predicted the absence of resonance at 6.73 MeV with $J^{\pi}=5/2$ in the
nucleon channel, or, in other words, the impossibility of the $M1$ transition
from this resonance to the GS. This has been confirmed by the new LUNA results
[9] and, indirectly, by the data of [30]. The width of the resonance peak at
9.29 MeV is taken from Table 7.10 of [30], although another state,
${}^{2}P_{1/2}$, is indicated therein. At 9.27 MeV, the given moment is $7/2$,
as per Table 7.7 in Ref. [30], so we infer the presence of an $F$ state.
However, this resonance leads to a minimal increase in cross-sections at the
$E2$ transition. It is negligible against the background of the $E1$ resonance
at 4.3 MeV with a transition from the ${}^{2}P_{3/2}$ scattering state for the
potential parameters #7 (Table 4) or #5 (Table 5), respectively.
In Fig. 1 are shown the doublet and quartet $S$ phase shifts of elastic
$p^{6}$Li scattering at low energies. The ${}^{2}S$ potential from Table 4
with a depth of 58 MeV has one FS and allows one to describe the phase shifts
of [14] up to 1 MeV, shown in Fig. 1 by solid circles. Moreover, it gives
phase shifts below 2 MeV that coincide with the phase shifts obtained with the
potential from Ref. [14], with a depth of 126 MeV and a width of 0.15 fm-2.
This early potential has two FSs and does not agree with our new
classification from Table 1. To compare the results, we construct a new
potential (depth 58 MeV) that gives the most overlap in phase shifts from
prior work [14]. The phase shift of the new potential is given in Fig. 1 by a
solid curve, while dash-dotted curves refers to results from Ref. [14].
## VII Astrophysical S-factor and reaction rate
A special feature of cross-sections of nuclear reactions with charged
particles at low and ultra-low energies is an extreme reduction by several
orders of a cross-section magnitude due to the decrease in transmission
probability through the Coulomb barrier. For practical purposes, the
astrophysical S-factor is introduced as
$S(E)=\dfrac{\sigma(E)}{P(E)}E,\ \ \ P(E)=e^{-2\pi\eta},$ (12)
where the factor $P(E)$ reflects the permeability of the Coulomb barrier. The
advantage of the astrophysical $S-$factor is that it shows a smooth energy
dependence at low energies.
Following the excellent manuscript by Christian Iliadis [39], we would like to
provide a brief discussion of the use of the $S-$factor in conventional
calculation schemes in order to clarify the current approach. The definition
of the $S-$factor (12) allows to write the following expression for the
reaction rate:
$N_{A}\langle\sigma\nu\rangle=\left(\dfrac{8}{\pi\mu}\right)^{1/2}N_{A}\left(\kappa
T_{9}\right)^{-3/2}\int\limits_{0}^{\infty}e^{-2\pi\eta}S(E)e^{-E/\kappa
T_{9}}dE.$ (13)
In Eq.(13) $\kappa$ is the Boltzmann constant and $N_{A}$ is Avogadro’s
number. A notable effort has been expended to bring the integral in (13) to
analytical form. This is possible only if the expansion of the $S(E)-$factor
in the $E$ series, given by
$S(E)\approx S(0)+S^{\prime}(0)E+S^{\prime\prime}E^{2}.$ (14)
is valid at low energies (c.f. Ref. [39], Section 3.2). Expression (14)
explains the active interest to determine the value $S(0)$ as a key one for
the calculation of the reaction rate in the form of (13) with its further
analytical parameterizations.
Table 6 presents the available experimental data of the astrophysical
$S-$factor for the 6Li$(p,\gamma)^{7}$Be reaction, as well as the extrapolated
$S(0)$ values. Results of previous theoretical calculations and the present
work are presented in Table 7. Our calculations are analyzed and interpreted
based on the experimental data [9] as one of the newest and most accurate.
In the present work we introduce some corrections to [14, 15] that allow us to
extend the energy interval for the cross-sections and corresponding
$S-$factors. It is also worth mentioning that in Ref. [18], the authors used
the calculation scheme based on [15]. The value $S(0)=95.0$ eV$\cdot$b is
obtained which is consistent with the LUNA experimental data [9]. However, the
astrophysical reaction rate is missing in Ref. [18].
Table 6: Experimental data on the astrophysical $S-$factor of6Li$(p,\gamma)^{7}$Be. $S(E)$ and $S(0)$ are given in eV$\cdot$b and $S(0)$ are extrapolated values. $E$, keV | $S(E)$ | $S(0)$ | Reference | Method/project
---|---|---|---|---
$135$ | $51(15)$ | — | Switkowski et al. [4], 1979 | $\gamma$-ray Ge(Li) spectrometers for proton bombarding energies 200 – 1200 keV
$340$ | $43(3)$ | — | Ostojic et al. [5], 1983 | Direct radiative capture
$40$ | $65$ | $65$ | Cecil et al. [40], 1992 | Thick-target $\gamma$-ray-to-charged-particle branching ratio measurements
$35$ | $40(14)$ | — | Bruss [6], 1993 |
— | — | $79(18)$ | Prior et al. [10], 2004 | Polarized proton beams, TUNL
$250$ | $95(10)$ | — | He et al. [8], 2013 | 320 keV platform with highly charged ions
$60$ | $92(6)$ | $95(9)$ | Piatti et al. [9], 2020 | LUNA collaboration
Table 7: Theoretical calculations of the astrophysical $S-$factor of 6Li$(p,\gamma)^{7}$Be. $S(0)$, eV$\cdot$b | Reference | Model
---|---|---
$106$ | Barker [41], 1980 | Direct-capture potential model
$105$ | Arai et al. [12], 2002 | Four-cluster microscopic model
$95,5$ | Huang et al. [13], 2010 | Single-particle model
$114$ | Dubovichenko et al. [14, 15], 2010, 2011 | Modified potential cluster model
$73^{+56}_{-11}$ | Xu et al. [16], 2013 | Direct-capture potential model
$88,34$ | Dong et al. [17], 2017 | Gamow shell model
$103,9$ | Gnech and Marcucci [18], 2019 | Potential cluster model
$96,5\pm 5,7$ | Kiss et al. [11], 2021 | Modified two-body potential method
$92\pm 12$ | Kiss et al. [11], 2021 | Modified two-body potential method
$98,3$ | Present work | Modified potential cluster model
The results of the present calculations of $S-$factors along with available
experimental data are shown in Figs. 4, 5, and 6. Fig. 4 shows the
astrophysical $S(E)-$factor of 6Li$(p,\gamma_{0})^{7}$Be capture to the GS of
the 7Be nucleus in an energy range up to 5 MeV. The solid red curve 2 and the
two dashed curves, blue 1 and green 3, show the calculation for all
transitions to the GS given in Table 4. Parameters $V_{0}$ (only integer
values) and $\alpha$ of the corresponding potentials as well as $C_{\text{w}}$
from Table 3 are indicated in the figures, and the parameters of the GS
potentials are taken from Table 4. The solid red curve 2 is the result for the
potential with the set of parameters #2 from Table 3, leading to the average
value of AC.
Figure 4: Astrophysical $S-$factor of the 6Li$(p,\gamma_{0})^{7}$Be capture to
GS. Experimental data are taken from $\medblacktriangleup$ – [4],
$\medblackcircle$ – [8], $\medblacksquare$ – [9], $\medblackdiamond$ – [7],
$\medblackstar$ – [42], $\medsquare$ – [3], $\medtriangledown$ – [5].
Parameters of the continuum potential are given in Table 4. A band shows the
sensitivity to changes in $C_{\text{w}}$.
In Fig. 5, similar curves show the results for transitions and potentials from
Table 5 to FES. FES potentials have three sets of parameters from Table 3. The
solid red curve 2 shows the results for capture with set # 5 from Table 3,
allowing us to determine the average value of AC. This result is in good
agreement with the experimental data [6] presented in Fig. 5. The two dashed
curves 1 and 3 almost completely cover the interval or band of cross-section
errors of the capture to the FES.
Figure 5: Astrophysical $S-$factor of the 6Li$(p,\gamma_{1})^{7}$Be capture to
FES. Experimental data $\medcircle$ are taken from [6]. Parameters of the
continuum potential are given in Table 5. A band shows the sensitivity to
changes in $C_{\text{w}}$.
In Fig. 6, similar curves show the astrophysical $S-$factor for the total
cross-sections corresponding to the transition to GS and FES. The two dashed
curves 1 and 3 show the range of $S-$factor values due to ambiguities in the
AC of the GS and FES. For the scattering potentials, the parameters from
Tables 4 and 5 are used.
Figure 6: Astrophysical $S-$factor of the 6Li$(p,\gamma_{0+1})^{7}$Be capture
to GS and FES. Experimental data for capture are from $\medblacktriangleup$ –
[4], $\medblackcircle$ – [8], $\medblacksquare$ – [9], $\medblackdiamond$ –
[7], $\medsquare$ – [3], $\medtriangledown$ – [5], $\medblackstar$ – [42],
$\medcircle$ – [6]. The parameters of GS and FES potentials are listed in the
figure. A band shows the sensitivity to changes in $C_{\text{w}}$.
The best agreement of the $S-$factor with experimental data is achieved for
the values of $C_{\text{w}}=2.74$ for the GS and $C_{\text{w}}=1.68$ for the
FES. We recommend these values as the most reliable benchmarks for future
experimental studies.
Fig. 6 shows that almost all experimental data lie between the solid red curve
2 and the green dashed curve 3. If we use the GS and FES potentials set of
parameters #3 and #4 from Table 3, the result is shown in Fig. 6 by the black
curve 4. In this case the experimental data [9] are reproduced entirely, and
$S-$factor at 10 keV is found to be 101 eV$\cdot$b. For the scattering
potentials, the data from Table 4 and Table 5 are used.
Due to the uncertainty of the $S-$factor that arises from the uncertainty of
the AC, it is desirable to select other options for the potentials of the GS
and FES to correctly describe the LUNA data [9]. This can be the subject of
future work if more accurate data is compiled for the AC of the ground and
first excited states of the 7Be nucleus.
The approximation of the $S-$factor shown by the black curve 4 in Fig. 6 has
an analytical form
$S(E)=S_{0}+S_{1}E+S_{2}E^{2}$ (15)
with parameters $S_{0}=98.31$ eV$\cdot$b, $S_{1}=-187.18$
MeV${}^{-1}\cdot$eV$\cdot$b and $S_{2}=442.51$ MeV${}^{-2}\cdot$eV$\cdot$b.
This approximation leads to $\chi^{2}=2.4\cdot 10^{-4}$ with the error of 5%
in the energy range of 30 to 100 keV. This shows that $S(0)=98.3$ eV$\cdot$b
and $S(30)=93.1$ eV$\cdot$b.
New experimental data from LUNA [9] can be approximated to the first order
$S(E)=S_{0}+S_{1}E$ (16)
with parameters $S_{0}=91.952$ eV$\cdot$b and $S_{1}=-75.471$
MeV${}^{-1}\cdot$eV$\cdot$b, leading to $\chi^{2}=0.6$ and $S(0)=92$
eV$\cdot$b.
To compare the calculated $S-$factor at zero energy (10 keV), we present the
known results for the total $S(0)$: $79(18)$ eV$\cdot$b [10], 105 eV$\cdot$b
(at 10 keV) [12] and 106 eV$\cdot$b [41].The $S-$factor for transitions to the
ground state in [40], 39 eV$\cdot$b is specified, and for the transition to
the first excited state, the $S-$factor value is equal to 26 eV$\cdot$b, the
total $S-$factor is 65 eV$\cdot$b. In our previous works [14, 15], a value of
114 eV$\cdot$b was obtained. The summary for $S-$factor experimental and
theoretical values are presented in Tables 6 and 7.
To sum up, our astrophysical $S-$factor is given in Fig. 7 with a solid red
curve, together with experimental data and theoretical calculations. The
$R-$matrix fit of the data from LUNA collaboration [9] and Switkowski et al.
[4] is represented with the solid blue curve. A solid green curve was obtained
by Kiss et al. [11] using the weighted means of the ANCs from the analysis of
the 6Li(3He,$d$)7Be transfer reaction within the modified two-body potential
method (MTBPM). In addition, [11] contains the results for the $S-$factor of
the 6Li$(p,\gamma)^{7}$Be reaction calculated within the MTBPM, using the
values of ANCs obtained from the analysis of the experimental astrophysical
$S-$factors of the 6Li$(p,\gamma)^{7}$Be reaction [9]. These results are given
in Fig. 7 with the solid black curve.
Figure 7: Comparison of 6Li$(p,\gamma)^{7}$Be reaction astrophysical
$S-$factors. Experimental data for capture to GS and FES are from
$\medblacktriangleup$ – [4], $\medblackcircle$ – [8], $\medblacksquare$ – [9],
$\medblackdiamond$ – [7], $\medsquare$ – [3], $\medtriangledown$ – [5],
$\medblackstar$ – [42]. Results of calculations: red curve – present work;
blue curve – Ref. [9]; black and green curves – Ref. [11].
To calculate the 6Li$(p,\gamma)^{7}$Be capture reaction rate in units of cm
3mol-1s-1, we used the expression [43] analogous to Eq. (13), but substituting
the corresponding constants values
$N_{A}\langle\sigma\nu\rangle=3.7313\cdot
10^{4}\mu^{-1/2}T_{9}^{-3/2}\int\limits_{0}^{\infty}\sigma(E)E\exp(-11.605E/T_{9})dE.$
(17a) In Eq. (17a) $E$ is given in MeV, the total cross-section $\sigma(E)$ is
taken in $\muup$b, and $\mu$ is the reduced mass in amu and $T_{9}=10^{9}$ K
[43]. Using real integration limits $E_{min}$ and $E_{max}$ Eq. (17a) becomes
$N_{A}\langle\sigma\nu\rangle=3.7313\cdot
10^{4}\mu^{-1/2}T_{9}^{-3/2}\int\limits_{E_{min}}^{E_{max}}\sigma(E)E\exp(-11.605E/T_{9})dE.$
(17b)
It is important to stress this fact as the choice of $E_{max}$ in Eq. (17b)
may have a significant impact on the final result for the reaction rate. The
reaction rate (17b) is calculated based on cross-sections, displayed in the
form of $S-$factors (12) in Figs. 4, 5 and 6 within the energy range
$E_{min}=1$ keV to $E_{max}=5$ MeV. The results of these calculations are
plotted in Fig. 8.
Figure 8: Total 6Li$(p,\gamma)^{7}$Be capture reaction rate. Curves indicate
the different sums of the capture rates to the GS and FES. Curves are
designated as in Fig. 6. The yellow band is taken from Ref. [16].
As noted above, curve 4 in Fig. 6 is in best agreement with all experimental
data of the $S-$factor. Therefore, the corresponding reaction rate, also
marked as curve 4 in Fig. 7 is most recommended description of the reaction
rate. Curve 4 can be approximated by a function of the form [44]
$\begin{gathered}N_{A}\langle\sigma\nu\rangle=a_{1}/T_{9}^{1/3}\exp\left(-a_{2}/T_{9}^{2/3}\right)\left(1+a_{3}T_{9}^{1/3}+a_{4}T_{9}^{2/3}+a_{5}T_{9}+a_{6}T_{9}^{4/3}+a_{7}T_{9}^{5/3}\right)+\\\
+a_{8}T_{9}^{2/3}\exp\left(-a_{9}/T_{9}^{1/3}\right).\end{gathered}$ (18)
The parameters of approximation (18) with an average value of $\chi^{2}=0.014$
and the error of 5% are given in Table 8.
Table 8: The reaction rate approximation parameters. $i$ | $a_{i}$ | $i$ | $a_{i}$ | $i$ | $a_{i}$
---|---|---|---|---|---
$1$ | $0.00319$ | $4$ | $-544516.9$ | $7$ | $85197.34$
$2$ | $4.16292$ | $5$ | $16401.8$ | $8$ | $924167.3$
$3$ | $3721.884$ | $6$ | $-8044.932$ | $9$ | $8.36494$
Comparing our results with the reaction rate presented in NACRE [43] and NACRE
II [16], we follow the format of Fig. 4 from [9]. We added our results for the
reaction rate, normalized to the reaction rate from NACRE [43]. The comparison
is shown in Figs. 10 and 10.
Figure 9: Comparison of the astrophysical reaction rates in the range 0.001 to
1 $T_{9}$ from [43, 16, 9] to present work, normalized to the NACRE rate [43].
Dotted curves represent the uncertainty of the NACRE [43] rate, while shaded
areas represent the uncertainties from LUNA [9] (pale red) and NACRE II [16]
(yellow).
Figure 10: Comparison of the astrophysical reaction rates in the range 0.001
to 10 $T_{9}$ from [43, 16, 9] to present work, normalized to the NACRE rate
[43]. Dotted curves represent the uncertainty of the NACRE [43] rate, while
shaded areas represent the uncertainties from LUNA [9] (pale red) and NACRE II
[16] (yellow).
As stated in Ref. [9], the LUNA ”thermonuclear reaction rate is 9% lower than
NACRE [43] and 33% higher than reported in NACRE II [16] at 2 MK, and the
reaction rate uncertainty has been significantly reduced”. Fig. 10 shows that
the deviation between the adopted reaction rate obtained in [9] and the
present calculations in the range of 0.01 to 1 $T_{9}$ does not exceed 5%.
Therefore, the present calculations confirm the above conclusion by Piatti et
al. [9].
Fig. 10 shows two of our results — the blue curve 1 is the reaction rate
calculated on the basis of the total cross sections in the energy range from 1
keV to 5 MeV, and the green one 2 shows the rate at the upper range of the
integration limit $E_{max}=0.6$ MeV, and this value corresponded to the upper
limit of energy in work [9]. The difference between these curves illustrates
the importance of the resonance region contribution for the cross sections in
the range of 0.6 MeV to 5 MeV at temperatures above 1 $T_{9}$.
## VIII Conclusion
We present the results of calculations and analyses of the $S-$factor and
astrophysical reaction rate for the 6Li$(p,\gamma)^{7}$Be reaction in the
framework of MPCM. It is demonstrated that the MPCM approach has only one
ambiguity arising from the accuracy of the experimentally determined
asymptotic constants. This effect manifests as bands in Figs. 4 – 6 for the
astrophysical $S-$factor. Precise LUNA experimental data played a role of the
criterion, in reducing the ANC ambiguity with theoretical simulations.
Comparing the $R-$matrix method which is constrained by the parameterization
of the experimental cross-sections data, MPCM enables to implement
calculations in wider energy ranges. We extended the energy interval for the
total cross-sections and $S-$factors up to 5 MeV, including resonances in the
continuum. The numerical signature of this extension is seen in Fig. 10 for
the reaction rate.
It was also shown in the present work that MPCM had predicted the absence of
resonance at 6.73 MeV in the nucleon channel [14, 15], which was confirmed by
the LUNA results [9], as well as, indirectly, by the data from [30].
We suggest that the NACRE [43] and NACRE II [16] databases should be updated
in light of LUNA data [9] and present calculations.
## Dedication
We dedicate this paper to the memory of our colleague, Dr. Albert Dzhazairov-
Kakhramanov, who recently passed away from COVID19.
## Acknowledgments
This work was supported by the grant of the Ministry of Education and Science
of the Republic of Kazakhstan #AP08855556 ”Study of additional thermonuclear
reactions flowing in the process of controlled thermonuclear fusion on lithium
isotopes” through the V.G. Fesenkov Astrophysical Institute of the ”National
Center of Space Research and Technology” of the Aerospace committee of the
Ministry of Digital Development, Innovations and Aerospace Industry of the
Republic of Kazakhstan.
## Appendix A
A solution of a two-body problem for a discrete energy spectrum with a given
potential requires finding the binding energy of the system and the wave
function of the state. This problem can be solved using the Variational Method
and the Finite Difference Method (FDM) [45]. If both methods are used for the
same system of particles, it is possible to control the correctness of the
search for the binding energy and WF of the state. We already have used such
an approach for $p^{2}$H and $p^{3}$H systems in [46, 28] and demonstrated
that the FDM provides more precise description of the systems. Below we
present the FDM approach.
The calculation of the binding energy of a two-cluster system by the FDM
relies on the representation of the Schr$\ddot{\text{o}}$dinger equation in
finite differences [47]. The radial equation for the central potential [45]
$u^{\prime\prime}_{L}(r)+\left[k^{2}-V(r)\right]u_{L}(r)=0$ (19)
with some boundary condition for $k^{2}<0$, ($k^{2}=2\mu E/\hbar$) takes the
form of a Sturm-Liouville type boundary value problem. Recasting the second
derivative in finite difference form, we obtain
$\begin{gathered}u^{\prime\prime}=\left[u_{n+1}-2u_{n}+u_{n-1}\right]/h^{2},\
\ \ u_{n}=u(r_{n})\end{gathered}$ (20)
and (19) becomes a closed system of linear algebraic equations. Thus, for a
certain $k_{0}$, $D_{N}(k)=0$
$D_{N}(k)=\begin{pmatrix}\thetaup_{1}&1&0&.&.&.&0\\\
\alphaup_{2}&\thetaup_{2}&1&0&.&.&0\\\ 0&\alphaup_{3}&\thetaup_{3}&1&0&.&0\\\
.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.\\\ 0&.&0&0&\alphaup_{N-1}&\thetaup_{N-1}&1\\\
0&.&0&0&0&\alphaup_{N}&\thetaup_{N}\end{pmatrix}=0.$ (21)
Eq. (21) allows one to determine the binding energy $E_{b}$ of a system of two
particles. The elements of the tridiagonal determinant (21) are defined as
follows:
$\displaystyle\alphaup_{n}=1,$
$\displaystyle\thetaup_{n}=k^{2}h^{2}-2-V_{n}h^{2},\ \ \ n=1,2,\dots,N-1,$
(22) $\displaystyle\alphaup_{N}=2,$
$\displaystyle\thetaup_{N}=k^{2}h^{2}-2-V_{N}h^{2}+2hf(\eta,L,Z_{N}),$
$\displaystyle Z_{n}=2kr_{n},$ $\displaystyle
f(k,\eta,L,Z_{n})=-k-\dfrac{2k\eta}{Z_{n}}-\dfrac{2k(L-\eta)}{Z_{n}^{2}}.$
Here $\eta$ is the Coulomb parameter, $k=|\sqrt{k^{2}}|$ is the wave number
expressed in fm-1 and determined by the energy of interacting particles in the
input channel, and $V_{n}=V(r_{n})$ is the interaction potential of clusters
at the point $r_{n}=nh$ from the interval of zero to $R$. The number of
equations $N$ or the dimension of the determinant, which usually turns out to
be in the range $100\ 000-1\ 000\ 000$ [45], $h=\Delta r/N$ is the step of the
finite difference grid and $\Delta r$ is the solution interval of the system
(usually from zero to $r_{N}=R$).
By writing $f(k,\eta,L,Z_{n})$ in the form given in Eq. (22) it is possible to
take the Coulomb interaction into account [33]. The form of the logarithmic
derivative of the WF in the external region can be obtained from the integral
representation of the Whittaker function [33]
$f(k,\eta,L,Z)=-k-\dfrac{2k\eta}{Z}-\dfrac{2k(L-\eta)}{Z^{2}}S(\eta,L,Z),$
(23)
where
$S(\eta,L,Z)=\dfrac{\int\limits_{0}^{\infty}t^{L+\eta+1}(1+t/Z)^{L-\eta-1}e^{-t}dt}{\int\limits_{0}^{\infty}t^{L+\eta}(1+t/Z)^{L-\eta}e^{-t}dt}.$
(24)
Calculations show that the value $S(\eta,L,Z)$ does not exceed 1.05, and its
effect on the binding energy of a two-particle system is negligible [45]. When
$f(k,0,0,Z)=-k$ in Eq. (23), the binding energy search process is noticeably
accelerated.
The calculation of the band determinant $D_{N}(k)$ for a given $k$ is carried
out using recurrent formulas of the form [47]
$\displaystyle D_{-1}=0,$ $\displaystyle
D_{n}=\thetaup_{n}D_{n-1}-\alphaup_{n}D_{n-2},$ (25) $\displaystyle D_{0}=1,$
$\displaystyle n=1,\dots,N.$
Any energy $E$ or wave number $k$ that leads to zero determinant
$D_{N}(k_{0})=0$ (26)
is an eigenenergy of the system $E_{b}$ or $k_{0}$, and the wave function at
this energy, determined by recurrent process below, is an eigenfunction of the
problem.
Methods for determining the zero of some functional of one variable $k$ are
well known [48]. The number $N_{D}$ of determinant values is determined
automatically from the accuracy condition of the binding energy value. The
latter one is usually set to the level $\epsilon\approx 10^{-5}$–$10^{-9}$MeV,
and $r_{N}=R$ is fixed on the range 20–30 fm [45].
After determining the eigenenergy $E_{b}$, the WF of this state is sought. To
find the shape of the eigenfunctions of bound states, the recurrent procedure
$\displaystyle u_{0}=0,$ $\displaystyle u_{n}=\thetaup_{n-1}u_{n-1}+u_{n-2},$
(27) $\displaystyle u_{1}=\text{const},$ $\displaystyle n=2,\dots,N.$
is carried out, where $u_{1}$ is an arbitrary number, usually fixed on the
range 0.01–0.1 [48].
For bound states, the determined WF is normalized to unity. Comparing it to
Whittaker asymptotics, one can find an asymptotic constant denoted by
$C_{\text{w}}$ (see Sec. IV).
The WF search area $R$ is usually of 20 to 30 fm, and the number of steps
$N_{WF}$ for the desired WF is fixed between 10 000 and 50 000. Only in the
case of a very low binding energy (0.1–0.2 MeV) the WF search area increased
to 100–200 fm or more.
The recurrence relation (27) is also used to search for WFs in the case of a
continuous spectrum of eigenvalues at predetermined positive energy
$(k^{2}>0)$ of interacting particles[45]. However, the WF must now be matched
with asymptotics of the form
$N_{L}u_{L}(r)\xrightarrow[r\rightarrow
R]{}F_{L}(kr)+\tan\left(\deltaup_{S,L}^{J}\right)G_{L}(kr).$ (28)
Matching the numerical solution $u_{L}(R)$ of Eq. (19) for two points at large
distances ($R$ on the order of 10–20 fm) with asymptotics (28), it is possible
to calculate the scattering phase shifts for each value of the momenta $JLS$
for a given energy of interacting particles, as well as the normalization of
the WF for scattering processes [45]. To calculate the WF, one can also use
the Numerov method [49]. When the number of steps exceeds 10 000, both methods
yield the same results within the typical required accuracy. Such results can
be compared by calculating the values of AC or charge radii for the BS or the
matrix elements for the scattering processes [45].
## Appendix B
Table 9: The astrophysical 6Li$(p,\gamma)^{7}$Be reaction rate in the range of 0.001 to 10 $T_{9}$ $T_{9}$ | Rate | $T_{9}$ | Rate | $T_{9}$ | Rate | $T_{9}$ | Rate
---|---|---|---|---|---|---|---
0.001 | $3.20\times 10^{-29}$ | 0.035 | $6.90\times 10^{-5}$ | 0.19 | $1.35\times 10^{0}$ | 2.25 | $7.80\times 10^{2}$
0.002 | $7.07\times 10^{-22}$ | 0.040 | $1.92\times 10^{-4}$ | 0.20 | $1.67\times 10^{0}$ | 2.50 | $9.07\times 10^{2}$
0.003 | $2.53\times 10^{-18}$ | 0.045 | $4.57\times 10^{-4}$ | 0.25 | $3.98\times 10^{0}$ | 2.75 | $1.03\times 10^{3}$
0.004 | $4.35\times 10^{-16}$ | 0.050 | $9.60\times 10^{-4}$ | 0.30 | $7.64\times 10^{0}$ | 3.00 | $1.16\times 10^{3}$
0.005 | $1.68\times 10^{-14}$ | 0.055 | $1.83\times 10^{-3}$ | 0.35 | $1.28\times 10^{1}$ | 3.25 | $1.29\times 10^{3}$
0.006 | $2.71\times 10^{-13}$ | 0.060 | $3.25\times 10^{-3}$ | 0.40 | $1.94\times 10^{1}$ | 3.50 | $1.42\times 10^{3}$
0.007 | $2.49\times 10^{-12}$ | 0.065 | $5.41\times 10^{-3}$ | 0.45 | $2.75\times 10^{1}$ | 3.75 | $1.56\times 10^{3}$
0.008 | $1.54\times 10^{-11}$ | 0.070 | $8.55\times 10^{-3}$ | 0.50 | $3.71\times 10^{1}$ | 4.0 | $1.69\times 10^{3}$
0.009 | $7.20\times 10^{-11}$ | 0.075 | $1.30\times 10^{-2}$ | 0.55 | $4.81\times 10^{1}$ | 4.5 | $1.95\times 10^{3}$
0.010 | $2.70\times 10^{-10}$ | 0.080 | $1.89\times 10^{-2}$ | 0.60 | $6.03\times 10^{1}$ | 5.0 | $2.22\times 10^{3}$
0.011 | $8.58\times 10^{-10}$ | 0.085 | $2.68\times 10^{-2}$ | 0.65 | $7.37\times 10^{1}$ | 5.5 | $2.50\times 10^{3}$
0.012 | $2.38\times 10^{-9}$ | 0.090 | $3.69\times 10^{-2}$ | 0.70 | $8.83\times 10^{1}$ | 6.0 | $2.77\times 10^{3}$
0.013 | $5.92\times 10^{-9}$ | 0.095 | $4.97\times 10^{-2}$ | 0.75 | $1.04\times 10^{2}$ | 6.5 | $3.05\times 10^{3}$
0.014 | $1.35\times 10^{-8}$ | 0.10 | $6.55\times 10^{-2}$ | 0.80 | $1.20\times 10^{2}$ | 7.0 | $3.32\times 10^{3}$
0.015 | $2.83\times 10^{-8}$ | 0.11 | $1.08\times 10^{-1}$ | 0.85 | $1.38\times 10^{2}$ | 7.5 | $3.59\times 10^{3}$
0.016 | $5.60\times 10^{-8}$ | 0.12 | $1.67\times 10^{-1}$ | 0.90 | $1.56\times 10^{2}$ | 8.0 | $3.86\times 10^{3}$
0.017 | $1.05\times 10^{-7}$ | 0.13 | $2.48\times 10^{-1}$ | 0.95 | $1.74\times 10^{2}$ | 8.5 | $4.13\times 10^{3}$
0.018 | $1.86\times 10^{-7}$ | 0.14 | $3.52\times 10^{-1}$ | 1.00 | $1.94\times 10^{2}$ | 9.0 | $4.38\times 10^{3}$
0.019 | $3.18\times 10^{-7}$ | 0.15 | $4.84\times 10^{-1}$ | 1.25 | $2.99\times 10^{2}$ | 9.5 | $4.63\times 10^{3}$
0.020 | $5.23\times 10^{-7}$ | 0.16 | $6.48\times 10^{-1}$ | 1.50 | $4.13\times 10^{2}$ | 10 | $4.88\times 10^{3}$
0.025 | $4.13\times 10^{-6}$ | 0.17 | $8.45\times 10^{-1}$ | 1.75 | $5.32\times 10^{2}$ | |
0.030 | $1.98\times 10^{-5}$ | 0.18 | $1.08\times 10^{0}$ | 2.00 | $6.55\times 10^{2}$ | |
## References
* [1] C. A. Barnes, D. D. Clayton, and D. N. Schramm, eds., _Essays in nuclear astrophysics: presented to William A. Fowler, on the occasion of his seventieth birthday_ (Cambridge University Press, New York, 1982) p. 562.
* [2] R. N. Boyd, C. R. Brune, G. M. Fuller, and C. J. Smith, Phys. Rev. D 82, 105005 (2010), arXiv:1008.0848.
* [3] S. Bashkin and R. R. Carlson, Phys. Rev. 97, 1245 (1955).
* [4] Z. E. Switkowski, J. C. Heggie, D. L. Kennedy, D. G. Sargood, F. C. Barker, and R. H. Spear, Nucl. Phys. A 331, 50 (1979).
* [5] R. Ostojic, K. Subotic, and B. Stepancic, Il Nuovo Cimento A 76, 73 (1983).
* [6] R. Bruss, in _Nuclei in the Cosmos: Proceedings of the Second International Symposium on Nuclear Astrophysics_ , edited by F. Kappeler and K. Wisshak (IOP Publishing Ltd, Karlsruhe, Germany, 1993) p. 648.
* [7] T. Paradellis (1999), unpublished, quoted by [12] as Ref. [25].
* [8] J. J. He, S. Z. Chen, C. E. Rolfs, S.W. Xu, J. Hu, X.W. Ma, M.Wiescher, R. J. DeBoer, T. Kajino, M. Kusakabe, L. Y. Zhang, S. Q. Hou, X. Q. Yu, N. T. Zhang, G. Lian, Y. H. Zhang, X. H. Zhou, H. S. Xu, G. Q. Xiao, and W. L. Zhan, Phys. Let. B 725, 287 (2013).
* [9] D. Piatti, T. Chillery, R. Depalo, M. Aliotta, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Cavanna, G. F. Ciani, P. Corvisiero, L. Csedreki, T. Davinson, A. Di Leva, Z. Elekes, F. Ferraro, E. M. Fiore, A. Formicola, Z. Fulop, G. Gervino, A. Gnech, A. Guglielmetti, C. Gustavino, G. Gyurky, G. Imbriani, M. Junker, I. Kochanek, M. Lugaro, L. E. Marcucci, P. Marigo, E. Masha, R. Menegazzo, V. Mossa, F. R. Pantaleo, V. Paticchio, R. Perrino, P. Prati, L. Schiavulli, K. Stockel, O. Straniero, T. Szucs, M. P. Takacs, and S. Zavatarelli, Phys. Rev. C 102, 052802 (2020).
* [10] R. M. Prior, M. C. Spraker, A. M. Amthor, K. J. Keeter, S. O. Nelson, A. Sabourov, K. Sabourov, A. Tonchev, M. Ahmed, J. H. Kelley, D. R. Tilley, H. R. Weller, and H. M. Hofmann, Phys. Rev. C 70, 10.1103 (2004).
* [11] G. G. Kiss, M. La Cognata, R. Yarmukhamedov, K. I. Tursunmakhatov, I. Wiedenhover, L. T. Baby, S. Cherubini, A. Cvetinovic, G. D’Agata, P. Figuera, G. L. Guardo, M. Gulino, S. Hayakawa, I. Indelicato, L. Lamia, M. Lattuada, F. Mudo, S. Palmerini, R. G. Pizzone, G. G. Rapisarda, S. Romano, M. L. Sergi, R. Spart‘a, C. Spitaleri, O. Trippella, A. Tumino, M. Anastasiou, S. A. Kuvin, N. Rijal, B. Schmidt, S. B. Igamov, S. B. Sakuta, Z. Fulop, G. Gyurky, T. Szucs, Z. Halasz, E. Somorjai, Z. Hons, J. Mrazek, R. E. Tribble, and A. M. Mukhamedzhanov, Phys. Rev. C 104, 015807 (2021).
* [12] K. Arai, D. Baye, and P. Descouvemont, Nucl. Phys. A 699, 963 (2002).
* [13] J. T. Huang, C. A. Bertulani, and V. Guimaraes, Atomic Data and Nuclear Data Tables 96, 824 (2010), arXiv:0810.3867.
* [14] S. B. Dubovichenko, N. Burtebaev, D. M. Zazulin, and A. S. Amar, Russian Physics Journal 53, 743 (2010).
* [15] S. B. Dubovichenko, N. Burtebaev, D. M. Zazulin, Z. K. Kerimkulov, and A. S. Amar, Physics of Atomic Nuclei 74, 984 (2011).
* [16] Y. Xu, K. Takahashi, S. Goriely, M. Arnould, M. Ohta, and H. Utsunomiya, Nucl. Phys. A 918, 61 (2013), arXiv:1310.7099.
* [17] G. X. Dong, N. Michel, K. Fossez, M. Płoszajczak, Y. Jaganathen, and R. M. Betan, Journal of Physics G 44, 1 (2017), arXiv:arXiv:1601.06660v1.
* [18] A. Gnech and L. E. Marcucci, Nucl. Phys. A 987, 1 (2019).
* [19] S. B. Dubovichenko and A. V. Dzhazairov-Kakhramanov, Nucl. Phys. A 941, 335 (2015).
* [20] P. Descouvemont, Frontiers in Astronomy and Space Sciences 7, 9 (2020).
* [21] T. A. Tombrello and P. D. Parker, Phys. Rev. 131, 2582 (1963).
* [22] K. Wildermuth and Y. C. Tang, _A Unified Theory of the Nucleus_ (Vieweg+Teubner Verlag, 1977).
* [23] V. I. Kukulin, V. G. Neudatchin, I. T. Obukhovski, and Y. F. Smirnov, in _Clusters as Subsystems in Light Nuclei_ (Vieweg+Teubner Verlag, Wiesbaden, 1983) pp. 1–155.
* [24] V. G. Neudatchin, V. I. Kukulin, V. N. Pomerantsev, and A. A. Sakharuk, Phys. Rev. C 45, 1512 (1992).
* [25] C. Itzykson and M. Nauenberg, Reviews of Modern Physics 38, 95 (1966).
* [26] A. Bohr and B. R. Mottelson, in _Nuclear Structure_ (World Scientific Publishing Company, 1998) pp. 137–307.
* [27] A. M. Mukhamedzhanov and R. E. Tribble, Phys. Rev. C 59, 3418 (1999).
* [28] S. B. Dubovichenko, Thermonuclear processes in Stars and Universe, 2nd ed. (Scholar’s Press, Saarbrucken, 2015) p. 332.
* [29] Physical Measurement Laboratory; http://physics.nist.gov/cgi-bin/cuu/Value?mud%7Csearch_for =atomnuc!
* [30] D. R. Tilley, C. M. Cheves, J. L. Godwin, G. M. Hale, H. M. Hofmann, J. H. Kelley, C. G. Sheu, and H. R. Weller, Energy levels of light nuclei A = 5, 6, 7 (2002).
* [31] V. Varlamov, B. Ishkhanov, and S. Komarov, Atomic Nuclei Map (2015); http://cdfe.sinp.msu.ru/services/ground/NuclChart_release.html
* [32] G. R. Plattner and R. D. Viollier, Nucl. Phys. A 365, 8 (1981).
* [33] M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical function, 10th ed. (U.S. Government Printing Office, Washington, 1972) p. 1046.
* [34] Sukhoruchkin S.I., Supplement to I/25 A-F (Springer Berlin Heidelberg, 2016).
* [35] K. M. Nollett and R. B. Wiringa, Phys. Rev. C 83, 10.1103 (2011).
* [36] N. K. Timofeyuk, Phys. Rev. C 88, 044315 (2013).
* [37] N. Burtebayev, J. Burtebayeva, N. Glushchenko, Z. Kerimkulov, A. Amar, M. Nassurlla, S. Sakuta, S. Artemov, S. Igamov, A. Karakhodzhaev, K. Rusek, and S. Kliczewski, Nucl. Phys. A 909, 20 (2013).
* [38] M. Skill, R. Baumann, G. Keil, N. Kniest, E. Pfaff, M. Preiss, G. Reiter, G. Clausnitzer, M. Haller, and W. Kretschmer, Nucl. Phys. A 581, 93 (1995).
* [39] C. Iliadis, Nuclear physics of stars, 2nd ed. (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2015) p. 672.
* [40] F. E. Cecil, D. Ferg, H. Liu, J. C. Scorby, J. A. McNeil, and P. D. Kunz, Nucl. Phys. A 539, 75 (1992).
* [41] F. Barker, Australian Journal of Physics 33, 159 (1980).
* [42] A. Amar and N. Burtebayev, Journal of Nuclear Sciences 1, 15 (2014).
* [43] C. Angulo, M. Arnould, M. Rayet, P. Descouvemont, D. Baye, C. Leclercq-Willain, A. Coc, S. Barhoumi, P. Aguer, C. Rolfs, R. Kunz, J.W. Hammer, A. Mayer, T. Paradellis, S. Kossionides, C. Chronidou, K. Spyrou, S. Degl’Innocenti, G. Fiorentini, B. Ricci, S. Zavatarelli, C. Providencia, H. Wolters, J. Soares, C. Grama, J. Rahighi, A. Shotter, and M. Lamehi Rachti, Nucl. Phys. A 656, 3 (1999).
* [44] G. R. Caughlan and W. A. Fowler, Thermonuclear reaction rates V (1988).
* [45] S. B. Dubovichenko, Methods for calculating nuclear characteristics. Nuclear and thermonuclear processes., 2nd ed. (Lambert Academic Publishing, Saarbrucken, 2012) p. 425.
* [46] S. B. Dubovichenko, A. V. Dzhazairov-Kakhramanov, and N. V. Afanasyeva, Nucl. Phys. A 963, 52 (2017).
* [47] G. Marchuk and V. Kolesov, Application of Numerical Methods to Neutron Cross-Section Calculations (Atomizdat, Moscow, 1970) p. 304\.
* [48] T. Korn and G. A. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New-York, 1968) p. 832.
* [49] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag Berlin Heidelberg, 1993) p. 528.
|
# Collective dynamics of evanescently coupled excitable lasers with saturable
absorber
M. Lamperti1 and A. M. Perego2
1,∗ Politecnico di Milano, Department of Physics and IFN-CNR, Via G. Previati
1/C, 23900, Lecco, Italy
2 Aston Institute of Photonic Technologies, Aston University, Birmingham B4
7ET, Aston Street, UK
<EMAIL_ADDRESS>
###### Abstract
We present a numerical study of the collective dynamics in a population of
coupled excitable lasers with saturable absorber. At variance with previous
studies where real-valued (lossy) coupling was considered, we focus here on
the purely imaginary coupling (evanescent wave coupling). We show that
evanescently coupled excitable lasers synchronize in a more efficient way
compared to the lossy coupled ones. Furthermore we show that out-of-diagonal
disorder-induced localization of excitability takes place for imaginary
coupling too, but it can be frustrated by nonvanishing linewidth enhancement
factor.
## 1 Introduction
The last decade has been characterized by the rise and spread of data storage
and analytics over many industrial and commercial fields, due to the
decreasing cost of data collection, storage and transmission. While it is
clear that large datasets are needed to obtain meaningful statistical
information about complex systems, the instruments used for such information
distillation have not evolved with the same speed as the data infrastructure,
due on one hand to our still rough understanding of large, complex systems,
and on the other hand to the exponentially increasing computing resources
needed to apply more and more complex models for data interpretation. A
promising framework for advanced data processing tasks is constituted by
neural networks [1], which currently see widespread usage to perform actions
that are traditionally a premise of human beings, such as pattern recognition,
natural language processing (information extraction from texts and
translation), and identification of objects in images. Although specialized
hardware and software has been developed to increment the efficiency of
implementing such framework with the currently most advanced computing
paradigma, based on a Von Neumann architecture executed on CMOS-based
integrated circuits, the level of complexity and efficiency of the human brain
is far from reach by many orders of magnitude.
The maturity and apparent limits of the current technology have pushed
scientists from many disciplines to propose new computing paradigms that could
enable a leap towards the higher processing power required for the
aforementioned purposes; the most promising results take inspiration from what
is considered to be the most advanced naturally evolved computer: the human
brain. The resulting field of neuromorphic computing aims at taking advantage
from systems that exhibit naturally the characteristics that make biological
neural networks so powerful and efficient, mapping the process paradigm to its
underlying dynamics rather than abstracting away from it, as it is currently
done with software implementation of neural networks on serialized, digital
hardware [2].
The two key ingredients for implementing neuromorphic computing are neuron-
like behavior and a large network of interconnections [1]. The first
ingredient is provided by excitable systems: excitability is an ubiquitous
process in nature mostly known in biology [3, 4] in the context of the
neuronal cell activity and can be defined as the generation of spike-like
behaviour in one or more system dynamical variables in response to external
perturbations whose magnitude exceeds a given threshold. In such regime the
system response does not depend on the perturbation strength and each spike
generation is followed by a refractory time during which the system remains
silent; after such refractory time the emission of another spike can take
place again. Excitability needs to be provided by a suitable neuron-like
system, which in turn will constitute the building block to perform simple
computations in a highly parallel fashion, making the system fast and
efficient. What is needed to perform complex computations is a large network
of interconnects that weight the outputs of previous neurons and route them to
other computation elements, i.e. other neuron-like blocks. A network of
interconnects is thus the key ingredient to transform a bunch of excitable
systems into a highly-efficient computer able to solve complex tasks.
Currently, the golden standard for electronics-based computing, CMOS
integrated circuits, is unable to provide these two elements while keeping the
power efficiency high [5]. As we approach the few-atom transistor, with bigger
and bigger challenges without any significant decrease of power consumption,
new directions have been probed in search for another platform that could
provide high integration, low power consumption and scalability. Thanks to
recent advances in optoelectronics integration, photonics technologies
constitute one of the most promising platforms for neuromorphic computing:
neuromorphic photonics is gaining momentum as a research field where emerging
photonics technologies are used to mimic neuronal dynamics and/or to perform
computational tasks based on brain inspired strategies [6]. Excitability in
photonics has been reported in a variety of different lasers and amplifiers
systems [7, 8, 9, 10, 11, 12, 13, 14, 15]. Possibly the first historically
studied example of a photonics system exhibiting excitability is the
semiconductor laser with saturable absorber [7]. Excitability in such laser is
enabled by a phase space portrait exhibiting a limit cycle close to a saddle
node bifurcation. In the regime where the absorption is the slow dynamical
variable, if the below, but close-to-threshold laser is perturbed strongly
enough, then the stimulated emission process builds up producing the emission
of giant light pulse. Such emission depletes the gain and is followed by a
refractory time after which the laser is ready to be excited again. Such
dynamics can be associated to a so called type III excitability [8].
Engineering optically excitable elements connectivities and studying their
collective properties and dynamics are both crucial tasks towards achieving a
general understanding of neuromorphic photonics systems. Focussing our
attention to the excitable laser with saturable absorber, it is important to
stress that coupling engineering of excitable semiconductor lasers with
saturable absorber has been shown to allow pattern recognition [14], neuronal
circuits design [16] and coincidence detection devices [17]. As far as the
collective dynamics is concerned, we have shown theoretically in two recent
works [18, 19] that temporal and intensity synchronization, array enhanced
coherence resonance, and even disorder-induced localization of excitability
can take place in arrays of excitable lasers with nearest neighbour real-
valued (lossy) coupling. In this work we extend our previous studies on
synchronization and disorder-induced localization of excitability to the case
of an ensemble of excitable lasers with saturable absorber coupled via a
nearest neighbour purely imaginary coupling coefficient. Imaginary valued
couplings describe the physical evanescent waves interaction between adjacent
laser cavities. Such interaction is most likely the relevant coupling
mechanism for micropillars lasers, where collective excitable dynamics could
be observed [20].
## 2 The model
We consider here a population of $n$ lasers with nearest neighbour coupling.
The following normalized Yamada model describes the $i$-th laser dynamics:
$\displaystyle\dot{F}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[G_{i}(1-i\alpha)-Q_{i}(1-i\beta)-1]F_{i}+\sigma_{i}$
$\displaystyle-$ $\displaystyle
i(K_{i,i+1}+K_{i,i-1})F_{i}+iK_{i+1,i}F_{i+1}+iK_{i-1,i}F_{i-1},$
$\displaystyle\dot{G}_{i}$ $\displaystyle=$
$\displaystyle\gamma(A-G_{i}-I_{i}G_{i}),$ $\displaystyle\dot{Q}_{i}$
$\displaystyle=$ $\displaystyle\gamma(B-Q_{i}-aQ_{i}I_{i}).$ (1)
Here $F_{i}$ denotes the electric field strength, $I_{i}=|F_{i}|^{2}$ its
intensity, $G_{i}$ and $Q_{i}$ gain and absorption respectively. $A$ is the
pump parameter, $B$ the background absorption, $a$ the differential absorption
relative to the differential gain, $\gamma$ is the absorber and gain decay
rate, $\alpha$ and $\beta$ denote the linewidth enhancement factors for the
gain and the absorber respectively; these parameters do not have subscript $i$
since they have been considered identical for all lasers. $\sigma_{i}$
describes a delta correlated Gaussian noise term of strength $D$ with
$\langle\sigma_{i}(t_{1})\sigma_{j}(t_{2})\rangle=\sqrt{2D}\delta(t_{1}-t_{2})\delta_{ij}$
providing the perturbations needed for excitable behaviour. The dot denotes
temporal derivative and the time variable has been normalized to the uncoupled
laser photon lifetime. $K_{i,j}$ denotes the nearest neighbour coupling
strength describing light coupling from the $i$-th to the $j$-th laser where
the reciprocity condition $K_{i,j}=K_{j,i}$ has been imposed. The identical
coupling case $K_{i,i+1}=K$ results in an effective discrete Laplace
diffraction operator in the array.
Periodic conditions at the array boundary have been applied. Across the whole
paper we have set $A=6.5$, $B=5.8$, $a=1.8$, $\gamma=10^{-3}$ while for
$\alpha$ and $\beta$ different values have been used and specified across the
paper.
## 3 Synchronization
We have first studied the effect of coupling on the synchronization of the
firing events in the lasers array as a function of the number of lasers
constituting the array itself and of the values of the linewidth enhancement
factors. To this purpose we have considered independent additive noise sources
to be present in all lasers and varied their amplitude, $D$, from 0 to 0.15.
The coupling strength, $K=K_{0}$, has been considered identical for all lasers
and has been varied from 0 to 1. For each pair $(D,K)$ we run one simulation
for a time duration $T=100000$. The pulse synchronization can be understood
both in space and time: in the first case it refers to neighbor lasers
emitting a pulse at the same time while in the second case it characterizes a
regular (i.e. equally spaced) emission of pulses from the single laser. To
give a qualitative flavour of temporally synchronous pulses (spikes) emission
of coupled lasers we have plotted in Fig. 1 the temporal traces corresponding
to coupled and uncoupled laser (left and right column, respectively).
Figure 1: In panels a) to e) the temporal traces of five different uncoupled
lasers have been plotted showing uncorrelated lasers firing. In panels f) to
j) the time traces of five randomly-picked up lasers in a chain of 50 coupled
elements have been depicted. Spatial synchronization is clearly evident in
presence of non vanishing coupling. Parameters used are $D=0.01$,
$\alpha=\beta=0$ for both the coupled and uncoupled case; $K_{0}=0$ in a) to
e) and $K=0.05$ in f) to j).
We have characterized the spatial synchronization by using an indicator ($S$)
that quantifies the phase slip between the firing events of nearest-neighbors
lasers [18, 21]. The phase of the $i$-th laser reads:
$\displaystyle\phi_{i}(t)=\frac{t-\tau_{k}}{\tau_{k+1}-\tau_{k}}+2k\pi$ (2)
where $\tau_{k}$ is the time of the $k$-th firing event, i.e. the position in
time of the $k$-th emitted pulse. We define then
$\displaystyle s_{i}=\sin\left(\frac{\phi_{i}-\phi_{i+1}}{2}\right)^{2}$ (3)
which characterizes the phase synchronization between the $i$-th and $i+1$-th
lasers. The average both across all the array elements and along temporal
duration of the time trace gives the $S$ indicator, which provides a measure
of the spatial synchronization of the whole system:
$\displaystyle
S=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\left(\frac{1}{n}\sum_{i=1}^{n}s_{i}\right)dt.$
(4)
The maximum synchronization occurs for $S=0$ while for the completely non
synchronized state $S=0.5$.
We have furthermore characterized the regular emission of pulses of the
individual laser (temporal synchronization) by means of the single laser pulse
jitter defined as $J=\sigma_{T}/\langle T\rangle$, where $\langle T\rangle$ is
the average time interval between two consecutive pulses and $\sigma_{T}$ is
the standard deviation. A value of $J$ close to 0 indicates regular pulses
emission, while values close or greater than 1 indicate poor regularity.
Figure 2: The single-laser temporal synchronization index, $J$, is plotted in
the $(D,K)$ space for different numbers of lasers (rows - 4, 20 and 150 from
top to bottom) and different values of the line enhancement factors (columns -
0, 1, 3 and 5 from left to right).
The panels in Fig. 2 and in Fig. 3 show the $J$ and $S$ synchronization
indicators, respectively, for different number of lasers constituting the
chain and different values of the line enhancement factors, versus the noise
and the coupling strength. The behavior observed here is qualitatively
different from the case of real (dissipative) coupling, showing no array-
enhanced coherence resonance [18]. Instead of enlarging the region of the
$(D,K)$ parameter space characterized by high synchronization, an increase in
the number of coupled lasers improves the temporal synchronization in a
limited region of the parameters space (the lower-left one) while worsening
the synchronization in all the remaining region. In other words, enlarging the
array creates a sharper separation between the regions of high and low
synchronization, without changing its area.
On the other hand, increasing the number of lasers in the array has the effect
of reducing the region of highest spatial synchronization, i.e. low value of
the $S$ indicator. This effect is well visible in the part of the $(D,K)$
space characterized by $D>0.05$, and manifests itself quickly with the
increase of lasers forming the chain (compare first and second row of Fig. 3).
It is worth noting that even in the worst synchronization cases, the $S$
indicator is smaller than $10^{-5}$, indicating good spatial synchronization
in all the $(D,K)$ space.
Figure 3: The logarithm of the spatial synchronization, $\log_{10}(S)$, is
plotted in the $(D,K)$ space for different numbers of lasers (rows - 4, 20 and
150 from top to bottom) and different values of the line enhancement factors
(columns - 0, 1, 3 and 5 from left to right).
The second control parameter we considered, the line enhancement factor,
differently from the number of lasers modifies the shape and size of the
synchronization region. For a small number of lasers (4, first row in Figs.
2-3), its progressive increase from 0 to 5 causes an enlargement of the region
of high synchronization towards larger values of the noise, both in time and
space. For a longer chain (20 and 150 lasers, two bottom rows), high
synchronization occurs in a more and more localized region towards small
values of $K$ and $D$; even in a strong coupling regime high temporal
synchronization of spikes emitted by the same laser results more difficult to
achieve.
As compared to the real coupling case [18] we can observe that a much smaller
value of the coupling strength is needed to achieve comparable
synchronization. Imaginary coupling is hence a much more efficient way to
achieve synchronization of excitability.
## 4 Disorder-induced localization
In our previous study for real-valued couplings we pointed out how randomness
in the laser array coupling strength is strikingly able to induce spatial
exponential localization of excitable behavior in a given area of the array
[19]. Such disorder-induced localization does not occur due to a trivial
breaking of the lasers chain but due to a dynamical process entailing both
scattering and dissipation. We want to stress that the localization we are
discussing here, although mediated by disorder, can not be assimilated
directly to the celebrated _Anderson localization_ [22]. The latter, although
it has inspired our work, is occurring in ideally conservative systems while
in our system dissipation plays of course a paramount important role. Still
inspired by the conspicuous literature on _Anderson localization_ we can
borrow some useful terminology. In the traditional studies of _Anderson
localization_ of electronic transport in solid state physics, the distinction
between diagonal and out-of-diagonal disorder is made when referring to
disorder on the individual atomic sites, or in the hopping terms between
neighbour atomic sites respectively [23]. In our system we identify clearly
the former as corresponding to randomness in some internal laser parameter
(e.g. pump parameter), while the latter to the coupling between the lasers. In
this paper we will consider out-of-diagonal disorder defining the coupling as:
$K_{i,i+1}=K_{0}+\rho_{i,i+1}$, being $K_{0}$ a value common to all lasers and
$\rho_{i,i+1}$ a random number constant in time drawn from a uniform
distribution in the interval $[-r,+r]$ which describes light coupling from the
$i$-th laser to nearest neighbour on the right. We recall also that the
reciprocity constraint implies $\rho_{i,i+1}=\rho_{i+1,i}$.
Figure 4: In a) the intensity dynamics in absence disorder: an excitability
wave propagates through the array. In b) an example of the temporal dynamics
in presence of disorder is shown: excitability is spatially localized.
Parameters used are: $\alpha=\beta=0$, $K_{0}=0.05$, $r=0.005$.
In order to investigate the disorder-induced localization we have considered
that the additive noise is present, without loss of generality, only in the
central laser of the array. If all the lasers are coupled with identical
coupling strength ($K_{i,i+1}=K_{0}$ $\forall$ $i$), then an excitability wave
propagates from the centre of the array both towards left and right (see Fig.
4). We have considered a chain of 150 lasers with an average coupling
$K_{0}=0.05$, a noise on the central laser with strength $D=0.1$ and a random
coupling distribution varying in the interval $r\in[0,0.002]$. We have
verified that in this condition, with noise acting only on one laser, we have
regular emission of pulses that propagate to all the lasers in the chain for
$r=0$.
Figure 5: The temporal average of 100 traces characterized by different
realizations of disorder with the same value of $r$ is plotted here together
with the fit to Eq. 5 used to extract the value of the localization exponent,
$\lambda$. To extract the correct localization exponent we discard the central
laser, whose average intensity is systematically too high due to the presence
of additive noise, and its nearest neighbors. Parameters used are the same of
Fig. 6, with $\alpha=\beta=0$ and $r=0.002$.
For each value of $r$, 100 numerical experiments with a time duration of
$T=50000$ has been carried out. For each disordered coupling configuration we
have computed the average intensity trace across the array. We have then
fitted the averaged intensity distribution after the 100 realizations of the
disorder with the same strength $r$, with the following exponential function:
$\displaystyle f=b+\exp\left(-\lambda|i-i_{0}|\right)$ (5)
where $i_{0}$ denotes the position of the central laser (see Fig. 5 for an
example). This procedure has been repeated 8 times, so that the average value
and standard deviation of the fit parameters can be extracted; the average
localization exponent $\langle\lambda\rangle$ versus the disorder strength $r$
is plotted in Fig. 6, together with error bars indicating the standard
deviation.
Figure 6: The average localization exponent $\langle\lambda\rangle$ has been
plotted versus randomness strength $r$ for 3 different values of the linewidth
enhancement factors $\alpha$ and $\beta$. For vanishing $\alpha$ and $\beta$ a
transition from ballistic to localized regime occurs, low values of $\alpha$
and $\beta$ require an higher threshold for localization. At large values of
the linewidth enhancement factor localization is lost. Parameters used are
$D=0.1$ and $K=0.05$.
Such disorder-induced localization occurs for coupling values such that if we
couple all the lasers with identical value of $K_{0}-r$ equal to the minimum
possible value generated by the random process, we would still have an
excitability wave spreading from the array center both towards left and right;
this ensures the fact that localization is not caused by a trivial breaking of
the laser chain but is indeed provoked by a dynamical effect. For vanishing
$\alpha$ and $\beta$ we can notice that localization occurs for coupling
strength much smaller than the ones needed in the real coupling case (about
0.3, see [19]).
We have furthermore investigated the impact of linewidth enhancement factor
both for the gain medium ($\alpha$) and for the saturable absorber ($\beta$)
on the disorder-induced localization of excitability. Physically $\alpha$ and
$\beta$ are responsible for coupling of amplitude to phase fluctuations. It is
interesting to appreciate that linewidth enhancement factor
($\alpha$,$\beta\neq 0$) causes a decrease (for $\alpha=\beta=1$) and even a
vanishing (for $\alpha=\beta=3$ or $5$) of the localization.
## 5 Conclusions
We have demonstrated with the help of numerical simulations that evanescently
coupled excitable lasers with saturable absorber synchronize in a more
efficient way compared to lossy coupled ones, but that they do not exhibit
array enhanced coherence resonance. We have furthermore demonstrated that
disorder-induced localization of excitability caused by randomness in the
coupling strength exists for imaginary coupling too. Taking into account the
linewidth enhancement factor both in the gain medium and in the saturable
absorber we have shown that localization is reduced and eventually quenched by
an increase of the linewidth enhancement factor. This fact may constitute a
serious obstacle in the experimental observation of the disorder-induced
localization of excitability. Our results shed further light on the collective
dynamics of coupled excitable lasers with saturable absorber and suggest
interesting directions for experimental studies.
## 6 Acknowledgements
The authors gratefully acknowledge useful discussion with Dr. Sylvain Barabay.
## References
* [1] S. Furber, Journal of neural engineering 13, 051001 (2016)
* [2] B.J. Shastri, A.N. Tait, T.F. de Lima, M.A. Nahmias, H.T. Peng, P.R. Prucnal, _Principles of neuromorphic photonics_ (2017), 1801.00016
* [3] J.D. Murray, _Mathematical Biology: I. An Introduction_ , Vol. 1 (Springer-Verlag, 3rd Edition Berlin Heidelberg, 2002)
* [4] E.M. Izhikevich, International Journal of Bifurcation and Chaos 10, 1171 (2000)
* [5] I.K. Schuller, R. Stevens, Tech. rep. (2015), https://science.energy.gov/$\sim$/media/bes/ pdf/reports/2016/NCFMtSA_rpt.pdf
* [6] P.R. Prucnal, B.V. Shastri, _Neuromorphic Photonics_ (Taylor & Francis, 1st Edition Boca Raton, 2017)
* [7] J.L. Dubbeldam, B. Krauskopf, D. Lenstra, Physical Review E 60, 6580 (1999)
* [8] S. Barbay, R. Kuszelewicz, A.M. Yacomotti, Optics letters 36, 4476 (2011)
* [9] S. Barland, O. Piro, M. Giudici, J.R. Tredicce, S. Balle, Physical Review E 68, 036209 (2003)
* [10] M. Giudici, C. Green, G. Giacomelli, U. Nespolo, J. Tredicce, Physical Review E 55, 6414 (1997)
* [11] F. Plaza, M. Velarde, F. Arecchi, S. Boccaletti, M. Ciofini, R. Meucci, EPL (Europhysics Letters) 38, 85 (1997)
* [12] M. Brunstein, A.M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, A. Levenson, Physical Review A 85, 031803 (2012)
* [13] M. Turconi, M. Giudici, S. Barland, Physical Review Letters 111, 233901 (2013)
* [14] B.J. Shastri, M.A. Nahmias, A.N. Tait, A.W. Rodriguez, B. Wu, P.R. Prucnal, Scientific Reports 6, 19126 (2016)
* [15] B. Romeira, R. Avó, J.M. Figueiredo, S. Barland, J. Javaloyes, Scientific reports 6, 19510 (2016)
* [16] B.J. Shastri, M.A. Nahmias, A.N. Tait, B. Wu, P.R. Prucnal, Optics express 23, 8029 (2015)
* [17] B.J. Shastri, A.N. Tait, M. Nahmias, B. Wu, P. Prucnal, _Coincidence detection with graphene excitable laser_ , in _CLEO: Science and Innovations_ (Optical Society of America, 2014), pp. STu3I–5
* [18] A.M. Perego, M. Lamperti, Physical Review A 94, 033839 (2016)
* [19] M. Lamperti, A.M. Perego, Physical Review A 96, 041803(R) (2017)
* [20] F. Selmi, R. Braive, G. Beaudoin, I. Sagnes, R. Kuszelewicz, S. Barbay, Opt. Lett. 40, 5690 (2015)
* [21] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 44, 874 (1997)
* [22] P.W. Anderson, Physical Review 109, 1492 (1958)
* [23] J.B. Pendry, J. Phys. C: Solid State Phys. 15, 5773 (1982)
|
††thanks: C. X. and H. Y. contributed equally to this manuscript.††thanks: C.
X. and H. Y. contributed equally to this manuscript.
# Three-nodal surface phonons in solid-state materials: Theory and material
realization
Chengwu Xie School of Physical Science and Technology, Southwest University,
Chongqing 400715, China; Hongkuan Yuan School of Physical Science and
Technology, Southwest University, Chongqing 400715, China; Ying Liu ying
<EMAIL_ADDRESS>(Y. L.); School of Materials Science and Engineering, Hebei
University of Technology, Tianjin 300130, China; Xiaotian Wang
<EMAIL_ADDRESS>(X. W.); School of Physical Science and Technology,
Southwest University, Chongqing 400715, China; Gang Zhang
<EMAIL_ADDRESS>(G. Z.); Institute of High Performance Computing,
Agency for Science, Technology and Research (A*STAR), 138632, Singapore
###### Abstract
This year, Liu et al. [Phys. Rev. B 104, L041405 (2021)] proposed a new class
of topological phonons (TPs; i.e., one-nodal surface (NS) phonons), which
provides an effective route for realizing one-NSs in phonon systems. In this
work, based on first-principles calculations and symmetry analysis, we
extended the types of NS phonons from one- to three-NS phonons. The existence
of three-NS phonons (with NS states on the $k_{i}$ = $\pi$ ($i$ = $x$, $y$,
$z$) planes in the three-dimensional Brillouin zone (BZ)) is enforced by the
combination of two-fold screw symmetry and time reversal symmetry. We screened
all 230 space groups (SGs) and found nine candidate groups (with the SG
numbers (Nos.) 19, 61, 62, 92, 96, 198, 205, 212, and 213) hosting three-NS
phonons. Interestingly, with the help of first-principles calculations, we
identified $P2_{1}$2121-type YCuS2 (SG No. 19), $Pbca$-type NiAs2 (SG No. 61),
$Pnma$-type SrZrO2 (SG No. 62), $P4_{1}$212-type LiAlO2 (SG No. 92),
$P4_{3}$212-type ZnP2 (SG No. 96), $P2_{1}$3-type NiSbSe (SG No. 198),
$Pa\bar{3}$-type As2Pt (SG No. 205), $P4_{3}$32-type BaSi2 (SG No. 212), and
$P4_{1}$32-type CsBe2F5 (SG No. 213) as realistic materials hosting three-NS
phonons. The results of our presented study enrich the class of NS states in
phonon systems and provide concrete guidance for searching for three-NS
phonons and singular Weyl point phonons in realistic materials.
## I Introduction
Topological quantum states of matter add1 ; add2 ; add3 are an important
topic in the field of modern condensed-matter physics. Over the past 15 years,
we have witnessed the emergence of many types of topological electronic
materials, such as topological insulators add4 ; add5 ; add6 ; add7 ,
topological crystalline insulators add8 ; add9 ; add10 , topological Kondo
insulators add11 ; add12 ; add13 , higher-order topological insulators add14 ;
add15 ; add16 ; add17 , topological semimetals add18 ; add19 ; add20 , and
higher-order topological semimetals add21 ; add22 ; add23 ; add24 . In
particular, the types and numbers of topological semimetals add20 are rapidly
increasing. In contrast to Dirac, Weyl, and Majorana fermions, which are
allowed in high-energy physics, the types of quasiparticles in topological
semimetals add25 are more diverse owing to fewer constraints imposed by the
space group (SG) symmetries of the crystal. Based on the dimensionality of the
band-crossings in the momentum space, the topological semimetals can be
classified into nodal point add26 ; add27 ; add28 ; add29 ; add30 , nodal line
add31 ; add32 ; add33 ; add34 ; add35 , and nodal surface (NS) add36 ; add37 ;
add38 ; add39 ; add40 semimetals with zero-, one-, and two-dimensional band-
crossings, respectively.
Three-dimensional topological semimetals with two-dimensional band-crossings
can host NS states in the Brillouin zone (BZ). Each point on the NS should be
a two-fold degenerate point with linear band dispersion along the surface
normal direction. Researchers hope that NS semimetals exhibit exotic physical
properties, such as stronger quantum oscillations and peculiar plasmon
excitations. Wu et al. add36 summarized an essential NS state dictated by
nonsymmorphic symmetry without spin-orbit coupling (SOC). The existence of a
series of NS semimetals in realistic electronic systems has been predicted,
including BaVS3 add40 , ZrSiS add41 ; add42 , K6YO4 add36 , FeB4 add43 , Ti3Al
add37 , and X(MoS)3 (X = K, Rb, and Cs) add44 . However, in general, SOC in
electronic materials cannot be ignored; thus, the proposed two-dimensional
nonsymmorphic symmetry-enforced NS states in electronic systems will usually
be destroyed or reduced to one-dimensional nodal lines when SOC is considered
add42 . Moreover, the NS states in some materials are far from the Fermi level
and exhibit large energy variations, which hinder their experimental
detection.
The proposed topological phonons (TPs) add45 ; add46 have renewed the
interest in topological quantum states; TPs are a basic kind of boson-type
quasiparticles; they are not affected by the Pauli exclusion principle and
SOC. Therefore, TPs can normally be observed in spinless phononic systems in
all frequency ranges. In addition to the proposed nodal point phonons add47 ;
add48 ; add49 ; add50 ; add51 ; add52 ; add53 ; add54 ; add55 and nodal line
phonons add56 ; add57 ; add58 ; add59 ; add60 ; add61 ; add62 ; add63 ; add64
, one-NS phonons add65 have been presented by Liu et al. based on symmetry
analysis and first-principles calculations. The researchers provided a
complete list of the one-NS phonons in the 230 SGs and discovered that RbTeAu
family materials with SG number (No.) 51 may contain one-NS states (on the
$k_{x}$ = $\pi$ plane). The occurrence of one-NS states is ensured by screw
rotation symmetry along the $i$ axis ($i$ = $x$, $y$, or $z$) and time-
reversal symmetry $\mathcal{T}$. Fig. 1(a) presents a schematic diagram of
one-NS phonons. Moreover, two more types of NS phonons should exist: two- and
three-NS phonons, as illustrated in Fig. 1(b) and Fig. 1(c), respectively.
In this study, we extended the class of NS phonons from one- to three-NS
phonons. For the three-NS phonons, the NS states are localized on the $k_{i}$
= $\pi$ ($i$ = $x$, $y$, $z$) planes in the three-dimensional BZ. We screened
all 230 SGs; the SGs with Nos. 19, 61, 62, 92, 96, 198, 205, 212, and 213 are
candidate groups that can obtain three-NS phonons. Because the three-NS
phonons in these SGs are symmetry-enforced, one can easily achieve three-NS
phonons in realistic materials with the previously presented SGs. For example,
in this work, we identified $P2_{1}$2121-type YCuS2 (SG No. 19), $Pbca$-type
NiAs2 (SG No. 61), $Pnma$-type SrZrO2 (SG No. 62), $P4_{1}$212-type LiAlO2 (SG
No. 92), $P4_{3}$212-type ZnP2 (SG No. 96), $P2_{1}$3-type NiSbSe (SG No.
198), $Pa\bar{3}$-type As2Pt (SG No. 205), $P4_{3}$32-type BaSi2 (SG No. 212),
and $P4_{1}$32-type CsBe2F5 (SG No. 213) as realistic materials that can host
three-NS phonons.
Figure 1: Schematic diagrams of (a) one-NS, (b) two-NS, (c) and three-NS
phonons, respectively.
## II Symmetry Analysis of Three-NS Phonons.
In this part, we searched all essential NSs, which are only dictated by
symmetries, in spinless systems add36 . Such an NS is protected by the
combination of time-reversal symmetry ($\mathcal{T}$) and two-fold screw
rotation symmetry ($S_{2i}$).
Without loss of generalization, we take two-fold screw rotation along the
z-direction as an example: $S_{2z}$:$(x,y,z)\to(-x,-y,z+\frac{1}{2})$ with a
half translation in the lattice constant along its rotation axis. It also
affects the momentum space:
$S_{2z}$:$(k_{x},k_{y},k_{z})\to(-k_{x},-k_{y},k_{z})$, thereby only
preserving the momentum along $k_{z}$. Without SOC,
$S_{2z}^{2}$=$T_{100}$=$e^{-ik_{z}}$, where $T_{100}$ is the translation along
the z-direction. For time-reversal symmetry, in spinless systems,
$\mathcal{T}^{2}$ = 1, which is antiunitary and inverses the momentum k.
Consequently, their combination $\mathcal{T}S_{2z}$ is also antiunitary.
Remarkably, on planes where $k_{z}$ = $\pm\pi$,
$(\mathcal{T}S_{2z})^{2}$=$e^{-ik_{z}}|_{k_{z}=\pm\pi}$ = $-1$, which suggests
Kramer-like degeneracy on these planes. Thereby, it leads to Kramer-like
degeneracy. Hence, the phonon bands on the $k_{i}$ = $\pi$ ($i$ = $x$, $y$,
$z$) planes must become two-fold degenerate, thereby forming three-NS phonons.
Furthermore, the presence of three two-fold rotation symmetries (i.e.,
$S_{2x}$, $S_{2y}$, and $S_{2z}$) leads to three NSs on the planes $k_{i}$ =
$\pm\pi$ ($i$ = $x$, $y$, $z$). In this study, we proposed all the three-NS
phonons by searching all 230 SGs in phonon systems. According to the results,
the SGs with Nos. 19, 61, 62, 92, 96, 198, 205, 212, and 213 (see Table 1) can
host three-NS phonons.
## III Computational Details
First-principles calculations based on density functional theory were
performed to study the ground states of $P2_{1}$2121-type YCuS2, $Pbca$-type
NiAs2, $Pnma$-type SrZrO2, $P4_{1}$212-type LiAlO2, $P4_{3}$212-type ZnP2,
$P2_{1}$3-type NiSbSe, $Pa\bar{3}$-type As2Pt, $P4_{3}$32-type BaSi2, and
$P4_{1}$32-type CsBe2F5 materials, as implemented in the Vienna Ab Initio
Simulation Package. The projector augmented wave method and generalized
gradient approximation add66 with Perdew–Burke–Ernzerhof functions were used
for the ionic potential and exchange-correlation interaction. In addition, a
plane wave cutoff energy of 500 eV was used for the structural relaxation. The
following $k$-mesh samples were used for YCuS2, NiAs2, SrZrO2, LiAlO2, ZnP2,
NiSbSe, As2Pt, BaSi2, and CsBe2F5: $9\times 7\times 5$, $5\times 5\times 3$,
$5\times 5\times 5$, $7\times 7\times 7$, $7\times 7\times 3$, $7\times
7\times 7$, $7\times 7\times 7$, $9\times 9\times 9$, and $5\times 5\times 5$,
respectively. All these materials are experimentally synthesized materials.
The phononic dispersions of the $2\times 2\times 1$ YCuS2, $2\times 2\times 1$
NiAs2, $2\times 1\times 2$ SrZrO2, $2\times 2\times 2$ LiAlO2, $2\times
2\times 1$ ZnP2, $2\times 2\times 2$ NiSbSe, $2\times 2\times 2$ As2Pt,
$2\times 2\times 2$ BaSi2, and $1\times 1\times 1$ CsBe2F5 cells were examined
with density functional perturbation theory and PHONOPY codes add67 .
Table 1: A complete list of three-NS phonons in 230 SGs. The first and second columns present the SG numbers and SG symbols, the third column lists the three-NSs along the symmetry paths, and the fourth column presents the corresponding realistic materials. Space group | Space group | Three-nodal | Realistic
---|---|---|---
numbers | symbols | surfaces | materials
19 | $P2_{1}$2121 | NSTYS, NSSXU, and NSUZT | YCuS2
61 | $Pbca$ | NSTYS, NSSXU, and NSUZT | NiAs2
62 | $Pnma$ | NSTYS, NSSXU, and NSUZT | SrZrO2
92 | $P4_{1}$212 | NSZRA, and NSMXR | LiAlO2
96 | $P4_{3}$212 | NSZRA, and NSMXR | ZnP2
198 | $P2_{1}$3 | NSRXM | NiSbSe
205 | $Pa\bar{3}$ | NSRXM | As2Pt
212 | $P4_{3}$32 | NSRXM | BaSi2
213 | $P4_{1}$32 | NSRXM | CsBe2F5
## IV Materials with Three-NS phonons
For phonon systems with SG Nos. 19, 61, 62, the three-NSs (i.e., NSTYS, NSSXU,
and NSUZT) appear on the planes $k_{y}$ = $\pi$, $k_{x}$ = $\pi$, and $k_{z}$
= $\pi$, respectively. Some realistic materials were selected as examples to
demonstrate that they host three-NSs in their phonon dispersions:
$P2_{1}$2121-type YCuS2 (SG No. 19) can be prepared add68 by fusing the high-
purity elements in evacuated quartz ampoules; Murray and Heyding add69
prepared $Pbca$-type NiAs2 (SG No. 61) by heating the elements in sealed
evacuated Vycor tubes; $Pnma$-type SrZrO2 powders (SG No. 62) were prepared
with the polymeric precursor method by Cavalcante et al. add70 . The crystal
structures of these three materials are shown in Fig. 2. They are completely
relaxed; their theoretically determined lattice constants and the previously
published experimentally determined data are listed in Table 2.
Figure 2: Crystal structures of $P2_{1}$2121-type YCuS2 (SG No. 19), $Pbca$-type NiAs2 (SG No. 61), and $Pnma$-type SrZrO2 (SG No. 62), respectively. Table 2: Theoretically and experimentally determined lattice constants of YCuS2, NiAs2, and SrZrO2. Materials | Theoretical lattice | Experimental lattice
---|---|---
| constants | constants add68 ; add69 ; add70
YCuS2 | a = 3.96 Å, b = 6.26 Å, | a = 3.97 Å, b = 6.27 Å,
| c = 13.48 Å | c = 13.38 Å
NiAs2 | a = 5.80 Å, b = 5.89 Å, | a = 5.77 Å, b = 5.83 Å,
| c = 11.50 Å | c = 11.41 Å
SrZrO2 | a = 5.91 Å, b = 8.29 Å, | a = 5.81 Å, b = 8.19 Å,
| c = 5.84 Å | c = 5.79 Å
Figure 3: (a) Three-dimensional BZ and symmetry points. Three-NS states (red,
green, and yellow) are localized on the $k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$)
planes in three-dimensional BZ. (b)–(d) Calculated phonon dispersions of
YCuS2, NiAs2, and SrZrO2, respectively. Three NS regions (i.e., NSTYS, NSSXU,
and NSUZT) are highlighted in green, red, and yellow, respectively. Figure 4:
(a), (c), and (e) Enlarged phonon dispersions of three regions (see Fig.
3(b)–(d)) of YCuS2, NiAs2, and SrZrO2, respectively. (b), (d), and (f) phonon
dispersions along $a$ (0.70, 0.25, 0.00) -$P$ (0.50, 0.25, 0.00) -$a^{\prime}$
(0.30, 0.25, 0.00), $b$ (0.00, 0.25, 0.7) -$Q$ (0.00, 0.25, 0.5) -$b^{\prime}$
(0.00, 0.25, 0.3), and $c$ (0.00, 0.7, 0.25) -$N$ (0.00, 0.5, 0.25)
-$c^{\prime}$ (0.00, 0.3, 0.25), respectively. All points at $P$, $Q$, and $N$
points are two-fold degenerate with linear phonon band dispersions.
The phonon dispersions of YCuS2, NiAs2, and SrZrO2 along the $\Gamma-X-
S-Y-\Gamma-Z-S-X-U-Z-T-Y-S-R$ paths (see Fig. 3(a)) are shown in Fig.
3(b)–(d), respectively. Three regions (highlighted in red, yellow, and green)
are of interests in this study. The enlarged figures of the phonon dispersions
of YCuS2, NiAs2, and SrZrO2 in the three regions are shown in Fig. 4(a), (c)
and (e), respectively. All the phonon bands along the $S$-$X$-$U$,
$U$-$Z$-$T$, and $T$-$Y$-$S$ planes have two-fold degeneracy. To explain this
in more detail, some symmetry lines (i.e., $a$-$P$-$a^{\prime}$,
$b$-$Q$-$b^{\prime}$, and $c$-$N$-$c^{\prime}$; see Fig. 3(a)) are selected,
they are perpendicular to the $S$-$X$, $T$-$Z$, and $Y$-$T$ symmetry lines,
respectively. Subsequently, we calculate the phonon dispersions along the
$a$-$P$-$a^{\prime}$, $b$-$Q$-$b^{\prime}$, and $c$-$N$-$c^{\prime}$ paths;
the results are presented in Fig. 4(b), (d), (f), respectively. Evidently, the
points (highlighted by red circles) at the $P$, $Q$, and $N$ symmetry points
are two-fold degenerate and have linear band dispersions. In addition, two-
fold Kramer-like degeneracy occurs at every point on the $S$-$X$-$U$,
$U$-$Z$-$T$, and $T$-$Y$-$S$ planes, thereby forming three-NSs on the $k_{i}$
= $\pi$ ($i$ = $x$, $y$, $z$) planes. These density functional theory results
agree well with the argument (Section II) that the antiunitary symmetry
$\mathcal{T}S_{2i}$ ensures the existence of three-NS phonons on the $k_{i}$ =
$\pm\pi$ ($i$ = $x$, $y$, $z$) planes.
Figure 5: (a) and (b) Crystal structures of $P4_{1}$212-type LiAlO2 (SG No.
92) and $P4_{3}$212-type ZnP2 (SG No. 96), respectively; (c) Three-dimensional
BZ and symmetry points. Three-NS states (highlighted in yellow and green) are
localized on $k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$) planes in three-dimensional
BZ. (b)–(d) Calculated phonon dispersions of LiAlO2 and ZnP2, respectively. NS
regions (i.e., NSMXR, and NSZRA) are highlighted in green and yellow,
respectively.
In the next step, some realistic materials with SG Nos. 92 and 96 and three-NS
phonons are presented. The first example is $P4_{1}$212-type LiAlO2 (SG No.
92); Remeika and Ballman add71 prepared these single crystals from a flux.
The second example is $P4_{3}$212-type ZnP2 (SG No. 96). Researchers add72
have reported that ZnP2 crystals can exist in an enantiomorphic form with SG
$P4_{3}$212 = $D_{4}^{8}$. The theoretically determined and previously
published experimental lattice constants are shown in Table 3. The crystal
structures of the two materials are shown in Fig. 5(a) and (b). The phonon
dispersions of the two materials along the $\Gamma-X-M-\Gamma-Z-R-A-M-X-R$
paths (see Fig. 5(c)) are presented in Fig. 5(d) and (e). To examine the
three-NS phonons in these two materials, we only focused on two paths:
$Z$-$R$-$A$ and $M$-$X$-$R$, which are highlighted in yellow and green in Fig.
5(d) and (e), respectively. The enlarged phonon dispersions of these two
regions for LiAlO2 and ZnP2 are shown in Fig. 6(a) and (c), respectively. All
the phonon bands along the $Z$-$R$-$A$ and $M$-$X$-$R$ paths are two-fold
degenerate. To present examples, we select two symmetry points $N$ and $Q$ on
the $k_{y}$ = $\pi$ and $k_{z}$ = $\pi$ planes, respectively. We construct the
two paths $b$-$Q$-$b^{\prime}$ and $c$-$N$-$c^{\prime}$, which vertically pass
through the $k_{z}$ = $\pi$ and $k_{y}$ = $\pi$ planes, respectively. The
obtained phonon dispersions along the $b$-$Q$-$b^{\prime}$ and
$c$-$N$-$c^{\prime}$ paths for LiAlO2 and ZnP2 are shown in Fig. 6(b) and (d),
respectively. There are two two-fold degenerate points at $Q$ and $N$, which
represent the NS states on the $k_{z}$ = $\pi$ and $k_{y}=\pi$ planes. Because
LiAlO2 and ZnP2 with SG Nos. 92 and 96 host four-fold screw rotation, $S_{4z}$
= $\\{C_{4z}|00\frac{1}{2}\\}$, there should be NSs on the $k_{x}$ = $\pm\pi$
planes.
Table 3: Theoretically and experimentally determined lattice constants of LiAlO2 and ZnP2. Materials | Theoretical lattice | Experimental lattice
---|---|---
| constants | constants add71 ; add72
LiAlO2 | a = b = 5.21 Å, | a = b = 5.17 Å,
| c = 6.30 Å | c = 6.59 Å
ZnP2 | a = b = 5.06 Å, | a = b = 5.10 Å,
| c = 18.53 Å | c = 18.62 Å
Figure 6: (a), (c) Enlarged phonon dispersions of two regions (see Fig. 5(d)
and (e)) of LiAlO2 and ZnP2, respectively. (b), (d) Phonon dispersions along
$b$ (0.00, 0.25, 0.7) -$N$ (0.00, 0.25, 0.5) -$b^{\prime}$ (0.00, 0.25, 0.3),
and $c$ (0.00, 0.7, 0.25) -$Q$ (0.00, 0.5, 0.25) -$c^{\prime}$ (0.00, 0.3,
0.25), respectively. All the points at the $Q$ and $N$ points are two-fold
degenerate points with linear phonon band dispersions.
Finally, some realistic materials with SG Nos. 198, 205, 212, and 213 are
presented, which host three-NS phonons, i.e., NSRXM. The first example is
$P2_{1}$3-type NiSbSe (SG No. 198). It was prepared by letting powders of
binary nickel chalcogenides react with the respective pnictogen component in
evacuated sealed silica tubes add73 . The second example is $Pa\bar{3}$-type
As2Pt (SG No. 205). Ramsdell add74 produced artificial PtAs2, which is
identical to natural sperrylite. The third example is $P4_{3}$32-type BaSi2
(SG No. 212). This compound is an interesting material add75 that can host
three types of polymorphs (orthorhombic, trigonal, and cubic crystal classes
with $Pnma$, $P\bar{3}m1$, and $P4_{3}32$ SGs) at up to 40 kbar and 1000
${}^{\circ}C$. $P4_{3}$32-type BaSi2 represents one kind of the high-pressure
phases. The fourth example is $P4_{1}$32-type CsBe2F5 (SG No. 213). Le Fur and
Al$\rm{\acute{e}}$onard add76 dissolved Cs2CO2 carbonate in a hydrofluoric
solution containing excess BeF2. A single CsBe2F5 crystal can be obtained via
evaporation at 55 ${}^{\circ}C$. The crystal structures of these materials are
shown in Fig. 7. We determined their lattice constants with structural-
relaxation calculations (see Fig. 7 and Table 4).
Figure 7: (a)–(d) Crystal structures of $P2_{1}$3-type NiSbSe (SG No. 198), $Pa\bar{3}$-type As2Pt (SG No. 205), $P4_{3}$32-type BaSi2 (SG No. 212), and $P4_{1}$32-type CsBe2F5 (SG No. 213), respectively. Table 4: Theoretically and experimentally determined lattice constants of NiSbSe, As2Pt, BaSi2, and CsBe2F5 Materials | Theoretical lattice | Experimental lattice
---|---|---
| constants | constants add73 ; add74 ; add75 ; add76
NiSbSe | a = b = c = 6.13 Å | a = b = c = 6.08 Å
As2Pt | a = b = c = 6.06 Å | a = b = c = 5.92 Å
BaSi2 | a = b = c = 6.77 Å | a = b = c = 6.71 Å
CsBe2F5 | a = b = c = 8.06 Å | a = b = c = 7.93 Å
Figure 8: (a) Three-dimensional BZ and symmetry points. Three-NS states (green
color) are localized on $k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$) planes in three-
dimensional BZ. (b)–(e) Calculated phonon dispersions of NiSbSe, As2Pt, BaSi2,
and CsBe2F5 materials, respectively. NS regions (i.e., NSRXM) are highlighted
in green.
We calculated the phonon dispersions of NiSbSe, As2Pt, BaSi2, and CsBe2F5
along the symmetry paths $\Gamma-X-M-\Gamma-R-X-M$ (see Fig. 8(a)); the
results are shown in Fig. 8(b)–(d). Let us focus on the two-fold degenerate
phonon bands along the $R$-$X$-$M$ paths (see Fig. 9(a), (c), (e), and (g)).
To prove that these bands are degenerated, we chose the path
$a$-$P$-$a^{\prime}$ that vertically passes through the $k_{y}$ = $\pi$ plane.
The obtained phonon dispersions along the $a$-$P$-$a^{\prime}$ path for these
materials are shown in Fig. 9(b), (d), (f), and (h), respectively. There are
two evident two-fold degenerate points at $P$ with linear band dispersions. We
can conclude that an NS phonon exists on the $k_{y}$ = $\pi$ plane on which
the two low-energy phonon bands cross linearly. Owing to $C_{3,111}$ symmetry,
equivalent NS phonons can be found on the $k_{x}$ = $\pi$ and $k_{y}$ = $\pi$
planes. We would like to point out that although symmetry requires the
occurrence of three-NS phonons and limits the possible positions on the
$k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$) planes, it does not limit the
frequencies and dispersions of three-NS phonons.
Figure 9: (a), (c), (e), (g) Enlarged phonon dispersions of $R$-$X$-$M$ path
(see Fig. 8(b)–(e)) of As2Pt, BaSi2, and CsBe2F5 materials, respectively. (b),
(d), (f), (h) Phonon dispersions along $a$ (0.00, 0.70, 0.25) -$P$ (0.00,
0.50, 0.25) -$a^{\prime}$ (0.00, 0.30, 0.25) path. All points at the $P$
symmetry point are two-fold degenerate points with linear phonon band
dispersions (indicated by red circles in (b), (d), (f), and (h)).
## V Summary and remarks
In conclusion, according to the symmetry analysis results, there are three-NS
phonons in the SGs with SG Nos. 19, 61, 62, 92, 96, 198, 205, 212, and 213 of
the 230 SGs. More interestingly, by performing first-principles calculations,
we discovered that the realistic materials $P2_{1}$2121-type YCuS2 (SG No.
19), $Pbca$-type NiAs2 (SG No. 61), $Pnma$-type SrZrO2 (SG No. 62),
$P4_{1}$212-type LiAlO2 (SG No. 92), $P4_{3}$212-type ZnP2 (SG No. 96),
$P2_{1}$3-type NiSbSe (SG No. 198), $Pa\bar{3}$-type As2Pt (SG No. 205),
$P4_{3}$32-type BaSi2 (SG No. 212), and $P4_{1}$32-type CsBe2F5 (SG No. 213)
include three-NS phonons in their phonon dispersions.
We present the following remarks: (i) Because phonons obey Bose–Einstein
statistics and are not limited by the Fermi energy, three-NS in the phonon
system may be more common in realistic materials; (ii) Unlike fermions in
electronic systems with heavy elements, SOC can be neglected for TPs in phonon
systems. Hence, three-NS phonons in phonon systems can be considered real NS
states without SOC-induced gaps; (iii) Although three-NS phonons in SGs 19,
61, 62, 92, 96, 198, 205, 212, and 213 can be determined by the combination of
two-fold screw symmetry and time reversal symmetry, the frequencies and
dispersions of three-NS phonons are not limited; (iv) One may ask what is the
difference between three-NS and one-/two-NS states? As we know, constrained by
the no-go theorem, among all the Weyl semimetals add77 discovered in
experiments before 2019, Weyl points always occur in pairs in the momentum
space, without exception. Interestingly, in 2019, as demonstrated by Yu et al.
add78 , the three-NS state is a good platform for realizing a singular Weyl
point by circumventing the no-go theorem. However, for two- and one-NS states,
although Weyl points and NS states can coexist, there must be more than one
Weyl point in the BZ. Fortunately, in this year, Ma et al. add77 observed a
singular Weyl point surrounded by three-NSs in PtGa with SG No. 198 in an
experiment.
Acknowledgments X.T.W. thanks Prof. Zhi-Ming Yu for his help regarding to this
manuscript. Y.L. is grateful for the support from the Nature Science
Foundation of Hebei Province (No. A2021202002). X.T.W. is grateful for the
support from the National Natural Science Foundation of China (No. 51801163)
and the Natural Science Foundation of Chongqing (No. cstc2018jcyjA0765).
## References
* (1) F. D. M. Haldane,?Rev. Mod. Phys. 89, 040502 (2017).
* (2) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
* (3) N. Goldman, J. C. Budich, and P. Zoller, Nat. Phys. 12, 639 (2016).
* (4) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
* (5) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
* (6) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
* (7) Y. Tokura, K. Yasuda, and A. Tsukazaki, Nat. Rev. Phys. 1, 126 (2019).
* (8) L. Fu, Phys. Rev. Lett. 106, 106802 (2011).
* (9) Y. Ando and L. Fu, Annu. Rev. Condens. Matter Phys. 6, 361 (2015).
* (10) T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nat. Commun. 3, 982 (2012).
* (11) M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev. Lett. 104, 106408 (2010).
* (12) M. Dzero, J. Xia, V. Galitski, and P. Coleman, Annu. Rev. Condens. Matter Phys. 7, 249 (2016).
* (13) M. Dzero, K. Sun, P. Coleman, and V. Galitski, Phys. Rev. B 85, 045130 (2012).
* (14) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. 4, eaat0346 (2018).
* (15) E. Khalaf, Phys. Rev. B 97, 205136 (2018).
* (16) M. Ezawa, Phys. Rev. Lett. 120, 026801 (2018).
* (17) R. Chen, C.-Z. Chen, J.-H. Gao, B. Zhou, and D.-H. Xu, Phys. Rev. Lett. 124, 036803 (2020).
* (18) A. A. Burkov, Nat. Mater. 15, 1145 (2016).
* (19) H. Gao, J. W. F. Venderbos, Y. Kim, and A. M. Rappe, Annu. Rev. Mater. Res. 49, 153 (2019).
* (20) B. Q. Lv, T. Qian, and H. Ding, Rev. Mod. Phys. 93, 025002 (2021).
* (21) D. C$\rm{\breve{a}lug\breve{a}ru,V.Juri\check{c}i\acute{c}}$, and B. Roy, Phys. Rev. B 99, 041301(R) (2019).
* (22) Z. Zhang, Z.-M. Yu, and S. A. Yang, Phys. Rev. B 103, 115112 (2021).
* (23) W. Wu, Z.-M. Yu, X. Zhou, Y. X. Zhao, and S. A. Yang, Phys. Rev. B 101, 205314 (2020).
* (24) Z.-M. Yu, W. K. Wu, X.-L. Sheng, Y. X. Zhao, and S. Y. A. Yang, Phys. Rev. B 99, 121106(R) (2019).
* (25) Z.-M. Yu, Z. Zhang, G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S. A. Yang, and Y. Yao, arXiv:2102.01517 (2021).
* (26) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
* (27) Q. D. Gibson, L. M. Schoop, L. Muechler, L. S. Xie, M. Hirschberger, N. P. Ong, R. Car, and R. J. Cava, Phys. Rev. B 91, 205128 (2015).
* (28) B.-J. Yang and N. Nagaosa, Nat. Commun. 5, 4898 (2014).
* (29) B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
* (30) A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B.?A. Bernevig, Nature (London) 527, 495 (2015).
* (31) C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B 92, 081201 (2015).
* (32) Q. Xu, R. Yu, Z. Fang, X. Dai, and H. Weng, Phys. Rev. B 95, 045136 (2017).
* (33) T. He, X. Zhang, Y. Liu, X. Dai, G. Liu, Z.-M. Yu, and Y. Yao, Phys. Rev. B 102, 075133 (2020).
* (34) L. Jin, X. M. Zhang, Y. Liu, X. F. Dai, X. N. Shen, L. Y. Wang, and G. D. Liu, Phys. Rev. B 102, 125118 (2020).
* (35) H. Zhang, X. Zhang, T. He, X. Dai, Y. Liu, G. Liu, L. Wang, and Y. Zhang, Phys. Rev. B 102, 155116 (2020).
* (36) W. Wu, Y. Liu, S. Li, C. Zhong, Z.-M. Yu, X.-L. Sheng, Y. X. Zhao, and S. A. Yang, Phys. Rev. B 97, 115125 (2018).
* (37) X. M. Zhang, Z.-M. Yu, Z. M. Zhu, W. K. Wu, S.-S. Wang, X.-L. Sheng, and S. A. Yang, Phys. Rev. B 97, 235150 (2018).
* (38) B.-B. Fu, C.-J. Yi, T.-T. Zhang, M. Caputo, J.-Z. Ma, X. Gao, B. Q. Lv, L.-Y. Kong, Y.-B. Huang, P. Richard, M. Shi, V. N. Strocov, C. Fang, H.-M. Weng, Y.-G. Shi, T. Qian, and H. Ding, Sci. Adv. 5, eaau6459 (2019).
* (39) S. Z. Chen, S. Li, Y. Chen, and W. Duan, Nano Lett. 20, 5400 (2020).
* (40) Q.-F. Liang, J. Zhou, R. Yu, Z. Wang, and H. Weng, Phys. Rev. B 93, 085427 (2016).
* (41) A. Topp, R. Queiroz, A. Grneis, L. M$\rm{\ddot{u}}$chler, A. W. Rost, A. Varykhalov, D. Marchenko, M. Krivenkov, F. Rodolakis, J. L. McChesney, B. V. Lotsch, L. M. Schoop, and C. R. Ast, Phys. Rev. X 7, 041073 (2017).
* (42) C. Chen, X. Xu, J. Jiang, S.-C. Wu, Y. P. Qi, L. X. Yang, M. X. Wang, Y. Sun, N. B. M. Schr$\rm{\ddot{o}}$ter, H. F. Yang, L. M. Schoop, Y. Y. Lv, J. Zhou, Y. B. Chen, S. H. Yao, M. H. Lu, Y. F. Chen, C. Felser, B. H. Yan, K. Liu, and Y. L. Chen, Phys. Rev. B 95, 125126 (2017).
* (43) F. Zhou, Y. Liu, J. H. Wang, M. Q. Kuang, T. Yang, H. Chen, X. T. Wang, and Z. X. Cheng, Phys. Rev. Materials 5, 074201 (2021).
* (44) T. Yang and X. Zhang, J. Mater. Chem. C 8, 9046 (2020).
* (45) O. Stenull, C. L. Kane, and T. C. Lubensky, Phys. Rev. Lett. 117, 068001 (2016).
* (46) Y. Liu, X. Chen, and Y. Xu, Adv. Funct. Mater. 30, 1904784 (2019).
* (47) T. T. Zhang, Z. D. Song, A. Alexandradinata, H. M. Weng, C. Fang, L. Lu, and Z. Fang, Phys. Rev. Lett. 120, 016401 (2018).
* (48) J. X. Li, Q. Xie, S. Ullah, R. H. Li, H. Ma, D. Z. Li, Y. Y. Li, and X.-Q. Chen, Phys. Rev. B 97, 054305 (2018).
* (49) H. Miao, T. T. Zhang, L. Wang, D. Meyers, A. H. Said, Y. L. Wang, Y. G. Shi, H. M. Weng, Z. Fang, and M. P. M. Dean, Phys. Rev. Lett. 121, 035302 (2018).
* (50) B. W. Xia, R. Wang, Z. J. Chen, Y. J. Zhao, and H. Xu, Phys. Rev. Lett. 123, 065501 (2019).
* (51) Q.-B. Liu, Y. Qian, H.-H. Fu, and Z. Wang, Npj Comput. Mater. 6, 95 (2020).
* (52) Q.-B. Liu, Z. Wang, and H.-H. Fu, Phys. Rev. B 103, L161303 (2021).
* (53) J. Liu, W. Hou, E. Wang, S. Zhang, J.-T. Sun, and S. Meng, Phys. Rev. B 100, 081204(R) (2019).
* (54) Y. J. Jin, Z. J. Chen, X. L. Xiao, and H. Xu, Phys. Rev. B 103, 104101 (2021).
* (55) J.-Y. You, X.-L. Sheng, and G. Su, Phys. Rev. B 103, 165143 (2021).
* (56) G. Liu, Y. J. Jin, Z. G. Chen, and H. Xu, Phys. Rev. B 104, 024304 (2021).
* (57) B. Zheng, B. Xia, R. Wang, Z. Chen, J. Zhao, Y. Zhao, and H. Xu, Phys. Rev. B 101, 100303(R) (2020).
* (58) Y. J. Jin, Z. J. Chen, B. W. Xia, Y. J. Zhao, R. Wang, and H. Xu, Phys. Rev. B 98, 220103(R) (2018).
* (59) R. Y. Wang, Z. J. Chen, Z. Q. Huang, B. W. Xia, and H. Xu, Phys. Rev. Materials 5, 084202 (2021).
* (60) B. B. Zheng, F. Y. Zhan, X. Z. Wu, R. Wang, and J. Fan, Phys. Rev. B 104, L060301 (2021).
* (61) J. J. Zhu, W. K. Wu, J. Z. Zhao, Hao. Chen, L. F. Zhang, S. Y. A. Yang, arXiv:2104.14816 (2021).
* (62) Q.-B. Liu, H.-H. Fu, and R. Q. Wu, Phys. Rev. B 104, 045409 (2021).
* (63) J. Li, J. Liu, S. A. Baronett, M. Liu, L. Wang, and R. Li, Y. Chen, D. Li, Q. Zhu, and X.-Q. Chen, Nat. Commun. 12, 1204 (2021).
* (64) T. T. Zhang, H. Miao, Q. Wang, J. Q. Lin, Y. Cao, G. Fabbris, A. H. Said, X. Liu, H. C. Lei, Z. Fang, H. M. Weng, and M. P. M. Dean, Phys. Rev. Lett. 123, 245302 (2019).
* (65) Q.-B. Liu, Z.-Q. Wang, and H.-H. Fu, Phys. Rev. B 104, L041405 (2021).
* (66) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
* (67) A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
* (68) L. D. Gulay, V. Ya. Shemet, and I. D. Olekseyuk, J. Alloys Compd. 388, 59 (2005).
* (69) J. J. Murray and R. D. Heyding, Can. J. Chem. 45, 2675 (1967).
* (70) L. S. Cavalcante, A. Z. Simoes, E. Longo, J. C. Sczancoski, R. Erlo, M. T. Escote, V. M. Longo, and J. A. Varela, Solid State Sci. 9, 1020 (2007).
* (71) J. P. Remeika and A. A. Ballman, Appl. Phys. Lett. 5, 180 (1964).
* (72) K. B. Aleinikova, A. I. Kozlov, S. G. Kozlova, and V. V. Sobolev, Phys. Solid State, 44, 1257 (2002).
* (73) A. J. Foecker and W. Jeitschko, J. Sol. State. Chem. 162, 69 (2001).
* (74) L. S. Ramsdell, Am. Mineral. 10, 281 (1925).
* (75) J. Evers, J. Solid State Chem. 32, 77 (1980).
* (76) Y. L. Fur and S. Al$\rm{\acute{e}}$onard, Acta Cryst. 28, 2115 (1972).
* (77) J.-Z. Ma, Q.-S. Wu, M. Song, S.-N. Zhang, E. B. Guedes, S. A. Ekahana, M. Krivenkov, M. Y. Yao, S.-Y. Gao, W.-H. Fan, T. Qian, H. Ding, N. C. Plumb, M. Radovic, J. H. Dil, Y.-M. Xiong, K. Manna, C. Felser, O. V. Yazyev, and M. Shi, Nat. Commun. 12, 1 (2021).
* (78) Z.-M. Yu, W. Wu, Y. X. Zhao, and S. A. Yang, Phys. Rev. B 100, 041118(R) (2019).
|
# Exemplar-based Pattern Synthesis with Implicit Periodic Field Network
Haiwei Chen1,2 Jiayi Liu1,2 Weikai Chen3 Shichen Liu1,2 Yajie Zhao1,2
1University of Southern California
2USC Institute for Creative Technologies
3Tencent Game AI Research Center
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Synthesis of ergodic, stationary visual patterns is widely applicable in
texturing, shape modeling, and digital content creation. The wide
applicability of this technique thus requires the pattern synthesis approaches
to be scalable, diverse, and authentic. In this paper, we propose an exemplar-
based visual pattern synthesis framework that aims to model the inner
statistics of visual patterns and generate new, versatile patterns that meet
the aforementioned requirements. To this end, we propose an implicit network
based on generative adversarial network (GAN) and periodic encoding, thus
calling our network the Implicit Periodic Field Network (IPFN). The design of
IPFN ensures scalability: the implicit formulation directly maps the input
coordinates to features, which enables synthesis of arbitrary size and is
computationally efficient for 3D shape synthesis. Learning with a periodic
encoding scheme encourages diversity: the network is constrained to model the
inner statistics of the exemplar based on spatial latent codes in a periodic
field. Coupled with continuously designed GAN training procedures, IPFN is
shown to synthesize tileable patterns with smooth transitions and local
variations. Last but not least, thanks to both the adversarial training
technique and the encoded Fourier features, IPFN learns high-frequency
functions that produce authentic, high-quality results. To validate our
approach, we present novel experimental results on various applications in 2D
texture synthesis and 3D shape synthesis.
## 1 Introduction
The synthesis of visual patterns, may that be a wooden texture for painting,
or a simulation of natural cave systems, is a technique that is applied
ubiquitously in computer-aided design and digital content creation. Visual
patterns can be understood as arts, shapes, or natural textures following
certain geometric structures. In an application context, let us start by
defining several characteristics that are desirable for an algorithm that
generates visual patterns:
* •
Authenticity. Probably the most prioritized quality of synthesized visual
patterns is its visual quality. When patterns are synthesized from an
exemplar, the quality is determined by whether they faithfully recreate the
source pattern.
* •
Diversity. It would be undesirable for a synthesizer to only copy patterns
from the source. Diversity is thus an equally important measurement that
evaluates whether the synthesized patterns vary from the source and each
other. We strive to achieve two different levels of diversity: the patterns
should be diversified both within a generated sample and across samples.
* •
Scalability. As patterns are usually demanded at different and potentially
large scales for many practical applications, we want a scalable synthesizer
to be able to efficiently generate patterns of arbitrary size. Scalability is
particularly valuable when it comes to the synthesis of 3D models, as the
extra dimension translates to a much larger amount of computations.
A scalable design choice leads us to formulate the synthesis problem as
generating patterns from a continuous, real coordinate space. This is
generally known as the implicit formulation, where a nonlinear function maps
points defined in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ to features that
represent the synthesized subjects. In particular, the implicit function has
been shown to be an efficient representation for synthesizing 3D volumes
[henzler2020learning, park2019deepsdf, mildenhall2020nerf].
Patterns that scale well to an infinitely large space, in general, possess a
stationary property - a shift-invariant structure that can be expanded by
tiling or stacking blocks of elements. We therefore develop our method by
pivoting on the fact that many types of natural and artistic patterns can be
analyzed and recreated in a stationary framework. The goal of synthesizing an
authentic and diverse stationary pattern from an exemplar, however, requires
careful modeling that is compatible with the underlying structure of the
pattern.
Generative adversarial networks (GAN) [goodfellow2014generative] is one of the
most promising techniques so far to model data distribution in an unsupervised
manner and has been frequently adapted to convolutional models that synthesize
visually authentic images [shaham2019singan, jetchev2016texture,
bergmann2017learning, zhou2018non, xian2018texturegan, shocher2019ingan]. How
would a GAN generator be leveraged to modeling stationary pattern? As all
stationary patterns contain a repeating structure with local variations that
“vivifies” its appearance, in an ideal situation, a stationary pattern can be
modeled by a discrete random field, where each random variable is associated
with a patch of the basic element. Thus a natural GAN formulation models image
patches with a spatially defined latent field [jetchev2016texture,
bergmann2017learning]. In a convolutional framework, however, problems arise
when fake samples generated from a discrete noise image are discriminated from
randomly sampled patches from a real image. The first problem is that the
sampled patch does not necessarily agree with the scale of the repeating
structure. The second problem is that the sampled patch can be arbitrarily
shifted from the center of a stationary element. A typical deconvolutional
network [dong2015image, zeiler2010deconvolutional] that upsamples from an
evenly-spaced noise image may not sufficiently address the previously
mentioned problems. To study their effects we designed a convolutional network
following the DCGAN architecture [radford2015unsupervised] to synthesize a
honeycomb pattern from a $2\times 2$ noise map, which is trained against
random patches sampled from a source image. The comparison between its result
and that synthesized by our generator network that is trained with an
identical discriminator is shown in Figure. 1. We found that the noise map
does not capture well the honeycomb structure as seams and intercepting
elements are visible from the synthesized image.
Though various techniques have been proposed in the past to address the
aforementioned issues (e.g. [jetchev2016texture, bergmann2017learning]), in
this paper, we consider a more natural way to model stationary patterns with
an implicit periodic generator. The core of the formulation is to match the
repeatable structure of a stationary pattern to the period of a learnable
continuous periodic function. Instead of modeling the pattern with a discrete
noise tensor, we define latent variables in a continuous space, where the
extent of each latent factor is learned to match with the extent of the
repeating elements. The benefits of this design align well with the desirable
characteristics for visual pattern synthesis: 1) learned periodicity of the
implicit field encourages latent factors to model the stationary variations
observed from the exemplar pattern; 2) a continuous representation provides
flexibility during training to learn a distribution from randomly shifted
patches cropped from the exemplar; 3) a Fourier encoding scheme learns high-
frequency details from the exemplar. This allows our model to synthesize
visually authentic results, and 4) multilayer perceptron (MLP) that implicitly
takes coordinates as input scales well to the generation of 3D shapes when
compared to 3D convolution. Based on these design choices, we term our network
the Implicit Periodic Field Network (IPFN).
We validate our proposed design by showing various applications in texture
image synthesis and 3D volume synthesis in Section. 4. Specifically, besides
synthesizing stationary patterns, we design a conditional formulation of our
model to tackle the synthesis of directional patterns and to provide
application in controllable shape generation. An ablation study is also
conducted to verify the effectiveness of our design choices.
Figure 1: Comparison between synthesized honeycomb from a DCGAN convolution
generator and the periodic MLP generator. Seams and intercepting patterns are
visible in the former result due to difficulty for the convolution generator
to capture the repeating structure.
## 2 Related Work
#### Pattern Synthesis
2D visual patterns are generally referred to as “texture” due to their
prevalent applications in computer-aided design. Well-known early attempts to
synthesize texture derive patterns from smoothly interpolated noise
[perlin1985image, perlin2002improving, worley1996cellular] and create
aesthetically pleasing materials that display a high level of randomness.
[henzler2020learning, portenier2020gramgan] are two recent works related to us
that utilize randomness from a continuously-defined Perlin noise
[perlin1985image] to synthesize exemplar-based textures in an implicit field.
Their works demonstrate the advantage of smooth noise field and implicit
formulation in efficiently and diversely generating a 3D texture field.
However, just like their procedural precedence, these methods have been shown
to be limited in synthesizing patterns that are more complicated in structures
(e.g. bricks, rocky surface) [portenier2020gramgan].
Later works in traditional image synthesis are generally divided into pixel-
based method (e.g. [efros1999texture, wei2003texture]), patch-based method
(e.g. [kopf2007solid, efros2001image, kwatra2003graphcut]) and optimization-
based method (e.g. [kopf2007solid, kwatra2005texture]) and have shared with us
important considerations for recreating new patterns from an exemplar. For
instance, synthesis on a patch level encourages preservation of fine local
details, and the efforts are focused on ”quilting” the discontinuity between
patches [efros2001image, kwatra2003graphcut] and encouraging global similarity
[kopf2007solid]. Early statistical model [efros1999texture,
portilla2000parametric] utilizes a random field representation that captures
variations in stationary patterns and synthesizes a variety of new patterns
under the Julesz’ conjecture. Our work is inspired by key ideas from the
traditional modeling of stationary textures, while we strive to unify
authenticity, diversity, and scalability with a neural representation to
overcome limitations that existed in the traditional approaches.
Compared to earlier parametric models, the artificial neural network is
powerful in its generalizability in pattern recognition [hansen1990neural].
Thus neural synthesis of texture images has marked the advances in recent
endeavors. How does a neural network learn the stylish representation of a
texture without simply reconstructing it? A milestone that unifies both
synthesized quality and sampling power adversarially trains a generator
network and a discriminator network to learn the mapping from a latent space
to the texture distribution [goodfellow2014generative]. We are particularly
interested in the generative adversarial networks (GANs) that models the inner
statistics of the exemplars. This is marked by a patch-based approach that
represents an image as a collection of smaller images: [li2016precomputed,
jetchev2016texture] formalizes patch-based synthesis in the GAN setting with
the concept of Markovian and spatial GAN. [bergmann2017learning] motivated us
with its periodic design in latent codes, which effectively captures
stationary patterns from an input exemplar. [zhou2018non] can be seen as a
complement to our design by focusing on addressing non-stationary texture
synthesis by learning to expand an image patch. In addition,
[shaham2019singan, shocher2019ingan] present multi-scale, global-focused
approaches to effectively recreate natural images. While the aforementioned
approaches all utilize convolutional designs, our work extends texture
synthesis to the continuous domain with an implicit representation of
textures, as we argue that such representation provides a more natural and
efficient way to synthesize stationary patterns.
The synthesis of 3D shapes is of particular interest in computer graphics and
thus has a rich history. To name a few, this includes volumetric field design
[zhang2006vector, palacios2016tensor, chen2016synthesis], procedural
generation [ijiri2008example, merrell2007example] and 3D texturing
[zhou2006mesh, bhat2004geometric]. However, very few works have considered the
synthesis of 3D patterns with neural networks, with the exception of
[henzler2020learning, portenier2020gramgan], which explores the generation of
3D solid textures.
#### Implicit Network
Implicit network refers to multilayer perceptron that learns a continuous
function in real coordinate space. In particular, the implicit network is
mainly utilized in the reconstruction of 3D shapes [park2019deepsdf,
saito2019pifu, liu2019learning, sitzmann2020implicit], where shapes are
represented by a distance function, or a radiance field [mildenhall2020nerf,
yu2021pixelnerf]. We are motivated by the signed distance representation of
shapes and the Fourier encoding explored in [mildenhall2020nerf,
tancik2020fourier] in our design of the implicit networks, and our work adopts
these features to a generative setting where novel 3D patterns are
synthesized.
## 3 Method
Our method is best introduced by expanding from the Wasserstein GAN value
function [gulrajani2017improved] constructed as the Earth-Mover distance
between the real data distribution $\mathbb{P}_{data}$ and the generator
distribution $\mathbb{P}_{g}$:
$\underset{G}{\min}\>\underset{D}{\max}\>\mathbb{E}_{\bm{x}\sim\mathbb{P}_{data}}[\log(D(\bm{x}))]-\mathbb{E}_{\bm{z}\sim\mathbb{P}_{Z}}[\log(D(G(\bm{z}))].$
(1)
Our first change of the above objective is to draw real-valued coordinates
$\bm{c}\in\mathbb{R}^{k}$ ($k=2$ for image and $k=3$ for volume) from a
distribution $\\{s\bm{c}\>|\>\bm{c}\sim\mathbb{P}_{\bm{c}}\\}$ as input to the
generator, where $s$ is a constant scalar. This underlies the implicit nature
of the generative network, as $G$ learns a mapping from the real coordinate
space in a stochastic process. Instead of being sampled from a prior
distribution $\mathbb{P}_{Z}$, latent variables are drawn from a random field
$f_{z}(\bm{c})$ defined on the real coordinate space. $G$ is therefore a
function of both the coordinates $\bm{c}\in\mathbb{R}^{k}$ and the latent
variables $f_{z}(\bm{c})\in\mathbb{R}^{d}$. This updates Eq.1 to
$\begin{split}\underset{G}{\min}\>\underset{D}{\max}\>&\mathbb{E}_{\bm{x}\sim\mathbb{P}_{data}}[\log(D(\bm{x}))]-\\\
&\mathbb{E}_{\bm{c}\sim\mathbb{P}_{c},\bm{z}\sim\mathbb{P}_{f_{z}(\bm{c})}}[\log(D(G(\bm{z},\bm{c}))].\end{split}$
(2)
In the implementation, our goal is to synthesize color, distance function,
normal vectors, etc. that are defined in a grid-like structure
$X:\mathbb{R}^{H\times W\times C}$ or $X:\mathbb{R}^{H\times W\times D\times
C}$ and thus we also sample grid coordinate input $C:\mathbb{R}^{H\times
W\times 2}$ or $C:\mathbb{R}^{H\times W\times D\times 3}$ whose center is
drawn from a uniform distribution $U(-s,s)$. The randomness to the grid center
position is critical in encouraging smooth and seamless transition between
blocks of patterns as it models the distribution of randomly sampled patches
from the input pattern.
In the following sections, key modules are discussed in detail: 1) a
deformable periodic encoding of the coordinates to model stationary patterns;
2) implementation of the latent random field; 3) a conditional variance of our
objective for the synthesis of directional exemplar and controllability of the
synthesized patterns, and 4) the overall network structure that completes our
pattern synthesis framework.
### 3.1 Periodic Encoding
As discussed in the introduction, it is critical to represent stationary
patterns from the input by a repeated structure that avoids simply
reconstructing the original exemplar. Simply mapping spatial coordinates to
the visual pattern does not satisfy this requirement: since each coordinate is
unique in the real-value space, the network would learn to overfit the
coordinates to their associated positions in the exemplar and therefore fail
to capture a repeatable structure.
The benefits of a periodic encoding to the coordinates are two-fold: Firstly,
it disentangles patch-level appearance from their specific position in the
exemplar, which allows the pattern synthesized by the generator to be shift-
invariant. Secondly, recent advances in implicit network [mildenhall2020nerf,
tancik2020fourier, sitzmann2020implicit] have found a Fourier feature mapping
with a set of sinusoids effective in learning high-frequency signals by
addressing the “spectral bias”[basri2020frequency] inherent to MLPs. In our
work, we use the following periodic mapping for the input coordinates:
$\begin{split}\gamma(\bm{c})=[\cos(2^{0}\pi a\bm{c}),\sin(2^{0}\pi
a\bm{c}),&\\\ \cdot\cdot\cdot,\cos(2^{i}\pi a\bm{c}),\sin(2^{i}\pi
a\bm{c})],\end{split}$ (3)
where the learnable parameter $a\in\mathbb{R}^{k}$ determines the period of
the encoding. This design allows the network to learn to match the period of
the encoding to the repeatable structure of the exemplar. It thus provides
robustness to the scale of the patches sampled in training as such scale no
longer dictates the period of the synthesized pattern.
### 3.2 Latent Random Field
In a generative framework, latent noise sampled from a prior distribution
model the variation in observations. A noise function that is smoothly defined
in the real coordinate space encourages smooth transition between the
synthesized patches. In a 2D example, we start with a discrete random field
that maps a uniform grid of coordinates to random variables
$\\{f_{z}(\bm{c})\>|\>\bm{c}:\mathbb{R}^{H\times W\times 2}\\}$. Then the
discrete random field is smoothly interpolated to form a smooth latent field
(see the visualization in Figure 2). In our implementation, we used an
exponential interpolation:
$f(\bm{x})=\sum_{i=1}^{K}w_{i}(\bm{x})f_{z}(\bm{c}_{i}),\>w_{i}(\bm{x})=\frac{\mathrm{e}^{\frac{||\bm{x}-\bm{c}_{i}||_{2}}{\sigma}}}{\sum_{j=1}^{K}\mathrm{e}^{\frac{||\bm{x}-\bm{c}_{j}||_{2}}{\sigma}}},$
(4)
where the latent code at spatial position $x$ is interpolated from $K=4$
latent vectors defined at the grid corners. In implementation, the discrete
grid used to define the random field has a spacing of 1. To match the extent
of a latent factor with the learned period $a$, we simply scale the uniform
grid of the discrete random field accordingly.
### 3.3 Conditional IPFN
Extending our GAN objective to be conditional enables many practical
applications. Assume each input patch is paired with a guidance factor
$\bm{g}$, the conditional objective is simply an extension:
$\begin{split}\underset{G}{\min}\>\underset{D}{\max}\>&\mathbb{E}_{\bm{x}\sim\mathbb{P}_{data}}[\log(D(\bm{x}|\bm{g}))]-\\\
&\mathbb{E}_{\bm{c}\sim\mathbb{P}_{c},\bm{z}\sim\mathbb{P}_{f_{z}(\bm{c})}}[\log(D(G(\bm{z},\bm{c}|\bm{g}))].\end{split}$
(5)
Here we outline two applications in pattern synthesis using the conditional
formulation:
* •
Synthesis of directional pattern: Many natural patterns have a directional
distribution that is oftentimes considered non-stationary. A typical example
is a leaf texture - a midrib defines the major direction that separates the
blade regions by half (see Figure 3). A conditional extension of our model is
able to model patch distribution along a specified direction. For simplicity,
we present a 2D formulation that can be easily extended to 3D. With a user-
defined implicit 2D line equation $ax+by+c=0$, the guidance factor is defined
as $\bm{g}(x,y)=ax+by+c$. Pixel coordinates $(p_{x},p_{y})$ from an input
texture image with width $w$ and height $h$ are transformed as
$(x,y)=(\frac{2p_{x}}{w}-1,\frac{2p_{y}}{h}-1)$ to be normalized to the value
range of $[-1,1]$. In our experiments we have found it sufficient to condition
our model on the horizontal ($y=0$) and vertical direction ($x=0$) for the
evaluated exemplars.
* •
Controlling synthesis of 3D shapes: In the modeling of geometric patterns, it
is often desirable for the synthesis algorithm to provide explicit control of
certain geometric properties such as density and orientation. These geometric
properties can be calculated from the exemplar shape. Let $g(x)$ be a shape
operator that defines the geometric property of interest, our conditional
model trained with the sample pair $(x,g(x))$ then learns a probabilistic
mapping from a guidance vector field to the target 3D shape. An intuitive
example of this application can be found in Section 4.5.
### 3.4 Network Structure
Figure 2: Overview of our network architecture discussed in Section 3.4.
The overall structure of IPFN is visualized in Figure 2. The Generator Network
$G$ is a 10-layer MLP with ReLU activations between layers and a sigmoid
function in the end. A grid of coordinates is sampled based on a randomly
shifted center. The coordinates are then passed to two separate branches: 1)
the periodic encoder, and 2) a projection on the latent field to obtain a
5-dim latent vector for each coordinate. The latent codes and the periodically
encoded coordinates are then concatenated as input to the generator mlp, which
outputs the fake sample. The discriminator $D$ discriminates between the
generated samples and randomly cropped patches from the real input. We
implement $D$ by following the DCGAN architecture [radford2015unsupervised]
with a stride of 2. For discriminating 3D volumes, 2D convolution layers are
replaced by 3D convolution.
Figure 3: Main results for 2D texture synthesis with comparisons to Henzler et
al. [henzler2020learning], Bergmann et al. [bergmann2017learning], and Zhou et
al. [zhou2018non] on synthesizing two stationary patterns (top four rows) and
two directional patterns (bottom four rows).
## 4 Experiments
We hypothesize that our approach is most suitable for synthesizing texture
patterns and 3D shapes with repeating structures and local variations in
appearance. To this end, we demonstrate our main results by applying IPFN to
the synthesis of 2D texture images and 3D structured shapes. In addition, IPFN
is adapted to two applications in 3D texturing and shape manipulation. To
evaluate the effectiveness of the proposed techniques, we have also conducted
an ablation study where several key designs are altered.
Evaluation metric While the quality for pattern synthesis is not easily
quantifiable, human eyes usually provide a reasonable qualitative assessment
for whether the synthesized patterns capture the aesthetics and structure of
the exemplar. In our evaluation, we present comparisons of visual results that
are self-manifesting, since the synthesized patterns bear obvious
characteristics of the underlying designs of the synthesizer. In addition, we
have provided quantifiable metrics in terms of Single Image Frechét Inception
Distance and inference time and memory.
Implementation details For all of our experiments, the network is optimized
under WGAN loss with gradient penalty [gulrajani2017improved]. Adam optimizer
[kingma2014adam] is used with a learning rate of $1e-4$ for both the
discriminator D and the generator G. In each iteration, both D and G are
updated for 5 steps sequentially. Input images and volumes are randomly
cropped to a smaller-scale patch. For positional encoding, we choose a
bandwidth $i=5$ as a wider kernel tends to produce sinusoidal artifacts,
whereas a narrower kernel produces blurry results. The input coordinates are
randomly shifted by an offset in the range $[-4,4]$ to accommodate for the
chance that the network may learn an increased period for the periodic
encoding. Accordingly, noises are interpolated from a $5\times 5$ grid
($5^{3}$ for 3D volume) discrete random field, where the point locations are
ranged between $[-5,5]$. A single-exemplar experiment is typically trained for
12,500 iterations with a batch size of 8 and runs on a single Nvidia GTX 1080
GPU, which takes about 6-9 hours to complete. Inference Time: IPFN only
requires 24 milliseconds to generate a $1024\times 1024$ image. 3D volumes are
generated iteratively and a large-scale $512^{3}$ volume takes only 22.9
seconds to be generated. Source code will be made publicly available upon
acceptance.
### 4.1 Texture Pattern Synthesis
Image sources selected from the Oxford Describable Textures Dataset (DTD)
[cimpoi2014describing] are shown in Figure 3. Specifically, we selected two
exemplars with stationary patterns (top 4 rows in Figure 3) and two exemplars
with directional patterns (bottom 4 rows in Figure 3) to demonstrate that IPFN
synthesizes visually similar patterns in both cases. During training, images
were randomly cropped into patches of size $128\times 128$. During inference,
the synthesized images were scaled up four times to a size of $512\times 512$.
Our results are compared to the three most relevant baseline generative
methods that synthesize texture patterns from a single image:
* •
Henzler et al. [henzler2020learning]: A method that similarly utilizes
implicit network and smooth noise field for texture synthesis. The synthesized
results were obtained from running the officially released code.
* •
Bergmann et al. [bergmann2017learning]: A convolutional method that combines
noise with periodic signals to synthesize stationary pattern. The synthesized
results were obtained from running the officially released code.
* •
Zhou et al. [zhou2018non]: A convolutional image expansion approach targeted
for non-stationary texture synthesis. Since [zhou2018non] expands from an
image input deterministically and does not utilize latent code, only one
synthesized result is shown per row. The synthesized results were obtained
from the authors.
Visual inspection is sufficient to show that IPFN provides promising results.
When compared to [henzler2020learning], IPFN synthesized results with obvious
structures as noise is not directly mapped to the output. While
[bergmann2017learning] synthesizes periodic samples that display diversity
across samples and similarity to the stationary exemplars, their synthesized
patterns lack variation within the image. In comparison, our synthesized
patterns show a higher level of local variations and adapt well to the
directional cases. [zhou2018non] has provided the most visually authentic
results among the baselines. However, in the stationary cases, radial
distortion is noticeable near the boundaries of its synthesized images.
Moreover, without requiring image input, IPFN provides a more direct approach
to synthesizing diversified samples from random noise.
| honey | crosshatch | rock | leaf
---|---|---|---|---
Henzler [henzler2020learning] | 332.66 | 310.49 | 351.23 | 225.11
Bergmann [bergmann2017learning] | 62.75 | 177.88 | 120.64 | 164.37
Zhou [zhou2018non] | 14.54 | 154.63 | 118.29 | 38.13
Ours | 10.15 | 130.83 | 113.81 | 103.6
Table 1: SIFID scores between the exemplars and the generated patterns from
ours and different baselines.
Single Image Frechét Inception Distance (SIFID) SIFID introduced in
[shaham2019singan] is a metric commonly used to assess the realism of
generated images. For the computation of SIFID, we have used a patch size of
$128\times 128$ in all experiments, where the synthesized patterns have the
same resolution as the original exemplars. Table 1 shows the SIFID comparisons
between ours and the baselines in various categories of exemplars. For Zhou et
al. [zhou2018non], only the generated (expanded) portion of the images were
used. The results show that our method can generate results that better
resemble the distribution of the real texture in the stationary categories
(honey, crosshatch, rock) as our generated patterns receive lower SIFID
scores. For the leaf category, a typical directional pattern, Zhou et al.
[zhou2018non] achieves the best performance as its method specifically targets
non-stationary expansions, while our method still performs better than other
baselines that have not taken into consideration the synthesis of non-
stationary patterns.
Time (ms) / memory (GB) | $128^{2}$ | $256^{2}$ | $512^{2}$ | $1024^{2}$
---|---|---|---|---
Henzler [henzler2020learning] | 218/1.38 | 278/1.62 | 328/2.72 | 458/6.45
Bergmann [bergmann2017learning] | 7/2.37 | 13/5.79 | 42/19.68 | 115/31.88
Zhou [zhou2018non] | 356/1.20 | 349/1.34 | 510/2.00 | 612/4.66
Ours | 8/0.76 | 11/0.85 | 15/1.23 | 24/2.81
Table 2: Comparisons of inference time and inference memory consumption,
measured in milliseconds (ms) / gigabytes (GB), when patterns of increasing
size (top row) are generated.
Inference Time and Memory Table 2 measures the inference time and memory
consumption of our network compared to the baselines when generating image at
different sizes. Our implicit formulation is shown to be significantly more
efficient in both time and space without the needs to rely on computation of
pseudo-random noise ([henzler2020learning]) or convolutional operations
([bergmann2017learning, zhou2018non]). This validates our claim on the
scalability of our design.
### 4.2 Ablation Study
Figure 4: Synthesized honeycomb textures for the ablation study. The blue
boxes represent the learned scale of periodic encoding in ours, where in w/o
deformation, the period is default to 1, which does not match with the
repeating structure of the honeycomb pattern and results in visual artifact.
To validate our design choices, we have conducted an ablation study by
removing two designs that we consider critical for our network to be
effective. The comparison results are shown in Figure 4. w/o deformation is a
network model that encodes input coordinates without the learnable parameters
$a$ described in Section 3.1. w/o shift is a model that is trained without
randomly shifting the input coordinates. The resulted patterns are indicative
of the effects of these designs: when coordinates are encoded at a fixed
scale, the w/o deformation model generates hexagons that are seemingly glued
together as the presumed scale does not match with the actual period of the
repeating structure. The w/o shift model synthesized distorted patterns as we
speculate that, without the random sampling of the input coordinates, the
network faces difficulty in matching the patch-based priors of the image
patches.
### 4.3 Volumetric Shape Synthesis
Figure 5: Main results for 3D volume synthesis. a. Exemplar porous structure.
b. Synthesized structure models interior tunnels. c. Global views of
synthesized porous structures. c. Exemplar foam structure. e. Two scales of
noise fields for the foam structure synthesis. f. Synthesized foam structures.
Larger scale of the noise field leads to more isotropic foam structures.
Figure 6: IPFN learns multi-channel textures that are applicable to seamless
3D texturing. The original 3D texture in this example is not symmetric and
therefore visible seams can be found on the texture-mapped surface and in the
closeup view (A in figure). As synthesized patterns learnt from this exemplar
can be tiled in any direction, the mapped surface (B in surface) is seamless.
For the evaluation on volumetric shape synthesis, We have obtained two 3D
models from turbosquid.com: a porous structure (Figure 5.a) and a foam
structure ((Figure 5.d). The 3D meshes are preprocessed into signed distance
fields by uniformly sampling points in a volumetric grid. For the porous
structure, we have sampled $256\times 256\times 256$ points and extracted
$64\times 64\times 64$ patches during training. For the foam structure, we
have sampled $200\times 200\times 128$ points and extracted $32\times 32\times
32$ patches during training. During inference, porous structures are
synthesized at their original resolution, while we scale the synthesized foam
structures to be twice as large as the original shape in the XY direction.
Figure 5.c and Figure 5.f show the synthesized shapes. For the porous
structures, both outer structures and interior structures are learned (see
Figure 5.b for zoom-in interior views) and the structures are diversified both
across and within samples. For the foam structures, we have shown different
results by varying the extent of the latent random field (see Figure 5.e). A
larger-scale random field encourages the synthesizer to generate globally
varied structures, whereas a smaller scale produces locally anisotropic
structures.
### 4.4 Application: Seamless 3D Texturing
Due to the periodic nature of the synthesized patterns, noise manipulation
allows IPFN to create textures that are mirror-symmetric. This property
provides an immediate application to seamless 3D texture mapping: in Figure 6,
the original 9-channel texture, composed of color, normal, and bump maps, is
tiled and directly mapped to a planar surface. Due to discrepancies on the
edges, as the textures are wrapped to create the tiled patterns, the mapped
surfaces show visible seams. We recreate this texture through our network
under a constant latent vector (Figure 6 B). When repeatedly mapped to the
surface, the symmetric texture is seamless while faithfully reflecting the
appearance and structure of the original texture.
Figure 7: Synthesized foam structures with controllable density. a. The grey
scale bar controls the synthesized structure from the highest density (white)
to the lowest (black). b. Smooth interpolation of the guidance factor allows
us to synthesize a foam structure with smoothly changing densities.
### 4.5 Application: 3D Foam with controllable density
The original foam shape used in our experiment contains holes of various
sizes, which corresponds to the density of the foam structure. This geometric
property $g$ can be easily approximated with an unsigned distance field
representation. For a patch $X^{\prime}$ of size $H^{\prime}\times
W^{\prime}\times D^{\prime}$, we estimate its density by:
$g(X^{\prime})=\frac{1}{m}\sum_{i=1}^{H^{\prime}}\sum_{j=1}^{W^{\prime}}\sum_{k=1}^{D^{\prime}}|sdf(X_{ijk})|$
(6)
, where m is a normalization factor. Figure 7 shows the synthesized foam
structures by gradually increasing the density factor (Figure 7.a) and by a
linearly interpolated density map ((Figure 7.b).
## 5 Limitations and Discussions
Figure 8: The examples demonstrating limitations of our network
The main limitation of our method is its emphasis on modeling stationary
patterns. While this is based on our observation that a broad range of natural
patterns is stationary or directional, our method does not provide a natural
way to address a class of patterns that are radial, which are exemplified by
web structures and spiral patterns (see the third column in Figure 8).
While our conditional formulation is in theory compatible with the synthesis
of landscape images, experiments found that the quality of the synthesized
landscapes are subpar - while the synthesized landscape appears globally
similar to the exemplar, some local regions contain ”fading-out” elements that
are blended with the background (see the first two columns in Figure 8). We
speculate that this phenomenon is due to an under-representation of these
elements in the exemplar.
The above limitations have inspired us to consider many potential improvements
of our methods in future works. A multi-scale synthesis approach marked in
[shaham2019singan, shocher2019ingan] strikes a good balance between learning
the distribution of global structure and local, high-frequency details of an
image. Different geometric encoding schemes may also extend our framework to
synthesize beyond stationary patterns. We believe there are still ample
opportunities for the extension of our methods to a broader range of 3D
applications.
## Acknowledgements
This research was sponsored by the Army Research Office and was accomplished
under Cooperative Agreement Number W911NF-20-2-0053, and by the U.S. Army
Research Laboratory (ARL) under contract number W911NF-14-D-0005. The views
and conclusions contained in this document are those of the authors and should
not be interpreted as representing the official policies, either expressed or
implied, of the Army Research Office or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation.
|
# Is First Person Vision Challenging for Object Tracking?
Matteo Dunnhofer∙
Antonino Furnari⋆ Giovanni Maria Farinella⋆ Christian Micheloni∙ ∙Machine
Learning and Perception Lab, University of Udine, Udine, Italy
⋆Image Processing Laboratory, University of Catania, Catania, Italy
###### Abstract
Understanding human-object interactions is fundamental in First Person Vision
(FPV). Tracking algorithms which follow the objects manipulated by the camera
wearer can provide useful cues to effectively model such interactions. Visual
tracking solutions available in the computer vision literature have
significantly improved their performance in the last years for a large variety
of target objects and tracking scenarios. However, despite a few previous
attempts to exploit trackers in FPV applications, a methodical analysis of the
performance of state-of-the-art trackers in this domain is still missing. In
this paper, we fill the gap by presenting the first systematic study of object
tracking in FPV. Our study extensively analyses the performance of recent
visual trackers and baseline FPV trackers with respect to different aspects
and considering a new performance measure. This is achieved through TREK-150,
a novel benchmark dataset composed of 150 densely annotated video sequences.
Our results show that object tracking in FPV is challenging, which suggests
that more research efforts should be devoted to this problem so that tracking
could benefit FPV tasks.
## 1 Introduction
Understanding the interactions between a camera wearer and the surrounding
objects is a fundamental problem in First Person Vision (FPV) [19, 87, 57, 33,
20, 73, 12, 4, 5, 13]. To model such interactions, the continuous knowledge of
where an object of interest is located inside the video frame is advantageous.
The benefits of tracking in FPV have been explored by a few previous works to
predict future active objects [32], analyze social interactions [2], improve
the performance of hand detection for rehabilitation purposes [83], locate
hands and capture their movements for action recognition [44] and human-object
interaction forecasting [57]. On a more abstract level, the features computed
after the frame by frame localization of objects have been increasingly used
for egocentric action recognition [87, 89, 62] and anticipation [33, 76, 78].
Figure 1: Qualitative examples of some sequences contained in the proposed
TREK-150 benchmark dataset. The white rectangle represents the ground-truth
bounding box of the target object. Each number in the top left corner
identifies the frame index. For each sequence, the action performed by the
camera wearer is also reported (verb in orange, noun in blue). As can be
noted, objects undergo significant appearance and state changes due to the
manipulation by the camera wearer, which makes the proposed setting
challenging for current trackers.
Despite the aforementioned attempts to leverage tracking in egocentric vision
pipelines, most approaches rely on object detection models that evaluate video
frames independently. This paradigm has the drawback of ignoring all the
temporal information coming from the object appearance and motion contained in
consecutive video frames and generally requires a higher computational cost
due to the repeated detection process on every frame. In contrast, visual
object tracking aims to exploit past information about the target to infer its
position and shape in the next frames of a video [63]. This process is subject
to different challenges including occlusions, appearance changes, illumination
variation, fast motion, and motion blur. Additionally, many practical
applications pose real-time constraints to the computation, which specifically
hold in FPV when the localization of objects is needed by higher-level real-
time algorithms. While the use cases of object tracking in egocentric vision
are manifold as previously discussed, it is clear that tracking is still not a
dominant technology in the FPV field. We experimentally show that this is
mainly due to the limited performance of current trackers in egocentric videos
due to the involved FPV challenges such as camera motion, persistent
occlusion, significant scale and state changes, as well as motion blur (see
Figure 1). Due to these challenges, previous works have proposed customized
approaches to track specific targets like people [3], people faces [1], or
hands [44, 83, 65, 37, 81] from the FPV perspective. A solution specifically
designed to track arbitrary objects in egocentric videos is still missing.
Instead, the computer vision community has made significant progress in the
visual tracking of generic objects. This has been possible thanks to
development of new and effective tracking principles [11, 40, 7, 21, 8, 16,
98, 18], and to the careful design of benchmark datasets [91, 66, 35, 52, 31,
41] and challenges [49, 48, 50, 47]. Nowadays, the state-of-the-art tracking
solutions achieve excellent results on a large variety of tracking domains
[91, 66, 35, 41, 50, 47]. However, all these research endeavours have taken
into account mainly the classic third person scenario in which objects are
observed from an external point of view and are not manipulated by the camera
wearer. Additionally, the performance of existing trackers has never been
evaluated in the FPV domain, which raises the question of whether current
solutions can be used “off-the-shelf” or more domain-specific investigations
should be carried out.
To answer the aforementioned questions, in this paper we aim to extensively
analyze the problem of visual object tracking in the FPV domain. Given the
lack of suitable benchmarks, we follow the standard practice of the visual
tracking community that suggests to build an accurate dataset for evaluation
[91, 56, 66, 52, 35, 50, 59]. Therefore, we propose a novel visual tracking
benchmark, TREK-150 (TRacking-Epic-Kitchens-150), which is obtained from the
large and challenging FPV dataset EPIC-KITCHENS-55 (EK-55) [19]. TREK-150
provides 150 video sequences densely annotated with the bounding boxes of a
target object the camera wearer interacts with. Additionally, sequences have
been labelled with attributes that identify the visual changes the object is
undergoing, the class of the target object and the action the person is
performing. Using the dataset, we present an in-depth study of the accuracy
and speed performance of both non-FPV and FPV visual trackers. A new
performance measure is also introduced to evaluate trackers with respect to
FPV scenarios.
In sum, the contributions of this paper are: (i) the first systematic analysis
of visual object tracking in FPV; (ii) the description and release of the new
TREK-150 dataset, which offers new challenges and complementary features with
respect to existing visual tracking benchmarks; (iii) two FPV baseline
trackers combining a state-of-the-art generic object tracker and FPV object
detectors; (iv) a new and improved measure to assess the tracker’s ability to
maintain temporal reference to targets.
Our results show that FPV offers challenging tracking scenarios for the most
recent and accurate trackers [18, 22, 80, 21, 9] and even for FPV trackers.
Considering the potential impact of tracking on FPV, we suggest that more
research efforts should be devoted to the considered task, for which we
believe the proposed TREK-150 benchmark will be a key research tool.
Annotations, trackers’ results, and code are available at
https://machinelearning.uniud.it/datasets/trek150/.
Table 1: Statistics of the proposed TREK-150 benchmark compared with other benchmarks designed for SOT evaluation. Benchmark | OTB-50 | OTB-100 | TC-128 | UAV123 | NUS-PRO | NfS | VOT2019 | CDTB | TREK-150
---|---|---|---|---|---|---|---|---|---
[90] | [91] | [56] | [66] | [52] | [35] | [50] | [59]
# videos | 51 | 100 | 128 | 123 | 365 | 100 | 60 | 80 | 150
# frames | 29K | 59K | 55K | 113K | 135K | 383K | 20K | 102K | 97K
Min frames across videos | 71 | 71 | 71 | 109 | 146 | 169 | 41 | 406 | 161
Mean frames across videos | 578 | 590 | 429 | 915 | 371 | 3830 | 332 | 1274 | 649
Median frames across videos | 392 | 393 | 365 | 882 | 300 | 2448 | 258 | 1179 | 484
Max frames across videos | 3872 | 3872 | 3872 | 3085 | 5040 | 20665 | 1500 | 2501 | 4640
Frame rate | 30 FPS | 30 FPS | 30 FPS | 30 FPS | 30 FPS | 240 FPS | 30 FPS | 30 FPS | 60 FPS
# target object classes | 10 | 16 | 27 | 9 | 8 | 17 | 30 | 23 | 34
# sequence attributes | 11 | 11 | 11 | 12 | 12 | 9 | 6 | 13 | 17
FPV | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓
# action verbs | n/a | n/a | n/a | n/a | n/a | n/a | n/a | n/a | 20
## 2 Related Work
### Visual Tracking in FPV.
There have been some attempts to tackle visual tracking in FPV. Alletto et al.
[3] improved the TLD tracker [43] with a 3D odometry based module to track
people. For a similar task, Nigam et al. [70] proposed a combination of the
Struck [38] and MEEM [95] trackers with a person re-identification module.
Face tracking was tackled by Aghaei et al. [1] through a multi-object tracking
approach termed extended-bag-of-tracklets. Hand tracking was studied in
several works [44, 83, 65, 37, 81]. Sun et al. [81] developed a particle
filter framework for hand pose tracking. Müller et al. [65] proposed a
solution based on an RGB camera and a depth sensor. Kapidis et al. [44] and
Visée et al. [83] proposed to combine the YOLO [74] detector trained for hand
detection with trackers. The former used the multi-object tracker DeepSORT
[88], whereas the latter employed the KCF [40] single object tracker. Han et
al. [37] exploited a detection-by-tracking approach on video frames acquired
with 4 fisheye cameras. All the presented solutions focused on tracking
specific targets (i.e., people, faces, or hands), and thus they are likely to
fail in generalizing to arbitrary target objects. Moreover, they have been
validated on custom designed datasets, which limits their reproducibility and
the ability to compare them to other works. In contrast, we focus on the
evaluation of algorithms for the generic object tracking task. We design our
evaluation to be reproducible and extendable by releasing TREK-150, a dataset
of 150 videos of different objects manipulated by the camera wearer, which we
believe will be useful to study object tracking in FPV. To the best of our
knowledge, ours is the first attempt to evaluate systematically generic object
tracking in the FPV context.
### Visual Tracking for Generic Settings.
In recent years, there has been an increased interest in developing accurate
and robust single object tracking (SOT) algorithms for generic targets and
domains. Preliminary trackers were based on mean shift algorithms [17], key-
point [64], part-based methods [14, 69], or SVM learning [38]. Later,
solutions based on correlation filters gained popularity thanks to their
processing speed [11, 40, 23, 6, 46]. More recently, algorithms based on deep
learning have been proposed to extract efficient image and object features.
This kind of representation has been used in deep regression networks [39,
30], online tracking-by-detection methods [68, 80], approaches based on
reinforcement learning [94, 15, 27, 36], deep discriminative correlation
filters [21, 22, 8, 24, 60, 9], and trackers based on siamese networks [7, 53,
86, 16, 98]. All these methods have been designed for tracking arbitrary
target objects in unconstrained domains. However, no solution has been studied
and validated on a number of diverse FPV sequences as we propose in this
paper.
Figure 2: (a) Distribution of the sequences within TREK-150 with respect to
the attributes. (b) Comparison of the distributions of common attributes in
different benchmarks. Distributions of (c) action verb labels, and (d) target
object categories (nouns).
### Visual Tracking Benchmarks.
Disparate bounding box level benchmarks are available today to evaluate the
performance of SOT algorithms. The Object Tracking Benchmarks (OTB) OTB-50
[90] and OTB-100 [91] are two of the most popular benchmarks in the visual
tracking community. They provide 51 and 100 sequences respectively including
generic targets like vehicles, people, faces, toys, characters, etc. The
Temple-Color 128 (TC-128) dataset [56] comprises 128 videos and was designed
for the evaluation of color-enhanced trackers. The UAV123 dataset [66] was
constructed to benchmark the tracking of 9 classes of target in 123 videos
captured by unmanned aerial vehicle (UAV) cameras. The NUS-PRO dataset [52]
contains 365 sequences and aims to benchmark human and rigid object tracking
with targets belonging to one of 8 categories. The Need for Speed (NfS)
dataset [35] provides 100 sequences with a frame rate of 240 FPS. The aim of
the authors was to benchmark the effects of frame rate variations on the
tracking performance. The VOT2019 benchmark [50] was the last iteration of the
annual Visual Object Tracking challenge that required bounding-boxes as target
object representation. This dataset contains 60 highly challenging videos,
with generic target objects belonging to 30 different categories. The Color
and Depth Tracking Benchmark (CDTB) dataset [59] offers 80 RGB sequences
paired with a depth channel. This benchmark aims to explore the use of depth
information to improve tracking performance. Following the increased
development of deep learning based trackers, large-scale generic-domain SOT
datasets have been recently released [67, 41, 31]. These include more than a
thousand videos normally split into training and test subsets. The evaluation
protocol associated with these sets requires the evaluation of the trackers
after they have been trained on the provided training set. Despite the fact
that all the presented benchmarks offer various tracking scenarios, limited
work has focused on FPV, with some studies tackling the problem of tracking
pedestrians or cars from a moving camera [77]. Some datasets of egocentric
videos such as ADL [72] and EK-55 [19] contain bounding-box object
annotations. But due to the sparse nature of such annotations (typically 1/2
FPS), these datasets cannot be used for the accurate evaluation of trackers in
FPV context. To the best of our knowledge, our proposed TREK-150 dataset is
the first benchmark for tracking objects which are relevant to (or manipulated
by) a camera wearer in egocentric videos. We believe that TREK-150 is
tantalizing for the tracking community because it offers complementary
tracking situations (which we characterize with a total of $17$ attributes)
and new target object categories (for a total of $34$) that are not present in
other tracking benchmarks. Since in this paper we aim to benchmark generic
approaches to visual tracking (that would not necessarily consider the deep
learning approach), we follow the practice of previous works [91, 56, 66, 52,
35, 50, 59] and set up a well described dataset for evaluation of generic SOT
algorithms. We believe that TREK-150 can be a useful research tool for both
the FPV and visual tracking research communities.
## 3 The TREK-150 Benchmark Dataset
The proposed TREK-150 dataset is composed of 150 video sequences. In each, a
single target object is labeled with a bounding box which encloses the visible
parts of the object. The bounding boxes are given for each frame in which the
object is visible (as a whole or in part). To be compliant with other tracking
challenges, every sequence is additionally labeled with one or more of 17
attributes describing the visual variability of the target in the sequence,
plus two additional action verb and noun attributes indicating the action
performed by the camera wearer and the class of the target. Qualitative
examples of the video sequences are shown in Figure 1, whereas Table 1 reports
key statistics of our dataset in comparison with existing benchmarks.111Please
see Appendix A of the supplementary material for additional motivations and
details.
Table 2: Selected sequence attributes. The first block of rows describes attributes commonly used by the visual tracking community. The last four rows describe additional attributes introduced in this paper to characterize FPV tracking sequences. Attribute | Meaning
---|---
SC | Scale Change: the ratio of the bounding-box area of the first and the current frame is outside the range [0.5, 2]
ARC | Aspect Ratio Change: the ratio of the bounding-box aspect ratio of the first and the current frame is outside the range [0.5, 2]
IV | Illumination Variation: the area of the target bounding-box is subject to light variation
SOB | Similar Objects: there are objects in the video of the same object category or with similar appearance to the target
RIG | Rigid Object: the target is a rigid object
DEF | Deformable Object: the target is a deformable object
ROT | Rotation: the target rotates in the video
POC | Partial Occlusion: the target is partially occluded in the video
FOC | Full Occlusion: the target is fully occluded in the video
OUT | Out Of View: the target completely leaves the video frame
MB | Motion Blur: the target region is blurred due to target or camera motion
FM | Fast Motion: the target bounding-box has a motion change larger than its size
LR | Low Resolution: the area of the target bounding-box is less than 1000 pixels in at least one frame
HR | High Resolution: the area of the target bounding-box is larger than 250000 pixels in at least one frame
HM | Head Motion: the person moves their head significantly thus causing camera motion
1H | 1 Hand Interaction: the person interacts with the target object with one hand for consecutive video frames
2H | 2 Hands Interaction: the person interacts with the target object with both hands for consecutive video frames
### Data Collection.
The videos have been sampled from EK-55 [19], which is a public, large-scale,
and diverse dataset of egocentric videos focused on human-object interactions
in kitchens. EK-55 provides videos annotated with the actions performed by the
camera wearer in the form of temporal bounds and verb-noun labels. The dataset
also contains sparse bounding-box references of manipulated objects annotated
at 2 frames per second in a temporal window around each action. To obtain a
suitable pool of video sequences interesting for object tracking, we cross-
referenced the original verb-noun temporal annotations of EK-55 to the sparse
bounding box labels. This allowed to select sequences in which the camera
wearer manipulates an object. Each sequence is composed of the video frames
contained within the temporal bounds of the action, extracted at the original
60 FPS frame rate and at the original full HD frame size [19]. According to
the authors of [19], this frame rate is necessary in FPV to contrast the fast
motion and motion blur happening due to the proximity of the main scene and
the camera point of view. From the initial pool, we selected 150 video
sequences which were characterized by attributes such as scale changes,
partial/full occlusion and fast motion, which are commonly considered in
standard tracking benchmarks [91, 66, 67, 31, 50]. The top part of Table 2
reports the $13$ attributes considered for the selection.
### Data Labeling.
After selection, the 150 sequences were associated to only 3000 bounding
boxes, due to the sparse nature of the object annotations in EK-55. Since it
has been shown that visual tracking benchmarks require dense and accurate
annotations [50, 66, 31, 82], we re-annotated the bounding boxes of the target
objects on the 150 sequences. Batches of sequences were delivered to
annotators who were explicitly instructed to perform the labeling. Such
initial annotations were then carefully checked and refined by a visual
tracking expert. This process produced 97296 frames labeled with bounding
boxes related to the position and visual presence of objects the camera wearer
is interacting with. Following the initial annotations, we employed axis-
aligned bounding boxes. This kind of representation is widely used in many FPV
pipelines [32, 34, 33, 19, 45, 83, 79], and thus it allows us to give
immediate results on the impact of trackers in such contexts. Moreover, the
recent progress of trackers on various benchmarks that use this state
representation [91, 66, 35, 59, 67, 31, 41] demonstrates that it provides
sufficient information about the target for consistent and reliable
performance evaluation.
Along with the bounding boxes, the sequences have been labeled considering 17
attributes which define the motion and visual appearance changes the target
object is subject. These include the aforementioned 13 standard tracking
attributes, plus 4 additional ones (High Resolution, Head Motion, 1-Hand
Interaction, 2-Hands Interaction) which have been introduced to characterize
FPV sequences and are summarized in the bottom part of Table 2. Figure 2(a)
reports the distributions of the sequences with respect to the 17 attributes.
Figure 2(b) compares the distributions of the most common SOT attributes in
TREK-150 and in other well-known benchmarks. Our dataset provides a larger
number of sequences affected by partial occlusions (POC), changes in scale
(SC) and/or aspect ratio (ARC), and motion blur (MB). We claim that these
peculiarities, which are complementary to those of existing datasets, are due
to the particular first person viewpoint, camera motion, and the human-object
interactions contained in the videos. Based on EK-55’s verb-noun labels,
sequences were also associated to 20 verb labels (e.g., “wash” - see Figure 1)
and 34 noun labels indicating the category of the target object (e.g., “box”).
Figures 2(c-d) show the distributions of the videos relative to verbs and
target nouns. As can be noted, TREK-150 reflects the EK-55’s long-tail
distribution of labels.
## 4 Trackers
We considered 33 trackers in our benchmark evaluation. 31 of these trackers
have been selected to represent different popular approaches to SOT, for
instance with respect to the matching strategy, type of image representations,
learning strategy, etc. Specifically, in the analysis we have included short-
term trackers [50] based on both correlation-filters with hand-crafted
features (MOSSE [11], DSST [23], KCF [40], Staple [6], BACF [46], DCFNet [85],
STRCF [54], MCCTH [84]) and deep features (ECO [21], ATOM [22], DiMP [8],
PrDiMP [24], KYS [9]). We also considered deep siamese networks (SiamFC [7],
GOTURN [39], DSLT [58], SiamRPN++ [53], SiamDW [97], UpdateNet [96], SiamFC++
[92], SiamBAN [16], Ocean [98]), tracking-by-detection methods (MDNet [68],
VITAL [80]), as well as trackers based on target segmentation representations
(SiamMask [86], D3S [60]), meta-learning (MetaCrest [71]), and fusion
strategies (TRASFUST [29]). The long-term [50] trackers SPLT [93], GlobalTrack
[42], and LTMU [18] have been also taken into account in the study. These
trackers are designed to address longer target occlusion and out of view
periods by exploiting object re-detection modules. All of the selected
trackers are state-of-the-art approaches published between the years
2010-2020.
In addition to the aforementioned generic object trackers, we developed 2
baseline FPV trackers that combine the LTMU tracker [18] with (i) the EK-55
trained Faster-R-CNN [19] and (ii) the Faster-R-CNN-based hand-object detector
[79]. We refer to them as LTMU-F and LTMU-H respectively. These baseline
trackers exploit the respective detectors as object re-detection modules
according to the LTMU scheme [18]. In short, the re-detection happens when a
verification module notices that the tracker is not following the correct
target. In such a case, the module triggers the execution of the respective
FPV detector which proposes candidate locations of the target object. Each of
the candidates is evaluated by the verification module, and the location with
highest confidence is used to re-initialize the tracker.222More details are
given in Appendix B of the supplementary material. The two modules implement
conceptually different strategies for FPV-based object localization. The first
aims to find objects in the scene, while the second looks for the interaction
between the camera wearer and objects.
## 5 Evaluation
### Evaluation Protocols.
We employed three standard protocols to perform our analysis.333See Appendix C
of the supplementary material for further details. The first is the one-pass
evaluation (OPE) protocol detailed in [91], which implements the most
realistic way to execute trackers. It consists in initializing a tracker with
the ground-truth bounding box of the target in the first frame and let the
tracker run on every subsequent frame until the end of the sequence.
To obtain a more robust evaluation [51], especially for the analysis over
sequence attributes and action verbs, we employ the recent protocol of [47]
which defines different points of initialization along a sequence. A tracker
is initialized with the ground-truth in each point and let run either forward
or backward in time (depending on the longest sub-sequence yielded by the
initialization point) until the end of the sub-sequence. This protocol allows
a tracker to better cover all the situations happening in the sequences,
ultimately leading to more robust evaluation scores. We refer to this setup as
multi-start evaluation (MSE).
Since many FPV tasks such as object interaction [20] and early action
recognition [34], or action anticipation [19], require real-time computation,
we evaluated the ability of trackers to provide their object localization in
such a setting. This was achieved by following the details given in [48, 55].
In short, this protocol, which we refer to as RTE, runs an algorithm
considering its running time. The protocol skips all the frames, considered to
occur regularly according to the frame rate, which appeared during the
interval between the algorithm’s execution start and end times.
Figure 3: Performance of the selected trackers on the proposed TREK-150
benchmark under the OPE protocol. The curves in solid colors report the
performance of the 33 benchmarked trackers on TREK-150, whereas the curves
overlaid in semi-transparent colors outline the performance obtained by the
same trackers on the standard OTB-100 [91] dataset. In brackets, next to the
trackers’ names, we report the SS, NPS and GR values achieved on TREK-150 (in
black) and on OTB-100 [91] (in gray). As can be noted, all the trackers
exhibit a significant performance drop when tested on our challenging FPV
benchmark. LTMU-H and LTMU-F achieve marginally better performance, while we
expect significant boosts to be achievable with a careful design of FPV
trackers. Figure 4: SS, NPS, and GSR of 17 of the benchmarked trackers on the
sequence attributes of proposed TREK-150 benchmark under the MSE protocol. The
red plain line highlights the average performance. (The results for POC are
not reported because this attribute is present in every sequence).
### Performance Measures.
To quantitatively assess the performance of the trackers on the proposed
dataset, we used different measures that compare all tracker’s predicted
bounding boxes with respect to the temporally aligned ground-truth bounding
boxes. To evaluate the localization accuracy of the trackers, we employ the
success plot [91], which shows the percentage of predicted bounding boxes
whose intersection-over-union with the ground-truth is larger than a threshold
varied from 0 to 1 (Figure 3 (a)). We also use the normalized precision plot
[67], that reports, for a variety of distance thresholds, the percentage of
bounding boxes whose center points are within a given normalized distance (in
pixels) from the ground-truth (Figure 3 (b)). As summary measures, we report
the success score (SS) [91] and normalized precision scores (NPS) [67], which
are computed as the Area Under the Curve (AUC) of the success plot and
normalized precision plot respectively.
Along with these standard metrics, we employ a novel plot which we refer to as
generalized success robustness plot (Figure 3 (c)). We take inspiration from
the robustness metric proposed in [47] which measures the normalized extent of
a tracking sequence before a failure. But differently from [47], which uses a
fixed overlap threshold to detect a collapse, we propose to use different
thresholds ranging in [0, 0.5]. This allows to assess the length of tracking
sequences for different application scenarios. We consider 0.5 as the maximum
threshold as higher overlaps are usually associated to positive predictions in
many computer vision tasks. Similarly as [91, 67], we use the AUC of the
generalized robustness plot to obtain an aggregate score which we refer to as
generalized success robustness (GSR). This new measure evaluates trackers’
capability of maintaining long temporal reference to targets. We think this
aspect is especially important in FPV as longer references to the target can
lead to a better modeling of the camera viewer’s actions and interactions with
objects.
Finally, we evaluate the trackers’ processing speed in frames per second (FPS)
to quantify their efficiency.
Figure 5: SS, NPS, and GSR performance of 17 among the 33 selected trackers
with respect to the action verbs (first row of plots) and target nouns (second
row of plots) in TREK-150. The red plain line highlights the average
performance.
## 6 Results
Table 3: Performance achieved by 17 of the benchmarked trackers on TREK-150 using the RTE protocol. Metric | Ocean | SiamBAN | SiamRPN++ | DiMP | KYS | ATOM | LTMU | D3S | ECO | GlobalTrack | Staple | MOSSE | LTMU-H | MetaCrest | LTMU-F | VITAL | KCF
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
FPS | 21 | 24 | 23 | 16 | 12 | 15 | 8 | 16 | 15 | 8 | 13 | 26 | 4 | 8 | 4 | 4 | 6
SS | 0.365 | 0.360 | 0.362 | 0.336 | 0.327 | 0.319 | 0.284 | 0.276 | 0.252 | 0.253 | 0.249 | 0.227 | 0.213 | 0.207 | 0.205 | 0.204 | 0.186
NPS | 0.358 | 0.366 | 0.356 | 0.331 | 0.317 | 0.312 | 0.257 | 0.263 | 0.231 | 0.227 | 0.236 | 0.190 | 0.174 | 0.175 | 0.161 | 0.165 | 0.157
GSR | 0.294 | 0.313 | 0.293 | 0.224 | 0.237 | 0.179 | 0.169 | 0.182 | 0.173 | 0.139 | 0.169 | 0.141 | 0.161 | 0.165 | 0.162 | 0.158 | 0.177
Table 4: Accuracy results on TREK-150 of a video-based hand-object detection solution which considers each of the considered trackers as localization method for the object involved in the interaction. As a baseline, we employ the object detection capabilities of the hand-object interaction solution Hands-in-contact [79]. Hands-in-contact [79] | LTMU-H | LTMU-F | ATOM | LTMU | Ocean | SiamBAN | SiamRPN++ | MetaCrest | D3S | DiMP | KYS | VITAL | GlobalTrack | MOSSE | ECO | Staple | KCF
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
0.354 | 0.368 | 0.367 | 0.361 | 0.354 | 0.340 | 0.340 | 0.311 | 0.293 | 0.292 | 0.292 | 0.279 | 0.253 | 0.251 | 0.231 | 0.230 | 0.197 | 0.177
### How Do the Trackers Perform in the FPV Scenario?
Figure 3 reports the performance of the selected trackers on TREK-150 using
the OPE protocol. For reference, we also report the performance of the
trackers on the popular OTB-100 [91] benchmark (semi-transparent curves - gray
numbers in brackets). It can be clearly noted that the overall performance of
the trackers is decreased across all measures when considering the challenging
FPV scenario of TREK-150. For example, the SS, NPS, and GSR scores of LTMU on
TREK-150 are 43.7% , 44.8%, and 43.1%, which are much lower than the
respective 69.6%, 76%, and 78%, achieved on OTB-100. With the MSE protocol,
LTMU achieves the respective scores of 46.9%, 48.3%, 38.6%.444See Appendix D
for the overall MSE results of all trackers. These results show that the
particular characteristics of FPV present in TREK-150 introduce challenging
scenarios for visual trackers. Some qualitative examples of the trackers’
performance are shown in Figure 11 of the Appendix.
Generally speaking, trackers based on deep learning (e.g. LTMU, TRASFUST,
ATOM, KYS, Ocean) perform better in SS and NPS than those based on hand-
crafted features (e.g. BACF, MCCTH, DSST, KCF). Among the first class of
trackers, the ones leveraging online adaptation mechanisms (e.g. LTMU, ATOM,
VITAL, ECO, KYS, DiMP) are more accurate than the ones based on single-shot
instances (e.g. Ocean, D3S, SiamRPN++). The generalized success robustness
plot in Figure 3(c) and the GSR results of Figure 10 of the supplementary
report a different rankings of the trackers, showing that more spatially
accurate trackers are not always able to maintain longer reference to targets.
Under both the OPE and MSE protocols, the proposed FPV trackers LTMU-H and
LTMU-F are largely better in SS and NPS, while they lose some performance in
GSR. Such outcome shows that adapting a state-of-the-art method to FPV allows
to marginally improve results, while we expect significant performance
improvements to be achievable by a tracker accurately designed to tackle the
FPV challenges introduced by this benchmark.
### In Which Conditions Do the Trackers Work Better?
Figure 4 reports the SS, NPS, and GSR scores, computed with the MSE protocol,
of 17 trackers with respect to the attributes introduced in Table 2.555The
analysis was restricted to 17 trackers for a better visualization of the
plots/tables. The 17 trackers were selected to represent various
methodologies. The results for all trackers are available in Appendix D. We do
not report results for the POC attribute as it is present in every sequence,
as shown in Figure 2 (a). It stands out clearly that full occlusion (FOC), out
of view (OUT) and the small size of targets (LR) are the most difficult
situations for trackers. The fast motion of targets (FM) and the presence of
similar objects (SOB) are also critical factors that cause drops in
performance. Trackers show to be less vulnerable to rotations (ROT) and to the
illumination variation (IV). Generally, tracking rigid objects (RIG) results
easier than tracking deformable ones (DEF). With respect to the new 4 sequence
attributes related to FPV, it results that tracking objects held with two
hands (2H) is more difficult than tracking objects held with a single hand
(1H). This is probably due to the additional occlusions generated in the 2H
scenario. Trackers are instead quite robust to head motion (HM) and seem to
cope better with objects appearing in larger size (HR).
### How Do the Trackers Perform With Respect to the Actions?
The first row of plots in Figure 5 reports the MSE protocol results of SS,
NPS, and GSR with respect to the associated verb action
labels.${}^{\ref{note1}}$ Actions that mainly cause a spatial displacement of
the target (e.g. “move”, “store”, “check”) generally have less impact on the
performance. Actions that change the state, shape, or aspect ratio of an
object (e.g. “remove”, “squeeze”, “cut”, “attach”) generate harder tracking
scenarios. Also the sequences characterized by the “wash” verb lead trackers
to poor performance. Indeed, the wash action can cause many occlusions and
make the object harder to track.
The second row of the same figure presents the performance scores of the
trackers with respect to the associated noun labels. Rigid, regular-sized
objects such as “pan”, “kettle”, “bowl”, “plate”, and “bottle” are among the
ones associated with high average scores. On the other hand, some rigid
objects such as “knife”, “spoon”, “fork” and “can” are harder to track,
probably due to their particularly thin shape and the light reflectance they
are easily subject to. Deformable objects such as “sponge”, “onion”, “cloth”
and “rubbish” are in general also difficult to track.
### How Fast Are the Trackers?
Table 3 reports the FPS performance of the trackers and the SS, NPS, and GSR
scores achieved under the RTE protocol.${}^{\ref{note1}}$ None of the trackers
achieve the frame rate speed of 60 FPS. We argue that this is due the full HD
resolution of frames which requires demanding image crop and resize operations
with targets of considerable size. Thanks to their non-reliance of online
adaptation mechanisms, trackers based on siamese networks (e.g. Ocean,
SiamBAN, SiamRPN++) emerge as the fastest trackers and exhibit a less
significant performance drop of the proposed scores. Trackers using online
learning approaches (e.g. ATOM, DiMP, ECO, KYS) generally achieve a below
real-time speed, consequently causing a major accuracy loss when deployed to
real-time scenarios. In general, we observe that the GSR score is the measure
on which all trackers present the major drop in the real-time setting,
suggesting that particular effort should be spent to better model actions and
interactions in such scenarios.
### Do Trackers Already Offer Any Advantage in FPV?
Despite we are demonstrating that FPV is challenging for current trackers, we
assess whether these already offer an advantage in the FPV domain to obtain
information about the objects’ locations and movements in the scene [87, 32,
33, 78, 79]. To this aim, we performed two experiments.666Details are given in
the Appendix C of the supplementary material. First, we evaluated the
performance of a Faster R-CNN [75] instance trained on EK-55 [19] when used as
a naive tracking baseline. Such a solution achieves an SS, NPS, and GSR of
0.323, 0.369, 0.044, by running at 1 FPS. Comparing these results with the
ones presented in Figure 3, we clearly notice that trackers, if properly
initialized by a detection module, can deliver faster, more accurate and much
more temporally long object localization than detectors.
As a second experiment, we evaluated the accuracy of a video-based hand-object
interaction detection solution [79] whose object localisation is given by a
tracker rather than a detector. The tracker is initialized with the object
detector’s predicted bounding-box at the first detection of the hand-object
interaction, and let run until its end. By this setting, we created a ranking
of the trackers which is presented in Table 4. The results demonstrate that
stronger trackers can improve the accuracy and efficiency of current
detection-based methodologies [79]. Interestingly, the trackers’ ranking
differs from what shown in Figure 3, suggesting that trackers can manifest
other capabilities when deployed into application scenarios.
Given these preliminary results, we hence expect that trackers will likely
gain more importance in FPV as new methodologies explicitly considering the
first person point of view are investigated.
## 7 Conclusions
In this paper, we proposed the first systematic evaluation of visual object
tracking in FPV. The analysis has been conducted with standard and novel
measures on the newly introduced TREK-150 benchmark, which contains 150 video
sequences extracted from the EK-55 [19] FPV dataset. TREK-150 has been densely
annotated with 97K bounding-boxes, 17 sequence attributes, 20 action verb
attributes and 34 target object attributes. The performance of 31 state-of-
the-art visual trackers and two baseline FPV trackers was analysed extensively
on the proposed dataset. The results show a generalized drop in accuracy with
respect to the performance achieved on existing tracking benchmarks.
Furthermore, our analysis provided insights about which scenarios and actions
cause the performance to change. Finally, we have shown that object tracking
gives an advantage in terms of object localization accuracy and efficiency
over object detection. These results suggest that FPV is a challenging
scenario for current trackers and that tracking will likely get more
importance in this domain as new FPV-specific solutions will be investigated.
Annotations, results, and code, are available at
https://machinelearning.uniud.it/datasets/trek150/.
Acknowledgements. Research at the University of Udine has been supported by
the ACHIEVE-ITN H2020 project. Research at the University of Catania has been
supported by MIUR AIM - Attrazione e Mobilita Internazionale Linea 1 \-
AIM1893589 - CUP: E64118002540007.
## References
* [1] Maedeh Aghaei, Mariella Dimiccoli, and Petia Radeva. Multi-face tracking by extended bag-of-tracklets in egocentric photo-streams. Computer Vision and Image Understanding, 149:146–156, aug 2016\.
* [2] Maedeh Aghaei, Mariella Dimiccoli, and Petia Radeva. With whom do i interact? Detecting social interactions in egocentric photo-streams. In Proceedings - International Conference on Pattern Recognition, volume 0, pages 2959–2964. Institute of Electrical and Electronics Engineers Inc., jan 2016.
* [3] Stefano Alletto, Giuseppe Serra, and Rita Cucchiara. Egocentric object tracking: an odometry-based solution. In International Conference on Image Analysis and Processing, pages 687–696. Springer, 2015.
* [4] Gedas Bertasius, Hyun Soo Park, Stella X. Yu, and Jianbo Shi. First-person action-object detection with egonet. In Proceedings of Robotics: Science and Systems, July 2017.
* [5] Gedas Bertasius, Hyun Soo Park, Stella X Yu, and Jianbo Shi. Unsupervised learning of important objects from first-person videos. In Proceedings of the IEEE International Conference on Computer Vision, pages 1956–1964, 2017.
* [6] Luca Bertinetto, Jack Valmadre, Stuart Golodetz, Ondrej Miksik, and Philip H.S. Torr. Staple: Complementary learners for real-time tracking. In IEEE Conference on Computer Vision and Pattern Recognition, volume 2016-Decem, pages 1401–1409, 2016.
* [7] Luca Bertinetto, Jack Valmadre, João F. Henriques, Andrea Vedaldi, and Philip H.S. Torr. Fully-convolutional siamese networks for object tracking. European Conference on Computer Vision, 9914 LNCS:850–865, 2016\.
* [8] Goutam Bhat, Martin Danelljan, Luc Van Gool, and Radu Timofte. Learning Discriminative Model Prediction for Tracking. In Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019.
* [9] Goutam Bhat, Martin Danelljan, Luc Van Gool, and Radu Timofte. Know Your Surroundings: Exploiting Scene Information for Object Tracking. In European Conference on Computer Vision, mar 2020.
* [10] Goutam Bhat, Felix Järemo Lawin, Martin Danelljan, Andreas Robinson, Michael Felsberg, Luc Van Gool, and Radu Timofte. Learning what to learn for video object segmentation. In Andrea Vedaldi, Horst Bischof, Thomas Brox, and Jan-Michael Frahm, editors, Computer Vision – ECCV 2020, pages 777–794, Cham, 2020. Springer International Publishing.
* [11] David S Bolme, J Ross Beveridge, Bruce A Draper, and Yui Man Lui. Visual object tracking using adaptive correlation filters. In IEEE Conference on Computer Vision and Pattern Recognition, pages 2544–2550. IEEE, 2010.
* [12] Minjie Cai, Kris M Kitani, and Yoichi Sato. Understanding hand-object manipulation with grasp types and object attributes. In Robotics: Science and Systems, volume 3. Ann Arbor, Michigan;, 2016.
* [13] Zhe Cao, Ilija Radosavovic, Angjoo Kanazawa, and Jitendra Malik. Reconstructing hand-object interactions in the wild. arXiv preprint arXiv:2012.09856, 2020.
* [14] Luka Čehovin, Matej Kristan, and Aleš Leonardis. Robust visual tracking using an adaptive coupled-layer visual model. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(4):941–953, 2013.
* [15] Boyu Chen, Dong Wang, Peixia Li, Shuang Wang, and Huchuan Lu. Real-time ’Actor-Critic’ Tracking. In European Conference on Computer Vision, pages 318–334, 2018\.
* [16] Zedu Chen, Bineng Zhong, Guorong Li, Shengping Zhang, and Rongrong Ji. Siamese Box Adaptive Network for Visual Tracking. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6667–6676, 2020.
* [17] Dorin Comaniciu, Visvanathan Ramesh, and Peter Meer. Real-time tracking of non-rigid objects using mean shift. IEEE Conference on Computer Vision and Pattern Recognition, 2:142–149, 2000.
* [18] Kenan Dai, Yunhua Zhang, Dong Wang, Jianhua Li, Huchuan Lu, and Xiaoyun Yang. High-Performance Long-Term Tracking With Meta-Updater. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6297–6306. Institute of Electrical and Electronics Engineers (IEEE), aug 2020.
* [19] D. Damen, H. Doughty, G. M. Farinella, S. Fidler, A. Furnari, E. Kazakos, D. Moltisanti, J. Munro, T. Perrett, W. Price, and M. Wray. Scaling egocentric vision: The epic-kitchens dataset. In European Conference on Computer Vision (ECCV), 2018.
* [20] Dima Damen, Teesid Leelasawassuk, and Walterio Mayol-Cuevas. You-do, i-learn: Egocentric unsupervised discovery of objects and their modes of interaction towards video-based guidance. Computer Vision and Image Understanding, 149:98–112, 2016.
* [21] Martin Danelljan, Goutam Bhat, Fahad Shahbaz Khan, and Michael Felsberg. ECO: Efficient Convolution Operators for Tracking. In IEEE Conference on Computer Vision and Pattern Recognition, nov 2017.
* [22] Martin Danelljan, Goutam Bhat, Fahad Shahbaz Khan, and Michael Felsberg. ATOM: Accurate Tracking by Overlap Maximization. In IEEE Conference on Computer Vision and Pattern Recognition, 2019\.
* [23] Martin Danelljan, Gustav Hager, Fahad Shahbaz Khan, and Michael Felsberg. Discriminative Scale Space Tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(8):1561–1575, 2017.
* [24] Martin Danelljan, Luc Van Gool, and Radu Timofte. Probabilistic Regression for Visual Tracking. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7181–7190, 2020.
* [25] Patrick Dendorfer, Aljosa Osep, Anton Milan, Konrad Schindler, Daniel Cremers, Ian Reid, Stefan Roth, and Laura Leal-Taixé. Motchallenge: A benchmark for single-camera multiple target tracking. International Journal of Computer Vision, 129(4):845–881, 2021\.
* [26] J. Deng, W. Dong, R. Socher, L. Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 248–255, 2009.
* [27] Matteo Dunnhofer, Niki Martinel, Gian Luca Foresti, and Christian Micheloni. Visual Tracking by means of Deep Reinforcement Learning and an Expert Demonstrator. In Proceedings of The IEEE/CVF International Conference on Computer Vision Workshops, 2019.
* [28] Matteo Dunnhofer, Niki Martinel, and Christian Micheloni. An exploration of target-conditioned segmentation methods for visual object trackers. In Adrien Bartoli and Andrea Fusiello, editors, Computer Vision – ECCV 2020 Workshops, pages 618–636, Cham, 2020. Springer International Publishing.
* [29] Matteo Dunnhofer, Niki Martinel, and Christian Micheloni. Tracking-by-Trackers with a Distilled and Reinforced Model. In Asian Conference on Computer Vision, 2020.
* [30] Matteo Dunnhofer, Niki Martinel, and Christian Micheloni. Weakly-supervised domain adaptation of deep regression trackers via reinforced knowledge distillation. IEEE Robotics and Automation Letters, 6(3):5016–5023, 2021.
* [31] Heng Fan, Liting Lin, Fan Yang, Peng Chu, Ge Deng, Sijia Yu, Hexin Bai, Yong Xu, Chunyuan Liao, and Haibin Ling. LaSOT: A High-quality Benchmark for Large-scale Single Object Tracking. In IEEE Conference on Computer Vision and Pattern Recognition, sep 2019.
* [32] Antonino Furnari, Sebastiano Battiato, Kristen Grauman, and Giovanni Maria Farinella. Next-active-object prediction from egocentric videos. Journal of Visual Communication and Image Representation, 49:401–411, nov 2017.
* [33] Antonino Furnari and Giovanni Farinella. Rolling-Unrolling LSTMs for Action Anticipation from First-Person Video. IEEE Transactions on Pattern Analysis and Machine Intelligence, pages 1–1, may 2020.
* [34] Antonino Furnari and Giovanni Maria Farinella. What Would You Expect? Anticipating Egocentric Actions With Rolling-Unrolling LSTMs and Modality Attention. In IEEE/CVF International Conference on Computer Vision, 2019.
* [35] Hamed Kiani Galoogahi, Ashton Fagg, Chen Huang, Deva Ramanan, and Simon Lucey. Need for Speed: A Benchmark for Higher Frame Rate Object Tracking. In Proceedings of the IEEE International Conference on Computer Vision, volume 2017-Octob, pages 1134–1143. Institute of Electrical and Electronics Engineers Inc., mar 2017.
* [36] Qing Guo, Ruize Han, Wei Feng, Zhihao Chen, and Liang Wan. Selective Spatial Regularization by Reinforcement Learned Decision Making for Object Tracking. IEEE Transactions on Image Processing, 29:2999–3013, 2020.
* [37] Shangchen Han, Beibei Liu, Randi Cabezas, Christopher D. Twigg, Peizhao Zhang, Jeff Petkau, Tsz Ho Yu, Chun Jung Tai, Muzaffer Akbay, Zheng Wang, Asaf Nitzan, Gang Dong, Yuting Ye, Lingling Tao, Chengde Wan, and Robert Wang. MEgATrack: Monochrome Egocentric Articulated Hand-Tracking for Virtual Reality. ACM Transactions on Graphics, 39(4):13, jul 2020.
* [38] Sam Hare, Stuart Golodetz, Amir Saffari, Vibhav Vineet, Ming Ming Cheng, Stephen L Hicks, and Philip H.S. Torr. Struck: Structured Output Tracking with Kernels. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38(10):2096–2109, 2016.
* [39] David Held, Sebastian Thrun, and Silvio Savarese. Learning to Track at 100 FPS with Deep Regression Networks. European Conference on Computer Vision, abs/1604.0, 2016.
* [40] Joao F. Henriques, Rui Caseiro, Pedro Martins, and Jorge Batista. High-speed tracking with kernelized correlation filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(3):583–596, 2015.
* [41] Lianghua Huang, Xin Zhao, and Kaiqi Huang. GOT-10k: A Large High-Diversity Benchmark for Generic Object Tracking in the Wild. IEEE Transactions on Pattern Analysis and Machine Intelligence, pages 1–1, oct 2019.
* [42] Lianghua Huang, Xin Zhao, and Kaiqi Huang. GlobalTrack: A Simple and Strong Baseline for Long-term Tracking. In AAAI Conference on Artificial Intelligence, dec 2020.
* [43] Zdenek Kalal, Krystian Mikolajczyk, and Jiri Matas. Tracking-learning-detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(7):1409–1422, 2012.
* [44] Georgios Kapidis, Ronald Poppe, Elsbeth Van Dam, Lucas Noldus, and Remco Veltkamp. Egocentric hand track and object-based human action recognition. In 2019 IEEE SmartWorld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computing, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld/SCALCOM/UIC/ATC/CBDCom/IOP/SCI), pages 922–929. IEEE, 2019.
* [45] Georgios Kapidis, Ronald Poppe, Elsbeth Van Dam, Lucas Noldus, and Remco Veltkamp. Egocentric hand track and object-based human action recognition. In Proceedings - 2019 IEEE SmartWorld, Ubiquitous Intelligence and Computing, Advanced and Trusted Computing, Scalable Computing and Communications, Internet of People and Smart City Innovation, SmartWorld/UIC/ATC/SCALCOM/IOP/SCI 2019, pages 922–929. Institute of Electrical and Electronics Engineers Inc., may 2019.
* [46] Hamed Kiani Galoogahi, Ashton Fagg, and Simon Lucey. Learning Background-Aware Correlation Filters for Visual Tracking. In IEEE Conference on Computer Vision and Pattern Recognition, 2017\.
* [47] Matej Kristan, Aleš Leonardis, Jiří Matas, Michael Felsberg, Roman Pflugfelder, Joni-Kristian Kämäräinen, Martin Danelljan, Luka Čehovin Zajc, Alan Lukežič, Ondrej Drbohlav, Linbo He, Yushan Zhang, Song Yan, Jinyu Yang, Gustavo Fernández, et al. The eighth visual object tracking vot2020 challenge results. In Adrien Bartoli and Andrea Fusiello, editors, Computer Vision – ECCV 2020 Workshops, pages 547–601, Cham, 2020. Springer International Publishing.
* [48] Matej Kristan, Ales Leonardis, Jiri Matas, Michael Felsberg, Roman Pflugfelder, Luka Cehovin Zajc, Tomas Vojir, Gustav Hager, Alan Lukezic, Abdelrahman Eldesokey, Gustavo Fernandez, et al. The Visual Object Tracking VOT2017 Challenge Results. In Proceedings of the IEEE International Conference on Computer Vision Workshops, pages 1949–1972. IEEE, oct 2017.
* [49] Matej Kristan, Jiri Matas, Ales Leonardis, Michael Felsberg, Luka Cehovin, Gustavo Fernandez, Tomas Vojir, Gustav Hager, Georg Nebehay, Roman Pflugfelder, et al. The Visual Object Tracking VOT2015 Challenge Results. In Proceedings of the IEEE International Conference on Computer Vision Workshops, pages 564–586. IEEE, dec 2015.
* [50] Matej Kristan, Jiří Matas, Aleš Leonardis, Michael Felsberg, Roman Pflugfelder, Joni-Kristian Kämäräinen, LukaČehovinLukaˇLukaČehovin Zajc, Ondrej Drbohlav, Alan Lukežič, Amanda Berg, Abdelrahman Eldesokey, Jani Käpylä, Gustavo Fernández, et al. The Seventh Visual Object Tracking VOT2019 Challenge Results. In Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops, 2019.
* [51] M. Kristan, J. Matas, A. Leonardis, T. Vojíř, R. Pflugfelder, G. Fernández, G. Nebehay, F. Porikli, and L. Čehovin. A novel performance evaluation methodology for single-target trackers. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38(11):2137–2155, 2016.
* [52] Annan Li, Min Lin, Yi Wu, Ming Hsuan Yang, and Shuicheng Yan. NUS-PRO: A New Visual Tracking Challenge. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38(2):335–349, feb 2016.
* [53] Bo Li, Wei Wu, Qiang Wang, Fangyi Zhang, Junliang Xing, and Junjie Yan. SIAMRPN++: Evolution of siamese visual tracking with very deep networks. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 2019-June, pages 4277–4286, 2019\.
* [54] Feng Li, Cheng Tian, Wangmeng Zuo, Lei Zhang, and Ming Hsuan Yang. Learning Spatial-Temporal Regularized Correlation Filters for Visual Tracking. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 4904–4913. IEEE Computer Society, mar 2018.
* [55] Mengtian Li, Yu-Xiong Wang, and Deva Ramanan. Towards streaming perception. In Andrea Vedaldi, Horst Bischof, Thomas Brox, and Jan-Michael Frahm, editors, Computer Vision – ECCV 2020, pages 473–488, Cham, 2020. Springer International Publishing.
* [56] Pengpeng Liang, Erik Blasch, and Haibin Ling. Encoding Color Information for Visual Tracking: Algorithms and Benchmark. IEEE Transactions on Image Processing, 24(12):5630–5644, dec 2015\.
* [57] Miao Liu, Siyu Tang, Yin Li, and James Rehg. Forecasting human object interaction: Joint prediction of motor attention and actions in first person video. In Proceedings of the European Conference on Computer Vision (ECCV), volume 2, 2020.
* [58] Xiankai Lu, Chao Ma, Bingbing Ni, Xiaokang Yang, Ian Reid, and Ming Hsuan Yang. Deep regression tracking with shrinkage loss. In European Conference on Computer Vision, volume 11218 LNCS, pages 369–386, 2018.
* [59] Alan Lukezic, Ugur Kart, Jani Kapyla, Ahmed Durmush, Joni Kristian Kamarainen, Jiri Matas, and Matej Kristan. CDTB: A color and depth visual object tracking dataset and benchmark. In Proceedings of the IEEE International Conference on Computer Vision, volume 2019-Octob, pages 10012–10021. Microsoft Research Asia, jul 2019\.
* [60] Alan Lukežič, Jiří Matas, and Matej Kristan. D3S – A Discriminative Single Shot Segmentation Tracker. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, nov 2020.
* [61] Alan Lukežič, Luka Čehovin Zajc, Tomáš Vojíř, Jiří Matas, and Matej Kristan. Now you see me: evaluating performance in long-term visual tracking, 2018\.
* [62] Minghuang Ma, Haoqi Fan, and Kris M Kitani. Going deeper into first-person activity recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1894–1903, 2016.
* [63] Dr Emilio Maggio and Dr Andrea Cavallaro. Video Tracking: Theory and Practice. Wiley Publishing, 1st edition, 2011.
* [64] Mario Edoardo Maresca and Alfredo Petrosino. MATRIOSKA: A multi-level approach to fast tracking by learning. In International Conference on Image Analysis and Processing, volume 8157 LNCS, pages 419–428, 2013.
* [65] Franziska Mueller, Dushyant Mehta, Oleksandr Sotnychenko, Srinath Sridhar, Dan Casas, and Christian Theobalt. Real-Time Hand Tracking Under Occlusion from an Egocentric RGB-D Sensor. In Proceedings - 2017 IEEE International Conference on Computer Vision Workshops, ICCVW 2017, volume 2018-Janua, pages 1284–1293, 2017.
* [66] Matthias Mueller, Neil Smith, and Bernard Ghanem. A Benchmark and Simulator for UAV Tracking. In European Conference on Computer Vision, pages 445–461. Springer, Cham, 2016.
* [67] Matthias Müller, Adel Bibi, Silvio Giancola, Salman Alsubaihi, and Bernard Ghanem. TrackingNet: A Large-Scale Dataset and Benchmark for Object Tracking in the Wild. In European Conference on Computer Vision, volume 11205 LNCS, pages 310–327. Springer Verlag, mar 2018.
* [68] Hyeonseob Nam and Bohyung Han. Learning Multi-domain Convolutional Neural Networks for Visual Tracking. IEEE Conference on Computer Vision and Pattern Recognition, 2016-Decem:4293–4302, 2016.
* [69] Hyeonseob Nam, Seunghoon Hong, and Bohyung Han. Online graph-based tracking. In European Conference on Computer Vision, volume 8693 LNCS, pages 112–126. Springer Verlag, 2014.
* [70] Jyoti Nigam and Renu M Rameshan. EgoTracker: Pedestrian Tracking with Re-identification in Egocentric Videos. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, volume 2017-July, pages 980–987, 2017.
* [71] Eunbyung Park and Alexander C. Berg. Meta-tracker: Fast and Robust Online Adaptation for Visual Object Trackers. In European Conference on Computer Vision, volume 11207 LNCS, pages 587–604. Springer Verlag, jan 2018.
* [72] Hamed Pirsiavash and Deva Ramanan. Detecting activities of daily living in first-person camera views. In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pages 2847–2854. IEEE, 2012.
* [73] Francesco Ragusa, Antonino Furnari, Salvatore Livatino, and Giovanni Maria Farinella. The meccano dataset: Understanding human-object interactions from egocentric videos in an industrial-like domain. In IEEE Winter Conference on Application of Computer Vision (WACV), 2020.
* [74] Joseph Redmon, Santosh Kumar Divvala, Ross B Girshick, and Ali Farhadi. You Only Look Once: Unified, Real-Time Object Detection. In IEEE Conference on Computer Vision and Pattern Recognition, pages 779–788, 2016.
* [75] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in Neural Information Processing Systems, pages 91–99, 2015.
* [76] Ivan Rodin, Antonino Furnari, Dimitrios Mavroedis, and Giovanni Maria Farinella. Predicting the future from first person (egocentric) vision: A survey. Computer Vision and Image Understanding, 2021.
* [77] Ricardo Sanchez-Matilla and Andrea Cavallaro. A predictor of moving objects for first-person vision. In 2019 IEEE International Conference on Image Processing (ICIP), pages 2189–2193. IEEE, 2019.
* [78] Fadime Sener, Dipika Singhania, and Angela Yao. Temporal aggregate representations for long-range video understanding. In European Conference on Computer Vision, pages 154–171. Springer, 2020.
* [79] Dandan Shan, Jiaqi Geng, Michelle Shu, and David F. Fouhey. Understanding human hands in contact at internet scale. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
* [80] Yibing Song, Chao Ma, Xiaohe Wu, Lijun Gong, Linchao Bao, Wangmeng Zuo, Chunhua Shen, Rynson W.H. Lau, and Ming Hsuan Yang. VITAL: VIsual Tracking via Adversarial Learning. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 8990–8999. IEEE Computer Society, apr 2018.
* [81] Li Sun, Ulrich Klank, and Michael Beetz. EYEWATCHME-3D Hand and object tracking for inside out activity analysis. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pages 9–16. Institute of Electrical and Electronics Engineers (IEEE), mar 2010.
* [82] Jack Valmadre, Luca Bertinetto, João F. Henriques, Ran Tao, Andrea Vedaldi, Arnold W.M. Smeulders, Philip H.S. Torr, and Efstratios Gavves. Long-Term Tracking in the Wild: A Benchmark. In European Conference on Computer Vision, volume 11207 LNCS, pages 692–707. Springer Verlag, mar 2018.
* [83] Ryan J. Visee, Jirapat Likitlersuang, and Jose Zariffa. An Effective and Efficient Method for Detecting Hands in Egocentric Videos for Rehabilitation Applications. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 28(3):748–755, mar 2020.
* [84] Ning Wang, Wengang Zhou, Qi Tian, Richang Hong, Meng Wang, and Houqiang Li. Multi-cue Correlation Filters for Robust Visual Tracking. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 4844–4853, 2018.
* [85] Qiang Wang, Jin Gao, Junliang Xing, Mengdan Zhang, and Weiming Hu. DCFNet: Discriminant Correlation Filters Network for Visual Tracking. apr 2017.
* [86] Qiang Wang, Li Zhang, Luca Bertinetto, Weiming Hu, and Philip H S Torr. Fast Online Object Tracking and Segmentation: A Unifying Approach. In IEEE Conference on Computer Vision and Pattern Recognition, 2019\.
* [87] Xiaohan Wang, Yu Wu, Linchao Zhu, and Yi Yang. Symbiotic attention with privileged information for egocentric action recognition. In AAAI Conference on Artificial Intelligence, 2020.
* [88] Nicolai Wojke, Alex Bewley, and Dietrich Paulus. Simple online and realtime tracking with a deep association metric. In Proceedings - International Conference on Image Processing, ICIP, volume 2017-Septe, pages 3645–3649. IEEE Computer Society, mar 2018.
* [89] Chao-Yuan Wu, Christoph Feichtenhofer, Haoqi Fan, Kaiming He, Philipp Krahenbuhl, and Ross Girshick. Long-term feature banks for detailed video understanding. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 284–293, 2019.
* [90] Yi Wu, Jongwoo Lim, and Ming Hsuan Yang. Online object tracking: A benchmark. In IEEE Conference on Computer Vision and Pattern Recognition, pages 2411–2418. IEEE Computer Society, 2013.
* [91] Yi Wu, Jongwoo Lim, and Ming Hsuan Yang. Object tracking benchmark. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(9):1834–1848, sep 2015.
* [92] Yinda Xu, Zeyu Wang, Zuoxin Li, Ye Yuan, and Gang Yu. SiamFC++: Towards Robust and Accurate Visual Tracking with Target Estimation Guidelines. In AAAI Conference on Artificial Intelligence, nov 2020.
* [93] Bin Yan, Haojie Zhao, Dong Wang, Huchuan Lu, and Xiaoyun Yang. ’Skimming-perusal’ tracking: A framework for real-time and robust long-term tracking. In Proceedings of the IEEE International Conference on Computer Vision, volume 2019-Octob, pages 2385–2393, 2019.
* [94] Sangdoo Yun, Jongwon Choi, Youngjoon Yoo, Kimin Yun, and Jin Young Choi. Action-decision networks for visual tracking with deep reinforcement learning. In IEEE Conference on Computer Vision and Pattern Recognition, volume 2017-Janua, pages 1349–1358. IEEE, jul 2017.
* [95] Jianming Zhang, Shugao Ma, and Stan Sclaroff. MEEM: Robust tracking via multiple experts using entropy minimization. In European Conference on Computer Vision, volume 8694 LNCS, pages 188–203. Springer Verlag, 2014.
* [96] Lichao Zhang, Abel Gonzalez-Garcia, Joost Van De Weijer, Martin Danelljan, and Fahad Shahbaz Khan. Learning the model update for siamese trackers. In Proceedings of the IEEE International Conference on Computer Vision, volume 2019-Octob, pages 4009–4018. Institute of Electrical and Electronics Engineers Inc., oct 2019.
* [97] Zhipeng Zhang and Houwen Peng. Deeper and Wider Siamese Networks for Real-Time Visual Tracking. IEEE Conference on Computer Vision and Pattern Recognition, jan 2019\.
* [98] Zhipeng Zhang, Houwen Peng, Jianlong Fu, Bing Li, and Weiming Hu. Ocean: Object-aware Anchor-free Tracking. In European Conference on Computer Vision, jun 2020.
* [99] L. Čehovin, A. Leonardis, and M. Kristan. Visual object tracking performance measures revisited. IEEE Transactions on Image Processing, 25(3):1261–1274, 2016.
Figure 6: Examples of target objects contained in TREK-150 that are difficult
to represent with more sophisticated representations (e.g. rotated bounding
box or segmentation mask). The first two images from the left show objects
such as “cheese” and “onion” which prevent the determination of the angle for
an oriented bounding box, or an accurate segmentation mask. The last two
images present objects which prevent a consistent definition of a
segmentation.
## Appendix A Motivations And Details Behind TREK-150
In this section, we provide more motivations and details behind the
construction of the TREK-150 benchmark dataset.
First of all, we remark that TREK-150 has been designed for the _evaluation_
of visual tracking algorithms in FPV regardless of their methodology. Indeed,
this paper does not aim to provide a large-scale dataset to improve the
performance of deep learning based trackers. Instead, its goal is to assess
the impact of the first-person viewpoint on current trackers and, to the best
of our knowledge, this analysis was never done before. Hence, as a _first
step_ towards providing an answer to such a point (which is also highlighted
in the title of the paper), we focused on benchmarking the tracking progress
made by the computer vision community in the last years.
### Video Collection.
The video sequences contained in TREK-150 have been sampled from the EPIC-
Kitchens-55 (EK-55) dataset [19]. This has been done because EK-55 is
currently the largest dataset for understanding human-object interactions in
FPV (it provides up to 55 hours of human-object interaction examples). Thanks
to its dimension, it is the only database that provides a significant amount
of diverse interaction situations between various people and several different
types of objects. Hence, it allowed us to select suitable diverse tracking
sequences that reflect the common scenarios tackled in FPV tasks.
### Bounding Box Annotations.
To represent the spatial localization of objects, we employed axis-aligned
bounding-boxes. This design choice for the TREK-150 benchmark is supported by
the fact that this representation is largely used in many FPV pipelines [32,
34, 33, 19, 45, 83, 79]. Therefore, computing performance results based on
such allows us to correlate them to the results of other FPV tasks that employ
the same object representation. Hence, we can better highlight the impact that
trackers would have in such contexts. Moreover, we would like to highlight the
difficulty that the FPV setting poses on the development of more sophisticated
annotations for object categories that appear commonly in FPV scenarios.
Figure 6 shows some examples of these. The first two images from the left show
the objects “cheese” and “onion” (these are considered as single objects
according to the EK-55 annotations [19]) which prevent the determination of
the angle for an oriented bounding-box, or an even accurate segmentation mask
due to their spatial sparsity. The two images on the right present objects for
which providing a segmentation is very ambiguous. Indeed, most of the pixels
in the image area of the knife (third image) belong actually to foam, while
the heavy motion blur happening on the object of the fourth image (where the
target is a bottle) prevents the definition of the actual pixels belonging to
the object. In all these scenarios, axis-aligned bounding-boxes result in
robust target representations that provide a consistent delineation of the
object. For these motivations, and to make representations and annotations
consistent across the whole dataset, we employed such annotation
representations.
Moreover, the latest progress of visual tracking algorithms on various
benchmarks that use this state representation [91, 66, 35, 59, 67, 31, 41]
demonstrates that it provides sufficient information about the target for
consistent and reliable performance evaluation. Furthermore, using more
sophisticated target representation would have restricted our analysis [86,
60, 28, 10] since the majority of state-of-the-art trackers output just axis-
aligned bounding boxes [11, 23, 40, 68, 6, 7, 39, 21, 46, 80, 54, 84, 71, 53,
97, 22, 8, 93, 24, 42, 92, 18, 16, 98, 9].
Finally, we point out that the proposed axis-aligned bounding-boxes have been
carefully and tightly drawn around the visible parts of the objects. Figure 7
shows some examples of the quality of the bounding box annotations of TREK-150
in contrast to the ones available in the popular OTB-100 tracking benchmark.
Figure 7: Examples of the quality of the bounding box annotations contained in
TREK-150 in comparison with the ones available in the popular OTB-100
benchmark. TREK-150 provides careful and high-quality annotations that tightly
enclose all the target objects.
### Frame Rate.
The videos contained in TREK-150 have a frame rate of 60 FPS. This is
inherited from the EK-55 dataset [19], from which videos are sampled.
According to the authors [19], EK-55 has been acquired with such a setting
because of the proximity of the camera point of view and the main scene (i.e.
manipulated objects), which causes very fast motion and heavy motion blur when
the camera wearer moves (especially when he/she moves the head).
We empirically evaluated the fast motion issue by assessing the average
normalized motion happening on the frames that include fast motion (FM) (we
computed them by considering the automatic procedure defined in [91, 67] to
assign the FM attribute). Such a motion quantity has been computed as the
distance between the center of two consecutive ground-truth bounding boxes
normalized by the frame size. Considering TREK-150 with the videos at 30 FPS,
such a value achieves 0.075. This is higher than the 0.068 obtained for
OTB-100, the 0.033 of UAV123, or the 0.049 of NfS considered at 30 FPS. These
comparisons demonstrate that the FPV scenario effectively includes challenging
scenarios due to the faster motion of targets/scene. Considering the 60 FPS
frame rate, the fast motion quantity of TREK-150 is reduced to 0.062, which is
comparable to the values obtained in other third-person tracking benchmarks.
### Sequence Labels.
To study the performance of trackers under different aspects, the sequences of
TREK-150 have been associated with one or more of 17 attributes that indicate
the visual variability of the target in the sequence (see Table 2 of the main
paper for the details). The extended usage of this practice [91, 66, 35, 52,
67, 31, 41] showed how this kind of labeling is sufficient to estimate the
trackers’ performance on particular scenarios. We therefore follow such an
approach to associate labels on TREK-150’s videos. However, we argue that, by
using this labeling setting, attention must be paid to how trackers are
evaluated. The standard OPE protocol, which has been generally used to perform
such evaluations, could lead to less accurate estimates. For example, it could
happen that a tracker would fail for some event described by an attribute
(e.g. FOC) in the first frames of a video, but that the sequence also contains
some other event (e.g. MB) in the end. With the score averaging procedure
defined by the OPE protocol, the low results achieved due to the first event
would set low scores also for the second event, while the tracker failed just
for the first one. Therefore, the performance estimate for the second
attribute would not be realistic. We believe a reasonable option is to use a
more robust evaluation protocol such as the multi-start evaluation (MSE).
Thanks to its points of initialization which generate multiple diverse sub-
sequences, this protocol allows a tracker to better cover all the possible
situations happening along the videos, both forward and backward in time. All
the results achieved on the sub-sequences are then averaged to obtain the
overall scores on a sequence. We think the scores computed in this way to be
more robust and accurate estimates of the real performance of the trackers.
Hence, in this work, we follow such an approach to evaluate trackers over
sequence attributes.
Figure 8: Comparison between TREK-150 (last column of plots) and other popular
visual tracking benchmarks on the distributions computed for different
bounding box characteristics. Each column of plots reports the distribution of
bounding box sizes, scale changes, and aspect ratio change (the x-axis of each
plot reports the range of the bounding box statistic).
### Single Object Tracking.
In this paper, we restricted our analysis to the tracking of a single object
per video. This has been done because in the FPV scenario a person interacts
through hands with one or two objects at a time in general [19] (if a person
interacts with two objects they can be still tracked by two single-object
trackers). Moreover, focusing on a single object allows us to analyze better
all the challenging and relevant factors that characterize the tracking
problem in FPV. We believe that future work could investigate the employment
of multiple object tracking (MOT) solutions [25] for a general understanding
of the position and movement of all objects visible in the scene. We think
that the study presented in this paper will give useful insights even for the
development of such methods.
### Differences With Other Tracking Benchmarks.
We believe that the proposed TREK-150 benchmark dataset offers _complementary_
features with respect to existing visual tracking benchmarks.
Table 1 and Figure 2(a) and (b) of the main paper show that TREK-150 provides
complementary characteristics to what is available today to study the
performance of visual trackers. Particularly, our proposed dataset offers
different distributions of the common challenging factors encountered in other
datasets. For example, TREK-150 includes a larger number of examples with
occlusions (POC), fast motion (FM), scale change (SC), aspect ratio change
(ARC), illumination variation (IV), and motion blur (MB), while it provides a
competitive number of scenarios for low resolution (LR), full occlusion (FOC),
deformable objects (DEF), and presence of similar objects (SOB). Additionally,
even though the 4 new attributes high resolution (HR), head motion (HM), one-
hand interaction (1H), two-hands interaction (2H), define particular FPV
scenarios, we think that they can be of interest even for the visual tracking
community. For example, as shown by the second row of images of Figure 7, 1H
and 2H can be considered as attributes that define different levels of
occlusion, as objects manipulated with two hands generally cause more extended
hiding of the targets. Besides these sequence-level features, TREK-150 offers
up to 34 target categories which, to the best of our knowledge, have never
been studied. As shown by the Figures 6 and 7, these objects have challenging
appearances (e.g. transparent or reflective objects like lids, bottles, or
food boxes) and shapes (e.g. knives, spoons, cut food) that change
dramatically due to the interaction or motion induced by the camera viewer.
We additionally computed some statistics on the bounding box ground-truth
trajectories contained in the proposed dataset. Figure 8 reports these
distributions. For comparison, we report the distributions computed on the
popular tracking benchmarks VOT2019, UAV123, OTB-100. As can be noted, our
dataset exhibits different distributions, and thus offers different behaviors
of the target appearances and motions. Particularly, observing the first plot
of the last column, it can be noted that TREK-150 has a wider distribution of
bounding box sizes, hence making it suitable for the evaluation of trackers
with targets of many different sizes. Particularly, TREK-150 has a larger
number of bounding boxes with greater dimension. The plot just below the first
shows that TREK-150 provides more references to assess the trackers’
capabilities in tracking objects that become smaller. Finally, the last plot
shows a wider distribution for the aspect ratio change, showing that TREK-150
offers a large variety of examples to evaluate the capabilities of trackers in
predicting the shape change of targets.
Additionally to these characteristics, we think TREK-150 is interesting
because it allows the study of visual object tracking in unconstrained
scenarios of _every-day_ situations.
Table 5: Details of the trackers involved in our evaluation. In the Image Representation column the acronyms stand for: CNN - Convolutional Neural Network; HOG - Histogram of Oriented Gradients; Pixel - Pixel Intensity; Color - Color Names or Intensity. Regarding the Matching strategy column the acronyms stand for: CF - Correlation Filter; CC - Cross Correlation; T-by-D - Tracking by Detection; Reg - Regression; Had - Hadamard Correlation. The ✓ symbol in the Model Update column expresses the target model update during the tracking procedure. The last column reports the tracking method class according to [61] (ST \- Short-Term trackers, LT \- Long-Term trackers). Tracker | Venue | Image Representation | Matching | Model Update | [61] Class
---|---|---|---|---|---
MOSSE [11] | CVPR 2010 | Pixel | CF | ✓ | $\text{ST}_{0}$
DSST [23] | BMVC 2014 | HOG+Pixel | CF | ✓ | $\text{ST}_{0}$
KCF [40] | TPAMI 2015 | HOG | CF | ✓ | $\text{ST}_{0}$
MDNet [68] | CVPR 2016 | CNN | T-by-D | ✓ | $\text{ST}_{1}$
Staple [6] | CVPR 2016 | HOG+Color | CF | ✓ | $\text{ST}_{0}$
SiamFC [7] | ECCVW 2016 | CNN | CC | ✗ | $\text{ST}_{0}$
GOTURN [39] | ECCV 2016 | CNN | Reg | ✗ | $\text{ST}_{0}$
ECO [21] | CVPR 2017 | CNN | CF | ✓ | $\text{ST}_{0}$
BACF [46] | ICCV 2017 | HOG | CF | ✓ | $\text{ST}_{0}$
DCFNet [85] | ArXiv 2017 | CNN | CF | ✓ | $\text{ST}_{0}$
VITAL [80] | CVPR 2018 | CNN | T-by-D | ✓ | $\text{ST}_{1}$
STRCF [54] | CVPR 2018 | HOG | CF | ✓ | $\text{ST}_{0}$
MCCTH [84] | CVPR 2018 | HOG +Color | CF | ✓ | $\text{ST}_{0}$
DSLT [58] | ECCV 2018 | CNN | CC | ✓ | $\text{ST}_{0}$
MetaCrest [71] | ECCV 2018 | CNN | CF | ✓ | $\text{ST}_{1}$
SiamRPN++ [53] | CVPR 2019 | CNN | CC | ✗ | $\text{ST}_{0}$
SiamMask [86] | CVPR 2019 | CNN | CC | ✗ | $\text{ST}_{0}$
SiamDW [97] | CVPR 2019 | CNN | CC | ✗ | $\text{ST}_{0}$
ATOM [22] | CVPR 2019 | CNN | CF | ✓ | $\text{ST}_{1}$
DiMP [8] | ICCV 2019 | CNN | CF | ✓ | $\text{ST}_{1}$
SPLT [93] | ICCV 2019 | CNN | CF | ✓ | $\text{LT}_{1}$
UpdateNet [96] | ICCV 2019 | CNN | CC | ✓ | $\text{ST}_{0}$
SiamFC++ [92] | AAAI 2020 | CNN | CC | ✗ | $\text{ST}_{0}$
GlobalTrack [42] | AAAI 2020 | CNN | Had | ✗ | $\text{LT}_{0}$
PrDiMP [24] | CVPR 2020 | CNN | CF | ✓ | $\text{ST}_{1}$
SiamBAN [16] | CVPR 2020 | CNN | CC | ✗ | $\text{ST}_{0}$
D3S [60] | CVPR 2020 | CNN | CF | ✗ | $\text{ST}_{0}$
LTMU [18] | CVPR 2020 | CNN | CF/CC | ✓ | $\text{LT}_{1}$
Ocean [98] | ECCV 2020 | CNN | CC | ✗ | $\text{ST}_{0}$
KYS [9] | ECCV 2020 | CNN | CF | ✓ | $\text{ST}_{1}$
TRASFUST [29] | ACCV 2020 | CNN | Reg | ✗ | $\text{ST}_{1}$
## Appendix B Tracker Details
### Generic Object Trackers Details.
Table 5 reports some additional information about the 31 considered generic-
object trackers such as: venue and year of publication; type of image
representation used; type of matching strategy; employment of target model
updates; and category of tracker according to the classification of [61]. For
each tracker, we used the code publicly available and adopted default
parameters for evaluation purposes.
### FPV Trackers Details.
In this section, we provide details on the LTMU-F and LTMU-H FPV trackers
considered as baselines in our study. For a better understanding, we briefly
recap the processing procedure of the LTMU tracker [18]. After being
initialized with the target in the first frame of a sequence, at every other
frame LTMU first executes the short-term tracker DiMP [8] that tracks the
target in a local area (based on the target’s last known position) of the
frame. The image patch extracted from the bounding box prediction of DiMP is
evaluated by an online-learned verifying module which outputs a probability
estimate for the target being contained in the patch. Such an estimate is
employed to decide if the short-term tracker is tracking the target or not. If
it is, the box predicted by the short-term tracker is given as output for the
current frame. In the other case, a re-detection module is executed to search
for the target in the global frame. The detector returns some candidate
locations to contain the target and each of these is checked by the
verification module. The candidate patch with the highest confidence is given
as output and used as a new target location to reset the short-term tracker.
Figure 9: Visual representation of the LTMU [18] scheme performed at every
frame that has been adapted for the development of the baseline FPV trackers
LTMU-F and LTMU-H.
In our setting, we employ FPV-based detectors to implement such a re-detection
module. For LTMU-F, we employed the EK-55 trained Faster R-CNN [19]. Among the
many detections given as output, this module has been set to retain the first
10, considering a ranking based on the scores attributed by the detector to
each detection. If no detection is given for a frame, the last available
position of the target is considered as candidate location. For LTMU-H, we
employ the object localization contained in the hand-object interaction
detections given by the FPV version of Hands-in-contact [79] to obtain the
target candidate locations. Such a solution [79] is implemented as an improved
Faster R-CNN which is set to learn to provide, at the same time, the
localization of hands and objects, and their state of interaction. As before,
if no detection is given for a frame, the last available position of the
target is considered as candidate location. For both methods, the original
pre-trained models (made available by the authors) which consider FPV data
have been used. The described setups, which the common scheme is presented in
Figure 9, give birth to two trackers that implement conceptually different
strategies for FPV-based object localization. Indeed, the first solution
reasons just to find objects in the scene, while the second reasons in terms
of the interaction happening between the camera wearer (i.e. hands) and the
objects. We would like to remark that other FPV trackers (such as the ones
described in Section 2 of the main paper) have not been tested on TREK-150
because their implementations are not available.
### Implementation Details.
The evaluations were performed on a machine with an Intel Xeon E5-2690 v4 @
2.60GHz CPU, 320 GB of RAM, and an NVIDIA TITAN V GPU. We considered the
Python publicly available implementations of each tracker and adopted default
parameters. Annotations, results of the trackers, and code are available at
https://machinelearning.uniud.it/datasets/trek150/.
## Appendix C Experimental Details
### Details On The Generalized Robustness.
The robustness measure has been first introduced in [51]. This metric was
defined as the number of drifts (i.e. the complete non-overlap between
predictions and ground-truths) performed by a visual tracking algorithm. In
the last iteration of the VOT challenge [47], such a robustness measure has
been revised and defined as the extent of a tracking sequence before the
tracker’s failure. Such an extent is determined as the number of frames
positively tracked normalized by the total number of frames in the sequence.
The failure event is triggered when the overlap between the predicted and
ground-truth bounding-boxes becomes lower than a fixed threshold (the value
0.1 is used in [47]). In simpler words, this measure expresses the fraction of
a tracking sequence that is correctly tracked from its beginning. We think
this measure is of special interest to the FPV community. Indeed, it can
assess the ability of a tracker to maintain reference in time to the target
objects. Since many FPV tasks are devoted to understand the action performed
by the camera viewer or its interaction with objects [32, 34, 33, 19, 73],
having solutions capable of maintaining temporally longer references to target
nouns can be advantageous to model such events. However, we believe that
having a single fixed threshold is restrictive, as different applications can
make different assumptions on the concept of tracking failure. Therefore,
following [91] which proposed to evaluate trackers with plots computed after
thresholding bounding-box overlaps with different values, we propose to build
a plot considering different overlap thresholds for the determination of
failure in the robustness measure [47]. This leads to the creation of the
Generalized Success Robustness Plot (Figure 3(c) of the main paper) which
reports the different robustness scores for the different thresholds. The
latters have been studied just in the range [0, 0.5] because it is common
practice, in the computer vision literature, to consider overlaps greater than
50% as positive predictions. Notice that failures could be also defined in
terms of center error. In this paper, we focused on overlap-based failures
since bounding-box overlap has been shown to be superior for target
localization accuracy [99], but future work will investigate the employment of
the center error as this kind of bounding box distance is used in FPV tasks
[79]. Moreover, to compare trackers with a single score, following [91] and
[67], we compute the AUC of the Generalized Success Robustness Plot which we
refer as to generalized success robustness (GSR). This value expresses the
average of all the scores obtained with the different thresholds. In other
words, the GSR score expresses the average successful extent of the
predictions of a tracker.
Figure 10: SS, NPS, and GSR performance of the 33 benchmarked trackers on the
proposed TREK-150 benchmark under the MSE protocol. The general low
performances confirms the conclusions achieved with the OPE protocol.
### Details On The Evaluation Protocols.
In this section, we give further details on the experimental protocols used
for the execution of the trackers.
The one-pass evaluation (OPE) protocol, which is detailed in [91], consists of
two main stages: (i) initializing a tracker with the ground-truth bounding box
of the target in the first frame; (ii) let the tracker run on every subsequent
frame until the end of the sequence and record predictions to be considered
for the evaluation. For each sequence, predictions and ground-truth bounding
boxes are compared according to the employed measures (only for frames where
ground-truths are present) to obtain the performance scores. The overall
scores, presented in brackets in Figure 3 of the main paper, are obtained by
averaging the scores achieved for every sequence.
For the implementation of the multi-start evaluation (MSE) protocol, we
followed the details given in [47]. For each sequence, different points of
initialization (called anchors) separated by 2 seconds (in our setting every
120 frames) are defined. Anchors are always set in the first and last frame of
a sequence. Some anchors are shifted forward for a few frames to obtain a more
consistent bounding-box for tracker initialization. A tracker is run on each
of the sub-sequences yielded by the anchor (in total 1032 sub-sequences are
generated), either forward or backward in time depending on the longest sub-
sequence the anchor generates. The tracker is initialized with the ground-
truth in the first frame of the sub-sequence and let run until its end. Then,
similarly as for the OPE, predicted and ground-truth bounding boxes are
compared to obtain the performance scores for each sub-sequence. Scores for a
single sequence are computed by a weighted average where the scores of each
sub-sequence are weighted by its length (as number of frames). Similarly, the
overall scores for the whole dataset (which are shown in Figure 10) are
obtained by a weighted average where each sequence’s score is weighted by the
number of frames in that sequence.
The real-time evaluation (RTE) protocol has been implemented following the
details in [48, 55]. Similar to the OPE protocol, a tracker is initialized
with the ground-truth in the first frame of a sequence. Then the tracker is
presented with a new frame only after its execution over the previous frame
has finished. The new presented frame is the last frame available for the time
instant in which the tracker becomes ready to be executed, considering that
frames occur regularly according to the frame rate of the video. In other
words, all the frames occurring in the time interval between the start and end
time instants of the tracker’s execution are skipped. For all the frames
skipped, the last bounding box given by the tracker is used as location for
the target in such frames. The sequence and overall scores are ultimately
obtained as for the OPE protocol.
### Experiments On The Impact of Trackers In FPV.
In this section, we report more details on the experiments performed to
evaluate the impact of trackers in FPV tasks (presented in the paragraph “Do
Trackers Already Offer Any Advantage in FPV?” of Section 6 of the main paper.)
In the first experiment, we assessed the capabilities of continuous object
localization of an object detector, as this method is usually exploited in
many FPV pipelines. We executed the EK-55 trained Faster-R-CNN [19] on all the
frames of TREK-150, by recording in each frame the bounding box of the
detection having the same class of the target and the highest confidence
score. Such bounding box predictions were then compared to the ground truth
annotations using the considered tracking evaluation measures. This
experimental strategy respects the evaluation procedure of the OPE protocol.
In this way, we can compare the performance of the tracking approach and the
detection approach in providing localization and temporal reference of/to
objects.
In the second experiment, we evaluated the impact of trackers in a video-based
hand-object interaction detection setting. Since this paper is focused on
objects (visual object tracking), we restricted our study in evaluating the
detection of the objects involved in the interactions. To this aim, we first
built tracks of hand-object interactions over the sequences of TREK-150. The
hand-object interaction detector Hands-in-contact [79] has been executed to
obtain sparse interaction detections that involved the object defined by the
ground-truths of TREK-150. Clusters of detections have been then set to form
separate tracks if the interval between two detections was longer than 30
frames. The missing detections within a cluster have been filled with the
TREK-150’s object ground-truth bounding boxes and the most frequent
interaction state (i.e. if the object was in interaction with the left hand,
the right hand, or both) appearing in the cluster. Once had these references,
we ran the trackers in an OPE-like fashion. Each tracker was initialized in
the first frame of a track with the object detection given by Hands-in-contact
[79], and then let run for the other frames of the track. We then evaluated
the performance in a track by the normalized count of frames having
intersection-over-union $\geq 0.5$ with the object’s ground-truth. The overall
result is obtained by averaging the outcomes of all tracks. This experimental
procedure gives us an estimate of the accuracy of the hand-object interaction
detection system if trackers would have been included in its pipeline. More
interestingly, it allows also to build a ranking of the trackers based on the
results of a downstream application.
Table 6: Performance achieved by the 33 benchmarked trackers on TREK-150 using the RTE protocol. Tracker | FPS | SS | NPS | GSR
---|---|---|---|---
Ocean | 21 | 0.365 | 0.358 | 0.294
SiamBAN | 24 | 0.360 | 0.366 | 0.313
SiamRPN++ | 23 | 0.362 | 0.356 | 0.293
PrDiMP | 13 | 0.352 | 0.349 | 0.243
DiMP | 16 | 0.336 | 0.331 | 0.224
SiamMask | 23 | 0.335 | 0.333 | 0.298
SiamFC++ | 45 | 0.330 | 0.331 | 0.308
SiamDW | 32 | 0.327 | 0.334 | 0.317
KYS | 12 | 0.327 | 0.317 | 0.237
ATOM | 15 | 0.319 | 0.312 | 0.179
UpdateNet | 21 | 0.311 | 0.297 | 0.295
DCFNet | 49 | 0.299 | 0.286 | 0.335
TRASFUST | 13 | 0.296 | 0.270 | 0.185
SiamFC | 34 | 0.293 | 0.295 | 0.280
LTMU | 8 | 0.284 | 0.257 | 0.169
D3S | 16 | 0.276 | 0.263 | 0.182
BACF | 9 | 0.276 | 0.262 | 0.234
SPLT | 8 | 0.265 | 0.247 | 0.203
STRCF | 10 | 0.264 | 0.250 | 0.218
DSLT | 7 | 0.260 | 0.234 | 0.211
ECO | 15 | 0.252 | 0.231 | 0.173
GlobalTrack | 8 | 0.253 | 0.227 | 0.139
MCCTH | 8 | 0.251 | 0.231 | 0.232
Staple | 13 | 0.249 | 0.236 | 0.169
GOTURN | 44 | 0.247 | 0.242 | 0.119
MOSSE | 26 | 0.227 | 0.190 | 0.141
LTMU-H | 4 | 0.213 | 0.174 | 0.161
MetaCrest | 8 | 0.207 | 0.175 | 0.165
LTMU-F | 4 | 0.205 | 0.161 | 0.162
VITAL | 4 | 0.204 | 0.165 | 0.158
DSST | 2 | 0.191 | 0.145 | 0.161
KCF | 6 | 0.186 | 0.157 | 0.177
MDNet | 1 | 0.185 | 0.140 | 0.161
## Appendix D Additional Results
### MSE Protocol Results.
Figure 10 reports the overall performance of the 33 benchmarked trackers on
TREK-150 using the MSE protocol. The overall low performances of all the
trackers confirm the conclusions achieved using the OPE protocol. The FPV
setting introduces challenging factors for current visual trackers.
### Qualitative Examples
The first 7 rows of images of Figure 11 present qualitative results of 10 of
the generic-object trackers in comparison with the ground-truth (which is
identified by the white rectangles). The action performed by the camera wearer
is also reported for each sequence. The remaining 4 rows show the qualitative
performance of the FPV baseline trackers LTMU-F and LTMU-H in comparison with
LTMU and the ground-truth. For a better visualization, a video can be found at
https://youtu.be/oX1nICHgEJM.
Figure 11: Qualitative results of some of the studied trackers on the proposed
TREK-150 dataset. The first 7 rows of images show the qualitative performance
of 10 of the selected generic-object trackers, while the last 4 rows show the
results of the baseline FPV trackers LTMU-F and LTMU-H in comparison with
LTMU. For a better visualization, a video can be found at
https://youtu.be/oX1nICHgEJM.
### Per Attribute/Action Results.
Figure 12 presents the SS, NPS, and GSR scores achieved by the 33 trackers
considering the attributes assigned to sequences. Similarly, Figures 13 and 14
report the results for the whole batch of trackers with respect to action
verbs and target nouns.
### RTE Protocol Results.
Table 6 reports the FPS, SS, NPS, and GSR performance of all 33 benchmarked
trackers obtained using the RTE protocol. As stated in the main paper, offline
siamese trackers emerge as the best solution in this scenario. Online deep
discriminative trackers achieve comparable results in SS and NPS, but
demonstrate a larger drop in performance in the GSR score, showing that online
learning mechanisms influence this performance in the real-time setting.
Table 7: Performance of the offline trackers SiamFC and SiamRPN++ on a subset of 50 sequences of TREK-150 without and with fine-tuning on the remaining 100 videos. Tracker | Fine-tuning | OPE | MSE
---|---|---|---
SS | NPS | GSR | SS | NPS | GSR
SiamFC | ✗ | 0.311 | 0.332 | 0.317 | 0.307 | 0.317 | 0.307
✓ | 0.267 | 0.275 | 0.278 | 0.287 | 0.305 | 0.292
SiamRPN++ | ✗ | 0.384 | 0.395 | 0.377 | 0.367 | 0.385 | 0.333
✓ | 0.348 | 0.407 | 0.313 | 0.336 | 0.406 | 0.314
### Adaptation Of Offline Trackers.
Many current visual trackers employ deep learning architectures. Among these,
trackers based on siamese neural networks emerged as the most popular
approaches nowadays. These trackers are said to be offline (e.g. SiamFC [7],
SiamRPN++ [53], SiamMask [86], SiamBAN [16]) because they are trained to track
objects on large-scale tracking dataset [26, 67, 41, 31], and do not use
online adaptation mechanisms at test time. In our evaluation, such trackers
have been employed as they are described and trained in their original paper.
Given their generally low performance, one could wonder how these trackers
perform if knowledge about the FPV domain is exploited for learning. Our
TREK-150 dataset, which is designed to evaluate the progress of visual
tracking solutions in FPV, does not provide a large-scale database of learning
examples as needed by these methods. Instead, it well aligns with real-world
datasets where millions of frames are not available for training. In such
scenarios, the reasonable options the machine learning community suggest are
to use trackers as they are because of their general knowledge, or to adapt
them through fine-tuning using a smaller training set. We tried the second
strategy by randomly splitting TREK-150 in a training and test set of 100 and
50 videos respectively. We fine-tuned the popular offline trackers SiamFC and
SiamRPN++ on the training set according to their original learning strategy.
We then tested the fine-tuned versions on the test set and the results are
reported in Table 7. It shows that fine-tuning leads to substantial
overfitting that cause the performance to drop in general. These outcomes
prove the decision to evaluate offline trackers as they are is the right one
given the current lack of large-scale FPV tracking datasets. Moreover, given
the overall results presented in this paper, we hypothesize that visual
tracking in FPV will require more than just large-scale training. We hope the
results presented in this section will encourage the community to work on
domain adaptation techniques for offline trackers that are currently starting
to be investigated [30].
Figure 12: SS, NPS, and GSR results per sequence attribute achieved by the 33
benchmarks on the TREK-150 benchmark. Figure 13: SS, NPS, and GSR results
achieved by the 33 benchmarks on the TREK-150 benchmark considering each verb
associated to the action performed by the camera wearer. Figure 14: SS, NPS,
and GSR results achieved by the 33 benchmarks on the TREK-150 benchmark
considering the different target categories.
|
Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.